Tải bản đầy đủ - 0 (trang)
5 Relaying with Multiple Antennas at Source, Relay and Destination

5 Relaying with Multiple Antennas at Source, Relay and Destination

Tải bản đầy đủ - 0trang

797



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.



799



Cooperative Communications



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



800



Digital Communications



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



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)



801



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





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



=







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







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)



802



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



L − 1 γL







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



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 =



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







Cm γ eq n



(14C.5)



803



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)



804



Digital Communications



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)



805



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







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



×







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.



References

[1] A. Scaglione, D. L. Goeckel, and J. N. Laneman,

Cooperative Communications in Mobile Ad Hoc

Networks, IEEE Signal Processing Magazine,

vol. 18 September 2006.

[2] R. Ahlswede, N. Cai, S.-Y. R. Li, and R. W. Yeung,

Network Information Flow, IEEE Trans. Information

Theory, vol. 46, no. 4, July 2000, pp. 1204–1216.

[3] X. Tao, X. Xu, and Q. Cui, An overview of cooperative communications, IEEE Communications Magazine, pp. 65–71, June 2012.

[4] K.B. Letaif and W. Zhang, Cooperative communications for cognitive radio networks, Proc. IEEE,

vol. 97, no. 5, pp. 878–893, May 2009.

[5] A. Ghasemi, and E. S. Sousa, Spectrum Sensing in

Cognitive Radio Networks: Requirements, Challenges and Design Trade-offs, IEEE Communications

Magazine, April 2008, pp. 32–39.

[6] C.-X. Wang et al., Cooperative MIMO Channel

Models: A Survey, IEEE Communications Magazine,

February 2010, pp. 80–87.

[7] N. Yang, M. Elkashlan, and J. Yuan, Impact of opportunistic scheduling on cooperative dual-hop relay networks, IEEE Trans. Communications, 2011, vol. 59,

no.3, pp. 689–694.

[8] T. A. Tsiftsis, G. K. Karagiannidis, P. T. Mathiopoulos, and S. A. Kotsopoulos, Nonregenerative dual-hop

cooperative links with selection diversity’, EURASIP

J. Wireless Communications and Networking, 2006,

pp. 1–8, DOI 10.1155/WCN/2006/17862.

[9] P. L. Yeoh, M. Elkashlan, and I. B. Collings, Exact

and asymptotic SER of distributed TAS/MRC in

MIMO relay networks, IEEE Trans. Wireless Communications, 2011, vol. 10, no. 3, pp. 751–756.



[10] D.B. da Costa, and S. Aissa, Amplify-and-forward

relaying in channel-noise assisted cooperative networks with relay selection, IEEE Communications

Letters, 2010, vol. 14, no. 7, pp. 608–610.

[11] M.O. Hasna, and M-S. Alouini, A performance study

of dual-hop transmissions with fixed gain relays,

IEEE Trans. Wireless Communications, 2004,

vol. 3, no. 6 pp. 1963–1968.

[12] M.O. Hasna, and M-S. Alouini: End-to-end performance of transmission systems with relays over

Rayleigh-fading channels, IEEE Trans. Wireless

Communications, Nov. 2003, vol. 2, no. 6, pp.

1126–1130.

[13] P. A. Anghel, and M. Kaveh, Exact symbol error

probability of a cooperative network in a

Rayleigh-fading environment, IEEE Trans. Wireless

Communications, vol. 3, no. 5, pp. 1416–1421,

September 2004.

[14] P.L. Yeoh, M. Elkashlan, and I. B. Collings, Selection

relaying with transmit beamforming: A comparison of

fixed and variable gain relaying, IEEE Trans. Commun., vol. 59, no. 6, pp. 1720–1730, June 2011.

[15] M. Torabi, and D. Haccoun, Capacity analysis of

opportunistic relaying in cooperative systems with

outdated channel information, IEEE Communication

Letters, vol. 14, no. 12, pp. 1137–1139, December 2010.

[16] P. Xin et al., Diversity analysis of transmit beamforming and the application in IEEE 802.11n systems,

IEEE Trans VT, vol. 57, no. 4, pp. 2638–2642,

July 2008.

[17] P. A. Dighe, R. K. Mallik, and S.S. Jamuar, “Analysis

of Transmit-Receive Diversity in Rayleigh Fading,”

IEEE Trans. Communications, vol. 51, pp. 694–

703, April 2003.

[18] S. Prakash and I. McLoughlin, Performance of dualhop multi-antenna system with fixed-gain AF relay

selection, IEEE Trans. Wireless Communications,

vol. 10, no. 6, pp. 1709–1714, June 2011.

[19] S. Loyka, and G. Levin, On outage probability and

diversity-multiplexing trade-off in MIMO relay channels, IEEE Trans. Communications, vol. 59, no. 6, pp.

1731–1741, June 2011.

[20] M. Torabi, W. Ajib, and D. Haccoun, Performance

analysis of amplify-and-forward cooperative networks with relay selection over Rayleigh fading channels, IEEE VTC, Spring 2009.

[21] A. Zanella, M. Chiani, and M. Z. Win, “On the marginal distribution of the eigenvalues of Wishart

matrices,” IEEE Trans. Commun., vol. 57, pp.

1050–1060, April 2009.

[22] A. Zanella, M. Chiani, and M. Z. Win, “Performance

of MIMO MRC in correlated Rayleigh fading

environments,” in Proc. IEEE Vehicular Technology

Conference (VTC 2005 Spring), Stockholm, Sweden,

pp. 1633–1637, vol. 3, May 2005.



807



Cooperative Communications



[23] P. J. Smith, P.-H. Kuo and L. M. Barth, Level Crossing

Rates for MIMO Channel Eigenvalues: Implications

for Adaptive Systems, in Conf. Rec. 2005 IEEE Int.

Conf. Commun. (ICC), vol. 4, pp. 2442–2446.

[24] S. H. Simon, A.L. Moustakas, and L. Marinelli”, Capacity and character expansions: moment-generating

function and other exact results for MIMO correlated

channels,” IEEE Trans. Information Theory, vol. 52,

pp. 5336–5351, December 2006.

[25] M. R. McKay, A. J. Grant, and I. B. Collings, “Performance analysis of MIMO-MRC in DoubleCorrelated Rayleigh Environments,” IEEE Trans.

Communications, vol. 55, pp. 497–507, March 2007.

[26] M. M. Fareed and M. Uysal, On Relay Selection for

Decode-and-Forward Relaying, IEEE Trans. Wireless

Communications, vol. 8, no. 7, p. 3341–3346, July 2009.

[27] T. Wang, A. Cano, G. B. Giannakis and J. N. Laneman, High-performance cooperative demodulation

with decode-and-forward relays, IEEE Trans. Communications, vol. 55, no. 7, pp. 1427–1438, July 2007.

[28] R. M. Radaydeh and M. M. Matalgah, Results for

Integrals Involving m-th Power of the Gaussian Qfunction Over Rayleigh Fading Channels with Applications, IEEE ICC 2007, pp. 5910–5914.

[29] I. Safak, E. Aktas, and A.O. Yılmaz, Error Rate Analysis of GF(q) Network Coded Detect-and-Forward

Wireless Relay Networks Using Equivalent Relay

Channel Models, IEEE Trans. Wireless Communications, 2013, vol. 12, no. 8, pp. 3908–3919.

[30] Y. Li, Distributed coding for cooperative wireless networks: an overview and recent advances, IEEE Communications Magazine, pp. 71–77, August 2009.

[31] A. Nosratinia, T. E. Hunter, and A. Hedayat, Cooperative Communication in Wireless Networks, IEEE

Communications Magazine, October 2004, pp. 74–80.

[32] L. Fan, X. Lei, and W. Li, Exact closed-form expression for ergodic capacity of amplify-and-forward

relaying in channel-noise-assisted cooperative networks with relay selection, IEEE Communications

Letters, vol. 15, no. 3, pp. 332–333, March 2011.

[33] H. A. Suraweera, P.J. Smith, A. Nallanathan, and J.S.

Thompson, Amplify-and-Forward Relaying with

Optimal and Suboptimal Transmit Antenna Selection,

IEEE Trans. Wireless Communications, vol. 10, no. 6,

pp. 1874–1885, June 2011.

[34] L. S. Gradshteyn, and L. M. Ryzhik, Table of Integrals, Series and Products (6th ed.), Academic Press:

San Diego, 2000, p. 17, 0.314.



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







0



1 −γ

e Fγ γ dγ





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



Tài liệu bạn tìm kiếm đã sẵn sàng tải về

5 Relaying with Multiple Antennas at Source, Relay and Destination

Tải bản đầy đủ ngay(0 tr)

×