5 Relaying with Multiple Antennas at Source, Relay and Destination
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Cooperative Communications
∞
For DF protocol, the cdf of the equivalent
SNR γ nrm, df of SRD link with n × r SR, and
r × m RD MIMO channels may be determined
using the pdf and the cdf of the maximum
eigenvalues of the corresponding MIMO channels in (14.89):
Fγ nrm, af γ = 1−
γ
∞
= 1−
0
0
γ
γ 2 U γ 1 λ1
fλ1 , n × r λ1 dλ1
(14.98)
where the pdf fλ1 , n × r λ1 and the cdf Fλ2 , r × m λ2
of the maximum eigenvalues are given by
(13.16)–(13.22) and (13.23), respectively.
Similarly, the cdf of the equivalent SNR for
AF relaying with multiple antennas at the
Source, the Relay and the Destination may be
derived as in (14.D2):
1
fλ1 , n × r λ1 dλ1
n = m = 1, r = 1
n = m = 2, r = 1
n = m = 3, r = 1
n = m = 4, r = 1
n = m = 6, r = 1
n = m = 2, r = 2
n = m = 2, r = 3
Cdf
0.1
γ γ 1 λ1
γ 2 γ 1 λ1 − γ
fγ 1 γ 1 dγ 1
Figure 14.38 shows the cdf of the equivalent
SNR of the SRD link with DF relaying protocol
for various combinations of the number of
antennas at the Source, the Relay and the Destination. Based on the equivalence of MRC and
MRT, for example, the curve for n = m = 3,
r = 1 also shows the performance of a SRD
relay system with single antennas at the Source
and Destination (n = m = 1), but three antennas
at the Relay transmitter and receiver (r = 3).
In each case, there are three independent propagation paths in both SR and RD links. The outage probability was observed to improve
rapidly with increasing values of n and m. In
view of the above, it might be of interest to
compare the outage performance of the
SRD link with n = m = r = 2 and with n = m = 4,
Pr γ 2 U γ 1 < γ γ 1 fγ1 γ 1 dγ 1
= Fλ2 , r × m
γγ 1
γ1 − γ
(14.99)
∞
∞
1− Fλ2, r × m
γ γ1
Fγnrm, df γ = Pr γ nrm, df < γ
=
1 −Fγ2
0.01
0.001
−15
−10
−5
0
5
10
15
20
SNR (dB)
Figure 14.38 Cdf of the Equivalent SNR For the SRD Link with DF Relaying For γ 1 = γ 2 = 10 dB. The
number of antennas at the Source, the Destination and the Relay are denoted by n, m, and r, respectively.
798
r = 1. At an outage probability of 10−3, the
SRD link with n = m = r = 2 performs 0.75 dB
worse than the link with n = m = 4, r = 1 (see
also Figure 13.18). Similarly, the SRD link
with n = m = 2, r = 3 performs 1.1 dB worse
than the link with n = m = 6, r = 1. This small
reduction in the performance may be attributed
to processing at multiple antennas on both sides
of the links. These results also show that,
depending on the system requirements and cost
and space limitations, the number of antennas
at the Source, the Relay and the Destination
should be chosen adaptively.
14.6 Coded Cooperation
Coded cooperation brings an additional dimension into cooperative communication, since it
introduces channel coding into cooperative
communications. This consists sending different portions of each user’s encoded signal
(codeword) via different relays with independently fading links. For example, the data of
a user, augmented with CRC for error detection, is encoded into a codeword of length
n = n1 + n2, that is partitioned into two segments, containing n1 bits and n2 bits, respectively. Puncturing this codeword down to n1
bits, we obtain the first segment, which itself
is a valid but weaker codeword and the remaining n2 bits consist of the puncture bits. The
transmission period is divided into two time
frames of n1-bit and n2-bit long, respectively.
In the first time frame, each user transmits n1
bits and checks whether the transmission of
its partner is correctly received. If reception is
successful (checking with the CRC code), then,
in the second frame, it calculates and transmits
n2 bits of its partner. Otherwise, the intended
user transmits its own n2-bit segment. The level
of cooperation is hence n2/n, the ratio of bits for
each codeword the user transmits for its partner.
The efficiency of cooperation is managed automatically through code design, with no feedback between the users. Such a scheme may
be typically realized by using a rate-compatible
Digital Communications
punctured convolutional (RCPC) codeword
punctured to rate 1/2. This codeword is subsequently repeated by the Relay, resulting in
an overall rate of 1/4. Coded cooperation performs better than AF and DF, albeit at the
expense of reduced throughput. [29][30][31]
Other scenarios for coded cooperation can be
considered.
Example 14.5 Network Coding.
Network coding is a method to improve the performance of wireless networks. Consider a
relay network as shown in Figure 14.39, where
S1 and S2 denote the two Sources transmitting
f(x1) and f(x2) for bits x1 and x2, respectively,
independently of each other. Let us consider
BPSK modulation where f x
1,0
+ Eb , − Eb . The transmission interval is
divided into three, namely, for receiving transmission of S1, S2 and retransmission of combined and encoded messages from S1 and S2
to the Destination. The Relay R is assumed to
decode the signals from the sources perfectly
and retransmits f(x1 x2) to the Destination.
Then, the Destination D jointly decodes x1
and x2 by using three independent observations
f(x1) + n1, f(x2) + n2, and f(x1 x2) + n3. The
noises n1, n2 and n3 are all assumed to have
zero-mean and variance σ 2.
S1
x1
S2
x2
R
x1⊕x2
x1
x2
D
Figure 14.39 Coded Cooperation with Two Sources
and a Single Relay.
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Such networks generally use iterative
decoding using soft-in soft-out decoders. The
following provides the basic log-likelihood
concepts, which are widely used in soft-in
soft-out decoding. The log-likelihood ratio
(LLR) of a binary random variable X = {0,1},
LX(x) is defined as
P y x=0 =
P y x=1 =
1
exp − y + Eb
2π σ
1
exp − y − Eb
2π σ
2
2σ 2
2
2σ 2
(14.103)
Inserting (14.103) into (14.102), one gets
P x=0
LX x = ln
P x=1
(14.100)
LX Y x y = −
where P(x) denotes the probability that the random variable X takes the value x. Note that
LX(x) has a soft value; the sign of LX(x) denotes
the hard decision and the magnitude |LX(x)|
shows the reliability of the decision.
Inserting P x = 1 = 1 − P x = 0 into (14.100),
one gets
2 Eb
y
σ2
(14.104)
Now consider the RD link, which is also
modeled by AWGN with noise variance σ2.
Assuming that x1 and x2 are statistically independent random variables, P(x1 x2 = 0) and
P(x1 x2 = 1) may be expressed as
P x1 x2 = 0 = P x1 = 0 P x2 = 0 + P x1 = 1 P x2 = 1
e LX x
P x=0 =
1 + e LX
x
1
, P x=1 =
1 + e LX x
(14.101)
The LLR of the binary random variable X
conditioned on a different random variable Y,
LX|Y(x|y), is obtained using the Bayes’ theorem
P x1 x2 = 1 = P x1 = 0 P x2 = 1 + P x1 = 1 P x2 = 0
(14.105)
Using (14.100) and (14.105), one gets the
LLR of x1 x2 as follows:
LX1
LX Y x y = ln
X2
x1 x2 = ln
P x=0 y
P x=1 y
= ln
= ln
P y x=0 P x=0 P y
P y x=1 P x=1 P y
= ln
P y x=0
P x=0
+ ln
P y x=1
P x=1
= LY X y x + LX x
P x1 x2 = 0
P x1 x2 = 1
1 + e LX x1 + LX x2
e LX x1 + e LX x2
(14.106)
and
Using
(14.106),
P(x1 x2 = 0)
P(x1 x2 = 1) may be expresed in terms of
the LLR of x1 and x2 as follows:
(14.102)
For P x = 0 = P x = 1 = 1 2, we have
LX(x) = 0. Assuming that all links are described
by AWGN with variance σ2, Y represents the
received noisy signal by the Destination in
S1D or S2D link, when equiprobable bits
X=1 or 0 are transmitted. The pdf of received
signals with mean ± Eb , and variance σ2
may then be written as:
P x1 x2 = 0 =
1 + e L x1 + L x2
1 + e L x1 1 + e L
x2
P x1 x2 = 1 =
e L x1 + e L x 2
1 + e L x1 1 + e L
x2
(14.107)
Assume that x1 = 1 and x2 = 0 is transmitted
by Source 1 and Source 2, respectively, and
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Table 14.4 Decoding in the Coded Cooperation.
S1D link
S2D link
RD link
Transmitted bit: xi
Transmitted signal: f xi
Noise: ni
Received signal at D: y = f xi + ni
Lx y x y = −2y
P x i = 0 = e L xi 1 + e L xi
x1 = 1
+1
−0.1
0.9
−1.8
0.142
x2 = 0
−1
0.7
−0.3
0.6
0.65
x3 = x1 x2 = 1
+1
−0.3
0.7
−1.4
0.198
P x i = 1 = 1 1 + e L xi
Decoded bit: xi
L x1 x3 = L x2
Decoded bit: xi (correction)
0.858
1
0.35
0?
0.927
0
0.802
1
1
Eb = σ 2 = 1. S1R and S2R links are assumed to be
perfect, that is, with no error. Table 14.4 shows
the steps for decoding the transmitted bits at the
Destination.
Based on the LLR, the Destination estimates
the transmitted bits as x1 = x2 = x3 = 1. Note
that the decisions for x1 and x3 are taken with
high confidence but the decision for x2 is
questionable because of its lower LLR. The
next step would be to check whether x2 , the
bit with the lowest LLR, is correct. This
can be achieved by determining the LLR
of x1 x3 = x2 :
Fγeq γ = Pr γ eq < γ
∞
Pr
=
0
∞
=
γ
=
x1 x3 = ln
1+e
e LX x1 + e LX
x3
= 0 927
The cdf of γ eq, the equivalent SNR of the SRD
link, may be written as
γ c + γ1
γ1
γ1 − γ
fγ1 γ 1 dγ 1
≡1
∞
+
γ
γ
=
Pr γ 2 <
γ c + γ1
γ1
γ 1 −γ
fγ1 γ 1 dγ 1
fγ1 γ 1 dγ 1
∞
1− exp −
+
∞
= 1−
CDF of γ eq and γ eq, 0
fγ1 γ 1 dγ 1
Pr γ 2 >
(14.108)
Appendix 14A
γ c + γ1
γ1
γ1 − γ
Pr γ 2 <
γ
Based on (14.101) and (14.108), the Destination decodes x1 x3 = x2 as zero since
P x1 x3 = 0 = 0 716 and P x1 x3 = 1 =
0 284. Consequently, with the help of network
coding, the Destination decodes as x2 = 0 (see
the last row of Table 14.4).
fγ1 γ 1 dγ 1
0
0
X3
γ1γ2
< γ γ1
γ1 + γ2 + c
0
LX x1 + LX x3
LX1
1
γ
exp −
= 1− exp −
∞
×
γ c + γ1
γ2 γ1 − γ
γ c + γ1
γ2 γ1 − γ
fγ1 γ 1 dγ 1
1
γ
exp − 1
γ1
γ1
dγ 1
1 1
γ
+
γ1 γ2
exp −
0
λ γ c+γ
−
γ1
γ2 λ
1
dλ
γ1
(14A.1)
where the variable transformation λ = γ 1 − γ is
applied in the last equation. Finally, the cdf
of γeq is found using (D.65):
Fγeq γ, α, β = 1 − β
γ c+γ
× exp − αγ K1 β
γ c+γ
(14A.2)
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Cooperative Communications
where K1(x) denotes the modified Bessel function of the second kind, α = 1 γ 1 + 1 γ 2 ,
β = 2 γ 1 γ 2 . The corresponding pdf is determined by taking the derivative of (14A.2)
and using (D.105):
c
+ γ K0 β
2
fγeq γ, α, β = βe − αγ β
+α
γ γ + c K1 β
γ γ+c
γ γ+c
(14A.3)
The cdf and the pdf for γ eq,0 are obtained
by inserting c = 1 into (14A.2) and (14A.3).
Similarly, the cdf and the pdf for γ eq are
found by inserting c = 0 into (14A.2) and
(14A.3) (see Table 14.1).
The pdf, cdf and the MGF for γ eq, 1 and γ eq, 2
are found as follows: [11]
Appendix 14B
Average Capacity of γ eq,0
The derivation of the average (ergodic) capacity for SRD link using the equivalent SNR
γ eq, 0 = γ 1 γ 2 γ 1 + γ 2 + 1 is based on [32]. Here,
γ i , i = 1, 2 are assumed to be Rayleigh distributed with mean values γ i , i = 1, 2. The ergodic
capacity may be written as follows:
1
Cav = E log2 1 + γ eq, 0
2
=
1
E ln
2 ln 2
1
=
2 ln 2
1 + γ1 1 + γ2
1 + γ1 + γ2
E ln 1 + γ 1 + E ln 1 + γ 2
− E ln 1 + γ 1 + γ 2
(14B.1)
Fγeq γ = Pr γ eq < γ
∞
=
Pr
0
∞
=
0
γ1 γ2
< γ γ2
γ2 + C
Pr γ 1 < γ C γ 2 + 1 γ 2 fγ 2 γ 2 dγ 2
∞
=
1−exp −
0
1
= 1− e − γ
γ2
= 1−β
fγeq γ =
1 −γ
e
γ1
× β
Mγeq s =
fγ2 γ 2 dγ 2
γ1
γ C γ2 + 1
γ1
∞
e
− γ 2 γ 2 − γC γ 1 γ 2
fγ 2 γ 2 dγ 2
Cm γ i =
dγ 2
0
Cγ e −γ
γ1
K1 β
=
Cγ
1
2 ln2
∞
ln 1 + γ i
0
1
γ
exp − i dγ i
γi
γi
1 1 γi
e E1 1 γ i , i = 1, 2
2 ln2
(14B.2)
γ1
Cγ K1 β
Cγ +
1
sC γ 1
+
1 + sγ 1 γ 2 1 + sγ 1
× exp
The factor ½ appearing in front of the
expectation operator accounts for the division
of the transmission period into two. One
may easily evaluate the first two integrals using
(D.73):
C
γ 2 1 + sγ 1
2C
K0 β
γ2
Cγ
2
E1
The random variable z = γ 1 + γ 2 , appearing in
the last integral, corresponds to the MRC combination of γ 1 and γ 2 . The pdf of z is given by
(11.216) and (11.218) for distinct and identical
mean SNRs, respectively:
C
γ 2 1 + sγ 1
(14A.4)
Note that C = γ 1 + 1 corresponds to γ eq,1 and
C = γ 1 corresponds to γ eq,2.
fZ z =
z −z
e
γ2
1
e−z
γ1 − γ2
γ1
γ
− e−z
γ1 = γ2
γ2
γ1
γ2
(14B.3)
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Digital Communications
Therefore, the last term in (14B.1) evaluated as
1
2 ln2
∞
ln 1 + z f z dz
0
1
1
+ Cm γ 1 1 −
2 ln2
γ1
γ1
γ2
Cm γ 1 −
Cm γ 2
γ1 − γ2
γ1 − γ2
=
γ1 = γ2
γ1
γ2
where the binomial expansion 1 −x K =
K
K
− x n is used in the first expression.
n=0 n
The BER for BPSK modulation for the SRD
link is given by
∞
2γ fγeq, R γ dγ
Pb = Q
0
(14B.4)
(14C.2)
K
K
−1
=
Finally, inserting (14B.2) and (14B.4) into
(14B.1), the average capacity of the SRD link
with single relay is found to be
n+1
I 1, γ eq n
n
n=1
where I(L,x) is defined by (D.70):
Cav =
−1
1
Cm γ 1
+ 1+
2 ln2
γ1
γ1
γ
Cm γ 2 − 2 Cm γ 1
γ1 − γ2
γ 1 −γ 2
γ1 = γ2
γ1
γ2
∞
I L, γ = Q
1 γL−1 −γ γ
e
dγ
L − 1 γL
2γ
0
(14B.5)
γ
1+γ
1 1
= −
2 2
L−1
2k
k=0
k
1
k
4 1+γ
(14C.3)
Appendix 14C
The mean SNR may be obtained from MGF
Rayleigh Approximation for
using (F.37) and (D.24):
Equivalent SNR with Relay Selection
14C.1 SRD Link
E γ eq, R = γ eq, R = −
Based on the Rayleigh approximation for SRD
link with a mean SNR γ eq , given by (14.16) for
AF and (14.86) for DF relaying, one may write
the cdf, pdf and MGF of the SNR γ s,R when one
of the K relays, with highest equivalent SNR
γ eq, is selected in the SRD link:
d
Mγ
s
ds eq, R
K
K
n=1
n
= γ eq
s=0
−1 n + 1
= γ eq
n
K
1
n
n=1
(14C.4)
The average capacity is found as follows:
Fγs, R γ = 1 −e −γ
γ eq K
K
K
=
n=0
K
K
fγ s, R
d
γ = Fγ γ =
dγ
n=1
Mγs, R s = E e −s γeq, R =
−1
n
K
n=1
n
K
n
− 1 n e −n γ
γ eq
n −n γ
e
γ eq
γ eq
n+1
−1
n+1
Cav, R =
K
1
1 + sγ eq n
(14C.1)
=
n=1
K
1
E ln 1 + γ eq, R
2 ln2
∞
K
1
n −nγ
n+1
−1
ln 1 + γ
e
2
ln2
γ
eq
n
0
K
=
n=1
n
−1
n+1
γ eq
dγ
Cm γ eq n
(14C.5)
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Cooperative Communications
where Cm(x), given by (14B.2), denotes the
average capacity of a link with mean SNR, x.
The cdf of the capacity may be written as
follows:
∞
2γ fγc, R γ dγ
Pb = Q
0
K
K
Cout, R Cth
1
= Prob log2 1 + γ eq, R < Cth
2
= I 1, γ 3 +
K
−1
−
= Fγeq , R 22Cth − 1
n+1
n+1
I 1,
n
n=1
I 1, γ eq n
n
n=1
K
= Prob γ eq, R < 22Cth − 1
−1
1
n γ eq + 1 γ 3
(14C.8)
(14C.6)
The mean SNR may be written from the
MGF using (F.37) and (D.24):
14C.2 Selection combined SRD
and SD links
K
E γ c, R = γ c, R = γ 3 + γ eq
The cdf, pdf and MGF of the SNR γ c, R =
max γ eq, R , γ 3 when one of the K relays, with
highest equivalent SNR, is selected in the SRD
link and is selection-combined with the SRD link
with mean SNR γ 3 :
Fγc, R γ = 1− e − γ
K
γ eq
−1
=
n=0
M γ c, R s =
n
γ3
− n γ γ eq
−e
− n γ eq + 1 γ 3 γ
Cav, c, R =
1
E ln 1 + γ c, R
2 ln2
K
−1
n=1
n −nγ
e
γ eq
γ eq
−1
n=1
n+1
n
−
K
= Cm γ 3 +
K
+
−1 n+1
n γ eq + 1 γ 3
(14C.9)
K
K
1
+
1 + sγ 3 n = 1
×
e
n=1
n
1 −γ
e
γ3
×
n
K
−
The average capacity is found as follows:
γ3
K
K
f γ c, R γ =
1− e − γ
K
1
n
n=1
n
1 −
+
e
γ eq γ 3
n γ eq + 1 γ 3 γ
−1
n=1
n+1
Cm
n
Cm
γ eq
n
1
n γ eq + 1 γ 3
(14C.10)
K
−1
n
K
K
−
n+1
where Cm(x) is defined by (B.2).
The cdf of the capacity may be written as
follows:
n+1
n
1
1
−
1 + sγ eq n 1 + s n γ eq + 1 γ 3
(14C.7)
The BER for BPSK modulation for the
selection combined SRD and SD links is
given by
Cout, c, R Cth = Prob
1
log 1 + γ c, R < Cth
2 2
= Prob γ c, R < 22Cth − 1
= Fγc , R 22Cth − 1
(14C.11)
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14C.3 MRC Combined SRD and
SD Links
The mean SNR may be obtained from MGF
using (F.37) and (D.24):
Now let us consider that the SRD links via K relays
are all approximated by the Rayleigh pdf, and
the SRD link with the highest equivalent SNR
is selected. The equivalent SNR of the selected
SRD link, γ s,R, is MRC combined with the SRD
link with SNR γ 3 . The cdf, pdf and MGF of the
SNR γ c, R, MRC = γ s, R + γ 3 are given by
Mγ c, R, MRC s = Mγ eq, R s
K
K
n=1
n
K
K
n=1
n
K
K
n=1
n
fγ c, R , MRC γ =
K
n=1
n
γ eq − nγ
e
n
n=1
n
Pc, R, MRC =
n+1
γ eq
n
γ eq
− e −γ
1
log 1 + γ c, R, MRC < Cth
2 2
= Prob γ c, R, MRC < 22Cth −1
γ3
= Fγc , R, MRC 22Cth − 1
(14C.16)
−1 n+1
γ eq n− γ 3
γ eq
− γ3 e − γ
1
n − γ3
γ eq
γ eq
Cm
− γ 3 Cm γ 3
n
n
(14C.15)
Cout, c, R, MRC Cth = Prob
Appendix 14D
γ3
where (D.23) is used. The BEP for BPSK
modulation when the SRD and SD links are
maximal-ratio combined is given by
K
−1
×
(14C.12)
K
K
where Cm(x) is defined by (14B.2). The cdf of
the capacity may be written as
− 1 n + 1 − nγ
e
γ eq n− γ 3
K
1
E ln 1 + γ c, R, MRC
2 ln2
n=1
−1 n + 1
1 + sγ eq n 1 + sγ 3
Fγ c, R, MRC γ = 1 −
×
Cav, c, R, MRC =
K
γ eq n
γ3
−
1 + sγ 3
1 + sγ eq n
×
The average capacity is found as follows:
=
−1 n+1
γ eq n− γ 3
1
n
n=1
(14C.14)
1
1 + sγ 3
=
=
K
E γ c, R, MRC = γ c, R, MRC = γ 3 + γ eq
−1 n+1
γ eq n − γ 3
γ eq
I 1, γ eq n − γ 3 I 1, γ 3
×
n
(14C.13)
CDF of γ eq,a
Using (F.115), the pdf of the SNR γ 1 of the SR
link with mean SNR γ 1 applying MRT with n
antennas and the pdf of the SNR γ 2 of the
RD link with mean SNR γ 2 applying MRC with
m antennas may be written as
fγ i z =
1
k−1
Fγi z = 1 − e −z
z
γi
γi
k−1
k−1
e − z γi
γ i i = 1, k = n
1
z γi
j
j=0
j
i = 2, k = m
(14D.1)
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Cooperative Communications
The cdf of γ eq,a, the equivalent SNR of the
SRD link, is given by
Fγeq, a γ = P γ eq < γ
∞
=
P
0
∞
fγ1 γ 1 dγ 1
γ γ1
γ
γ1 − γ 1
fγ1 γ 1 dγ 1
0
γ
=
j + n −1
γ1 γ2
< γ γ1
γ1 + γ2
P γ2 <
=
P γ2 >
0
∞
γ γ1
γ
γ 1 −γ 1
γ
P γ2 <
∞
= 1−
γ
γ
= 1−
fγ1 γ 1 dγ 1
γ γ1
γ
γ1 − γ 1
Using (14D.1), (14D.2) may be written as
m −1
1
j
j=0
× exp −
∞
γ
γ γ1
γ2 γ1 − γ
e −α γ m −1 1 γ
n− 1 γ n1 j = 0 j γ 2
= 1−
∞
×
0
e −q
λ −pλ
j
γ 1 γ 1 n−1 e −γ1 γ1
dγ 1
n− 1
γ1
γ γ1
γ2 γ1 − γ
γ
r
e−q
λ
2e −α γ
n− 1
j + n −1
βγ
2
dλ
j + n −1
r=0
r
1 γ
j
γ2
j=0
j
r 2
γ2
γ1
Kn− r βγ
(14D.4)
For n = m = 1, (14D.4) reduces to (14A.2),
as expected. Here Kv(x) denotes the modified Bessel function of the second kind,
α = 1 γ 1 + 1 γ 2 , β = 2 γ 1 γ 2 , q = γ 2 γ 2 and
p = 1 γ 1 . The corresponding pdf is determined by taking the derivative of (14D.4) and
using (D.105):
fγeq γ,α, β =
2e − αγ
n− 1
j
βγ
2
n m −1
1 γ
j
γ2
j=0
j + n−1
j + n− 1
r=0
r
×
dλ
n m−1
×
j + n−1
λ+γ
λj
λ− pλ n − r − 1
fγ1 γ 1 dγ 1
(14D.2)
Fγeq, a γ = 1−
∞
×
fγ1 γ 1 dγ 1
γ γ1
γ
γ 1 −γ 1
1 −Fγ2
j + n −1
r
r=0
γ γ1
P γ2 >
γ fγ γ dγ 1
γ 1 −γ 1 1 1
∞
= 1−
×
j
0
≡1 since γ 1 − γ < 0
+
e − α γ m −1 1 γ
n− 1 γ n1 j = 0 j γ 2
Fγeq, a γ = 1 −
(14D.3)
×
α−
γ2
γ1
j
r 2
j+r
Kn −r βγ + βKn− r −1 βγ
γ
(14D.5)
where the variable transformation λ = γ 1 − γ
is applied in the last equation. Using the binoR r R−r
R
mial expansion λ + γ R =
γ λ
r=0 r
above, (14D.3) reduces to
For n = m = 1, (14D.5) reduces to (14A.3), as
expected.
The MGF corresponding to (14D.4) and
(14D.5) may easily be found using (F.39)
and (D.67):
806
Digital Communications
∞
Mγ eq, a s = s
0
e −s z Fγ eq, a z dz
−2n
= 1−
×
s2
π
n− 1
2
γ1
× 2F1
r
m−1
j=0
2γ 2
j
− j j + n− 1
r=0
j + n −1
r
Γ j + r + 1 Γ 2n− r + j + 1
s+α
r+j+1
Γ n+j+3 2
j+r+1 j+r+2
3
β2
,
;n + j + ;1−
2
2
2
s+α
2
(14D.6)
For n = m = 1, the MGF given by (14D.6)
reduces to (14.15), as expected.
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Problems
0
ln 1 + αγ f γ dγ =
2. Using the derivative of the Q function
given by (B.13) and integration by parts,
prove the following identity [33]
∞
Pe =
∞
0
α
1 − F γ dγ
1 + αγ
2γ fγ γ dγ
Q
0
=
1
2π
∞
0
1 −γ
e Fγ γ dγ
2γ
3. Derive the pdf in (14.22) by taking the
derivative of the corresponding cdf.
4. Plot and compare mean equivalent SNRs,
given by (14.24), of the selection-combined
and maximal-ratio-combined SRD and
SD links.
5. Noting that (14.27)–(14.31) correspond to
selection combining of two direct links,
that is, γ D = max γ 2 , γ 3 , derive the expressions corresponding to (14.27)–(14.31)
for maximal ratio combining, that is,
γ MRC = γ 2 + γ 3 , where γ 2 and γ 3 are
assumed to be independent of each other.
6. The equivalent SNR of the SRD link with
Rayleigh approximation and relay selection is given by (14C.1). Using (14C.8)
and (14C.13), plot and compare the BEP
performances when the SNRs of SRD
and SD links are selection- and maximal
ratio combined.
7. Show that (14.35) is identical to γ c =
max γ eq, 1 , γ eq, 2 ,…, γ eq, K , γ 3 .
8. The pdf and cdf of a random variable with
central chi-square distribution are given
below (see (F.115)):
fγ z =
1. Prove the following
∞
where f γ and F γ denote respectively
the pdf and the cdf of γ.
zm − 1
−z
exp
m − 1 γm
γ
Fγ z = 1 − exp
−z
γ
m −1
1 z
k
γ
k=0
k