Spectrum Sensing and Access Strategies for Markovian Primary Users

Spectrum Sensing and Access Strategies for
Markovian Primary Users
Ardavan Salehi Nobandegani
Department of Electrical & Computer Engineering
McGill University
Montréal, Canada
September 2011
A thesis submitted to McGill University in partial fulfillment of the requirements for the
degree of M.Eng.
c 2011 Ardavan Salehi Nobandegani
ii
Abstract
As a large portion of currently assigned spectrum is underutilized [1], the concept of cognitive radio is put forward as a promising solution to enhance the utilization of the available
spectrum resources. Spectrum sensing, as a sole means for the Secondary User (SU) to
gain awareness regarding his surrounding radio activities, is a cornerstone of the cognitive
radio concept. In this thesis, the problem of spectrum sensing for the case of a single SU in
the presence of an un-slotted Primary User (PU) whose channel usage pattern is modelled
by a Continuous-Time Markov Chain (CTMC) is being studied and two sensing schemes,
namely, Steady-State Occupancy Probability Approach (SS-OPA) and Bayesian Recursive
Method (BRM) are discussed. As it takes the SU some time to process the observations he
makes for the purpose of spectrum sensing, the issue of Sensing Delay (SD) is also addressed
and the required modifications have been applied on our proposed sensing scheme, BRM,
so as to obtain an SD-aware sensing scheme. As soon as the SU finds the channel free, it
gains access to it and start transmitting for a length of time which we refer to as a SU’s
transmission period. The problem of SU’s transmission period optimization is being considered and two schemes are proposed as to strike a balance between maximizing the SU’s
spectrum utilization and the PU’s protection against interference induced by the SU. The
two SU’s channel access strategies are developed so as to incorporate the effect of sensing
error in their operation; henceforth the impractical assumption of perfect sensing has been
dropped. Last but not least, the problem of uncertainty regarding the CTMC’s parameters
is studied and a sensing scheme, namely, Sequential Restricted-Minimax (SRM) scheme
has been proposed so as to minimize, at each sensing instant, the worst-case cost incurred
by the sensing rule. This study is motivated by the fact that the parameters involved in
the PU’s CTMC are not known a-priori and have to be replaces by imperfect estimates.
iii
Sommaire
Étant donné qu’une portion importante du spectre assigné est actuellement sous-utilisée [1],
le concept de radio cognitive apparait comme une solution prometteuse dans l’enrichissement
de l’utilisation du spectre disponible. La détection du spectre, en tant que sole moyen pour
l’utilisateur secondaire (SU) de reconnaitre des activités radio environnantes est un des
piliers du concept radio cognitive. Cette Thése étudie le probléme de la détection du spectre dans le cas d’un unique US en présence d’un utilisateur primaire (PU) sans crenéau
temporel un-slotted et dont la modélisation de l’utilisation de chaine est acquise par une
chaine de temps continu de Markov (CTMC). Ceci est analysé à partir de deux schémas, la
méthode de probabilité d’occupation l’état stationnaire (SS-OPA) et la méthode récursive
bayésienne (BRM). Puisque le SU requiert d’un certain temps afin de traiter les observations qu’il fait pour détecter le spectre, le problème du retardement de la détection (dit
problème SD) fera aussi l’objet de cette étude. De même, les modifications requises ont
été appliquées dans le schéma BRM proposé afin d’obtenir un schéma de détection sensible
au SD. Dés que le SU trouve une chaine de libre il en gagne l’accs et commence à émettre
pour une période de temps que l’on appellera période de transmission du SU. Le problème
de l’optimisation du période de transmission du SU est aussi considérée et deux schémas
sont proposés afin d’atteindre un équilibre entre la maximisation de l’utilisation du spectre
par le SU et la protection du PU contre l’interfrence induite par le SU. Toutes les deux
stratégies d’accs de chaine ont été développées dans le but d’incorporer l’effet de détection
d’erreur dans leur opération; dès lors, on abandonne hypothèse fictive, de la détection parfaite. Enfin, le problème d’incertitude en relation aux paramètres du CTMC a été étudié
et on a proposé le restreint-minimax sq́uentiel (SRM) comme schéma de minimisation à
chaque instant de détection où le cout du pire scenario est engagé par la règle de détection.
Cet étude est motivée par le fait que les paramètres impliqués dans le CTMC du PU ne
sont pas connus apriori et doivent être remplacés par des estimations imparfaites.
iv
Acknowledgments
I would like to express my gratitude towards my supervisor, Prof. Ioannis Psaromiligkos,
for his invaluable guidance and advice. I am highly indebted to my family for their constant
supports and encouragements.
v
Contents
1 Introduction
1.1 Challenges in Spectrum sensing . . . . . . . . . . . . . . . . . . . . . . . .
1.2 Using the Past to Predict the Future: PU’s Channel Usage Pattern . . . .
1.3 Thesis Organization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
2
4
4
2 Related Work
7
3 Mathematical Background
3.1 Bayesian Decision Making Framework . . . . . . . . . . . . .
3.1.1 Bayesian Decision Rule for Binary-Hypothesis Testing
3.2 Minimax Decision Making Framework: . . . . . . . . . . . .
3.3 Generalization of the Minimax Decision Making Framework:
3.4 PU’s Channel Usage Model: Continuous Time Markov Chain
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4 Spectrum Sensing for Markovian Primary Sources
4.1 General Objective, System Model and Problem Statement . . . . . . . . .
4.2 Steady-State Occupancy Probability Approach (SS-OPA) . . . . . . . . . .
4.3 Spectrum Sensing of CTMC Primary Users Using Bayesian-Recursive Method
(BRM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4 The Effect of Processing Delay on BRM . . . . . . . . . . . . . . . . . . .
4.5 Simulations and Numerical Results . . . . . . . . . . . . . . . . . . . . . .
4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.1 Gaussian Assumption . . . . . . . . . . . . . . . . . . . . . . . . . .
4.6.2 Assumption of Mutual Independence Among The Received Signal
Samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
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vi
5 SU’s Transmission Period Optimization
5.1 Case I: Ideal Scenario . . . . . . . . . . . . . . .
5.1.1 Minimum Transmission Period Problem
5.1.2 Discussion . . . . . . . . . . . . . . . . .
5.2 Case II: Bounded Single-Interference Scenario .
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6 Uncertainty In The PU’s Channel Access Pattern Parameters
6.1 Two-channel Uncertainty Scenario . . . . . . . . . . . . . . . . . . . . . . .
6.2 General Case of Uncertainty . . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.1 Discussion on General Case: Tracking the Propagation of Uncertainty
6.3 Numerical Results and Simulations . . . . . . . . . . . . . . . . . . . . . .
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7 Conclusion and Future Work
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A Necessary and Sufficient Condition for Minimizing E[C (ti ) ]
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B Recursive Algorithm to be Used in BRM
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C Comments on the Convergence of Average Cost Series E[< Cost ttn1 >]
73
References
77
vii
List of Figures
3.1
3.2
3.3
Case (ii) of Proposition 1. . . . . . . . . . . . . . . . . . . . . . . . . . . .
Case (i) of Proposition 1. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
CTMC channel usage model. . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
4.2
4.3
Steady-state occupancy probability of the channel usage pattern. . . . . . .
28
tnN
Finite horizon average cost, i.e., Cost tN , for the SS-OPA and BRM schemes. 36
Convergence of finite horizon average cost, i.e., Cost ttnN
, as n → ∞ for the
N
BRM scheme. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
37
5.1
5.2
The general form of outcomes belonging to the event A. . . . . . . . . . . .
Randomize-Selection methodology has been applied along the AB line among
point A and point B. As can be seen, the average Transmission period of
Case I
has been achieved. . . . . . . . . . . . . . . . . . . . . . . .
Tavg > Tmax
The general form of outcomes belonging to the event B. . . . . . . . . . . .
The objective of Case II is illustratively depicted. . . . . . . . . . . . . . .
5.3
5.4
6.1
6.2
Infinite horizon average cost, i.e., Cost ∞
tN , for the SRM scheme in twochannel uncertainty setting. . . . . . . . . . . . . . . . . . . . . . . . . . .
Infinite horizon average cost, i.e., Cost ∞
tN , for the SRM scheme in multichannel (four-channel) uncertainty setting. . . . . . . . . . . . . . . . . . .
20
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viii
ix
List of Acronyms
ADC
Analog-to-Digital Converter
AWGN Additive White Gaussian Noise
BDR
Bayes Decision Rule
BHT
Binary Hypothesis Testing
BRM
Bayesian Recursive Method
BS
Block-Sensing
BST
Bayesian Sequential Testing
CFA
Complete Forward Algorithm
CR
Cognitive Radio
CRB
Cramer-Rao-Bound
CSS
Cooperative Spectrum-Sensing
CTMC Continuous-Time Markov Chain
DSP
Digital Signal Processors
ECC
Error-Correcting Code
FPGA
Field-Programmable Gate Arrays
GMM
Generalized-Minimax
HMM
Hidden Markov Model
x
List of Figures
HMP
Hidden Markov Process
ID-SU
Ideal-Detector SU
LHS
Left-Hand Side
LR
Likelihood-Ratio
MAP
Maximum A-Posteriori
NID-SU Non Ideal-Detector SU
NP
Neyman-Pearson
PDF
Probability Density Function
PU
Primary User
QD
Quickest Detection
QoS
Quality of Service
RHS
Right-Hand Side
RMM
Restricted-Minimax
RP
Random Process
RS
Randomized-Selection
RV
Random Variable
SD
Sensing Delay
SN
Secondary Network
SNR
Signal-to-Noise Ratio
SPRT
Sequential-Probability-Ratio Test
SRM
Sequential Restricted-Minimax
SS-OPA Steady-State Occupancy Probability Approach
List of Figures
SS-SA Selective-Sensing Selective-Access
SU
Secondary User
WLAN Wireless Local-Area-Network
xi
xii
1
Chapter 1
Introduction
The more bandwidth available, the higher the data rate achievable. This statement is a
direct result of Shannon’s capacity equation for an Additive White Gaussian Noise (AWGN)
channel. As the current static frequency allocation scheme can not meet the requirements
of an ever-increasing number of high data rate devices [2], state-of-the art techniques which
are capable of exploiting the available spectrum opportunities need to be investigated.
Cognitive Radio (CR) is a promising approach for alleviating the problem of spectral
congestion by utilizing, in an opportunistic manner, the frequency bands which are not
heavily occupied by a licensed/primary user [3],[4]. As of now, no precise definition of CR
has been widely adopted, however, the concept has evolved recently so as to encompass
various meanings in different contexts [5],[2]. According to the Federal Communications
Commission (FCC) [4], a cognitive radio is: “A radio or system that senses its operational electromagnetic environment and can dynamically and autonomously adjust its radio
operating parameters to modify system operation, such as maximize throughput, mitigate
interference, facilitate interoperability, access secondary markets.” Accordingly, one of the
main ideas behind CR technology is to autonomously exploit the locally unused spectrum.
This key idea makes CR an outstanding candidate to perform the role of a backbone to
future communication systems. The main philosophy behind the CR concept is to enable
communication devices to sense their environment and henceforth gain awareness with regard to the radios’ operating environment, user requirements and applications, available
networks (infrastructures) and nodes, local policies and other operating restrictions [2].
In CR terminology, the users having a higher priority or legacy right to use a particular
2
Introduction
bandwidth1 are referred to as the Primary Users (PUs). On the other hand, the users
having a lower priority to access that specific part of the spectrum are called Secondary
Users (SUs). The SUs therefore ought to seek for opportunistic access to this bandwidth
such that they do not induce any interference2 to the PUs. To achieve this, SUs need to
be capable of sensing the spectrum accurately and adapt their transmission parameters
accordingly.
Spectrum sensing, the main subject of this thesis, is therefore of great importance for the
realization of CR. Basically, spectrum sensing is to learn about the spectrum usage of the
PUs and their existence in a given geographical area through observing the spectrum and
analysing the collected observations. The knowledge pertaining to the existence of PUs in a
given environment and bandwidth can be acquired by employing the databases containing
the information regarding radio activities in a given geo-location, by taking advantage of
beacons transmitted by the active PUs in the area, or eventually by performing spectrum
sensing task at the SUs [6–8]. By employing beacons, possible SUs are informed regarding
the state of a channel, that is whether it is occupied by the PUs at the moment or it is free
and hence can be used by the SUs. In other words, the notion of beacons provides the SUs
with a sort of side-information with regard to the occupancy status of a channel [2].
1.1 Challenges in Spectrum sensing
In this section, some of the most important challenges pertaining to the task of spectrum
sensing are addressed. Many of the challenges are due to the requirements of CR technology,
such as [2]
(i) High sampling rate capacity: If a wide bandwidth is being observed/sensed by the
SU (this process is also referred to as a Wideband Spectrum Sensing3 ) and in order
for the SU to detect any change in the occupancy state of channel quickly.
(ii) High resolution Analog-to-Digital Converter (ADC): In order to suppress the harmful
effect of quantization error.
1
Part of a spectrum.
Causing no interference whatsoever, is an idealistic and theoretical concept. Practically speaking, the
amount of interference caused by the SUs to the PU must be kept below a given threshold. This threshold
is chosen according to the specifications of the PU such as its Quality of Service (QoS) requirements.
3
This is a very common scenario, as in practice, the SU requires to observe a wide range of spectrum
so as to increase the chance identifying transmission opportunities and thereby achieve a higher data rate.
2
1.1 Challenges in Spectrum sensing
3
(iii) ADC’s with large dynamic range: In order for SUs to be capable of sensing very week
signals (low Signal-to-Noise Ratio (SNR) condition) as well as strong signals (high
SNR condition).
(iv) High speed signal processors, e.g., Field-Programmable Gate Arrays (FPGAs) and
Digital Signal Processors (DSPs): In order to perform computationally demanding
signal processing tasks as quickly as possible and without introducing delay.
As the SUs have to operate over a wide range of frequencies, they require a large
operating bandwidth which, in turn, imposes heavy requirements on the Radio-Frequency
(RF) components [2].
The hidden PU problem is another critical issue in CR technology. This problem refers
to the setting under which the SU(s) can not detect the presence of the PU(s), either
due to their geographical location with respect to the PU(s) or due to the multipath
fading and shadowing effect. In those cases, the SU(s) may unintentionally introduce
interference to the PU(s) due to not being aware of their presence. As a solution to
this problem, Cooperative Spectrum Sensing has been proposed in the literature [9–11]
which is discussed next. Cooperative Spectrum-Sensing (CSS) involves performing the
task of spectrum sensing collaboratively by combining the observations of multiple SUs
so as to increase the sensing accuracy. CSS is a promising approach to spectrum sensing
under noise uncertainty conditions and also when the SU(s) are suffering from multi-path
fading/shadowing effects. CSS improves the accuracy of spectrum sensing and, in addition,
it can decrease the sensing time required to achieve a given level of accuracy/reliability [9–
11]. CSS is very effective when each of the collaborating SUs observe independently faded
or shadowed signals and, therefore, can fully exploit the spatial diversity gain [11],[12].
The loss in performance of spectrum sensing under the correlated shadowing condition is
studied in [13],[14].
Another difficulty for the task of spectrum sensing arises in the case of detecting PUs
employing spread spectrum signalling. Since in the spread spectrum scheme the transmission power is distributed over a wide range of frequencies, the power spectral density is
relatively small and hence it is more difficult to decide upon the presence of PU(s) in such
a scenario [2],[15].
4
Introduction
1.2 Using the Past to Predict the Future: PU’s Channel Usage
Pattern
In order for the SU to be able to highly utilize the opportunities, it needs to be capable of
predicting the PU’s channel usage pattern and thereby, adapt its operation accordingly so
as to make the most out of the opportunities while preventing any interference to the PU.
Thus, the SU needs to investigate the pattern according to which the PU appears and disappears through the observation it makes. As stated in [16][17], on one hand, the employed
PU’s channel usage pattern should be accurate enough so that the PU’s activities could be
predicted well enough accordingly. On the other hand, the employed model for the PU’s
channel usage pattern should be simple enough to avoid extreme computational complexity.
Hence, the PU’s channel usage model which the SU employs as a side-information for the
purpose of spectrum sensing should strike a balance between accuracy and complexity.
1.3 Thesis Organization
Here, we provide a summary of a thesis as follows.
In Chapter 2, the related work in the literature will be reviewed and also compared to
our work.
In Chapter 3, the mathematical backgrounds required for this thesis are discussed.
Moreover, the channel usage pattern of the PU (also referred to as PU’s channel access
pattern) and its mathematical properties are introduced.
In Chapter 4, we study the spectrum sensing problem for the case of unslotted4 Markovian PU modelled by a Continuous-Time Markov Chain (CTMC). It is assumed that a
single SU is observing a given channel whose PU’s channel access pattern is according to
a known CTMC. The SU collects samples and then attempts to make the best decision
with regard to the current state of the channel/PU at time instant t through using all the
available samples up to and including time instant t. We also address the issue of SU’s
processing delay [18] and how it could be incorporated in the sensing rule.
In Chapter 5, the problem of SU’s transmission period optimization is considered. If
the SU senses the given channel and finds it free, it occupies it and transmits for a period of
4
“Slotted” refers to the case where the PU’s channel access scheme is time-slotted. In other words,
the PU occupies the channel for a duration of T seconds that is an integer multiple of the basis time slot
corresponding to the minimum channel usage duration.
1.3 Thesis Organization
5
time which we refer to as the SU’s transmission period. As the SU has to avoid interfering
with the PU, the optimization of transmission period is of great importance. The longer
the transmission period, the larger amount of data transmitted by the SU, but at the same
time, the higher is the probability of interfering with the PU. The transmission period
optimization aims to strike a balance between the two. We adapt two channel access
strategies for the SU based on the Bayesian Recursive Method (BRM) sensing scheme
proposed in Chapter 4. Our channel access strategies take into account the accuracy of the
employed sensing scheme, i.e., they do not make the non-practical assumption of perfect
sensing. In other words, we incorporate our level of confidence with regard to the state of
channel in the proposed SU’s channel access schemes and the SU’s transmission period as
well.
In Chapter 6, we turn our attention to the case when there is uncertainty with regard
to the parameters of PU’s channel usage model, i.e., the parameters of the corresponding
CTMC. In practice, the channel usage parameters have to be estimated. However, no
estimator is perfectly accurate5 which brings up the important question of how we could
sense the channel when the parameters of the CTMC modelling the PU’s channel access
pattern are not perfectly known. We will elaborate on this problem in Chapter 6.
In the end, we conclude the thesis in Chapter 7 and outline possible future research
directions.
5
This problem is more noticeable under low-SNR condition.
6
7
Chapter 2
Related Work
In the following, the related work in the literature will be reviewed and compared to our
work as well.
The authors in [19] have considered a scheme to jointly sense, track and exploit spectrum
opportunities in unslotted PU. A Neyman-Pearson (NP) test has been deployed so as to
sense the state of the channel whereas we will base our method on the Bayesian framework
in Chapter 4. Nevertheless, their formulation for the aforementioned joint problem is in
general intractable and a closed-form solution could not be derived. Also, in the formulation
of the problem, the authors in [19] assumed a fixed transmission period for the SU whereas
we propose an adaptive transmission period for the SU in Chapter 5. Moreover, the problem
of uncertainty with regard to the CTMC’s parameters is not considered in [19].
In [20], the authors have developed a general sensing/transmission structure for SUs in a
cognitive radio network for the scenario where the PU’s channel access pattern is unslotted
and modelled by a CTMC. However, the scheme in [20] is optimal only when the SU has
perfect knowledge of PU’s channel access pattern and, more importantly, the SU is able to
detect perfectly the start of PU’s free period.
In [21], inspired by Sequential-Probability-Ratio Test (SPRT) and Quickest Detection
(QD) methods, the authors have developed a joint sensing/transmission algorithm for a
single SU so as to achieve a minimum detection delay and low sensing overhead. However,
they have presented no proof for the optimality of the sensing rule scheme being deployed.
In addition, [21] assumes that the PU’s channel usage model is perfectly known to the SU.
Finally, the method in [21] suffers from the same fundamental problem as SPRT and QD,
that is that, the amount of time required to make a decision with regard to the state of
8
Related Work
channel is not deterministic but random. This fact makes these methods inappropriate for
Cooperative/Collaborative Spectrum Sensing (CSS). On the contrary, the Block-Sensing
(BS) approach, which we elaborate on in Chapter 4, allows all SUs to make use of the entire
time they have available to sense the channel and make a decision with regard to the state
of channel at the very last moment of the block.1
In [22], the author has proposed an adaptive sensing scheme for a Markovian PU where
the PU’s channel access pattern is modelled by a perfectly known CTMC. However, [22]
deploys an Energy-Detection based sensing rule that does not take into account the fact that
the PU can switch its state frequently and postulates that the state of PU can change only
once during the sensing session. In addition, [22] employs a fixed transmission period for
the SU. Also, [22] considers a multi-channel scenario, where the SUs can exploit spectrum
opportunities in N channels. However, [22] assumes that N is very large such that it takes a
long time for a SU to visit and sense the same channel again. Hence, when a SU returns to
a channel for the second time, it is assumed that the probability of channel to be free/busy
has converged to its steady-state value. This very assumption prevents the scheme in [22]
to be adopted for a single-channel scenario that we are considering in this thesis.
The critical problem with the Discrete-Time Markov Chain (DTMC) model of a slotted
PU is that it is not valid under some practical settings, e.g., in a Wireless Local-AreaNetwork (WLAN) where the transmissions do not show a slotted structure but a continuous
structure [23]. However, the channel usage pattern of the existing wireless access applications (e.g., 802.11 WiFi) can be well approximated by a CTMC [24][25]. The analysis
of slotted-PU modelled by a DTMC is out of the scope of the thesis, however, a number
of works exist in literature which have studied the problem of SUs’ sensing/transmission
scheme under the presence of slotted-PU, for instance [26–29].
In [30], the authors have considered the problem of SU’s channel sensing/access under
the Hidden Markov Process (HMP) setting which is applicable to a slotted PU.2 In [30], it
is assumed that all the parameters involved in the problem, namely, transition probabilities
1
Practically speaking, in cognitive radio technology there are a number of SUs referred to as a SecondaryNetwork (SN) that they would like to communicate with one another in a harmless manner to the PU
(or PUs). The SUs in the SN will employ a collaborative sensing scheme so as to improve the sensing
performance under the fading/shadowing and hidden terminal settings. However, collaborative sensing
requires accurate synchronization among the SUs to be implemented. SPRT and QD due to random
detection time can not provide SUs with the required synchronization.
2
An HMP is a “discrete-time finite-state homogeneous Markov chain observed through a discrete-time
memoryless time-invariant channel” [31].
9
and signal and noise statistics, are unknown beforehand and need to be estimated recursively using the Baum estimation algorithm.3 However, [30] does not elaborate on how the
threshold against which the proposed test statistics should be compared, is derived in order
to meet the given constraint on the interference level between the SU and the PU.
The author of [32] proposes a particular type of Bayesian Sequential Testing (BST)
for which it is assumed that all collected observations in a sensing block are identically
distributed (or equivalently the state of the channel has remained the same throughout
the sensing block.)4 As this assumption is not very practical, [32] also uses a time-domain
Hidden Markov Model (HMM) as a classifier in order to detect the time instants at which
the state of channel changes and then group the observations that belong to the same state
of channel. By performing the BST on these groups as in [32], the sensing performance
improves.
Coulson in [33] adopts the same scheme to study the effect of the fading channel model,
(more specifically, Rayleigh fading) and compare the performance of BST approach under
Rayleigh channel with that of non-fading channel and demonstrates that the choice of the
channel model (i.e., fading or non-fading) has a small effect on the detection performance
of the proposed scheme. In [34], the authors have developed a Selective-Sensing SelectiveAccess (SS-SA) scheme in a multi-channel setting so as to maximize SU’s channel utilization
while limiting its interference to the PUs. In the proposed SS-SA algorithm each channel
is sensed periodically with its own period and as soon as the SU finds a channel to be free
it gains access to the channel and starts transmitting until the next time the channel is
sensed again. However, [34] assumes perfect sensing and also postulates that the sensing
time is negligibly short and can be ignored accordingly.
In much of the available literature on SU’s channel access schemes, i.e., the schemes
that control how the SU has to gain access to the channel so as to meet the constraint
on the amount of interference caused to the PU, the SU’s channel access scheme has been
formulated given the assumption that the SU’s sensing scheme can always detect perfectly
the state of channel. In other words, the SU’s channel access scheme is developed independently from the SU’s sensing strategy. For instance, in [35] the authors have developed a
channel access scheme for CR technology to coexist with multiple parallel WLAN channels
3
For more information on the Baum estimation algorithm, the reader is referred to [31].
We will make the same assumption for our first proposed sensing rule in Chapter 4, namely, the SteadyState Occupancy Probability Approach (SS-OPA) scheme. However, for the case of our second proposed
sensing rule, Bayesian Recursive Method (BRM), we drop this assumption.
4
10
Related Work
while satisfying an interference constraint. However, the sensing rule performance has not
been incorporated into the derivation of the scheme. The same holds for [24],[36].
In [37], the authors have considered the sensing problem for a slotted-PU whose channel access pattern is modelled according to a DTMC. They have proposed two sequential
sensing algorithms in a Bayesian framework, namely, Sequence Maximum A-Posteriori
(Sequence-MAP) scheme and symbol-by-symbol (symbol-wise) sensing algorithm. The solution to the former can be obtained using the Viterbi Algorithm and that of the latter
through the forward-backward algorithm. The former scheme attempts to find the most
probable channel state sequence which could produce the observation sequence collected at
the SU, while the latter scheme intends to detect the state of channel at any time instant
an observation has made. However, decision with regard to the state of the channel in the
past is of no use. More importantly, due to the memoryless property of CTMC/DTMC,
revealing/detecting the past states of channel gives no information about the behaviour of
PU in the future. Hence, we only need to decide upon the current state of the channel given
all the available observations and simply abandon any attempt to detect the past states
of a channel whose usage pattern is either DTMC or CTMC, and therefore memoryless.
The authors in [37], inspired by the above fact, proposed a sensing algorithm namely, a
Complete Forward Algorithm (CFA) which solely attempts to detect the current state of
channel. As the authors state in [37], the CFA could have a finite or infinite memory, i.e., it
can employ part of or all the available observations so as to sense the current state of channel, respectively. In [37], it is claims that CFA does not exploit fully the memory inherent
in the underlying Markov process, however, this is not true from the CR’s application point
of view. Using the available observations in order to make a decision on the state of the
channel at a time instant which is already gone is of no use. What indeed matters to CR
application is to make the best decision pertaining to the current state of the channel.
The CFA is, in fact, the BRM sensing algorithm we have developed in Chapter 4.
However, we have looked at the sensing problem from a different perspective compared to
[37] as we have set a different but more sensible objective to meet and then we have proved
the optimality of our proposed sensing rule, BRM, accordingly. The CFA in [37] is proposed
for a slotted-PU modelled by DTMC whereas we consider the unslotted-PU modelled by
CTMC. The work in [37] does not consider the effect of SU’s processing delay on the
sensing performance. It also assumes that perfect knowledge of the transition probabilities
of the DTMC is available to the SU and does not investigate the effect of uncertainty on
11
the proposed sensing rule. Finally, [37] has only concentrated on the problem of channel
sensing and has not introduced any channel access scheme according to its proposed sensing
algorithms.
12
13
Chapter 3
Mathematical Background
In this chapter, the basic mathematical background required for this thesis, namely, the
Bayesian decision making framework, the Minimax and Restricted Minimax decision rule
are introduced to pave the way for understanding the rest of the thesis. Furthermore,
the channel usage pattern of the PU (also referred to as the channel access pattern) is
introduced and its mathematical properties are discussed.
3.1 Bayesian Decision Making Framework
Here we briefly review very quickly the Bayesian Decision Making framework for the Binary
Hypothesis Testing (BHT) scenario. BHT refers to the case where we have to decide among
two possible hypotheses (or states) based on some observed data. For the channel sensing
problem, the two aforementioned hypotheses are, “PU is not active in the given channel”
and “PU is active/ON in the given channel”, labelled H0 and H1 , respectively.
3.1.1 Bayesian Decision Rule for Binary-Hypothesis Testing
As mentioned before, there are only two possible states. Thus, as the decision is made
with regard to the correct hypothesis, a correct decision would correspond to one of the
following two cases:
1. H0 is true and we choose H0 ;
2. H1 is true and we choose H1 .
14
Mathematical Background
Of course, we may make a wrong decision which corresponds to one of the following two
cases:
3. H0 is true and we choose H1 ;
4. H1 is true and we choose H0 .
We assume that different costs are assigned to each of the above four cases such that Cij
is the cost associated to the case where hypothesis Hj is true and we choose hypothesis
Hi . Thus, equivalently, the cost incurred by the decision rule can be modelled by a RV C
defined as follows,


C11 ,



 C ,
00
C=

C10 ,



 C ,
01
H0
H1
H0
H1
is
is
is
is
true
true
true
true
and
and
and
and
we
we
we
we
choose
choose
choose
choose
H0 ,
H1 ,
H1 ,
H0 .
(3.1)
The key assumption we make in the Bayesian Decision making framework is that the
a-priori probabilities of hypotheses H0 and H1 , denoted by π0 and π1 , respectively, where
π0 + π1 = 1, are known. We further assume that under hypothesis Hi , i = 0, 1 the
Probability Density Function (PDF) of the observation x is given by f (x|Hi ). A decision
rule δ for the BHT problem is a mapping from the observation set1 Γ into the decision
set {H0 , H1 } such that the observation value x is mapped by decision rule δ to H1 or
H0 with probability αδ (x) or 1 − αδ (x), respectively. In other words, αδ (x) denotes the
probability of deciding in favour of hypothesis H1 upon observing the value x. Indeed, we
have αδ (x) ∈ [0, 1], ∀x ∈ Γ. Thus, decision rule δ can be defined as,
(
δ(x) =
H1
H0
with probability αδ (x)
with probability 1 − αδ (x)
(3.2)
where δ(x) denotes the decision for the observation x assigned by decision rule δ. The
average cost is given by
E(C) =
1 X
1
X
j=0 i=0
1
πj Cij Pr(Hi |Hj )
(3.3)
The observation set Γ is the set containing all possible observations, e.g. it could be the set of natural
numbers N or the set of complex numbers C, if the observed data are real or complex, respectively.
3.1 Bayesian Decision Making Framework
15
where Pr(Hi |Hj ) denotes the probability of deciding in favour of hypothesis Hi given that
the hypothesis Hj is true. Equation (3.3) can be re-written as follows,
E(C) =
1
X
πj
X
1
j=0
i=0
Cij Pr(Hi |Hj ) .
(3.4)
P
For a given index j = 0, 1, we can define Rj (δ) , 1i=0 Cij Pr(Hi |Hj ) as the conditional
risk for the hypothesis Hj [38]. Specifically, the conditional risk Rj (δ) shows what would
be the average cost incurred by the decision rule δ when the hypothesis Hj is true. Hence,
(3.4) can be reconfigured as follows,
E(C) =
1
X
πj Rj (δ).
(3.5)
j=0
We have,
Pr(H1 |H1 ) =
Pr(H0 |H0 ) =
Pr(H1 |H0 ) =
Pr(H0 |H1 ) =
Z
αδ (x)f (x|H1 )dx,
ZΓ
Γ
(1 − αδ (x))f (x|H0 )dx,
Z
(3.6)
αδ (x)f (x|H0 )dx,
ZΓ
Γ
(1 − αδ (x))f (x|H1 )dx.
Using the above equations, we can expand (3.5) as follows,
E(C) = π1 R1 (δ) + π0 R0 (δ)
= π0 [C00 Pr(H0 |H0 ) + C10 Pr(H1 |H0 )] + π1 [C01 Pr(H0 |H1 ) + C11 Pr(H1 |H1 )]
Z
Z
= π0 [C00 (1 − αδ (x))f (x|H0 )dx + C10 αδ (x)f (x|H0 )dx]
ZΓ
ZΓ
+ π1 [C01 (1 − αδ (x))f (x|H1 )dx + C11 αδ (x)f (x|H1 )dx]
Γ
Γ
(3.7)
16
Mathematical Background
After some simplifications, we have
E(C) = π0 C00 + π1 C01 +
Z
Γ
αδ (x) · Ξ(x)dx,
(3.8)
where
Ξ(x) , π0 f (x|H0 )(C10 − C00 ) − π1 f (x|H1 )(C01 − C11 ).
(3.9)
Thus, in order to minimize E(C), the optimum choice for αδ (x) is given by,
(
αδ (x) =
when Ξ(x) ≤ 0
when Ξ(x) > 0.
1
0
(3.10)
According to (3.8), the region {x ∈ Γ | Ξ(x) = 0} does not contribute to the average cost
and thus, it makes absolute no difference in favour of which hypothesis we decide when we
observe x for which Ξ(x) = 0.
The Bayes decision rule δBayes which minimizes the average cost E(C) is given by,
(
δBayes (x) =
H1
H0
if Ξ(x) ≤ 0,
if Ξ(x) > 0.
(3.11)
Let us define the set Γ1 , {x ∈ Γ| Ξ(x) ≤ 0} and the set Γ0 , {x ∈ Γ| Ξ(x) > 0}. Then,
we have the following properties for the sets Γ1 and Γ0 ,
Γ0 ∪ Γ1 = Γ and Γ0 ∩ Γ1 = ∅
(3.12)
and the Bayes decision rule δBayes can be written as,
(
δBayes (x) =
H1
H0
if x ∈ Γ1
if x ∈ Γ0 .
(3.13)
3.2 Minimax Decision Making Framework:
17
Thus, for the set Γ1 we have,
Γ1 = {x ∈ Γ | Ξ(x) ≤ 0}
= {x ∈ Γ | π0 f (x|H0 )(C10 − C00 ) − π1 f (x|H1 )(C01 − C11 ) ≤ 0}
f (x|H1 )
π
(C
−
C
)
0
10
00
≥
= x ∈ Γ f (x|H0 )
π1 (C01 − C11 )
(3.14)
where for the last equality, we assumed that C01 − C11 > 0 or equivalently C01 > C11 . This
assumption is quite reasonable as, in practice, we would like to assign a higher cost to a
wrong decision than a correct decision. Similarly, we can show that
Γ1
f (x|H1 )
π0 (C10 − C00 )
= x ∈ Γ|
<
f (x|H0 )
π1 (C01 − C11 )
If we define the threshold η ,
by,
(3.15)
π0 (C10 − C00 )
, then, the Bayes decision rule δBayes is given
π1 (C01 − C11 )
δBayes (x) =



H1 ,



if
f (x|H1 )
≥η
f (x|H0 )




 H0 ,
if
f (x|H1 )
< η.
f (x|H0 )
The Bayes decision rule δBayes can be shown in the following format as well,
f (x|H1 ) H1
R η.
f (x|H0 ) H0
(3.16)
3.2 Minimax Decision Making Framework:
As we saw in the previous section, in order to derive the Bayes decision rule, perfect
knowledge of the a-priori probabilities π0 and π1 = 1 − π0 is essential. In this section, we
will consider the scenario for which the a-priori probabilities are totally unknown. Such a
scenario arises very often in practice due to the fact that the designer of the decision rule
(also known as decision maker) either has no control or access to the mechanism generating
the state of nature [38].
18
Mathematical Background
In such a scenario, as the a-priori probabilities π0 and π1 = 1 − π0 are unknown, the
average cost E(C), cannot be minimized. As an alternative approach, one may aim to find
a decision rule which minimizes, over all δ’s, the maximum of the conditional risks, R0 (δ)
and R1 (δ), i.e., the decision rule which minimizes the following,
max{R0 (δ), R1 (δ)}.
(3.17)
Such a decision rule is referred to as the Minimax Decision Rule (MDR). As discussed
in [38], the MDR is the Bayes decision rule conducted for the least-favourable a priori
probabilities. By definition, the least-favourable prior probabilities π0L and π1L = 1 − π0L are
the ones for which the minimum Bayes risk, i.e., the minimum average cost, is maximum.
If π1L 6= 0, 1 then it can be shown [38] that
R0 (δπ1L ) = R1 (δπ1L ),
(3.18)
where δπ1L denotes the Bayes decision rule conducted for the pair a-priori probabilities π1L
and π0L = 1 − π1L , i.e., the Bayes Decision Rule (BDR) whose threshold η is given by,
η=
π0L (C10 − C00 )
.
π1L (C01 − C11 )
(3.19)
3.3 Generalization of the Minimax Decision Making Framework:
In the previous section, the concept of minimax decision making was discussed. As discussed, the minimax decision making methodology in its basic structure has been developed
for the case where the priori probabilities π0 and π1 are totally unknown. In other words,
the actual πj , j = 0, 1 could take any value in the interval Ωπj = [0, 1] j = 0, 1. We
refer to the interval Ωπj , j = 0, 1 as the uncertainty intervals/regions associated to the
a-priori probabilities πj , j = 0, 1. In this section, we generalize the concept of minimax
decision making for the case where Ωπj ⊆ [0, 1], j = 0, 1. Basically, we are considering the
case where the uncertainty pertaining to the a-priori probabilities is not absolute2 as is the
case for the basic minimax decision rule. It can be easily shown that the solution to this
generalized minimax decision rule is given by the following proposition:
2
The absolute uncertainty for a priori probabilities is the case where Ωπj = [0, 1], j = 0, 1
3.3 Generalization of the Minimax Decision Making Framework:
19
Proposition 1 Suppose that the uncertainty region corresponding to the a-priori probability π1 is given by Ωπ1 ⊆ [0, 1], where π1max = sup(Ωπ1 ) and π1min = inf(Ωπ1 ) then the
decision rule which minimizes, over all δ’s, the quantity
max {π1 R1 (δ) + (1 − π1 )R0 (δ)},
π1 ∈Ωπ1
(3.20)
is the Generalized-Minimax (GMM) Decision Rule (or may be referred to as the RestrictedMinimax (RMM) Decision rule). Then,
(i) If π1min ≤ π1L ≤ π1max , then the RMM decision rule is the Bayes decision rule conducted
for the least-favourable a-priori probability pair of π1L and π0L = 1 − π1L , i.e.,
L(r) =
f (r|H1 ) H1 π0L (C10 − C00 )
≷
f (r|H0 ) H0 π1L (C01 − C11 )
(3.21)
(ii) If π1L > π1max or π1L < π1min , then the RMM decision rule is the Bayes decision rule
conducted for the a-priori probability pair of π0L,RM M and π1L,RM M = 1 − π1L,RM M given by
π1L,RM M = argx∈Ωπ1 min |x − π1L |.
(3.22)
where |.| indicates the absolute value operator. That is,
f (r|H1 ) H1 π0L,RM M (C10 − C00 )
L(r) =
≷
f (r|H0 ) H0 π1L,RM M (C01 − C11 )
(3.23)
2
As can be seen, the above proposition considers all three scenarios for which either the set
Ωπ1 is a discrete or a continuous interval or a mixture of the two. Here, we provide an
illustrative and intuitive proof for it.
We define δπ1 as the BDR conducted for the a-priori probability π1 .3 Also, we define
the function r(π1 , δ) as follows,
r(π1 , δ) = π1 R1 (δ) + (1 − π1 )R0 (δ).
(3.24)
where δ is an arbitrary decision rule and R1 (δ) and R0 (δ) are its associated conditional
risks. As can be seen, for a given decision rule δ, r(π1 , δ) as a function of π1 is simply a
3
Or we may say “conducted for the pair of π1 and π0 = 1 − π1 .”
20
Mathematical Background
straight line for which we have r(0, δ) = R0 (δ) and r(1, δ) = R1 (δ). Let V (π1 ) , r(π1 , δπ1 ),
that is V (π1 ) is the minimum average cost achieved by BDR for the a-priori π1 . It can be
shown [38] that V (π1 ) is a continuous concave function of π1 , ∀π1 ∈ [0, 1].
We present the proof for the case (ii) first and later for the case (i). According to the
R1 (δπ1min )
r(π1 , δπ1min )
R1 (δπ1max )
r(π1 , δπ1max )
A
B
R0 (δπ1max )
R0 (δπ1min )
V (π1 )
C00
π 1 ∈ Ωπ 1
0
π1min
C11
π1max π1L
Fig. 3.1
π1
1
Case (ii) of Proposition 1.
BDR, if the a-priori probabilities are known to be π1∗ and π0∗ = 1−π1∗ , then the decision rule
minimizing the average cost is δπ1∗ . In other words, if any other decision rule is employed
in such a case, the achieved average cost will be greater than r(π1∗ , δπ1∗ ). For case (ii), the
uncertainty region Ωπ1 is either on the left-hand side or on the right-hand side of the leastfavourable a-priori π1L . A typical example for case (ii) is depicted in Fig. 3.1, where the
uncertainty region Ωπ1 is on the left-hand side of π1L and therefore V (π1max ) = max V (π1 ),
(π1max )
π1 ∈Ωπ1
prior π1max
as can be seen in Fig. 3.1. The decision rule δ
achieves V
for
and
max
max
r(π1 , δπ1max ) < V (π1 ) for any π1 ∈ Ωπ1 and π1 6= π1
according to Fig. 3.1. As for
any decision rule δ 6= δπ1max we have r(π1max , δ) > V (π1max ) = r(π1max , δπ1max ), therefore, the
π1max
3.3 Generalization of the Minimax Decision Making Framework:
21
RMM decision rule is δπ1max for the aforementioned setting and π1max = argx∈Ωπ1 min |x−π1L |;
hence, π1max = π1L,RM M . If the uncertainty region Ωπ1 is entirely on the right-hand side of
π1L , then, following a similar methodology, the RMM decision rule is δπ1min where π1min =
argx∈Ωπ1 min |x − π1L | and hence π1min = π1L,RM M .
R0 (δ
π1max
R1 (δπ1min )
)
r(π1 , δπ1min )
r(π1 , δπ1L )
R0 (δπ1L )
A
r(π1 , δ
π1max
)
R1 (δπ1L )
B
R1 (δπ1max )
R0 (δπ1min )
V (π1 )
C00
π 1 ∈ Ωπ 1
0
π1min
Fig. 3.2
π1L
C11
π1max
1
π1
Case (i) of Proposition 1.
For case (i), where π1min ≤ π1L ≤ π1max , i.e., the uncertainty region is partially on the
right-hand side of π1L and partially on the left-hand side of π1L , the RMM decision rule is
simply the BDR for the prior π1L . This is due to the fact that maxπ1 ∈Ωπ1 V (π1 ) = V (π1L ), as
can be seen in Fig. 3.2. The decision rule δπ1L achieves V (π1L ) for prior π1L where V (π1L ) is
the lowest achievable average cost for the prior π1L and it is achieved by decision rule δπ1L .
As for any decision rule δ 6= δπ1L we have r(π1max , δ) > V (π1L ) = r(π1L , δπ1L ), therefore, the
RMM decision rule for the setting of case (ii) is δπ1L .
22
Mathematical Background
3.4 PU’s Channel Usage Model: Continuous Time Markov Chain
In this section, we will present the Continuous Time Markov Chain (CTMC) channel usage
model. Let us denote the state of channel at time instant t by the RV S(t) where S(t) = 1
indicates that the channel is busy because the PU is transmitting at time t and S(t) = 0
indicates that the channel is free at time t as the PU is not active at that time. Therefore,
the collection of RVs S(t) indexed by t ≥ 0 forms a continuous-time two-state Random
Process (RP). Then, we assume the following scenario. The RP S(t) switches it states with
probability one after spending an exponentially distributed amount of time in a state. The
ith time the RP S(t) is at state 1, ∀i ≥ 1, it spends an exponentially distributed amount
of time there denoted by yi , i.e., yi ∼ Exp(λb ) where E(yi ) = λb −1 . Similarly, the ith time
the RP S(t) takes the state 0, ∀i ≥ 1, it spends an exponentially distributed amount of
time there denoted by xi , i.e., xi ∼ Exp(λf ) where E(xi ) = λf −1 . Based on the above
properties, the RP S(t), t ≥ 0 is equivalent to a CTMC [39], as depicted in Fig. 3.3.
with probability 1
S(t) = 1
S(t) = 0
PU active
PU inactive
(Channel busy)
(Channel free)
y ∼ Exp(λb )
x ∼ Exp(λf )
with probability 1
Fig. 3.3
CTMC channel usage model.
3.4 PU’s Channel Usage Model: Continuous Time Markov Chain
23
Let us define the following transition probabilities,
= Pr(S(t + ∆T ) = 1|S(t) = 1),
A∆T
11
(3.25)
= Pr(S(t + ∆T ) = 1|S(t) = 0),
A∆T
10
(3.26)
∆T
= Pr(S(t + ∆T ) = 0|S(t) = 1),
B01
(3.27)
∆T
= Pr(S(t + ∆T ) = 0|S(t) = 0),
B00
(3.28)
for which we have [40],
1
(λf + λb e−(λf +λb )·∆T ),
λf + λb
1
A∆T
(λf − λf e−(λf +λb )·∆T ),
10 =
λf + λb
1
∆T
(λb − λb e−(λf +λb )·∆T ),
B01
=
λf + λb
1
∆T
B00
=
(λb + λf e−(λf +λb )·∆T ).
λf + λb
A∆T
11 =
(3.29)
(3.30)
(3.31)
(3.32)
As can be seen, the transition probabilities are solely functions of transition duration ∆T
and totally independent of starting time t. Such a CTMC is called homogeneous. Thus,
we have

λf


,
 lim A∆T
11 =
∆T →∞
λf + λb
(3.33)
λf

∆T

,
lim
A
=
 ∆T →∞ 10
λf + λb
and,


∆T

=
 lim B01
∆T →∞

∆T

B00
 ∆Tlim
→∞
λb
,
λf + λb
λb
=
.
λf + λb
(3.34)
As can be inferred from (3.33), as t → +∞, regardless of the initial state of S(t), the
λf
. Similarly,
probability of observing S(t) at the state 1 converges to π1 (∞) ,
λf + λb
as can be inferred from (3.34), as t → +∞, regardless of the initial state of S(t), the
24
Mathematical Background
λb
. We refer to
λf + λb
π0 (∞) and π1 (∞) as the steady-state occupancy probability of channel to be free and busy,
respectively.
probability of observing S(t) at the state 0 converges to π0 (∞) ,
25
Chapter 4
Spectrum Sensing for Markovian
Primary Sources
4.1 General Objective, System Model and Problem Statement
As discussed in Chapter 3, we assume that the channel occupancy is modelled according to
a Continuous-Time Markov Chain (CTMC). At each sensing session a total of N samples
are taken. Let the Random Variable (RV) S(t) denote the state of the channel at time
instant t where S(t) = 1 indicates the channel is free and S(t) = 0 indicates the channel is
busy. Then, the sample xt taken at time instant t is given by
S(t) = 0 (Channel is free) :
x t = nt
S(t) = 1 (Channel is busy) : xt = st + nt
(4.1)
where st and nt are the signal sample and the noise sample, respectively. The ith sample is
collected at time instant ti at the SU. Thus, in the ith sensing session the sample sequence
(xt(i−1)N +1 , xt(i−1)N +2 , · · · , xtiN ) is collected at the SU receiver and the decision with regard
to the state of the channel is made at time instant tiN , i ≥ 1. Also, let us define X ti ,
(xt1 , xt2 , · · · , xti ), as the sequence of observations made prior and including time instant ti .
Let us denote by Ŝ(t) the decision regarding the state of channel at time instant t.
Similar to S(t), Ŝ(t) = 0 refers to declaring the state of channel “free” and Ŝ(t) = 1 refers
to declaring the state of channel “busy”. Let also the RV C (ti ) represent the cost associated
26
Spectrum Sensing for Markovian Primary Sources
with the decision at time ti with regard to the state of channel. More specifically,
C (ti )


C00 ,



 C ,
01
=

C10 ,



 C ,
11
Ŝ(ti ) = 0
Ŝ(ti ) = 0
Ŝ(ti ) = 1
Ŝ(ti ) = 1
and
and
and
and
S(ti ) = 0
S(ti ) = 1
S(ti ) = 0
S(ti ) = 1,
(4.2)
where Cij ≥ 0 i, j = 1, 2. Then, the overall accumulated cost due to decision making up
to time instant tnN is given by
CttNnN =
n
X
C (tiN ) .
(4.3)
i=1
where the subscript indicates the very first instant of time for which we make a decision
and the superscript indicated the very last one. Then, its time average over the finite time
horizon (up to time instant tiN ) is
n
<
CttNnN
1 X (tiN )
C
.
>=
n i=1
(4.4)
Our objective is to minimize
E[< CttNnN >].
(4.5)
In the above, the expectation is with respect to any random quantities involved in the
operand.
Thus, the procedure of sensing is to observe a block of N samples at each sensing
session and then, using all the available samples made in the current and the past sessions,
make a decision with regard to the state of channel at the time instant at which the last
sample is taken. In other words, assuming that the current sensing session is the ith one,
the observation sequence of (xt(i−1)N +1 , xt(i−1)N +2 , · · · , xtiN ) is collected at the SU during the
current session, and then, a decision is to be made with regard to the state of channel at
time instant tiN , i.e., S(tiN ), using all the available observations made at the SU up to
and including time instant tiN , i.e., X tiN . Following the procedure discussed above, the
objective is to make decisions with respect to the state of channel at time instants tiN
4.2 Steady-State Occupancy Probability Approach (SS-OPA)
27
where i ∈ N such that (4.5) is minimized.
In the following, we will describe two approaches to achieve our objective. The first one
is called the Steady-State Occupancy Probability Approach (SS-OPA) and the second is
called the Bayesian Recursive Method (BRM). The first approach seeks simplicity at the
cost of losing performance whereas the second one aims to achieve optimality at the cost
of increased complexity.
4.2 Steady-State Occupancy Probability Approach (SS-OPA)
One may not fully utilize the knowledge of the channel occupancy model1 but, instead, use
it only to the extent of deriving the steady-state occupancy probabilities, namely π1 (∞)
and π0 (∞) = 1 − π1 (∞). The philosophy behind this approach is to derive a simple sensing
rule at the cost of decreased performance.
According to Section 3.4 of Chapter 3,
π1 (∞) =
λ−1
b
.
λ−1
+
λ−1
b
f
(4.6)
In Fig, 4.1, it is shown, illustratively, how the channel reaches its steady-state occupancy
probability as time progresses. As can be seen in Fig. 4.1, if we wait sufficiently long and
sense the channel at time instance t, the probability of the channel to be busy at time t
gets arbitrarily close to the steady-state value given in (4.6) regardless of the initial state
of the channel occupancy model.
Thus, if we wait for a sufficiently long time T , then the probability of the channel to be
busy is very close to π1 (∞) and it will remain so afterwards. In other words, we will have
Pr(S(t∗ ) = 1) ∼
= π1 (∞) ∀t∗ > T,
where T → ∞.
(4.7)
Thus, regardless of the initial channel state, we could assume that the a-priori probabilities
of being busy/free are equal to the steady-state occupancy probabilities π1 (∞) and π0 (∞),
respectively, and they can be used to make decisions at any instant of time after T . Let
us pictorially explain the case in Fig. 4.1. Based on the concept of steady-state occupancy
probability of a CTMC, regardless of initial state of the CTMC, the probability of observing
1
We may refer to it also as the channel access pattern.
28
Spectrum Sensing for Markovian Primary Sources
1
At11 = Pr(S(t) = 1|S(0) = 1)
π1 (∞)
At10 = Pr(S(t) = 1|S(0) = 0)
T
0
Fig. 4.1
t∗
t(sec)
Steady-state occupancy probability of the channel usage pattern.
the channel to be busy after a long period of time T , is π1 (∞), no matter at what instant
of time after T we observe the channel. Therefore, if we look at the sensing problem from
such an angle, we just need to let the CTMC get close enough to its steady-state and
then run the following BDR for all sensing times after time instant T , assuming that all
the observations made in the ith sensing session/phase are independent of each other2 , and
2
For more discussion, please refer to Section 4.6 of Chapter 4.
4.3 Spectrum Sensing of CTMC Primary Users Using Bayesian-Recursive
Method (BRM)
29
also, the state of channel remains unchanged during the ith sensing session/phase:3
N
Y
iN
LR(Xtt(i−1)N
),
+1
j=1
N
Y
j=1
f (xt(i−1)N +j |S(t(i−1)N +j ) = 1)
Ŝ(tiN )=1
≷
f (xt(i−1)N +j |S(t(i−1)N +j ) = 0)
Ŝ(tiN )=0
π0 (∞)(C10 − C00 )
π1 (∞)(C01 − C11 )
(4.8)
As we can see, the threshold with which we need to compare the Likelihood-Ratio (LR) is
π0 (∞)(C10 − C00 )
constant and is given by η(∞) ,
.
π1 (∞)(C01 − C11 )
The advantage of this approach is its simplicity. However, the knowledge of channel occupancy model is not fully exploited. More precisely, the knowledge of channel occupancy
model is used here solely to derive the steady-state behaviour of the channel; the fact that
the a-priori probability of the channel to be busy (or free) at time instant t varies according the observations made prior to t is overlooked. This very fact inspires the Bayesian
Recursive Method (BRM) we discuss next.
4.3 Spectrum Sensing of CTMC Primary Users Using
Bayesian-Recursive Method (BRM)
In this section, we present another approach to the problem of spectrum sensing that fully
exploits the knowledge of CTMC channel occupancy model.
According to Section (4.1), our objective is to minimize E[< CttNnN >] given in (4.5).
Using the linearity property of expectation operator E(·), we have
n
n
1 X (tiN )
1X
C
]=
E[C (tiN ) ].
E[
n i=1
n i=1
(4.9)
Let us define
n
Cost ttnN
,
N
3
1X
E[C (tiN ) ].
n i=1
(4.10)
Clearly, the shorter the sensing session is, the lower the probability of a change occurring during the
sensing session. Nonetheless, this assumptions, i.e., “the state of channel remains unchanged during the
ith sensing session/phase”, is not valid all the times and in our next sensing approach introduced in the
next section, it will be dropped.
30
Spectrum Sensing for Markovian Primary Sources
Hence, the optimum decision rule δ ∗ must minimize4 Cost ttnN
.
N
As it can be inferred from (4.10), the necessary and sufficient condition to minimize
Cost ttnN
is to minimize5 E[C (tiN ) ], ∀i ∈ N. Thus, having assumed that we have observed
N
∗
the channel up to some time instant t∗ and collected the observation sequence X t , the
objective is to make the optimum decision with regard to S(t∗ ) in a sense that it minimizes
∗
E[C (t ) ]. It is shown in Appendix A that for any i ∈ N, E[C (tiN ) ] is minimized if and only
if E[C (tiN ) |X t(iN −1) ] is minimized. Thus, the optimum decision rule is the one minimizing
E[C (tiN ) |X t(iN −1) ], ∀i ∈ N, which is given by
E[C (tiN ) |X t(iN −1) ] =
t
(t |X (iN −1) ) π1 iN
C01 Pr(Ŝ(tiN ) = 0|S(tiN ) = 1, X t(iN −1) ) + C11 Pr(Ŝ(tiN ) = 1|S(tiN ) = 1, X t(iN −1) ) +
t
(t |X (iN −1) ) C00 Pr(Ŝ(tiN ) = 0|S(tiN ) = 0, X t(iN −1) ) + C10 Pr(Ŝ(tiN ) = 1|S(tiN ) = 0, X t(iN −1) ) ,
π0 iN
(4.11)
(t
|X
t(iN −1)
)
(t
|X
t(iN −1)
)
where π1 iN
and π0 iN
are the a-priori probabilities for the channel to be
busy and free at time instant tiN given all the observations prior (not including) time
instant tiN , i.e., X t(iN −1) . More specifically,
(t
π1 iN
|X
(t |X
π0 iN
t(iN −1)
)
= Pr(S(tiN ) = 1|X t(iN −1) ),
(4.12)
t(iN −1)
)
= Pr(S(tiN ) = 0|X t(iN −1) ).
(4.13)
In order to find the optimum decision rule, we expand (4.11) as follows,
Pr(Ŝ(tiN ) = 0|S(tiN ) = 1, X
t(iN −1)
Z
) =
Γ0
t
X (iN −1)
f (xtiN |S(tiN ) = 1, X t(iN −1) )dxtiN
(4.14)
In the above, Γ0 t(iN −1) is the decision region for the 0 hypothesis, i.e., if xtiN lies in Γ0 t(iN −1)
X
4
X
If a detection scheme happens to be optimum in minimizing the given criterion (which is defined over
a finite duration), it is indeed optimum in minimizing it over an infinite horizon, i.e., when n → ∞.
5
This is due to the fact that, the process of decision making at some time instant, say tp , is carried
totally independently of that of time instant tq where tp > tq . This concept will be clearer to the reader
as he/she proceeds with Section 4.3.
4.3 Spectrum Sensing of CTMC Primary Users Using Bayesian-Recursive
Method (BRM)
31
we decide in favour of hypothesis Ŝ(tiN ) = 0, that is we claim the channel to be free. Letting
Γ1 t(iN −1) be the decision region of hypothesis Ŝ(tiN ) = 1, we have also
X
Pr(Ŝ(tiN ) = 1|S(tiN ) = 1, X
t(iN −1)
Z
) =
Γ1
t
X (iN −1)
f (xtiN |S(tiN ) = 1, X t(iN −1) )dxtiN
(4.15)
Z
Pr(Ŝ(tiN ) = 0|S(tiN ) = 0, X t(iN −1) ) =
Γ0
t
X (iN −1)
f (xtiN |S(tiN ) = 0, X t(iN −1) )dxtiN
(4.16)
Z
Pr(Ŝ(tiN ) = 1|S(tiN ) = 0, X t(iN −1) ) =
Γ1
t
X (iN −1)
f (xtiN |S(tiN ) = 0, X t(iN −1) )dxtiN
(4.17)
As Γ0 t(iN −1) ∪ Γ1 t(iN −1) = R, Γ0 t(iN −1) ∩ Γ1 t(iN −1) = ∅, (4.14) and (4.16) can be rewritten
X
X
X
X
as follows:
Z
t(iN −1)
f (xtiN |S(tiN ) = 1, X t(iN −1) )dxtiN
Pr(Ŝ(tiN ) = 0|S(tiN ) = 1, X
) =
Γ0
t
X (iN −1)
= 1−
Z
Γ1
t
X (iN −1)
f (xtiN |S(tiN ) = 1, X t(iN −1) )dxtiN
(4.18)
and, similarly,
Pr(Ŝ(tiN ) = 0|S(tiN ) = 0, X
t(iN −1)
)=1−
Z
Γ1
t
X (iN −1)
f (xtiN |S(tiN ) = 0, X t(iN −1) )dxtiN .
(4.19)
Substituting (4.15),(4.17),(4.18) and (4.19) in (4.11) and after some simplifications, we
get to the conclusion that in order to minimize the E[C (tiN ) |X t(iN −1) ] the decision regions
32
Spectrum Sensing for Markovian Primary Sources
Γ0 t(iN −1) and Γ1 t(iN −1) for the optimum decision rule must be as follows
X
X
t
tiN |X (iN −1)
t(iN −1)
π
(C10 − C00 )
f
(x
|S(t
)
=
1,
X
)
tiN
iN
0
1
>
, (4.20)
ΓX t(iN −1) = xtiN ∈ R
t
t |X (iN −1)
f (xtiN |S(tiN ) = 0, X t(iN −1) )
(C01 − C11 )
π1iN
t
tiN |X (iN −1)
t(iN −1)
f
(x
|S(t
)
=
1,
X
)
π
(C10 − C00 )
t
iN
0
iN
0
ΓX t(iN −1) = xtiN ∈ R
<
, (4.21)
t
t |X (iN −1)
f (xtiN |S(tiN ) = 0, X t(iN −1) )
π iN
(C − C )
01
1
11
where R is the set of real numbers, and accordingly, the optimum decision rule δ ∗ which
we refer to as the Bayesian Recursive Method (BRM) is given by
f (xtiN |S(tiN ) = 1, X t(iN −1) )
f (xtiN |S(tiN ) = 0, X t(iN −1) )
Ŝ(tiN )=1
≷
Ŝ(tiN )=0
t(iN −1)
t
|X
t
t
|X (iN −1)
π0iN
π1iN
(C10 − C00 )
.
(4.22)
(C01 − C11 )
The expressions required for the recursive update of the BRM decision rule are provided
in Appendix B.
The key fact behind BRM is that the decision regarding the current state of the channel
S(t) is based solely on all available observations up to the current time, i.e., X t , and not
based on the past decisions made about the state of the channel at time instants prior to
t. In other words, if we make a wrong decision with respect to the state of the channel
∗
at some instant of time say t∗ based on X t , then the decision about the state of channel
at time instant t where t > t∗ will not be affected by this fact. Thus, there is no error
propagation due to wrong decisions made in previous steps.
4.4 The Effect of Processing Delay on BRM
Practically speaking, it takes some time for the SU to perform all the required processes6
and calculations in order to decide upon current state of the channel7 . Let us assume that
it takes the SU Td seconds in order to eventually reach a decision regarding the state of the
channel. The effect of this delay is that, if the SU obtains the last observation at time ti ,
then, it will be able to reach a decision regarding the state of channel S(ti ) no sooner than
6
By processes we mean fetching/loading the data/observations stored in the memory into the processor
unit.
7
This delay is even more noticeable in a case of CSS, as all the SUs require to send their observations
to the Fusion Center so that it makes the decision regarding the channel state.
4.4 The Effect of Processing Delay on BRM
33
time instant ti + Td . Of course, there is no guarantee that the channel state has remained
the same, i.e., S(ti + Td ) = S(t).
In this section, we would like to address the effect of the delay in decision making of SU
and explain how this delay can be accommodated in the BRM in advance. By modifying
the BRM to decide with regard to the state of channel at time instant ti + Td instead of ti ,
we have,
Pr(S(ti + Td ) = 1|X ti )
Pr(S(ti + Td ) = 0|X ti )
Ŝ(ti +Td )=1
≷
Ŝ(ti +Td )=0
(C10 − C00 )
(C01 − C11 )
(4.23)
Let us define the Left-Hand Side (LHS) of (4.23) as the Posterior-Likelihood-Ratio (PLR)
at time instant ti + Td denoted by P LR(ti + Td ). Thus, for P LR(ti + Td ) we have the
following,
P1
P LR(ti + Td ) =
=
=
j=0
P1
j=0
P1
j=0
P1
j=0
Pr(S(ti + Td ) = 1|X ti , S(ti ) = j)Pr(S(ti ) = j|X ti )
Pr(S(ti + Td ) = 0|X ti , S(ti ) = j)Pr(S(ti ) = j|X ti )
Pr(S(ti + Td ) = 1|S(ti ) = j)Pr(S(ti ) = j|X ti )
Pr(S(ti + Td ) = 0|S(ti ) = j)Pr(S(ti ) = j|X ti )
,
,
(4.24)
(4.25)
Pr(S(ti + Td ) = 1|S(ti ) = 1)P LR(ti ) + Pr(S(ti + Td ) = 1|S(ti ) = 0)
.
Pr(S(ti + Td ) = 0|S(ti ) = 1)P LR(ti ) + Pr(S(ti + Td ) = 0|S(ti ) = 0)
(4.26)
Let us define the following transition probabilities
A∆T
= Pr(S(t + ∆T ) = 1|S(t) = 1),
11
(4.27)
A∆T
= Pr(S(t + ∆T ) = 1|S(t) = 0),
10
(4.28)
∆T
B01
= Pr(S(t + ∆T ) = 0|S(t) = 1),
(4.29)
∆T
= Pr(S(t + ∆T ) = 0|S(t) = 0).
B00
(4.30)
Then using (4.27), (4.28), (4.29) and (4.30) in (4.26) leads to the following equality,
P LR(ti + Td ) =
AT11d P LR(ti ) + AT10d
.
Td
Td
B01
P LR(ti ) + B00
(4.31)
Thus, the optimum decision rule accommodating the delay of Td in decision making is given
34
Spectrum Sensing for Markovian Primary Sources
by
AT11d P LR(ti ) + AT10d
Td
Td
P LR(ti ) + B00
B01
Ŝ(ti +Td )=1
≷
Ŝ(ti +Td )=0
(C10 − C00 )
.
(C01 − C11 )
(4.32)
Thus, instead of using
P LR(ti )
Ŝ(ti +Td )=1
≷
Ŝ(ti +Td )=0
(C10 − C00 )
,
(C01 − C11 )
(4.33)
The SU must use the sensing rule in (4.32) to decide upon the state of channel at time
instant ti + Td . According to Section 3.4 of Chapter 3, as we have
lim A∆T
11 = 1,
∆T →0
and
∆T
lim B00
= 1,
(4.34)
∆T →0
and since A10 = 1 − B00 and B01 = 1 − A11 , we see that
∆T
lim A∆T
10 = 1 − lim B00 = 0,
∆T →0
∆T →0
and
∆T
lim B01
= 1 − lim A∆T
11 = 0,
∆T →0
∆T →0
(4.35)
and hence,
lim P LR(ti + Td ) = P LR(ti ).
Td →0
(4.36)
Simply, (4.36) implies that if the delay in decision making approaches to zero, then the
optimum decision rule reduces to what we derived previously in (4.33). This is of course
intuitively obvious.
4.5 Simulations and Numerical Results
In this section we present the simulation results for the SS-OPA and the BRM schemes.
We assume that the signal samples, sti ’s, and the noise samples, nti ’s, both follow zeromean independent and identically distributed Gaussian random sequence8 [37],[41], i.e.,
iid
iid
nti ∼ N (0, σn2 ) and sti ∼ N (0, σs2 ). Thus, under hypothesis H0 we have xti ∼ N (0, σ02 )
where σ02 = σn2 . Similarly, under hypothesis H1 we have xti ∼ N (0, σ12 ) where σ12 = σn2 + σs2 .
8
More discussion is given in Section (4.6) of Chapter 4.
4.5 Simulations and Numerical Results
35
We also assume that σn2 and σs2 are known at the SU. Under the assumption of the mutual
independence of the observations9 , i.e., assuming that the distribution of xti does not
depend on the observations made prior to time instant ti , the BRM decision rule given in
(4.22) can be simplified further as follows
f (xtiN |S(tiN ) = 1)
f (xtiN |S(tiN ) = 0)
Ŝ(tiN )=1
≷
Ŝ(tiN )=0
t(iN −1)
t
|X
t
t
|X (iN −1)
π0iN
π1iN
(C10 − C00 )
.
(4.37)
(C01 − C11 )
Without loss of generality, we assume σn2 = 1 and, for the decision costs, we assume
C01 = C10 = 1 and C11 = C00 = 0. For the CTMC modelling the PU’s channel usage
−1
pattern, we assume λ−1
f = 4 and λb = 1. Furthermore, we assume that there is no sensing
delay10 , i.e., Td = 0.
In Fig. 4.2, the simulation results for the SS-OPA and the BRM schemes have been
shown where the vertical axis presents the finite horizon average cost, i.e., Cost ttnN
N
σs2
and the horizontal axis denotes the Signal-to-Noise Ratio (SNR) defined as 2 . Here
σn
3
−2
we have n = 2 × 10 and for the sampling period we have ∆T = 10 seconds. The
performance of SS-OPA scheme for the case of N = 1, 2, 3, 4, 20 has been plotted where
N indicates the number of samples collected in each sensing session. As can be seen, the
SS-OPA’s performance starts to improve as we increase N , however, when N gets large,
e.g. N = 20 in Fig. 4.2, the SS-OPA’s performance starts to degrade. As we increase N , we
categorize larger number of samples under the same channel state, i.e., we assume all of N
samples belong to the same state of channel and thus channel is either free or busy for all of
N samples collected in the sensing session. However, the larger the N , the more likely it is
to see a change in the state of channel in a block of N samples and thus our basic assumption in the derivation of SS-OPA scheme, i.e., all the samples collected in a block belong
to the same state of channel, will be more and more unrealistic. That is why we see, for
the SS-OPA scheme, that the performance improves as we increase N (N = 1, 2, 3, 4), but
this behaviour does not continue forever and as N gets relatively large11 , the performance
for the SS-OPA scheme starts to degrade.
9
For more discussion, please refer to Section 4.6 of Chapter 4.
As the SD-aware BRM scheme developed in the Section 4.4 of Chapter 4 can perfectly incorporate the
effect of sensing delay of Td seconds, in case of non-zero SD, the performance will be exactly the same as
the case where we have zero sensing delay.
11
This value indeed depends on the CTMC’s parameters, i.e., λf and λb .
10
36
Spectrum Sensing for Markovian Primary Sources
Fig. 4.2 Finite horizon average cost, i.e., Cost ttnN
, for the SS-OPA and BRM
N
schemes.
In Fig. 4.2, the simulation results for the BRM scheme has been shown for N = 1, 2, 3, 4
as well. As can be seen, the BRM scheme provides much better performance in terms of the
finite horizon average cost compared to that of SS-OPA. However, it might seem surprising
that the curve remains the same for N = 1, 2, 3, 4. In fact, if the decision costs, namely,
C01 , C10 , C11 , C00 are fixed and we only change the value of N , as far as the finite horizon
average cost, i.e., Cost ttnN
is concerned and we have a large n, the performance does not
N
change. However, practically speaking, the increase of N means longer time we need to wait
so as to decide upon the state of channel or equivalently more delay. Thus, such a longer
delay should be incorporated into the decision costs. Thus, if we increase the decision costs
as we increase N , then we will see that the curves for different N ’s will be different as well.
4.5 Simulations and Numerical Results
37
Fig. 4.3 Convergence of finite horizon average cost, i.e., Cost ttnN
, as n → ∞
N
for the BRM scheme.
The convergence behaviour of the BRM sensing scheme has been investigated12 in Fig.
4.3 where the vertical axis presents the finite horizon average cost, Cost ttnN
, and the horizonN
tal axis presents the total number of observations made at the SU. Also we have ∆T = 10−2 ,
σ2
SNR= s2 = 5, and the CTMC parameters are the same as what we had for Fig. 4.3. Let
σn
us define
n
Cost ∞
tN
, lim
n→+∞
Cost ttnN
N
1X
= lim
E[C (tiN ) ],
n→+∞ n
i=1
(4.38)
as an infinite horizon average cost corresponding to the BRM sensing scheme. As stated in
Section 4.3, this very fact that the BRM sensing scheme is optimum in finite setting indeed
12
The analytical proof of convergence for two special cases is given in Appendix C for the BRM sensing
scheme.
38
Spectrum Sensing for Markovian Primary Sources
makes it optimum for the infinite setting as well.
4.6 Discussion
Here we address several questions that may arise naturally.
4.6.1 Gaussian Assumption
It is worth noting that, for given mean and variance, among all the distributions with
infinite support, the normal distribution N (µ, σ 2 ) is the one having maximum Shannon
entropy and hence it is the least informative prior distribution [42]. It is quite practically
reasonable to assume that the mean of the transmitted process by the PU(s), at a given
time, is equal to zero and hence µ = 0. Under the no-fading condition and for a fixed PU’s
transmission power, the receiver (the SU), after the sampling step, obtains a sequence of
RV’s each of which has an almost similar13 variance of σs2 .
Another supportive argument for the assumption of st ∼ N (0, σs2 ) is as follows. The
SU filters the received signal at its front end and obtains the observations xt ’s by sampling
the filtered process. The SU is not synchronized with the PU14 , i.e., it is not aware of the
fact that at which time instants it needs to sample the received signal so as to avoid the
Inter-Symbol Interference (ISI)15 . Consequently, the SU experiences ISI and hence the RV
representing the signal sample collected at time instant t, i.e., st , depends on the adjacent
random symbols. Thus, all the adjacent symbols contribute to st and, inspired by the
Central Limit Theorem (CLT), one would expect for st to be normally distributed.
4.6.2 Assumption of Mutual Independence Among The Received Signal
Samples
As described above, due to the ISI effect, typically, the st ’s are not mutually independent. It
depends on the statistical properties of the PU’s transmitted signal to answer the question
of how significantly st depends on the observations made prior or after it, however, one
could expect to see a decrease in the level of dependence between st ’s as the sampling
13
Here, we are ignoring the edge-effect, i.e., when the observation is made, at the SU, at the very
beginning or at the end of one of the PU’s transmission phases.
14
In other words, the SU performs an asynchronous receiving.
15
Multipath fading, by itself, leads to ISI effect, too.
4.6 Discussion
39
period ∆T at the SU increases. Nonetheless, one should be careful abut the choice of ∆T
as, indeed, there is a trade-off between the BRM’s performance and the amount of ∆T ,
i.e., the sampling period. We should note that the power of the BRM sensing scheme
comes from the fact that it exploits the variation in the state of channel which is modelled
according to a CTMC. If the samples are taken far apart in time, the BRM scheme reduces
to a simple test which compares the LR evaluated for the current observation against a
π0 (∞)
. This loss in the power of the BRM decision rule corresponds
fixed threshold η =
π1 (∞)
to the simple fact that, if observations are made far apart in time, then, knowing the state
of channel at a given time, gives us little information or almost no information regarding
the state of channel at a later time.
40
41
Chapter 5
SU’s Transmission Period
Optimization
Regardless of the employed decision rule, the SU will initiate a transmission the moment
it detects the channel to be free. If the SU senses a busy channel, it will sense the channel
again later until it finds the channel free. A question that arises naturally is for how long
the SU should transmit after detecting the channel to be free at a given time instant. We
refer to the length of time the SU transmits after deciding that the channel is free as the
SU’s transmission period and we denote it by T .
Ideally, the SU would like to keep transmitting as long as the channel is free to fully
exploit the opportunity. However, the SU does not know when the PU will return to the
channel. In other words, the SU does not know in advance how much time is available
for transmission. However, what is clear is that SU should not continue its transmission
session after the return of PU to the channel in order to avoid interfering with the PU.
There is another issue which is of great importance and needs to be taken into account.
No matter what sensing rule the SU employs to sense the channel, it is never certain that
the decision is absolutely correct. If it happens that the SU makes a wrong decision with
regard to the state of channel by claiming it to be free when in fact it is busy then, as soon
as the SU initiates transmission, it will interfere with the PU. Therefore, the performance
of the sensing rule directly affects the amount of interference the SU imposes on the PU.
We will call the imaginary SU that can always detect perfectly the current state of
channel as the Ideal-Detector SU (ID-SU). Suppose that the regulation is set such that SU
42
SU’s Transmission Period Optimization
must vacate the channel no later than δ seconds after the PU returns.1 Then in order to
meet this requirement, as the SU never knows when the PU may return, the ID-SU should
stop each transmission session after δ seconds and sense the channel again; if the ID-SU
finds the channel free, it would resume transmitting. This interruption in the transmission
phase reduces the performance and the QoS of the SUs. Unfortunately, for a Non IdealDetector SU (NID-SU) which may wrongly detect the current state of channel, such a
strict requirement, i.e., vacating the channel after δ seconds of return of the PU, can not
be satisfied. Consider the scenario where a NID-SU transmits for δ seconds, then stops
transmitting and, after sensing the channel, it wrongly claims the state of channel to be
free and continues its transmission for another δ seconds. This simple example explains
this very fact that we can never put a deterministic bound on the amount of interference
the SU imposes on the PU.
In this chapter, we will consider two cases for the problem of SU’s transmission period
optimization. The first one aims to, for each SU’s transmission session, introduce no
interference to the PU with a high probability. The second, using the fact that PU is
employing a type of Error-Correcting Coding and thus is resistant to interference for a
duration of δ seconds, tries to increase the SU’s transmission period.
5.1 Case I: Ideal Scenario
Ideally, what we would like to happen is (i) that the SU correctly claims the channel to be
free and then starts its transmission phase immediately, and (ii) the PU does not return
within the transmission time and therefore there is no interference between the SU and the
PU. This case is illustrated in Fig. 5.1. Suppose that as the SU claims a free channel at
time instant ti , the posterior probability of the channel to be free given all the past and
t|X t
current data, namely X t , is π0 . If, for a specific transmission session starting at time
t, we define the event A as “the channel is free at the beginning of the transmission phase
and no interference happens during the transmission phase,” then, we would like to have
the probability of occurrence of A given all the collected observations up to time instant t,
1
For example, IEEE 802.22 dictates that the channel must be vacated before 2 seconds elapse after the
return of the PU.
5.1 Case I: Ideal Scenario
43
to be high, i.e.,
Pr(A|X t ) ≥ 1 − ,
0 < 1.
(5.1)
The general form of events belonging to the set A is depicted in Fig. 5.1. Thus, if the RV
SU is transmitting
T
PU
active
PU
active
t
Fig. 5.1
t+T
PU
active
T ime
The general form of outcomes belonging to the event A.
x∼Exp(λf ) denotes the duration of a free period, we should have
t|X t
π0
t|X t
π0
· Pr(x > T ) ≥ 1 − ,
or
(5.2)
· e−λf T ≥ 1 − ,
which implies,
t|X t
T ≤ λf
−1
π
· log( 0 ),
1−
(5.3)
and hence, the maximum transmission period would be
t|X t
Case I
Tmax
= λf
−1
π
· log( 0 ).
1−
(5.4)
t|X t
Case I
As can be seen, Tmax
depends on the posterior probability π0 of the channel to be
free given all the available observations up to time instant t. Also, as it can be inferred
t|X t
Case I
from (5.4), the maximum transmission period Tmax
is a decreasing function of π0 which
indeed makes sense. Basically, eq. (5.4) dictates the SU to transmit for a shorter amount
t|X t
of time as the certainty of the SU with regard the state of channel to be free, namely π0 ,
decreases.
44
SU’s Transmission Period Optimization
5.1.1 Minimum Transmission Period Problem
In this section, we will consider the case where the SU requires a minimum amount of time,
say Tmin , to be available for each of its transmission sessions to take place. Hence, we need
to have
Case I
Tmax
> Tmin
(5.5)
which leads to
t|X (t)
π0
Since the function g(x) =
have
> (1 − )eλf Tmin ,
1−x
,
x
t|X (t)
π0
(5.6)
0 < x ≤ 1, is a monotonically decreasing function, we
t|X (t)
1 − π0
0 < 1.
<
1 − (1 − )eλf Tmin
.
(1 − )eλf Tmin
(5.7)
(5.7) implies that in order to satisfy the requirement in (5.1), we need to use the following
detection rule
t|X (t) Ŝ(t)=0
1 − π0
t|X (t)
π0
≶
Ŝ(t)=1
1 − (1 − )eλf Tmin
.
(1 − )eλf Tmin
(5.8)
According to Chapter 4, the general form of sensing rule is,
t|X (t) Ŝ(t)=1
π1
≷
t|X (t)
π0
Ŝ(t)=0
Thus, if we define C ,
1 − (1 − )eλf Tmin
.
(1 − )eλf Tmin
(C10 − C00 )
.
(C01 − C11 )
(5.9)
(C10 − C00 )
, then, according to (5.7), we just need to have C =
(C01 − C11 )
5.1.2 Discussion
t|X t
As depicted in Fig. 5.2, ψ(T ) = π0 · e−λf T , T > 0, is monotonically decreasing and
strictly convex function of T for T ∈ [0, +∞). Thus, in order to meet the requirement in
5.1 Case I: Ideal Scenario
t|X t
ψ(T ) = π0
45
· e−λf T
t|X t
π0
t|X t
π0
· e−λf T1
A
Threshold line
1−
t|X t
π0
B
· e−λf T2
Case I
Tmax
T1
Tavg = αT1 T1 + αT2 T2
T2
T (sec)
Fig. 5.2 Randomize-Selection methodology has been applied along the AB
line among point A and point B. As can be seen, the average Transmission
Case I has been achieved.
period of Tavg > Tmax
(5.1) we could also employ a Randomized-Selection (RS) method. Suppose that instead
Case I
of transmitting for Tmax
seconds, we choose the transmission length randomly among
two choices; T1 (with probability αT1 ) and T2 (with probability αT2 = 1 − αT1 ) where
Case I
T1 < Tmax
< T2 , such that the following holds,
t|X t
αT1 π0
t|X t
Pr(x > T1 ) + αT2 π0
Pr(x > T2 ) = 1 − ,
1.
(5.10)
Then, we can achieve an average transmission period of length Tavg = αT1 T1 + αT2 T2 which
Case I
is even greater than Tmax
as shown in Fig. 5.2. It is straightforward to check that (5.10)
implies
αT1 = 1 −
e−λf T1 −
e−λf T1
1−
t|X t
π0
.
− e−λf .T2
(5.11)
46
SU’s Transmission Period Optimization
The philosophy behind the RS methodology is that the SU can achieve an average
transmission time of Tavg which could be arbitrarily large if it admits the risk of transmitting
Case I
. In other words, in case
with some probability2 , say αT1 for a shorter duration than Tmax
of employing the RS approach, the SU must be prepared to transmit for an amount of time
Case I
T1 where T1 < Tmax
with non-zero probability of αT1 .
Case I
Mathematically speaking, Tmax
derived in (5.4) is the Max-min solution to requirement in (5.1) due to the fact that if the SU employs any form of RS approach to achieve
Case I
by randomly combining different candidate transmission periods {T1 , T2 , · · · , TM },
Tavg > Tmax
then the SU has to admit the fact that it has to transmit with some probability for a duraCase I
. Basically, In order for the SU to achieve an average transmission
tion shorter than Tmax
Case I
, it has to apply the RS approach on a set of candidate
period of length Tavg > Tmax
transmission periods {T1 , T2 , · · · , TM } for which we have the following,
Case I
min{T1 , T2 , · · · , TM } = T1 < Tmax
.
(5.12)
Thus, the RS method provides the SU with a more flexible way of deciding with regard to
its transmission duration. However, this flexibility comes at the cost of having to transmit
Case I
.
with some non-zero probability for a duration of time less than Tmax
5.2 Case II: Bounded Single-Interference Scenario
In this section, we will consider a relaxed version of the ideal case discussed previously.
It may well be that if the amount of overlap between a single busy period of the PU and
the SU transmission session is less than δ seconds, the PU data can still be decoded using
Error-Correcting Codes (ECC) which is quite a common setting in practice. In that case the
PU can tolerate a limited amount of interference and successfully retrieve the transmitted
data. Let t indicate the initial moment that the SU’s transmission period starts and S(t)
denote the state of channel at that moment. For a specific transmission session starting at
time t, we define the event B as
B=“ (i) the state of the channel at the beginning of the transmission session is free, i.e.,
S(t) = 0, (ii) the amount of overlap between a single busy period and the SU transmission
session is less than or equal to δ, and (iii) no busy period gets totally distorted by the SU
2
It is straightforward to show that the higher Tavg the SU aims to achieve, the larger this probability
αT1 must be, given that the SU has to meet the requirement in (5.1).
5.2 Case II: Bounded Single-Interference Scenario
47
transmission session.” 3
The general form of events belonging to B is depicted in Fig. 5.3. Then, one valid
SU is transmitting
T
Overlap
PU
active
PU
active
t
Fig. 5.3
PU
active
t+T
T ime
The general form of outcomes belonging to the event B.
candidate for the problem of Transmission Period Optimization is to have the probability
of occurrence of B given all the available observations up to time instant t, that is Pr(B|X t )
to be high. Mathematically speaking, we would like to have,
Pr(B|X t ) ≥ 1 − ,
0 < 1.
(5.13)
Conditioning on the initial state of channel at the beginning the transmission session,
namely S(t), then (5.13) can be written as follows,
t|X t
Pr(B|X t ) = Pr(B|X t , S(t) = 0)Pr(S(t) = 0|X t ) = Pr(B|X t , S(t) = 0)π0
.
(5.14)
Let us, with slight abuse of notation, define Bof f (T ) , Pr(B|X t , S(t) = 0), then we have
(
Bof f (T ) =
Pr(x > T ) + Pr(x < T, x + y > T )
Pr(x > T ) + Pr(x < T, x + y > T, T − x < δ)
if 0 < T ≤ δ
if T ≥ δ
(5.15)
where x1 ∼ Exp(λf ) and y1 ∼ Exp(λb ) denoting the duration of a free period and a busy
period of the PU, respectively. As the RVs x and y are independent, their joint PDF is
3
The requirement of S(t) = 0, prevents the SU from repeatedly interfering with the PU in consecutive
transmission sessions.
48
SU’s Transmission Period Optimization
given by f (x1 , y1 ) = f (x1 )f (y1 ) = λf e−λf x1 λb e−λb y1 for x1 , y1 ≥ 0. Thus, for (5.15) we have
(
Bof f (T ) =
R T R +∞
e−λf T + 0 T −x1 λf e−λf x1 λb e−λb y1 dy1 dx1
R T R +∞
e−λf T + T −δ T −x1 λf e−λf x1 λb e−λb y1 dy1 dx1
if 0 < T ≤ δ
if T ≥ δ
(5.16)
Then, using (5.16) in (5.14), the objective of Case II as given in (5.13) can be reformulated
as follows,
t|X t
Pr(B|X t ) = π0
Bof f (T ) > 1 − ,
0 < 1.
(5.17)
The objective is illustrated in Fig. 5.4, where, for a given value of δ, Pr(B|X t ; T ) as a
function of the SU’s transmission period, T , has been plotted.
Pr(B|X t ; T )
t|X t
π0
t|X t
π0
· Bof f (δ)
Threshold line
1−
Bof f (T )
0
T =δ
Fig. 5.4
Case II
Tmax
T (sec)
The objective of Case II is illustratively depicted.
As Bof f (T ) is a monotonically decreasing function of T for a given δ (where T > δ > 0),
if we denote the minimum transmission period for this case of study by Tmin , then we
5.2 Case II: Bounded Single-Interference Scenario
49
should have
t|X t
π0
Bof f (Tmin ) ≥ 1 − ,
0<1
(5.18)
or
t|X t
π0
As g(x) =
>
1−
,
Bof f (Tmin )
0 < 1.
(5.19)
1−x
is a monotonically decreasing function of x for x > 0, we have
x
t|X t
π1
t|X t
π0
=
1
t|X t
− π0 Ŝ(t)=0
<
t|X t
π0
1−
Bof f (Tmin )
.
1−
Bof f (Tmin )
1−
(5.20)
Thus, the sensing rule for this case is given by
t|X t Ŝ(t)=1
π1
≷
t|X t
π0 Ŝ(t)=0
Bof f (Tmin ) − (1 − )
(1 − )
(5.21)
As can be seen in Fig. 5.4, the RS methodology introduced in Case I could be applied for
Case II, in the same manner.
50
51
Chapter 6
Uncertainty In The PU’s Channel
Access Pattern Parameters
As discussed previously, the channel access pattern is modelled according to a 2-state
CTMC as depicted in Fig. 3.3 of Chapter 3. The zero state (S = 0) corresponds to a free
period and the one state (S = 1) corresponds to a busy period of the channel. The length
of a free period is an exponentially distributed RV with parameter λf denoted by x with
E(x) = λ−1
f . Similarly, the length of a busy period is an exponentially distributed RV with
parameter λb denoted by y with E(y) = λ−1
b . The RVs x and y are independent of each
other.
According to (4.22) in Chapter 4, in order to make a decision on the state of channel
at time ti , the following decision rule is used,
t |X ti Ŝ(t )=1
i
π1i
ti |X ti
π0
≷
Ŝ(ti )=0
(C10 − C00 )
,
(C01 − C11 )
(6.1)
which is equivalent to,
f (xti |S(ti ) = 1)
f (xti |S(ti ) = 0)
Ŝ(ti )=1
≷
Ŝ(ti )=0
t |X
π0i
t
π1i
t(i−1)
t
|X (i−1)
(C10 − C00 )
.
(C01 − C11 )
(6.2)
t |X
t(i−1)
t |X
t(i−1)
In order to employ the sensing rule in (6.2), we first need to evaluate π0i
and π0i
through the recursive method discussed in Appendix B which indeed requires the perfect
52
Uncertainty In The PU’s Channel Access Pattern Parameters
knowledge of the values of the parameters λ∗f and λ∗b .
In practice, the parameters λf and λb are unknown and must be estimated by the SU(s).
However, no estimator is perfectly accurate and an estimate could deviate from the true
value of the parameter. In estimation theory, the uncertainty pertaining to an unbiased
estimator is characterized by the Cramer-Rao Bound (CRB).1 Now, the question is how to
employ the BRM in the presence of uncertainty with regard to parameters λf and λb . In
other words, in the scenario for which our best knowledge with regard to the parameters
λf and λb is that their true values namely, λ∗f and λ∗b are located within two sets of δλf and
δλb respectively, how could we get close (in some sense that we will define shortly) to our
ultimate objective given in (4.5) in Chapter 4?
In this chapter we study the performance of BRM when the exact values of the parameters λf and λb are not perfectly known to the SU. However, we assume that the true value
of parameter λf (denoted by λ∗f ) is within an uncertainty interval δλf and similarly, that
the true value of parameter λb (denoted by λ∗b ) is within an uncertainty interval δλb . The
intervals δλf and δλb are defined as follows,
max
δλf , [λmin
],
f , λf
and δλb , [λmin
, λmax
],
b
b
(6.3)
max
where the interval end points, namely, λmin
, λmin
and λmax
are known.
f , λf
b
b
The key issue arising under uncertainty pertaining to CTMC’s parameters λf and λb
is that the uncertainty with regard to the true values of parameters λf and λb leads to an
uncertainty region with regard to the a-priori probability of channel to be ON (or OFF)
t
t
t |X (i−1)
t |X (i−1)
), respectively. In order to grasp this
at time instant ti denoted by π1i
(or π0i
concept, let us follow the BRM step by step. First, we consider a simple scenario and later
on we will turn our attention back to the general form of the problem.
6.1 Two-channel Uncertainty Scenario
Before we begin, let us quickly introduce the notation we will use. Hereafter, we refer to a
CTMC with parameters λf and λb where λf is the parameter of exponentially distributed
RV modelling a free period and similarly λb is the parameter of exponentially distributed
RV modelling a busy period, as CT M C(λf , λb ).
1
The CRB imposes a lower bound on the variance of an unbiased estimator. The estimator that achieves
this lower bound is called an efficient estimator [43].
6.1 Two-channel Uncertainty Scenario
53
In this section, we assume that the SU’s uncertainty with regard to the CTMC modelling
the channel access pattern is of a particular type, namely, that the channel access pattern of
PU is either CT M C(λf1 , λb1 ) or CT M C(λf2 , λb2 ), where λf1 , λf2 , λb1 , λb2 are known. Since
the SU does not know which of CT M C(λf1 , λb1 ) or CT M C(λf2 , λb2 ) is the correct one, it is
not able to minimize the average cost at the sensing time ti . At each sensing instant, say ti ,
the SU has two candidates/choices for the a-priori probability of channel to be ON, namely,
t |X ti−1
t |X ti−1
(λf2 , λb2 ). Thus, the set of uncertainty corresponding to the
(λf1 , λb1 ) and π1i
π1i
a-priori probability of channel to be ON given all the past observations at time instant ti
is given by,
t |X ti−1
δ[π1i
t |X ti−1
] = {π1i
t |X ti−1
(λf1 , λb1 ), π1i
(λf2 , λb2 )},
(6.4)
t |X ti−1
where π1i
(λf1 , λb1 ) is the a-priori probability of channel to be ON at ti given all
the observations made prior to time instant ti and based on the assumption that the
t |X ti−1
CT M C(λf1 , λb1 ) is the true model for PU’s channel access pattern. Similarly, π1i
(λf2 , λb2 )
is the a-priori probability of channel to be ON at ti given all the observations made prior
to time instant ti assuming that the true PU’s channel access pattern is CT M C(λf2 , λb2 ).
t |X ti−1
Essentially, (6.4) implies that the true value of π1i
t
∗ ti |X i−1
π1
t |X ti−1
∈ δ[π1i
t
∗ ti |X i−1
, π1
].
t |X ti−1
, lies in δ[π1i
], i.e.,
(6.5)
As we have the following for E[C (ti ) |X ti−1 ],
(t |X ti−1 )
C01 Pr(Ŝ(ti ) = 0|S(ti ) = 1, X ti−1 ) + C11 Pr(Ŝ(ti ) = 1|S(ti ) = 1, X ti−1 ) π1 i
(t |X ti−1 )
+ C00 Pr(Ŝ(ti ) = 0|S(ti ) = 0, X ti−1 ) + C10 Pr(Ŝ(ti ) = 1|S(ti ) = 0, X ti−1 ) π0 i
E[C (ti ) |X t(i−1) ] =
minimizing E[C (ti ) |X t(i−1) ] is simply not achievable due to the lack of knowledge with regard
t
∗ ti |X i−1
to π1
. Instead, in order to sense the state of channel at time instant ti , we propose to
use a sensing/decision rule Dti which is optimum in a sense that it minimizes the following
quantity,
max
t
t
t |X i−1
t |X i−1
π1i
∈δ[π1i
]
E[C (ti ) |X t(i−1) ]. ∀i ≥ 1.
(6.6)
54
Uncertainty In The PU’s Channel Access Pattern Parameters
Mathematically speaking, the decision rule for sensing the state of channel at time instant
ti is given by,
min
t
D
i
max
t
t
t |X i−1
t |X i−1
]
∈δ[π1i
π1i
E[C
(ti )
|X
t(i−1)
] ,
∀i ≥ 1.
(6.7)
Thus, the sensing rule Dti minimizes the worst-case scenario for the expected cost of our
decision regarding the state of channel at time instant ti . We refer to this sensing methodology as the Sequential Restricted Minimax (SRM) approach. The term “Restricted” refers
t |X ti−1
t |X ti−1
]. The term
is restricted to fall in the uncertainty set of δ[π1i
to the fact that π1i
“Minimax” emphasises the fact that we intend to minimize the worst-case scenario in each
sensing instant. Finally, the term “Sequential” in SRM indicates the very need for updating
t |X ti−1
our knowledge of set δ[π1i
] at each sensing time ti sequentially.
To clarify the SRM approach, let us follow it step by step in the following. Consider
the decision making with regard to the very first sensing instant using SRM, since the SU
has no prior data, it uses π0 (∞) and π1 (∞) as the a-priori probability of channel to be
ON and OFF at time instant t1 , respectively.2 At time instant t1 , the SU is unaware of
λ−1
whether π1 (∞; λf1 , λb1 ) , −1 b1 −1 is the true a-priori probability of channel to be ON
λb1 + λf1
−1
λ
or π1 (∞; λf2 , λb2 ) , −1 b2 −1 .
λb2 + λf2
The expected cost of the decision at time instant t1 , E[C (t1 ) |X t0 = ∅] = E[C (t1 ) ], is
given by
E[C (t1 ) ] =
C01 Pr(Ŝ(t1 ) = 0|S(t1 ) = 1) + C11 Pr(Ŝ(t1 ) = 1|S(t1 ) = 1) π1 (∞)
+ C00 Pr(Ŝ(t1 ) = 0|S(t1 ) = 0) + C10 Pr(Ŝ(t1 ) = 1|S(t1 ) = 0) π0 (∞). (6.8)
However, the SU is unaware of whether it has to use π1 (∞|λf1 , λb1 ) or π1 (∞|λf2 , λb2 ) instead
of π1 (∞) in (6.8). This very confusion of the SU with regard to the true value of a-priori
probability of channel to be ON at time instant t1 prevents it from being able to minimize
the quantity in (6.8). Hence, we employ the SRM methodology as described earlier in order
2
As before, we assume that the very first observation made by the SU from the channel takes place at
time instant t1 , and also that the PU has been active for a long time and thus the CTMC modelling the
PU’s channel access pattern has reached to its steady-state occupancy probabilities.
6.1 Two-channel Uncertainty Scenario
55
to make a decision with respect to the state of channel at time instant t1 . In this case, the
t |X t0
uncertainty set regarding π11
is,
t |X t0
δ[π11
]
=
π1 (∞|λf1 , λb1 ), π1 (∞|λf2 , λb2 ) ,
(6.9)
and the SRM sensing rule at time instant t1 , namely, Dt1 is given by
max
min
t
D
t |X t0
t |X t0
]
∈δ[π11
π11
1
E[C
(t1 )
|X ] .
t0
(6.10)
Using the recursive method discussed in Appendix B and assuming that N , i.e., the number
of observation made in a sensing session, is equal to 1, then, the a-priori probability of
t |X t1
channel at time instant t2 given the observation made at t1 , namely, π12
can be obtained
as follows,
t |X t1
π12
(λf1 , λb1 ) =
M(t1 |λf1 , λb1 )
,
f (xt1 )
(6.11)
where
M(t1 |λf1 , λb1 ) =
1
X
j=0
f (xt1 |S(t1 ) = j)Pr(S(t2 ) = 1|S(t1 ) = j)
|t −t1 |
= A112
|t −t1 |
A102
(λf1 , λb1 )f (xt1 |S(t1 ) = 1)π1 (∞|λf1 , λb1 )+
(6.12)
(λf1 , λb1 )f (xt1 |S(t1 ) = 0)π1 (∞|λf1 , λb1 ),
Similarly,
t |X t1
π12
(λf2 , λb2 ) =
M(t1 |λf2 , λb2 )
f (xt1 )
(6.13)
where
M(t1 |λf2 , λb2 ) =
1
X
j=0
f (xt1 |S(t1 ) = j)Pr(S(t2 ) = 1|S(t1 ) = j)
|t −t1 |
= A112
|t −t1 |
A102
(λf2 , λb2 )f (xt1 |S(t1 ) = 1)π1 (∞|λf1 , λb1 )+
(λf2 , λb2 )f (xt1 |S(t1 ) = 0)π1 (∞|λf2 , λb2 ).
(6.14)
56
Uncertainty In The PU’s Channel Access Pattern Parameters
|t −t |
|t −t |
|t −t |
|t −t |
In the above, A102 1 (λfj , λbj ), A102 1 (λfj , λbj ), A102 1 (λfj , λbj ) and A102 1 (λfj , λbj ) are
the transition probabilities defined in Section 3.4 of Chapter 3 for a transition duration of
length |t2 − t1 | and also based on the assumption that the channel evolves according to
CT M C(λfj , λbj ), j = 1, 2.
As before, if the SU intends to make a decision with regard to the state of channel at
time instant t2 through BRM, it will have two candidates for the a-priori probability of
t |X t1
t |X t1
channel to be ON at time instant t2 given X t1 , namely, π12
(λf1 , λb1 ) and π12
(λf2 , λb2 ).
t2
As a result, the SU needs to employ the SRM rule D as to sense the channe at time instant
t2 as follows,
min
t
D
2
E[C
max
t |X t1
π12
t |X t1
∈δ[π12
]
(t2 )
|X ] ,
t1
(6.15)
where,
t |X t1
δ[π12
]
t2 |X t1
t2 |X t1
= π1
(λf1 , λb1 ), π1
(λf2 , λb2 ) .
(6.16)
The same methodology will be deployed to sense the state of channel at time instant
ti , i ≥ 3.
In the following, we turn our attention to the general case we discussed at the beginning
of this chapter.
6.2 General Case of Uncertainty
In this part, we turn our attention to the general setting of our problem, that is we have
the following,
max
λ∗f ∈ δλf = [λmin
] and λ∗b ∈ δλb = [λmin
, λmax
].
f , λf
b
b
(6.17)
Since, λ∗f and λ∗b are now restricted to the intervals δλf and δλb , respectively, the cant |X t0 =∅
didates for the a-priori probability of channel to be ON at time instant t1 , π11
, form
the set
λ−1
t1 |X t0
b
(6.18)
δ[π1
],
−1 λf ∈ δλf , λb ∈ δλb .
λ−1
b + λf
6.2 General Case of Uncertainty
t |X t0
57
t |X t0
In other words, π11
∈ δ[π11
].
∆
∆
∆
Similarly, if we define the four-tuple (A∆
11 (λf , λb ), A10 (λf , λb ), B01 (λf , λb ), B00 (λf , λb ))
as the sequence of transition probabilities corresponding to CT M C(λf , λb ) and length of
transition of ∆ seconds, then, we have the following candidates for the sequence of transition
probabilities,
∆
∆
∆
{(A∆
11 (λf , λb ), A10 (λf , λb ), B01 (λf , λb ), B00 (λf , λb ))|λf ∈ δλf , λb ∈ δλb }.
(6.19)
∆
∆
∆
Let us denote the set in (6.19) by δ[(A∆
11 , A10 , B01 , B00 )], Mathematically speaking, there
is a unique one-to-one and onto mapping between {(λf , λb )|λf ∈ δλf , λb ∈ δλb } and
∆
∆
∆
δ[(A∆
11 , A10 , B01 , B00 )]. In other words, each ordered-pair of λf and λb , namely (λf , λb ),
for which λf ∈ δλf and λb ∈ δλb specifies a unique 4-tuple transition probabilities and
vice-versa.
If we take the pair (λ̂f , λ̂b ) ∈ {(λf , λb )|λf ∈ δλf , λb ∈ δλb } and its unique 4-tuple corresponding sequence of transition probabilities
∆
∆
∆
∆
∆
∆
∆
(Â∆
11 (λf , λb ), Â10 (λf , λb ), B̂01 (λf , λb ), B̂00 (λf , λb )) ∈ δ[(A11 , A10 , B01 , B00 )], then, assuming
that the a-priori probability of channel to be ON at the very first sensing instant is
π1 (∞; λ̂f , λ̂b ) and evolving it through its corresponding sequence of transition probabilities,
∆
∆
∆
(Â∆
11 (λf , λb ), Â10 (λf , λb ), B̂01 (λf , λb ), B̂00 (λf , λb )) and all the collected observations prior to
time instant ti , that is X ti−1 by using the recursive method given in the Appendix B, we
t |X ti−1
reach to π̂1i
as a a-priori probability of channel to be ON given all the observations
made prior to time instant ti for which indeed we have,
t |X ti−1
π̂1i
t |X ti−1
∈ δ[π1i
],
∀i ≥ 1.
(6.20)
t |X ti−1
Hence, given the δ[π1i
], in order to sense the state of channel at time instant ti , we
employ the SRM decision rule Dti at time instant ti which is given by,
min
t
D
i
max
t
t
t |X i−1
t |X i−1
π1i
∈δ[π1i
]
E[C |X
ti
ti−1
] ,
∀i ≥ 1.
(6.21)
As it can be inferred from (6.21), the SRM decision rule deployed as to sense the state of
channel at time instant ti , namely, Dti is indeed a time-varying decision rule, that is Dti
and Dtj for i 6= j are potentially different.
58
Uncertainty In The PU’s Channel Access Pattern Parameters
Before we proceed to the simulation results, we need to put forward a discussion on the
general case.
6.2.1 Discussion on General Case: Tracking the Propagation of Uncertainty
In order to solve the general case in which δλf and δλb are continuous intervals, an infinite
t |X ti−1
amount of memory is required to keep track of all the possible candidates for π1i
for sensing the state of channel at time instant ti . Thus, in order to make the problem
tractable, we ought to discretize the continuous intervals δλf and δλb .
The continuous interval A = [amin , amax ] ⊂ R can be discretized into N evenly-spaced
points between and including amin and amax , and thus, form the set ÃN . We refer to ÃN as
the discrete counterpart of continuous interval A with precision N . Obviously, the higher
the precision of discretization, the larger number of channels to deal with and henceforth
more computationally expensive the procedure of sensing the state of channel will be.
However, the larger the N is, the more accurate approximation of set AN the set ÃN can
provide us with. Thus, if we discretize δλf and δλb into the sets δ̃λNf and δ̃λMb , respectively,
then, we need to keep track of N × M different CTMC’s as the probable candidates for the
true CTMC which models the PU’s channel access pattern. Hence we have,
(1)
(2)
(N )
(6.22)
(M )
, λb },
(6.23)
δ̃λNf = {λf , λf , · · · , λf },
δ̃λMb =
(1)
(N )
(1)
(2)
{λb , λb , · · ·
(1)
(M )
where λf = λmin
= λmax
, λb = λmin
and λb = λmax
. Then, the set of N × M
f , λf
f
b
f
CTMC candidates denoted by δN M [CT M C] is given by,
(1)
(1)
(1)
(2)
(N )
(M )
{CT M C11 (λf , λb ), CT M C12 (λf , λb ), · · · , CT M CN M (λf , λb
)},
(6.24)
and we have
CT M C(λ∗f , λ∗b ) ∈ δN M [CT M C].
(6.25)
t |X ti−1
The set δN M [CT M C] gives rise to the following uncertainty set with regard to π1i
t |X ti−1
δ[π1i
t |X ti−1
] = {π1i
(1)
(1)
t |X ti−1
(λf , λb ), π1i
(1)
(2)
t |X ti−1
(λf , λb ), · · · , π1i
(N )
(M )
(λf , λb
)}.
,
(6.26)
6.3 Numerical Results and Simulations
59
Therefore, the SRM decision rule Dti deployed to sense the state of channel at time instant
ti is given by,
min
t
D
i
max
t
t
t |X i−1
t |X i−1
]
∈δ[π1i
π1i
E[C |X
ti
ti−1
]
(6.27)
6.3 Numerical Results and Simulations
In this section, we present the numerical results for our proposed sensing scheme, i.e., SRM.
In Fig. 6.1 the simulation result for the case of two-channel uncertainty has been shown.
Fig. 6.1 Infinite horizon average cost, i.e., Cost
two-channel uncertainty setting.
∞,
tN
for the SRM scheme in
The vertical axis presents the infinite horizon average cost, i.e., Cost ∞
tN , and the horizontal
60
Uncertainty In The PU’s Channel Access Pattern Parameters
σs2
. Without loss of generality, we
σn2
assume σn2 = 1. We also assume that the signal samples, sti ’s, and the noise samples, nti ’s,
both follow zero-mean independent and identically distributed Gaussian random processes
[37],[41]. We also assume that N = 1 and for the sampling period we have ∆T = 10−2
seconds.
For the case of two-channel uncertainty, it is assumed that the SU knows that either
2
CT M C(λf1 = 0.25, λb1 = 1) or CT M C(λf2 = 2, λb2 = ) is the PU’s channel usage
9
pattern, however, does not know which one it the true PU’s CTMC channel usage pattern.
As can be seen in Fig. 6.1, for the case that CT M C(λf1 = 0.25, λb1 = 1) is the true CTMC,
the SRM gives us a lower infinite horizon average cost than that of the BRM conducted
2
for the CT M C(λf2 = 2, λb2 = ).
9
axis denotes the Signal-to-Noise Ratio (SNR) defined as
Fig. 6.2 Infinite horizon average cost, i.e., Cost
multi-channel (four-channel) uncertainty setting.
∞,
tN
for the SRM scheme in
6.3 Numerical Results and Simulations
61
In Fig. 6.2, the simulation results for the case of multi-channel uncertainty has been
depicted. The candidates for the PU’s channel usage pattern are CT M C(λf1 = 1, λb1 =
1
0.5), CT M C(λf2 = 0.25, λb2 = 0.25), CT M C(λf3 = 0.5, λb3 = ) and CT M C(λf4 =
6
1
1
, λb = ). We assume that the true CTMC is CT M C(λf1 = 1, λb1 = 0.5), however, the
3 4
8
SU does not know it. As can be seen in Fig. 6.2, the infinite horizon average cost achieved
by the SRM is close to that of the BRM conducted for the CT M C(λf1 = 1, λb1 = 0.5).
Due to the optimality of the BRM, the BRM conducted for the CT M C(λf1 = 1, λb1 = 0.5)
achieves the minimum infinite horizon average cost. Thus, the SRM scheme can closely
follow the BRM in term of the performance, i.e., the infinite horizon average cost. However,
in contrast to the BRM, the SRM scheme does not need the perfect knowledge of PU’s
channel usage pattern.
62
63
Chapter 7
Conclusion and Future Work
In this chapter, we conclude the thesis and outline promising directions for the future work.
In Chapter 4, we studied the spectrum sensing problem for the case of an unslotted
CTMC PU. We assumed that the CTMC’s parameters are perfectly known to the SU
and based on this assumption, we proposed two sensing schemes, namely, SS-OPA and
BRM. The former seeks simplicity at the cost of decreased performance due to not fully
exploiting the knowledge of the PU’s channel access pattern whereas the latter aims at
achieving optimality at the cost of increased complexity. We also proved the optimality of
the BRM scheme. Furthermore, the issue of the SU’s processing delay was addressed and
an SD-aware BRM was developed as a solution to the problem.
In Chapter 5, the problem of determining an appropriate transmission period for the
SU was studied. We adapted two SU’s channel access strategies to our proposed sensing
scheme, termed BRM. In other words, we incorporated the effect of sensing scheme into the
SU’s channel access strategy and thereby we avoided the impractical assumption of perfect
sensing when optimizing the SU’s transmission period. We attempted to strike a balance
between SU’s channel utilization and the PU’s protection against the interference induced
by the SU.
The BRM sensing scheme proposed in Chapter 4 required the perfect knowledge of PU’s
usage pattern, i.e., the CTMC. However, in Chapter 6, we proposed a new sensing scheme,
SRM scheme, which is able to operate in the setting where there is uncertainty with regard
to the parameters of PU’s channel usage model, i.e., the parameters of the corresponding
CTMC. More specifically, the SRM sensing scheme minimized the worst-case cost incurred
by the sensing rule at each sensing time. Simulation results were presented when the SRM
64
Conclusion and Future Work
scheme was employed for the case of two-channel uncertainty and multi-channel uncertainty.
It was shown that the performance of the SRM scheme is quite promising in the channeluncertainty setting.
Nonetheless, the following issues/topics were not addressed in this thesis and thus could
be subject to further studies in the future.
The CTMC model is proposed as a good approximation for the channel usage pattern
of Wireless LAN (WLAN) [23]. However,
(i) More accurate models such as Semi-Markov Model [23] can be employed to model
the PU’s channel usage pattern and henceforth the performance of sensing scheme
can be improved further.
(ii) There is a fundamental practical issue with the CTMC model that, theoretically, in
a given amount of time, the CTMC could switch between its states arbitrary many
times. However, this can not be the case for a band-limited PU [24]. In other words,
the PU whose channel usage pattern is according to a CTMC must occupy infinite
amount of spectrum or equivalently must poses infinite bandwidth.
For the two proposed SU’s channel access schemes, also referred to as SU’s transmission
period optimization, the following performance metrics [36] cane be derived theoretically
and/or examined through simulations:
No. of collisions in [0, T ]
.
T →∞ No. of busy periods of PU in [0, T ]
(i) Collision probability observed by the SU = lim
No. of collisions in [0, T ]
.
T →∞ No. of busy periods of PU in [0, T ]
(ii) Collision probability observed by the PU = lim
Length of overlap in [0, T ]
.
T →∞
T
(iii) Percentage of overlap observed by the PU = lim
Similar to our proposed approach for the case of uncertainty with regard to the PU’s
channel access pattern modelled by a CTMC, the problem of PU’s signal uncertainty and
noise uncertainty could be studied. We can employ the SRM decision rule to minimize the
worst cost incurred by the decision rule under the existence of uncertainty in noise power,
PU’s signal power and channel model’s parameters all together. More specifically, for the
65
case of uncertainty in noise and signal power, we can consider a similar binary hypothesis
testing problem to the one discussed in Section 4.1 of Chapter 4, that is
H0 (PU is inactive) : x(i) = n(i),
H1 (PU is active) :
x(i) = s(i) + n(i),
(7.1)
with the following modifications,
θ0 , σn2 ∈ δθ0 , [θ0min , θ0max ].
(7.2)
θ1 , σs2 ∈ δθ1 , [θ1min , θ1max ].
(7.3)
and, similarly,
Then, we can employ the SRM decision rule Dti so as to make decision regarding the state
of channel at time instant ti . Thus, the decision rule Dti will be given by
min
t
D
i
max
t
t |X i−1
π1i
t
t |X i−1
∈δ[π1i
θ1 ∈δθ
1
θ0 ∈δθ
0
E[C |X
ti
]
ti−1
] ,
∀i ≥ 1.
(7.4)
66
67
Appendix A
Necessary and Sufficient Condition
for Minimizing E[C (ti)]
Suppose up to time instant ti the observation sequence of X ti = (xt1 , xt2 , · · · , xti ) is made
where xtj is the observation made at time instant tj . The iterative rule for expectation
operator states that,
E[X] = EY [EX|Y [x|y]]
(A.1)
where X and Y are general RVs and x and y are their corresponding realizations, respectively. Then, using the above property for E[C ti ], we have,
E[C (ti ) ] = EX ti−1 [EC (ti ) |X ti−1 [C (ti ) |X ti−1 ]]
(A.2)
Thus,
E[C
(ti )
Z
+∞
]=
−∞
EC (ti ) |X ti−1 [C (ti ) |X ti−1 ]f (X ti−1 )dX ti−1
(A.3)
where f (X ti−1 ) is the joint PDF of the observation sequence of X ti−1 = (xt1 , xt2 , · · · , xti−1 ).
Hence, for any given observation sequence of X ti−1 , i.e., the observations made prior to
the time instant ti , minimizing the quantity E[C ti ] is equivalent to minimizing the quantity EC (ti ) |X ti−1 [C (ti ) |X ti−1 ]. Thus, we can conclude that, E[C (ti ) ] is minimized, if and only
if, for any given X ti−1 , EC (ti ) |X ti−1 [C (ti ) |X ti−1 ] is minimized. Or equivalently, minimizing
68
Necessary and Sufficient Condition for Minimizing E[C (ti ) ]
EC (ti ) |X ti−1 [C (ti ) |X ti−1 ], for any given X ti−1 , is a necessary and sufficient condition to minimize the quantity E[C (ti ) ].
69
Appendix B
Recursive Algorithm to be Used in
BRM
Here, we prove the the following propositions,
t |X ti
(a)
π1i
t |X ti
π0i
?
=
t |X ti−1
(b)
π1i
t |X ti−1
π0i
|t −ti−1 |
where in the latter, A11i
given by,
f (xti |S(ti ) = 1, X ti−1 ) Pr(S(ti ) = 1|X ti−1 )
,
f (xti |S(ti ) = 0, X ti−1 ) Pr(S(ti ) = 0|X ti−1 )
|t −ti−1 | ti−1 |X ti−1
π
?
= |t −t | 1t |X ti−1
B01i i−1 π1i−1
A11i
|t −ti−1 |
, A10i
|t −ti−1 |
, B01i
|t −ti−1 | ti−1 |X ti−1
π0
|t −t
| t
|X ti−1
B00i i−1 π0i−1
+ A10i
+
|t −ti−1 |
and B00i
(B.1)
.
are the transition probabilities
A∆T
= Pr(S(t + ∆T ) = 1|S(t) = 1) ∀t ≥ 0,
11
A∆T
= Pr(S(t + ∆T ) = 1|S(t) = 0) ∀t ≥ 0,
10
∆T
B01
= Pr(S(t + ∆T ) = 0|S(t) = 1) ∀t ≥ 0,
∆T
= Pr(S(t + ∆T ) = 0|S(t) = 0) ∀t ≥ 0,
B00
(B.2)
(B.3)
(B.4)
(B.5)
70
Recursive Algorithm to be Used in BRM
Proof for B.1(a):
t |X ti
π1i
t
π0i
|X ti
=
=
=
=
=
=
Pr(S(ti ) = 1|X ti )
Pr(S(ti ) = 0|X ti )
Pr(S(ti ) = 1, X ti )
Pr(S(ti ) = 0, X ti )
Pr(S(ti ) = 1, xti , X ti−1 )
Pr(S(ti ) = 0, xti , X ti−1 )
f (xti |S(ti ) = 1, X ti−1 )Pr(S(ti−1 ) = 1, X ti−1 )
f (xti |S(ti ) = 0, X ti−1 )Pr(S(ti−1 ) = 0, X ti−1 )
f (xti |S(ti ) = 1, X ti−1 )Pr(S(ti−1 ) = 1|X ti−1 )f (X ti−1 )
f (xti |S(ti ) = 0, X ti−1 )Pr(S(ti−1 ) = 0|X ti−1 )f (X ti−1 )
f (xti |S(ti ) = 1, X ti−1 )Pr(S(ti−1 ) = 1|X ti−1 )
.
f (xti |S(ti ) = 0, X ti−1 )Pr(S(ti−1 ) = 0|X ti−1 )
(B.6)
The proof is completed for B.1(a).
Proof for B.1(b):
t |X ti−1
π1i
t |X ti−1
π0i
Pr(S(ti ) = 1|X ti−1 )
Pr(S(ti ) = 0|X ti−1 )
P1
ti−1
)
j=0 Pr(S(ti ) = 1, S(ti−1 ) = j|X
= P1
ti−1 )
j=0 Pr(S(ti ) = 0, S(ti−1 ) = j|X
P1
ti−1
)Pr(S(ti−1 ) = j|X ti−1 )
j=0 Pr(S(ti ) = 1|S(ti−1 ) = j, X
= P1
ti−1 )Pr(S(t
ti−1 )
i−1 ) = j|X
j=0 Pr(S(ti ) = 0|S(ti−1 ) = j, X
P1
ti−1
)
?
j=0 Pr(S(ti ) = 1|S(ti−1 ) = j)Pr(S(ti−1 ) = j|X
= P1
ti−1 )
j=0 Pr(S(ti ) = 0|S(ti−1 ) = j)Pr(S(ti−1 ) = j|X
=
=
|t −ti−1 | ti−1 |X ti−1
π1
|ti −ti−1 | ti−1 |X ti−1
B01
π1
A11i
(B.7)
|t −ti−1 | ti−1 |X ti−1
π0
|ti −ti−1 | ti−1 |X ti−1
B00
π0
+ A10i
+
In the step indicated by ?, the following Markovian property has been employed,
Pr(S(ti )|S(ti−1 ), X ti−1 ) = Pr(S(ti )|S(ti−1 )),
(B.8)
which implies, given that the state of channel at time instant ti−1 is S(ti−1 ), the probability
71
that the channel takes the state S(ti ) at time ti is independent of the past observations
X ti−1 .
72
73
Appendix C
Comments on the Convergence of
Average Cost Series E[< Cost ttn1 >]
In the followings, we prove the convergence of the average cost series, i.e.,
E[< Cost
tn
t1
n
1 X
>] = {
E[C (ti ) ]}
n i=1
(C.1)
as n approaches to infinity, i.e.,
lim E[< Cost ttn1 >]
n→+∞
(C.2)
for the following two asymptotic cases,
(i) ∆t , |ti+1 − ti | too large, i.e., ideally, ∆t → +∞.
(ii) ∆t too small, i.e., ideally, ∆t ≈ 0.
Proof Case (i)
Intuitive proof : When |ti+1 −ti | is too large, detection of the channel state at time instant
ti , i.e., S(ti ), provides us with no information regarding S(ti+1 ); since by the time ti+1 , the
channel reaches to its steady-state occupancy probability and hence Pr(S(ti+1 )|S(ti )) =
Pr(S(ti+1 )) = Pr(S(t = ∞)).
Comments on the Convergence of Average Cost Series E[< Cost ttn1 >]
74
Analytical proof : It can be easily verified that
Pr(S(ti ) = 1|X (ti−1 ) ) = πon (∞),
and Pr(S(ti ) = 0|X (ti−1 ) ) = πof f (∞).
iid
Thus, ∀i ∈ N , C ti ∼ Bernoulli(p), where p = E[C t1 ]. Therefore, we have
lim E[< Cost ttn1 >] = E[C t1 ] = p.
n→+∞
Proof Case (ii)
Analytical proof :
According to Chapter 4, we have
E[C t1 ] = πof f (∞)Pr(Ŝ(t1 ) = 1|S(t1 ) = 0) + πon (∞)Pr(Ŝ(t1 ) = 0|S(t1 ) = 1)
iid
iid
For the case of nti ∼ N (0, σ02 = 1) and sti ∼ N (0, σ12 − σ02 ) we get
√
√
A1
A1
E[C ] = πof f (∞){2Q(
)} + πon (∞){1 − 2Q(
)},
σ0
σ1
t1
where Q(x) = Pr(z > x) and z ∼ N (0, 1) and A1 is given by
A1 =
where η1 =
2
1
1
− 2
2
σ0 σ1
log(η1
σ1
),
σ0
πof f (∞)
. Then, for E[C t2 ] we have
πon (∞)
√
√
A2
A2
E[C ] = πof f (∞) 2Ext1 |S(t1 )=0 [Q(
)] + πon (∞) 1 − 2Ext1 |S(t1 )=1 [Q(
)] ,
σ0
σ1
t2
where
A2 = A1 + log(
σ1 2
)x .
σ0 t1
75
Following the same methodology, we get
√
Ai
)]
E[C ] = πof f (∞) 2Exti−1 |S(ti−1 )=0 [...[Ext1 |S(t1 )=0 [Q(
σ0
√
Ai
+ πon (∞) 1 − 2Exti−1 |S(ti−1 )=0 [...[Ext1 |S(t1 )=1 [Q(
)] ,
σ1
ti
where Ai = Ai−1 + log(
σ1 2
)x . As the sequences,
σ0 ti−1
√
√
A1
A2
Ai
), Ext1 |S(t1 )=0 [Q(
)], ... , Exti−1 |S(ti−1 )=0 [...[Ext1 |S(t1 )=0 [Q(
)], ...
S1 : Q(
σ0
σ0
σ0
√
and
√
√
A1
A2
Ai
S2 : Q(
), Ext1 |S(t1 )=1 [Q(
)], ... , Exti−1 |S(ti−1 )=1 [...[Ext1 |S(t1 )=0 [Q(
)], ...
σ1
σ1
σ1
√
are both decreasing and bounded from below (as they are both non-negative sequences),
thus, their convergences are granted. Now, one may be interested to know what values the
above sequences converge to. Using the Craig’s formula [44], i.e.,
Z π
1 2
x2
Q(x) =
exp(−
) dθ,
π 0
2 sin2 (θ)
it can be shown that both of the sequences S1 and S2 converge to zero.
(C.3)
76
77
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