To support the rapidly increasing number of
mobile users and mobile multimedia services,
and the related demands for bandwidth, wireless
communication technology is facing a potentially scarcity of radio spectrum resources. However, spectrum measurement campaigns have
shown that the shortage of radio spectrum is due
to inefficient usage and inflexible spectrum allocation policies. Thus, to be able to meet the
requirements of bandwidth and spectrum utilization, spectrum underlay access, one of the
techniques in cognitive radio networks (CRNs),
has been proposed as a frontier solution to deal
with this problem. In a spectrum underlay network, the secondary user (SU) is allowed to simultaneously access the licensed frequency band
of the primary user (PU) as long as the interference caused by the SU to the PU is kept below a
predefined threshold. By doing so, the spectrum
utilization can be improved significantly. Moreover, the spectrum underlay network is not only
considered as the least sophisticated in implementation, but also can operate in dense areas
where the number of temporal spectrum holes is
small. Inspired by the above discussion, this thesis provides a performance analysis of spectrum
underlay networks which are subject to interference constraints.
The thesis is divided into an introduction part
and five parts based on peer-reviewed international research publications. The introduction part
provides the reader with an overview and background on CRNs. The first part investigates the
performance of secondary networks in terms of
outage probability and ergodic capacity subject
to the joint outage constraint of the PU and the
peak transmit power constraint of the SU. The
second part evaluates the performance of CRNs
with a buffered relay. Subject to the timeout
probability constraint of the PU and the peak
transmit power constraint of the SU, system performance in terms of end-to-end throughput,
end-to-end transmission time, and stable transmission condition for the relay buffer is studied.
The third part analyzes a cognitive cooperative
radio network under the peak interference power constraint of multiple PUs with best relay
selection. The obtained results readily reveal
insights into the impact of the number of PUs,
channel mean powers of the communication and
interference links on the system performance.
The fourth part studies the delay performance
of CRNs under the peak interference power constraint of multiple PUs for point-to-point and
point-to-multipoint communications. A closedform expression for outage probability and an
analytical expression for the average waiting
time of packets are obtained for point-to-point
communications. Moreover, the outage probability and successful transmission probability for
packets in point-to-multipoint communications
are presented. Finally, the fifth part presents
work on the performance analysis of a spectrum
underlay network for a general fading channel.
A lower bound on the packet timeout probability and the average number of transmissions per
packet are obtained for the secondary network.
Performance Analysis of Cognitive
Radio Networks with Interference Constraints
ABSTRACT
Hung Tran
ISSN 1653-2090
ISBN: 978-91-7295-249-2
2013:03
2013:03
Performance Analysis of Cognitive
Radio Networks with Interference
Constraints
Hung Tran
Blekinge Institute of Technology
Doctoral Dissertation Series No. 2013:03
School of Computing
Performance Analysis of Cognitive Radio
Networks with Interference Constraints
Hung Tran
Blekinge Institute of Technology doctoral dissertation series
No 2013:03
Performance Analysis of Cognitive Radio
Networks with Interference Constraints
Hung Tran
Doctoral Dissertation in
Telecommunication Systems
School of Computing
Blekinge Institute of Technology
SWEDEN
2013 Hung Tran
School of Computing
Publisher: Blekinge Institute of Technology,
SE-371 79 Karlskrona, Sweden
Printed by Printfabriken, Karlskrona, Sweden 2013
ISBN: 978-91-7295-249-2
ISSN 1653-2090
urn:nbn:se:bth-00550
v
Abstract
To support the rapidly increasing number of mobile users and mobile multimedia services, and the related demands for bandwidth, wireless communication technology is facing a potentially scarcity of radio
spectrum resources. However, spectrum measurement campaigns have
shown that the shortage of radio spectrum is due to inefficient usage
and inflexible spectrum allocation policies. Thus, to be able to meet the
requirements of bandwidth and spectrum utilization, spectrum underlay access, one of the techniques in cognitive radio networks (CRNs),
has been proposed as a frontier solution to deal with this problem. In
a spectrum underlay network, the secondary user (SU) is allowed to
simultaneously access the licensed frequency band of the primary user
(PU) as long as the interference caused by the SU to the PU is kept
below a predefined threshold. By doing so, the spectrum utilization can
be improved significantly. Moreover, the spectrum underlay network is
not only considered as the least sophisticated in implementation, but
also can operate in dense areas where the number of temporal spectrum
holes is small. Inspired by the above discussion, this thesis provides a
performance analysis of spectrum underlay networks which are subject
to interference constraints.
The thesis is divided into an introduction part and five parts based
on peer-reviewed international research publications. The introduction
part provides the reader with an overview and background on CRNs.
The first part investigates the performance of secondary networks in
terms of outage probability and ergodic capacity subject to the joint
outage constraint of the PU and the peak transmit power constraint
of the SU. The second part evaluates the performance of CRNs with a
buffered relay. Subject to the timeout probability constraint of the PU
and the peak transmit power constraint of the SU, system performance
in terms of end-to-end throughput, end-to-end transmission time, and
stable transmission condition for the relay buffer is studied. The third
part analyzes a cognitive cooperative radio network under the peak interference power constraint of multiple PUs with best relay selection.
The obtained results readily reveal insights into the impact of the number of PUs, channel mean powers of the communication and interference links on the system performance. The fourth part studies the delay
performance of CRNs under the peak interference power constraint of
multiple PUs for point-to-point and point-to-multipoint communications. A closed-form expression for outage probability and an analytical expression for the average waiting time of packets are obtained for
point-to-point communications. Moreover, the outage probability and
successful transmission probability for packets in point-to-multipoint
communications are presented. Finally, the fifth part presents work on
the performance analysis of a spectrum underlay network for a general
fading channel. A lower bound on the packet timeout probability and
vi
the average number of transmissions per packet are obtained for the
secondary network.
vii
Preface
This thesis summarizes my work within the field of performance analysis for
cognitive radio networks over fading channels. The work has been performed
at the School of Computing, Blekinge Institute of Technology, Karlskrona,
Sweden. The thesis comprises of an introduction section and five parts, which
are:
Part I
Impact of Primary Networks on the Performance of Secondary Networks
Part II
Performance Analysis of a Cognitive Cooperative Radio Network with
a Buffered Relay
Part III
Cognitive Cooperative Networks with DF Relay Selection under Interference Constraints of Multiple Primary Users
Part IV
Delay Performance of Cognitive Radio Networks for Point-to-Point and
Point-to-Multipoint Communications
Part V
Performance of Cognitive Radio Networks over General Fading Channels
ix
Acknowledgements
Over four years working towards my Ph.D. degree, I was very lucky to receive
the help of many people. Without their kind support and advice, I would not
have been able to complete this thesis. It is a great opportunity for me to
thank all of them.
First of all, I would like to express the deepest gratitude to my principal supervisor Prof. Hans-Jürgen Zepernick for his admission and invaluable advices
in all aspects of professional life. His patience, positive thinking, and excellent
supervision have encouraged me to overcome challenging times. Without his
expert guidance and knowledge, it would have not been possible to come to
an end with this dissertation. I feel very lucky and absolutely happy to work
with him. I would also like to thank my co-supervisor Prof. Markus Fiedler
for his advice on knowledge of queuing theory and quality of experience.
I am grateful to Dr. Trung Duong Quang, Dr. Viet Vu Thuy, Dr.
Muhammad Immran Iqbal, Dr. Thomas Sjögren, Dr. Trinh Pham Quang,
Hoc Phan, Nam Hoang Le, Lap Bui Tien, Charles Kabiri, Louis Sibomana,
and Chinh Chu Thi My for all their helps, cooperation, and many fruitful
discussions. Also, I am thankful to all colleagues at the Blekinge Institute of
Technology for being so helpful, friendly and providing a great work environment during my stay in Sweden.
I would like to thank the National Institute of Education Management and
the Vietnam International Education Development. Without their admission
and support, I would not have been able to go to the Blekinge Institute of
Technology.
Finally, I want to express my hearty gratitude to my parents Huy Tran
Van and Thao Bui Thi, my younger sister Hien Tran Thi Thu, and other
family members for their unmeasurable love and support which helped me to
overcome the difficulties of living far away from my country. Last but certainly
not least, I want to thank my fiancé, Lan Dang Thu, for her continuous
encouragement, faithfulness, and endless love. Thank you so much!
Hung Tran
Karlskrona, Spring 2013
x
To my parents Huy Tran Van and Thao Bui Thi
xi
Publications
Part I is published as:
H. Tran, M. A. Hagos, M. Mohamed, and H.-J. Zepernick, “Impact of Primary
Networks on the Performance of Secondary Networks,” in Proc. International Conference on Computing, Management and Telecommunications, Ho
Chi Minh City, Vietnam, Jan. 2013, pp. 1-6.
Part II is published as:
H. Tran, H.-J. Zepernick, H. Phan, and L. Sibomana, “Performance Analysis
of a Cognitive Cooperative Radio Network with a Buffered Relay”, IEEE
Transaction on Vehicular Technology, Feb. 2013, under review.
Part III is published as:
H. Tran, H.-J. Zepernick, and H. Phan, “Cognitive Cooperative Networks
with DF Relay Selection under Interference Constraints of Multiple Primary
Users”, Wireless Communications and Mobile Computing, Feb. 2013, under
minor revision.
Based on:
H. Tran, T. Q. Duong, and H.-J. Zepernick, “Performance Analysis of Cognitive Relay Networks Under Power Constraint of Multiple Primary Users,”
in Proc. IEEE Global Telecommunications Conference, Houston, U.S.A., Dec.
2011, pp. 1–6.
Part IV is based on the publications as:
H. Tran, T. Q. Duong, and H.-J. Zepernick, “Delay Performance of Cognitive Radio Networks for Point-to-Point and Point-to-Multipoint Communications,” EURASIP Journal on Wireless Communications and Networking,
vol. 2012, no. 1, 2012.
H. Tran, T. Q. Duong, and H.-J. Zepernick, “Queuing Analysis for Cognitive
Radio Networks Under Peak Interference Power Constraint,” in Proc. IEEE
International Symposium on Wireless and Pervasive Computing, Hong Kong,
xii
China, Feb. 2011, pp. 1–5.
H. Tran, “Numerical Results on the Delay Performance of Cognitive Radio
Networks for Point-to-Point and Point-to-Multipoint Communications”, BTH
Technical Report, No. 2012:06, Dec. 2012.
Part V is based on the publications as:
H. Tran, T. Q. Duong, and H.-J. Zepernick, “On the Performance of Spectrum
Sharing Systems Over α-µ Fading Channel for Non-identical µ Parameter,” in
Proc. IEEE International Symposium on Wireless Communication Systems,
Aachen, Germany, Nov. 2011, pp. 1-6.
H. Tran, H.-J. Zepernick, H. Phan, and M. Fiedler,“Outage Probability, Average Transmission Time, and Quality of Experience for Cognitive Radio Networks over General Fading Channels,” in Proc. EURO-NF Conference on
Next Generation Internet, Karlskrona, Sweden, Jun. 2012, pp. 9–15.
Other publications in conjunction with the thesis but not included:
Journals
H. Phan, H.-J. Zepernick, and H. Tran, “Impact of Interference Constraint on
Multi-hop Cognitive Amplify-and-Forward Relay Networks over Nakagami-m
Fading Channels,” IET Communications, Jan. 2013, accepted.
H. Tran, H.-J. Zepernick, and H. Phan, “Cognitive Proactive and Reactive
DF Relaying Schemes under Joint Outage and Peak Transmit Power Constraints,” IEEE Communications Letters, Nov. 2012, under revision.
H. Phan, H.-J. Zepernick, T. Q. Duong, H. Tran, and T. M. C. Chu, “Cognitive AF Relay Networks with Beamforming under Primary User Power Constraint over Nakagami-m Fading Channels,” Wireless Communications and
Mobile Computing, DOI: 10.1002/wcm.2317, Oct. 2012.
T. Q. Duong, V. N. Q. Bao, H. Tran, G.C. Alexandropoulos, and H.-J. Zepernick, “Effect of Primary Network on Performance of Spectrum Sharing AF
Relaying,” Electronics Letters, vol. 48, no. 1, pp. 25–27, Jan. 2012.
xiii
Conferences
H. Tran, H.-J. Zepernick, and H. Phan, “Impact of the Number of Antennas
and Distances Among Users on Cognitive Radio Networks,” in Proc. International Conference on Advanced Technologies for Communications, Hanoi,
Vietnam, Nov. 2012, pp. 1–5.
C. Kabiri, H. Tran, and H.-J. Zepernick, “Outage Probability and Ergodic Capacity of a Spectrum Sharing System with Multiuser Diversity,” in Proc. International Conference on Advanced Technologies for Communications, Hanoi,
Vietnam, Nov. 2012, pp. 1–5.
H. Tran and H.-J. Zepernick, “Impact of Secondary Network on the Security of Primary Network,” Swedish Communication Technologies Workshop,
Poster Presentation, Lund, Sweden, Oct. 2012.
L. Sibomana, H.-J. Zepernick, H. Tran, and C. Kabiri, “Timeout Probability for Cognitive Radio Networks,” Swedish Communication Technologies
Workshop, Poster Presentation, Lund, Sweden, Oct. 2012.
H. Tran, T. Q. Duong, and H.-J. Zepernick, “Performance of Spectrum Sharing System over Nakagami-m Fading Channels,” in Proc. International Conference on Signal Processing and Communication Systems, Gold Coast, Australia, Dec. 2010, pp. 1–5.
H. Tran, T. Q. Duong, and H.-J. Zepernick, “Average Waiting Time of Packets with Different Priorities in Cognitive Radio Networks,” in Proc. IEEE
International Symposium on Wireless Pervasive Computing, Modena, Italy,
May 2010, pp. 122–127.
xv
Contents
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . v
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
Acknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix
Publications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2.
Fundamentals of Cognitive Radio Networks . . . . . . . . . . . . . . . . . 2
3.
4.
2.1.
Spectrum Holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
2.2.
Cognitive Radio Characteristics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
Cognitive Radio Network Paradigms . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.1.
Interweave Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
3.2.
Underlay Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
3.3.
Overlay Paradigm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
Propagation Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
4.1.
Path Loss Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
4.2.
Statistical Multipath Channel Models . . . . . . . . . . . . . . . . . . . . . . . . 22
5.
Applications of Cognitive Radio Networks . . . . . . . . . . . . . . . . . 27
6.
Challenges of Cognitive Radio Networks . . . . . . . . . . . . . . . . . . . 28
7.
Thesis Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Part I
Impact of Primary Networks on the Performance of Secondary Networks
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
xvi
Part II
Performance Analysis of a Cognitive Cooperative Radio Network with
a Buffered Relay . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
Part III
Cognitive Cooperative Networks with DF Relay Selection under Interference Constraints of Multiple Primary Users . . . . . . . . . . . . . . . . . . . . 101
Part IV
Delay Performance of Cognitive Radio Networks for Point-to-Point and
Point-to-Multipoint Communications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
Part V
Performance of Cognitive Radio Networks over General Fading Channels
. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
Introduction
1
Motivation
The electromagnetic radio spectrum is a naturally limited and precious resource in wireless communications. In the last couple of decades, the emergence of a large number of new wireless systems and the related increasing demand on bandwidth has resulted in frequency spectrum becoming exhausted.
Surprisingly, many measurement campaigns have shown that almost all of the
allocated spectrum is being used inefficiently most of the time [1–3]. As such,
the spectrum scarcity is not only caused by inflexible spectrum management,
but also due to inefficient usage. In efforts to improve the bandwidth utilization, the concept of cognitive radio networks (CRNs) has been proposed as a
key enabling solution for next generations of wireless networks [4–6].
The basic idea behind CRNs is to efficiently utilize the temporary unused
licensed spectrum at a specific time or a specific geographic area for communications. In particular, users in a CRN are classified into primary users
(PUs) and secondary users (SUs). A PU, also called licensed user, has the
highest priority to access the spectrum and should not be subject to harmful
interference from other users. On the other hand, an SU or cognitive user
deploys advanced radio access techniques along with dynamic spectrum allocation policies to coexist with the PU provided that the interference caused
by the SU does not degrade the PU performance [7]. With this approach, a
CRN can overcome the shortage of radio spectrum.
Based on the CRN concept, three main spectrum access approaches have
been proposed, known as interweave, overlay, and underlay [7]. With interweave spectrum access, an SU opportunistically accesses the licensed spectrum
when it is not being occupied by the PU. As a consequence, the SU has to
use sensing techniques to exactly select the unused spectrum for its own communications. In the overlay paradigm, the SU can simultaneously access the
licensed frequency bands given that it is able to exploit PU information such
as codebooks and messages. Upon these information, the SU carefully splits
its transmit power into two parts; one part is used to support the communication of the PU while the remainder is used for its own communication. In
1
2
Introduction
contrast to the aforementioned paradigms, in the underlay approach, the SU
and PU can concurrently access the frequency band as long as the interference
caused by the SU to the PU is kept below a predefined threshold.
In the past few years, the underlay spectrum access has obtained great attention in the research community, since its requirements for implementation
are less complicated than for the overlay and interweave networks. Moreover,
the SU can operate in crowded areas where the number of spectrum holes are
small. However, the SU must be subject to the interference constraints to not
degrade the PU performance, which results in short-range communication
and low transmission rate for the SU. To overcome this drawback, optimal
power allocation policies and cognitive cooperative radio networks (CCRNs)
have been proposed to enhance transmission rate, spectrum utilization, and
to expand the SU coverage range. In light of these motivations, this thesis
focuses on the performance evaluation of cognitive underlay networks for the
point-to-point, point-to-multipoint, and CCRN models, in which the SU is
subject to various interference power constraints such as the peak transmit
power constraint of the SU, and the outage constraint and peak interference
power constraint of the PU. Furthermore, the considered channels are assumed to undergo fading that can be described by a Rayleigh, Nakagami-m,
or α-µ distribution. On this basis, a performance analysis in terms of outage probability, symbol error probability, ergodic capacity, throughput, stable
transmission condition, as well as delay of packet transmission is conducted
for the considered systems.
2
Fundamentals of Cognitive Radio Networks
The temporary unused spectra of traditional wireless communications are referred to as spectrum holes or vacant spectrum which is considered as one of
the key ideas in the development of CRNs. Clearly, CRNs differ from traditional wireless communication networks, since they have cognitive capabilities
to sense, to analyze, and to adapt to the randomness of the communication
environment where the spectra have been allocated to the PU. With these
cognitive capabilities, the SU can monitor activities of different users on the
specific spectrum, and then exploit the temporal vacant frequency band to
obtain a reliable communication. In this section, the spectrum hole concepts
and background on CRNs are provided.
2.1
Spectrum Holes
A spectrum hole is originally defined as a band of frequencies which are readily assigned to a PU, however, it may not be always used by the PU at a
3
Introduction
Power
Time
Frequency
Frequency occupied by primary users
Spectrum white space which secondary users can utilize
Figure 1: Example of a spectrum allocation graph.
specific time or a geographic area [5]. Depending on the communication environment, the spectrum holes can be identified following frequency and time
(see Figure 1), or space as [8, 9]:
• Frequency spectrum hole is a contiguous frequency band in which
activities of the SU do not cause any harmful interference to the PUs.
• Temporal spectrum hole is a frequency band that is not occupied by
a PU for a period of time. By using advanced spectrum sensing techniques, an SU can detect spectrum holes and opportunistically access it
without degrading any quality of service (QoS) of the PU.
• Spatial spectrum hole is a frequency band in a specific geographic
area where the PU transmission is being occupied. The SU can utilize
this band if it is outside this area (see Figure 2).
Additionally, spectrum holes may also be classified into so-called spaces as
follows [5]:
• Black spaces where high power interferers dominate for some period
of time.
• Gray spaces where low power interferers partially dominate.
• White spaces where interferers do not exist, only natural noises such
as broadband thermal noise and impulsive noise are present.
4
Introduction
Protection area of primary user
PU-Rx
PU-Tx
SU-Tx
d
SU-Rx
Figure 2: An example of a spatial spectrum hole where the SU is not allowed
to operate in the PU protection area.
According to the space classification, the SU can transmit in the gray and
white spaces, but it is prohibited to operate in the black space once the PU
is active.
On the basis of spectrum hole concepts, an important definition of cognitive radio (CR), which is generally accepted by the research community, has
been given in [5]:
“CR is an intelligent wireless communication system that is aware of its
surrounding environment (i.e., outside world), and uses the methodology of
understanding-by-building to learn from the environment and adapt its internal states to statistical variations in the incoming radio frequency (RF)
stimuli by making corresponding changes in certain operating parameters (e.g.,
transmit-power, carrier-frequency, and modulation strategy) in real-time, with
two primary objectives in mind:
1) Highly reliable communications whenever and wherever needed;
2) Efficient utilization of the radio spectrum.”
Clearly, the awareness and adjustment according to the fluctuations of the
radio environment to create reliable communications and efficient spectrum
utilization are the most important criteria in a CRN.
2.2
Cognitive Radio Characteristics
Cognitive capabilities are the most different characteristics of a CRN from
traditional wireless communication networks. These capabilities allow an SU
Introduction
5
to observe the surrounding radio environment such as available frequency,
interference temperature, noise power, distance, and so on. Depending on
the collected information, the SU will make decisions about the selected frequency, transmit power level, or modulation scheme, to achieve an optimal
performance. In fact, to implement the CRN in practice, it should have main
characteristics as follows [8]:
• An SU should be equipped with a unified cross-layer architecture in
order to meet different QoS demands.
• An SU should take advantage of efficient spectrum sensing and analysis
techniques so that the SU can maintain continuous spectrum and keep
a reliable communication.
• An SU should utilize dynamic spectrum access approaches which can
adapt to the fluctuating nature of the CRN.
• An SU should share the spectrum information with other users and
coordinate communication to cause minimal interference or no collisions
to the PUs occupying the same frequency bands.
3
Cognitive Radio Network Paradigms
In this section, the key concepts attached with the interweave and underlay
paradigm are presented. Also, a brief discussion about the overlay paradigm
is provided.
3.1
Interweave Paradigm
In the interweave paradigm, an SU intelligently detects the spectrum holes,
and then opportunistically communicates over the vacant spectrum without
causing any interference to the PU (see Figure 3). To let an SU efficiently
communicate through the spectrum holes, an accurate sensing of the presence of the PU in the different frequency bands is required. Thus, advanced
spectrum sensing techniques to detect spectrum holes over a wide bandwidth
are crucial. It should be noted that there is no concurrent transmission of
the SU and PU on the same spectrum, and the SU must vacate the spectrum as soon as the PU reoccupies it. Accordingly, the communication of
the SU is forced to terminate if the SU cannot detect new spectrum holes to
handover. An advantage of the interweave access is that an SU can transmit
with maximum power once a spectrum hole has been detected without caring
about interference to the PU. However, it is not easy to exactly detect the
6
Introduction
Secondary user
Power
Primary user
Frequency
Figure 3: An example of interweave spectrum access where SUs access unused
spectrum of the PU.
spectrum holes in a highly fluctuating radio environment. As such, a missed
detection can make serious interference to the PU. In addition, in dense areas
with high spectrum occupancy and hence small number of spectrum holes,
the interweave approach is not efficient.
3.1.1
Spectrum Sensing Techniques
Spectrum sensing is one of the most important functions in a CRN, and it
is also a must-have component in the interweave approach. This does not
only help the SU to be aware of the surrounding radio environment, but
also provides information for communication decisions such that the system
performance is enhanced. In this section, well-known sensing techniques along
with their advantage and disadvantage are introduced.
• Direct signal detection from the PU
As the primary transmitter (PU-Tx) communicates with the primary
receiver (PU-Rx), it uses a certain power level to transmit signals in a
specific spectrum band. Thus, if the SU wants to know whether the PU
is active or not, it needs to observe the radio signal in the spectrum
band. Accordingly, the received signal y(t) at the SU can be given by
c(t)x(t) + n(t), H1
y(t) =
(1)
n(t),
H0
where x(t), n(t), and c(t) denote the PU signal, additive white Gaussian
noise (AWGN), and channel coefficient from the PU to the SU, respectively. The terms H0 and H1 are the two hypotheses which express the
absence and presence of the PU signal, respectively. From the received
signal y(t), the SU will make a decision between H0 and H1 .
7
Introduction
In order to examine the detection performance of the SU, the detection probability PD and false alarm probability PF , are typically used.
More specifically, PD is the probability that the SU decision for hypothesis H1 is correct. On the other hand, PF is the probability that the
SU decision is H1 while it actually should be H0 .
• Energy detection
Energy detection is a low-complexity spectrum sensing technique which
is applicable for wide-band spectrum sensing [8, 10]. As the name suggests, the energy detector observes the average energy within M samples
of the received signal, given by
Y =
M
1 X
|y(t)|2
M t=1
By comparing the average energy Y
decision of the energy detector may
H1 ,
θ=
H0 ,
(2)
with a predefined threshold λ, the
be expressed as
if
if
Y >λ
Y <λ
(3)
On this basis, the detection probability and false alarm probability are
formulated, respectively, as
PD = Pr{Y > λ|H1 }
(4)
PF = Pr{Y > λ|H0 }
(5)
Besides the advantages of the energy detector such as low cost, easy
implementation, short detection time, and no requirements of a priori
information about the PU, it also faces some challenges. For example,
the noise power may change over time and could be very high in some
cases. As such, it is difficult to exactly detect the presence of a PU in
real time. Furthermore, an energy detector cannot differentiate between
a PU signal and other interference sources as it simply compares its
received signal with a given threshold. Thus, the false alarm probability
may be high [11–13].
• Cyclostationary detection
Cyclostationary detection is a special sensing technique that allows the
energy detector to distinguish the PU signal from the interference and
noise [14]. Particularly, signals of wireless devices generally are modulated and generated following a periodicity. They can be recognized by
8
Introduction
analyzing the cyclic autocorrelation function of received signals, given
as
Ry(α) (τ ) = E[y(t + τ )y ∗ (t − τ )ej2παt ]
(6)
where E[·] denotes expectation, ∗ stands for complex conjugate, and α
represents the cyclic frequency. By using Fourier series expansion, (6)
can be rewritten in the form of a cyclic spectrum density (CSD) function
as [15]
S(f, α) =
∞
X
Ryα (τ )e−j2πf τ
(7)
τ =−∞
Since the primary signal is often modulated, the CSD function, S(f, α),
exhibits peaks when the cyclic frequency, α, equals the fundamental
frequencies of the transmitted signal [16]. Otherwise, the CSD does not
have any peak because the AWGN is a non-cyclostationary signal. In
practice, cyclostationary detection can perform better than the energy
detector as it not only can differentiate the PU signal, but also can
detect in the low signal-to-noise ratio (SNR) regime [17]. However,
cyclostationary detection demands long observation time and complex
computation.
• Matched filter detection
Radio transmission techniques often use pilot signals, a preamble, or a
training sequence to estimate channel state information (CSI). If an SU
has such information about PU signals, then this can be used efficiently
for the sensing process. More specifically, the SU senses the signals for
a short time, and then compares the received signals with the available
PU information. If the characteristics of the received signal and the
information about the PU signal match, a PU occupies the spectrum.
Otherwise, the spectrum is considered as being free. The main advantages of matched filter detection are short sensing time and high detection
performance. However, if the PU information is provided incorrectly to
the matched filter detector, the sensing performance degrades rapidly.
Inspired by the above techniques, many recent techniques have been proposed
to enhance the sensing performance, e.g., covariance-based sensing [18, 19],
filter-based sensing, fast sensing [20,21], learning/reasoning-based sensing [22,
23] and measurement-based sensing and modeling [24, 25]. More details on
these techniques can be found in [10–14, 26] and the references therein.
9
Introduction
PU-Tx
SU-Rx
PU-Rx
SU-Tx
Figure 4: An example of a hidden PU where the SU-Tx cannot detect the
appearance of the PU-Tx due to an obstacle.
3.1.2
Cooperative Spectrum Sensing
Noise, multi-path fading, and shadowing are natural characteristics of wireless
communications, which effect the received signal strength. For example, if a
PU is far away from the SU, or the PU signal is blocked by a large obstacle,
then the received signal may be very low at the SU. Accordingly, it is difficult
to exactly detect the presence of a PU. Figure 4 illustrates a scenario in which
the PU-Tx is hidden by an obstacle such that the secondary transmitter (SUTx) cannot detect the PU-Tx signal. Therefore, the SU-Tx may cause harmful
interference to the PU-Rx, as it commences using the licensed spectrum to
communicate with the secondary receiver (SU-Rx).
To overcome such problems, cooperative spectrum sensing has been proposed [27–29]. It has been shown that the advantages of spatial diversity
and independent fading channels of multiple users in cooperative networks
can be used to enhance the detection probability and decrease the sensing
time [28, 29]. An example scenario for cooperative spectrum sensing is shown
in Figure 5. The SU-Tx can detect the PU-Tx through the help of two secondary relays (SRs), SR1 and SR2 .
The basic structure of a cooperative spectrum sensing system includes a
secondary base station (SBS) or an access point, and a number of SUs as
depicted in Figure 6. In this context, the SUs gather information about the
PU-Tx and send this to the SBS through a common control channel. Based
on the provided information, the SBS will process and make a decision about
the presence or absence of a PU, and then send an announcement to the
10
Introduction
PU-Tx
PU-Rx
SU-Rx
SR1
SU-Tx
SR2
Figure 5: Two SRs support the SU-Tx in detecting a hidden PU-Tx.
SUs. Furthermore, to enhance the decision precision, the SBS can process
the received information from the SUs by using a soft combination or hard
combination technique as follows.
• Soft combination
The SUs forward their original sensing information to the SBS. Then,
the SBS is responsible for the calculation and making a final decision
[5, 30, 31]. A main advantage of soft combination is its high detection
performance. However, it requires large overhead information to be
exchanged between the SUs and the SBS. In addition, if there exists a
large number of SUs joining the detection process, the SBS may take a
long time to quantize information.
• Hard combination
The SUs make their own decisions about the presence or absence of
a PU-Tx in a specific spectrum, and send these to the SBS. As the
SBS receives the decisions from the SUs, it will apply a specific rule
such as AND rule, OR rule, or K-out-of-N rule to make a final decision
[28,29,32]. For example, N SUs independently detect the PU-Tx, where
each SUn , n = 1, . . . , N , has its own decision dn , dn ∈ {0, 1}. If the SBS
receives a number of K decisions dn = 1, 0 < K ≤ N , the SBS can
broadcast an announcement to all SUs that the frequency band is being
occupied by a PU. This is known as the K-out-of-N rule.
Besides the aforementioned cooperative sensing schemes, the SUs can act as
relays and apply amplify-and-forward (AF) or detect-and-relay protocols to
enhance the detection performance. Further information about these protocols can be found in [28, 33].
11
Introduction
SU1
SU2
PU-Tx
SBS
SU3
PU-Rx
Figure 6: Example of a cooperative spectrum sensing system with an SBS
and three SUs.
3.2
Underlay Paradigm
In underlay spectrum access, the SU is allowed to simultaneously access the
spectrum licensed to the PU, provided that its interference to the PU does not
violate a given interference constraint, e.g. peak interference power, average
interference power, or outage constraint [34,35]. As a result, these constraints
not only shorten the communication range, but also limit the transmission
rate of the SU. An illustration of the underlay paradigm is given in Figure 7
where the SU transmit power must be kept below a threshold given by the
PU. In order to improve the performance of the secondary network, the SU
should utilize advanced spread spectrum techniques such as ultra-wideband
communications. Also, it may use smart multiple antenna solutions to guide
the interference away from the PU-Rx. Generally, the activity of an SU in
underlay networks is based on the interference temperature and interference
temperature limit concepts which will be presented in more detail in the following.
3.2.1
Interference Temperature Concepts
In the highly time-variant radio environment, the SU detection techniques
are not always completely reliable. This may cause serious interference to
the PU-Rx when a missed detection takes place. Moreover, a rapid increase
of wireless devices may cause unpredictable interference sources which can
lead to performance degradation. In order to avoid performance loss due to
missed detection and unpredictable noise, the interference limit should be set
12
Introduction
Secondary user
Power
Primary user
Interference constraint
Frequency
Figure 7: Example of underlay spectrum access.
at the receiver side. Hence a new measure, named interference temperature,
has been introduced by the Federal Communications Commission (FCC) [36].
The interference temperature is a measure of the radio frequency power at a
receiving antenna generated by other transmitters and noise sources and can
be formulated as [37]
TI (fc , B) =
PI (fc , B)
kB
(8)
where PI (fc , B) is the mean interference power (Watts) at the carrier frequency fc , B is the bandwidth (Hertz), and k = 1.38 × 10−23 J/K is the
Boltzmann constant.
In light of the interference temperature concept, FCC further introduced
the specification of an interference temperature limit. It is defined as the
maximum amount of tolerable interference in a particular frequency band for
a given geographic location, where the QoS demands of the receiver can be
satisfied. As a result, if any SU-Tx wants to access the licensed band, its
transmit power plus other noise and interference must be kept below the interference temperature limit. In case that the PU is outside the coverage range
of the SU-Tx, the SU-Tx can transmit with maximal power to enhance its
performance. The results presented in [38], have shown that the interference
temperature limit is a feasible approach to reduce the collision probability
with the PU communication if the location of the PU is known to the SU, in
which the PU position can be provided by the help of a location system [39].
A main advantage of underlay networks is their ability to operate in dense
areas where the number of temporal spectrum holes is small, i.e., the interweave paradigm may not operate efficiently. However, the capacity and
coverage range of the underlay paradigm are limited due to the transmit
13
Introduction
PU-Rx
g1
SU-Tx
g0
SU-Rx
Figure 8: A basic model of an underlay CRN (the solid line is the SU-Tx→SURx communication link; the dashed line is the SU-Tx→PU-Rx interference
link).
power constraints. As a consequence, optimal power allocation to improve
transmission rate and to expand communication range has become one of the
most important problems [34, 40–45]. The related optimal power allocation
policies should be based on the knowledge of the PU-Rx location, and the CSI
between the SU-Tx→PU-Rx and PU-Tx→PU-Rx links. In fact, the CSI can
be estimated by the SU or exchanged between the primary networks and the
secondary networks over a common control channel. However, the signaling
overhead is increased and the signal processing at the SU may become more
sophisticated.
3.2.2
Power Control Under Interference Constraints
A common method to protect the PU communication is to establish an interference constraint at the PU-Rx, in which the SU-Tx should have an appropriate power allocation policy to keep the total amount of interference
received at the PU-Rx below a prescribed threshold. A basic system model of
an underlay CRN is shown in Figure 8 where the SU coexists with the PU on
the same frequency band. This model has been widely adopted to investigate
the performance of underlay CRNs [35,40,46–49]. More specifically, the channel power gains of the SU-Tx→SU-Rx and SU-Tx→PU-Rx links are denoted
by g0 and g1 , respectively. The SU-Tx must adapt its transmit power on the
basis of CSI to satisfy interference power constraints. For instance, the SU-Tx
transmit power may be subject to a short-term interference power constraint
(peak interference power constraint) as [34]
P (g0 , g1 )g1 ≤ Qpk
(9)
14
Introduction
or under a long-term interference power constraint (average interference power
constraint) [35, 49] as
E[P (g0 , g1 )g1 ] ≤ Qav
(10)
where Qpk and Qav , respectively, denote the peak interference power and average interference power that the PU-Rx can tolerate. Moreover, the transmit
power is limited in practice. Thus, the SU-Tx transmit power may be subject
to additional constraints such as peak or average transmit power, given by
P (g0 , g1 ) ≤ Ppk
E[P (g0 , g1 )] ≤ Pav
(11)
(12)
where Ppk and Pav stand for the peak and average transmit power, respectively.
3.2.3
Power Control Under Outage Constraint
The interference constraints given by the PU-Rx are considered as practical
approaches to protect the PU communication and have also been studied
thoroughly. However, there are only a few studies investigating the impact of
the PU-Tx→PU-Rx and PU-Tx→SU-Rx links on the power allocation policy
and performance of CRNs. In an effort to improve capacity of the SU and
to protect the PU communication, a power control strategy under the outage
constraint of the PU-Rx has been proposed to exploit the CSI of the PUTx→PU-Rx and PU-Tx→SU-Rx links in [44, 50–52]. Basically, the SU-Tx
transmit power is controlled so that the SU can obtain optimal performance,
and the outage probability of the PU does not exceed a certain outage constraint.
Consider the system model as shown in Figure 9 where the SU and the PU
share the same frequency band. Channel power gains of the PU-Tx→PU-Rx,
PU-Tx→SU-Rx, SU-Tx→SU-Rx, and SU-Tx→PU-Rx links are denoted by
h0 , h1 , g0 , and g1 , respectively. For PU communication, the PU-Tx uses a
transmit power Pp to communicate with the PU-Rx without considering the
existence of the SU. In addition, the QoS of PU is guaranteed if the outage
probability of the PU is kept below a predefined threshold, ǫ. Without the
existence of the SU in the licensed band, the outage constraint of the PU is
formulated by
Pp h0
p
(13)
< C0 ≤ ǫ
Pout = Pr log2 1 +
N0
where N0 and C0 stand for the noise power and outage transmission rate,
respectively. On the other hand, if the SU accesses the licensed spectrum, it
15
Introduction
PU-Rx
PU-Tx
h0
g1
h1
SU-Tx
g0
SU-Rx
Figure 9: A simple model of an underlay CRN.
must have a suitable power control policy to satisfy the outage constraint, ǫ.
In other words, the outage constraint of the PU in the presence of the SU is
given by
Pp h0
p
Pout
= Pr log2 1 +
(14)
< C0 ≤ ǫ
Ps g1 + N0
where Ps is the transmit power of the SU-Tx. In practice, the SU-Tx is
constrained by the peak transmit power, Ppk . Therefore, the capacity of the
SU-Tx can be expressed as
Ps g0
(15)
C = max log2 1 +
Pp h1 + N0
p
Subject to: Pout
≤ǫ
(16)
Ps ≤ Ppk
(17)
In view of (15), the SU should have an appropriate power control policy based
on the available CSI of both the secondary and primary network to achieve
maximal capacity.
Clearly, the model shown in Figure 9 is more practical than the one depicted in Figure 8, as the interactions between PU and SU on the same frequency
band are considered. However, the power control policy of the SU-Tx under
the outage constraint requires more information from the PU-Rx compared
to the peak interference power constraint.
3.2.4
Cooperative Communications in Underlay Networks
In traditional wireless technologies, point-to-point or point-to-multipoint communications have been widely used. In contrast to point-to-point communic-
16
Introduction
Relay
Source
Destination
Figure 10: A basic model of relay network.
ations, recent advances in cooperative communications allow different mobile
users or terminals to cooperate and to share resources in a distributed manner. In light of this idea, a large number of recent publications have proved
that cooperative communication not only increases the multiplexing gain but
also combats the detrimental effects of severe fading environments [53–60].
More importantly, the capacity is enhanced significantly by applying suitable
cooperative schemes [53, 54]. A basic cooperative communication system is
shown in Figure 10 where the source communicates with the destination by
the help of a single relay node. In the cooperation with the source, the relay
can employ different processing schemes such as decode-and-forward (DF), AF
or compress-and-forward (CF) to enhance system performance. For the DF
scheme, the relay decodes the received message from the source, re-encodes
it, and then forwards it together with the source. Alternatively, in the AF
scheme, the relay only amplifies the received message from the source with a
certain scalar and sends it to the destination. Taking advantage of multiple
user diversity, the considered basic model has been extended to multiple relays where a group of relays can aid the source to forward a message to the
destination [56–61] (see Figure 11).
Thanks to cooperative communication techniques, limitations in underlay
CRNs such as low transmission rate and short range communication can be
overcome. In a CCRN, the SU-Tx communicates with the SU-Rx through the
help of one or several SRs in which both the SU-Tx and SRs have to adapt
their transmit power to satisfy the constraint of the PUs communication in
their vicinity. In Figure 12, a simple CCRN is presented where the SU-Tx
communicates with the SU-Rx through the help of a single SR. The SR and
SU-Rx are subject to the interference from the PU-Tx, while the SU-Tx and
SR must control their transmit power to keep the interference at the PU-Rx
at an acceptable level. Similar to traditional cooperative radio networks, a
17
Introduction
Relay 1
Relay 2
Source
Destination
Relay N
Figure 11: A multiple relay network model where the source communicates
with the destination through the help of multiple relays.
CCRN with multiple SRs utilizes diversity and improves performance. In
Figure 13, an example of a CCRN with multiple SRs is depicted. In this
example, SRN is selected to support the communication between the SU-Tx
and SU-Rx since it is far away from the PU-Tx and PU-Rx. Thus, the SRN
does not suffer from PU-Tx interference or causes interference to the PU-Rx.
In summary, benefits of the CCRN are as follows:
• The interference caused by the SU to the PU can be reduced by selecting
an appropriate SR.
• The coverage range of a CRN is expanded by the help of one or many
SRs.
• By implementing relaying protocols such as DF, AF or CF at the SR,
the transmission reliability of the CRN is improved even if the direct
transmission from the SU-Tx to the SU-Rx undergoes severe fading.
3.2.5
An Overview of Underlay CRN
Inspired by all of the above power constraints, a great deal of optimal power
transmission strategies and performance analysis for CRNs have been studied
[40, 44, 46, 48, 62–68]. In [40], subject to an average received power constraint,
the capacity for non-fading channel has been investigated for an underlay
CRN. As an extension of the work reported in [40], in [34], the optimal power
18
Introduction
PU-Rx
PU-Tx
SR
SU-Rx
SU-Tx
Interference link
Communication link
Figure 12: A basic model of cognitive cooperative communication.
allocation strategy has been analyzed to obtain the maximal capacity over
fading channel. In [62], subject to various combinations of power constraints,
an optimal power allocation policy to minimize the outage probability and
to maximize the ergodic capacity has been analyzed. Regarding the outage
constraint, optimal power allocation policies for the SU-Tx have been studied
in [51], in which the outage capacity constraint of the PU were converted
into an equivalent form of the peak interference power constraint to derive
the maximal SU capacity. Later, in [44], subject to the maximal acceptable
outage probability of the PU, convex optimization techniques to analyze power
allocation policies have been applied, and then these techniques have been
used to derive the ergodic capacity and outage capacity.
Recently, the performance analysis of underlay CRNs for fading channels
have been well investigated for point-to-point communications [34, 42, 43, 69,
70]. In [34], it has been shown that ergodic capacity under Rayleigh fading is
higher than for the non-fading channel. In [42, 43], a more practical analysis
for the fundamental capacity limit, where channel knowledge is imperfect,
has been presented for Rayleigh fading channel. Closed-form expressions for
the ergodic capacity and outage probability have been derived. The results
reported in [43] have revealed that the capacity is reduced significantly due
to the influence of an imperfect CSI. Later, in [69, 70], the outage probability
and ergodic capacity have been analyzed for a general fading model, named
as α-µ fading. The results confirm that deep fading of the SU-Tx→PU-Rx
19
Introduction
PU-Tx
PU-Rx
SR1
SU-Tx
SU-Rx
SRi
SRN
Figure 13: A CCRN model with multiple SRs, where the relay SRN is selected
to forward the source messages to the SU-Rx.
link can be exploited to increase the SU-Tx transmit power which in turn
improves the performance of the secondary network.
In a multi-access channel, the SU-Tx must sense the radio environment
and select several sub-channels to communicate with the SU-Rx. Similar
to the single channel communication, the SU-Tx must adapt its transmit
power on each selected sub-channel to keep the interference given by the
PUs below a predefined threshold, which can be peak/average interference
power, peak/average transmit power, or outage probability constraint [71–74].
In particular, in [71], an optimal power allocation policy to maximize the
capacity under the outage probability constraint has been analyzed. Also,
in [72], subject to the peak transmit power, peak interference power, and
average interference power constraints, the capacity maximization problem
has been investigated. The results indicate that system performance under
the average interference power constraint outperforms the one subject to the
peak interference power constraint.
The SU can be equipped with multiple antennas to exploit advantages
of both multiplexing and diversity gain in order to improve the performance
of a CRN and to reduce the interference to the PU [75–85]. In particular,
in [77], a single-input multiple-output (SIMO) multiple access channel of a
CRN under the peak interference power constraint has been studied. Given
this setting, the signal-to-interference-plus-noise ratio (SINR) balancing problem and the sum-rate maximization problem have been analyzed. In [78], sub-
20
Introduction
ject to a joint average and peak interference power constraint of a single PU,
a SIMO CRN has been investigated, and an optimal power allocation policy
and beamforming weights of the SU to minimize the outage probability have
been derived for Nakagami-m fading. In [79], given the peak interference
power constraint, ergodic capacity and outage capacity for a multiple-input
multiple-output (MIMO) CRN have been investigated. Besides the aforementioned approaches, game theory applied for the MIMO CRNs have been
investigated to optimize power allocation in [86,87]. More recent results about
multiple antenna techniques for CRNs can be found in [88–90] and the references therein.
Taking advantage of diversity, the performance analysis of a CCRN has
obtained great attention [91–97]. For instance, in [98], outage probability and
ergodic capacity of a CCRN, subject to the peak interference power constraint
of multiple PUs, have been analyzed for Nakagami-m fading channels. The
results have shown the impact of the fading parameter and the number of
the PUs on the system performance. In [96], the asymptotic outage behavior
for three cooperative schemes, selective AF, selective DF and AF with partial
relay selection, have been studied. More information about CCRNs can be
found in [94, 98–108] and the references therein.
3.3
Overlay Paradigm
In an overlay system, the SU-Tx is assumed to have information about the
PU such as codebooks and messages. The SU can obtain these information by
advanced decoding technique when the PU-Tx starts its transmission. Using
the obtained information from the PU, the SU-Tx can apply a special coding
scheme such as dirty paper coding [109], to aid the PU communication and
to achieve its own transmission rate [110–113]. This approach is a promising
solution to improve the performance of both secondary and primary networks.
However, the assumption of the SU having full messages of the PU at the start
of its transmission may be unreasonable in practice. This is because the PUTx may stay far away from the SU, or it is hidden by obstacles so that the
SU cannot decode the PU-Tx messages at the beginning of a transmission
phase.
4
Propagation Models
The wireless channel is an electromagnetic environment in which signals from
the transmitter antennas can propagate to the receiver antennas. This environment includes all physical objects which can influence the propagation
path between transmitter and receiver. Accordingly, these factors directly
Introduction
21
affect the performance of a CRN such as detection probability, false alarm
probability, transmit power control policy, channel capacity, and so on. In
this section, main characteristics of channel models with emphasis on path
loss and fading, which will be used in this thesis, are presented.
4.1
Path Loss Model
In practice, free-space conditions for wireless communications rarely exist due
to the presence of obstacles as shown in Figure 14. Therefore, the signal is attenuated by distance and impaired by large objects between transmitter and
receiver antennas such as buildings, hills, forest, and so on. In addition, movements of smaller objects such as cars, trains, users, etc., can cause rapid signal
variations of the propagating radio waves. As a result, the wireless channel
becomes more random and more difficult to predict than a wired channel. As
such, a propagation model should consider parameters such that it can reveal the effect of a radio environment on the received power. For example, a
generic description of received power in linear scale can be formulated as [114]
ν d0
ζs
(18)
Pr = Pt a
d
where Pt and Pr denote transmitted power, and received power, respectively;
a is the adjustment parameter depending on antenna gains, frequency, and
other physical factors. Furthermore, ν is called path loss exponent which
depends on the radio environment (see Table 1). The symbols d and d0
denote the path length and a reference distance, respectively. Finally, ζ and s
are small-scale and large-scale fading factors of the propagation environment,
respectively, which are described as follows:
• Small-scale fading (multipath fading): Received signal amplitude and
phase change rapidly due to user movements over a short distance. This
happens when the transmitted signal arrives at the receiver through
different paths.
• Large-scale fading (shadowing): Average received power changes gradually due to user movements over large distances. This phenomenon is
caused by large obstructions such as hills and high buildings located
between transmitter and receiver antennas.
From (18), the average path loss (PL) in linear and logarithmic domain
may be expressed, respectively, as
ν
Pt
d
1
PL =
=
(19)
Pr
aζs d0
22
Introduction
Defraction
Scattering
Transmission
Receiver
Transmitter
Reflection
Figure 14: Propagation mechanisms for wireless communications where the
transmitted signals travel through different paths to arrive at the receiver.
Table 1: Path loss exponents for different radio environments [115].
Environment
ν range
Free space
2.0
Urban microcells
2.7-3.5
Urban macrocells
3.7-6.5
Office Building (same floor)
1.6-3.5
Office Building (multiple floors)
4.0-6.0
Factory
1.6-3.3
and
PL[dB] = 10 log10
Pt
Pr
= A + 10ν log
d0
d
, d ≥ d0
(20)
where A = −10 log10 a − 10 log10 ζ − 10 log10 s.
4.2
Statistical Multipath Channel Models
Generally, the term fading is used to describe the fluctuations in the received
power and is classified as large-scale and small-scale fading as mentioned
above. Large-scale fading is used to characterize the average received signal
strength that decays over relatively large separation distances. On the other
hand, the rapid fluctuations of received signal strength and phase over very
short distance or a short period of time is known as small-scale fading [116].
Introduction
23
Note that a precise deterministic channel model for wireless communications is
either impossible or difficult to obtain. In this case, statistical models become
more appropriate to characterize the fading channels.
4.2.1
Input/Output Model of a Wireless Channel
In view of [115], the transmitted signal with zero initial phase offset can be
modeled as
s(t) = R{u(t) exp(j2πfc t)} = R{u(t)} cos(2πfc t) − I{u(t)} sin(2πfc t) (21)
where u(t) is the equivalent low pass signal of s(t) with bandwidth B, and fc
is the carrier frequency. R{·} and I{·} denote real part and imaginary part
of a complex signal, respectively.
The transmitted signal often travels through different paths before reaching at the receiver. This phenomenon is caused by reflections, scattering,
diffractions, and other factors in the radio propagation environment (see Figure 14). The effect of multipath results in variations of the received signal.
Therefore, at the receiver side, the received signal may be formulated as a
sum of line-of-sight (LOS) path and all multipath components, given by
X
r(t) = R
(22)
αn (t)u(t − τn (t)) exp[j2πfc (t − τn (t)) + ΦDn ]
n
where τn (t), ΦDn , and αn (t) are the delay of multipath component, Doppler
phase shift, and amplitude.
For simplicity, we can set Φn (t) = 2πfc τn (t) − ΦDn , and then rewrite the
received signal as
X
r(t) = R
αn (t) exp[−jΦn (t)]u(t − τn (t)) exp(j2πfc t)
(23)
n
Assume that the delay spread of a channel is much smaller than the inverse
of the signal bandwidth B, and u(t − τn (t)) ≈ u(t). Then, the received signal
given in (23) can be rewritten as [115, Eq. (3.14)]
r(t) = rI (t) cos(2πfc t) − rQ (t) sin(2πfc t)
(24)
where rI (t) and rQ (t) are in-phase and quadrature components, given by
X
rI (t) =
αn (t) cos Φn (t)
(25)
n
rQ (t) =
X
αn (t) sin Φn (t)
(26)
n
If the number of multipath components is large enough, rI (t) and rQ (t) can
be approximated as Gaussian random processes [115].
24
4.2.2
Introduction
Rayleigh Fading
Let the in-phase rI (t) and quadrature component rQ (t) be Gaussian random
variables (RVs). As the phase Φn (t) is uniformly distributed, the terms rI (t)
and rQ (t) are zero-mean Gaussian RVs having a common variance σ 2 . The
signal amplitude can be expressed as
q
2 (t)
(27)
z(t) = |r(t)| = rI2 (t) + rQ
and |r(t)| is an RV with Rayleigh distribution given as
2
2z
−z
, z≥0
fZ (z) =
exp
Ω
Ω
P
where Ω = E[α2n ] = 2σ 2 is the average received power of the signal.
(28)
n
Moreover, we can find the power distribution by changing of variables
h = z 2 (t) = |r(t)|2 (h is also called channel power gain or channel gain) as
x
1
fh (x) = exp −
, x≥0
(29)
Ω
Ω
The cumulative distribution function (CDF) of h can be easily found as
x
Fh (x) = 1 − exp −
(30)
Ω
In fact, the Rayleigh distribution is an efficient model to characterize multipath fading in which the signal is scattered before it arrives at the receiver,
i.e., there is no LOS path between transmitter and receiver antenna.
4.2.3
Rician Fading
If there exists a LOS path between transmitter and receiver antennas, the
signal amplitude is distributed following a Rician distribution, given by [115,
117]
z
zs
(z 2 + s2 )
fZ (z) = 2 exp −
I0
,z ≥ 0
(31)
σ
2σ 2
σ2
P
where s2 = α0 denotes the power in the LOS component and 2σ 2 = n≥1 E[α2n ]
is the average power in the non-LOS multipath components. The function
I0 (·) is the modified Bessel function of the 0-th order. The average received
power for this fading channel is given by [115, Eq. (3.35)]
Ω=
Z∞
0
z 2 fZ (z)dz = s2 + 2σ 2
(32)
25
Introduction
By setting K =
s2
2σ2 ,
we can rewrite (31) in terms of K and Ω as
(K + 1)z 2
2z(K + 1)
I0
exp −K −
fZ (z) =
Ω
Ω
2z
r
K(K + 1)
Ω
!
, z≥0
(33)
where K is the fading parameter given in the range [0, ∞). For K = 0,
there is no LOS component, i.e., the channel follows Rayleigh fading, while
for K = ∞, multipath components do not exist.
Consequently, the power distribution for Rician fading can be formulated
as
(K + 1)
(1 + K)x
fh (x) =
I0
exp −K −
Ω
Ω
2
s
K(1 + K)x
Ω
!
(34)
Rician fading is often observed in microcellular urban, suburban land-mobile,
and factory environment. Also, this model may be applicable to analyze
the performance of satellite and ship-to-ship communications where dominant
LOS component do exist [117].
4.2.4
Nakagami-m Fading
Measurement campaigns have shown that Rayleigh and Rician fading models
sometime do not fit well with experimental data. Thus, an alternative fading
model was developed to fit a variety of empirical measurements, named as
Nakagami-m fading. Its probability density function (PDF) of the amplitude
is formulated as [115, 117]
mz 2
2mm z 2m−1
, z≥0
exp −
fZ (z) =
Ωm Γ(m)
Ω
(35)
where m is called fading severity parameter which is in the range of [0.5, ∞).
Nakagami-m includes special cases of fading models such as Rayleigh (m = 1),
and one-sided Gaussian (m = 0.5). Specifically, as m → +∞ the Nakagamim fading channel becomes a non-fading channel, i.e., AWGN channel. In
addition, it can approximate Rician fading with parameter K by setting m =
(K + 1)2 /(2K + 1). It should be noted that Nakagami-m fading becomes less
severe when m is increased.
Furthermore, the channel power gain of Nakagami-m fading is character-
26
Introduction
ized by the gamma distributions with PDF and CDF, respectively, given by
mx mm xm−1
, x ≥ 0, m ≥ 0.5
exp
−
Ωm Γ(m)
Ω
Γ(m, mx
Ω )
Fh (x) = 1 −
, x≥0
Γ(m)
fh (x) =
(36)
(37)
where Γ(·) and Γ(·, ·) are gamma and incomplete gamma functions which are
defined, respectively, as
Γ(m) =
Z∞
tm−1 e−t dt
(38)
0
Γ(a, x) =
Z∞
ta−1 e−t dt
(39)
x
In practice, Nakagami-m fading is applicable to evaluate the performance
of multipath propagation for land-mobile systems [118] and indoor-mobile
systems [119].
4.2.5
Weibull Fading
Weibull fading is another effective statistical model for wireless communications. It is used to characterize the amplitude in both indoor and outdoor
environments [120, 121]. Specifically, it is appropriate for mobile systems operating in the 800-900 MHz frequency range. The PDF of the amplitude for
Weibull fading is expressed as
" a
a2 #
Γ(1 + 2/a) 2 a−1
z2
, z ≥ 0 (40)
Γ (1 + 2/a)
fZ (z) = a
z
exp −
Ω
Ω
where a is the fading parameter. By varying fading parameter a, the Weibull
fading becomes a specific distribution. For example, for a = 1, it describes an
exponential distribution, and for a = 2 it characterizes a Rayleigh distribution.
The PDF and CDF of the channel power gain can be presented, respectively, as
a Γ(1 + 2/a)
x a2
a
−1
2
fh (x) =
, x≥0
(41)
exp − Γ(1 + 2/a)
x
2
Ω
Ω
x a2
(42)
Fh (x) = 1 − exp − Γ(1 + 2/a)
Ω
27
Introduction
4.2.6
α − µ Fading
Recently, a new general distribution, which is used to model fading channels,
has been introduced by M. D Yacoub, named as α − µ distribution [122]. It
is known as a general, flexible, and easy tractability model. Furthermore, it
includes some other important distributions such as Rayleigh, Nakagami-m,
Weibull, one-sided Gaussian, Gamma, and exponential distribution. According to [122], the PDF of the channel amplitude can be formulated as
zα
αµµ z αµ−1
exp −µ α , z > 0, α, µ > 0,
fZ (z) = αµ
(43)
ẑ Γ(µ)
ẑ
where
α is the power parameter, ẑ is an α-root mean value given by ẑ =
p
α
E(Z α ), and µ is the inverse of the normalized variance of Z α given by
µ=
E2 (Z α )
V(Z α )
(44)
where E(·) and V(·) are expectation and variance, respectively. Accordingly,
the PDF and CDF of channel power gain can be expressed, respectively, as
follows [123]:
x α/2
αxαµ/2−1
exp
−
αµ/2
ϑ
2Γ(µ)ϑ
i
h
α/2
Γ µ, ϑx
Fh (x) = 1 −
Γ(µ)
fh (x) =
(45)
(46)
and Ω = E[Z 2 ]. By varying fading parameters (α; µ) in
where ϑ = Ω Γ(µ+2/α)
Γ(µ)
(43), we can obtain different distributions such as Rayleigh (α = 2; µ = 1),
Nakagami-m (α = 2; µ = m), Weibull (µ = 1), exponential (α = 1; µ = 1),
and one-sided Gaussian distribution (α = 1; µ = 0.5).
5
Applications of Cognitive Radio Networks
CRNs have intelligent capabilities to adapt with the change of radio environment such that it can enhance the spectral efficiency and obtain a reliable
communication. After many years of effort to establish rules for unlicensed TV
bands, in September 2010 [124], the FCC has announced official approaches
to use the spectrum holes in TV bands, for which, the access fashions (centralized or distributed fashions) can be designed to utilize the temporary free
radio spectrum.
28
Introduction
In military, the cognitive characteristics become one of the most important
communication techniques. With cognitive capabilities, the SUs can increase
more transmission opportunities, secure their communications, and may detect the enemies information. Moreover, the spectrum congestion can reduce
significantly by using dynamic spectrum access and efficient bandwidth allocation. Recently, the department of defense, U.S.A., has supported two
research projects, next Generation (XG) and SPEAKeasy radio system [125],
to study about the advantages of the CR technology.
Moreover, under severe conditions such as terrorist attack, natural disaster
or accidents, the existing communication infrastructure could be destroyed.
Hence, an emergency network, which can support the communication between
survivors and responders, is invaluable. Currently, the infrastructure of wireless networks is inadequate to meet the future requirements of emergency
events. Therefore, the SU with cognitive capabilities can recognize available spectrum and reconfigure itself to provide a reliable emergency network
with minimal information delay [126]. Additionally, intelligent applications
equipped in the SU can increase the searching ability significantly and locate
the survivors more precisely in extreme environments.
Especially, CRN is considered as a key enabling wireless technology in
the commercial markets. Because the SU can be aware of the optimal channel conditions and automatically switch to an unoccupied channel, it can
provide additional bandwidth for data application and avoid spectrum conflict between users. Additionally, new intelligent radio management software
can be integrated into cognitive devices. As such, configuration and service
bandwidth can be updated easily by new versions of radio software, and enduser hardware does not need to be updated.
Most recently, femto-cells, a concept of small cells with inexpensive, low
power base stations, has been proposed to overcome the traffic growth and indoor coverage range problem [127, 128]. However, the centralized interference
management is a difficult problem. With cognitive capabilities such as searching ability and estimating the available spectrum bands, the coverage range
of femto-cells have been improved significantly. Moreover, the interference to
the others cells can also be avoided.
6
Challenges of Cognitive Radio Networks
Overall, the benefits of the CRN are obvious. However, there are many challenging problems that need to be solved before CRNs can be implemented in
pratice such as [129]:
• Common control channel: A common control channel supports many
Introduction
29
functionalities of a CRN. It is an efficient approach to exchange information during spectrum sensing and communication of the CR. However,
this channel is not always available due to the randomness appearance
of the PU. It can be occupied by the PU at an unpredictable time. In
this context, a fixed common control channel implemented for the CRN
is infeasible. In order to properly operate in a CRN, the common control channel setup and its maintainance mechanism are expected to need
more advanced investigations.
• Channel estimation: To protect the communication of the PU and enhance the performance of the CRNs, channel information between the
SU and the PU are very important. In practice, it is difficult to obtain
exactly the CSI due to fading, path loss, delay, and so on.
• Joint sensing and access: The sensing and spectrum access modules
are usually designed separately. Though they should be integrated to
optimize the SU sensing time and transmit power [130]. One of the
most concerned problems in CRNs is how to sense multiple channels
and utilize these multiple random channels efficiently.
• Location information: Knowing distances between the PU and the SU
are crucial in a CRN. Based on the distance among users, the SU can regulate its communication parameters to cancel interference and enhance
system performance. Existing works often assume that the information
about distance and PU transmit power are available for simple investigation. However, this assumption may not be always true in a practical
CRN.
• CRN architectures: To implement a CRN with full characteristics of
CRN prototypes, a cross-layer architecture of the SUs is essential. An
expected cross-layer architecture for the CR should be flexible to meet
different QoS requirements. This is a complicated problem and desires
more investigations.
7
Thesis Overview
This thesis has aimed at investigating CRN with spectrum underlay access in
which the SU transmit power is subject to practical constraints such as peak
interference power and outage constraint of the PU, and peak transmit power
constraint of the SU. The thesis consists of five parts based on a number
of peer-reviewed conference papers, and one published and two submitted
journal articles as follows. Part I studies the impact of interference from the
30
Introduction
PU on the performance of the SU in a CRN, in which the SU-Tx transmit
power is subject to both the outage constraint of the PU-Rx and the peak
transmit power constraint of the SU-Tx. In Part II, the performance analysis
of a CCRN under the joint timeout probability constraint of the PU and the
peak transmit power constraint of the SU is investigated, where a single SR
operating in full-duplex mode is used to assist the SU communication. In
Part III, subject to the peak interference power constraint of multiple PUs,
a performance evaluation for a CCRN with multiple DF SRs is provided.
Part IV investigates the delay performance for point-to-point and point-tomultipoint communications of spectrum sharing networks, where the SU-Tx
is subject to the peak interference power constraint of multiple PUs. Part
V studies the performance of an underlay network over α-µ fading channels,
where the delay of acknowledgement (ACK) and fading parameters of the
feedback channel are taken into account. In the following, a summary of the
contribution of each part is provided.
Part I - Impact of Interference from Primary
Networks on the Performance of Secondary Networks
In this part, we investigate the performance of a single-input and multipleoutput cognitive radio network over Rayleigh fading. In particular, we assume
that SU-Tx and PU-Tx are equipped with a single antenna while SU-Tx
and PU-Rx have multiple antennas. Additionally, the SU-Tx transmit power
is subject to outage constraint of the primary network and peak transmit
power constraint of the secondary network. Given these settings, an adaptive
transmit power allocation policy for the SU-Tx, a closed-form expression for
outage probability, and an approximation for ergodic capacity are obtained.
These formulas will be used to examine the impact of the PU-Tx transmit
power, the number of antennas at receivers, and channel mean powers on
the performance of the secondary network. More importantly, our results
reveal that the SU-Tx using the power allocation policy can obtain optimal
performance.
Part II - Performance Analysis of a CCRN with
a Buffered Relay
In this paper, we analyze the packet transmission time in a CCRN where a
SU-Tx sends packets to a SU-Rx through the help of a SR. In particular,
Introduction
31
we assume that the SU-Tx and SR are subject to the joint constraint of the
timeout probability of the PU and the peak transmit powers of the secondary
users. On this basis, we investigate the impact of the transmit power of
the PUs and channel mean powers on the packet transmission time of the
CCRN. Utilizing the concept of timeout, adaptive transmit power allocation
policies for the SU-Tx and SR are considered. More importantly, analytical
expressions for the end-to-end throughput, end-to-end packet transmission
time, and stable condition for the SR operation are obtained. Our results
indicate that the second hop of the considered CCRN is not a bottleneck if
the channel mean powers of the interference links among the networks are
small and the SR peak transmit power is set to a high value.
Part III - Cognitive Cooperative Networks with
Decode-and-Forward Relay Selection under Interference Constraints of Multiple Primary
Users
In this part, we study the performance of cognitive cooperative radio networks
under the peak interference power constraints of multiple primary users. In
particular, we consider a system model where the secondary user communication is assisted by multiple secondary relays that operate in the decode-andforward mode to relay the signal from a secondary transmitter to a secondary
receiver. Moreover, we assume that the transmit powers of the secondary
transmitter and the secondary relays are subject to the peak interference
power constraints of the multiple primary users that operate in their coverage
range. Given this system setting, we first derive the cumulative distribution function of the instantaneous end-to-end signal-to-noise ratio. Then, we
obtain a closed-form expression for the outage probability and an exact expression for the symbol error probability of the considered network. These
tractable formulas enable us to examine the impact of the presence of multiple primary users on the performance of the considered spectrum sharing
system. Furthermore, our numerical results show that system performance
is improved significantly when the number of secondary relays increases or
the channel mean power from the secondary user to the primary users is low.
Also, any increase in the number of primary users in the coverage range of
the secondary transmitter or the secondary relays leads to degradation in system performance. Finally, Monte Carlo simulations are provided to verify the
correctness of our analytical results.
32
Introduction
Part IV - Delay Performance of Cognitive Radio Networks for Point-to-Point and Point-toMultipoint Communications
In this part, we analyze the packet transmission time in spectrum sharing
systems where a SU simultaneously accesses the spectrum licensed to PUs.
In particular, under the assumption of an independent identical distributed
Rayleigh block fading channel, we investigate the effect of the peak interference power constraint imposed by multiple PUs on the packet transmission
time of the SU. Utilizing the concept of timeout, exact closed-form expressions of outage probability and average packet transmission time of the SU
are derived. In addition, employing the characteristics of the M/G/1 queuing
model, the impact of the number of PUs and their peak interference power
constraint on the stable transmission condition and the average waiting time
of packets at the SU are examined. Moreover, we then extend the analysis for
point-to-point to point-to-multipoint communications allowing for multiple
SUs and derive the related closed-form expressions for outage probability and
successful transmission probability for the best channel condition. Numerical
results are provided to corroborate our theoretical results and to illustrate applications of the derived closed-form expressions for performance evaluation
of cognitive radio networks.
Part V - Performance of Cognitive Radio Networks over General Fading Channels
In this part, the performance analysis of cognitive radio networks over α-µ
fading channels is investigated. In particular, we consider a scenario where the
SU-Tx sends a data packet to the SU-Rx and waits for an ACK to transmit the
next packet. If a given round-trip-time (RTT) is expired, the data packet is
retransmitted. In addition, all operations of the secondary users are assumed
to be subject to the peak interference power constraint of a primary user.
Given these settings, the cumulative distribution function of packet transmission time is derived. More importantly, performance metrics in terms of a
lower bound of timeout probability and average number of transmissions per
packet are obtained by utilizing the timeout concept. These formulae allow
us to examine the impact of the delay of data packet transmission and ACK
transmission, and distances among users on the system performance for various fading channels, such as one-sided Gaussian, Rayleigh, Nakagami-m, and
Weibull channels.
Introduction
33
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Part I
Part I
Impact of Primary Networks on the
Performance of Secondary Networks
Part I is published as:
H. Tran, M.-A. Hagos, M. Mohamed, and H.-J. Zepernick, “Impact of Primary Networks on the Performance of Secondary Networks, in Proc. International Conference on Computing, Management and Telecommunications, Ho
Chi Minh City, Vietnam, Jan. 2013, pp. 1–6.
Impact of Primary Networks on the
Performance of Secondary Networks
Hung Tran, Maarig Aregawi Hagos,
Marshed Mohamed, and Hans-Jürgen Zepernick
Abstract
In this paper, we investigate the performance of a single-input and
multiple-output cognitive radio network over Rayleigh fading. In particular, we assume that secondary transmitter (SU-Tx) and primary
transmitter (PU-Tx) are equipped with a single antenna while secondary receiver (SU-Rx) and primary receiver (PU-Rx) have multiple
antennas. Additionally, the SU-Tx transmit power is subject to outage
constraint of the primary network and peak transmit power constraint
of the secondary network. Given these settings, an adaptive transmit power allocation policy for the SU-Tx, a closed-form expression
for outage probability, and an approximation for ergodic capacity are
obtained. These formulas will be used to examine the impact of the PUTx transmit power, the number of antennas at receivers, and channel
mean powers on the performance of the secondary network. More importantly, our results reveal that the SU-Tx using the power allocation
policy can obtain optimal performance.
1
Introduction
The rapid growth of wireless services has led to radio resources getting exhausted. However, measurement campaigns have shown that almost all allocated spectrum is under-utilized at a specific geographical area, time, and
frequency [1,2]. This implies that the shortage of the radio spectrum is mainly
49
50
Part I
due to inefficient usage rather than real scarcity. In order to overcome this
problem, cognitive radio networks (CRNs) have been proposed as a promising
solution [3].
In a CRN, secondary users (SUs) are allowed to access the licensed radio
frequency band of primary users (PUs) as long as the quality of service (QoS)
of the primary network is assured. According to [4], CRNs are classified into
three main techniques, spectrum overlay, interweave, and spectrum underlay.
In spectrum overlay, the SU can transmit simultaneously with the PU, given
that the SU knows channel state information (CSI), codebook, and messages
of PUs to keep the transmission rate of the primary network stable. In the interweave approach, the secondary transmitter (SU-Tx) can access the licensed
frequency if and only if it is not occupied by the PU. In contrast to interweave
and overlay, the SU in spectrum underlay may cause limited interference to
the PU as long as this interference is kept below a predefined level.
Recently, the multiple antennas technique has been considered to improve
the performance of underlay networks. Accordingly, many works have studied
the impact of multiple antenna terminals on the system performance under
various interference constraints of primary networks, e.g. [5–11] and the references therein. More particularly, in [5], the maximum sum-rate of a singleinput multiple-output (SIMO) CRN, where the SU-Tx is subject to the peak
transmit power and interference constraint of the PU, has been analyzed. Regarding the capacity of spectrum underlay CRN, fundamental capacity limits
for a multiple-input multiple-output (MIMO) system has been investigated
in [6]. In [7], the mean uplink capacity of a MIMO CRN under average interference power constraint has been derived. The results reported in [7], have
shown the impact of the number of PUs and SUs on system performance.
In [12], the performance of a SIMO CRN, in which the SU-Tx is subject to
the combined constraint of average interference power and peak interference
power of the PU, has been investigated for Nakagami-m fading. Following
this study, an optimal power allocation, beamforming weights, and outage
probability of a secondary network have been derived. Later, in [11], a MIMO
CRN with transmit antenna selection has been studied, and the system performance in terms of ergodic capacity and outage capacity has been analyzed.
More recently, in [13], a pre-coding scheme has been introduced for a MIMO
CRN in order to minimize interference to the PU and maximize the sum capacity. Our most recent works, reported in [14], have examined the impact
of the number of antennas, and distances between SU and PU on the performance of a SIMO CRN. The results have shown that subject to the peak
interference power constraint, the system performance degrades significantly
as the number of antennas of the primary receiver (PU-Rx) increases. However, the impact of interference from the primary transmitter (PU-Tx) to the
Impact of Primary Networks on the Performance of Secondary Networks
51
secondary receiver (SU-Rx) on the performance of the secondary network has
not been investigated.
In this paper, we analyze the performance of a SIMO CRN in which the
impact of interference from the primary network on the system performance
of the secondary network is studied. In particular, we assume that the PU-Tx
and SU-Tx are equipped with a single antenna while the PU-Rx and SU-Rx
are equipped with multiple antennas. Additionally, the SU-Tx transmit power
is subject to the outage constraint of the PU as well as the peak transmit
power constraint of the SU. We further assume that both PU-Rx and SU-Rx
use selection combining (SC) to process the received signal, and all channels
undergo Rayleigh fading. The contributions of this paper are summarized as
follows:
• An adaptive transmit power allocation policy for the SU-Tx is derived.
Utilizing this policy, the cumulative distribution function (CDF) and
probability density function (PDF) of the signal-to-interference-plusnoise ratio (SINR) are also obtained.
• A closed-form expression for the outage probability and an approximate
expression for the ergodic capacity are obtained. These formulas will be
used to examine the impact of the PU-Tx transmit power, the number of
antennas at the receivers, and channel mean powers on the performance
of the secondary network.
• The results indicate that the performance of the secondary network is
improved significantly as the number of antennas of the PU-Rx or SU-Rx
increases.
The remainder of this paper is organized as follows. In Section 2, the
system model and assumptions are introduced. In Section 3, the performance
analysis in terms of outage probability and ergodic capacity are provided.
Numerical results are presented in Section 4. Finally, conclusions are given in
Section 5.
2
System Model
Let us consider a spectrum underlay network as shown in Figure 1 in which the
SU shares the same frequency band with the PU. The SU-Tx and PU-Tx are
equipped with a single antenna while the SU-Rx and PU-Rx have N and M
antennas, respectively. Both SU-Rx and PU-Rx use SC to process the received
signal. For mathematical modeling, we denote the instantaneous channel
power gains of the SU-Tx→SU-Rx and PU-Tx→PU-Rx links by gi and hj ,
52
Part I
h1
PU-Tx
hj
hM
PU-Rx
1
j
M
f1
fi
SU-Tx
g1
gi
fN
gN
SU-Rx
Interference links
Communication links
Figure 1: System model of a CRN in which SU-Tx and PU-Tx have a single
antenna while the SU-Rx and PU-Rx are equipped with N and M antennas,
respectively.
i = 1, 2, . . . , N , j = 1, 2, . . . , M , respectively. The instantaneous channel
power gains of the PU-Tx→SU-Rx and SU-Tx→PU-Rx interference links are
denoted, respectively, by fi and βj . We assume that all channels undergo
Rayleigh fading and system bandwidth is normalized to one. Accordingly,
the instantaneous channel power gains, hj , gi , βj , and fi are independent
exponentially distributed random variables (RVs) with channel mean powers
denoted by Ωh , Ωg , Ωβ , and Ωf , respectively.
It is noted that the PU allows the SU to access its licensed frequency
band as long as the PU QoS is not compromised by the interference from
the SU. In the considered system model, the QoS of the PU is assured if the
outage probability is kept below a predefined constraint. This condition may
be formulated as
p
≤ γth } ≤ ε
Ppout = Pr {γsc
(1)
where γth = 2rp − 1, rp and ε are outage transmission rate and outage conp
straint of the primary network, respectively. The symbol γsc
represents the
instantaneous SINR at the PU-Rx defined as
Pp hj
p
(2)
γsc
= max
j=1,2,...,M
βj Ps + N0
where Pp , N0 , and Ps are average transmit power of the PU-Tx, average
noise power, and instantaneous transmit power of the SU-Tx, respectively. It
Impact of Primary Networks on the Performance of Secondary Networks
53
is important that the SU-Tx transmit power must be controlled such that the
condition given in (1) is satisfied.
Moreover, the transmit power is usually limited in practice. Thus, the
SU-Tx transmit power is subject to an additional constraint as
Ps ≤ Ppk
(3)
where Ppk is the peak transmit power of the SU-Tx. In the next section, the
combined constraint given in (1) and (3) will be used to derive the adaptive
transmit power allocation policy for the SU-Tx.
3
Performance Analysis
In this section, we derive the transmit power allocation policy, outage probability, and approximate expression for the ergodic capacity. For this purpose,
let us commence with the following lemma.
Lemma 1. Assume Xj and Zj , j = 1, 2, . . . , L are independent exponentially
distributed RVs with mean Ωx and Ωz respectively, and a1 , a2 , a3 are positive
constants. A random variable Y is defined as
Y =
max
j=1,2,...,M
a1 X j
a2 Z j + a3
(4)
with CDF and PDF, respectively, as
L
exp (−yC)
FY (y) = 1 −
yD + 1
L−1
X L − 1
fY (y) =L
(−1)k exp (−yC(k + 1))
k
k=0
C
D
×
+
(1 + Dy)k+1
(1 + Dy)k+2
where C =
a3
a1 Ωx
and D =
(5)
(6)
a2 Ωz
a1 Ωx .
Proof: See Appendix A.
54
3.1
Part I
Transmit Power Allocation Policy of the SU-Tx
Substituting (2) into (1), we can derive a closed-form expression for the outage
probability of the primary network by using (5) in Lemma 1 as
Pp hj
(7)
≤ γth
Ppout = Pr
max
j=1,2,...,M
βj Ps + N0
M
N0 γth
Pp Ωh
exp −
= 1−
≤ε
γth Ps Ωβ + Pp Ωh
Pp Ωh
From (7), the maximum transmit power of the SU-Tx under the outage constraint can be obtained after some manipulations as
Pp Ωh
1
N0 γth
√
Ps =
−
1
(8)
exp
−
γth Ωβ 1 − M ε
Pp Ωh
Moreover, combining (8) with (3) yields the adaptive transmit power allocation policy for the SU-Tx as



N0 γth


exp
−
Pp Ωh
Pp Ωh 

√
−
1
P = min Ppk ,
(9)


γth Ωβ
1− M ε
By dividing both side of (9) by N0 , we can rewrite (9) in terms of transmit
signal-to-noise ratio (SNR) as



γth


γP T Ωh  exp − γP T Ωh

√
−
1
(10)
γST = min γpk ,


γth Ωβ
1− M ε
where γP T = Pp /N0 and γpk = Ppk /N0 denote the PU-Tx transmit SNR and
the SU-Tx peak transmit SNR, respectively.
3.2
CDF and PDF of the SINR at the SU-Rx
Now, the SU-Tx uses the power allocation policy given in (9) to transmit the
signal to SU-Rx. Accordingly, the SINR at each antenna branch of the SU-Rx
is expressed as
γis =
Pgi
Pp fi + N0
As the SU-Rx uses SC, the SINR at the SU-Rx is formulated as
Pgi
s
γsc
= max
i=1,2,...,N
Pp fi + N0
(11)
(12)
Impact of Primary Networks on the Performance of Secondary Networks
55
s
Consequently, the CDF and PDF of γsc
can be obtained by applying Lemma 1
as
N
Pgi
exp (−Gγ)
sc
(13)
Fγs (γ) = Pr
max
≤γ = 1−
i=1,2,...,N
Pp fi + N0
(1 + Aγ)
N
−1 X
N −1
fγssc (γ) =N
(−1)k exp (−γG(k + 1))
(14)
k
k=0
A
G
+
×
(1 + Aγ)k+1
(1 + Aγ)k+2
where A =
3.3
γP T Ωf
γST Ωg
, G=
1
γST Ωg .
Outage Probability of the Secondary Network
Outage probability of the secondary network is defined as the probability that
the instantaneous SINR is dropped below a given threshold. Using (13), the
outage probability can be easily obtained as
Psout = Pr {γssc ≤ βth }
N
exp (−Gβth )
= Fγssc (βth ) = 1 −
(1 + Aβth )
(15)
where βth = 2rs − 1 and rs is the outage transmission rate of the secondary
network.
3.4
Ergodic Capacity of the Secondary Network
Ergodic capacity is defined as the maximum long-term achievable rate over
all channel state information and can be expressed as
Cer =
Z∞
log2 (1 + γ)fγssc (γ)dγ = N
k=0
0
×
Z∞
N
−1 X
N −1
(−1)k
k
G exp (−γG(k + 1)) log2 (1 + γ)
dγ
(1 + Aγ)k+1
0
+
Z∞
0
A log2 (1 + γ) exp (−γG(k + 1))
dγ
(1 + Aγ)k+2
(16)
56
Part I
To the best of our knowledge, a closed-form expression for the ergodic capacity
(16) is not available. However, we can derive an approximate ergodic capacity
to examine the system performance by applying the well-known Taylor series
expansion of the logarithm function as
ln(1 + γ) ≈ ln(1 + γ̄) +
(γ − γ̄)2
γ − γ̄
+ o[(γ − γ̄)2 ]
−
1 + γ̄
2(1 + γ̄)2
(17)
where o[γ n ] is defined as Ra polynomial function with power equal or greater
∞
than n and γ̄ = E[γ] = 0 γfγssc (γ)dγ. Then, the ergodic capacity can be
approximated by using (17) as
E[γ 2 ] − γ̄ 2
(18)
Cer = E[log2 (1 + γ)] ≈ log2 (e) ln(1 + γ̄) −
2(1 + γ̄)2
Clearly, the approximate ergodic capacity (18) can be obtained if the mean of
SINR γ̄ and mean square SINR E[γ 2 ] are found. By definition, the first and
second moment of SINR can be written, respectively, as
E[γ] =
Z∞
γfγssc (γ)dγ = N
k=0
0
2
E[γ ] =
Z∞
N
−1 X
2
γ fγssc (γ)dγ = N
N
−1 X
k=0
0
N −1
(−1)k (I0 + I1 )
k
N −1
(−1)k (I2 + I3 )
k
(19)
(20)
where
I0 = G
Z∞
γ exp(−γG(k + 1))
dγ
(1 + Aγ)k+1
(21)
I1 = A
Z∞
γ exp(−γG(k + 1))
dγ
(1 + Aγ)k+2
(22)
I2 = G
Z∞
γ 2 exp(−γG(k + 1))
dγ
(1 + Aγ)k+1
(23)
I3 = A
Z∞
γ 2 exp (−γG(k + 1))
dγ
(1 + Aγ)k+2
(24)
0
0
0
0
In order to solve the integrals I0 , I1 , I2 , and I3 , let us consider the following
lemma.
Impact of Primary Networks on the Performance of Secondary Networks
57
Lemma 2. Given 1 ≤ n ≤ 2, a > 0, b > 0, and m > 0, we have
I=
Z∞
Γ(1 + n)
b
tn exp (−bt)
dt =
Φ(1 + n, 2 − m + n, )
(1 + at)m
an+1
a
(25)
0
where Φ(·, ·, ·) and Γ(·) are the confluent hypergeometric function [15, eq.
(9.211.4)] and gamma function [15, eq. (8.310.1)], respectively.
Proof: See Appendix B.
Applying Lemma 2 for the integrals given in (21), (22), (23), and (24),
yields
G
G(k + 1)
I0 = 2 Φ 2, 2 − k,
(26)
A
A
G(k + 1)
1
(27)
I1 = Φ 2, 1 − k,
A
A
2G
G(k + 1)
I2 = 3 Φ 3, 3 − k,
(28)
A
A
G(k + 1)
2
(29)
I3 = 2 Φ 3, 2 − k,
A
A
Substituting (26) and (27) into (19), we find an exact expression for the first
moment of the SINR as
γ̄ = N
N
−1
X
k=0
N −1
G(k + 1)
k G
(−1)
Φ 2, 2 − k,
k
A2
A
1
G(k + 1)
+ Φ 2, 1 − k,
A
A
(30)
Similarly, an exact expression for the second moment of the SINR is obtained
by substituting (28) and (29) into (20) as
2
E[γ ] = N
N
−1
X
k=0
G(k + 1)
N −1
k 2G
Φ 3, 3 − k,
(−1)
A3
A
k
2
G(k + 1)
+ 2 Φ 3, 2 − k,
A
A
(31)
Eventually, an approximate expression for the ergodic capacity is established
by substituting (30) and (31) into (18).
58
Part I
8
= 4 dB
pk
SU-Tx Transmit SNR,
ST
(dB)
0
-8
-16
N = M = 3
=
h
-24
=1,
=4,
h
=
=2
g
=
h
f
=
f
=
g
=
g
=2
f
=2
-32
-6
-4
-2
0
2
PU-Tx Transmit SNR,
4
6
PT
8
10
(dB)
Figure 2: SU-Tx transmit SNR versus PU-Tx transmit SNR, γP T , with different channel mean powers.
4
Numerical Results
In this section, numerical results for the outage probability and ergodic capacity are presented. We set the outage transmission rate of the primary network
rp = 0.4 bit/s, outage transmission rate of the secondary network rs = 0.1
bit/s, outage constraint of the primary network ǫ = 0.01, and the SU-Tx peak
transmit SNR γpk = 4 dB.
Figure 2 plots the SU-Tx transmit SNR as a function of the PU-Tx transmit SNR. We can see that as the PU-Tx transmit SNR increases, the SU-Tx
transmit SNR increases also. However, as the PU-Tx transmit SNR increases
further, the SU-Tx transmit SNR is saturated at the SU-Tx peak transmit
SNR, γpk = 4 dB.
Figure 3 shows the outage probability as a function of PU-Tx transmit
SNR. We can see that the outage probability firstly decreases to a minimum
value, and then increases as the PU-Tx transmit SNR, γP T , increases. This
is thought to be due to the fact that the SU-Tx transmit SNR is controlled
following the policy given in (10). Thus, an increasing PU-Tx transmit SNR
γP T leads to an increase of the SU-Tx transmit SNR γST (see Figure 2). Ac-
Impact of Primary Networks on the Performance of Secondary Networks
59
0
10
Ana.
=
h
=1,
-1
Outage Probability
10
f
=1,
h
g
=4,
=4,
g
h
=
=
=
h
f
f
=
=
f
g
g
=2, Sim.
=
f
=2, Sim.
=
=2, Sim.
=
=2, Sim.
=
=2, Sim.
g
h
-2
10
-3
10
pk
= 4 dB
N = M = 3
-4
10
-6
-4
-2
0
2
PU-Tx Transmit SNR,
4
PT
6
8
10
(dB)
Figure 3: Outage probability versus the PU-Tx transmit SNR, γP T , with
different channel mean powers.
cordingly, the SU-Rx SINR is increased, i.e., the outage probability decreases.
However, as the PU-Tx transmit SNR increases beyond a certain value, e.g.,
γP T > 5 dB, the SU-Tx cannot adapt to higher transmit SNR of the PU-Tx
due to the limitation of peak transmit SNR, γpk = 4 dB (see Figure 2). In this
context, if the PU-Tx transmit SNR increases further, the SU-Rx will suffer
strong interference from the PU-Tx. As a result, the system performance decreases. Additionally, as expected, the system performance can be improved
as the channel mean powers of the SU-Tx→PU-Rx and PU-Tx→SU-Rx interference links decrease from Ωf = Ωβ = 2 to Ωf = 1 or Ωβ = 1.
Figure 4 illustrates that the ergodic capacity increases to a maximum
value, and then decreases as the PU-Tx transmit SNR, γP T , increases. Clearly,
the approximation tightly matches with the simulation. These results are also
in line with the observations for the outage probability depicted in Figure 3.
This is because of the same reasons explained above. Hence, an increase of the
PU-Tx transmit SNR firstly leads to an increasing SU-Tx transmit SNR, i.e.,
the SU-Tx transmission rate is increased. On the other hand, as the SU-Tx
transmit SNR is restricted due to the peak transmit SNR γpk , it is not able
to adapt its transmit SNR following a further increase of the PU-Tx transmit
SNR. As a result, the PU-Tx transmit SNR causes strong interference to the
SU-Rx, i.e., the SU transmission rate is decreased.
60
Part I
5.0
=4,
f
h
4.5
=
=
=2
g
pk
Approx.
Ergodic Capacity (bps/Hz)
3.5
=4,
f
g
3.0
2.5
h
2.0
=
=
h
=
h
=
g
Sim.
=2
=1,
=
h
f
=
g
Approx.
Approx.
Sim.
Sim.
=
=
g
=2
f
Approx.
N=M= 3
Sim.
4.0
=1,
= 4 dB
=2
=2
f
Approx.
Sim.
1.5
1.0
0.5
0.0
-6
-4
-2
0
2
4
PU-Tx Transmit SNR,
6
PT
8
10
(dB)
Figure 4: Ergodic capacity versus the PU-Tx transmit SNR, γP T , with different channel mean powers.
Figure 5 and Figure 6 show the impact of the number of antennas on the
system performance in terms of outage probability and ergodic capacity. As
expected, the system performance can be improved as the number of antennas
N or M increases. This is due to the fact that the diversity of received signals
at the SU-Rx and PU-Rx is increased as the number of antennas increases.
Accordingly, the SU-Rx and PU-Rx have higher probability to decode the
received signals from the SU-Tx and PU-Tx. Interestingly, in contrast to
the results reported in [14], the results presented in this paper indicates that
increasing the number of antennas of the PU-Rx does not degrade the performance of the secondary network. This is because of three reasons: 1) In [14],
the impact of the communication links and interference of the primary network on the secondary network have not been included. 2) The SC of the
PU-Rx has not been considered. 3) The SU-Tx in [14] is only subject to the
peak interference power constraint of the PU-Rx. In other words, the considered system model has more advantage than the one reported in [14] as
it reveals conditions for optimal performance. However, the SU-Tx requires
more information from the primary network to be able to improve the system
performance.
Impact of Primary Networks on the Performance of Secondary Networks
61
0
10
-1
Outage Probability
10
pk
= 4 dB
=
h
g
=
f
=2
-2
10
Ana.
N=1, M=1, Sim.
N=2, M=2, Sim.
-3
10
N=2, M=4, Sim.
N=3, M=3, Sim.
N=4, M=3, Sim.
-4
10
-6
-4
-2
0
2
4
PU-Tx Transmit SNR,
6
PT
8
10
(dB)
Figure 5: Outage probability versus the PU-Tx transmit SNR, γP T , with
different number of antennas.
4.0
N = 1, M =1, Approx.
3.5
N = 1, M =1, Sim.
pk
Ergodic Capacity (bps/Hz)
N = 2, M = 2, Approx.
3.0
= 4 dB
h
N = 2, M = 2, Sim.
=
g
=
f
=2
N = 2, M = 4, Approx.
N = 2, M = 4, Sim.
2.5
N = 3, M= 3, Approx.
N = 3, M = 3, Sim.
2.0
N = 4, M = 3, Approx.
N = 4, M = 3, Sim.
1.5
1.0
0.5
0.0
-6
-4
-2
0
2
4
PU-Tx Transmit SNR,
6
PT
8
10
(dB)
Figure 6: Ergodic capacity versus PU-Tx transmit SNR, γP T , with different
number of antennas.
62
5
Part I
Conclusions
In this paper, we have examined the performance of a SIMO CRN based on
the outage constraint of the primary network and the peak transmit power
constraint of the secondary network. An adaptive transmit power allocation
policy for the SU-Tx, a closed-form expression for the outage probability, and
an approximation for the ergodic capacity have been obtained.
Appendix
Appendix A: Proof for the Lemma 1
According to the probability definition, we can define the CDF of Y in (4) as
a1 X j
≤y
(32)
FY (y) = Pr
max
j=1,2,...,L a2 Zj + a3
Because the RVs Xj and Zj are independent, FY (y) in (32) can be rewritten
as
L Z∞
Y
a1 X j
Pr
(33)
FY (y) =
≤ y fZj (z)dz
a2 z + a3
j=1 0 |
{z
}
K
Additionally, Zj and Xj are exponentially distributed RVs with mean Ωz and
Ωx . Thus, the PDF of Zj and K are expressed, respectively, as
1
z
fZj (z) =
exp −
(34)
Ωz
Ωz
y(a2 z + a3 )
K = 1 − exp −
(35)
a 1 Ωx
Substituting (34) and (35) into (33), and after some manipulations yields the
CDF of Y as
exp − z
L Z∞ Y
Ωz
y(a2 z + a3 )
dz
1 − exp −
FY (y) =
a 1 Ωx
Ωz
j=1
0
L
a3 y
a 1 Ωx
exp −
a 2 Ωz y + a 1 Ωx
a 1 Ωx
L
exp (−yC)
= 1−
yD + 1
= 1−
(36)
Impact of Primary Networks on the Performance of Secondary Networks
63
Ωz
.
where C = a1aΩ3 x and D = aa12 Ω
x
Differentiating (36) with respect to y and then applying the binomial expansion yields the PDF of Y as
L−1
X
L−1
(−1)k exp (−yC(k + 1))
k
k=0
C
D
×
+
(1 + Dy)k+1
(1 + Dy)k+2
fY (y) =L
(37)
Appendix B: Proof for the Lemma 2
By setting x = at and applying an exchange of variables in the integral (25)
yields
I=
1
an+1
Z∞
0
xn
b
exp
−
x
dx
(1 + x)m
a
Moreover, using [15, eq. (9.211.4)], an exact expression for I is obtained as
b
Γ(1 + n)
(38)
I = n+1 Φ 1 + n, 2 − m + n,
a
a
where Φ(·, ·, ·) and Γ(·) are the confluent hypergeometric function [15, eq.
(9.211.4)] and gamma function [15, eq. (8.310.1)], respectively.
References
[1] “Facilitating opportunities for flexible, efficient, and reliable spectrum use
employing cognitive radio technologies,” Federal Communications Commission (FCC), Tech. Rep. 03-108, Mar. 2005.
[2] “Cognitive radio technology - A study for Ofcom,” 2010. [Online].
Available: http://www.ofcom.org.uk/research/
[3] J. Mitola and G. Q. Maguire, “Cognitive radio: Making software radios
more personal,” IEEE Personal Commun. Mag., vol. 6, no. 4, pp. 13–18,
Aug. 1999.
[4] A. Goldsmith, S. Jafar, I. Maric, and S. Srinivasa, “Breaking spectrum
gridlock with cognitive radios: An information theoretic perspective,”
Proc. IEEE, vol. 97, no. 5, pp. 894–914, May 2009.
64
Part I
[5] L. Zhang, Y.-C. Liang, and Y. Xin, “Joint beamforming and power allocation for multiple access channels in cognitive radio networks,” IEEE
J. Sel. Areas Commun., vol. 26, no. 1, pp. 38–51, Jan. 2008.
[6] S. Sridharan and S. Vishwanath, “On the capacity of a class of MIMO
cognitive radios,” IEEE J. Sel. Topics Signal Process., vol. 2, no. 1, pp.
103–117, Feb. 2008.
[7] X. Hong, C.-X. Wang, H.-H. Chen, and J. Thompson, “Performance
analysis of cognitive radio networks with average interference power constraints,” in Proc. IEEE International Conference on Communications,
Beijing, China, May 2008, pp. 3578–3582.
[8] H. Dhillon and R. Buehrer, “Cognitive MIMO radio: Incorporating dynamic spectrum access in multiuser MIMO network,” in Proc. IEEE
Global Telecommunications Conference, Hawaii, U.S.A., Dec. 2009, pp.
1–6.
[9] M. Sarkar, T. Ratnarajah, M. Sellathurai, and C. Cowan, “Outage performance of SIMO multiple access cognitive radio channel,” in Proc. International Workshop on Cognitive Wireless Systems, New Delhi, India,
Dec. 2010, pp. 1–5.
[10] F. Negro, I. Ghauri, and D. Slock, “Beamforming for the underlay cognitive MISO interference channel via UL-DL duality,” in Proc. International Conference on Cognitive Radio Oriented Wireless Networks Communications, Cannes, France, Jun. 2010, pp. 1–5.
[11] Asaduzzaman and H. Kong, “Ergodic and outage capacity of interference
temperature-limited cognitive radio multi-input multi-output channel,”
IET Commun., vol. 5, no. 5, pp. 652–659, May 2011.
[12] Y. Yang, X. Liu, Q. Wu, and J. Wang, “Outage capacity analysis for
SIMO Nakagami-m fading channel in spectrum sharing environment,” in
Proc. IEEE International Conference on Information Theory and Information Security, Beijing, China, Dec. 2010, pp. 1058–1063.
[13] M. Jung, K. Hwang, and S. Choi, “Interference minimization approach
to precoding scheme in MIMO-based cognitive radio networks,” IEEE
Commun. Lett., vol. 15, no. 8, pp. 789–791, Aug. 2011.
[14] H. Tran, H.-J. Zepernick, and H. Phan, “Impact of the number of antennas and distances among users on cognitive radio networks,” in Proc.
International Conference on Advanced Technologies for Communications,
Hanoi, Vietnam, Oct. 2012, pp. 1–5.
Impact of Primary Networks on the Performance of Secondary Networks
65
[15] I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products,
7th ed. Elsevier, 2007.
Part II
Part II
Performance Analysis of a Cognitive
Cooperative Radio Network with a Buffered
Relay
Part II is published as:
H. Tran, H.-J. Zepernick, H. Phan, and L. Sibomana, “Performance Analysis of a Cognitive Cooperative Radio Network with a Buffered Relay,” IEEE
Transactions on Vehicular Technology, Feb. 2013, under review.
Performance Analysis of a Cognitive
Cooperative Radio Network with a Buffered
Relay
Hung Tran, Hans-Jürgen Zepernick, Hoc Phan,
and Louis Sibomana
Abstract
In this paper, we analyze the packet transmission time in a cognitive
cooperative radio network (CCRN) where a secondary transmitter (SUTx) sends packets to a secondary receiver (SU-Rx) through the help of
a secondary relay (SR). In particular, we assume that the SU-Tx and
SR are subject to the joint constraint of the timeout probability of
the primary user (PU) and the peak transmit powers of the secondary
users. On this basis, we investigate the impact of the transmit power of
the PUs and channel mean powers on the packet transmission time of
the CCRN. Utilizing the concept of timeout, adaptive transmit power
allocation policies for the SU-Tx and SR are considered. More importantly, analytical expressions for the end-to-end throughput, end-to-end
packet transmission time, and stable condition for the SR operation are
obtained. Our results indicate that the second hop of the considered
CCRN is not a bottleneck if the channel mean powers of the interference
links among the networks are small and the SR peak transmit power is
set to a high value.
1
Introduction
Recently, cognitive radio networks (CRNs) have been considered as a promising technology to deal with the under-utilization of spectrum resources in
wireless communications [1]. The key idea behind CRNs is to let the secondary user (SU) exploit the licensed frequency bands which are readily as71
72
Part II
signed to the primary users (PUs) without degrading the performance of the
PUs. Specifically, three main spectrum access methods have been proposed,
namely, overlay, interweave, and underlay [2]. In the overlay paradigm, the
SU can simultaneously access a licensed frequency band with the PUs by using a sophisticated coding technique (e.g., dirty paper coding) to cancel the
interference to the PU. In the interweave system, the SU needs to detect the
temporarily unused spectrum before accessing this free spectrum, and must
vacate it when it is reoccupied by the PU. On the other hand, in the underlay
system, the SU is allowed to simultaneously access the licensed frequency band
of the PU as long as the interference caused by the SU to the primary receiver
(PU-Rx) is below a given threshold. As a result, the secondary transmitter
(SU-Tx) transmit power is often kept at a low level, and a direct communication link between the SU-Tx and the secondary receiver (SU-Rx) can only
be maintained for a short range.
In order to overcome such coverage limitations in underlay CRNs, the
concept of cognitive cooperative radio network (CCRN) has been introduced
as a promising solution [3–19]. It has been shown that the coverage range
and link reliability of CRNs can be increased significantly through the help
of secondary relays (SRs). More specifically, in [3], the authors have focused
on the cooperation between the SU and PU, where the SU-Tx operates as
a relay to support the packet transmission of the primary networks. Given
this setting, a power allocation strategy has been derived for the SU-Tx, and
the stable throughput for the secondary network has been analyzed. In [4],
Sadek et al. have proposed two protocols for cognitive multiple access by
cooperation, for which the maximal throughput region and delay performance
have been analyzed. In [15], the authors have investigated the diversity order
of a CCRN with decode-and-forward (DF) relaying and studied the impact
of the distances between the terminals on outage performance. In [17, 20],
the asymptotic outage behavior and the effect of interference from the PU on
the outage performance of CCRNs with amplify-and-forward (AF) and DF
relaying have been examined. In [13, 16], the outage performance of a CCRN
with a single DF relay over Nakagami-m fading channels has been considered.
It has been shown that the outage probability depends on both the fading
severity parameters and the number of active PUs that are present in the
vicinity of the SU. In the aforementioned works and the references therein,
the authors have investigated CCRN scenarios in which the relay operates in
a half-duplex mode, i.e., the relay cannot receive and transmit packets at the
same time. Most recently, in [21], the optimal power allocation and outage
performance for a CCRN with full-duplex relaying have been investigated.
Motivated by all of the above, in this paper, we study the performance
of a CCRN with the SR operating in full-duplex mode. More specifically,
Performance Analysis of a Cognitive Cooperative Radio Network
73
we consider that the SR receives packets from the SU-Tx, decodes, and then
forwards them to the SU-Rx over orthogonal channels. To assure the desired
performance of the PU, the SU-Tx and SR must control their transmit powers
to meet both the timeout probability constraint of the PUs and the peak
transmit power constraint of the SUs. Given these settings, the performance
analysis for the considered system is investigated. Main contributions in this
paper are summarized as follows:
• Adaptive transmit power control policies for the SU-Tx and SR are
proposed.
• Using the timeout concept, cumulative distribution function (CDF) and
probability density function (PDF) for the packet transmission time,
and timeout probability are derived for the secondary network.
• By employing the GI/G/1 (general independent interarrival time, general service time, and a single server) queueing model, analytical expressions for the end-to-end throughput, end-to-end packet transmission
time, and stable condition for the SR operation are derived.
• Our results show that if the peak transmit power of the SR is set to a
high value and the channel mean powers of the interference links among
networks are low, the SR→SU-Rx link is not a bottleneck.
The remainder of this paper is organized as follows. In Section 2, the
system model and assumptions for the CCRN are introduced. In Section 3,
adaptive transmit power policies for the SU-Tx and SR under the joint peak
transmit power constraint of the SU and the timeout probability constraint of
the PU are analyzed. Accordingly, expressions for the PDF and CDF of packet
transmission time, timeout probability, end-to-end transmission time, end-toend throughput, and stable condition for the SR operation are derived. In
Section 4, numerical results and discussions are provided. Finally, conclusions
are presented in Section 5.
Notation: The CDF and PDF of a random variable (RV) Y are denoted
by FY (·) and fY (·), respectively. The terms Pr{·} and Pr{·|·} denote the
probability and conditional probability, respectively. E[·] and Var[·] stand
for expectation and variance, respectively. X (a,b) denotes a variable that
belongs to the a-th network and the b-th hop where a ∈ {s, p} and b ∈ {1, 2}.
Acronyms s and p represent the secondary network and primary network,
respectively.
74
Part II
PU-Rx1
PU-Tx1
PU-Rx2
0
g1
h1
1
g0
SU-Tx
ȕ0
PU-Tx2
ȕ1
h0
SR
Secondary user
Interference link
Primary user
Communication link
SU-Rx
Figure 1: System model of a CCRN where the SR receives a packet from the
SU-Tx in the first hop, decodes, and forwards it to the SU-Rx in the second
hop. The SU-Tx→SR link utilizes the licensed frequency band of the PUs in
Region I, while the SR→SU-Rx link exploits the licensed frequency band of
the PUs in Region II.
2
2.1
System Model of the Considered CCRN
System Model
Let us consider a dual-hop CCRN as shown in Figure 1 in which the SU-Tx
has a continuous stream of packets to transmit to the SU-Rx through the help
of a full-duplex DF relay. In particular, in the first hop, the communication
over the SU-Tx→SR link utilizes the licensed spectrum of the PUs in Region
I. In the second hop, the communication over the SR→SU-Rx link uses the
licensed frequency band of the PUs in Region II. Frequencies in the regions
are assumed to be orthogonal and the direct communication between SU-Tx
and SU-Rx is not available due to shadowing. Accordingly, the SUs and PUs
may cause mutual interference in the same region, but they do not cause
interference to users in the other region. Note that the considered scenario
is applicable in practice where the SR may act as a base station while the
SU-Tx→SR and SR→SU-Rx links are uplink and downlink, respectively.
2.2
Channel Model
As for the radio links between different users, we assume independent but
not necessarily identically distributed (i.n.i.d.) Rayleigh block fading chan-
Performance Analysis of a Cognitive Cooperative Radio Network
75
nels. Therefore, the channels are considered as constant during the transmission time of one packet but they may change independently to different
values thereafter. The channel power gains of the SU-Tx→SR, SR→SU-Rx,
primary transmitter (PU-Tx)1 →PU-Rx1 , and PU-Tx2 →PU-Rx2 communication links are denoted by g0 , h0 , α0 , and β0 , respectively. The channel power gains of the SU-Tx→PU-Rx1 , SR→PU-Rx2 , PU-Tx1 →SR, and
PU-Tx2 →SU-Rx interference links are denoted, respectively, by g1 , h1 , α1 ,
and β1 . In addition, the channel mean powers of exponentially distributed
RVs g0 , g1 , h0 , h1 , α0 , α1 , β0 , and β1 are expressed, respectively, by
Ωg0 , Ωg1 , Ωh0 , Ωh1 , Ωα0 , Ωα1 , Ωβ0 , and Ωβ1 . As the PUs have the highest
priority to access their spectrum, they can use an arbitrary power level for
the communication without caring about the existence of the SUs. On the
other hand, the existence of the SUs should not degrade the performance of
the PUs. Hence, the SUs should have full channel state information (CSI) to
control their transmit powers. The CSI of the SU-Tx→SR and SR→SU-Rx
links can be obtained by direct feedback from the SR and SU-Rx, respectively. The CSI of the interference links among networks can be acquired
from a band manager between the primary and secondary networks [22], or
SUs and PUs can cooperate and exchange channel knowledge by using the
methods proposed in [23, 24].
2.3
Timeout Probability Constraint of the PUs
Let the PU-Txm , m = 1, 2 transmits packets of length Lp bits to the corresponding PU-Rxm . Since the PUs share their licensed spectrum with the SUs,
they may be subject to interference caused by the SUs. As such, the performance of the PU is guaranteed if the probability of dropped packets due to
timeout is smaller than a predefined threshold. In the follow, we will formulate
this setting mathematically in terms of a timeout probability constraint.
For the PU-Tx1 →PU-Rx1 communication, the PU-Tx1 uses transmit power
P(p,1) to send a packet to the PU-Rx1 . The time taken by the PU-Tx1 to
transmit a packet to the PU-Rx1 can be formulated as [25]
T (p,1) =
B̃p
Lp
,
B log2 (1 + γ (p,1) )
ln(1 + γ (p,1) )
(1)
where B̃p = Lp ln(2)/B, B is the system bandwidth, and γ (p,1) is the instantaneous signal-to-interference-plus-noise ratio (SINR) at the PU-Rx1 , given
by
γ (p,1) =
P(p,1) α0
P(s,1) g1 + N0
(2)
76
Part II
where P(s,1) and N0 stand for the transmit power of the SU-Tx and noise
power, respectively.
A packet is considered as being successfully transmitted if its transmission
time, T (p,1) given in (1), is below a predefined timeout threshold tpout . In other
words, the probability of unsuccessful packet reception for PU-Rx1 , known as
(p,1)
timeout probability Pout , can be formulated as
(p,1)
Pout
= Pr{T
(p,1)
≥
tpout }
= Pr B̃p ln 1 +
P(p,1) α0
P(s,1) g1 + N0
−1
≥
tpout
(3)
Accordingly, the performance of the PU-Tx1 →PU-Rx1 in the presence of the
SU-Tx→SR communication is assured if and only if the timeout probability,
(p,1)
Pout given in (3), satisfies the condition
(p,1)
(p,1)
Pout ≤ θth
(4)
(p,1)
where θth is the timeout probability constraint for the PU-Rx1 .
Similarly, the packet transmission time of the PU-Tx2 →PU-Rx2 link is
expressed as
T (p,2) =
Lp
B̃p
,
(p,2)
B log2 (1 + γ
)
ln(1 + γ (p,2) )
(5)
where γ (p,2) is the instantaneous SINR at the PU-Rx2 given by
γ (p,2) =
P(p,2) β0
P(s,2) h1 + N0
(6)
where P(p,2) and P(s,2) are transmit powers of the PU-Tx2 and SR, respectively. Then, the timeout probability of the PU-Rx2 in the presence of the
SR→SU-Rx communication is formulated as
−1
P(p,2) β0
(p,2)
Pout = Pr{T (p,2) ≥ tpout } = Pr B̃p ln 1 + (s,2)
≥ tpout
P
h1 + N 0
(7)
Moreover, the condition to not degrade the PU performance is interpreted
in terms of the timeout probability constraint as
(p,2)
(p,2)
Pout ≤ θth
(p,2)
where θth
is the timeout probability constraint of the PU-Rx2 .
(8)
Performance Analysis of a Cognitive Cooperative Radio Network
2.4
77
CCRN Communication
In the first hop, the SU-Tx regulates its transmit power to send a packet
of length Ls bits to the SR. As the SR receives the packet successfully, it is
stored in a buffer at the SR. Then, the SR feeds back a short acknowledgement
(ACK) without delay to the SU-Tx. Upon the reception of the ACK, the SUTx transmits the next packet. The time consumed to send a packet in the
first hop is given by
T (s,1) =
B̃s
Ls
,
B log2 (1 + γ (s,1) )
ln(1 + γ (s,1) )
(9)
where B̃s = Ls ln(2)/B and γ (s,1) is the SINR at the SR, defined by
γ (s,1) =
P(s,1) g0
+ N0
P(p,1) α1
(10)
It is noted that the SU-Tx must control its transmit power, P(s,1) , to satisfy
(p,1)
the timeout probability constraint, θth , of the PU-Rx1 given in (4). In
addition, the SU-Tx transmit power is limited in practice, and hence it may
(s,1)
be constrained by a peak transmit power Ppk as
(s,1)
P(s,1) ≤ Ppk
(11)
In other words, to send a packet to the SR, the SU-Tx should control its power
to jointly meet the timeout probability constraint of the PU-Rx1 given in (4)
and the peak transmit power constraint of the SU-Tx given in (11).
In the second hop, the SR also adjusts its power to forward the packets
in its buffer to the SU-Rx, where packets are served in first come first serve
order. It is noted that the buffer length of commercial devices is often very
large nowadays. This fact can be adopted to assume that the length of the
SR buffer is infinite. Similar to the first hop, the time it takes to transmit a
packet in the second hop is expressed as
T (s,2) =
B̃s
Ls
,
B log2 (1 + γ (s,2) )
ln(1 + γ (s,2) )
(12)
where γ (s,2) is the SINR at the SU-Rx, and is given by
γ (s,2) =
P(s,2) h0
P(p,2) β1 + N0
(13)
78
Part II
The SR transmit power, P(s,2) , is also restricted by the timeout probability
(s,2)
constraint of the PU-Rx2 given in (8) and a peak transmit power Ppk as
(s,2)
P(s,2) ≤ Ppk
(14)
Without loss of generality, each packet in the secondary or primary network is considered as successfully transmitted if and only if the transmission
time from the source to the intended receiver is less than a given timeout
threshold. As a consequence, the event in which the packet is transmitted
successfully is defined by
(a,b)
Tsuc
= {T (a,b) |T (a,b) < taout }
2.5
(15)
Queueing Model for the SR Buffer
We have assumed that all the channels are constant for the duration of a packet
transmission, but may change independently thereafter. As a consequence,
interarrival time of packets at the SR buffer are identically and independently
distributed (i.i.d.) RVs with a general distribution, and so are the packet
transmission times. Therefore, the traffic for the SR buffer can be modeled
as a GI/G/1 queueing system [25, 26].
Let the interarrival time of packets and transmission time per packet be
denoted by the RVs A and T , respectively. Although, an exact closed-form expression for the average waiting time of a packet in the buffer is not available,
it can be approximated as [26]
E[W ] ≈
2
ρξ(CT2 + CA
) E[T ]
2(1 − ρ)
(16)
where W is an RV denoting the waiting time of a packet in the SR buffer and
Var[X]
E[T ]
2
, X = {T, A}
, CX
=
E[A]
(E[X])2

2 2

−2(1−ρ)(1−CA
)
2

<1
,
CA
exp
2 +C 2 )
3ρ(CT
A
ξ=
2

C −1
2

, CA
≥1
exp −(1 − ρ) C 2 A
+4C 2
ρ=
A
(17)
(18)
T
According to [27], a queueing system is stable if and only if the average transmission time is less than the average interarrival time, i.e.,
ρ=
where ρ is the channel utilization.
E[T ]
<1
E[A]
(19)
Performance Analysis of a Cognitive Cooperative Radio Network
3
79
Performance Analysis
In this section, a performance analysis for the considered system model is
presented. We first consider adaptive transmit power policies for the SU-Tx
and SR. Specifically, the CDF, PDF, and moments of the packet transmission
time for the first and second hop are derived. Finally, performance in terms
of end-to-end throughput, end-to-end transmission time, and stable condition
for the SR operation are presented.
3.1
Adaptive Transmit Power Policies
In this section, we derive the adaptive transmit power policies for the SU-Tx
and the SR. Let us commence considering a lemma as follows.
Lemma 1. Let a, b, c, and d be positive constants. Further, let X1 and X2 be
independent exponentially distributed RVs with mean Ω1 and Ω2 , respectively.
Then, the RV Z defined by
−1
aX1
(20)
Z = d ln 1 +
bX2 + c
has CDF and PDF, respectively, given by
aΩ1
c
d
FZ (z) =
−1
(21)
exp
exp −
bΩ2 [exp(d/z) − 1] + aΩ1
aΩ1
z
abdΩ1 Ω2
cd
fZ (z) =
+
z 2 [bΩ2 exp(d/z) + aΩ1 − bΩ2 ]
z 2 [bΩ2 exp(d/z) + aΩ1 − bΩ2 ]2
c
d
c
d
× exp
+
(22)
exp
−
z
aΩ1
z
aΩ1
Proof: See Appendix A.
3.1.1
Adaptive Transmit Power Policy for the SU-Tx
The task of the SU-Tx is to select a transmit power level such that it can
exploit the licensed spectrum of Region I as much as possible but does not
cause harmful interference to the PU-Rx1 . Given the related constraints in
(4) and (11), an adaptive transmit power policy for the SU-Tx is derived as
follows.
From (3), the timeout probability of a packet transmitted by the PU-Tx1
can be rewritten as
−1
P(p,1) α0
(p,1)
< tpout
(23)
Pout = 1 − Pr B̃p ln 1 + (s,1)
P
g 1 + N0
80
Part II
(p,1)
Applying Lemma 1 to (23), a closed-form expression for the Pout
obtained as
P(p,1) Ωα0
N 0 Φp
(p,1)
Pout = 1 − (s,1)
exp
−
P
Ωg1 Φp + P(p,1) Ωα0
P(p,1) Ωα0
can be
(24)
where Φp is defined by
B̃p
Φp = exp p
−1
tout
(25)
Substituting (24) into (4), we obtain the maximum transmit power of the
SU-Tx after some manipulations as
P(s,1) =
P(p,1) Ωα0
χ1
Ωg1 Φp
(26)
where
χ1 = max 0,
1
(p,1)
1 − θth
N 0 Φp
exp − (p,1)
−1
P
Ωα0
(27)
Combining (26) with (11), we can formulate an adaptive power allocation
policy for the SU-Tx as
(p,1)
P
Ωα0
(s,1)
P (s,1) = min
χ1 , Ppk
(28)
Ωg1 Φp
3.1.2
Adaptive Transmit Power for the SR
Similar to the derivation for the first hop, the adaptive transmit power for the
SR in the second hop is derived as follows.
From (8), the timeout probability of a packet transmitted by the PU-Tx2
is expressed by
−1
P(p,2) β0
(p,2)
< tpout
(29)
Pout = 1 − Pr B̃p ln 1 + (s,2)
P
h1 + N 0
Using the same manipulations for the SU-Tx as in the previous section, an
adaptive transmit power policy for the SR can be obtained as
(p,2)
P
Ωβ 0
(s,2)
(30)
χ2 , Ppk
P (s,2) = min
Ωh 1 Φp
where
χ2 = max 0,
1
(p,2)
1 − θth
N 0 Φp
exp − (p,2)
−1
P
Ωβ 0
(31)
81
Performance Analysis of a Cognitive Cooperative Radio Network
3.2
Performance Analysis for the First Hop
3.2.1
Statistics for packet transmission time in the first hop
In this subsection, we derive the timeout probability, the PDF of packet transmission time, as well as the first and the second moment of packet transmission
time from the SU-Tx to the SR.
In the first hop, the SU-Tx transmits packets to the SR by using the power
(s,1)
allocation policy given in (28). As such, the probability, Pout , that a packet
is dropped due to timeout can be expressed as
(s,1)
Pout
= Pr{T
(s,1)
≥
tsout }
P (s,1) g0
P(p,1) α1 + N0
= 1 − Pr B̃s ln 1 +
−1
<
tsout
(32)
where tsout is the timeout threshold for the secondary network. Applying (21)
(s,1)
to (32), we obtain a closed-form expression for the timeout probability, Pout ,
in the first hop as
(s,1)
Pout =
P (s,1) Ωg0
N 0 Φs
exp
−
P(p,1) Ωα1 Φs + P (s,1) Ωg0
P (s,1) Ωg0
(33)
where Φs is defined by
B̃s
−1
Φs = exp s
tout
(34)
In addition, the PDF of T (s,1) can be easily derived by using (22) as
N0
B̃s
N0
B̃s
+ (s,1)
fT (s,1) (x) = exp
− (s,1)
exp
x
x
P
Ωg0
P
Ωg0



P (s,1) P(p,1) B̃s Ωg0 Ωα1
×
i2
h

 x2 P(p,1) Ωα exp(B̃s /x) + P (s,1) Ωg − P(p,1) Ωα
1
0
1


N0 B̃s
i
h
+

x2 P(p,1) Ω exp(B̃ /x) + P (s,1) Ω − P(p,1) Ω
α1
s
g0
(s,1)
(35)
α1
Furthermore, the probability for the event Tsuc = {T (s,1) |T (s,1) < tsout }
that a packet from the SU-Tx to the SR is transmitted successfully, can be
82
Part II
expressed by using Bayes’ rule as
Pr{T (s,1) |T (s,1) < tsout } =
=
Pr{T (s,1) , T (s,1) < tsout }
Pr{T (s,1) < tsout }
Pr{T (s,1) , T (s,1) < tsout }
(s,1)
(s,1)
where Pout is given by (33). Hence, the CDF of Tsuc
using (36) as
FT (s,1) (x) =
suc
1
(s,1)
1 − Pout
Zx
(36)
(s,1)
1 − Pout
can be formulated by
fT (s,1) (t)dt, 0 ≤ x < tsout
(37)
0
(s,1)
Differentiating (37) with respect to x, we obtain the PDF of Tsuc
fT (s,1) (x) =
suc
fT (s,1) (x)
1−
(s,1)
Pout
1
=
(s,1)
as
, 0 ≤ x < tsout
N0
B̃s
N0
B̃s
+ (s,1)
exp
− (s,1)
exp
x
x
P
Ωg0
P
Ωg0
1 − Pout



P (s,1) P(p,1) B̃s Ωg0 Ωα1
×
i2
h

 x2 P(p,1) Ωα exp(B̃s /x) + P (s,1) Ωg − P(p,1) Ωα
1
0
1


N0 B̃s
i
h
+

x2 P(p,1) Ω exp(B̃ /x) + P (s,1) Ω − P(p,1) Ω
s
α1
(38)
α1
g0
and fT (s,1) (x) = 0, x ≥ tsout .
suc
Furthermore, we can determine the first and second moments of the successful packet transmission time as stated in the following Lemma 2.
(s,1)
Lemma 2. The first and second moments of Tsuc
(s,1) i
E[(Tsuc
)]=
N0 B̃s I1 (i)
1−
(s,1)
Pout
+
are given by
P (s,1) P(p,1) B̃s Ωg0 Ωα1 I2 (i)
(s,1)
1 − Pout
, i = 1, 2
(39)
where Ik (i), k = 1, 2, is defined by
Ik (i) =
Z∞
Φs
N0
exp − P (s,1) Ωg t
0
B̃s1−i [ln(t
Proof: See Appendix B.
+
1)]i (P(p,1) Ω
α1 t
+ P (s,1) Ωg0 )k
dt
(40)
Performance Analysis of a Cognitive Cooperative Radio Network
83
Due to the impairments caused by the wireless channel, a packet may be
transmitted successfully or unsuccessfully. As such, the moments of packet
transmission time should be calculated on the packet transmission time with
and without being timed out [25]. Therefore, the moment of T (s,1) can be
obtained by using the law of total expectation as
(s,1)
(s,1)
(s,1) i
E[(T (s,1) )i ] = (1 − Pout ) E[(Tsuc
) ] + (tsout )i Pout
(41)
(s,1)
where E[(Tsuc )i ] is given in Lemma 2.
3.2.2
Inter-arrival time of packets at the SR
As a packet is transmitted, the SU-Tx waits for an ACK. If the SU-Tx does
not receive an ACK before tsout , the packet is considered as lost or dropped.
In this case, the packet is retransmitted, and the interarrival time of packets
at the SR buffer can be expressed as
(s,1)
A = Tsuc
+ (N (s,1) − 1)tsout
(42)
where N (s,1) is a geometric RV that denotes the number of transmissions
between two successful packet arrivals in the SR buffer. Its probability can
be calculated as
(s,1)
(s,1)
Pr{N (s,1) = ℓ} = (Pout )ℓ−1 (1 − Pout ), ℓ ≥ 1
(43)
Additionally, we can derive the first and second moment of N (s,1) as in the
following Lemma 3.
Lemma 3. The first and second moment of the number of transmissions
between two successful packet arrivals in the SR buffer can be calculated, respectively, as
E[N (s,1) ] =
∞
X
ℓ Pr{N (s,1) = ℓ} =
(s,1)
(s,1)
ℓ(Pout )ℓ−1 (1 − Pout )
ℓ=1
ℓ=1
=
∞
X
1
(s,1)
1 − Pout
∞
∞
X
X
(s,1)
(s,1)
E[(N (s,1) )2 ] =
ℓ2 Pr{N (s,1) = ℓ} =
ℓ2 (Pout )ℓ−1 (1 − Pout )
ℓ=1
=
(s,1)
1 + Pout
(s,1)
(1 − Pout )2
Proof: See Appendix C.
(44)
ℓ=1
(45)
84
Part II
Lemma 4. The mean and variance of the interarrival time of packets at the
SR buffer are, respectively, given by
(s,1)
E[A] =
N0 B̃s I1 (1) + P (s,1) P(p,1) B̃s Ωg0 Ωα1 I2 (1) + tsout Pout
(s,1)
1 − Pout
(46)
(s,1)
(s,1) 2
(s,1) 2
Var[A] = E[(Tsuc
) ] − (E[Tsuc
]) +
(tsout )2 Pout
(s,1)
(1 − Pout )2
(47)
(s,1)
where E[(Tsuc )i ] is determined by (39).
Proof: See Appendix D.
3.3
3.3.1
Performance Analysis for the Second Hop
Statistics for packet transmission time in the second hop
Similar to the first hop, in the second hop, the SR uses the adaptive transmit
power policy given in (30) to transmit packets to the SU-Rx. The timeout
probability for a packet transmitted in the second hop can be derived as
−1
P (s,2) h0
s
<
t
out
P(p,2) β1 + N0
N 0 Φs
P (s,2) Ωh0
exp − (s,2)
= (p,2)
P
Ωβ1 Φs + P (s,2) Ωh0
P
Ωh 0
(48)
(s,2)
Pout = Pr{T (s,2) ≥ tsout } = 1 − Pr B̃s ln 1 +
We can now apply similar manipulations as in the first hop to derive the
(s,2)
PDF and the moment of Tsuc in the second hop as follows.
(s,2)
The PDF of packet transmission time without timeout, Tsuc , can be given
by
fT (s,2) (y) =
suc
1
(s,2)
N0
N0
B̃s
B̃s
+ (s,2)
− (s,2)
exp
exp
y
y
P
Ωh 0
P
Ωh 0
1 − Pout



P (s,2) P(p,2) B̃s Ωh0 Ωβ1
×
i2
h

 y 2 P(p,2) Ωβ exp(B̃s /y) + P (s,2) Ωh − P(p,2) Ωβ
1
0
1


N0 B̃s
i , 0 ≤ y < tsout
h
+
y 2 P(p,2) Ω exp(B̃ /y) + P (s,2) Ω − P(p,2) Ω 
β1
s
h0
β1
(49)
Performance Analysis of a Cognitive Cooperative Radio Network
85
and fT (s,2) (y) = 0, y ≥ tsout .
suc
The first and second moment of T (s,2) are obtained by
(s,2)
(s,2)
(s,2) i
E[(T (s,2) )i ] = (1 − Pout ) E[(Tsuc
) ] + (tsout )i Pout
(50)
where
(s,2) i
E[(Tsuc
)]=
N0 B̃s J1 (i) + P (s,2) P(p,2) B̃s Ωh0 Ωβ1 J2 (i)
(s,2)
1 − Pout
, i = 1, 2
(51)
k dt
(52)
and Jk (i), k = 1, 2, is defined by
Jk (i) =
Z∞
Φs
3.4
N0
exp − P (s,2) Ωh t
0
B̃s1−i [ln(t
+ 1)]i P(p,2) Ωβ1 t + P (s,2) Ωh0
Performance Measures
Given the results from Sections 3.2 and 3.3, we are now in the position to
derive performance measures for the CCRN such as the stable condition for the
SR operation, end-to-end throughput, and end-to-end average transmission
time as follows.
Lemma 5. The SR operation is stable if and only if
(s,1)
N0 B̃s I1 (1) + P (s,1) P(p,1) B̃s Ωg0 Ωα1 I2 (1) + tsout Pout
(s,1)
1 − Pout
(s,2)
> N0 B̃s J1 (1) + P (s,2) P(p,2) B̃s Ωh0 Ωβ1 J2 (1) + tsout Pout
(53)
Proof: Because the interarrival time of packets are i.i.d. with a general
distribution, so is the transmission time of packets. The packet traffic at the
SR can be modeled as a GI/G/1 queueing system. Using the stable condition
given in (19), we can deduce the desired stable condition for the SR buffer as
E[A] > E[T (s,2) ]
Substituting (46) and (50) for i = 1 into (54), we obtain (53).
(54)
The end-to-end throughput for the considered CCRN can be determined
by the following Lemma 6.
86
Part II
Lemma 6. The throughput of the considered CCRN in the stable and unstable
transmission condition can be formulated as
(s,2)
R=
1 − Pout
(packets/sec)
max{E[T (s,2) ], E[A]}
(55)
Proof: If the SR operation is stable, i.e. E[A] > E[T (s,2) ], then the average
transmission rate of the SR is equal to the average arrival rate of packets at
the SR buffer. Hence, the end-to-end throughput can be determined as
1
1
=
E[A]
max{E[T (s,2) ], E[A]}
(56)
On the other hand, if the SR operation is unstable, i.e. E[A] ≤ E[T (s,2) ],
then the average transmission rate of the SR is equal to the average transmission rate of the SR→SU-Rx link because the SR becomes a bottleneck. Hence,
the average transmission rate of the SR can be expressed as
1
1
=
E[T (s,2) ]
max{E[T (s,2) ], E[A]}
(57)
(s,2)
Moreover, in both cases (stable and unstable condition), a fraction of Pout
of packets is dropped due to timeout in the SR→SU-Rx link. Therefore, the
SU-Rx receives packets at an average rate as given in (55).
Finally, the end-to-end transmission time from the SU-Tx to the SU-Rx
over the SR buffer can be calculated as
(s,1)
(s,2)
D = E[Tsuc
] + E[W ] + E[Tsuc
]
(58)
(s,2)
(s,1)
where E[Tsuc ] and E[Tsuc ] are given, respectively, in (39) and (51) for i = 1.
The average waiting time of a packet in the SR buffer E[W ] is given by
E[W ] ≈
2
ρξ(CT2 (s,2) + CA
) E[T (s,2) ]
2(1 − ρ)
(59)
where
Var[X]
E[T (s,2) ]
2
, X = {T (s,2), A}
, CX
=
E[A]
(E[X])2

2 2

−2(1−ρ)(1−CA
)
2

,
CA
<1
exp

2)
3ρ(C 2 (s,2) +CA
T
ξ=
2

CA
−1
2

, CA
≥1
exp −(1 − ρ) C 2 +4C
2
ρ=
A
T (s,2)
(60)
(61)
Performance Analysis of a Cognitive Cooperative Radio Network
4
87
Numerical Results
In this section, we present simulation and analysis results for the considered
system. In particular, we study the impact of the PU-Tx transmit power and
the channel mean powers on the end-to-end throughput, end-to-end average
packet transmission time, and stable condition for the SR operation. Unless
otherwise stated, the following system parameters are used for both simulation
and analysis:
System bandwidth: B= 2 MHz
Packet size: Ls = Lp = 224 bits (28 bytes)
Timeout: tsout = tpout = 0.06 seconds
(p,1)
(p,2)
Outage constraints: θout = θout = 1%
4.1
End-to-end Throughput of CCRN
Figure 2 shows the throughput as a function of PU-Tx2 transmit power for
(s,2)
different values of the SR peak transmit power Ppk = 1, 3, 4, 8 dB and identical channel mean powers Ωα0 = Ωα1 = Ωh0 = Ωh1 = Ωβ0 = Ωβ1 = Ωg0 =
Ωg1 = 2. For brevity, we discuss only the scenario where the PU-Tx1 transmit
(s,1)
power P(p,1) = 2 dB and the SU-Tx peak transmit power Ppk = 4 dB in the
first hop, and distinguish the following two cases:
(s,2)
= 4 dB.
(s,2)
= 8 dB.
• Case 1: SR peak transmit power is set to Ppk
• Case 2: SR peak transmit power is set to Ppk
(s,1)
The other cases, with P(p,1) = 4 dB and Ppk = 4 dB where the SR peak
transmit power is given as 1, 3 dB, show a similar progression of the end-to-end
throughput versus transmit power of the PU-Tx2 .
As can clearly be seen from the figure, simulations and analytical results
match very well. In all cases, the throughput is first kept at a constant level
in the low regime of the PU-Tx2 transmit power, P(p,2) , and then decreases
in the high regime of P(p,2) . In particular, for Case 1, the throughput is
unchanged for P(p,2) < 2 dB, and it is degraded for P(p,2) > 2 dB. These
results are thought to be due to the fact that in the low regime of the PU-Tx2
transmit power, the PU-Tx2 does not cause much interference to the SU-Rx.
Thus, the SR can adapt its transmit power, P (s,2) , to the change of P(p,2) ,
following the strategy given in (30). However, in the high regime of the PUTx2 transmit power, the SR cannot increase its transmit power further due to
88
Part II
2200
End-to-end throughput (packets/sec)
2000
(p,1)
P
(s,1)
=2 dB, P
pk
=4 dB
1800
1600
1400
1200
Ana.
(s,2)
P
1000
pk
=4 dB (Sim.)
(s,2)
P
800
pk
=8 dB (Sim.)
Ana.
600
(s,2)
P
pk
400
=1 dB (Sim.)
(s,2)
P
pk
(p,1)
=3 dB (Sim.)
P
(s,1)
=4 dB, P
pk
=4 dB
200
-6
-5
-4
-3
-2
-1
0
1
2
3
4
(p,2)
Transmit Power of PU-Tx , P
2
5
6
7
8
(dB)
Figure 2: Impact of the PU-Tx2 transmit power in Region II, P(p,2) , on the
throughput of the CCRN for identical channel mean powers, Ωα0 = Ωα1 =
Ωh0 = Ωh1 = Ωβ0 = Ωβ1 = Ωg0 = Ωg1 = 2.
the peak transmit power constraint. As a consequence, the PU-Tx2 transmit
power now becomes strong interference to the SU-Rx. This results in a large
increase in the number of dropped packets due to timeout in the second hop,
which degrades the end-to-end throughput of the CCRN. On the other hand,
in Case 2, the throughput becomes more stable in the high regime of the PUTx2 transmit power, and is only degraded as the PU-Tx2 increases beyond
(s,2)
6 dB. In other words, if the SR peak transmit power, Ppk , is set to a high
value, the SR→SU-Rx link in the second hop may not become a bottleneck
as the PU-Tx2 transmit power increases to a high value.
Figure 3 depicts the impact of channel mean powers of the interference
links on the CCRN throughput. In particular, we have set identical channel
mean powers for all communication links of both the primary network and
the secondary network, i.e., Ωg0 = Ωh0 = Ωβ0 = Ωα0 = 2. Then, the following
four cases are considered:
• Case 3 : The primary network and the secondary network do not cause
severe interference to each other due to low channel mean powers of the
89
Performance Analysis of a Cognitive Cooperative Radio Network
3300
End-to-end throughput (packets/sec)
Case 3
Ana.
3000
2700
h
2400
h
2100
h
1800
h
=
1
=
1
=
1
=
1
g
g
g
g
=
=
= 0.5 (Sim.)
1
1
=
1
=
= 2 (Sim.)
1
=
1
1
=
= 4 (Sim.)
1
1
=0.5,
=
1
1
=5 (Sim.)
1
1500
(s,1)
P
1200
pk
(s,2)
=P
pk
(p,1)
=4 dB, P
Case 4
=4 dB
900
600
Case 6
Case 5
300
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
6
(p,2)
Transmit Power of PU-Tx , P
2
7
8
9
10
(dB)
Figure 3: Impact of the PU-Tx2 transmit power, P(p,2) , on the throughput of
the CCRN for different channel mean powers of interference links.
interference links, i.e., Ωh1 = Ωg1 = Ωβ1 = Ωα1 = 0.5.
• Case 4 : This case shall be considered as a reference where all channel
mean powers for interference links and communication links are identical, i.e., Ωh1 = Ωg1 = Ωβ1 = Ωα1 = 2.
• Case 5 : The primary network and the secondary network can cause
strong interference to each other due to high channel mean powers of
the interference links, i.e., Ωh1 = Ωg1 = Ωβ1 = Ωα1 = 4.
• Case 6 : The primary network causes strong interference to the secondary network due to high channel mean powers of the PU-Tx1 → SR
and PU-Tx2 → SU-Rx interference links, i.e., Ωβ1 = Ωα1 = 5, while the
secondary network causes weak interference to the primary network due
to low channel mean powers of the SU-Tx→PU-Rx1 and SR→PU-Rx2
interference links, i.e., Ωh1 = Ωg1 = 0.5.
We can observe from the figure that the throughput of Case 3 outperforms
the Reference Case 4. This is because the interference between the primary
90
Part II
network and the secondary network is reduced as the channel mean powers
of the interference links are decreased from Ωh1 = Ωg1 = Ωβ1 = Ωα1 = 2
to Ωh1 = Ωg1 = Ωβ1 = Ωα1 = 0.5. Accordingly, the timeout probability
constraint of the primary network may not be violated. Thus, the SU-Tx and
the SR can increase their transmit power to reduce the packet transmission
time. As such, the number of packets that suffers a timeout in the secondary
network may be decreased, and hence the end-to-end throughput of the CCRN
is increased.
In Case 5, where the channel mean powers of the interference links are set
to high values, i.e., Ωh1 = Ωg1 = Ωβ1 = Ωα1 = 4, the interference between the
primary network and the secondary network can increase as well. Accordingly,
the timeout probability constraints of the primary network may be easily
violated with a low power level of the SU-Tx and SR. Therefore, the SU-Tx
and SR must reduce its transmit power to protect the communication of the
primary network. As a consequence, the packet transmission time and the
number of packets being timed out increase. Therefore, the throughput of
Case 5 is lower than that of Case 4.
Finally, Case 6 results in the worst performance compared to the other
considered cases. This can be explained by the fact that the SR and SU-Rx
suffer from strong interference caused by the PU-Tx1 and PU-Tx2 , respectively. Therefore, there may be many packets in both hops of the CCRN
being timed out, which leads to low end-to-end throughput.
4.2
End-to-end Transmission Time
In Figure 4, the end-to-end transmission time is plotted as a function of the
PU-Tx2 transmit power P(p,2) for different values of the SU-Tx peak transmit
(s,1)
power in the first hop, Ppk = 1, 4, 6 dB. It can be seen from the figure that
the end-to-end transmission time increases as the PU-Tx2 transmit power
increases. Typically, the end-to-end transmission time increases rapidly as
(s,1)
the SU-Tx peak transmit power is set to a high value such as Ppk = 4, 6 dB.
This can be understood by the fact that as the SU-Tx peak transmit power
(s,1)
is set to a high value, e.g., Ppk = 6 dB, the SU-Tx can adjust its transmit
power following the policy given in (28) to send more packets to the SR,
i.e., the number of packets arriving at the SR buffer increases. On the other
hand, in the second hop, increasing PU-Tx2 transmit power, P(p,2) , leads to
a decrease of the SU-Rx SINR given in (13). Hence, the packet transmission
time is increased. As a consequence, the number of packets arriving at the SR
buffer increases while the number of packets leaving the SR buffer decreases.
Therefore, the number of packets waiting in the SR buffer and the end-to-end
transmission time increase.
91
Performance Analysis of a Cognitive Cooperative Radio Network
End-to-end transmission time (seconds)
0.4
(p,1)
P
(s,2)
=4 dB, P
pk
=4 dB,
Ana.
0.3
(s,1)
P
pk
(s,1)
P
pk
(s,1)
P
0.2
pk
=1 dB (Sim.)
=4 dB (Sim.)
=6 dB (Sim.)
0.1
0.0
-5
-4
-3
-2
-1
0
1
2
3
(p,2)
Transmit Power of PU-Tx , P
2
4
5
(dB)
Figure 4: Impact of the PU-Tx2 transmit power in Region II, P(p,2) , on the
end-to-end transmission time of packets for identical channel mean powers
Ωα0 = Ωα1 = Ωh0 = Ωh1 = Ωβ0 = Ωβ1 = Ωg0 = Ωg1 = 2.
Figure 5 depicts the end-to-end transmission time as a function of the
PU-Tx2 transmit power P(p,2) for different channel mean powers of the PUTx1 →SR and PU-Tx2 →SU-Rx interference links. In particular, we set the
channel mean powers Ωg0 = Ωh0 = Ωβ0 = Ωα0 = Ωh1 = Ωg1 = 2, and consider
the following two cases:
• Case 7 : The channel mean power of the PU-Tx1 →SR interference link
is set to Ωα1 = 2 while the channel mean power of the PU-Tx2 →SU-Rx
interference link is set to Ωβ1 = 0.5, 2, 3.
• Case 8 : The channel mean power of the PU-Tx2 →SU-Rx interference
link is set to Ωβ1 = 3 while the channel mean power of the PU-Tx1 →SR
interference link is set to Ωα1 = 2, 5.
As expected, the end-to-end transmission time increases with increasing
the PU-Tx2 transmit power P(p,2) . Moreover, for Case 7, we can observe that
the end-to-end transmission time increases as the channel mean power of the
PU-Tx2 →SU-Rx interference link increases from Ωβ1 = 0.5 to Ωβ1 = 3. Due
92
Part II
End-to-end transmission time (seconds)
0.10
Ana.
0.09
=2,
=0.5 (Sim.)
1
0.08
1
=2,
=2 (Sim.)
1
1
0.07
=2,
=3 (Sim.)
1
1
0.06
=5,
=3 (Sim.)
1
1
0.05
0.04
0.03
P
(p,1)
=P
(s,1)
(s,2)
pk
pk
= P
=6 dB
0.02
0.01
0.00
-5
-4
-3
-2
-1
0
1
2
3
(p,2)
Transmit Power of PU-Tx , P
2
4
5
(dB)
Figure 5: Impact of the PU-Tx2 transmit power, P(p,2) , on the end-to-end
transmission time of packets for channel mean powers Ωα0 = Ωg0 = Ωg1 =
Ωh0 = Ωβ0 = Ωh1 = 2.
to the same reason as for the throughput results, the SU-Rx may be subject
to strong interference from the PU-Tx2 when the channel mean power of the
PU-Tx2 →SU-Rx interference link is high, e.g. Ωβ1 = 3. Accordingly, the SURx SINR given in (13) decreases and packet transmission time in the second
hop increases. Then, the remaining packets have to stay for a long duration
in the SR buffer, leading to an increasing end-to-end transmission time.
In Case 8, the end-to-end transmission time decreases as the channel mean
power of the PU-Tx1 →SR interference link in the first hop increases from
Ωα1 = 2 to Ωα1 = 5. In this case, the SR is subject to strong interference
from the PU-Tx1 and the SR SINR given (10) decreases. Therefore, the packet
transmission time in the first hop increases. As a consequence, the number of
packet arriving at the SR buffer decreases, which in turn reduces the waiting
time in the SR buffer and the end-to-end transmission time.
93
Performance Analysis of a Cognitive Cooperative Radio Network
1.4
=0.5
1
=2
1
1.2
Channel utilization,
=3
1
1.0
(p,1)
P
(s,1)
=P
pk
(s,2)
= P
pk
=4 dB
0.8
0.6
0.4
=
g
0
0.2
-4
-3
-2
-1
0
=
g
1
1
=
2
=
=
h
0
h
1
3
0
4
5
(p,2)
Transmit Power of PU-Tx , P
2
=
0
=2
1
6
7
8
(dB)
Figure 6: Impact of channel mean powers on the stability of the SR operation.
4.3
Stability of the SR Operation
Finally, we examine the impact of the channel mean power of the PU-Tx2 →SURx interference link on the stability of the SR operation as shown in Figure
6. As expected, the SR operation with the low channel mean power of the
PU-Tx2 →SU-Rx interference link, Ωβ1 = 0.5, is more stable than for the
higher channel mean powers. Specifically, the channel utilization ρ is kept
below 1 for the entire range of considered PU-Tx2 transmit power. On the
other hand, the SR operation becomes unstable (ρ ≥ 1) for Ωβ1 = 2, 3 once
the PU-Tx2 transmit power progresses towards high values, e.g., P(p,2) > 4
dB. In case of these high channel mean powers, the PU-Tx2 causes strong
interference to the SU-Rx when the PU-Tx2 transmit power P(p,2) increases.
Hence, the packet transmission time increases in the second hop, which leads
the channel utilization ρ = E[T (s,2) ]/ E[A] to become greater than 1, causing
the SR operation to be unstable.
94
5
Part II
Conclusions
In this paper, we have analyzed the performance of a CCRN with a buffered
relay. In particular, we have assumed that packets in the primary network
and secondary network are subject to timeout constraints. In addition, we
have assumed that the communication channels and interference channels undergo Rayleigh fading. The transmit powers of the SU-Tx and SR are subject
to both the peak transmit power constraint and the timeout probability constraint of the PUs. On this basis, adaptive transmit power policies for the
SU-Tx and the SR have been investigated. The CDF and PDF, timeout probability, and moments for packet transmission time in each hop have been derived. Moreover, by employing the GI/G/1 queueing model, the performance
analysis in terms of the end-to-end throughput, end-to-end transmission time,
and stable condition for the SR operation has been provided for the considered system. Numerical results have been presented to quantify the impact
of PU-Tx transmit power and channel mean powers of the interference links
on the performance of the CCRN.
Appendices
Appendix A: Proof of Lemma 1
A.1 The CDF of Z
Following the probability definition, the CDF of Z can be defined as
−1
aX1
<z
FZ (z) = Pr d ln 1 +
bX2 + c
d
aX1
−1
< exp
=1 − Pr
bX2 + c
z
(62)
Using the same approach as given in [28, Eq. (14)] for (62), we finally obtain
the CDF of Z as
FZ (z) =
aΩ1
c
d
−1
exp
exp −
bΩ2 [exp(d/z) − 1] + aΩ1
aΩ1
z
(63)
Performance Analysis of a Cognitive Cooperative Radio Network
95
A.2 The PDF of Z
The PDF of Z is obtained by differentiating (63) with respect to z as
cd
fZ (z) =
2 + z 2 [bΩ exp(d/z) + aΩ − bΩ ]
2
1
2
z 2 [bΩ2 exp(d/z) + aΩ1 − bΩ2 ]
d
c
c
d
× exp
+
(64)
exp
−
z
aΩ1
z
aΩ1
abdΩ1 Ω2
With (63) and (64), Lemma 1 is proved.
Appendix B: Proof of Lemma 2
The moments of packet transmission time without timeout can be expressed
as
Ztout
(s,1) i
xi fT (s,1) (x)dx, i = 1, 2
E[(Tsuc
)]=
s
suc
(65)
0
Substituting fT (s,1) (x) given in (38) into (65), we have
suc
(s,1) i
E[(Tsuc
)]=
N0
B̃s
N0
B̃s
exp
+
exp
−
x
x
P (s,1) Ωg0
P (s,1) Ωg0
P (s,1) P(p,1) B̃s Ωg0 Ωα1
i2 dx
h
(s,1)
B̃s
1 − Pout
(p,1)
(s,1)
(p,1)
2−i
Ωα1
Ωg0 − P
P
Ωα1 exp( x ) + P
0 x
N0
N0
tsout
Z
exp B̃xs + P (s,1)
exp B̃xs − P (s,1)
Ωg0
Ωg0
N0 B̃s
i dx
h
+
(66)
(s,1)
B̃s
(s,1) Ω − P(p,1) Ω
2−i P(p,1) Ω
1 − Pout
)
+
P
exp(
x
α
g
α
1
0
1
x
0
Ztout
s
For brevity, let us define
B̃s
B̃s
N0
N0
tZsout
exp x − P (s,1) Ωg exp x + P (s,1) Ωg
0
0
Ik (i) =
h
ik dx
B̃s
2−i P(p,1) Ω
(p,1) Ω
(s,1) Ω
exp(
x
−
P
)
+
P
α1
α1
g0
0
x
Changing variable t = exp (B̃s /x) − 1 for (67) yields
N0
Z∞
t
exp − P (s,1)
Ωg0
dt
Ik (i) =
B̃s1−i [ln(t + 1)]i (P(p,1) Ωα1 t + P (s,1) Ωg0 )k
Φs
(67)
(68)
96
Part II
(s,1)
where Φs = exp tB̃s s − 1. Finally, an expression for E[(Tsuc )i ] given in
out
(65) is obtained as
N0 B̃s I1 (i)
(s,1) i
E[(Tsuc
)]=
1−
(s,1)
Pout
+
P (s,1) P(p,1) B̃s Ωg0 Ωα1 I2 (i)
(s,1)
1 − Pout
(69)
Lemma 2 is proved.
Appendix C: Proof of Lemma 3
Let us first consider the series given by [29, Eq. (0.112)]
M
X
xn = x
n=1
1 − xM
, x < 1, M ≥ 1
1−x
(70)
Differentiating (70) with respect to x yields
M
X
n−1
nx
n=1
x 1 − xM
M xM
1 − xM
=−
+
+
1−x
1−x
(1 − x)2
(71)
Multiplying both sides of (71) with (1 − x) and let M → ∞, we reach
∞
X
nxn−1 (1 − x) =
n=1
1
1−x
(72)
Secondly, we differentiate (71) with respect to x as
M
X
n(n − 1)xn−2 =
n=1
−M xM−1 (1 + M )
2M xM
−
1−x
(1 − x)2
2 1 − xM
2x 1 − xM
+
+
(1 − x)2
(1 − x)3
(73)
Multiplying both sides of (73) with x(1 − x) yields
M
X
n=1
n2 xn−1 (1 − x) =
M
X
nxn−1 (1 − x) − M xM (1 + M )
n=1
2x 1 − xM
2x2 1 − xM
2M xM+1
+
+
−
1−x
1−x
(1 − x)2
(74)
97
Performance Analysis of a Cognitive Cooperative Radio Network
Now, let M → ∞ in (74), which yields
∞
X
n2 xn−1 (1 − x) =
n=1
1+x
(1 − x)2
(75)
(s,1)
Finally, by substituting x = Pout into (72) and (75), Lemma 3 is proved.
Appendix D: Proof of Lemma 4
From (42), the first moment of the interarrival time A can be expressed as
(s,1)
E[A] = E[Tsuc
] − tsout + tsout E[N (s,1) ]
(76)
Substituting (39) with i = 1 and E[N (s,1) ] given in (44) into (76) and after
performing some manipulations, a tractable expression for the first moment
of interarrival time is given in (46).
Moreover, we can also present the variance of interarrival time from (42)
as
(s,1)
Var[A] = Var[Tsuc
] + (tsout )2 Var[N (s,1) ]
(77)
(s,1)
where Var[Tsuc ] is determined as
(s,1) 2
(s,1) 2
(s,1)
])
) ] − (E[Tsuc
Var[Tsuc
] = E[(Tsuc
(78)
and
(s,1)
Var[N (s,1) ] = E[(N (s,1) )2 ] − (E[N (s,1) ])2 =
Pout
(s,1)
(1 − Pout )2
(79)
Substituting (78) and (79) into (77) gives (47). Lemma 4 is proved.
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Part II
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Performance Analysis of a Cognitive Cooperative Radio Network
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Part III
Part III
Cognitive Cooperative Networks with DF
Relay Selection under Interference Constraints
of Multiple Primary Users
Part III is published as:
H. Tran, H.-J. Zepernick, and H. Phan, “Cognitive Cooperative Networks
with DF Relay Selection under Interference Constraints of Multiple Primary
Users”, Wireless Communications and Mobile Computing, Feb. 2013, under
minor revision.
based on:
H. Tran, T. Q. Duong, and H.-J. Zepernick, “Performance Analysis of Cognitive Relay Networks Under Power Constraint of Multiple Primary Users,” in
IEEE Global Telecommunications Conference, Houston, U.S.A., Dec. 2011,
pp. 1–6.
Cognitive Cooperative Networks with DF Relay
Selection under Interference Constraints of
Multiple Primary Users
Hung Tran, Hans-Jürgen Zepernick, and Hoc Phan
Abstract
In this paper, we study the performance of cognitive cooperative radio networks under the peak interference power constraints of multiple
primary users. In particular, we consider a system model where the
secondary user communication is assisted by multiple secondary relays
that operate in the decode-and-forward mode to relay the signal from
a secondary transmitter to a secondary receiver. Moreover, we assume
that the transmit powers of the secondary transmitter and the secondary relays are subject to the peak interference power constraints of
the multiple primary users that operate in their coverage range. Given
this system setting, we first derive the cumulative distribution function
of the instantaneous end-to-end signal-to-noise ratio. Then, we obtain
a closed-form expression for the outage probability and an exact expression for the symbol error probability of the considered network. These
tractable formulas enable us to examine the impact of the presence of
multiple primary users on the performance of the considered spectrum
sharing system. Furthermore, our numerical results show that system
performance is improved significantly when the number of secondary
relays increases or the channel mean power from the secondary user
to the primary users is low. Also, any increase in the number of primary users in the coverage range of the secondary transmitter or the
secondary relays leads to degradation in system performance. Finally,
Monte Carlo simulations are provided to verify the correctness of our
analytical results.
105
106
1
Part III
Introduction
Cognitive radio networks (CRNs) have emerged as a promising technique to
improve the spectrum utilization in wireless communication systems [1–5].
Generally, in a CRN, a secondary user (SU) is allowed to access the licensed
spectrum band of a primary user (PU) provided that reliable communication of the PU is guaranteed. In particular, two main approaches, known as
opportunistic spectrum access and spectrum sharing, have been proposed to
enhance spectrum utilization. In opportunistic spectrum access, the SU performs spectrum sensing to detect the presence of PUs and to identify which
portion of the spectrum is not occupied by PUs. Unlike the opportunistic
spectrum access, in the spectrum sharing technique, the SU coexists with the
active PU in the licensed spectrum as long as the maximal interference at the
PU caused by the SU does not exceed a tolerable threshold.
Given the restrictions on the transmit power of secondary networks in
the spectrum sharing approach, reliability and coverage range of an SU may
not always be guaranteed. In this context, cognitive cooperative techniques
have been considered as a powerful solution to overcome severe multipath
fading and to improve the performance of the secondary system. Recently,
a number of cognitive cooperative protocols have been investigated in terms
of outage probability, ergodic capacity and bit error probability for spectrum
sharing systems [5–15]. In particular, a cognitive cooperative transmission
protocol has been proposed in [5,6]. According to these studies, the secondary
transmitter (SU-Tx) may act as a relay to aid the communication of the PU.
Moreover, on the basis of the cooperative strategy suggested in [6], an optimal
outage probability has been determined and used to investigate the system
performance. In [7], a cognitive cooperative radio network (CCRN) with
a single decode-and-forward (DF) relay has been studied and a closed-form
expression for the symbol error probability (SEP) has been derived. In [8, 9],
upper and lower bounds on the outage probability of a CCRN under the peak
interference power constraint of a PU have been derived. Similar to [8, 9],
Guo et al. have examined the outage behavior of a CCRN under both the
peak interference power constraint of a single primary receiver (PU-Rx) and
the peak transmit power constraint of an SU [10]. Later in [11], the three
relay selection strategies of selective amplify-and-forward (AF), selective DF,
and AF with partial relay selection have been analyzed. Also, the respective
asymptotic outage behaviors have been investigated for these three strategies.
In [12], Sagong et al. have compared the capacity of a CCRN with reactive
DF and proactive DF. The results have shown that the capacity of the CCRN
with reactive and proactive DF is the same as long as the PU is far away from
the SU-Tx. In [13], a comparison between a conventional relay network and
Cognitive Cooperative Networks with DF Relay Selection under Interference Constraints
of Multiple Primary Users
107
a CCRN has been presented. The results indicated that although the outage
probability of the CCRN is higher than that of conventional relay networks
due to the interference constraint, the SU can still utilize the spectrum to
remain an acceptable transmission rate. In [14], a single DF relay network
under the peak interference power constraint of a single PU over Nakagami-m
fading channel has been studied and the impact of fading severity parameters
on the outage performance has been examined. Our recent work, [15], has
analyzed the performance of a CCRN in terms of outage probability and
ergodic capacity under the peak interference power constraints of multiple
PUs for independent and identically distributed Nakagami-m fading channels.
In particular, we derive the cumulative distribution function (CDF) and the
probability density function (PDF) of the instantaneous end-to-end signal-tonoise ratio (SNR). Then, we use these two functions to derive a closed-form
expression for the outage probability and an approximation for the ergodic
capacity. It is shown that system performance does not only depend on the
fading severity parameters but also the number of active PUs. However, for
analytical simplicity, the work only considered a CCRN with a single relay
and identical number of PUs around the SU-Tx and the secondary relay (SR).
Interestingly, during our studies reported in [15], we recognized that directly
applying the result of conventional cooperative networks as given in [16] for a
spectrum sharing CCRN does not provide an exact result when the number of
relays is larger than one and the peak interference power is in the high regime.
This is due to the fact that the end-to-end SNR of a spectrum sharing CCRN
depends on both interference and communication links.
As an extension of the system model considered in [15], in this paper, we
analyze the end-to-end performance of a spectrum sharing CCRN in which
the SU-Tx and multiple SRs operate under the peak interference power constraints of multiple PUs. In particular, we assume that the SU-Tx communicates with the secondary receiver (SU-Rx) through the help of multiple SRs
while adhering to the interference constraints of multiple PUs at the SU-Tx
and SRs. It should be mentioned that the number of PUs in the coverage
range of the SU-Tx is not necessarily identical to the number of PUs around
the SRs. Additionally, we assume that the SRs operate in DF mode using
best relay selection and that all channels are subject to independent Rayleigh
fading. Based on these assumptions, we first derive the CDF of the instantaneous end-to-end SNR. Then, we derive a closed-form expression for the
outage probability and an exact expression for the SEP of the considered
CCRN. Those expressions allow us to analyze the impact of the number of
SRs, the number of PUs in the coverage range of the SU-Tx and/or SRs, and
the channel mean powers on the performance of the considered CCRN. To
the best of the authors’s knowledge, there are no previous works addressing
108
Part III
this problem.
The remainder of this paper is organized as follows. In Section 2, the
system model and assumptions for the CCRN under the peak interference
power constraints of multiple PUs are introduced. In Section 3, the CDF
of the instantaneous end-to-end SNR is derived. On this basis, tractable
formulas for the outage probability and the SEP are obtained. In Section 4, we
provide numerical results and discussions. Finally, conclusions are presented
in Section 5.
2
System Model
Let us consider a spectrum sharing system in which multiple DF SRs are
employed to aid the SU’s communication. Our model consists of an SU-Tx
and an SU-Rx located in the proximity of multiple PU-Rxs and multiple DF
SRs as shown in Figure 1. In particular, there exist a number of N and K
PU-Rx operating in the coverage range of the SU-Tx and M SRs, respectively. The instantaneous channel power gains of the SU-Tx→PU-Rxn and
SRm →PU-Rxk links, respectively, are denoted by αn , n = 1, 2, . . . , N , and
βmk , m = 1, 2, . . . , M , k = 1, 2, . . . , K. The instantaneous channel power
gains of the SU-Tx→SRm and SRm →SU-Rx links, respectively, are represented by h1m and h2m , m = 1, 2, . . . , M . The direct link between SU-Tx and
SU-Rx is not available due to severe shadowing. Therefore, communication
on the secondary network is provided only through the SRs in half-duplex
mode. Note that in the considered spectrum sharing system, the primary
transmitters (PU-Tx) are assumed to be far away from the SRs and SU-Rx,
while the SU-Tx and SRs are close to the PU-Rx. Therefore, only the SU-Tx
and SRs cause interference to the PU-Rx while interference from the PU-Tx
to SRm and PU-Tx to SU-Rx is very small and may be considered as Gaussian
noise [17, 18].
In this way, during the first hop, the SU-Tx broadcasts the signal to all
SRs. As the N PU-Rx and the SU-Tx share the same spectrum band, the
SU-Tx is permitted to access the licensed frequency band provided that the
interference does not compromise the quality of service of any PU-Rx. According to [17], we can formulate the instantaneous SNR at the mth SR as
γ1m =
PS (h11 , h12 , . . . , h1M ; α1 , α2 , . . . , αN )h1m
, m = 1, 2, . . . , M
N0
(1)
where PS (h11 , h12 , . . . , h1M ; α1 , α2 , . . . , αN ) is the instantaneous transmit power
of the SU-Tx and N0 denotes the noise power. Moreover, the SU-Tx needs to
regulate its transmit power on the basis of joined channel state information
Cognitive Cooperative Networks with DF Relay Selection under Interference Constraints
of Multiple Primary Users
109
PU-Rx1
PU-Rxk
M1
11
Mk
1k
1K
PU-RxK
PU-RxN
SR1
N
PU-Rxn
11
MK
n
SU-Tx
21
1
PU-Rx1
1M
SU-Rx
SRM
2M
Figure 1: Model of the considered cognitive cooperative radio network with
multiple SRs and multiple PU-Rxs.
(CSI) to constrain the interference to the N PU-Rxs in its coverage range.
Therefore, the interference power constraint of the SU-Tx with respect to
PU-Rxn is given by
PS (h11 , h12 , . . . , h1M ; α1 , α2 , . . . , αN )αn ≤ Q(S)
n , n = 1, 2, . . . , N
(2)
(S)
where Qn is the peak interference power tolerated by the PU-Rxn in the
coverage area of the SU-Tx.
Similar to the first-hop transmission, in the second hop, one out of the M
SRs, say SRm , is selected to decode the received signal from the SU-Tx, and
then to forward the output to the SU-Rx. Thus, the instantaneous SNR at
the SU-Rx can be expressed as
ȕMK
γ2m =
PRm (h2m ; βm1 , βm2 , . . . , βmK )h2m
, m = 1, 2, . . . , M
N0
(3)
where PRm (h2m ; βm1 , βm2 , . . . , βmK ) is the instantaneous transmit power of
the SRm . It is noted that the SRm also adjusts its transmit power to not
cause harmful interference to any of the K PU-Rx around, i.e.,
(R)
PRm (h2m ; βm1 , βm2 , . . . , βmK )βmk ≤ Qk , k = 1, 2, . . . , K
(R)
(4)
where Qk is the peak interference power that the PU-Rxk in the proximity
of the SRs can tolerate.
110
Part III
The CSI of the SU-Tx→SRm and SRm →SU-Rx links can be obtained
through feedback of the SRm and SU-Rx, respectively. Similarly, the CSI of
the SU-Tx→PU-Rxn and SRm →PU-Rxmk links can be provided by feedback
from the PUs or indirect feedback from a band manager who is responsible
for the shared spectrum [4].
Assuming perfect CSI, the relay SRm is selected such that the end-toend SNR for the SU-Tx→SRm →SU-Rx links is maximized. Applying the
best relay selection protocol of [16], the instantaneous end-to-end SNR can be
given by
γs =
max
m=1,2,...,M
{min{γ1m , γ2m }}
(5)
We consider that the channels undergo independent Rayleigh fading with
channel mean powers of the SU-Tx→PU-Rxn link, the SRm →PU-Rxmk link,
the SU-Tx→SRm links and the SRm →SU-Rx link, respectively, given by ΩSP ,
ΩRP , ΩSR , and ΩRD .
Furthermore, we assume that the tolerable peak interference power is the
(R)
(S)
same for all PU-Rxs, i.e., Qn = Qk = Q for n = 1, 2, . . . , N and k =
1, 2, . . . , K. Thus, the transmit powers of the SU-Tx and SRm must satisfy
the following peak interference power constraints:
PS (h11 , h12 , . . . , h1m ; α1 , α2 , . . . , αN ) ≤
Q
max {αn }
(6)
n=1,2,...,N
PRm (h2m ; βm1 , βm2 , . . . , βmK ) ≤
Q
max
k=1,2,...,K
{βmk }
(7)
Given perfect CSI for all the links, SU-Tx and SRm can operate with maximal
instantaneous SNR as defined by (6) and (7), respectively. As a result, the
instantaneous end-to-end SNR in (5) can be formulated by substituting (6)
and (7) into (1) and (3), respectively, giving



 Q


h2m
h1m
(8)
,
γs =
max
min
m=1,2,...,M 
 max {αn }
max {βmk }  N0
n=1,2,...,N
3
k=1,2,...,K
Performance Analysis
In this section, based on the peak interference power constraints of multiple
PUs, we derive the CDF of the instantaneous end-to-end SNR. Accordingly,
this function will be applied to investigate outage probability and SEP for the
considered CCRN.
Cognitive Cooperative Networks with DF Relay Selection under Interference Constraints
of Multiple Primary Users
111
Let us commence with a theorem stating the CDF of the instantaneous
end-to-end SNR, γs as follows.
Theorem 1. The CDF of the instantaneous end-to-end SNR γs is given by
Fγs (γ) =
M N
−1 K−1
X
X
X M N − 1K − 1
l
i
j
i=0 j=0
l=0
×
(−1)l+i+j N K
ΩSP ΩRP (Al γ + Bi )(Cl γ + Dj )
(9)
where
Al =
N0 l
N0 l
i+1
j+1
, Cl =
, Bi =
, Dj =
ΩSR Q
ΩSP
ΩRD Q
ΩRP
(10)
Proof: For brevity, let us denote
α=
βm =
max
{αn }
(11)
max
{βmk }
(12)
n=1,2,...,N
k=1,2,...,K
Then, the instantaneous end-to-end SNR in (8) is rewritten as
γs = Z
Q
N0
(13)
where
h1m h2m
,
min
Z=
max
m=1,2,...,M
α βm
(14)
It should be noted that the random variables, α and βm , appear in the denominators of the parameters in (14). As M > 1, the CDF of Z cannot be
obtained by using the order statistic theorem directly due to the interdependence among the random variables h1m /α and h2m /βm , m = 1, 2, . . . , M , i.e.,
the CDF of Z cannot be obtained by directly using the result of conventional
relay networks reported in [16]. Instead, the CDF of Z can be derived by using
the law of total probability as
FZ (z) =
Z∞Z∞
0
0
h1m h2m
Pr
max
min
,
< z fβm (y)dy fα (x)dx
m=1,2,...,M
x
y
(15)
112
Part III
In view of the order statistics theory, we have
FZ (z) =
Z∞ Z∞ Y
M
0
=
0
h1m h2m
Pr min
< z fα (x)fβm (y)dxdy
,
x
y
m=1
Z∞ Z∞
0
0
M
1 − 1 − Pr{h1m < xz} 1 − Pr{h2m < yz}
× fα (x)fβm (y)dxdy
(16)
Since the channels are modeled as Rayleigh fading, the channel power gains
follow an exponential distribution. Therefore, the terms Pr{h1m < xz} and
Pr{h2m < yz} in (16) are given by
xz
Pr{h1m < xz} = 1 − exp −
ΩSR
yz
Pr{h2m < yz} = 1 − exp −
ΩRD
(17)
(18)
On the other hand, the PDF of α and βm in (11) and (12), respectively,
can be found by using the order statistics
theory
and the binomial theorem
P
in [19, eq. (1.111)], i.e., (a + x)n = nk=0 nk xk an−k as follows:
fα (x) =N fαn (x){Fαn (x)}N −1
N −1 N X N −1
(i + 1)x
i
=
(−1) exp −
ΩSP i=0
ΩSP
i
fβm (y) =Kfβmk (y){Fβmk (y)}K−1
K−1 (j + 1)y
K X K −1
j
(−1) exp −
=
j
ΩRP j=0
ΩRP
(19)
(20)
Accordingly, the CDF of Z can be determined by substituting (17), (18), (19)
Cognitive Cooperative Networks with DF Relay Selection under Interference Constraints
of Multiple Primary Users
113
and (20) into (16), giving
FZ (z) =
M N −1 K−1 N −1 K −1
NK X X X M
(−1)l+i+j
l
i
j
ΩSP ΩRD
i=0 j=0
l=0
×
Z∞ Z∞
0
=
0
lz
i+1
j+1
lz
exp −
x exp −
y dxdy
+
+
ΩSR
ΩSP
ΩRD
ΩRP
M N
−1 K−1
X
X
X
M
l
l=0 i=0 j=0
×
ΩSP ΩRP
N −1 K −1
i
j
(−1)l+i+j N K
i+1
lz
lz
ΩSR + ΩSP
ΩRD +
Finally, the CDF of γs
Fγs (γ) = FZ (γ NQ0 ).
j+1
ΩRP
is obtained as shown in
(21)
(9) by using
In the subsequent sections, we use the derived CDF of γs to investigate
the performance of the CCRN with best relay selection.
3.1
Outage Probability
The outage probability Pout is defined as the probability that the instantaneous end-to-end SNR falls below a predefined threshold γth . Using (9), the
outage probability is given by
Pout = Fγs (γth )
(22)
where γth = 22rs − 1, rs is outage rate.
3.2
Symbol Error Probability
We now consider the SEP of the CCRN described in Section 2. By making
use of a result in [20, eq. (20)], the SEP can be expressed in terms of the CDF
of γs as
√ Z∞
η θ
e−θγ
Pe = √
Fγs (γ) √ dγ
2 π
γ
(23)
0
where η and θ are constants that depend on the specific modulation scheme
[21]. For example, for M -PSK these parameters are given by η = 2 and
θ = sin2 (π/M ).
114
Part III
Substituting (9) into (23), the expression for the SEP is obtained as
√
M N
−1 K−1
X
X
X M N − 1K − 1 (−1)l+i+j N Kη θ
√
Pe =
2 πΩSP ΩRP
l
i
j
i=0 j=0
l=0
×
Z∞
0
e−θγ dγ
√
γ(Al γ + Bi )(Cl γ + Dj )
(24)
Clearly, the analytical expression of Pe is found by solving the integral in (24).
Therefore, let us consider the following lemma.
Lemma 1. Given a, c ≥ 0 and α, b, d > 0, the integral
I(α, a, b, c, d) =
Z∞
0
e−αt dt
√
t(at + b)(ct + d)
can be solved as follows:
• If a = 0 and c = 0, then
1
I(α, a, b, c, d) =
bd
r
π
α
• If a = 0 and c > 0, then
"
r !#
αd
ec π
αd
I(α, a, b, c, d) = √
1−Q
c
b dc
• If a > 0 and c = 0, then
"
r !#
αb
eaπ
αb
I(α, a, b, c, d) = √
1−Q
a
d ab
• If a > 0, c > 0 and a =
bc
d,
then
"
r !# αd
√
e c dπ(c − 2αd)
απ
αd
+ 1−Q
I(α, a, b, c, d) =
3
bc
c
2b(cd) 2
• If a > 0, c > 0 and c =
ad
b ,
then
"
r !# αb
√
απ
αb
e a bπ(a − 2αb)
I(α, a, b, c, d) =
+ 1−Q
3
ad
a
2d(ab) 2
(25)
Cognitive Cooperative Networks with DF Relay Selection under Interference Constraints
of Multiple Primary Users
115
• If a > 0, c > 0 and bc 6= ad, then
(
r !#
r "
1
c
αd
dα
I(α, a, b, c, d) =
ec π
1−Q
bc − ad
d
c
r !#)
r "
a
αb
bα
1−Q
−e a π
b
a
where Q(x) is the error function defined as Q(x) =
√2
π
R∞
x
2
e−t dt.
Proof: The detailed proof is provided in the Appendix.
Finally, using Lemma 1, the SEP is given by
Pe =
(26)
√
M N
−1 K−1
X
X
X M N − 1K − 1 (−1)l+i+j N Kη θ
√
I(Al , Bi , Cl , Dj )
l
i
j
2 πΩSP ΩRP
i=0 j=0
l=0
(27)
4
Numerical Results
In this section, numerical results on the performance of the considered CCRN
are presented and discussed. Without loss of generality, we set the outage
rate rs = 1 bit/s. Also, we select 8-PSK modulation giving parameters η = 2
and θ = sin (π/8) in the calculation of SEP with respect to (23). Monte-Carlo
simulations are provided to verify the correctness of the analytical results.
We first examine the impact of the number of relays on the system performance while setting the number of PU-Rxs to N = K = 1. Figure 2 and
Figure 3 show the outage probability and the SEP as a function of peakinterference-power-to-noise ratio (PIP) Q/N0 , respectively. Clearly, the analytical results match very well with the simulations. Furthermore, the results
shown in these figures reveal that both outage probability and SEP decrease
significantly as the number of SRs increases, i.e., M = 1, 3, 5, 7, 9. In other
words, the more available relays assist the SU-Tx, the lower the probability
that all relay links are in deep fading due to the randomness and independence of the relaying channels. This leads to the observed performance improvements when increasing the number of relays. It can also be seen that
additional performance improvements are not significant once the number of
SRs increases beyond M = 5 indicating that additional SRs may be saved.
116
Part III
Outage Probability
1
0.1
Analysis
M=1, Simulation
M=3, Simulation
M=5, Simulation
M=7, Simulation
M=9, Simulation
0.01
-6
-4
-2
0
2
Q/N
0
4
6
8
10
(dB)
Figure 2: Outage probability for a single PU-Rx at both SU-Tx and SRs, i.e.,
N = K = 1, ΩSP = ΩRP = ΩSD = ΩRD = 1.
Symbol Error Probability
1
Analysis
M=1, Simulation
0.1
M=3, Simulation
M=5, Simulation
M=7, Simulation
M=9, Simulation
-6
-4
-2
0
2
Q/N
0
4
6
8
10
(dB)
Figure 3: Symbol error probability for a single PU-Rx at both SU-Tx and
SRs, i.e., N = K = 1, channel mean powers ΩSP = ΩRP = ΩSD = ΩRD = 1,
and 8-PSK modulation.
Cognitive Cooperative Networks with DF Relay Selection under Interference Constraints
of Multiple Primary Users
117
1
0.9
0.8
0.7
Outage Probability
N=1, K=3
0.6
N=1, K=5
Simulation
0.5
0.4
N=5, K=3
N=7, K=3
0.3
Simulation
M=5 Relays
0.2
-6
-4
-2
0
2
Q/N
0
4
6
8
10
(dB)
Figure 4: Outage probability for multiple PU-Rxs in the proximity of SU-Tx
and SRs, and channel mean powers ΩSP = ΩRP = ΩSD = ΩRD = 1.
Secondly, we examine the effect of multiple PU-Rxs on the system performance as presented in Figure 4 and Figure 5. As can clearly be seen from these
figures, outage probability and SEP increase when the number of PU-Rxs increases either in the coverage range of the SU-Tx or the SRs. This is thought
to be due to the fact that increasing the number of PU-Rxs also increases
the peak interference power constraints to both the SU-Tx and SRs. This in
turn limits the maximum transmit power of the SU-Tx and SRs resulting in
decreased system performance.
Thirdly, we analyze the impact of channel mean powers on the system
performance. Specifically, we set the number of PU-Rxs at the SU-Tx and
SRs to N = K = 3, and the number of SRs to M = 5. Figure 6 and
Figure 7 illustrate that outage probability and SEP decrease as the channel
mean powers of the SU-Tx→SRm or SRm →SU-Rx links increase. For the
cases of having the same channel mean powers on the SU-Tx→PU-Rxn and
SU-Rxm →PU-Rxk links, for example ΩSP = ΩRP = 1, outage probability
and SEP for the scenario of ΩSR = ΩRD = 0.75 outperform the performance
obtained for the scenario with the lower channel mean powers of ΩSR =
ΩRD = 0.5. It may be conjectured that the higher the channel mean powers
on the SU-Tx→SRm or SRm →SU-Rx links are, the better the performance
118
Part III
Symbol Error Probability
1
M= 5 Relays
N=1 K=2
M=5
N=3 K=2
0.1
N=5 K=2
N=5 K=5
Simulation
-6
-4
-2
0
2
Q/N
0
4
6
8
10
(dB)
Figure 5: Symbol error probability for multiple PU-Rxs in the proximity of
the SU-Tx and SRs, channel mean powers ΩSP = ΩRP = ΩSD = ΩRD = 1,
and 8-PSK modulation.
of the considered CCRN gets. Alternatively, we examine the role of channel
mean powers between the SU-Tx and PU-Rxs by fixing ΩSR = ΩRD = 1 and
varying ΩSP = ΩRP = {0.5, 0.75}. In this setting, outage probability and
SEP obtained for the higher channel mean powers of ΩSP = ΩRP = 0.75 are
inferior compared to the performance obtained for the lower channel mean
powers of ΩSP = ΩRP = 0.5. As the channel mean powers between the
SU-Tx and PU-Rxs become low, the interference from the SU-Tx and SRs
to the PU-Rxs is also kept to a relatively low level. Accordingly, the SU-Tx
and SRs may increase their transmit powers and in this way improve system
performance.
Fourthly, we evaluate the impact of PIP Q/N0 on the SEP using different
modulation schemes, i.e., BPSK, QPSK, 8-PSK, and 16-PSK modulation as
shown in Figure 8. Similar to the results for 8-PSK modulation shown in
Figure 5 and Figure 7, simulations and analysis are in good agreement. Additionally, SEP is improved with increased PIP for all the considered modulation schemes. Accordingly, with the PUs tolerating higher interference levels,
the SU-Tx and SR can transmit with higher power, which in turn leads to
improvement in signal reliability.
Cognitive Cooperative Networks with DF Relay Selection under Interference Constraints
of Multiple Primary Users
119
Outage Probability
1
M=5 Relays
N=K=3 PUs
=
=0.5,
SP
=
SP
=
0.1
SP
=
SP
RP
=0.75,
RP
=1,
=
=1
SR
RD
=
=1
SR
RD
=
RP
=0.75
SR
=1,
RD
=
RP
=0.5
SR
RD
Simulation
-6
-4
-2
0
2
Q/N
0
4
6
8
10
(dB)
Figure 6: Outage probability for multiple PU-Rxs at both SU-Tx and SRs
with different channel mean powers.
1
M=5 Relays
Symbol Error Probability
N=K=3 PUs
SP
SP
RP
0.75
SR= RD=1
RP
0.5
SR= RD=1
Simulation
0.1
SP
RP
1
SR= RD=0.5
SP
RP
1
SR= RD=0.75
Simulation
-6
-4
-2
0
2
Q/N
0
4
6
8
10
(dB)
Figure 7: Symbol error probability for multiple PU-Rxs at both SU-Tx and
SRs, i.e., N = K = 3, number of SRs M = 5, and 8-PSK modulation.
120
Part III
1.0
M=5 Relays
0.9
N=K=3 PUs
Symbol Error Probability
0.8
0.7
0.6
0.5
0.4
0.3
Analysis
0.2
BPSK, Simulation
QPSK, Simulation
0.1
8-PSK, Simulation
0.0
16-PSK, Simulation
-6
-4
-2
0
2
Q/N
0
4
6
8
10
(dB)
Figure 8: Symbol error probability for multiple PU-Rxs at both SU-Tx and
SRs, i.e., N = K = 3, ΩSP = ΩRP = ΩSD = ΩRD = 1, number of SRs
M = 5, and BPSK, QPSK, 8-PSK and 16-PSK modulation.
5
Conclusions
In this paper, we have studied the performance of a DF CCRN with best relay
selection. Specifically, we have considered the case of the SU being subject to
the peak interference power constraints imposed by multiple PUs. We have
derived an expression for the CDF of the instantaneous end-to-end SNR and
utilized it for developing exact expressions for outage probability and SEP of
the considered CCRN. The obtained analytical expressions allow us to examine the impact of the number of relays, the number of PUs, and the channel
mean powers on the system performance. In particular, the numerical results
show that increasing the number of SRs leads to an improvement of system
performance. On the other hand, increasing the number of PUs in the coverage range of the SU-Tx or SRs results in a degradation of system performance
in the considered scenarios. It is also illustrated that having low channel mean
powers between SUs and PUs could improve CCRN performance by allowing
the SUs to transmit with higher power levels. Finally, it should be noted that
Monte-Carlo simulations match well with analytical results, thus, verifying
our analytical expressions.
Cognitive Cooperative Networks with DF Relay Selection under Interference Constraints
of Multiple Primary Users
121
Appendix
In this appendix, we prove Lemma 1 as follows:
I(α, a, b, c, d) =
Z∞
0
e−αt dt
√
t(at + b)(ct + d)
(28)
• If a = 0 and c = 0, by using [19, eq. (3.361.2)], we obtain
r
Z∞ −αt
1
e dt
1 π
√
I(α, a, b, c, d) =
=
bd
bd α
t
(29)
0
• If a = 0 and c > 0, then
1
I(α, a, b, c, d) =
bc
Z∞
0
e−αt dt
√
t(t + dc )
(30)
Changing the variable in (30) by setting x = t + dc and using [19,
eq. (3.363.1)], we have
"
r !#
∞
αd
αd Z
e−αx dx
ec π
ec
αd
q
(31)
= √
1−Q
I(α, a, b, c, d) =
bc
c
d
b
dc
x x−
d
c
c
• If a > 0 and c = 0, then we apply the same method for (30), yielding
"
r !#
Z∞
αb
e−αt dt
αb
eaπ
√
I(α, a, b, c, d) =
= √
(32)
1−Q
a
d t(at + b)
d ab
0
• If a > 0, c > 0 and a = bc/d, then
d
I(α, a, b, c, d) = 2
bc
Z∞
e−αt dt
√
t(t + dc )2
0
|
{z
}
(33)
I1
Then, we apply the method of partial integration for I1 as follows:
u = e−αt ⇒ du = −αe−αt dt
vdv = √
dt
⇒v=
t(t + dc )2
(34)
√
arctg
t
+
3
d
d
d 2
c (t + c )
q c
ct
d
(35)
122
Part III
where v in (35) is simplified by using [19, eq. (2.213.4)] and [19, eq. (2.211)].
Thus, I1 can be expressed as
r !
Z∞ √ −αt
Z∞
αc
te dt
ct
α
e−αt dt
I1 =
arctg
+ d 3
(36)
d
d
d
t+ c
(c )2
0
0
|
|
{z
}
{z
}
I21
I22
Let us first calculate the integral I21 by changing the variable x = t + dc
as
q
r !#
r "
r
Z∞ e αd
c
x − dc e−αx
π
d
αd
αd
I21 =
1−Q
ec
dx =
−π
x
α
c
c
d
c
(37)
The integral (37) is obtained with the help of [19, eq. (3.363.1)].
Secondly, we can rewrite the integral I22 by using the method of partial
integration as
q
r
d
ct
c dt
u = arctg(
) ⇒ du = √
d
2 t(t + dc )
vdv = e−αt dt ⇒ v = −
e−αt
α
(38)
In addition, we have
I22 =
q
d
c
2α
Z∞
0
e−αt dt
√
t(t + dc )
Changing the variable in (39) to x = t + dc gives
q
"
r !#
d dα
Z∞ −αx
αd
c
e
dx
πe c
αd
ce
q
I22 =
1−Q
=
2α
2α
c
d
x x− c
d
(39)
(40)
c
Note that the integral (40) is obtained with the help of [19, eq. (3.363.2)].
By substituting (37) and (40) into (36), we finally obtain an exact expression for (33) as
"
r !# αd
√
απ
αd
e c dπ(c − 2αd)
I(α, a, b, c, d) =
(41)
+ 1−Q
3
bc
c
2b(cd) 2
Cognitive Cooperative Networks with DF Relay Selection under Interference Constraints
of Multiple Primary Users
123
• If a > 0, c > 0 and c = ad/b, then
I(α, a, b, c, d) =
b
a2 d
Z∞
0
e−αt dt
√
t(t + ab )2
"
r
√
πα
=
+ 1−Q
da
bα
a
!#
e
bα
a
πb(a − 2bα)
3
2(ab) 2 d
(42)
The integral (42) can be solved by using the same approach as for (33).
• If a > 0, c > 0 and bc 6= ad, then
I(α, a, b, c, d) =
Z∞
0
e−αt
√
t(at + b)(ct + d)



Z∞
Z∞ −αt
e dt 
e−αt dt
ac 


=
− √

 √
ad − cb 
t(t + ab )
t(t + dc ) 

0
0
{z
} |
{z
}
|
J1
(43)
J2
By changing the variable as x = t + b/a and using [19, eq. (3.363.2)], an
exact expression for J1 is obtained as
r !#
r "
Z∞ αb −αx
eae
αb
dx
a
αb
q
J1 =
1−Q
(44)
= πe a
b
a
b
x x−
a
b
a
Similar to J1 , we can express J2 as
J2 =
Z∞
d
c
αd
c
αd
e−αx dx
q
= πe c
x x − dc
e
r !#
r "
c
αd
1−Q
d
c
(45)
Finally, we obtain the exact expression for I(α, a, b, c, d) in (43) by substituting (44) and (45) into (43) as
r !#
r "
ac
αb
a
αb
I(α, a, b, c, d) =
πe a
1−Q
ad − cb
b
a
r !#!
r "
αd
c
αd
−πe c
1−Q
(46)
d
c
124
Part III
References
[1] J. Mitola, “Cognitive radio for flexible mobile multimedia communications,” in Proc. IEEE International Workshop on Mobile Multimedia
Communication, San Diego, U.S.A., Nov. 1999, pp. 3–10.
[2] J. Mitola and G. Q. Maguire, “Cognitive radio: Making software radios
more personal,” IEEE Personal Commun. Mag., vol. 6, no. 4, pp. 13–18,
Aug. 1999.
[3] S. Haykin, “Cognitive radio: Brain-empowered wireless communications,” IEEE J. Sel. Areas Commun., vol. 23, no. 2, pp. 201–220, Feb.
2005.
[4] J. M. Peha, “Approaches to spectrum sharing,” IEEE Personal Commun.
Mag., vol. 43, no. 2, pp. 10–12, Feb. 2005.
[5] Y. Han, A. Pandharipande, and S. H. Ting, “Cooperative decode-andforward relaying for secondary spectrum access,” IEEE Trans. Wireless
Commun., vol. 8, no. 10, pp. 4945–4950, Oct. 2009.
[6] Y. Han, S. H. Ting, and A. Pandharipande, “Cooperative spectrum sharing protocol with secondary user selection,” IEEE Trans. Wireless Commun., vol. 9, no. 9, pp. 2914–2923, Sep. 2010.
[7] V. Asghari and S. Aissa, “Cooperative relay communication performance
under spectrum-sharing resource requirements,” in Proc. IEEE International Conference on Communications, Cape Town, South Africa, May
2010, pp. 1–6.
[8] Y. Guo, G. Kang, N. Zhang, W. Zhou, and P. Zhang, “Outage performance of relay-assisted cognitive-radio system under spectrum-sharing
constraints,” IET Electronics Letters, vol. 46, no. 2, pp. 182–184, Feb.
2010.
[9] L. Luo, P. Zhang, G. Zhang, and J. Qin, “Outage performance for cognitive relay networks with underlay spectrum sharing,” IEEE Commun.
Lett., vol. 15, no. 7, pp. 710–712, Jul. 2011.
[10] Y. Guo, G. Kang, S. Qiaoyun, M. Zhang, and P. Zhang, “Outage performance of cognitive-radio relay system based on the spectrum-sharing
environment,” in Proc. IEEE Global Communications Conference, Miami, U.S.A., Dec. 2010, pp. 1–5.
Cognitive Cooperative Networks with DF Relay Selection under Interference Constraints
of Multiple Primary Users
125
[11] H. Ding, J. Ge, D. B. da Costa, and Z. Jiang, “Asymptotic analysis of
cooperative diversity systems with relay selection in a spectrum-sharing
scenario,” IEEE Trans. Veh. Technol., vol. 60, no. 2, pp. 457–472, Feb.
2011.
[12] S. Sagong, J. Lee, and D. Hong, “Capacity of reactive DF scheme in cognitive relay networks,” IEEE Trans. Wireless Commun., vol. 10, no. 10,
pp. 3133–3138, Oct. 2011.
[13] J. Lee, H. Wang, J. Andrews, and D. Hong, “Outage probability of cognitive relay networks with interference constraints,” IEEE Trans. Wireless
Commun., vol. 10, no. 2, pp. 390–395, Feb. 2011.
[14] C. Zhong, T. Ratnarajah, and K.-K. Wong, “Outage analysis of decodeand-forward cognitive dual-hop systems with the interference constraint
in Nakagami-m fading channels,” IEEE Trans. Veh. Technol., vol. 60,
no. 6, pp. 2875–2879, Jul. 2011.
[15] H. Tran, T. Q. Duong, and H.-J. Zepernick, “Performance analysis of cognitive relay networks under power constraints of multiple primary users,”
in Proc. IEEE Global Communications Conference, Houston, U.S.A.,
Dec. 2011.
[16] A. Bletsas, H. Shin, and M. Win, “Cooperative communications with
outage-optimal opportunistic relaying,” IEEE Trans. Wireless Commun.,
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[17] A. Ghasemi and E. S. Sousa, “Fundamental limits of spectrum-sharing
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[19] I. Gradshteyn and I. Ryzhik, Table of Integrals, Series, and Products,
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Part IV
Part IV
Delay Performance of Cognitive Radio
Networks for Point-to-Point and
Point-to-Multipoint Communications
Part IV is based on the publications as:
H. Tran, T. Q. Duong, and H.-J. Zepernick, “Delay Performance of Cognitive Radio Networks for Point-to-Point and Point-to-Multipoint Communications,” EURASIP Journal on Wireless Communications and Networking,
vol. 2012, no. 1, 2012.
H. Tran, T. Q. Duong, and H.-J. Zepernick, “Queuing analysis for cognitive
radio networks under peak interference power constraint,” in IEEE International Symposium on Wireless and Pervasive Computing, Hong Kong, China,
Feb. 2011, pp. 1–5.
H. Tran, ”Numerical Results on the Delay Performance of Cognitive Radio
Networks for Point-to-Point and Point-to-Multipoint Communications”, BTH
Technical Report, No. 2012:06, Dec. 2012.
Delay Performance of Cognitive Radio Networks
for Point-to-Point and Point-to-Multipoint
Communications
Hung Tran, Trung Q. Duong, and Hans-Jürgen Zepernick
Abstract
In this paper, we analyze the packet transmission time in spectrum
sharing systems where a secondary user (SU) simultaneously accesses
the spectrum licensed to primary users (PUs). In particular, under
the assumption of an independent identical distributed Rayleigh block
fading channel, we investigate the effect of the peak interference power
constraint imposed by multiple PUs on the packet transmission time of
the SU. Utilizing the concept of timeout, exact closed-form expressions
of outage probability and average packet transmission time of the SU are
derived. In addition, employing the characteristics of the M/G/1 queuing model, the impact of the number of PUs and their peak interference
power constraint on the stable transmission condition and the average
waiting time of packets at the SU are examined. Moreover, we then
extend the analysis for point-to-point to point-to-multipoint communications allowing for multiple SUs and derive the related closed-form
expressions for outage probability and successful transmission probability for the best channel condition. Numerical results are provided to
corroborate our theoretical results and to illustrate applications of the
derived closed-form expressions for performance evaluation of cognitive
radio networks.
1
Introduction
Radio spectrum is one of the most precious and limited resources in wireless
communications. It has become scarce due to the rapid growth of a variety
131
132
Part IV
of mobile devices and the emerging of many new mobile services. However,
recent measurement campaigns conducted by the Federal Communications
Commission in the United States have revealed that vast portions of the allocated spectrum are heavily under-utilized [1]. Clearly, the scarcity of the
spectrum is due to its inefficient usage rather than a shortage of spectrum
resources. As a consequence, the spectrum utilization problem has become
more crucial and has stimulated new research such as extensive work on cognitive radio networks (CRN) [2]. In CRNs, there are two types of users who
are referred to as primary user (PU) and secondary user (SU). The PU licenses the spectrum while a SU may access the spectrum owned by the PU
provided that it does not compromise the quality of service (QoS) delivered to
the PU. Therefore, a major challenge with the design of CRNs is to maintain
the desirable QoS at the PU while offering a sufficiently high transmission
rate to the SUs.
Recently, the spectrum sharing approach is considered as a promising solution to utilize the licensed radio frequency. Particularly, the SU and the
PU can transmit simultaneously as long as the interference caused by the SU
to the PU is lower than a predefined threshold. In [3], considering different
fading channels, the ergodic capacity of the spectrum sharing system is investigated for either peak interference power constraint or average received
interference power at the PU-Rx. This work has revealed that if the link
from the secondary transmitter (SU-Tx) to the primary receiver (PU-Rx) resides in a deep fade, the power of the SU-Tx can be increased to improve
the link to the SU-Rx without compromising the peak interference power
constraint. Later, the fundamental capacity limits with imperfect channel
knowledge have been studied in [4, 5]. In [6], the authors have considered
a new sophisticated approach for spectrum sharing systems where the impact of channel knowledge on the performance of a secondary user has been
studied. The results show that the channel knowledge of the PU-Tx→PURx link is important to mitigate the interference from the SU-Tx→PU-Rx
link while the channel knowledge of the SU-Tx→PU-Rx link has little impact
on the SU capacity. In [7], different notions of capacity are investigated for
the Rayleigh fading channel subject to both the peak and average interference
power constraints. Especially, the ergodic capacity and outage capacity which
are considered suitable for delay-insensitive and delay-sensitive applications
are studied. In [8–10], the novel concept of effective capacity has been introduced to investigate the QoS requirements such as delay constraint in wireless
communication systems. In particular, the effective capacity is defined as the
maximum constant arrival rate that can be provided by the channel while the
delay constraint of the spectrum sharing system is satisfied [9]. The results
in [9] have also shown that for a given peak and average interference power
Delay Performance of Cognitive Radio Networks for Point-to-Point and
Point-to-Multipoint Communications
133
constraint at the PU-Rx, the maximal effective capacity is achieved under
the optimal power control policy. In relation to the delay constraint in the
spectrum sharing system, in [11, 12], we have used another approach, which
is based on the packet transmission time to investigate the performance of
CRN. These results have revealed the impact of the peak interference power
constraint on the delay of packets for different types of fading channels. However, we analyzed the spectrum sharing system with peak interference power
constraint only for a single PU.
In this paper, we therefore extend our previous work [11] to consider the
more realistic case of a CRN under the peak interference power constraint in
the presence of multiple PUs. Specifically, we examine the delay performance
for two scenarios, point-to-point and point-to-multipoint communications. In
the latter scenario, we extend the investigation from multiple PUs to also
allow for multiple SUs at the receiving end. We assume that each packet of
the SU-Tx has a delay constraint. In order to not cause harmful interference
to any surrounding PU-Rx, the SU-Tx needs to adapt its transmit power and
commence transmission before the packet delay threshold is reached. Given
this setting, in the point-to-point scenario, we derive the probability density
function (PDF) and cumulative density function (CDF) for the packet transmission time, outage probability and average transmission time of packets at
the SU-Tx. Furthermore, assuming that packet arrivals at the SU-Tx follow a Poisson process, the queueing model for point-to-point scenario can be
described as an M/G/1 system in which packet inter-arrival times are exponentially distributed, service time is a general distribution and traffic is
processed by a single server. In the point-to-multipoint scenario, also known
as multicast, a secondary base station (SBS) transmits a common packet to
all SU-Rx while keeping the peak interference power to the surrounding PURx below a given threshold. By applying the obtained PDF and CDF for the
point-to-point scenario, a closed-form expression for the outage probability
that the SBS cannot transmit the common packet successfully to a number of
SU-Rx are obtained. Moreover, a closed-form expression for the probability
that the SBS can transmit the common packet successfully to all SU-Rx, i.e.
the best channel condition, is also achieved.
The rest of the paper is organized as follows. In Section 2, the system
model and assumptions for the point-to-point and point-to-multipoint scenarios are introduced. In Section 3, analytical formulations for the point-topoint scenario such as the PDF and CDF of the packet transmission time, the
outage probability, and the moment of packet transmission time is derived.
On this basis, queueing theoretical conclusions are drawn. In Section 4, we
present the delay performance for the point-to-multipoint scenario. Section 5
provides numerical results and discussions. Finally, conclusions are presented
134
Part IV
in Section 6.
2
System Model
In the sequel, we introduce the point-to-point and point-to-multipoint scenarios in the context of a spectrum sharing system where the SU operates in the
area of multiple PUs. As for the radio links between the different entities,
we assume identical and independent distributed (i.i.d.) Rayleigh block fading channels with unit-mean in the presence of additive white Gaussian noise
(AWGN). As the SU and the PUs may transmit simultaneously, the interference caused by the SU to the PUs should not exceed a certain threshold.
2.1
Point-to-point Scenario
Let us consider point-to-point communications in which an SU-Tx is transmitting packets to an SU-Rx while a number M of PU-Rx are operating on
the primary network as shown in Figure 1. The channel power gain of the SUTx→SU-Rx link is denoted by h1 . Similarly, the interference channel power
gain of the SU-Tx→PU-Rxm link is denoted by gm , m = 1, 2, . . . , M . Note
that channel state information (CSI) of the secondary system can be provided
to the SU-Tx through feedback from the SU-Rx while CSI of the SU-Tx to
the PU-Rx can be exchanged using a dedicated common control channel [13].
In our study, we follow the assumption given in [3, 5, 14, 15] that the SU-Tx is
close to the PU-Rx but the SU-Rx is far away from the primary transmitters
(PU-Tx). Therefore, only the SU-Tx causes interference to the PU-Rx while
interference caused by the PU-Tx to the SU-Rx is lumped with the AWGN.
2.1.1
Peak Interference Power Constraint
In order to process the offered traffic, the SU-Tx is equipped with a buffer
which stores incoming packets of the same size. The SU-Tx transforms the
stored packets into bit streams and adopts its transmission power based on
the joined CSI which shall be denoted as (M + 1)-tuple (g1 , g2 , . . . , gM ; h1 ).
The main objective for the considered spectrum sharing system may be posed
as to minimize the transmission time of packets at the SU while not causing
harmful interference to the PU-Rx. Following [16], the time taken by an
SU-Rx to decode L bits information of a packet can be expressed as
T =
e
L
B
,
B log2 (1 + γ)
loge (1 + γ)
(1)
Delay Performance of Cognitive Radio Networks for Point-to-Point and
Point-to-Multipoint Communications
135
1
2
1
2
M
1
M
Figure 1: Point-to-point communication of the considered spectrum sharing
system with multiple PUs (solid line: communication from SU-Tx to SU-Rx;
dashed line: interference from SU-Tx to surrounding PU-Rx).
e = L log (2)/B, and γ is the signal-towhere B is the system bandwidth, B
e
noise ratio (SNR) at the SU-Rx given by
γ=
h1 P (g1 , g2 , . . . , gM ; h1 )
N0
(2)
In (2), N0 represents the noise power and P (g1 , g2 , . . . , gM ; h1 ) is the power
allocation policy for the SU-Tx corresponding to the joined CSI given as
(g1 , g2 , . . . , gM ; h1 ). According to [3], the transmission power of the SU-Tx
with respect to PU-Rxm should be adjusted to be lower than an allowable
level:
gm P (g1 , g2 , . . . , gM ; h1 ) ≤ Qm
pk ,
m = 1, 2, . . . , M
(3)
where Qm
pk is the peak interference power that the PU-Rxm can tolerate without scarifying QoS. Furthermore, let us assume that the tolerable peak interference power is the same for all PU-Rx, i.e. Qm
pk = Qpk for m = 1, 2, . . . , M .
In order to not cause harmful interference to any PU-Rx in the primary system, the transmission power of the SU-Tx must then satisfy the peak interference power constraint given as
P (g1 , g2 , . . . , gM ; h1 ) ≤
Qpk
max {gm }
m
(4)
136
Part IV
SU-Tx
SU-Rx
T1> tout
tout
T2 < tout
ACK
tout
T3 < tout
tout
ACK
Figure 2: Example of a timing diagram for point-to-point communication
between SU-Tx and SU-Rx.
2.1.2
Delay Constraint
As far as the transmission time of packets is concerned, this is clearly nondeterministic due to the fading channel. In the sequel, the transmission of a
packet is considered as successful if the packet transmission time is less than a
predefined threshold, tout , referred to as timeout. Figure 2 shows an example
of a timing diagram of packet transmission for point-to-point communication
between SU-Tx and SU-Rx. Recall that the SU-Tx receives packets from
higher layers which it will convert into bit streams at the lower layer prior to
transmission over the fading channel. Once the SU-Rx has received a sufficient
number of bits and decoded the related packets successfully, it will respond
with an acknowledgement (ACK) packet that is assumed to be error-free and
incurs negligible delay to the SU-Tx. This ACK indicates the SU-Tx that
it can eliminate the corresponding packet at the head of the buffer and may
continue with transmitting the subsequent packets. In the example shown
in Fig. 2, the first packet is transmitted unsuccessfully as the SU-Tx does
not receive an ACK within tout , i.e. T1 ≥ tout . In this case, the SU-Tx
considers the packet as dropped. In contrast, the second and third packet are
transmitted successfully as their transmission times are less than the timeout
tout , i.e. T2 , T3 < tout .
Delay Performance of Cognitive Radio Networks for Point-to-Point and
Point-to-Multipoint Communications
2.1.3
137
Queuing Model for Point-to-point Communications
The packets arriving at the SU are stored in a buffer and served in first-in
first-out (FIFO) order. Assuming that the packet arrival follows a Poisson
process with arrival rate λ, the considered point-to-point scenario may be
modeled as an M/G/1 queueing system [17–19] with service time given as
general distribution and the system being equipped with a single server [20].
From (1), (2) and the peak interference power constraint (4), we can conclude that the packet transmission time depends on both the channel gain and
the peak interference power constraint. Clearly, once the distribution of transmission time is determined, the average waiting time of packets at the SUTx can be calculated by applying the Pollaczek-Khinchin’s equation [20, Eq.
(8.34)] as follows
E[W ] = E[T ] + E [Tq ]
(5)
where E[W ] is the total average waiting time of packets at the SU-Tx and
E [Tq ] is the average waiting time of packets in the buffer. It is noted that
E [Tq ] can be formulated as
λE T2
(6)
E [Tq ] =
2(1 − ρ)
where ρ = λ E[T ] is referred to as channel utilization and E[T i ], i = 1, 2
denotes the first and second moment of packet transmission time, respectively.
Furthermore, the following result from queueing theory can be applied for the
stability of transmission of the SU.
Stability condition [20]: Transmission of the SU-Tx is stable if and only if
the average arrival rate λ is less than the average transmission rate µ, that is
λ<µ
(7)
where average transmission rate is defined by the inverse of the average transmission time as
µ=
2.2
1
E[T ]
(8)
Point-to-multipoint Scenario
In this scenario, we consider a spectrum sharing system as shown in Fig. 3 in
which a secondary base station (SBS) transmits a common packet to a number
N of SU-Rx in its coverage range. This scenario is also known as mobile
multicast network in which the base station transmits common information
to multiple receivers over broadcast channels [21, 22].
138
Part IV
1
1
1
1
2
2
M
2
N
2
N
M
Figure 3: Point-to-multipoint communication of the considered spectrum
sharing system with multiple PUs and multiple SUs (solid line: communication from SU-Tx to surrounding SU-Rx; dashed line: interference from
SU-Tx to surrounding PU-Rx).
2.2.1
Peak Interference Power Constraint
In this spectrum sharing scenario, the power allocation problem becomes more
complicated as the SBS must not only adjust its power to guarantee successful packet transmission to all SU-Rx in the secondary system but must also
limit the interference power caused to the active PU-Rx in the primary system. Clearly, the transmission time of a common packet will vary among the
different SU-Rxn due to the involved i.i.d. Rayleigh fading channels. Similar
to (1), the transmission time of a packet to an SU-Rxn can be expressed as
Tn ,
e
B
,
loge (1 + γn )
n = 1, 2, . . . , N
(9)
and γn is the SNR at the nth SU-Rxn which can be formulated as
γn =
hn P (g1 , g2 , . . . , gM ; h1 , h2 , . . . , hN )
,
N0
n = 1, 2, . . . , N
(10)
where hn is the channel gain from the SBS to the SU-Rxn while the optimal transmission power P (g1 , g2 , . . . , gM ; h1 , h2 , . . . , hN ) of the secondary
base station is given with respect to the joined CSI denoted as (M + N )tuple (g1 , g2 , . . . , gM ; h1 , h2 , . . . , hN ). The transmission power policy of the
SBS with respect to the PU-Rxm should then satisfy the following condition:
gm P (g1 , g2 , . . . , gM ; h1 , h2 , . . . , hN ) ≤ Qm
pk ,
m = 1, 2, . . . , M
(11)
Delay Performance of Cognitive Radio Networks for Point-to-Point and
Point-to-Multipoint Communications
SU-Rx2
SBS
139
SU-Rx1
T1,1
T2,1
ACK
ACK
tout
T2,2
T1,2
ACK
tout
Figure 4: Example of a timing diagram for communication between the SBS
and two SU-Rx.
Similar to the point-to-point scenario, we assume Qm
pk = Qpk which leads to
the condition for the instantaneous transmission power of SBS as
P (g1 , g2 , . . . , gM ; h1 , h2 , . . . , hN ) ≤
Qpk
max {gm }
(12)
m
2.2.2
Delay Constraint
In the point-to-multipoint scenario, the SBS tries to broadcast common packets to all SU-Rx in its coverage range. Each common packet has a time-to-live
which should be less than tout . If an SU-Rx receives a common packet, it feeds
back an ACK to the SBS before tout . This means that the SU-Rx has received
the common packet successfully. Otherwise, the SBS implies that the SU-Rx
has not received the transmitted packet. Figure 4 shows an example of a
timing diagram where the SBS transmits common packets to two SU-Rx. In
particular, the SBS transmits the first packet successfully as both transmission times T1,1 and T2,1 corresponding to SU-Rx1 and SU-Rx2 , respectively,
are less than the timeout tout . It is noted that T1,1 may be different from
T2,1 due to the different fading channel and spatial separation of SU-Rx1 and
SU-Rx2 . In contrast, the second common packet is transmitted unsuccessfully
to SU-Rx2 as the SBS does not receive an ACK from SU-Rx2 before timeout
tout .
Clearly, if the SBS receives ACKs from all SU-Rx before tout , it can be
considered as the best channel condition. On the other hand, the SBS may
140
Part IV
not transmit the common packet successfully to all SU-Rx due to the fading
environment.
3
Performance Analysis for Point-to-point Communications
In this section, we derive closed-form expressions for the PDF and CDF of
packet transmission time as well as outage probability. Based on these results,
we not only quantify the first and second moment of packet transmission time
but also investigate the queueing theoretical characteristics of the considered
spectrum sharing system.
3.1
PDF of packet transmission time
In this scenario, the SU-Tx wants to transmit with maximum transmission
rate in order to reduce dropped packets due to timeout. On the other hand,
the SU-Tx not only needs to adjust its transmission power in response to
changes of the transmission environment but also guarantee the QoS of any
PU-Rx around. Given perfect CSI, the maximum instantaneous transmission
power of the SU-Tx in (4) can be expressed with equality as
P (g1 , g2 . . . , gM ; h1 ) =
Qpk
max {gm }
(13)
m
By substituting (13) into (2), we can rewrite (1) as
T =
loge 1 +
e
B
Qpk
h1
max{gm } N0
m
(14)
It is easy to see that the packet transmission time, T , now turns out to
be a function of multiple random variables, i.e. h1 , gm , m = 1, 2, . . . , M .
Therefore, in order to investigate the delay performance, we need to derive
the PDF of T in the sequel.
Let us start with the CDF of g0 = max {gm } where gm is the channel gain.
m
Because the channel coefficients undergo Rayleigh fading, the channel gain,
gm , is a random variable distributed following an exponential distribution
with unit-mean, given by
Fgm (y) = 1 − e−y
(15)
Delay Performance of Cognitive Radio Networks for Point-to-Point and
Point-to-Multipoint Communications
141
Using order statistics, we can easily obtain the CDF and PDF of g0 , respectively, as follows:
Fg0 (y) = 1 − e−y
fg0 (y) = M e
−y
M
1−e
(16)
−y M−1
(17)
For convenient derivation, let us denote Z = h1 /g0 . The PDF of Z can be
obtained by applying the method presented in [23] as
fZ (z) =
M−1
X
m=0
M −1
(−1)m M
m
(1 + m + z)2
On the other hand, the CDF of T can be formulated as
e
B
N0
FT (x) = Pr{T < x} = 1 − Pr Z < (e x − 1)
Qpk
e
B
M−1
X M −1
(−1)m (e x − 1)M
=1−
e
B
mQ
m
(1 + m)(e x + N0pk + G)
m=0
(18)
(19)
and the PDF of T can be derived by differentiating (19) with respect to x as
e /x
M−1
exp B
X M − 1 (−1)m B
e M Qpk
fT (x) =
2 , x ≥ 0
N0
m
e /x + mQpk + G x2
m=0
exp B
N0
(20)
Q
where G = Npk
− 1 is introduced for brevity. It is noted that (20) exactly
0
leads to the PDF of [11, Eq. (10)] for the peak interference power constraint
of a single PU-Rx by setting M = 1.
In the subsequent sections, the important result in (20) will be used to
investigate the outage probability, the average transmission time and the average waiting time of packets.
3.2
Outage Probability
Given the channel conditions and the peak interference power constraint, the
outage probability Pout is defined as the probability that the packet transmission time T exceeds the interval tout :
Pout = Pr(T ≥ tout )
(21)
142
Part IV
From (19), we can easily obtain the closed-form expression for the outage
probability as
Pout = 1 − P r(T < tout ) = 1 − FT (tout )
=
e /tout − 1
exp
B
M − 1 (−1) M
1 + m exp B
m
e /tout + mQpk + G
N0
M−1
X
m=0
m
(22)
On the other hand, let Tsuc denote the transmission time of a packet given
that it is not dropped, i.e.,
Tsuc = {T |T < tout }
(23)
Accordingly, applying Bayes’ rule, the probability that the event Tsuc takes
place can be expressed as
P {T |T < tout } =
P {T, T < tout }
P {T, T < tout }
=
P {T < tout }
1 − Pout
(24)
Based on (24), we can express the CDF of Tsuc as follows:
1
FTsuc (x) =
1 − Pout
Zx
fT (t)dt, 0 ≤ x < tout
(25)
0
and FTsuc (x) = 0 for x ≥ tout . Differentiating both sides of (25) with respect
to x, the PDF of the packet transmission time without being timed out can
be presented as
fTsuc (x) =
fT (x)
d
, 0 ≤ x < tout
FTsuc (x) =
dx
1 − Pout
(26)
and fTsuc (x) = 0 for x ≥ tout . Substituting (20) into (26), the PDF of packet
transmission time without being timed out can be obtained as
e /x
M−1
exp B
X M − 1 (−1)m B
e M Qpk
fTsuc (x) =
h
i2 ,
m
(1 − Pout ) N0
e /x + mQpk + G x2
m=0
exp B
N0
0 ≤ x < tout
(27)
while fTsuc (x) = 0 for x ≥ tout . In the following, the PDF fTsuc (x) given in
(27) will be used to derive the moment of packet transmission time.
Delay Performance of Cognitive Radio Networks for Point-to-Point and
Point-to-Multipoint Communications
3.3
143
Moment of Packet Transmission Time
Let us recall that a transmitted packet can be received successfully or not due
to the fading channel. Therefore, examining average transmission time shall
consider both packet transmission time without and with timeout.
Let us start with the average transmission time of packet without timeout
as follows
E [Tsuc ] =
tout
Z
0
M−1
X M − 1
M Qpk
xfTsuc (x)dx =
(−1)m
m
(1 − Pout ) N0 m=0
tout
Z
e /x
e exp B
B
×
i2 dx
h
mQpk
e
+
G
x
exp
B
/x
+
0
N0
(28)
e /x) and applying an exchange of variables in the integral
By setting t = exp(B
of (28), we finally obtain the first moment of packet transmission time without
timeout as
M Qpk
e /tout , G
(29)
E [Tsuc ] =
ψ1 exp B
(1 − Pout ) N0
where
ψ1 (a, b) =
Z∞
M −1
m
(−1)
m
M−1
X
m=0
a
e
B
(loge t) t +
mQpk
N0
+b
2 dt
(30)
Similarly, we can calculate the second moment of packet transmission time
without timeout as follows
2 =
E Tsuc
tout
Z
x2 fTsuc (t)dt =
0
M−1
X M − 1
M Qpk
(−1)m
m
(1 − Pout ) N0 m=0
tout
Z
e exp B
e /x
B
×
h
i2 dx
mQpk
e
exp
B
/x
+
+
G
0
N0
(31)
Using similar exchange of variables as above for (31), we obtain the second
moment of Tsuc as
2 M Qpk
e /tout , G
(32)
E Tsuc
=
ψ2 exp B
(1 − Pout ) N0
144
Part IV
where
ψ2 (a, b) =
Z∞
M −1
m
(−1)
m
M−1
X
m=0
a
e2
B
h
(loge t)2 t +
mQpk
N0
+b
i2 dt
(33)
Finally, by applying the law of total expectation, the first and the second
moment of packet transmission time (including dropped packets) can be given
by
i + tiout Pout , i = 1, 2
(34)
E[T i ] = (1 − Pout ) E Tsuc
i
where Pout is given by (22) and E[Tsuc
], i = 1, 2 can be calculated by (29)
and (32), respectively.
3.4
Queuing Theoretical Characteristics
Firstly, the expression for the average waiting time of packets in the buffer of
SU-Tx can be obtained by substituting (34) with respect to i = 1, 2 into (5)
and (6) as
e /tout , G
N0 tout Pout + M Qpk ψ1 exp B
E[W ] =
N0
e /tout , G + λN0 t2 Pout
λM Qpk ψ2 exp B
out
i (35)
+ h
e /tout , G − λtout Pout
2 N0 − λM Qpk ψ1 exp B
Secondly, the transmission of an SU is stable if and only if the average arrival
rate is less than the average transmission rate. Thus, we can make a statement
about the stable transmission condition as follows:
Remark: Given the channel state information and the peak interference
power constraint of M PUs, the transmission of the SU is stable if and only
if the average arrival rate of packet, λ, satisfies the condition
λ<
1
(1 − Pout ) E [Tsuc ] + tout Pout
(36)
The inequality (36) is derived by substituting (34) for i = 1 into (7).
Finally, by using the Little theorem [20, Eq. (8.2)], the average number of
packets waiting in the SU-Tx can be formulated as
NSU = λ E[W ]
where E[W ] is given by (35).
(37)
Delay Performance of Cognitive Radio Networks for Point-to-Point and
Point-to-Multipoint Communications
4
145
Performance Analysis for Point-to-multipoint
Communications
In this section, we consider point-to-multipoint communications, in which
both SU and PU links undergo Rayleigh fading. We first derive the exact
closed-form expression for the outage probability of the secondary network,
and then we consider the probability for the special case that the SBS can
transmit the common packet successfully to all SU-Rx in its coverage range.
4.1
Outage Probability
In the point-to-multipoint scenario, the SBS transmits common packets to
SU-Rx in its coverage. Some SU-Rx may not receive the common packets
successfully due to fading environment. In order to analyze the performance
of this scenario, we will calculate the probability that k out of the total of
N SU-Rx cannot receive the common packets successfully, known as outage
probability.
Similar to point-to-point communications, the event that the SU-Rxn cannot receive a packet successfully is formulated as Tn ≥ tout where Tn is an
i.i.d. random variable distributed following the CDF given by (19). Therefore,
the outage probability in this case can be formulated as
N
N −k
Prk {Tn ≥ tout } (1 − Pr{Tn ≥ tout })
k
N
−k X
N
N −k
(−1)j
=
k
j
j=0
"M−1 #k+j
X M − 1
e /tout ) − 1)
(−1)m M (exp(B
×
e /tout ) + mQpk + G)
m
(1 + m)(exp(B
m=0
N0
k
Pout
=
(38)
where (38) is obtained by using the binomial theorem and the help of (19).
4.2
Best Channel Condition
For point-to-multipoint communications, the SBS may transmit common packets successfully to all SU-Rx if the channel condition is ideal. This is known
as the best channel condition which can be expressed as the longest transmission time for one common packet to be less than tout , i.e., {max{Tn } < tout }.
n
Therefore, the probability that the SBS transmits the common packet to N
146
Part IV
0.40
Analysis
0.36
Q=1 dB, Simulation
Q=5 dB, Simulation
0.32
Q=10 dB, Simulation
Q=15 dB, Simulation
Outage probability
0.28
0.24
0.20
L=2048 bits
0.16
0.12
0.08
0.04
0.00
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Number of PUs, M
Figure 5: Impact of the number of PUs M on the outage probability Pout for
different values of the PIP Q.
SU-Rx with the best channel condition can be given as
Pr{max{Tn } < tout } =
n
N
Y
Pr{Tn < tout }
n=1
#N
"M−1 X M − 1
e /tout ) − 1)
(−1)m M (exp(B
=
e /tout ) + mQpk + G)
m
(1 + m)(exp(B
m=0
N0
(39)
where (39) can be calculated with the help of (19).
5
Numerical Results
In this section, numerical results are provided to examine the performance of
the considered cognitive radio networks for the point-to-point and point-tomultipoint communications. In particular, we first study the impact of the
peak-interference-power-to-noise ratio (PIP) Q = Qpk /N0 , packet size L, and
the number of PUs M on the outage probability, average transmission time,
and queuing characteristics of the point-to-point scenario. Then, numerical
Delay Performance of Cognitive Radio Networks for Point-to-Point and
Point-to-Multipoint Communications
147
results for point-to-multipoint communications are presented. More specifically, the probability that k out of N PUs cannot receive the common packet
successfully and the probability that the SBS transmits the common packet
successfully under the best channel condition are examined. Without loss of
generality, we can set system parameters as follows: System bandwidth B = 1
MHz and timeout threshold tout = 10 ms.
5.1
Point-to-point Communications
In the sequel, we consider the impact of the PIP, the number of the PUs, and
the packet size on the performance of an SU.
5.1.1
Outage Probability
Figure 5 plots the outage probability as a function of the number of PUs for
given PIP of Q = 1, 5, 10, 15 dB and packet size L = 2048 bits. Clearly, we can
see that the simulation matches very well with the analysis in all cases of the
PIP Q. The outage probability increases fast as the number of PUs increases
for low values of the PIP such as Q = 1 dB. On the other hand, increase of
outage probability is slow when the PIP is set to high values, Q = 5, 10, 15
dB and reaches a constant level fast for Q = 10, 15 dB. This can be explained
by the fact that the PUs can tolerate more interference from the SU-Tx when
the PIP is set to a high value. Accordingly, the SU-Tx can use higher power
to increase its own transmission rate without causing harmful interference to
any PU, and hence the time consumed for packet transmission is kept short.
As a result, the outage performance is enhanced. On the other hand, for a
fixed value of the PIP, e.g. Q = 1 dB, the SU-Tx transmit power is subject to
more constraints as the number of PUs in the SU-Tx coverage range increases.
This leads to a decrease of the SU-Tx transmit power, and hence the outage
performance is degraded.
Figure 6 shows the outage probability as a function of the PIP, Q, for
given packet size L = 1024, 2048 bits and the number of PUs M = 2, 4, 6.
As can clearly be observed from the figure, the outage probability degrades
significantly as the PIP increases to the high regime, e.g. Q > 3 dB. This
happens due to the same reason explained above, i.e., increasing PIP, Q,
leads to a decreasing of the packet transmission time, and hence the outage
performance is improved. In addition, for a given number of active PUs in
the coverage range of the SU-Tx, the outage performance of packets having
relatively small size, e.g. L = 1024 bits, outperforms the one having larger
size, e.g., L = 2048 bits. It is easy to understand that packets having smaller
size consume shorter transmission time, thus the outage probability decreases.
148
Part IV
1.0
Analysis
M=2, L=1024 bits, Simulation
M=2, L=2048 bits, Simulation
0.8
M=4, L=1024 bits, Simulation
Outage probability
M=4, L=2048 bits, Simulation
M=6, L=1024 bits, Simulation
M=6, L=2048 bits, Simulation
0.6
0.4
0.2
0.0
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
Q (dB)
Figure 6: Impact of the PIP Q on the outage probability Pout for different
values of packet size L and number of PUs M .
0.010
L=2048 bits
Average transmission time (seconds)
0.009
Analysis
Simulation
0.008
0.007
0.006
0.005
0.004
Number of PUs
0.003
M=1, 3, 5, 8
0.002
0.001
0.000
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
Q (dB)
Figure 7: Impact of the PIP Q on the average transmission time of packets
for different number of PUs M .
Delay Performance of Cognitive Radio Networks for Point-to-Point and
Point-to-Multipoint Communications
5.1.2
149
Average Transmission Time
Figure 7 displays the average transmission time of packets at the SU-Tx as
a function of the PIP, Q, for the number of PUs M = 1, 3, 5, 8 and packet
size L = 2048 bits. It can be observed from the figure that the average
transmission time for packets reduces when the PIP, Q, increases. Specifically,
the average packet transmission time reduces quite fast in the relative high
PIP regime, Q > 4 dB. This can be explained by the same reason discussed
above for the outage probability, i.e., the SU-Tx can transmit with high power
as the PUs tolerates a high interference level. This leads to an increase of
transmission rate, which in turn decreases the packet transmission time.
The results depicted in Figure 8 allow us to further examine the impact
of the number of PUs and the packet size on the average transmission time of
packets. It can be clearly seen that the number of active PUs in the coverage
range of the SU-Tx has a significant influence on the average transmission
time at low values of the PIP, Q = 5 dB and large packet size L = 2048
bits, and it increases quite fast with increasing M . In contrast, for higher
values of the PIP and smaller packet size such as Q = 10 dB and L = 1024
bits, an increase of the number of the PUs only causes a slow increase of the
average transmission time for M > 6. These results are consistent with the
observations for the outage probability given in Figure 5.
0.005
Analysis
Average transmission time (seconds)
L=2048 bits, Simulation
L=1024 bits, Simulation
0.004
L=2048 bits, Simulation
L=1024 bits, Simulation
0.003
Q=5 dB
0.002
Q=10 dB
0.001
0.000
1
2
3
4
5
6
7
8
9
10
11
12
13
14
Number of PUs, M
Figure 8: Average transmission time of packets versus number of PUs M .
150
Part IV
Average waiting time (seconds)
0.014
0.012
0.010
M=1, L=2048 bits,
packets/s
M=5, L=2048 bits,
packets/s
M=5, L=2048 bits,
packets/s
M=5, L=1024 bits,
packets/s
M=1, L=1024 bits,
packets/s
M=5, L=1024 bits,
packets/s
0.008
0.006
0.004
0.002
0.000
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
Q (dB)
Figure 9: Average waiting time of packets versus PIP Q for different number
of PUs M (analysis).
9
packets/s, L=1024 bits
Average number of packets in system
8
M=5
packets/s, L=2048 bits
packets/s, L=1024 bits
7
packets/s, L=2048 bits
6
5
4
3
2
1
0
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
Q (dB)
Figure 10: Impact of the PIP Q on the number of packets in the SU-Tx with
different arrival rate λ (analysis).
Delay Performance of Cognitive Radio Networks for Point-to-Point and
Point-to-Multipoint Communications
5.1.3
151
Queueing Theoretical Results
In the following, queuing characteristics of the SU-Tx under the peak interference power constraint (4) are shown in Figures 9, 10, and 11. In particular, we
have set the packet size L = 1024, 2048 bits and observe the average waiting
time, average number of packets, and channel utilization for different values
of average arrival rate.
Figures 9 indicates that the average waiting time increases as the number
of PUs, packet size and arrival rate increase. Clearly, these results are consistent with the observations for the outage probability and average transmission
time and can be understood as follows. For a fixed value of the PIP, Q, an
increasing number of PUs and packet size lead to an increase of average transmission time due to the same reason discussed above, and hence this causes
an increase of average waiting time. Additionally, when the arrival rate of
packets increases, the number of packets arriving at the buffer increases as
well (see Figure 10) and await transmission. On the other hand, the transmission rate is restricted due to the peak interference power constraint, and
hence the packets have to wait a long time in the SU-Tx buffer before they
are transmitted.
1.0
M=5
0.9
0.8
L=2048 bits,
packets/s
L=1024 bits,
packets/s
L=2048 bits,
packets/s
L=1024 bits,
packets/s
Channel utilization
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0.0
-8
-7
-6
-5
-4
-3
-2
-1
0
1
2
3
4
5
Q (dB)
Figure 11: Impact of the PIP Q on the channel utilization ρ with different
packet size L and arrival rate λ (analysis).
152
Part IV
Figure 11 shows the stable transmission condition as a function of the
PIP, Q, with the number of PUs given as M = 5, packet size L = 1024, 2048
bits, and arrival rate being λ = 50, 100 packets/s. The result reveals that
for a given packet size L and fixed value of PIP Q, the channel utilization
ρ = λ E[T ] for arrival rate λ = 50 packets/s is relative small. Furthermore,
the channel utilization outperforms the results for λ = 100 packets/s. In other
words, the significant lower channel utilization for λ = 50 packets/s compared
to λ = 100 packets/s provides a more stable transmission with respect to the
service rate µ in terms of the stable condition formulated in (36). For fixed
values of packet size L and PIP Q, the service rate µ is restricted due to the
transmit power constraint while a higher arrival rate leads to more packets
arriving at the buffer, i.e., more packets expect to be transmitted timely.
Therefore, the average arrival rate to average service rate ratio, relating to
the stable transmission condition ρ = λ/µ < 1, must be carefully considered
in order to not exceed the capacity of the secondary system. It can also be
seen from the figure that in the high regime of the PIP and small packet size,
e.g., Q ≥ 3 dB and L = 1024 bits, the stable transmission condition can be
easily satisfied due to the sufficiently low channel utilization, ρ < 1.
0
10
-1
10
Outage probability
-2
10
-3
10
-4
10
Analysis
k=1, L=2048 bits (simulation)
-5
k=1, L=1024 bits (simulation)
10
k=2, L=2048 bits (simulation)
k=2, L=1024 bits (simulation)
-6
10
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
Q (dB)
Figure 12: Impact of the PIP Q on the probability that k out of N SUs
cannot receive the common packet successfully for the point-to-multipoint
communications, for number of PUs and SUs set to N = M = 5, and packet
size L = 1024, 2048 bits.
Delay Performance of Cognitive Radio Networks for Point-to-Point and
Point-to-Multipoint Communications
153
1.00
Succesful transmission probability
0.95
0.90
0.85
0.80
M=8
0.75
Analysis, L=4096 bits
0.70
N=2, Simulation
N=3, Simulation
0.65
N=4, Simulation
0.60
Analysis, L=1024 bits
0.55
N=2, Simulation
0.50
N=3, Simulation
N=4, Simulation
0.45
0.40
10
11
12
13
14
15
16
17
18
19
20
Q (dB)
Figure 13: Impact of the PIP Q on the successful transmission probability for
the best channel condition (Number of PUs M = 8, number of SUs N = 2, 3, 4,
packet size L = 1024, 4096 bits).
5.2
Point-to-multipoint Communications
Figure 12 shows the impact of the PIP on the outage probability of the pointto-multipoint communications. In particular, we set the number of SUs and
PUs as N = M = 5, packet size L = 1024, 2048 bits and plot the outage
probability as a function of the PIP, Q, under the condition that k = 1, 2 out
of a total of N = 5 SU-Rxs cannot receive the common packets successfully.
Clearly, we can see from the results that the probability of exactly k out of
the N = 5 SU-Rx not being able to receive the common packet successfully
decreases as k increases. Furthermore, the outage performance is enhanced
as the value of Q increases and the packet size L decreases as expected.
Figure 13 shows the probability of all SU-Rx receiving the common packets
successfully as a function of the PIP with the number of PUs fixed to M = 8
and the number of SUs given as N = 2, 3, 4. This scenario is known as the
best channel condition as outlined in Section 4.2. We can see that in the high
regime of the PIP Q ≥ 18 dB, the probability of the SBS transmitting the
common packets successfully to all SUs is very high (above 0.925) for packets
having small size L = 1024 bits, and is quite high (above 0.85) for packets
having large size L = 4096 bits. The figure also indicates that the probability
154
Part IV
of successful transmission decreases with an increase of the number of SUs,
N = 2, 3, 4. It may be conjectured that the SBS power is restricted by the
interference constraint of the PUs, thus its communication range is restricted.
On the other hand, when the number of SUs increases, some of them may be
distributed far away from the SBS, and difficult to receive the common packet
successfully. As a result, the successful transmission probability for the best
channel condition is degraded.
6
Conclusions
In this paper, we have analyzed the delay performance of spectrum sharing
systems for point-to-point and point-to-multipoint communications. In particular, we have assumed that each packet has a delay threshold, transmission
channels undergo Rayleigh fading, SUs posses perfect CSIs and ACKs are
transmitted without error and delay. Closed-form expressions for the outage
probability and average transmission time for point-to-point communications
are obtained. In addition, we have utilized the M/G/1 queuing model to
analyze the queueing characteristics of such systems including the average
transmission time, the packet waiting time and the stable transmission condition of an SU. Based on the analytical framework established for point-topoint communications, we have also derived closed-form expressions for the
outage probability and the successful transmission probability for point-tomultipoint communications under best channel conditions. Numerical results
for representative scenarios have been provided to quantify the impact of an
increase of the number of SUs and PUs on system performance. In particular, it has been shown that an increasing number of SUs or PUs significantly
increases packet delay if the peak interference power is constraint by the PUs
to be low while small performance degradation is observed if the PUs tolerate
sufficiently large peak interference power. Accordingly, the developed analytical framework for point-to-point and point-to-multipoint communications in
spectral sharing systems may serve to efficiently examine system performance.
For example, it may be used to deduce a trade-off between QoS requirements
of the secondary system and interference constraints posed by the primary
system.
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Delay Performance of Cognitive Radio Networks for Point-to-Point and
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155
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Part V
Part V
Performance of Cognitive Radio Networks over
General Fading Channels
Part V is based on the publications as:
H. Tran, T. Q. Duong, and H.-J. Zepernick, “On the Performance of Spectrum
Sharing Systems Over α-µ Fading Channel for Non-identical µ Parameter,” in
Proc. IEEE International Symposium on Wireless Communication Systems,
Aachen, Germany, Nov. 2011, pp. 1-6.
H. Tran, H.-J. Zepernick, H. Phan, and M. Fiedler,“Outage Probability, Average Transmission Time, and Quality of Experience for Cognitive Radio Networks over General Fading Channels,” in Proc. EURO-NF Conference on
Next Generation Internet, Karlskrona, Sweden, Jun. 2012, pp. 9–15.
Performance of Cognitive Radio Networks over
General Fading Channels
Hung Tran, Hans-Jürgen Zepernick, Trung Q. Duong,
Markus Fiedler, and Hoc Phan
Abstract
In this study, the performance analysis of cognitive radio networks
over α-µ fading channels is investigated. In particular, we consider a
scenario where the secondary transmitter (SU-Tx) sends a data packet
to the secondary receiver (SU-Rx) and waits for an acknowledgement
(ACK) to transmit the next packet. If a given round-trip-time (RTT)
is expired, the data packet is retransmitted. In addition, all operations of the secondary users are assumed to be subject to the peak
interference power constraint of the primary user. Given these settings, the cumulative distribution function of packet transmission time
is derived. More importantly, performance metrics in terms of a lower
bound of timeout probability and average number of transmissions per
packet are obtained by utilizing the timeout concept. These formulae
allow us to examine the impact of the delay of data packet transmission and ACK transmission, and distances among users on the system
performance for various fading channels, such as one-sided Gaussian,
Rayleigh, Nakagami-m, and Weibull channels.
1
Introduction
In the last couple of decades, the rapid growth of wireless devices, services, and
demands for bandwidth have resulted in a shortage of spectrum. However,
many measurement campaigns have shown that the allocated spectrum is
often under-utilized due to inefficient usage and management. Hence, there
163
164
Part V
is a need to develop a new technology which can utilize the spectrum more
efficiency than the existing technologies. Therefore, cognitive radio networks
(CRNs) have emerged as a frontier solution to overcome the scarce spectrum
problem [1,2]. In a CRN, users are classified into two types, named as primary
user (PU) and secondary user (SU). The PU licenses the spectrum while the
SU uses its cognitive capabilities to access the PU spectrum as long as the
quality of service (QoS) of the PU does not suffer from the SU communication.
Specifically, three main paradigms have been proposed, known as interweave,
overlay, and underlay spectrum access, to enhance the efficiency of spectrum
usage [3, 4].
Recently, the underlay approach has obtained great attention in the research community due to its robust spectrum utilization [4–9]. Basically, in
an underlay system, the SU is allowed to concurrently access the PU spectrum
given that the interference caused by the SU to the PU is kept below a certain
threshold. As a consequence, the communication range and transmission rate
of the SU are often limited. In particular, the works of [7–9] have shown that
the QoS of the secondary system in terms of delay constraint is not easy to
assure due to the PU interference constraints and fading effects. In an effort to enhance the QoS for the spectrum underlay system [8], an adaptive
modulation scheme under delay constraints has been proposed for the pointto-point communications. An optimum power allocation to achieve maximal
effective capacity of the SU has been derived. Similarly, in [9], by assuming
that the SUs are subject to a specific delay constraint, the performance of
a cognitive relay network in terms of effective capacity has been considered.
Also, power allocation policies to maximize the capacity over Rayleigh fading channels have been obtained. Regarding packet transmission time in a
CRN, our previous works reported in [7, 10] have studied the packet delay
for spectrum underlay systems over Rayleigh and Nakagami-m channels. In
this context, the outage probability, average transmission time, and average
waiting time of packets by employing the M/G/1 queuing model have been
derived. However, the impact of the delay of acknowledgement (ACK) on
the performance of the CRN has not been considered due to mathematical
complexity.
Motivated by all of the above, in this study, we investigate the performance
of the well-known stop-and-wait automatic repeat request (ARQ) protocol
for a spectrum underlay network, in which the impact of ACK delay on the
system performance is analyzed. In particular, the secondary transmitter (SUTx) sends a data packet to the secondary receiver (SU-Rx) and waits for a
feedback ACK to transmit the next data packet. If the SU-Tx does not receive
the ACK before a timeout threshold, the data packet is retransmitted. Both
the SU-Tx and SU-Rx are subject to the peak interference power constraint
Performance of Cognitive Radio Networks over General Fading Channels
165
of a single PU, and the channels are assumed to undergo independent not
necessary identically distributed (i.n.i.d) α-µ fading channels [11]. It should
be mentioned that the α-µ fading channel is a very general model, and can
describe other fading channels by changing parameters (α, µ) such as onesided Gaussian channel (α = 1.5, µ = 0.5), Rayleigh channel (α = 2, µ = 1),
Nakagami-m channel (α = 2, µ = m), and Weibull channel (α = m, µ = 1).
Given these settings, the main contributions of this part can be summarized
as follows:
• A cumulative distribution function (CDF) of packet transmission time
and an upper bound for the CDF of the round-trip-time (RTT) are
derived for the considered CRN over general fading channels.
• Utilizing the above results, a lower bound for the timeout probability
and average number of transmissions per packet are obtained.
• Numerical results indicate that the fading severity of the feedback channel and close distance between SUs and primary receiver (PU-Rx) can
degrade the performance of the secondary network.
Notations: In the following, the CDF and probability density function (PDF)
of a random variable (RV) X are represented by FX (·) and fX (·), respectively. The probability and statistical average are denoted by Pr{·} and E[·],
respectively. Γ(·) is the gamma function [12, eq. (8.350.1)] and Γ(·, ·) denotes
the incomplete gamma function [12, eq. (8.350.2)]. Finally, 2 F1 (·, ·; ·; ·) is the
Gauss hypergeometric function [12, eq. (9.100)].
The remainder of this paper is organized as follows. In Section 2, the
system model, channel model, and communication protocol are introduced.
Section 3 presents derivations of the CDF, PDF for different RVs, and their
application to obtain a lower bound on the timeout probability and average
number of transmissions per packet. Section 4 provides and discusses about
numerical examples. Finally, conclusions are drawn in Section 5.
2
2.1
System and Channel Model
System Model
Let us consider a spectrum underlay network as shown in Figure 1 where there
exists a PU in the coverage range of the SU-Tx→SU-Rx communication. The
SU is allowed to access the licensed frequency band of the PU as long as
the SU transmit power does not exceed the maximum tolerable interference
given by the PU. We denote the channel power gains of the SU-Tx→SU-Rx
and SU-Rx→SU-Tx communication links by gsd and hds , respectively. The
166
Part V
PU-Rx
dp
sp
sd
SU-Rx
SU-Tx
ds
Communication link
Interference link
Figure 1: A model of a spectrum sharing system where the SU-Tx sends data
packets to the SU-Rx. If the SU-Rx receives a data packet successfully, it
sends an ACK to the SU-Tx.
channel power gains of the SU-Tx→PU-Rx and SU-Rx→PU-Rx interference
links are denoted, respectively, by gsp and hdp . The distances between SUTx↔SU-Rx, SU-Tx↔PU-Rx, and SU-Rx↔PU-Rx are represented by d, dsp ,
and ddp , respectively.
According to [13], the composite channel power gains with path-loss and
fading of the SU-Tx→SU-Rx and SU-Tx→PU-Rx links are given, respectively,
by
gsd
dν
gsp
= ν
dsp
g̃sd =
(1)
g̃sp
(2)
where ν is the path-loss exponent which depends on the transmission environment. For example, the path-loss exponent of ν is typically in the range of 2-6
for office buildings (multiple floors) [14]. The composite channel power gains
of the SU-Rx→SU-Tx and the SU-Rx→PU-Rx links can be, respectively, expressed as
hds
˜
hds = ν
d
hdp
˜
hdp = ν
ddp
(3)
(4)
Performance of Cognitive Radio Networks over General Fading Channels
2.2
167
Channel Model
In this study, all channel coefficients are assumed to be α-µ distributed RVs,
and the PDF of an α-µ RV R is formulated as [11, 15]
rα
αµµ rαµ−1
(5)
exp −µ α
fR (r) = αµ
r̂ Γ (µ)
r̂
p
where r̂ = α E [Rα ] is the α-root mean value, α > 0 is an arbitrary fading
parameter, and µ > 0 is the inverse of the normalized variance of Rα . The
parameter µ is obtained by µ = E2 [Rα ]/ V[Rα ], where E[·] and V[·] stand for
expectation and variance, respectively. It is noted that the α-µ distribution
is a general fading model and it can present quite precisely the small-scale
variation of fading signals in a non line-of-sight environment [11].
According to [16], the PDF and CDF of channel power gain, β = |R|2 ,
may be formulated, respectively, as follows:
x α/2
αxαµ/2−1
(6)
exp
−
fβ (x) =
αµ/2
Ω
2Ω
Γ(µ)
x α/2
Γ µ, Ω
(7)
Fβ (x) =
Γ(µ)
Γ(µ)
where Ω = β¯ Γ(µ+2/α)
and β¯ is defined as
Γ(µ + 2/α)
β¯ = E[β] = r̂2 2/α
µ Γ(µ)
(8)
Without loss of generality, we use the term X ∼ G(α, µ, Ω) to state that
X is a random variable with PDF and CDF given in (6) and (7), respectively.
Thus, all channel coefficients of the considered model are assumed to be i.n.i.d.
α-µ RVs. Accordingly, the channel power gains, gsd , gsp , hds , and hdp are
distributed, respectively, as G(αD , µsd , Ωsd ), G(αD , µsp , Ωsp ), G(αA , µds , Ωds ),
and G(αA , µdp , Ωdp ).
2.3
Communication Protocol
The SU-Tx first sends a data packet of length LD bits to the SU-Rx and waits
for an ACK from the SU-Rx before sending the next packet. The channels
are assumed to be constant during the transmission time of one packet but
they may change independently thereafter. According to [17], the time taken
to transmit a data packet can be formulated as
TD =
LD
B̃D
,
B log2 (1 + γD )
ln(1 + γD )
(9)
168
Part V
where B is the system bandwidth, B̃D = LD ln(2)/B, and γD is the signalto-noise ratio (SNR) at the SU-Rx given by
γD =
PS (g̃sd , g̃sp )g̃sd
N0
(10)
in which PS (g̃sd , g̃sp ) and N0 stand for the SU-Tx transmit power and noise
power, respectively. Here, we assume that the SU-Tx and SU-Rx have perfect
channel state information (CSI). The CSI can be obtained in several ways.
For example, the CSI of the SU-Tx↔SU-Rx links may be inserted into the
packets. Thus, as the SUs decode the packets successfully, they also obtain
the CSI. Additionally, the CSI of the SU-Tx→PU-Rx and SU-Rx→PU-Rx
interference links can be acquired from a band manager between the primary
and secondary networks [18].
Based on the availability of CSI, the SU-Tx can adjust its transmit power
to not cause harmful interference to the PU. This can be expressed by the
following constraint as
PS (g̃sd , g̃sp )g̃sp ≤ Q
(11)
where Q is the peak interference power that the PU-Rx can tolerate [19].
When a data packet is received by the SU-Rx, it will feedback the ACK
with LA bits to announce that the SU-Tx can send the next packet. Similar to
the data packet transmission, the time taken to feedback the ACK is presented
as
TA =
B̃A
ln(1 + γA )
(12)
where B̃A = LA ln(2)/B, and γA is the SNR at the SU-Tx given by
γA =
˜ ds , h
˜ dp )h
˜ ds
PR (h
N0
(13)
˜ ds , h
˜ dp ) is the SU-Rx transmit power.
in which PR (h
To feedback the ACK, the SU-Rx must control its transmit power to not
cause harmful interference to the PU, i.e., the SU-Rx transmit power should
satisfy the condition
PR (h̃ds , h̃dp )h̃dp ≤ Q
(14)
It is noted that whenever a data packet is transmitted, the SU-Tx starts a
timer and waits for an ACK. If an ACK arrives at the SU-Tx before the
Performance of Cognitive Radio Networks over General Fading Channels
169
timeout threshold, the timer is stopped and the SU-Tx transmits the next
packet. On the other hand, if the timer expires, the SU-Tx retransmits the
previous data packet which is declared as being lost or corrupted.
Overall, a successful transmission can be expressed as the summation of
the data packet transmission time TD and its own ACK transmission time
TA , which is strictly smaller than a timeout threshold tout , given by
TR = TD + TA < tout
(15)
where TR is RTT.
3
Performance Analysis
In this section, the distribution of packet transmission time for the considered
system model is derived. Then, it is applied to derive the lower bound of
timeout probability for the RTT and average number of transmissions per
packet. Let us first consider Lemma 1 and Lemma 2 as follows.
Lemma 1. Assuming that X1 ∼ G(α, µ1 , Ω1 ) and X2 ∼ G(α, µ2 , Ω2 ), the
PDF and CDF of Y = X1 /X2 are formulated, respectively, as follows:
α
fY (y) =
Ω1
Ω2
αµ2 2
2B[µ1 , µ2 ]
FY (y) =2 F1
y
α
2
+
y
αµ1
2
Ω1
Ω2
−1
µ1 + µ2 , µ1 ; 1 + µ1 ; −
αµ1
y 2
×
µ1 B(µ1 , µ2 )
Ω2
Ω1
(16)
α2 µ1 +µ2
Ω2
Ω1
α2
y
α
2
!
αµ2 1
(17)
where 2 F1 (·, ·; ·; ·) denotes the Gaussian hypergeometric function and B(a, b) =
Γ(a)Γ(b)/Γ(a + b).
Proof: See Appendix A.
Lemma 2. Assume that the random variable T is a function of two independent RVs X1 and X2 as
T =
ξ
ln(1 + ρX1 /X2 )
where ξ, ρ > 0 are constants, X1 ∼ G(α, µ1 , Ω1 ), and X2 ∼ G(α, µ2 , Ω2 ).
(18)
170
Part V
Then, the CDF of T is formulated as
αµ1 /2
αµ1 /2 exp ξt − 1
Ω2
ρΩ1
µ1 B(µ1 , µ2 )
α/2 α/2 Ω2
ξ
−1
(19)
exp
× 2 F1 µ1 + µ2 , µ1 ; 1 + µ1 ; −
ρΩ1
t
FT (t) =1 −
Proof: Following the probability definition, the CDF of random variable
T is defined as
FT (t) = Pr{T < t} = 1 − Pr
(
exp( ξt ) − 1
X1
<
X2
ρ
)
Applying (17) to (20), the CDF of T is easily found as shown in (19).
3.1
3.1.1
(20)
CDF of data packet and ACK transmission time
CDF of the data packet transmission time TD
As the SU-Tx can regulate its power on the basis of full CSI to send a data
packet, the maximum transmit power of the SU-Tx given in (11) can be
expressed as
PS (g̃sd , g̃sp ) =
Qdνsp
gsp
(21)
Substituting (1) and (21) into (10), the data packet transmission time TD
given in (9) can be rewritten as
TD =
where RSD =
B̃D
ln 1 +
ν
dsp /d
and γ̃ = Q/N0 .
gsd
gsp γ̃RSD
(22)
We can see that the data packet transmission time TD given in (22) has
the same form as (18). Thus, the CDF of TD can be found by applying Lemma
2 in which parameters are set as ξ = B̃D , ρ = γ̃RSD , X1 = gsd , X2 = gsp ,
Performance of Cognitive Radio Networks over General Fading Channels
171
gsd ∼ G(αD , µsd , Ωsd ), and gsp ∼ G(αD , µsp , Ωsp ). Thus, we obtain
αD µsd /2
B̃D
ϑαD µsd /2
exp
−1
FTD (t) = Pr{TD < t} = 1 −
µsd B(µsd , µsp )
t
αD /2 B̃D
× 2 F1 µsd + µsp , µsd ; 1 + µsd ; −ϑαD /2 exp
−1
(23)
t
where ϑ =
3.1.2
Ωsp
γ̃RSD Ωsd .
CDF of the ACK transmission time TA
Similar to the data packet transmission, the SU-Rx can also control its transmit power to feedback the ACK. Then, the maximum transmit power of the
SU-Rx given in (14) can be written as
˜ ds , h
˜ dp ) =
PR (h
Qdνdp
hdp
(24)
Consequently, substituting (24) into (12), the ACK packet transmission time
TA can be written as
TA =
where RDP
ν
= ddp /d .
B̃A
ln 1 +
hds
hdp γ̃RDP
(25)
Using Lemma 2 for TA given in (25) with ξ = B̃A , ρ = γ̃RDP , X1 = hds ,
X2 = hdp , hds ∼ G(αA , µds , Ωds ), and hdp ∼ G(αA , µdp , Ωdp ), we find the CDF
of TA as
αA µds /2
ψ αA µds /2
B̃A
exp
−1
FTA (t) = 1 −
µds B(µds , µdp )
t
αA /2 B̃A
αA /2
−1
(26)
exp
× 2 F1 µds + µdp , µds ; 1 + µds ; −ψ
t
where ψ =
3.2
Ωdp
γ̃RDP Ωds .
CDF of Round-trip-time TR
Following the probability definition, the CDF of TR can be formulated as
FTR (t) = Pr{TR = TD + TA < t}
(27)
172
Part V
Because the distributions of TA and TD are complicated, an exact closed-form
expression for the CDF of RTT TR given in (27) is not available. Therefore,
we will derive an upper bound of FTR (t) using the following Lemma 3.
Lemma 3. Assuming Z1 and Z2 are two independent non-negative RVs with
optional distributions, the upper bound probability for the RV Z = Z1 + Z2 is
given by
Pr{Z1 + Z2 < z} ≤ Pr{Z1 < z} Pr{Z2 < z}
Proof: See Appendix B.
(28)
Applying Lemma 3 to (27), we have
Pr{TR = TA + TD < t} ≤ Pr{TD < t} Pr{TA < t}
|
{z
} |
{z
}|
{z
}
(29)
FTuR (t) = FTD (t)FTA (t)
(30)
FTR (t)
FTD (t)
FTA (t)
where the upper bound for the CDF of the RTT TR is expressed as
and FTD (t) and FTA (t) are given in (23) and (26), respectively.
3.3
Timeout Probability
The timeout probability is defined as the probability that the RTT is greater
than or equal to a timeout threshold tout , i.e., Pr{TR ≥ tout }. Using (30), we
can deduce a lower bound for the timeout probability as
Pr{TR ≥ tout } = 1 − Pr{TR < tout }
= 1 − FTR (tout ) ≥ 1 − FTuR (tout )
(31)
As a result, the lower bound of the timeout probability is given by
Pout = 1 − FTuR (tout )
3.4
(32)
Average Number of Transmissions per Packet
Due to the fading effect and other channel impairments, a data packet may
take several retransmissions until it is received successfully. Therefore, the
probability that the SU-Tx takes at least ℓ transmissions, ℓ ≥ 1, for sending
the data packet, can be given as
ℓ−1
Pr{N = ℓ} = (Pout )
(1 − Pout )
(33)
Performance of Cognitive Radio Networks over General Fading Channels
173
0
10
-1
Timeout Probability
10
-2
10
One-sided Gaussian (
-3
10
Rayleigh (
Weibull (
A
A
2,
D
3,
D
Nakagami-m (
A
A
sp
sd
sp
D
1.5,
D
sd
2,
sp
dp
dp
sp
sd
sd
ds
0.5)
1)
ds
ds
dp
1)
dp
ds
3)
Simulation
-4
10
-8
-4
0
4
Q/N
0
(dB)
8
12
16
Figure 2: Lower bound of the timeout probability for symmetric fading channel and identical distances dsp = ddp = d = 1.
where Pout is calculated by (32).
As a consequence, the average number of transmissions per packet is obtained as
E[N ] =
∞
X
ℓ−1
ℓ(Pout )
ℓ=1
4
(1 − Pout ) =
1
1 − Pout
(34)
Numerical Results
In this section, we provide numerical results for the considered system. In
particular, we set the system bandwidth B = 1 MHz and timeout tout = 10
ms as in [17]. The size of a data packet and an ACK are set to LD = 512 bytes
and LA = 29 bytes, respectively, in which the first 20 bytes of a packet contain
the header and the remainder are optional data. The path loss exponent is
set to ν = 4. The distance between SU-Tx↔SU-Rx is normalized to one, i.e.,
d = 1. This is also considered as the largest distance compared to the ones
among SU-Tx↔PU-Rx and SU-Rx→PU-Rx, i.e., dsp , ddp ≤ d.
In Figure 2, we plot the timeout probability as a function of the peakinterference-power-to-noise ratio (PIP), Q/N0 , for various fading channels and
174
Part V
10
Timeout Probability
10
0
-1
Rayleigh
10
10
-2
d
=0.5, d
=1 (Ana.)
sp
dp
d
=0.5, d
=1 (Sim.)
sp
dp
d
=d
=1 (Ana.)
sp
dp
-3
Nakagami-m
10
10
-4
d
=0.5, d
=1 (Ana.)
sp
dp
d
=0.5, d
=1 (Sim.)
sp
dp
d
=d
=1 (Ana.)
sp
dp
-5
-8
-4
0
Q/N
4
0
(dB)
8
12
16
Figure 3: Lower bound of the timeout probability for symmetric fading channels and non-identical distances.
identical distances among users d = dsp = ddp = 1. In particular, we consider
the case that the channels for data packet and ACK transmissions are subject
to the same fading effects, such as one-sided Gaussian, Nakagami-m, Rayleigh,
and Weibull. It is clear to see that the lower bounds of the timeout probability
tightly match with the simulations for all fading channels. Furthermore, the
timeout probability decreases as the PIP Q/N0 increases beyond −4 dB. This
is thought to be due to the fact that the PU-Rx can tolerate a high interference
level with a high PIP Q/N0 . Accordingly, the SU-Tx and SU-Rx can increase
their transmit power, and hence increase their transmit rate to send the data
packet and ACK. As a result, the transmission time of the data packet and
ACK are reduced, and then the RTT is reduced which leads to a decrease in
the timeout probability.
Figure 3 shows the timeout probability as a function of the PIP, Q/N0 with
non-identical distances among users. We can observe from the figure that the
timeout probability curves with non-identical distances (dsp = 0.5, ddp = 1)
among users are above the ones of identical distances (dsp = ddp = 1). In
other words, the timeout probability increases as the distance between the
SU-Tx and the PU-Rx is short, e.g., dsp = 0.5. This is due to the fact that
the PU-Rx is easy to be interfered as it is closed to the SU-Tx. Thus, the
Performance of Cognitive Radio Networks over General Fading Channels
175
0
10
Rayleigh (
A
D
Nakagami-m (
2
A
sp
D
sd
2
dp
sp
sd
ds
dp
1)
ds
3
-1
Timeout Probability
10
-2
10
-3
10
Sim.
D
D
2,
2,
sp
sp
sd
sd
1
3
A
A
1.5
dp
2
dp
ds
ds
0.5 (Ana.)
1 (Ana.)
-4
10
-8
-4
0
Q/N
4
0
(dB)
8
12
16
Figure 4: Lower bound of the timeout probability for asymmetric fading channels and identical distances dsp = ddp = d = 1.
SU-Tx has to reduce its transmit power, i.e., the SU-Tx takes a long duration
to transmit the data packet. Accordingly, the RTT TR increases, and hence
the timeout probability increases.
Figure 4 shows the impact of asymmetric fading channels on the timeout
performance of the considered system. Again, we can see from the figure that
the lower bounds of timeout probability match tightly with simulation results.
In particular, the timeout probability increases as the feedback channel is
subject to severe fading. For example, the timeout probability with fading
parameters (αD = 2, µsp = µsd = 3, αA = 2, µdp = µds = 1) is higher than
that of the Nakagami-m fading channel (αA = αD = 2, µsp = µsd = µdp =
µds = 3) for Q/N0 ≥ 0 dB. This can be explained by the fact that the channel
for transmission of ACK may drop into deep fades with (αA = 2, µdp = µds =
1). Hence, it cannot support a high transmission rate, i.e., the SU-Rx takes a
long duration to transmit the ACK. Accordingly, the RTT TR and therefore
the timeout probability increase.
In Figure 5, the average number of transmissions per packet is depicted
as a function of the PIP Q/N0 for different fading channels and non-identical
distances. As expected, we can see that the average number of transmis-
176
Part V
4.0
Average Number of Transmissions
One-sided Gaussian (
Rayleigh (
3.5
A
Nakagami-m (
3.0
D
2
A
D
2.5
D
D
sp
D
D
A
2
1.5
2
2
1.5
sd
sp
sp
dp
sd
sd
sp
sd
sp
sd
1
3
sp
sd
ds
dp
0.5
A
A
dp
0.5)
ds
1)
ds
A
3
2,
1.5,
1.5,
dp
ds
dp
ds
dp
ds
10
0.5
0.5
2.0
1.5
1.0
d
sp
=1, d
dp
=0.5
0.5
-4
-2
0
2
4
6
8
10
12
14
16
Q/N
0
Figure 5: Average number of transmissions for asymmetric fading parameters
and non-identical distances among users.
sion is reduced with increasing Q/N0 , and it is quite small as Q/N0 > 14
dB. Moreover, in the low regime of the PIP, Q/N0 < 4 dB, the average
number of transmissions is degraded significantly as the channels for the data
packet transmission or ACK are not suffering from severe fading. For example,
the average number of transmissions with the asymmetric fading parameters
(αD = 1.5, µdp = µds = 0.5; αA = 2, µdp = µds = 10) is below the one of the
symmetric fading parameters (αD = αA = 1.5, µsp = µsd = µdp = µds = 0.5).
This is because the channels for the data packet or ACK transmissions are
not in deep fades and therefore can support a high transmission rate. Thus,
the duration taken to transmit the data packet or the ACK is reduced, and
hence the RTT is decreased. As a consequence, the timeout probability is
decreased, and then the number of transmissions per packet is also decreased.
This result is consistent with the observation shown in (34) where the average
number of transmissions depends on the timeout probability.
5
Conclusions
We have presented a performance analysis of a CRN over i.n.i.d. α-µ fading
channels. In particular, we have assumed that the SU-Tx and SU-Rx transmit
Performance of Cognitive Radio Networks over General Fading Channels
177
packets (data and ACK) under the peak interference power constraint of a
single PU. Then, an upper bound for the CDF of the RTT has been derived.
Thereafter, the CDF has been applied to analyze the system performance
in terms of timeout probability, and average number of transmissions per
packet. The obtained formulae for the general fading channel can be used to
investigate system performance for various fading channels such as one-sided
Gaussian, Rayleigh, Nakagami-m, and Weibull. The numerical results have
shown that the system performance of the secondary network is degraded
when the distance between the SUs and the PU-Rx is close or the feedback
channel is in deep fade.
Appendix
Appendix A: Proof of Lemma 1
In this appendix, the PDF and CDF for RV Y = X1 /X2 with X1 ∼ G(α, µ1 , Ω1 )
and X2 ∼ G(α, µ2 , Ω2 ) are derived.
1
Let us denote W = ln X
X2 = ln X1 − ln X2 , then the characteristic function
of W can be expressed as [20]
ΦW (t) = E[X1jt ] E[X2−jt ] =
Γ(µ1 + 2jt
α )Γ(µ2 −
Γ(µ1 )Γ(µ2 )
2jt
α )
Ω1
Ω2
jt
(35)
Taking the inverse transformation of the characteristic function in (35), the
PDF of W can be derived as
1
fW (w) =
2π
+∞
Z
−∞
Γ(µ1 + 2jt
α )Γ(µ2 −
Γ(µ1 )Γ(µ2 )
2jt
α )
Ω1
Ω2
jt
e−jtw dt
Changing the variable t in the integral (36) by setting z = −(µ2 −
A
fW (w) =
2πj
−µZ
2 +j∞
zα
exp −
2
−µ2 −j∞
(36)
2jt
α )
yields
Ω1
w − log
Γ(µ1 + µ2 + z)Γ(−z)dz
Ω2
(37)
where
αµ2
A=
αe− 2 w
2Γ(µ1 )Γ(µ2 )
Ω1
Ω2
αµ2 2
(38)
178
Part V
Using [12, Eq.(6.422.3)] for the integral given in (37), the PDF of W is found
as
i
h
αµ2
Ω1
log
−
w
exp
2
Ω2
α
(39)
fW (w) =
h
iµ1 +µ2
2B[µ1 , µ2 ]
Ω1
α
1 + exp α2 log Ω
w
−
2
2
where B(·, ·) is the Beta function. Moreover, the PDF of Y = eW = X1 /X2
can be easily obtained as
α
fY (y) =
Ω1
Ω2
αµ2 2
2B[µ1 , µ2 ]
α
y2 +
y
αµ1
2
Ω1
Ω2
−1
α2 µ1 +µ2
(40)
Note that by setting α = 2, Ω1 = Ω2 , and µ1 = µ2 = 1, the PDF given in
(40) is reduced to [19, Eq. (11)] where X1 and X2 are exponential distributed
RVs. Similarly, by setting α = 2, Ω1 = Ω2 , and µ1 = µ2 = m, it is simplified
to [19, Eq. (17)] where X1 and X2 are Gamma distributed RVs.
Finally, the CDF of Y can be found by integrating (40) with the help
of [12, Eq.(3.194.5)] as
!
αµ1
α2
αµ2 1
y 2
Ω2
α
Ω2
FY (y) =2 F1 µ1 + µ2 , µ1 ; 1 + µ1 ; −
y2
Ω1
µ1 B(µ1 , µ2 ) Ω1
(41)
where 2 F1 (·, ·; ·; ·) is the Gaussian hypergeometric function.
This completes the proof of Lemma 1.
Appendix B: Proof of Lemma 3
In this appendix, the upper bound for the probability of Z = Z1 + Z2 is
derived.
Since Z1 and Z2 are independent non-negative RVs, we have
Pr{Z = Z1 + Z2 < z} =
Zz
Pr{Z1 < z − u} Pr{Z2 ∈ du}
(42)
0
On the other hand, {Z1 < z − u} ⊆ {Z1 < z} which implies that
Pr{Z1 < z − u} ≤ Pr{Z1 < z}
(43)
Performance of Cognitive Radio Networks over General Fading Channels
179
Therefore, we obtain
Pr{Z1 + Z2 < z} ≤ Pr{Z1 < z}
Zz
Pr{Z2 ∈ du} = Pr{Z1 < z} Pr{Z2 < z}
0
(44)
and Lemma 3 is proved.
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To support the rapidly increasing number of
mobile users and mobile multimedia services,
and the related demands for bandwidth, wireless
communication technology is facing a potentially scarcity of radio spectrum resources. However, spectrum measurement campaigns have
shown that the shortage of radio spectrum is due
to inefficient usage and inflexible spectrum allocation policies. Thus, to be able to meet the
requirements of bandwidth and spectrum utilization, spectrum underlay access, one of the
techniques in cognitive radio networks (CRNs),
has been proposed as a frontier solution to deal
with this problem. In a spectrum underlay network, the secondary user (SU) is allowed to simultaneously access the licensed frequency band
of the primary user (PU) as long as the interference caused by the SU to the PU is kept below a
predefined threshold. By doing so, the spectrum
utilization can be improved significantly. Moreover, the spectrum underlay network is not only
considered as the least sophisticated in implementation, but also can operate in dense areas
where the number of temporal spectrum holes is
small. Inspired by the above discussion, this thesis provides a performance analysis of spectrum
underlay networks which are subject to interference constraints.
The thesis is divided into an introduction part
and five parts based on peer-reviewed international research publications. The introduction part
provides the reader with an overview and background on CRNs. The first part investigates the
performance of secondary networks in terms of
outage probability and ergodic capacity subject
to the joint outage constraint of the PU and the
peak transmit power constraint of the SU. The
second part evaluates the performance of CRNs
with a buffered relay. Subject to the timeout
probability constraint of the PU and the peak
transmit power constraint of the SU, system performance in terms of end-to-end throughput,
end-to-end transmission time, and stable transmission condition for the relay buffer is studied.
The third part analyzes a cognitive cooperative
radio network under the peak interference power constraint of multiple PUs with best relay
selection. The obtained results readily reveal
insights into the impact of the number of PUs,
channel mean powers of the communication and
interference links on the system performance.
The fourth part studies the delay performance
of CRNs under the peak interference power constraint of multiple PUs for point-to-point and
point-to-multipoint communications. A closedform expression for outage probability and an
analytical expression for the average waiting
time of packets are obtained for point-to-point
communications. Moreover, the outage probability and successful transmission probability for
packets in point-to-multipoint communications
are presented. Finally, the fifth part presents
work on the performance analysis of a spectrum
underlay network for a general fading channel.
A lower bound on the packet timeout probability and the average number of transmissions per
packet are obtained for the secondary network.
Performance Analysis of Cognitive
Radio Networks with Interference Constraints
ABSTRACT
Hung Tran
ISSN 1653-2090
ISBN: 978-91-7295-249-2
2013:03
2013:03
Performance Analysis of Cognitive
Radio Networks with Interference
Constraints
Hung Tran
Blekinge Institute of Technology
Doctoral Dissertation Series No. 2013:03
School of Computing