IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 2010
2413
SER Analysis and PDF Derivation for
Multi-Hop Amplify-and-Forward Relay Systems
Chris Conne, MinChul Ju, Zhihang Yi, Hyoung-Kyu Song, and Il-Min Kim, Senior Member, IEEE
Abstract—An amplify-and-forward, multi-branch, multi-hop
relay system with 𝐾 relays, in which the relays broadcast
to other relays as well as the destination, is analyzed. An
approximate symbol-error-rate (SER) expression, which is valid
for any number of relays and for several modulation schemes,
is found for the multi-hop system. Also, the cumulative density
function (CDF) and probability density function (PDF) are found
for the random variable, 𝑍 = 𝑋𝑌 /(𝑋 + 𝑌 + 𝑐), where 𝑋 and
𝑌 are sums of independent, Erlang random variables, and 𝑐 is a
constant. The moment generating function (MGF) of 𝑍 is found
for the special case in which 𝑐 = 0. It is shown that these results
are generalizations of previously published results for special
cases of 𝑍. The MGF of 𝑍 is used to develop the approximate
SER expression. Results for the analytic SER expression are
included and compared with simulation results for various values
of 𝐾, for various modulation schemes, and for two choices of
system parameters (channel variances). Results for the multi-hop
system are also compared to results for the two-hop system (in
which relays transmit only to the destination).
Index Terms—Amplify-and-forward, cooperative diversity networks, multi-hop, symbol-error-rate.
I. I NTRODUCTION
I
N this paper, the following system is analyzed: a multiplerelay, amplify-and-forward (AF), multi-hop, cooperative
diversity wireless communications system, in which the transmitters broadcast to the other terminals, the receivers use
maximum-ratio-combining (MRC), and the channels experience Rayleigh fading. In a cooperative diversity system,
multiple single-antenna radios relay each other’s symbols to
a destination. In this way, the destination receives multiple
copies of the symbols from multiple antennas over independent channels and, therefore, spatial diversity is achieved. It
has been shown that a system that uses cooperative diversity
has a higher capacity than one in which the users do not
cooperate with one another (see [1], [2], for example.) In [3],
Laneman et al. developed several protocols for the cooperative
diversity network, including the AF protocol, in which the
relays simply amplify the noisy symbols that they receive and
Paper approved by J. N. Laneman, the Editor for Cooperative Relaying
and Diversity Techniques of the IEEE Communications Society. Manuscript
received November 18, 2008; revised November 18, 2009.
This research was supported in part by the Natural Sciences and Engineering Research Council of Canada (NSERC), and by the Ubiquitous
Computing and Network (UCN) Project, Knowledge and Economy Frontier
R&D Program of the Ministry of Knowledge Economy (MKE) in Korea as
a result of UCN’s subproject 10C2-C2-12T.
C. Conne, M. Ju, Z. Yi, and I.-M. Kim are with the Department of
Electrical and Computer Engineering, Queen’s University, Kingston, ON,
Canada, K7M 2A8 (e-mail: [email protected]; [email protected];
[email protected]; [email protected]).
H.-K. Song is with the Department of Information and Communication
Engineering, Sejong University, Seoul, Korea (e-mail: [email protected]).
Digital Object Identifier 10.1109/TCOMM.2010.062510.080615
then re-transmit them. It was shown in [3] that an AF relay
system with one relay provides full transmit diversity order
of two (the two transmit antennas being the source and the
relay). For an AF relay system with 𝐾 relays, the diversity
order is 𝐾 + 1 (see [4], for example).
AF relay systems have been analyzed in many previous publications for various numbers of relays and various
configurations of branches and hops, where the number of
branches refers to the number of parallel paths from the
source to the destination, and the number of hops refers to
the number of serial jumps along a branch. Multi-branch twohop networks were considered in [4], [5], single-branch twohop networks were considered in [6]–[8], and multi-branch
multi-hop networks were considered in [9]–[13]. The network
considered in [4] uses 𝐾 relays in an AF system in which
the relays transmit only to the destination and not to other
relays, thereby making it a multi-branch two-hop system. It is
worth pointing out that the system in [4] uses an orthogonal
protocol (only one terminal transmits in a time slot) so that it
takes 𝐾 +1 time slots to transmit a symbol. The system in this
paper differs from the one in [4] because its relays broadcast
to other relays as well as the destination, thereby making it
a multi-branch, multi-hop system. The multi-hop system also
uses 𝐾 + 1 time slots to transmit a symbol.1 The system that
uses what Boyer et al. call multi-hop channels with diversity
in [11] is actually identical to the system considered here, as
are the systems in [12], [13].
Recently, in [5], many results for the two-hop system (called
the source-only system in [5]), which has been analyzed
extensively, were summarized. However, for the multi-hop
system (called the MRC-based system in [5]), which has
not been studied extensively, only a brief, limited discussion
was presented.2 For the multi-hop system, a very useful approximation for the instantaneous end-to-end3 signal-to-noise
ratio (SNR) was given in [11] and [12]. In [12], the system
equations and an expression for the exact instantaneous end-to1 The performance of the multi-hop system can be compared fairly to the
performance of the orthogonal two-hop system, by using the same modulation
scheme for the two systems, since they both use 𝐾 + 1 time slots to transmit
a symbol. The same comparison may not be made, however, between our
multi-hop AF system requiring 𝐾 + 1 time slots and a multiple-relay DF
system requiring two time slots which possibly uses a distributed space-time
code.
2 In [5, p. 223], during a discussion on the performance analysis of the
multi-hop system, the authors state that, “The SER analysis of the protocol
is very complicated and a close-form analysis is not tractable.”
3 The instantaneous SNR is the SNR conditioned on the event that the
random channel coefficients are given, that is, are considered to be constant.
For systems with relays, the end-to-end SNR refers to the overall receive SNR
at the destination terminal after all of the signals that it has received from the
source and the relays have been combined.
c 2010 IEEE
0090-6778/10$25.00 ⃝
2414
end SNR were developed. In [13], two relay-ordering schemes
were developed for the multi-hop system, and approximate
expressions for the symbol error rate (SER) were found for
the special case where the system uses a relay-ordering scheme
and the number of relays is restricted to 𝐾 = 2. To the best of
our knowledge, however, an approximate SER expression for
an arbitrary number of relays, 𝐾, has never been developed
for this system.
In this paper, a general approximate SER expression, which
is valid for several important modulation schemes, is found
for the multi-hop system for an arbitrary number of relays,
𝐾. This is one of two main contributions presented in this
paper. While several approximations are required to develop
this expression, it is shown to be very accurate for some
important, practically useful cases. On the other hand, one of
the approximations involves ignoring the correlation between
various SNR random variables (RVs). For the cases in which
these RVs are highly correlated, and the assumption that
the correlation can be ignored is not valid, the results for
the approximate SER expression can be poor. Therefore, the
usefulness of the approximate SER expression developed in
this paper is somewhat limited. Plots comparing the approximate analytical SER expression to the SER found from
simulations are generated for 𝐾 = 2, 3, 4, 6, and 8 relays; for
the following modulation schemes: binary phase-shift-keying
(BPSK), quadrature phase-shift keying (QPSK), 8-ary phaseshift-keying (8-PSK), and 16-ary quadrature amplitude modulation (16-QAM); and for two choices of channel variances
for the system. Most of the plots generated display very good
results for the approximate SER expression.
An important issue that arises during the development of the
SER expression for the multi-hop system is that the instantaneous SNR expressions that are considered and used to calculate the SER contain RVs of the form 𝑍 = 𝑋𝑌 /(𝑋 + 𝑌 + 𝑐),
where 𝑋 and 𝑌 are sums of exponential RVs, and 𝑐 is a
constant. In this paper, for the case where 𝑋 and 𝑌 are
sums of arbitrary numbers of independent exponential RVs
with no limitations on their means, the cumulative density
function (CDF) and the probability density function (PDF) of
𝑍, as well as the moment generating function (MGF) of 𝑍
for the special case where 𝑐 = 0, are derived. These new
results are the second main contribution of this paper. These
functions are then used in the process of deriving the SER
expression for the multi-hop system. RVs of the form given
by 𝑍 have been analyzed in many other publications including
[3], [4], [6]–[8], [14], and [15], since SNRs of this form
arise in various different relay systems. However, in all other
publications special cases for 𝑍 were considered, whereas in
this paper a more general case for 𝑍 is considered. The results
in this paper, concerning the CDF, PDF, and MGF of 𝑍, are
a generalization of the results in [4], [7], [8], [14], and [15].
The rest of the paper will be organized in the following
manner: In Section II, the model for the multi-hop system
is described and the system equations are developed. The
exact instantaneous end-to-end SNR of the system, as well as
approximate SNR expressions that are simple enough to make
the SER calculations manageable, are developed in Section III.
Expressions for the CDF, PDF, and MGF of 𝑍 are found in
Section IV. An approximate SER expression for the system is
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 2010
[t]
RELAY K
Time
Slot 2
Time
Slot 0
Time
Slot K
RELAY 2
Time
Slot 2
Time
Slot 1
Time
Slot 0
SOURCE
RELAY 1
Time Slot 0
Time
Slot 1
DESTINATION
Fig. 1. Block Diagram of the Multi-Hop System. The terminals involved,
the signals that they transmit, and the order in which they transmit them are
shown. Each point-to-point channel from terminal 𝑖 to terminal 𝑗 is a Rayleigh
fading channel with fading coefficient ℎ𝑖,𝑗 and additive Gaussian noise 𝑛𝑖,𝑗 .
developed in Section V. Numerical results, with plots showing
the SER found from the approximate SER expression and from
Monte Carlo simulations, for various system configurations,
are presented in Section VI. Concluding remarks are given in
Section VII.
II. S YSTEM M ODEL
The block diagram of the system is shown in Fig. 1.
A discrete-time, baseband equivalent model will be used to
describe this system. The source transmits symbols to the
destination with the help of the 𝐾 relays, using a timeorthogonal AF protocol over 𝐾 + 1 time slots (one for
the source and one for each of the relays). All channels
are Rayleigh fading channels with additive white Gaussian
noise (AWGN). The channel coefficient for the channel from
transmitting terminal 𝑖 to receiving terminal 𝑗 is denoted
by ℎ𝑖,𝑗 where 𝑖 = 𝑠 for the source; 𝑖 = 1, 2, . . . , 𝐾 for
relay 𝑖; 𝑗 = 1, 2, . . . , 𝐾 for relay 𝑗; and 𝑗 = 𝑑 for the
destination. Channel coefficient ℎ𝑖,𝑗 is a zero-mean complex
Gaussian RV with variance Ω𝑖,𝑗 /2 per dimension, and this
distribution is denoted by ℎ𝑖,𝑗 ∼ 𝒞𝒩 (0, Ω𝑖,𝑗 ). Since ℎ𝑖,𝑗 is
complex Gaussian with variance Ω𝑖,𝑗 , it follows that ∣ℎ𝑖,𝑗 ∣2
is an exponential RV with a mean of Ω𝑖,𝑗 . The channels
are considered to be slow fading, so that the coefficients are
constant over the time duration of one symbol, that is, over
one time slot. The AWGN component at terminal 𝑗, added
to the signal transmitted from terminal 𝑖, is denoted by 𝑛𝑖,𝑗 .
It is also a zero-mean complex Gaussian RV, its variance
per dimension is 𝑁𝑖,𝑗 /2, and its distribution is denoted by
𝒞𝒩 (0, 𝑁𝑖,𝑗 ). The channel coefficients and AWGN signals are
all mutually independent.
√
In time slot 0, the source broadcasts the symbol 𝜀𝑜 𝑥,
where 𝜀𝑜 is the average source power, 𝑥 ∈ 𝑆, and 𝑆 is
the set of possible signals for the source to transmit. The
relays (𝑗 = 1, 2, . . . , 𝐾) and destination (𝑗 = 𝑑) receive the
√
signals 𝑦𝑠,𝑗 = 𝜀𝑜 ℎ𝑠,𝑗 𝑥 + 𝑛𝑠,𝑗 for 𝑗 = 1, 2, . . . , 𝐾, 𝑑. It is
√
convenient to define the parameters, 𝛽𝑠,𝑗 = 𝜀𝑜 ℎ𝑠,𝑗 /𝑁𝑠,𝑗 for
𝑗 = 1, 2, . . . , 𝐾, 𝑑. Then the received signals can be expressed
as 𝑦𝑠,𝑗 = 𝑁𝑠,𝑗 𝛽𝑠,𝑗 𝑥 + 𝑛𝑠,𝑗 .
CONNE et al.: SER ANALYSIS AND PDF DERIVATION FOR MULTI-HOP AMPLIFY-AND-FORWARD RELAY SYSTEMS
In time slot 1, relay 1 broadcasts 𝑥1 = 𝛼1 𝑦𝑠,1 to terminals
𝑗 = 2, . . . , 𝐾, 𝑑, where 𝛼1 is chosen so that the average
power used by relay 1 is equal to 𝜀1 , a predetermined desired
value of[ power.
transmit
power of relay 1 is
] The average
]
[
𝜀1 = E ∣𝑥1 ∣2 = 𝛼21 E ∣𝑦𝑠,1 ∣2 = 𝛼21 (∣ℎ𝑠,1 ∣2 𝜀𝑜 + 𝑁𝑠,1 ) =
2
𝛼21 (𝑁𝑠,1
∣𝛽𝑠,1 ∣2 + 𝑁𝑠,1 ), where E [⋅] denotes an expectation.
The amplifying gain, 𝛼1 , is given by
√
√
𝜀1
𝜀1
𝛼1 =
=
.
(1)
2 ∣𝛽
2
∣ℎ𝑠,1 ∣2 𝜀𝑜 + 𝑁𝑠,1
𝑁𝑠,1
𝑠,1 ∣ + 𝑁𝑠,1
The relays and destination receive 𝑦1,𝑗 = ℎ1,𝑗 𝑥1 + 𝑛1,𝑗 =
𝛼1 𝑁𝑠,1 𝛽𝑠,1 ℎ1,𝑗 𝑥 + 𝜂1,𝑗 for 𝑗 = 2, . . . , 𝐾, 𝑑, where the total
received noise signal at terminal 𝑗 is 𝜂1,𝑗 = ℎ1,𝑗 𝛼1 𝑛𝑠,1 +
𝑛1,𝑗 . It will again be convenient to define the parameters,
𝛽1,𝑗 = 𝛼1 𝑁𝑠,1 𝛽𝑠,1 ℎ1,𝑗 /𝒩1,𝑗 for 𝑗 = 2, . . . , 𝐾, 𝑑, where
𝒩1,𝑗 = E[∣𝜂1,𝑗 ∣2 ] = 𝛼21 ∣ℎ1,𝑗 ∣2 𝑁𝑠,1 +𝑁1,𝑗 is the variance of the
noise signal 𝜂1,𝑗 . The received signals can now be expressed
as 𝑦1,𝑗 = 𝒩1,𝑗 𝛽1,𝑗 𝑥 + 𝜂1,𝑗 .
In time slot 𝑗, for 𝑗 = 2, . . . , 𝐾, relay 𝑗 broadcasts to
terminals 𝑘 = 𝑗 + 1, . . . , 𝐾 + 1, where the destination is
represented here by 𝑘 = 𝐾 + 1, instead of 𝑘 = 𝑑, for convenience. Before transmission, relay 𝑗 first combines the signals
that it has received using MRC.4,5 At time slot 𝑗, relay 𝑗 has
received 𝑦𝑠,𝑗 = 𝑁𝑠,𝑗 𝛽𝑠,𝑗 𝑥 + 𝑛𝑠,𝑗 and 𝑦𝑖,𝑗 = 𝒩𝑖,𝑗 𝛽𝑖,𝑗 𝑥 + 𝜂𝑖,𝑗
for 𝑖 = ∑
1, . . . , 𝑗 − 1. The
signal obtained after combining
∑𝑗−1
∗
𝜉
𝑦
=
is 𝑦𝑗 = 𝑗−1
𝑖=0 𝑖,𝑗 𝑖,𝑗
𝑖=0 𝛽𝑖,𝑗 𝑦𝑖,𝑗 , where the source is
represented here by 𝑖 = 0, instead of 𝑖 = 𝑠, for convenience.
The MRC coefficient 𝜉𝑖,𝑗 is the ratio of the conjugate of
the signal component coefficient to the noise power, that is,
∗
∗
∗
∗
/𝑁𝑠,𝑗 = 𝛽𝑠,𝑗
, and 𝜉𝑖,𝑗 = 𝒩𝑖,𝑗 𝛽𝑖,𝑗
/𝒩𝑖,𝑗 = 𝛽𝑖,𝑗
.
𝜉𝑠,𝑗 = 𝑁𝑠,𝑗 𝛽𝑠,𝑗
and
𝑦
into
the
equation
Substituting the equations
for
𝑦
𝑠,𝑗
𝑖,𝑗
)
(
∑𝑗−1
for 𝑦𝑗 yields 𝑦𝑗 = 𝑁𝑠,𝑗 ∣𝛽𝑠,𝑗 ∣2 + 𝑖=1 𝒩𝑖,𝑗 ∣𝛽𝑖,𝑗 ∣2 𝑥 +
(
)
∑𝑗−1 ∗
∗
𝛽𝑠,𝑗
𝑛𝑠,𝑗 + 𝑖=1 𝛽𝑖,𝑗
𝜂𝑖,𝑗 . Then relay 𝑗 broadcasts 𝑥𝑗 =
𝛼𝑗 𝑦𝑗 to terminals 𝑘 = 𝑗 + 1, . . . , 𝐾 + 1, where 𝛼𝑗 is chosen
so that the average power used by relay 𝑗 is 𝜀𝑗 . The transmit
power of relay 𝑗 is
{(
)2
𝑗−1
∑
[
]
2
2
2
2
𝜀𝑗 = E ∣𝑥𝑗 ∣ = 𝛼𝑗
𝒩𝑖,𝑗 ∣𝛽𝑖,𝑗 ∣
𝑁𝑠,𝑗 ∣𝛽𝑠,𝑗 ∣ +
𝑖=1
⎡
2 ⎤ }
𝑗−1
∑
∗
∗
+ E ⎣𝛽𝑠,𝑗
𝑛𝑠,𝑗 +
𝛽𝑖,𝑗
𝜂𝑖,𝑗 ⎦ .
𝑖=1
√
(2)
𝜀𝑗
,
𝐴2𝑗 +𝐴𝑗
The amplifying gain used for relay 𝑗 is 𝛼𝑗 =
where
∑
𝑗−1
𝐴𝑗 = 𝑁𝑠,𝑗 ∣𝛽𝑠,𝑗 ∣2 + 𝑖=1 𝒩𝑖,𝑗 ∣𝛽𝑖,𝑗 ∣2 . The received signals at
terminals 𝑘 = 𝑗 + 1, . . . , 𝐾 + 1 are 𝑦𝑗,𝑘 = ℎ𝑗,𝑘 𝑥𝑗 + 𝑛𝑗,𝑘 =
the noise) signals are given by 𝜂𝑗,𝑘 =
ℎ𝑗,𝑘 𝛼𝑗 𝐴
(𝑗 𝑥 + 𝜂𝑗,𝑘 , where
∑𝑗−1 ∗
∗
ℎ𝑗,𝑘 𝛼𝑗 𝛽𝑠,𝑗
𝑛𝑠,𝑗 + 𝑖=1 𝛽𝑖,𝑗
𝜂𝑖,𝑗 + 𝑛𝑗,𝑘 . By letting 𝛽𝑗,𝑘 =
𝛼𝑗 𝐴𝑗 ℎ𝑗,𝑘 /𝒩𝑗,𝑘 , where 𝒩𝑗,𝑘 = E[∣𝜂𝑗,𝑘 ∣2 ], the received signals
can then be expressed as 𝑦𝑗,𝑘 = 𝒩𝑗,𝑘 𝛽𝑗,𝑘 𝑥 + 𝜂𝑗,𝑘 .
4 Since the relays and destination use MRC, terminal 𝑗 requires knowledge
of the channel coefficients, ℎ𝑗1 ,𝑗2 , for 0 ≤ 𝑗1 < 𝑗2 ≤ 𝑗 where
𝑗 = 1, . . . , 𝐾 + 1. Therefore, more thorough and sophisticated channel
training is required for the multi-hop system as compared to the two-hop
system.
5 Note that MRC may not be the optimal combining method since the signals
received by terminal 𝑗 are not independent. A search for the optimal combiner
is beyond the scope of this paper.
2415
III. E XACT AND A PPROXIMATE I NSTANTANEOUS
E ND - TO -E ND SNR E XPRESSIONS
Now that the transmitted signals, noise signals, and received signals have been determined for all terminals in
the system, the instantaneous end-to-end SNR of the system can be found. After time slot 𝐾, the destination has
received the following signals: 𝑦𝑠,𝑑 = 𝑁𝑠,𝑑 𝛽𝑠,𝑑 𝑥 + 𝑛𝑠,𝑑
and 𝑦𝑚,𝑑 = 𝒩𝑚,𝑑 𝛽𝑚,𝑑 𝑥 + 𝜂𝑚,𝑑 for 𝑚 = 1, 2, . . . , 𝐾.
The destination combines
the signals using MRC to obtain
∑
∗
∗
𝑦𝑑 = 𝛽𝑠,𝑑
𝑦𝑠,𝑑 + 𝐾
𝑚=1 𝛽𝑚,𝑑 𝑦𝑚,𝑑 = 𝐴𝑑 𝑥 + 𝜂𝑑 , where
∑
𝐾
𝐴𝑑 = 𝑁𝑠,𝑑 ∣𝛽𝑠,𝑑 ∣2 + 𝑚=1 𝒩𝑚,𝑑 ∣𝛽𝑚,𝑑 ∣2 and the overall noise
∑𝐾
∗
∗
𝑛𝑠,𝑑 + 𝑚=1 𝛽𝑚,𝑑
𝜂𝑚,𝑑 . The essignal component is 𝜂𝑑 = 𝛽𝑠,𝑑
timate, 𝑥
ˆ, of the transmitted signal is obtained using maximum
likelihood (ML) detection. That is, 𝑥
ˆ = arg min ∣𝑦𝑑 − 𝐴𝑑 𝑠∣.
𝑠∈𝑆
Theorem 1 ([12]): The exact instantaneous end-to-end
SNR, Γ, of the multi-hop AF relay system, first given in [12],
is given by
[
]2
∑𝐾
𝑁𝑠,𝑑 ∣𝛽𝑠,𝑑 ∣2 + 𝑚=1 𝒩𝑚,𝑑 ∣𝛽𝑚,𝑑 ∣2
𝐴2𝑑
Γ=
=
.
(3)
𝒩𝑑
𝒩𝑑
The variance, 𝒩𝑚,𝑘 , of the total noise signal at terminal 𝑘,
due to transmission from terminal 𝑚, is given by
𝒩𝑚,𝑘 = E[∣𝜂𝑚,𝑘 ∣2 ] = 𝑁𝑚,𝑘 +
𝑗−1
𝑚 ∑
∑
∣𝜆(𝑖, 𝑗, 𝑚, 𝑘)∣2 𝑁𝑖,𝑗
𝑗=1 𝑖=0
(4)
for 𝑚 = 1, . . . , 𝐾, and 𝑘 = 𝑚+1, . . . , 𝐾 +1. The 𝜆(𝑖, 𝑗, 𝑚, 𝑘)
parameters are given by
{∑
𝑚−1 𝑘
𝑙=𝑗 𝜇𝑙,𝑚 𝜆(𝑖, 𝑗, 𝑙, 𝑚) for 1 ≤ 𝑗 < 𝑚,
𝜆(𝑖, 𝑗, 𝑚, 𝑘) =
𝑘
𝜇𝑖,𝑚
for 𝑗 = 𝑚,
(5)
where 𝑖 = 0, . . . , 𝑗 − 1. The 𝜇𝑘𝑖,𝑚 parameters are defined as
∗
for 𝑚 =
𝜇𝑘0,1 = ℎ1,𝑘 𝛼1 for 𝑚 = 1; and 𝜇𝑘𝑖,𝑚 = ℎ𝑚,𝑘 𝛼𝑚 𝛽𝑖,𝑚
2, . . . , 𝐾; where 𝑘 = 𝑚 + 1, . . . , 𝐾 + 1 and 𝑖 = 0, . . . , 𝑚 − 1.
As before, it is convenient here to let 𝑖 = 0 replace 𝑖 = 𝑠 and
𝑘 = 𝐾 + 1 replace 𝑘 = 𝑑.
Also, the variance, 𝒩𝑑 , of the overall noise signal at the
destination is given by
𝒩𝑑 = E[∣𝜂𝑑 ∣2 ] = ∣𝛽𝑠,𝑑 ∣2 𝑁𝑠,𝑑 +
𝐾
∑
∣𝛽𝑚,𝑑 ∣2 𝑁𝑚,𝑑
𝑚=1
+
𝑗−1
𝐾 ∑
𝑚 ∑
∑
∣𝛽𝑚,𝑑 ∣2 ∣𝜆(𝑖, 𝑗, 𝑚, 𝑑)∣2 𝑁𝑖,𝑗 .
(6)
𝑚=1 𝑗=1 𝑖=0
□
Proof: See Appendix A.6
It is worth mentioning here that by letting ℎ𝑖,𝑗 = 0 for
1 ≤ 𝑖 < 𝑗 ≤ 𝐾, the system reduces to the multi-branch
two-hop system that was analyzed in [4]. It was shown in
[12] that by setting ℎ𝑖,𝑗 = 0 for 1 ≤ 𝑖 < 𝑗 ≤ 𝐾 in the
system equations for the multi-hop system, the end-to-end
SNR expression given in (3) reduces to the end-to-end SNR
expression given in previous publications for the multi-branch
two-hop system (see [4], [7], [9]). Therefore, the result for the
6 This
proof has not previously been published.
2416
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 2010
SNR of the multi-hop system given in (3) is a generalization
of previously published results for the two-hop system. See
Section III-A in [12] for details.
A useful approximate expression for Γ of (3) can be found
by ignoring the dependency among the 𝜂𝑚,𝑑 terms found
in the expression for 𝜂𝑑 . Then the variance of the overall
noise signal at the destination becomes 𝒩𝑑 ≈ 𝑁𝑠,𝑑 ∣𝛽𝑠,𝑑 ∣2 +
∑𝐾
2
𝑚=1 𝒩𝑚,𝑑 ∣𝛽𝑚,𝑑 ∣ , which is equal to 𝐴𝑑 . Substituting this
expression into (3) leads to the following approximate instantaneous end-to-end SNR expression: Γ ≈ 𝑁𝑠,𝑑 ∣𝛽𝑠,𝑑 ∣2 +
∑𝐾
2
𝑚=1 𝒩𝑚,𝑑 ∣𝛽𝑚,𝑑 ∣ . In the following corollary, an approximate end-to-end SNR expression is given in a more useful
recursive formula.
Corollary 1 ([12]): An approximate expression, Γ𝑎𝑝𝑝 , for
the instantaneous end-to-end SNR is given as
and 𝛾𝑠,𝑖,𝑚 = 𝛾𝑠,𝑚 for 𝑖 = 𝑚.
Γ𝑎𝑝𝑝 = 𝛾𝑠,𝑑 +
𝐾
∑
𝐴𝑚 𝛾𝑚,𝑑
,
𝐴 + 𝛾𝑚,𝑑 + 1
𝑚=1 𝑚
(7)
where 𝛾𝑚,𝑘 is the instantaneous SNR for the channel from
terminal 𝑚 to terminal 𝑘 and is defined as
{
𝜀𝑚 ∣ℎ𝑚,𝑘 ∣2
𝑚 = 0, 1 . . . , 𝐾;
𝛾𝑚,𝑘 =
for
(8)
𝑁𝑚,𝑘
𝑘 = 𝑚 + 1, . . . , 𝐾 + 1.
Also, 𝐴𝑚 can be written as
𝐴𝑚 = 𝛾𝑠,𝑚 +
𝑚−1
∑
𝑖=1
𝐴𝑖 𝛾𝑖,𝑚
𝐴𝑖 + 𝛾𝑖,𝑚 + 1
for 𝑚 ≥ 2,
(9)
and 𝐴1 is given by 𝐴1 = 𝛾𝑠,1 .
Proof: The way that Γ𝑎𝑝𝑝 is derived is presented in
□
Appendix B.7
Note that, since ∣ℎ𝑚,𝑘 ∣2 is an exponential RV with mean equal
to Ω𝑚,𝑘 , it follows that 𝛾𝑚,𝑘 is an exponential RV with mean
equal to 𝛾 𝑚,𝑘 = 𝜀𝑚 Ω𝑚,𝑘 /𝑁𝑚,𝑘 . It is also interesting to note
that, although a different approach was taken in [11] towards
the analysis of the same multi-hop system, (7) is identical to a
recursive equation given there for the instantaneous end-to-end
SNR [11, eq. (13)].
A further approximation for the instantaneous end-to-end
SNR, which will be in a form that can be used to derive
an approximate SER expression for the multi-hop system, is
presented next. We introduce such approximation in order to
make the analysis tractable; without such approximation, we
conjecture that the analysis is not tractable.
Theorem 2: Another approximation, Γ′ , for the instantaneous end-to-end SNR is given as follows:
Γ𝑎𝑝𝑝 ≈ Γ′ = 𝛾𝑠,𝑑 +
𝐾
∑
𝛾𝑠,𝑚,𝑑 ,
(10)
𝑚=1
where
𝛾𝑠,𝑟𝑚 𝛾𝑚,𝑑
for 𝑚 = 1, 2, . . . , 𝐾,
𝛾𝑠,𝑟𝑚 + 𝛾𝑚,𝑑 + 1
𝑚
∑
=
𝛾𝑠,𝑖,𝑚 for 𝑚 = 1, 2, . . . , 𝐾,
𝛾𝑠,𝑚,𝑑 =
𝛾𝑠,𝑟𝑚
(11)
(12)
𝑖=1
𝛾𝑠,𝑖,𝑚 = min(𝛾𝑠,𝑖 , 𝛾𝑖,𝑖+1 , . . . , 𝛾𝑚−1,𝑚 ) for 𝑖 = 1, . . . , 𝑚 − 1,
(13)
7 The
explanation given in Appendix B concerning Γ𝑎𝑝𝑝 is more detailed
than the one given in [12] concerning the same approximate SNR expression.
8
□
By comparing (10) and (11) with (7), it can be seen that
𝛾𝑠,𝑟𝑚 in (11) is an approximation for 𝐴𝑚 in (7). Also note
that 𝛾𝑠,𝑚,𝑑 is in the form of a standard AF SNR, as seen in (7)
and in [3, eqs. (12), (13)] for a two-hop system. In (12), the
multi-hop source-to-relay-𝑚 SNR, 𝛾𝑠,𝑟𝑚 , is represented by a
sum of 𝑚 SNRs, one (𝛾𝑠,𝑚,𝑚 = 𝛾𝑠,𝑚 ) for the signal received
directly from the source, and one (𝛾𝑠,𝑖,𝑚 ) for the signal that
travels from the source to relay 𝑖 and then gets forwarded to
relay 𝑚, for each relay 𝑖 = 1, . . . , 𝑚 − 1. This expression was
derived intuitively and is in a form that is expected since relay
𝑚 receives 𝑚 signals, one from the source and one from each
of 𝑚 − 1 relays, and then combines them using MRC.
In (13), 𝛾𝑠,𝑖,𝑚 is an expression that was also derived
intuitively. It represents an SNR at relay 𝑚 due to a signal
that travels over the path from the source to relay 𝑖 to relay
𝑖 + 1 . . . to relay 𝑚 − 1 to relay 𝑚. It is comparable to
using min(𝛾𝑠,𝑟 , 𝛾𝑟,𝑑 ) as an approximation for the SNR of
a two-hop AF path (which has been used in many previous
publications, as in [4], for example), where 𝛾𝑠,𝑟 is the SNR
for the source-to-relay channel and 𝛾𝑟,𝑑 is the SNR for the
relay-to-destination channel. For the case of (13), 𝛾𝑠,𝑖,𝑚 is the
minimum of 𝑚− 𝑖 + 1 single-channel SNRs since it represents
the SNR of an AF path that consists of 𝑚 − 𝑖 + 1 hops.
As mentioned, the approximation, Γ′ , of the instantaneous
end-to-end SNR will be used to derive an approximate SER
expression for the multi-hop system. Before deriving the SER
expression, the CDF, PDF, and MGF of an RV, which is of the
same form as that given by 𝛾𝑠,𝑚,𝑑 in (11), will be derived in
the next section. Those results will then be used in Section V,
in which the desired SER expression will be found.
IV. D EVELOPMENT OF CDF, PDF, AND MGF OF 𝑍
In this section, the CDF and PDF will be developed for RVs
that have the following general form:
𝑋𝑌
,
(14)
𝑋 +𝑌 +𝑐
∑𝑚
∑𝑛
where 𝑋 = 𝑚′ =1 𝑋𝑚′ ; 𝑌 = 𝑛′ =1 𝑌𝑛′ ; 𝑐 is a constant;
and the terms, 𝑋𝑚′ and 𝑌𝑛′ , are all independent, exponential
RVs. The RVs, 𝑋𝑚′ , have mean 𝑥𝑚′ for 𝑚′ = 1, . . . , 𝑚; and
the RVs, 𝑌𝑛′ , have mean 𝑦𝑛′ for 𝑛′ = 1, . . . , 𝑛. The MGF of
𝑍 will also be found for the special case of 𝑐 = 0. It will
be shown in Section V that the terms, 𝛾𝑠,𝑖,𝑚 , of (13) are also
exponential RVs. Therefore, the terms, 𝛾𝑠,𝑚,𝑑 , of (11) are all of
the form given by (14) with 𝑍 = 𝛾𝑠,𝑚,𝑑 ; 𝑋 = 𝛾𝑠,𝑟𝑚 ; 𝑋𝑚′ =
𝛾𝑠,𝑖,𝑚 for 𝑖 = 𝑚′ = 1, . . . , 𝑚; 𝑛 = 1; 𝑌 = 𝑌1 = 𝛾𝑚,𝑑 ; and
𝑐 = 1 for 𝑚 = 1, 2, . . . , 𝐾. This provides the motivation for
analyzing the RV, 𝑍, in detail. The CDF, PDF, and MGF of 𝑍
developed in this section will be applied to the terms, 𝛾𝑠,𝑚,𝑑 ,
during the process of finding the SER of the multi-hop system
with instantaneous SNR approximated by Γ′ .
In [14], the CDF and PDF were found for an RV with the
same form as 𝑍, for the case where the 𝑥𝑚′ are all identical
and the 𝑦 𝑛′ are also all identical (but 𝑥𝑚′ ∕= 𝑦 𝑛′ in general),
𝑍=
8 Note that the expression, Γ′ , for the approximate SNR of the multi-hop
system reduces to the expression for the exact instantaneous end-to-end SNR
of the two-hop system for that case, in which 𝛾𝑖,𝑗 = 0 for 1 ≤ 𝑖 < 𝑗 ≤ 𝐾.
CONNE et al.: SER ANALYSIS AND PDF DERIVATION FOR MULTI-HOP AMPLIFY-AND-FORWARD RELAY SYSTEMS
which was appropriate for the conditions and the system analyzed there. For the multi-hop system discussed in this paper,
however, the means of the exponential RVs representing the
individual channel SNRs between two terminals are dissimilar
in general (although some may be similar), as are the means
of the RVs, 𝛾𝑠,𝑖,𝑚 . For that reason, in this section, the CDF,
PDF, and MGF of 𝑍 will be developed for the case in which
the means, 𝑥𝑚′ and 𝑦 𝑛′ , of the exponential RVs are dissimilar
in general.
A. The PDFs for 𝑋 and 𝑌
The first step in determining the CDF of 𝑍 is to find the
PDFs of 𝑋 and 𝑌 . Let 𝐽 be the number of distinct means in
the group of the 𝑚 individual RVs found in the summation for
𝑋. Denote these means as 𝑥1 , . . . , 𝑥𝑗 , . . . , 𝑥𝐽 . Let 𝑟𝑗 be the
number of RVs that have mean 𝑥𝑗 , for 𝑗 = 1, . . . , 𝐽. The total
∑𝐽
number of RVs is given by 𝑚 = 𝑗=1 𝑟𝑗 . Similarly, let 𝑄 be
the number of distinct means in the group of the 𝑛 individual
RVs found in the summation for 𝑌 . Denote these means as
𝑦 1 , . . . , 𝑦𝑞 , . . . , 𝑦 𝑄 . Let 𝑡𝑞 be the number of RVs that have
mean 𝑦𝑞 , for 𝑞 = 1, . . . , 𝑄. The total number of RVs is given
∑𝑄
by 𝑛 = 𝑞=1 𝑡𝑞 .
Lemma 1: The PDFs of 𝑋 and 𝑌 are
𝑓𝑋 (𝑥)
=
𝑟𝑗
𝐽 ∑
∑
𝑘𝑥,𝑖,𝑗 𝑖−1 −𝑥/𝑥𝑗
𝑥 𝑒
,
(𝑖 − 1)!
𝑗=1 𝑖=1
𝑡𝑞
𝑄 ∑
∑
𝑘𝑦,𝑝,𝑞 𝑝−1 −𝑦/𝑦𝑞
𝑦
𝑒
,
𝑓𝑌 (𝑦) =
(𝑝
− 1)!
𝑞=1 𝑝=1
2417
Theorem 3: The CDF, 𝐹𝑧 (𝛾), for the most general form of
𝑍, is given by
𝑟𝑗 𝑖−1 𝑄 𝑡𝑞 𝑝−1 𝑘 (
𝐽 ∑
∑
∑ ∑ ∑ ∑ ∑ 𝑝 − 1)( 𝑘 )
𝐹𝑧 (𝛾) = 1 − 2
𝑗′
𝑖′
′
𝑞=1 𝑝=1 ′
𝑗=1 𝑖=1
𝑗 =0 𝑖 =0
𝑘=0
𝑘𝑥,𝑖,𝑗 𝑘𝑦,𝑝,𝑞 [𝑖−(𝑖′ +𝑗 ′ +𝑘+1)/2] 𝜈/2 [𝑝+(𝑖′ −𝑗 ′ +𝑘−1)/2]
𝑥
𝑦𝑞 𝛾
⋅
𝑘!(𝑝 − 1)! 𝑗
( √
)
(
)
𝛾(𝛾 + 𝑐)
𝜎𝑗,𝑞 𝛾
(−𝑖′ +𝑗 ′ +𝑘+1)/2
⋅ (𝛾 + 𝑐)
exp −
,
𝐾𝜈 2
𝜌𝑗,𝑞
𝜌𝑗,𝑞
(19)
where 𝜎𝑗,𝑞 = 𝑥𝑗 + 𝑦 𝑞 , 𝜌𝑗,𝑞 = 𝑥𝑗 ⋅ 𝑦 𝑞 , 𝜈 = 𝑖′ + 𝑗 ′ − 𝑘 + 1, and
𝐾𝜈 (⋅) is the modified Bessel function of the second kind and
order 𝜈. The PDF of 𝑍, 𝑓𝑧 (𝛾), is given by
𝑓𝑧 (𝛾) = 2
𝑟𝑗 𝑖−1 𝑄 𝑡𝑞 𝑝−1 𝑘
𝐽 ∑
∑
∑∑∑ ∑ ∑
𝑗=1 𝑖=1 𝑘=0 𝑞=1 𝑝=1 𝑗 ′ =0 𝑖′ =0
[𝑖−(𝑖 +𝑗 ′ +𝑘+1)/2]
⋅𝑥𝑗
′
(
𝜎𝑗,𝑞 𝛾
⋅ exp −
𝜌𝑗,𝑞
′
𝑦 𝜈/2 𝛾 [𝑝+(𝑖 −𝑗
′
(
𝑝−1
𝑗′
)( )
𝑘 𝑘𝑥,𝑖,𝑗 𝑘𝑦,𝑝,𝑞
𝑖′ 𝑘!(𝑝 − 1)!
+𝑘−3)/2]
′
′
(𝛾 + 𝑐)(−𝑖 +𝑗 +𝑘−1)/2
( √
)
𝛾(𝛾 + 𝑐)
𝛾(𝛾 + 𝑐)
(2𝛾 + 𝑐)𝐾𝜈−1 2
𝜌𝑗,𝑞
𝜌𝑗,𝑞
( √
)]
𝛾(𝛾 + 𝑐)
,
(20)
+ 𝑔(𝛾)𝐾𝜈 2
𝜌𝑗,𝑞
𝑞
)[√
𝜎
(15)
where the expressions for the constants, 𝑘𝑥,𝑖,𝑗 and 𝑘𝑦,𝑝,𝑞 , are
given by
]
1
d𝑛 [
𝑀𝑋 (𝑠) ⋅ (𝑐𝑥,𝑗 − 𝑠)𝑟𝑗 𝑠=𝑐𝑥,𝑗
𝑛
𝑛
(−1) 𝑛! d𝑠
for 𝑖 = 1, . . . , 𝑟𝑗 ; 𝑗 = 1, . . . , 𝐽; 𝑛 = 𝑟𝑗 − 𝑖, (16)
]
d𝑛 [
1
𝑘𝑦,𝑝,𝑞 =
𝑀𝑌 (𝑠) ⋅ (𝑐𝑦,𝑞 − 𝑠)𝑡𝑞 𝑠=𝑐𝑦,𝑞
𝑛
𝑛
(−1) 𝑛! d𝑠
for 𝑝 = 1, . . . , 𝑡𝑞 ; 𝑞 = 1, . . . , 𝑄; 𝑛 = 𝑡𝑞 − 𝑝,
(17)
𝑘𝑥,𝑖,𝑗 =
where 𝑀𝑋 (𝑠) and 𝑀𝑌 (𝑠) are the MGFs of 𝑋 and 𝑌 ,
respectively, and they are given by
𝑗,𝑞
𝛾(𝛾 +
where 𝑔(𝛾) = (𝑖′ −𝑘)𝛾 + (−𝑝+𝑗 ′ −𝑘+1)(𝛾 +𝑐) + 𝜌𝑗,𝑞
𝑐).
Proof: See Appendix C.
□
Corollary 2: The CDF of 𝑍, 𝐹𝑧 (𝛾), and the PDF of 𝑍,
𝑓𝑧 (𝛾), for the special case where all of the means in 𝑋 are
dissimilar and all of the means in 𝑌 are also dissimilar, are
found as follows: Set 𝐽 = 𝑚; 𝑟𝑗 = 1 for 𝑗 = 1, . . . , 𝑚;
𝑄 = 𝑛; and 𝑡𝑞 = 1 for 𝑞 = 1, . . . , 𝑛 into (19) and (20),
respectively.
□
The special case considered in [14], where all of the
exponential RVs in the summation for 𝑋 have the same mean,
𝑥, and all of the exponential RVs in the summation for 𝑌 have
the same mean, 𝑦, is represented by letting 𝐽 = 1, 𝑟1 = 𝑚,
𝑄 = 1, and 𝑡1 = 𝑛. In this case, 𝑋 and 𝑌 become Erlang
RVs with the following distributions [16]:
𝑓𝑋 (𝑥)
𝑓𝑌 (𝑦)
1/𝑥𝑚 𝑚−1 −𝑥/𝑥
𝑒
,
𝑥
(𝑚 − 1)!
𝑛
1/𝑦
𝑦 𝑛−1 𝑒−𝑦/𝑦 .
=
(𝑛 − 1)!
=
(21)
𝑡𝑞
𝑄 ∑
By comparing equations (15) with equations (21), it can be
∑
𝑘𝑥,𝑖,𝑗
𝑘𝑦,𝑝,𝑞
,
𝑀
(𝑠)
=
𝑌
seen that, in this case, 𝑘𝑥,𝑖,1 = 0 for 𝑖 ∕= 𝑚, 𝑘𝑥,𝑚,1 = 𝑥 −𝑚 ,
(𝑐𝑥,𝑗 − 𝑠)𝑖
(𝑐𝑦,𝑞 − 𝑠)𝑝
𝑞=1 𝑝=1
𝑗=1 𝑖=1
𝑘𝑦,𝑝,1 = 0 for 𝑝 ∕= 𝑛, and 𝑘𝑦,𝑛,1 = 𝑦 −𝑛 .
(18)
Corollary 3: The CDF of 𝑍, 𝐹𝑧 (𝛾), and the PDF of 𝑍,
𝑓𝑧 (𝛾), for the special case where 𝑋 is a sum of exponential
and where 𝑐𝑥,𝑗 = 1/𝑥𝑗 and 𝑐𝑦,𝑞 = 1/𝑦𝑞 .
RVs with identical means and 𝑌 is also a sum of exponential
Proof: See [16], [17].
□ RVs with identical means, are found as follows: Set 𝐽 = 1,
𝑟1 = 𝑚, 𝑥1 = 𝑥, 𝑄 = 1, 𝑡1 = 𝑛, 𝑦 1 = 𝑦, 𝑘𝑥,𝑖,1 = 0 for 𝑖 ∕= 𝑚,
𝑘𝑥,𝑚,1 = 𝑥 −𝑚 , 𝑘𝑦,𝑝,1 = 0 for 𝑝 ∕= 𝑛, and 𝑘𝑦,𝑛,1 = 𝑦 −𝑛 into
B. The CDF and PDF of 𝑍
(19) and (20), respectively.
□
The CDF and PDF found by using Corollary 3 are identical
With the PDFs for 𝑋 and 𝑌 , the CDF of 𝑍 can be found.
The PDF of 𝑍 can always be found by differentiating the CDF to the CDF and PDF, respectively, derived in [14, eqs. (11)
and (12)] for the same RV.
of 𝑍.
𝑀𝑋 (𝑠) =
𝑟𝑗
𝐽 ∑
∑
2418
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 2010
Corollary 4: The CDF of 𝑍, 𝐹𝑧 (𝛾), and the PDF of 𝑍,
𝑓𝑧 (𝛾), for the special case where 𝑋 and 𝑌 are both single
exponential RVs with means equal to 𝑥 and 𝑦, respectively,
are found as follows: Set 𝐽 = 𝑚 = 1, 𝑟1 = 1, 𝑄 = 𝑛 = 1,
and 𝑡1 = 1 into (19) and (20), respectively, and note that
𝑘𝑥,1,1 = 𝑥 −1 and 𝑘𝑦,1,1 = 𝑦 −1 for this special case.
□
For the special case where 𝑐 = 0, the CDF and PDF
found by using Corollary 4 are identical to the CDF and PDF,
respectively, derived in [4, eqs. (8) and (12)] for the same RV.
𝛾2
𝛾 𝑚𝑖𝑛,2 = 𝛾𝛾 1+𝛾
. It is easy to show that for 𝑃 independent
1
2
exponential RVs, 𝛾1 , 𝛾2 , . . . , 𝛾𝑃 , the minimum of these RVs,
𝛾𝑚𝑖𝑛,𝑃 = min(𝛾1 , 𝛾2 , . . . , 𝛾𝑃 ), is also an exponential RV and
its mean is given by
∏𝑃
𝛾 𝑚𝑖𝑛,𝑃 −1 𝛾 𝑃
𝑝=1 𝛾 𝑝
𝛾 𝑚𝑖𝑛,𝑃 =
= ∑𝑃 ∏ 𝑃
for 𝑃 ≥ 2 ,
𝛾 𝑚𝑖𝑛,𝑃 −1 + 𝛾 𝑃
𝑝=1, 𝛾 𝑝
𝑞=1
C. The MGF of 𝑍
Theorem 4: The MGF of 𝑍, 𝑀𝑧 (𝑠), for the special case
where 𝑐 = 0, is given by
𝑟𝑗 𝑖−1 𝑄 𝑡𝑞 𝑝−1 𝑘 (
𝐽 ∑
∑
∑ ∑ ∑ ∑ ∑ 𝑝 − 1)( 𝑘 )
𝑀𝑧 (𝑠) = 2
𝑗′
𝑖′
′
𝑞=1 𝑝=1 ′
𝑗=1 𝑖=1
𝑘=0
𝑗 =0 𝑖 =0
𝑘𝑥,𝑖,𝑗 𝑘𝑦,𝑝,𝑞 [𝑖−(𝑖′ +𝑗 ′ +𝑘+1)/2] 𝜈/2
𝑥
𝑦𝑞
𝑘!(𝑝 − 1)! 𝑗
[
]
2
𝜎𝑗,𝑞
′
′
⋅ √
ℐ1 +
ℐ2 + (𝑖 + 𝑗 − 2𝑘 − 𝑝 + 1)ℐ3 , (22)
𝜌𝑗,𝑞
𝜌𝑗,𝑞
∫ ∞ 𝑝+𝑘 −𝛼𝛾
𝑒
𝐾𝜈−1 (𝛽𝛾) d𝛾,
where the integrals
0 𝛾
∫ ∞ 𝑝+𝑘ℐ1 −𝛼𝛾=
ℐ
=
𝛾
𝑒
𝐾
(𝛽𝛾)
d𝛾,
and
ℐ3
=
2
𝜈
∫ ∞ 𝑝+𝑘−1 −𝛼𝛾0
𝛾
𝑒
𝐾
(𝛽𝛾)
d𝛾,
with
𝛼
=
(𝜎
/𝜌
)
−
𝑠
𝜈
𝑗,𝑞
𝑗,𝑞
0
√
and 𝛽 = 2/ 𝜌𝑗,𝑞 , can be solved by using [18, eqs. 6.611.3,
6.611.9, 6.621.3, or 6.624.1]. The forms of the integrals
depend on the values of 𝑝, 𝑘, and 𝜈.
Proof: Set 𝑐 = 0 in (20)
∫ ∞ and use that form of the PDF
□
in the equation, 𝑀𝑧 (𝑠) = 0 𝑒𝑠𝛾 𝑓𝑧 (𝛾) d𝛾.
Corollary 5: The MGF of 𝑍, 𝑀𝑧 (𝑠), for the special case
where all of the means in 𝑋 are dissimilar, all of the means
in 𝑌 are also dissimilar, and 𝑐 = 0, is given by
[
𝑚 ∑
𝑛
2
∑
𝜌𝑗,𝑞 + 𝜎𝑗,𝑞 𝜌2𝑗,𝑞 𝑠
4𝜌2𝑗,𝑞 − 𝜎𝑗,𝑞
𝑘𝑥,1,𝑗 𝑘𝑦,1,𝑞
𝑀𝑧 (𝑠) =
4𝜌𝑗,𝑞 − (𝜎𝑗,𝑞 − 𝜌𝑗,𝑞 𝑠)2
𝑗=1 𝑞=1
)
(
𝜎 −𝜌𝑗,𝑞 𝑠 ]
√
4𝜌3𝑗,𝑞 𝑠 arccos 𝑗,𝑞
2 𝜌𝑗,𝑞
+
.
(23)
[4𝜌𝑗,𝑞 − (𝜎𝑗,𝑞 − 𝜌𝑗,𝑞 𝑠)2 ]3/2
⋅
Proof: See Appendix D.
□
Corollary 6: The MGF of 𝑍, 𝑀𝑧 (𝑠), for the special case
where 𝑋 and 𝑌 are both single exponential RVs with means
equal to 𝑥 and 𝑦, respectively, and 𝑐 = 0, is found by setting
𝑚 = 1 and 𝑛 = 1 into (23) and noting that 𝑘𝑥,1,1 = 𝑥 −1 and
𝑘𝑦,1,1 = 𝑦 −1 for this special case.
□
The result found by using Corollary 6 is identical to the
result derived in [4, eq. (7)] for the MGF of the same RV.
V. A PPROXIMATE SER E XPRESSION FOR M ULTI -H OP AF
S YSTEM W ITH 𝐾 R ELAYS
The approximate, instantaneous, end-to-end SNR, Γ′ , which
will be used to find the approximate SER of the multi-hop
system, is summarized by the equations given in Theorem
2. The first step in finding an approximate SER expression
for the multi-hop system is to recognize that the terms,
𝛾𝑠,𝑚,𝑑 , from (11) are RVs of the form given by 𝑍. From
[4], if 𝛾1 and 𝛾2 are independent exponential RVs, then
𝛾𝑚𝑖𝑛,2 = min(𝛾1 , 𝛾2 ) is also an exponential RV with mean
𝑝∕=𝑞
(24)
where 𝛾𝑚𝑖𝑛,𝑃 −1 = min(𝛾1 , 𝛾2 , . . . , 𝛾𝑃 −1 ). From (13), 𝛾𝑠,𝑖,𝑚
is the minimum of a set of exponential RVs for the 𝑖 < 𝑚
case, which makes it an exponential RV as well. By using
(24), its mean is found to be
∏𝑚−1
𝛾 𝑠,𝑖 𝑝=𝑖 𝛾 𝑝,𝑝+1
(
)
𝛾 𝑠,𝑖,𝑚 =
∑𝑚−1 ∏𝑚−1
∏𝑚−1
𝛾 𝑠,𝑖
𝛾
𝑝=𝑖, 𝑝,𝑝+1 +
𝑞=𝑖
𝑝=𝑖 𝛾 𝑝,𝑝+1
𝑝∕=𝑞
𝛾 𝑠,𝑖 𝛾 𝑖,𝑚
𝛾 𝑠,𝑖 +𝛾 𝑖,𝑚
(25)
for 𝑖 = 𝑚 − 1. From
for 𝑖 ≤ 𝑚 − 2 and 𝛾 𝑠,𝑖,𝑚 =
Theorem 2, 𝛾𝑠,𝑖,𝑚 is also an exponential RV for the 𝑖 = 𝑚
case, with mean 𝛾 𝑠,𝑖,𝑚 = 𝛾 𝑠,𝑚 , since 𝛾𝑠,𝑚,𝑚 = 𝛾𝑠,𝑚 .
From (12) and from the above discussion regarding 𝛾𝑠,𝑖,𝑚 ,
it follows that 𝛾𝑠,𝑟𝑚 is the sum of 𝑚 exponential RVs for
𝑚 = 1, 2, . . . , 𝐾. It then follows from (11) that the terms,
𝛾𝑠,𝑚,𝑑 , are of the same form as 𝑍 given in (14), with
𝑍 = 𝛾𝑠,𝑚,𝑑 ; 𝑋 = 𝛾𝑠,𝑟𝑚 ; 𝑌 = 𝛾𝑚,𝑑 ; 𝑛 = 1; 𝑋𝑖 = 𝛾𝑠,𝑖,𝑚
for 𝑖 = 1, . . . , 𝑚; 𝑌1 = 𝛾𝑚,𝑑 ; and 𝑐 = 1. Note that, since
the results for 𝑍 in the previous section were derived under
the restriction that the variables in the summations of 𝑋 and
𝑌 all be independent, it is implied here that the following
approximation is being made: the terms, 𝛾𝑠,𝑖,𝑚 , are treated
as though they are independent (although they are not). As
was the case for the approximation of the end-to-end SNR,
we introduce this approximation in order to make the analysis
tractable; without such approximation, we conjecture that the
analysis is not tractable. Since this approximation is not always
valid, the approximate SER expression will not always be
accurate and, therefore, its usefulness is limited. However,
it will be seen in Section VI that this approximation is
accurate for some important practical cases, and that the SER
expression is accurate for those cases.
The next step in finding the approximate SER expression of
the multi-hop system is to obtain the MGF of 𝛾𝑠,𝑚,𝑑 for 𝑚 =
1, . . . , 𝐾. In the expression for 𝛾𝑠,𝑚,𝑑 in (11), the constant 𝑐
is equal to 1. In this case, it is not possible to solve for the
MGF of 𝛾𝑠,𝑚,𝑑 in closed form. For that reason, at this point the
approximation 𝑐 = 0 is used. This type of approximation has
been used in many other publications as well [4], [6]–[8]. As
mentioned in [12] and [13], it leads to an SER approximation
that is very tight to the actual SER throughout the entire SNR
range.
Lemma 2: The MGF of 𝛾𝑠,𝑚,𝑑 , 𝑀𝛾𝑠,𝑚,𝑑 (𝑠), for the special
case where 𝑐 = 0, is given by
[
𝑚
2
𝜌𝑖,1 + 𝜎𝑖,1 𝜌2𝑖,1 𝑠
4𝜌2𝑖,1 − 𝜎𝑖,1
1 ∑
𝑘𝑥,1,𝑖
𝑀𝛾𝑠,𝑚,𝑑 (𝑠) =
𝛾 𝑚,𝑑 𝑖=1
4𝜌𝑖,1 − (𝜎𝑖,1 − 𝜌𝑖,1 𝑠)2
)
(
𝜎 −𝜌𝑖,1 𝑠 ]
√
4𝜌3𝑖,1 𝑠 arccos 𝑖,1
2 𝜌𝑖,1
+
,
(26)
[4𝜌𝑖,1 − (𝜎𝑖,1 − 𝜌𝑖,1 𝑠)2 ]3/2
CONNE et al.: SER ANALYSIS AND PDF DERIVATION FOR MULTI-HOP AMPLIFY-AND-FORWARD RELAY SYSTEMS
= 𝛾1
𝑠,1
𝛾 𝑠,𝑖,𝑚
𝑙=1, 𝛾
−𝛾 𝑠,𝑙,𝑚
𝑙∕=𝑖 𝑠,𝑖,𝑚
where 𝑘𝑥,1,1
∏𝑚
1
𝛾 𝑠,𝑖,𝑚
when 𝑚
=
1; 𝑘𝑥,1,𝑖
=
𝜎𝑖,1 = 𝛾 𝑠,𝑖,𝑚 + 𝛾 𝑚,𝑑 ; and 𝜌𝑖,1 = 𝛾 𝑠,𝑖,𝑚 𝛾 𝑚,𝑑 .
Proof: Set 𝑍 = 𝛾𝑠,𝑚,𝑑 and 𝑛 = 1 into (23), and note that
𝑘𝑦,1,1 = 𝛾 𝑚,𝑑 −1 for this special case. The terms, 𝑘𝑥,1,𝑖 , are
solved using (16). 9
□
With the MGF for 𝛾𝑠,𝑚,𝑑 , it is now possible to find an
approximate SER expression for the multi-hop system. In
the derivation of this SER expression, the following approximation is made: the terms, 𝛾𝑠,𝑚,𝑑 , are treated as though
they are independent (although they are not), just as was
the case for the terms, 𝛾𝑠,𝑖,𝑚 . As before, we introduce this
approximation in order to make the analysis tractable; without
such approximation, we conjecture that the analysis is not
tractable. The discussion regarding the approximation used
with the terms, 𝛾𝑠,𝑖,𝑚 , applies to this approximation used with
the terms, 𝛾𝑠,𝑚,𝑑 , as well. The approximate SER expression
is given in the following theorem.
Theorem 5: An approximate SER expression, P𝑠 (𝐸), for
the multi-hop system is given by
𝑎
P𝑠 (𝐸) =
𝜋
∫
𝜋/2
𝜃=0
RELAY 3
for 𝑖 = 1, . . . , 𝑚 when 𝑚 ≥ 2;
(
)
𝐾
∏
−𝑏
sin2 𝜃
𝑀𝛾𝑠,𝑚,𝑑
d𝜃,
sin2 𝜃 + 𝑏𝛾 𝑠,𝑑 𝑚=1
sin2 𝜃
(27)
where 𝑀𝛾𝑠,𝑚,𝑑 (𝑠) is given by (26), and 𝑎 and 𝑏 are
modulation-dependent constants. 10,11
Proof: See Appendix E.
□
The SER can now be solved with only one numerical
integration for any value of 𝐾 by using (26) and (27). (It
is worth mentioning here that the CDF of the most general
form of 𝑍 given in Theorem 3 can always be used to find the
exact SER of a system with end-to-end SNR represented by
𝑍, for any combination of individual channel SNR means, in
closed form for the case where 𝑐 = 0 and with one numerical
integration for the case where√𝑐 ∕=∫ 0. The equation that makes
∞ −𝑏𝛾
this possible is P𝑠 (𝐸) = 2𝑎√𝜋𝑏 0 𝑒√𝛾 𝐹𝑧 (𝛾) d𝛾, which is
found in [14, eq. (15)] and [20, eq. (20)]. While this result
is noteworthy, it does not apply to the multi-hop system being
discussed in this paper, since the SNR of the multi-hop system
consists of a sum of RVs of the form given by 𝑍, and not just
a single 𝑍 term.)
9 This result is also valid for the two-hop system, for which case only the
𝑖 = 𝑚 term of the summation in (26) is non-zero. For this case, Corollary 6
is used and the result for 𝑀𝛾𝑠,𝑚,𝑑 (𝑠) is identical to [4, eq. (7)].
10 For binary phase-shift-keying (BPSK), 𝑎 = 𝑏 = 1; for 𝑀 -ary pulseamplitude-modulation (𝑀 -PAM), 𝑎 = 2(𝑀 − 1)/𝑀, 𝑏 = 3/(𝑀 2 − 1);
for 𝑀 -ary phase-shift-keying (𝑀 -PSK), an approximate expression for the
SER is found by using (27) with 𝑎 = 2, 𝑏 = sin2 (𝜋/𝑀 ); and for 𝑀 -ary
quadrature-amplitude modulation (𝑀 -QAM),
expression for
√
√ an approximate
the SER is found by using (27) with 𝑎 = 4( 𝑀 − 1)/ 𝑀 , 𝑏 = 1.5/(𝑀 −
1). See [14], [19].
11 This result is also valid for the two-hop system (for the two-hop system,
𝑀𝛾𝑠,𝑚,𝑑 (𝑠) is found using Corollary 6 as discussed in Lemma 2), which is
not surprising since it was pointed out in Theorem 2 that Γ′ , the SNR used to
find P𝑠 (𝐸), reduces to the expression for the exact instantaneous end-to-end
SNR of the two-hop system for that case. For the two-hop system, setting
𝑐 = 0 when solving for 𝑀𝛾𝑠,𝑚,𝑑 (𝑠) is the only approximation that is used
in the process of solving for P𝑠 (𝐸).
2419
Strong
Paths
Weak
Paths
RELAY 2
RELAY 1
DESTINATION
SOURCE
Fig. 2. Block diagram of the multi-hop system for 𝐾 = 3 relays. The
labeling of the channels as either strong or weak corresponds to the channel
variances chosen for the systems with results shown in Figs. 5, 7, 10, and 12.
(For the cases of these plots, the pattern of strong and weak channels is similar
for the systems with other values of 𝐾 as well.) This visual representation of
the system provides an intuitive explanation as to why the multi-hop system
performs much better than the two-hop system for the cases of Figs. 4, 5, 7,
8, 10, and 12. (The two-hop results are shown only in Figs. 4, 5, 7, and 8.)
VI. N UMERICAL A NALYSIS
Plots for the SER versus the transmit SNR, 𝜀𝑜 /𝑁𝑜 , where
𝑁𝑖,𝑗 = 𝑁𝑜 for 0 ≤ 𝑖 < 𝑗 ≤ 𝐾 + 1, were generated for
𝐾 = 2, 3, 4, 6, and 8 relays; for BPSK, QPSK, 8-PSK, and
16-QAM modulation schemes; for both the two-hop and the
multi-hop systems; and for two choices of channel variances.
In all cases, the transmit power, 𝜀𝑗 , used by relay 𝑗 was 𝜀𝑗 =
𝜀𝑜 /𝐾 for 1 ≤ 𝑗 ≤ 𝐾, so that the total transmit power used
by any system was always equal to 2𝜀𝑜 . For the first choice
of channel variances, Ω𝑖,𝑗 = 1 for 0 ≤ 𝑖 < 𝑗 ≤ 𝐾 + 1,
that is, for all channels. For the second choice, the channel
variances were chosen so that some of the channels are strong
(Ω𝑖,𝑗 is large) and some of the channels are weak (Ω𝑖,𝑗 is
small). The channels that were chosen to be strong are as
follows: those channels between terminals 𝑖 and 𝑗 for 𝑖 = 0
and 1 ≤ 𝑗 ≤ 𝐾 − 1; for 1 ≤ 𝑖 < 𝑗 ≤ 𝐾; and for 𝑖 = 𝐾 and
𝑗 = 𝐾 + 1. The channels that were chosen to be weak are as
follows: those channels between terminals 𝑖 and 𝑗 for 𝑖 = 0
and 𝐾 ≤ 𝑗 ≤ 𝐾 +1; and for 1 ≤ 𝑖 ≤ 𝐾 −1 and 𝑗 = 𝐾 +1. A
visual representation depicting which channels are strong and
which are weak is given for the case of 𝐾 = 3 in Fig. 2. It will
be seen that for this choice of channel variances, the multi-hop
system significantly outperforms the two-hop system.12
The plots in Figs. 3, 4, and 5 are for BPSK modulation; the
plots in Figs. 6, 7, and 8 are for QPSK modulation; the plots
in Figs. 9 and 10 are for 8-PSK modulation; and the plots in
Figs. 11 and 12 are for 16-QAM modulation. For the plots in
Figs. 3, 6, 9, and 11, the first choice of channel variances was
used. It can be seen that for this choice of channel variances,
the approximate SER expression for the multi-hop system,
given in (27), is quite accurate for all modulation schemes
and all values of 𝐾. (For an SER of 10−4 , the analytic result
12 The comparison between a two-hop system and a multi-hop system that
use the same modulation scheme and the same amount of transmit power is
fair in terms of the amount of bandwidth required. For the multi-hop system,
however, more thorough and sophisticated channel training is required, and
MRC is required at the receivers.
2420
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 2010
SER vs SNR −− Multi−hop, BPSK, Ω = 1
0
10
−1
10
−1
−2
10
10
−2
SER
10
SER
SER vs SNR −− BPSK, Var Ω
0
10
−3
10
K=3
K=4
−3
10
K=2
K=6
K=3
−4
10
−4
10
K=4
K=6
−5
10
−6
10
−20
Equation
Simulation
−15
−10
−5
10
K=8
−5
0
5
10
Single channel SNR, ε / N (dB)
o
−6
15
20
−15
−10
o
K=4
K=6
−5
0
5
10
Single channel SNR, ε / N (dB)
o
Fig. 3. The SER for the multi-hop system, determined from the approximate
SER equation given in (27) and from simulations, is shown for 𝐾 = 2, 3,
4, 6, and 8 relays. BPSK modulation is used, and the channel variances are
Ω𝑖,𝑗 = 1 for all terminals 𝑖 and 𝑗.
SER vs SNR −− K = 2, BPSK, Var Ω
0
10
−20
K=3
Eqn − Multi−hop
Sim − Multi−hop
Eqn − Two−hop
Sim − Two−hop
10
−1
10
15
20
o
Fig. 5. The SER for the multi-hop system and for the two-hop system,
determined from the approximate SER equation given in (27) and from
simulations, is shown for 𝐾 = 3, 4, and 6 relays. BPSK modulation is used.
For 𝐾 = 3, the channel variances are as follows: Ω𝑠,1 = 10, Ω𝑠,2 = 11,
Ω𝑠,3 = 0.2, Ω𝑠,𝑑 = 0.19, Ω1,𝑑 = 0.18, Ω2,𝑑 = 0.17, and Ω3,𝑑 = 15
for both systems; Ω1,2 = 12, Ω1,3 = 13, and Ω2,3 = 14 for the multi-hop
system; and Ω𝑖,𝑗 = 0 for 1 ≤ 𝑖 < 𝑗 ≤ 3 for the two-hop system. For
𝐾 = 4 and 𝐾 = 6, the channel variances are as follows: 10 ≤ Ω𝑠,𝑗 ≤ 12
for 1 ≤ 𝑗 ≤ 𝐾 − 1, and Ω𝑠,𝐾 = 0.2, and 0.16 ≤ Ω𝑖,𝑑 ≤ 0.195 for
0 ≤ 𝑖 ≤ 𝐾 − 1, and Ω𝐾,𝑑 = 20 for both systems; 12.5 ≤ Ω𝑖,𝑗 ≤ 19.5 for
1 ≤ 𝑖 < 𝑗 ≤ 𝐾 for the multi-hop system; and Ω𝑖,𝑗 = 0 for 1 ≤ 𝑖 < 𝑗 ≤ 𝐾
for the two-hop system.
−2
SER
10
10
−4
10
−1
10
−6
10
−20
Eqn − Multi−hop
Sim − Multi−hop
Eqn − Two−hop
Sim − Two−hop
−15
−10
−5
0
5
10
Single channel SNR, εo / No (dB)
−2
10
15
20
SER
−5
10
SER vs SNR −− Multi−hop, QPSK, Ω = 1
0
−3
10
−3
10
K=2
−4
10
Fig. 4. The SER for the multi-hop system and for the two-hop system,
determined from the approximate SER equation given in (27) and from
simulations, is shown for 𝐾 = 2 relays. BPSK modulation is used, and
the channel variances are as follows: Ω𝑠,𝑑 = 0.15, Ω𝑠,1 = 6, Ω𝑠,2 = 0.2,
Ω1,𝑑 = 0.1, and Ω2,𝑑 = 3. For the multi-hop system, Ω1,2 = 10, whereas
for the two-hop system, Ω1,2 = 0.
is within 0.57 dB of the simulation result for any modulation
scheme and any value of 𝐾.) However, for this choice of
channel variances, the multi-hop system outperforms the twohop system only very slightly and there is only a very small
separation between the plot for the multi-hop system and the
plot for the two-hop system. Therefore, it is difficult to justify
the usefulness of the approximate SER expression based on
this case alone. For this reason, it is necessary to consider the
results for other choices of channel variances as well.
The second choice of channel variances was used for the
results in Figs. 4, 5, 7, 8 10, and 12. From Figs. 4, 5, 7, and
8, where the results for both the multi-hop system and the
two-hop system are plotted for BPSK modulation (Figs. 4
and 5) and QPSK modulation (Figs. 7 and 8), it can be
seen that the multi-hop system significantly outperforms the
K=3
K=4
−5
10
−6
10
−15
Equation
Simulation
−10
−5
K=6
K=8
0
5
10
15
Single channel SNR, εo / No (dB)
20
25
Fig. 6. The SER for the multi-hop system, determined from the approximate
SER equation given in (27) and from simulations, is shown for 𝐾 = 2, 3,
4, 6, and 8 relays. QPSK modulation is used, and the channel variances are
Ω𝑖,𝑗 = 1 for all terminals 𝑖 and 𝑗.
two-hop system. (For an SER of 10−4 , the multi-hop system
outperforms the two-hop system by at least 2.5 dB for any
value of 𝐾, and by as much as 4.2 dB for the case of
QPSK modulation and 𝐾 = 3.) Similarly, for this choice of
channel variances, it can be shown that the multi-hop system
significantly outperforms the two-hop system for the cases of
8-PSK and 16-QAM as well. (However, the plots for the twohop system were omitted from Figs. 10 and 12 so as not to
clutter the graphs.) It can be seen that for all cases where
𝐾 ≥ 3, and for all modulation schemes, the approximate
CONNE et al.: SER ANALYSIS AND PDF DERIVATION FOR MULTI-HOP AMPLIFY-AND-FORWARD RELAY SYSTEMS
SER vs SNR −− QPSK, Var Ω
0
10
−1
10
−1
−2
10
10
−2
SER
10
SER
SER vs SNR −− QPSK, Var Ω
0
10
2421
−3
10
−3
10
K=3
10
−6
10
−15
Eqn − Multi−hop
Sim − Multi−hop
Eqn − Two−hop
Sim − Two−hop
−10
−5
K=8
−4
10
K=3
Eqn − Multi−hop
Sim − Multi−hop
Eqn − Two−hop
Sim − Two−hop
−5
10
K=6
−6
0
5
10
15
Single channel SNR, ε / N (dB)
o
20
10
−15
25
−10
−5
o
Fig. 7. The SER for the multi-hop system and for the two-hop system,
determined from the approximate SER equation given in (27) and from
simulations, is shown for 𝐾 = 3 and 𝐾 = 6 relays. QPSK modulation
is used. For 𝐾 = 3, the channel variances are as follows: Ω𝑠,1 = 10,
Ω𝑠,2 = 11, Ω𝑠,3 = 0.2, Ω𝑠,𝑑 = 0.19, Ω1,𝑑 = 0.18, Ω2,𝑑 = 0.17, and
Ω3,𝑑 = 15 for both systems; Ω1,2 = 12, Ω1,3 = 13, and Ω2,3 = 14 for
the multi-hop system; and Ω𝑖,𝑗 = 0 for 1 ≤ 𝑖 < 𝑗 ≤ 3 for the two-hop
system. For 𝐾 = 6, the channel variances are as follows: 10 ≤ Ω𝑠,𝑗 ≤ 12
for 1 ≤ 𝑗 ≤ 5, and Ω𝑠,6 = 0.2, and 0.17 ≤ Ω𝑖,𝑑 ≤ 0.195 for 0 ≤ 𝑖 ≤ 5,
and Ω6,𝑑 = 20 for both systems; 12.5 ≤ Ω𝑖,𝑗 ≤ 19.5 for 1 ≤ 𝑖 < 𝑗 ≤ 6
for the multi-hop system; and Ω𝑖,𝑗 = 0 for 1 ≤ 𝑖 < 𝑗 ≤ 6 for the two-hop
system.
SER expression is very accurate. It is almost identical to the
SER obtained from simulations for all values of SER less
than about 10−3 . (For an SER of 10−4 , the analytic result
is within 0.13 dB of the simulation result for BPSK, within
0.28 dB for QPSK, within 0.15 dB for 8-PSK, and within
0.195 dB for 16-QAM.) These results demonstrate that the
approximate SER expression is useful for an important case –
one in which the multi-hop system outperforms the two-hop
system by a substantial amount. It is a case for which we
would like to use the multi-hop system instead of the two-hop
system and would benefit from having this useful analytical
expression for the SER. The plots in Fig. 4 also show, however,
that the results for the approximate SER expression of the
multi-hop system are poor for the case of 𝐾 = 2 relays and
the given choice of channel variances. (For an SER of 10−4 ,
the analytic result differs from the simulation result by 1.87
dB.) Therefore, there are limitations to the usefulness of the
approximate SER expression found for the multi-hop system
in this paper.
K=4
K=8
0
5
10
15
Single channel SNR, ε / N (dB)
o
20
25
o
Fig. 8. The SER for the multi-hop system and for the two-hop system,
determined from the approximate SER equation given in (27) and from
simulations, is shown for 𝐾 = 4 and 𝐾 = 8 relays. QPSK modulation
is used. For 𝐾 = 4, the channel variances are as follows: 10 ≤ Ω𝑠,𝑗 ≤ 12
for 1 ≤ 𝑗 ≤ 3, and Ω𝑠,4 = 0.2, and 0.16 ≤ Ω𝑖,𝑑 ≤ 0.19 for 0 ≤ 𝑖 ≤ 3,
and Ω4,𝑑 = 20 for both systems; 13 ≤ Ω𝑖,𝑗 ≤ 18 for 1 ≤ 𝑖 < 𝑗 ≤ 4 for
the multi-hop system; and Ω𝑖,𝑗 = 0 for 1 ≤ 𝑖 < 𝑗 ≤ 4 for the two-hop
system. For 𝐾 = 8, the channel variances are as follows: 10 ≤ Ω𝑠,𝑗 ≤ 12
for 1 ≤ 𝑗 ≤ 7, and Ω𝑠,8 = 0.2, and 0.16 ≤ Ω𝑖,𝑑 ≤ 0.195 for 0 ≤ 𝑖 ≤ 7,
and Ω8,𝑑 = 21.3 for both systems; 12 ≤ Ω𝑖,𝑗 ≤ 21 for 1 ≤ 𝑖 < 𝑗 ≤ 8
for the multi-hop system; and Ω𝑖,𝑗 = 0 for 1 ≤ 𝑖 < 𝑗 ≤ 8 for the two-hop
system.
SER vs SNR −− Multi−hop, 8−PSK, Ω = 1
0
10
−1
10
−2
10
SER
−5
10
K=4
K=6
−4
−3
10
K=2
−4
K=3
10
K=4
−5
10
Equation
Simulation
−6
10
−10
−5
0
K=6
K=8
5
10
15
20
Single channel SNR, εo / No (dB)
25
30
Fig. 9. The SER for the multi-hop system, determined from the approximate
SER equation given in (27) and from simulations, is shown for 𝐾 = 2, 3,
4, 6, and 8 relays. 8-PSK modulation is used, and the channel variances are
Ω𝑖,𝑗 = 1 for all terminals 𝑖 and 𝑗.
VII. C ONCLUSION
A multi-branch, multi-hop AF relay system with 𝐾 relays was analyzed. An approximate SER expression, valid
for an arbitrary number of relays and for several important
modulation schemes, was found for the multi-hop system.
Plots of the SER of the multi-hop system were generated
for various values of 𝐾, various modulation schemes, and
two choices of channel variances. A comparison was made
between the analytic results (obtained using the approximate
SER expression) and simulation results for the SER of the
multi-hop system. It was seen that the analytic results for the
SER were quite accurate when compared to the simulation
results for most, but not all, of the cases considered. The fact
that the analytic results may not be accurate (depending on
the system’s channel variances) for certain cases indicates that
the usefulness of the approximate SER expression developed
in this paper is limited.
A comparison between the multi-hop system and the twohop system was also made for certain cases. It was demonstrated that, for certain choices of channel variances, the multihop system can substantially outperform the two-hop system.
2422
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 2010
SER vs SNR −− Multi−hop, 8−PSK, Var Ω
0
A PPENDIX A
P ROOF OF T HEOREM 1
10
−1
10
−2
SER
10
−3
10
−4
10
K=3
K=4
−5
10
Equation
Simulation
−6
10
−10
−5
0
K=8
5
10
15
20
Single channel SNR, ε / N (dB)
o
25
30
o
Fig. 10. The SER for the multi-hop system, determined from the approximate
SER equation given in (27) and from simulations, is shown for 𝐾 = 3, 4,
and 8 relays. 8-PSK modulation is used. The channel variances are the same
as those given in Figs. 7 and 8 for the multi-hop systems.
for 𝑚 > 1. By using 𝜇𝑘0,1 and 𝜇𝑘𝑖,𝑚 , the noise signals can be
written as 𝜂𝑚,𝑘 = 𝑛1,𝑘 + 𝜇𝑘0,1 𝑛0,1 for 𝑚 = 1; and 𝜂𝑚,𝑘 =
∑
𝑘
𝑛𝑚,𝑘 + 𝑚−1
𝑙=0 𝜇𝑙,𝑚 𝜂𝑙,𝑚 for 𝑚 = 2, . . . , 𝐾, and 𝑘 = 𝑚 +
1, . . . , 𝐾 + 1. Since 𝜂𝑚,𝑘 consists of the terms 𝑛𝑖,𝑗 for 𝑖 =
0, . . . , 𝑗 − 1 and 𝑗 = 1, . . . , 𝑚, as well as the term 𝑛𝑚,𝑘 , it
can also be written as
𝜂𝑚,𝑘 = 𝑛𝑚,𝑘 +
𝑗−1
𝑚 ∑
∑
𝜆(𝑖, 𝑗, 𝑚, 𝑘)𝑛𝑖,𝑗 ,
(28)
𝑗=1 𝑖=0
for 𝑚 = 1, 2, . . . , 𝐾, and 𝑘 = 𝑚 + 1, . . . , 𝐾 + 1. The
coefficients, 𝜆(𝑖, 𝑗, 𝑚, 𝑘), are found as follows: For 𝑚 =
1, equating 𝜂1,𝑘 = 𝑛1,𝑘 + 𝜇𝑘0,1 𝑛0,1 with (28) leads to
𝜆(0, 1, 1, 𝑘) = 𝜇𝑘0,1 for 𝑘 = 2, . . . , 𝐾 + 1. For 𝑚 = 2, . . . , 𝐾,
using (28) to substitute
∑𝑚−1 for the terms, 𝜂𝑙,𝑚 , in the equation,
𝜂𝑚,𝑘 = 𝑛𝑚,𝑘 + 𝑙=0 𝜇𝑘𝑙,𝑚 𝜂𝑙,𝑚 , leads to
SER vs SNR −− Multi−hop, 16−QAM, Ω = 1
0
Equation (3) follows directly from the fact that the received
signal is 𝑦𝑑 = 𝐴𝑑 𝑥 + 𝜂𝑑 , so that the signal power is 𝐴2𝑑 and
the noise power is 𝒩𝑑 = E[∣𝜂𝑑 ∣2 ].
In Section II, it was found that the total noise component,
𝜂𝑚,𝑘 , of the signal received at terminal 𝑘, and transmitted from
terminal 𝑚, was given(by 𝜂1,𝑘 = ℎ1,𝑘 𝛼1 𝑛𝑠,1 +𝑛1,𝑘 for
) 𝑚 = 1;
∑
∗
∗
𝑛𝑠,𝑚 + 𝑚−1
𝛽
𝜂
and 𝜂𝑚,𝑘 = ℎ𝑚,𝑘 𝛼𝑚 𝛽𝑠,𝑚
𝑖,𝑚 𝑖,𝑚 + 𝑛𝑚,𝑘
𝑖=1
10
−1
10
−2
SER
10
𝜂𝑚,𝑘 = 𝑛𝑚,𝑘 +
−3
10
K=3
K=4
10
K=6
−5
10
Equation
Simulation
−6
10
−5
0
5
K=8
10
15
20
25
Single channel SNR, εo / No (dB)
30
𝜇𝑘𝑙,𝑚 𝑛𝑙,𝑚 +
35
Fig. 11. The SER for the multi-hop system, determined from the approximate
SER equation given in (27) and from simulations, is shown for 𝐾 = 2, 3, 4,
6, and 8 relays. 16-QAM modulation is used, and the channel variances are
Ω𝑖,𝑗 = 1 for all terminals 𝑖 and 𝑗.
In addition, the CDF, PDF, and MGF of the RV, 𝑍 =
𝑋𝑌 /(𝑋 +𝑌 +𝑐), were developed. (They were then used in the
derivation of the approximate SER expression.) In this paper,
𝑋 and 𝑌 are sums of independent, exponential RVs with no
limitations placed on the number of RVs or the parameters
of the RVs. (A sum of independent, exponential RVs with
identical means is an Erlang RV. Therefore, in general, 𝑋 and
𝑌 are sums of independent, Erlang RVs.) It was shown that the
results for the CDF, PDF, and MGF of 𝑍 found in this paper
are generalizations of previously published results regarding
special cases of 𝑍.
𝜇𝑘𝑙,𝑚 𝜆(𝑖, 𝑗, 𝑙, 𝑚)𝑛𝑖,𝑗 ,
(29)
where 𝑘 = 𝑚 + 1, . . . , 𝐾 + 1. Then the terms, 𝜆(𝑖, 𝑗, 𝑚, 𝑘),
are found by equating (28) with (29) and by matching the
coefficients of the noise variable terms. The results are given
in (5).
The overall noise signal at the destination can be expressed
in terms of the independent AWGN variables, 𝑛𝑖,𝑗 , by substituting (28) into the expression for 𝜂𝑑 given in Section III. The
result is
∗
𝜂𝑑 = 𝛽𝑠,𝑑
𝑛𝑠,𝑑 +
𝐾
∑
∗
𝛽𝑚,𝑑
𝜂𝑚,𝑑 =
𝑚=1
∗
𝛽𝑠,𝑑
𝑛𝑠,𝑑 +
It was also seen that the approximate SER expression found
in this paper for the multi-hop system is a generalization
of previously published results for the approximate SER
expression of the two-hop system.
𝑗−1
𝑙 ∑
𝑚−1
∑∑
𝑙=1 𝑗=1 𝑖=0
𝑙=0
K=2
−4
𝑚−1
∑
𝐾
∑
∗
𝛽𝑚,𝑑
𝑛𝑚,𝑑 +
𝑚=1
𝑗−1
𝐾 ∑
𝑚 ∑
∑
𝑚=1 𝑗=1 𝑖=0
∗
𝛽𝑚,𝑑
𝜆(𝑖, 𝑗, 𝑚, 𝑑)𝑛𝑖,𝑗 .
(30)
The noise variances are then found from 𝒩𝑚,𝑘 = E[∣𝜂𝑚,𝑘 ∣2 ]
and 𝒩𝑑 = E[∣𝜂𝑑 ∣2 ] as shown in (4) and (6), respectively.
A PPENDIX B
P ROOF OF C OROLLARY 1
√
By using 𝛽𝑠,𝑑 = 𝜀𝑜 ℎ𝑠,𝑑 /𝑁𝑠,𝑑 and 𝛾𝑠,𝑑 = 𝜀𝑜 ∣ℎ𝑠,𝑑 ∣2 /𝑁𝑠,𝑑 ,
it follows that 𝑁𝑠,𝑑 ∣𝛽𝑠,𝑑 ∣2 = 𝛾𝑠,𝑑 . By using 𝛽1,𝑑 =
𝛼1 𝑁𝑠,1 𝛽𝑠,1 ℎ1,𝑑 /𝒩1,𝑑 ; 𝒩1,𝑑 = 𝛼21 ∣ℎ1,𝑑 ∣2 𝑁𝑠,1 + 𝑁1,𝑑 ; equa𝜀 ∣ℎ𝑚,𝑘 ∣2
tion (1) for 𝛼1 ; and 𝛾𝑚,𝑘 = 𝑚𝑁𝑚,𝑘
; it follows that
𝛾
𝛾
𝐴 𝛾
1 1,𝑑
1,𝑑
𝒩1,𝑑 ∣𝛽1,𝑑 ∣2 = 𝛾𝑠,1𝑠,1
+𝛾1,𝑑 +1 = 𝐴1 +𝛾1,𝑑 +1 . For 𝑚 ≥ 2,
an approximation for 𝒩𝑚,𝑘 is found by neglecting the
dependency of the 𝜂𝑖,𝑚 terms in the equation 𝜂𝑚,𝑘 =
CONNE et al.: SER ANALYSIS AND PDF DERIVATION FOR MULTI-HOP AMPLIFY-AND-FORWARD RELAY SYSTEMS
(
)
∑
∗
∗
+ 𝑛𝑚,𝑘 when calℎ𝑚,𝑘 𝛼𝑚 𝛽𝑠,𝑚
𝑛𝑠,𝑚 + 𝑚−1
𝑖=1 𝛽𝑖,𝑚 𝜂𝑖,𝑚
culating 𝒩𝑚,𝑘 = E[∣𝜂𝑚,𝑘 ∣2 ]. The result is 𝒩𝑚,𝑘 ≈
𝜀 ∣ℎ
∣2
∣ℎ𝑚,𝑘 ∣2 𝛼2𝑚 𝐴𝑚 + 𝑁𝑚,𝑘 = 𝑚𝐴𝑚𝑚,𝑘
+ 𝑁𝑚,𝑘 , where 𝑘 =
+1
2
𝑚 + 1, . . . , 𝐾 + 1, and the equations
𝐴
𝑚 = 𝑁𝑠,𝑚 ∣𝛽𝑠,𝑚 ∣ +
√
∑𝑚−1
𝜀𝑚
2
𝑖=1 𝒩𝑖,𝑚 ∣𝛽𝑖,𝑚 ∣ and 𝛼𝑚 =
𝐴2 +𝐴𝑚 were used. Then,
2423
SER vs SNR −− Multi−hop, 16−QAM, Var Ω
0
10
−1
10
−2
10
by using 𝛽𝑚,𝑑 = 𝛼𝑚 𝐴𝑚 ℎ𝑚,𝑑 /𝒩𝑚,𝑑 , along with the approximation for 𝒩𝑚,𝑑 , it follows that
SER
𝑚
𝛼2𝑚 𝐴2𝑚 ∣ℎ𝑚,𝑑 ∣2
𝒩𝑚,𝑑
𝐴𝑚 𝛾𝑚,𝑑
𝜀𝑚 𝐴𝑚 ∣ℎ𝑚,𝑑 ∣2
=
. (31)
≈
2
𝜀𝑚 ∣ℎ𝑚,𝑑 ∣ + 𝑁𝑚,𝑑 (𝐴𝑚 + 1)
𝐴𝑚 + 𝛾𝑚,𝑑 + 1
−3
10
−4
10
𝒩𝑚,𝑑 ∣𝛽𝑚,𝑑 ∣2 =
A PPENDIX C
P ROOF OF T HEOREM 3
The CDF of 𝑍, 𝐹𝑧 (𝛾), given in (19),
( is found as) fol𝑋𝑌
lows: 𝐹𝑧 (𝛾) = P(𝑍 ≤ 𝛾) = P 𝑋+𝑌
=
+𝑐 ≤ 𝛾
(
)
∫∞
𝑋𝑦
𝑦=0 P 𝑋+𝑦+𝑐 ≤ 𝛾 𝑓𝑦 (𝑦) d𝑦, where P(⋅) denotes a probability (see [4]). The argument of the probability function can
if 𝑦 > 𝛾 and 𝑋 ≥ (𝑦+𝑐)𝛾
if
be expressed as 𝑋 ≤ (𝑦+𝑐)𝛾
𝑦−𝛾
𝑦−𝛾
𝑦 < 𝛾. Then, the CDF is evaluated as
(
)
∫ 𝛾
(𝑦 + 𝑐)𝛾
P 𝑋≥
𝐹𝑧 (𝛾) =
𝑓𝑦 (𝑦) d𝑦
𝑦−𝛾
𝑦=0
)
∫ ∞ (
(𝑦 + 𝑐)𝛾
P 𝑋≤
(32)
+
𝑓𝑦 (𝑦) d𝑦.
𝑦−𝛾
𝑦=𝛾
By using the PDFs of 𝑋 and 𝑌 from (15) in equation (32),
straightforward integrations and algebraic manipulations lead
to the following expression for the CDF of 𝑍:
𝐹𝑧 (𝛾) = 1−
𝑟𝑗
𝐽 ∑
∑
∫
𝑗=1 𝑖=1
∞
𝑤=0
𝑘𝑥,𝑖,𝑗
(
)
𝑡𝑞
𝑖−1 𝑖−𝑘 𝑘 ∑
∑
𝑥𝑗 𝛾 𝑄 ∑
𝜎𝑗,𝑞 𝛾
𝑘𝑦,𝑝,𝑞
exp −
𝑘! 𝑞=1 𝑝=1 (𝑝 − 1)!
𝜌𝑗,𝑞
𝑘=0
(
)
𝑤
𝛾(𝛾 + 𝑐)/𝑥𝑗
(𝑤 + 𝛾 + 𝑐)𝑘 (𝑤 + 𝛾)𝑝−1
−
exp
−
d𝑤.
𝑤𝑘
𝑤
𝑦𝑞
(33)
The integral in (33) is solved in the same manner as in
the appendix of [14], where the
is applied
( 𝑘 ) theorem
∑binomial
𝑘
𝑖′
𝑘−𝑖′
𝑤
(𝛾
+
𝑐)
and
to obtain (𝑤 + 𝛾 + (𝑐)𝑘 )=
𝑖′ =0 𝑖′′
∑
′
𝑝−1
𝑝−1
𝑝−1
𝑗 𝑝−1−𝑗
= 𝑗 ′ =0 𝑗 ′ 𝑤 𝛾
. Then, the remaining
(𝑤 + 𝛾)
integral is solved using [18, eq. 3.471.9]. The CDF given by
(19) is found by substituting the result for the integral into
(33).
The PDF of 𝑍, 𝑓𝑧 (𝛾), is found by differentiating (19) and
𝜈 (𝑧)
using ∂𝐾∂𝑧
= −𝐾𝜈−1 (𝑧) − 𝜈𝑧 𝐾𝜈 (𝑧). (See [21].)
K=4
Equation
Simulation
−6
10
Equation (7) is obtained by substituting these results into the
approximate expression for Γ given by Γ ≈ 𝑁𝑠,𝑑 ∣𝛽𝑠,𝑑 ∣2 +
∑𝐾
2
𝑚=1 𝒩𝑚,𝑑 ∣𝛽𝑚,𝑑 ∣ .
Using similar arguments, it follows that 𝑁𝑠,𝑚 ∣𝛽𝑠,𝑚 ∣2 =
𝐴𝑖 𝛾𝑖,𝑚
𝛾𝑠,𝑚 and 𝒩𝑖,𝑚 ∣𝛽𝑖,𝑚 ∣2 ≈ 𝐴𝑖 +𝛾
for 𝑖 = 1, . . . , 𝑚 − 1 and
𝑖,𝑚 +1
𝑚 = 2, . . . , 𝐾. Equation (9) is obtained by substituting these
results into the expression for 𝐴𝑚 given earlier for 𝑚 ≥ 2.
K=3
−5
10
−5
0
5
K=8
10
15
20
25
Single channel SNR, ε / N (dB)
o
30
35
o
Fig. 12. The SER for the multi-hop system, determined from the approximate
SER equation given in (27) and from simulations, is shown for 𝐾 = 3, 4, and
8 relays. 16-QAM modulation is used. The channel variances are the same
as those given in Figs. 7 and 8 for the multi-hop systems.
A PPENDIX D
P ROOF OF C OROLLARY 5
Set 𝐽 = 𝑚 and 𝑟𝑗 = 1 for 𝑗 = 1, . . . 𝑚, and set 𝑄 = 𝑛
and 𝑡𝑞 = 1 for 𝑞 = 1, . . . 𝑛 into (22).[ For this special
]
∑𝑚 ∑𝑛
2𝜎
√ 𝑗,𝑞
case, 𝑀𝑧 (𝑠) =
𝑗=1
𝑞=1 𝑘𝑥,1,𝑗 𝑘𝑦,1,𝑞 4 ℐ1 + 𝜌𝑗,𝑞 ℐ2 .
The integrals, ℐ1 and ℐ2 , were solved using the Symbolic
R from Matlab ⃝.
R The results are
Toolbox ⃝
∫ ∞
ℐ1 =
𝛾𝑒−𝛼𝛾 𝐾0 (𝛽𝛾) d𝛾
0
( )
𝛼
1
𝛼
= 2
− 2
arccos
,
(34)
2
2
3/2
𝛽 −𝛼
𝛽
(𝛽 − 𝛼 )
∫ ∞
𝛾𝑒−𝛼𝛾 𝐾1 (𝛽𝛾) d𝛾
ℐ2 =
0
( )
𝛼
𝛼/𝛽
𝛽
=− 2
+
arccos
.
(35)
𝛽 − 𝛼2
𝛽
(𝛽 2 − 𝛼2 )3/2
After substituting these results into the expression for 𝑀𝑧 (𝑠)
and rearranging terms, the result in Corollary 5 is obtained.
A PPENDIX E
P ROOF OF T HEOREM 5
equation
The SER, [P𝑠 (𝐸),
)] by using
[ the
)[19]),
]
(√is found
( (see
∫ 𝜋/2
′
𝑎
−𝑏Γ
′
= EΓ′ 𝜋 0 exp sin2 𝜃 d𝜃 ,
P𝑠 (𝐸) = E 𝑎𝑄
2𝑏Γ
where 𝑄(⋅) is the Gaussian 𝑄-function, Γ′ is the approximate
instantaneous end-to-end SNR from (10), EΓ′ [⋅] denotes that
the expectation is done with respect to Γ′ , and the expression
within the expectation on the right-hand side of the equation is
Craig’s Formula from [22]. The following result is obtained:
P𝑠 (𝐸) =
=
𝑎
𝜋
𝑎
=
𝜋
∫
∫
𝛾=0
𝜋/2
𝜃=0
∫
𝜋/2
𝜃=0
∞
∫
𝑎
𝜋
∞
𝛾=0
∫
0
∞
∫
𝜋/2
exp𝑠𝛾 d𝜃 𝑓Γ′ (𝛾) d𝛾
( [
])
𝐾
∑
exp 𝑠 𝛾𝑠,𝑑 +
𝛾𝑠,𝑚,𝑑
𝑓Γ′ (𝛾) d𝛾 d𝜃
𝜃=0
⋅⋅⋅
∫
0
𝑚=1
∞
𝐾 + 1 integrals
𝑒
𝑠𝛾𝑠,𝑑
𝑓𝛾𝑠,𝑑 (𝛾𝑠,𝑑 ) d𝛾𝑠,𝑑
2424
IEEE TRANSACTIONS ON COMMUNICATIONS, VOL. 58, NO. 8, AUGUST 2010
⋅
=
𝑎
𝜋
𝑎
=
𝜋
∫
∫
𝐾
∏
[21] Wolfram Functions Site [Online], developed with Mathematica, Wolfram Research Inc., updated Nov. 3, 2008 [cited Nov. 4, 2008]. Available:
http://functions.wolfram.com/Bessel-TypeFunctions/BesselK/20/01/02/
𝑒𝑠𝛾𝑠,𝑚,𝑑 𝑓𝛾𝑠,𝑚,𝑑 (𝛾𝑠,𝑚,𝑑 ) d𝛾𝑠,𝑚,𝑑 d𝜃
𝑚=1
𝜋/2
𝜃=0
𝜋/2
𝜃=0
𝑀𝛾𝑠,𝑑 (𝑠)
𝐾
∏
[22] J. W. Craig, “A new, simple and exact result for calculating the
probability of error for two-dimensional signal constellations,” in Proc.
IEEE MILCOM, Nov. 1991, pp. 571–575.
𝑀𝛾𝑠,𝑚,𝑑 (𝑠) d𝜃
𝑚=1
(
)
𝐾
∏
−𝑏
sin2 𝜃
𝑀
d𝜃,
𝛾
𝑠,𝑚,𝑑
sin2 𝜃 + 𝑏𝛾 𝑠,𝑑 𝑚=1
sin2 𝜃
(36)
2
where 𝑠 = −𝑏/ sin 𝜃 was
∏ used. Note that it was assumed
that 𝑓Γ′ (𝛾) = 𝑓 (𝛾𝑠,𝑑 ) 𝐾
𝑚=1 𝑓 (𝛾𝑠,𝑚,𝑑 ), implying that any
dependencies between the RVs, 𝛾𝑠,𝑚,𝑑 , were neglected.
R EFERENCES
[1] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversty–part
I: system description,” IEEE Trans. Commun., vol. 51, pp. 1927–1938,
Nov. 2003.
[2] —-, “User cooperation diversty–part II: implementation aspects and
performance analysis,” IEEE Trans. Commun., vol. 51, pp. 1939–1948,
Nov. 2003.
[3] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity
in wireless networks: efficient protocols and outage behaviour,” IEEE
Trans. Inf. Theory, vol. 50, pp. 3062–3080, Dec. 2004.
[4] P. A. Anghel and M. Kaveh, “Exact symbol error probability of a
cooperative network in a Rayleigh-fading environment,” IEEE Trans.
Wireless Commun., vol. 3, pp. 1416–1421, Sep. 2004.
[5] K. J. R. Liu, A. K. Sadek, W. Su, and A. Kwasinski, Cooperative
Communications and Networking. New York: Cambridge University
Press, 2009.
[6] M. O. Hasna and M.-S. Alouini, “Harmonic mean and end-to-end
performance of transmission systems with relays,” IEEE Trans. Commun.,
vol. 52, pp. 130–135, Jan. 2004.
[7] M. O. Hasna and M.-S. Alouini, “End-to-end performance of transmission
systems with relays over Rayleigh-fading channels,” IEEE Trans. Wireless
Commun., vol. 2, pp. 1126–1131, Nov. 2003.
[8] M. O. Hasna and M.-S. Alouini, “Performance analysis of two-hop
relayed transmission over Rayleigh-fading channels,” in Proc. IEEE
Vehicular Technology Conf. (VTC), Sep. 2002, pp. 1992–1996.
[9] A. Ribeiro, X. Cai, and G. B. Giannikis, “Symbol error probabilities
for general cooperative links,” IEEE Trans. Wireless Commun., vol. 4,
pp. 1264–1273, May 2005.
[10] A. K. Sadek, W. Su, and K. J. Ray Liu, “Multinode cooperative
communications in wireless networks,” IEEE Trans. Signal Process.,
vol. 55, pp. 341–351, Jan. 2007.
[11] J. Boyer, D. D. Falconer, and H. Yanikomeroglu, “Multihop diversity in
wireless relaying channels,” IEEE Trans. Commun., vol. 52, pp. 1820–
1830, Oct. 2004.
[12] I.-M. Kim, Z. Yi, M. Ju, and H.-K. Song, “Exact SNR analysis in multihop cooperative diversity networks,” in Proc. IEEE CCECE, May 2008,
pp. 843–846.
[13] Z. Yi, M. Ju, H.-K. Song, and I.-M. Kim, “Relay ordering in a multihop cooperative diversity network,” IEEE Trans. Commun., vol. 57, pp.
2590-2596, Sep. 2009.
[14] R. Louie, Y. Li, and B. Vucetic, “Performance analysis of beamforming
in two hop amplify and forward relay networks,” in Proc. IEEE Int. Conf.
Commun. (ICC), May 2008, pp. 4311–4315.
[15] L. L. Yang and H. H. Chen, “Error probability of digital communications
using relay diversity over Nakagami-𝑚 fading channels,” IEEE Trans.
Wireless Commun., vol. 7, pp. 1806–1811, May 2008.
[16] A. Leon-Garcia, Probability and Random Processes for Electrical
Engineering, 2nd edition. Reading, MA: Addison-Wesley, 1994.
[17] A. Papoulis, Probability, Random Variables, and Stochastic Processes,
3rd edition. New York: McGraw-Hill, 1991.
[18] I. S. Gradshteyn and I. M. Ryzhik, Table of Integrals, Series, and
Products, 6th edition. San Diego, CA: Academic Press, 2000.
[19] M. K. Simon and M. S. Alouini, Digital Communication over Fading
Channels, 2nd edition. Hoboken, NJ: Wiley, 2005.
[20] M. R. McKay, A. Grant, and I. B. Collings, “Performance analysis of
MIMO-MRC in double-correlated Rayleigh environments,” IEEE Trans.
Commun., vol. 55, pp. 497–507, Mar. 2007.
Chris Conne received his B.Eng from the Department of Electrical and
Computer Engineering, Ryerson University, Toronto, Canada, in June 2007. In
Sept. 2007, he joined the Wireless Information Transmission Lab (WITL) of
the Department of Electrical and Computer Engineering, Queen’s University,
Kingston, Canada, where he received his M.Sc (Eng) degree in Oct. 2009.
Mr. Conne’s research interests include cooperative diversity networks, digital
communication theory, and MIMO communication systems.
MinChul Ju received the B.S. degree in electrical engineering from Pohang University of Science
and Technology (POSTECH), Pohang, Korea, in
1997, and the M.S. degree in electrical engineering from the Korea Advanced Institute of Science
and Technology (KAIST), Taejon, Korea, in 1999.
He is now a Ph.D. student at Queen’s University,
Kingston, Canada. From 1999 to 2005, he was
a researcher at the Korea Electronics Technology
Institute (KETI), Korea. During this period, he was
involved in many projects related to WPAN systems
such as Bluetooth, IEEE802.11, IEEE802.15.3, and HomeRF. His research
interests are in the areas of MIMO communications, cooperative diversity,
and synchronization in communications.
Zhihang Yi received his B.Eng. degree in information science and electrical
engineering from Zhejiang University, China, in 2003, and the M.Sc. and
Ph.D. degrees from Queen’s University in 2005 and 2009, respectively. For his
doctoral research at Queen’s, Dr. Yi investigated innovative relaying, MIMO,
and OFDM technologies in wireless communication systems. Mr. Yi was the
recipient of several research excellence awards during his graduate studies,
including a NSERC Industrial R&D Fellowship, a NSERC Visiting Fellowship at Canadian Government Laboratories, Ontario Graduate Scholarships,
and the IEEE Kingston Section Ph.D. Research Excellence Award (Honorable
Mention).
Hyoung-Kyu Song was born in CungCheongBukdo,
Korea on May 14 in 1967. He received B.S., M.S.,
and Ph.D. degrees in electronic engineering from
Yonsei University, Seoul, Korea, in 1990, 1992, and
1996, respectively. From 1996 to 2000 he was a
managerial engineer at the Korea Electronics Technology Institute (KETI), Korea. Since 2000 he has
been an assistant professor at the Department of Information and Communications Engineering, Sejong
University, Seoul, Korea. His research interests include digital and data communications, information
theory, and their applications with an emphasis on mobile communications.
Il-Min Kim received the B.S. degree in electronics
engineering from Yonsei University, Seoul, Korea, in
1996, and the M.S. and Ph.D. degrees in electrical
engineering from the Korea Advanced Institute of
Science and Technology (KAIST), Taejon, Korea,
in 1998 and 2001, respectively. From October 2001
to August 2002 he was with the Dept. of Electrical Engineering and Computer Sciences at MIT,
Cambridge, USA, and from September 2002 to June
2003 he was with the Dept. of Electrical Engineering
at Harvard, Cambridge, USA, as a Postdoctoral
Research Fellow. In July 2003, he joined the Dept. of Electrical and Computer
Engineering at Queen’s University, Kingston, Canada, where he is currently
an associate professor. His research interests include cooperative diversity
networks, bidirectional communications, CoMP, femto cells, and green communications. He currently serves as an editor for the IEEE T RANSACTIONS
ON W IRELESS C OMMUNICATIONS and the Journal of Communications and
Networks (JCN).