Parameter Optimization for Amplify-and-Forward Relaying Systems with Pilot Symbol Assisted Modulation Scheme

doi:10.4236/wsn.2009.11003

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Wireless Sensor Network, 2009, 1, 1-60

Published Online April 2009 in SciRes (http://www.SciRP.org/journal/wsn/).

Parameter Optimization for Amplify-and-Forward Relaying

Systems with Pilot Symbol Assisted Modulation Scheme

Yi WU, Matthias PÄTZOLD

Faculty of Engineering and Science, University of Agder, Grimstad, Norway

E-mail: {yi.wu, matthias.paetzold}@uia.no

Received March 7, 2009; revised March 16, 2009; accepted March 18, 2009

Abstract

Cooperative diversity is a promising technology for future wireless networks. In this paper, we consider a

cooperative communication system operating in an amplify-and-forward (AF) mode with a pilot symbol as-

sisted modulation (PSAM) scheme. It is assumed that a linear minimum mean square estimator (LMMSE) is

used for the channel estimation at the receiver. A simple and easy-to-evaluate asymptotical upper bound

(AUB) of the symbol-error-rate (SER) is derived for uncoded AF cooperative communication systems with

quadrature amplitude modulation (QAM) constellations. Based on the AUB, we propose a criterion for the

parameter optimization in the PSAM scheme. We discuss how the pilot spacing and the length of the Wiener

ﬁlter should be chosen under the constraint of a tradeoff between pilot overhead, estimation accuracy, and

receiver complexity. We also formulate an power allocation problem for the considered system. It is shown

that the power allocation problem can optimally be solved by means of a gradient search method. Numerical

simulations are presented to verify the correctness of the theoretical results and to demonstrate the benefits of

the parameter optimization.

Keywords: Amplify-and-Forward, Cooperative Communication System, Imperfect Channel Estimations,

Parameter Optimization

1. Introduction

Recently, a new form of spatial diversity called “coop-

erative diversity” has attracted much research interest

because it provides effective diversity benefits for those

devices that cannot be equipped with multiple antennas

due to their size, complexity, and cost. The main idea

behind cooperative diversity is that the mobile users in

the neighborhood share the use of their antennas to assist

each other with data transmission. Cooperative diversity

is realized by collaboration among the users. If the

channel fading from a user to the destination terminal is

severe, then the information might be successfully

transmitted through the cooperative users. Different co-

operative protocols have been proposed to exploit the

diversity that cooperative systems offer (see, e.g., [1-4]

and the references therein). One commonly used protocol

is the amplify-and-forward (AF) protocol, in which the

relay terminal simply re-transmits a scaled version of the

received signal to the destination terminal. Depending on

the scaling factor, the AF relaying scheme can be further

divided into two types which are called fixed gain AF

systems and variable gain AF systems [5].

The performance of AF cooperative systems has been

studied in the past from different perspectives. For ex-

ample, [4,6,7] analyzed the performance of AF systems

in terms of the outage probability and diversity gain un-

der different assumptions for the amplifier gain. On the

other hand, the authors of [8-12] derived exact expres-

sions for the symbol-error-rate (SER) and presented

various SER bounds for AF cooperative communication

systems. However, all these papers have assumed that

the perfect channel state information (CSI) is available at

both the relay and destination terminal. More recently, [5]

and [13] have studied the performance of AF cooperative

communication systems with channel estimation errors

by means of Monte Carlo simulations. The accurate SER

expression for cooperative communication systems with

a pilot symbol assisted modulation (PSAM) scheme em-

ploying a linear minimum mean square estimator

(LMMSE) is derived in [14]. To the authors’ best

knowledge, no research has been conducted to solve the

This paper will be presented in part at the 69th IEEE Semiannual Ve-

hicular Technology Conference, VTC2009-Spring, Barcelona, Spain,

April 2009.

16 Y. WU ET AL.

problem of parameter optimization and optimum power

allocation for variable gain AF cooperative communica-

tion systems with a PSAM scheme.

The difficulty in optimizing the system design lies in

the fact that the accurate SER expression is given in form

of a double integral [14]. This prevents us from mini-

mizing the SER directly. In this paper, we propose to use

the asymptotic upper bound (AUB) of the SER to over-

come these difficulties. In particular, we derive a tight

expression for the AUB of the SER for the AF coopera-

tive communication system with a PSAM scheme. The

derived AUB has a simple form and no integrating op-

eration is involved. Using the AUB of the SER, we pre-

sent the criterion for the parameter choice in the PSAM

scheme and show that two parameters used in this

scheme, i.e., pilot spacing and Wiener filter length, can

be chosen in a tradeoff between system performance,

pilot overhead, and receiver complexity. With the de-

rived tight AUB, an optimum power allocation problem

is also formulated for the AF cooperative communication

system. Since the optimization of the power allocation is

very complicated, as it is related to many terms, and ob-

taining an analytical solution is unlikely, we propose to

find the optimum power allocation by means of a gradi-

ent search over a continuous range.

The rest of the paper is organized as follows. In Sec-

tion 2, we describe the system model and some prelimi-

naries of the AF cooperative system with a PSAM

scheme. In Section 3, we derive an AUB of the SER for

an AF cooperative communication system with LMMSE.

In Section 4, we first deal with the parameter optimiza-

tion for the PSAM scheme. Then, an optimum power

allocation problem is formulated. A gradient search al-

gorithm is also proposed to find the solution of the opti-

mization problem. Various simulation results and their

discussions are presented in Section 5. Finally, Section 6

contains the conclusions.

The following notation is used throughout the paper:

)(⋅

, T

)(⋅

, H

)(⋅, and 1

)( −

⋅

denote the complex con-

jugate, vector (or matrix) transpose, conjugate transpose,

and matrix inverse, respectively. The symbol ][

⋅

Ede-

notes the expectation operator, || z represents the ab-

solute value of a complex number

, and the complex

Gaussian distribution with mean m and covariance

is denoted by ),( PmNC .

2. System Model

We consider an AF cooperative communication system

which consists of a source, relay, and destination termi-

nal. The block diagram of the system is shown in Figure 1.

We assume that each terminal is equipped with a single

transmit and receive antenna and operates in a

half-duplex mode, i.e., it cannot transmit and receive

simultaneously. We adopt the so-called Protocol II pro-

posed by Nabar et al. [6] as the user cooperative protocol.

This means that two time slots are used to transmit one

data symbol. The source terminal communicates with the

relay and destination terminal during the first time slot.

In the second time slot, only the relay terminal commu-

nicates with the destination terminal. This protocol re-

alizes a maximum degree of broadcasting and exhibits

no receive collisions [6]. To simplify the following

analysis, we consider a symbol-by-symbol transmission,

so that the time slot index 1 and 2 can be dropped.

Throughout this paper, we assume that the system op-

erates in a Rayleigh flat fading environment with per-

fect synchronization, and imperfect channel estimation

is assumed at the receiver. As in [5], we use a PSAM

scheme for the channel estimation. Pilot symbols are

periodically inserted in data symbols with an insertion

period of L symbols. Since the design of an optimal

channel estimator is very complex, we resort to a

suboptimal LMMSE. We further assume that the data

information symbols are equally probable over a con-

stellation set composed of quadrature amplitude modu-

lation (QAM) symbols of size

, and the pilot sym-

bols are selected from a binary phase-shift keying

(BPSK) constellation.

With these assumptions, let us look at the received

signals corresponding to the k th transmitted symbol.

The received signals in the first time slot at the destina-

tion terminal and the relay terminal are given by

()=()() ()

SDS SDSD

rkPhkxk nk+ (1)

()=()() ()

SRS SRSR

rkPhkxk nk+ (2)

respectively, where S

P is the average power of the

transmitted signal at the source terminal, ()

hk and

)( khSR are the channel coefficients from the source

terminal to the destination terminal with distribution

)(0, 2

NC and from the source terminal to the relay

terminal with distribution )(0, 2

NC, respectively.

The symbol )(kx is the kth transmitted symbol from

the source terminal, and )(kn SD and )(knSR are the

additive receiver noises at the destination terminal and

the relay terminal, respectively, with the same distribu-

tion )(0, 0

NNC. Throughout this paper, we assume

Figure 1. Block diagram of the AF cooperative communica-

tion system.

Relay Terminal

hSR hRD

hSD

Source TerminalDestination Terminal

Relay Terminal

Y. WU ET AL. 17

that 1=]|)([| 2

kxE , i.e., the transmitted symbols have

an average energy of 1. According to the Protocol II, the

relay terminal will first normalize the received signal by

a factor of 2

(|() |)

Er k (to ensure the unity of

average energy). Then, the normalized signal will be

amplified and forwarded to the destination terminal

during the second time slot. Therefore, the received

signal at the destination terminal within the second time

slot is given by

()=() ()()

|()|

RDRD SRRD

SSR

rkhkrk nk

Ph kN

+ (3)

where

P is the average power of the transmitted signal

at the relay terminal, )(khRD is the channel coefficient

from the relay terminal to the destination terminal with

distribution )(0, 2

NC , and )(knRD is the additive

receiver noise at the destination terminal with distribution

)(0, 0

NNC. Using (2), we can rewrite (3) as

()=() ()()()

|()|

RDSR RDRD

SSR

rkhkhkxk nk

Ph kN

′

(4)

where

()=()()()

|()|

RDRD SRRD

SSR

nk hknknk

Ph kN

′+

(5)

Assuming that ()

nk and ()

nk are independent,

it can be shown that the noise term ()

′ is a com-

plex Gaussian random variable with distribution

)1)|)(|/((0, 00

2NNkhPP SRSR++NC . Since the

PSAM scheme is used for the channel estimation, the

packed transmission can be divided into blocks by pilot

symbols. In each block, there are L symbols in which

the first time slot is assigned to a pilot symbol and the

remaining 1−L symbols are assigned to data symbols.

The channel estimation at each symbol position in a

block is obtained using 1

N pilot symbols on the left-

hand side of the symbol position and 2

N pilot symbols

on the right-hand side of the symbol position. Therefore,

=NNN+ pilot symbols are used to estimate the

channel coefficient of the desired symbol position.

Let us denote the pilot symbols employed to estimate

the channel gain )(khSD of the desired data symbol

)(kx as an 1×N vector 1

= [((1)),...,

SD xk LNl−−−p

( ),(),...,()]

klxkLlxk LNl−+− +−, where =1, 2,...,l

1L− is the offset of the desired data symbol )(kx to

the closest pilot symbol on its left side. Using (1), we

obtain the received signal vector SD

r, corresponding to

the transmitted pilot vector SD

p, at the destination ter-

minal as

=()

SDSSD SDSD

Pdiag +rphn (6)

where 1

= [((1)),...,(),

SD SDSD

hkLNlhkl

−

−− −h

(),..., ()]T

SD SD

hkLlhkLNl+−+− and

= [((1)),...,(),

SD SDSD

nkLNl nkl

−

−− −n

(),...,( )]

SD SD

nkLl nkLNl+−+ − are the channel

coefficient and noise component at the pilot symbols’

position for estimating )(khSD , respectively.

Without loss of generality, we assume that positive

unit energy symbols are transmitted as pilot symbols, i.e.,

p is an all-one vector. Then, (6) simplifies to

SDS SDSD

P+rhn (7)

With these observations, the channel estimate for

)(khSD can be obtained by the LMMSE as [15]

()=

SDSD SD

hk wr (8)

where 1

=()

SD hSDSDSD

l−

wc C is an 1N× LMMSE

ﬁlter vector, =[ ]

SD SD

SD E

Crr and ,()=

hSDSD l

[()]

SD SD

Eh kr are the autocorrelation matrix of SD

and cross-correlation vector ()

hk and SD

r, respec-

tively. From the LMMSE theory [15], we know that

()

is distributed as

(0,()

hSDSDl

()

−

,())

hSD SDl

c. Let us define the discrete autocorrelation

function of )(khSD as ()=[ ()

SD SD

REhk

()]

Then, using the system model under consideration of the

channel properties described above, we can finally ex-

press SD

C and ()

hSDSD l

c as

(0)( )((1) )

()(0)(( 2))

((1) )((2) )(0)

SSDSSD SSD

SSD SSDSSD

RN PRLPRNL

RLPRNPRNL

PR NLPRNLPRN

+−

⎡

⎤

⎢

⎥

+−

⎢

⎥

⎢

⎥

⎢

⎥

−− +

⎣

⎦

MMOM

(9)

()=[((1)),,(),

((1))]

hSSD SSD

SD SD

SSD

lPRLNlPRLl

PR LNl

−

−−−

−−

(10)

From the LMMSE ﬁlter vector SD

w, we can see that

each data symbol position in a block requires a different

estimator. However, due to the periodic pilot insertion,

an identical estimator will be adopted at the same data

symbol positions across all blocks in a packet. Therefore,

without loss of generality, we will only consider 1

−

18 Y. WU ET AL.

different estimators for the data symbol positions in one

particular block in the following analysis and employ the

index

instead of k to distinguish them. With this in

mind, we can express the estimation error of the

th es-

timator as

()= ()()

SDSD SD

el hlhl

− (11)

Furthermore, the estimation error )(leSD is distrib-

uted as ))((0,2

SDe

NC , where 22

())= ()

eSDSD h

SD SD

σσ

−r

() ()

SDSD SD l

−

Cc . From (11) it follows that we can

model the channel gain )(lhSD as the sum of the chan-

nel estimate )(

ˆlhSD and the estimation error )(leSD , i.e.,

()= ()()

SDSD SD

hl hlel+ (12)

Similarly, we can model the channel gain from the

source terminal to the relay terminal )(lhSR and the

channel gain from the relay terminal to the source termi-

nal )(lhRD as

()= ()()

SRSR SR

hl hlel

+ (13)

()= ()()

RDRD RD

hlhlel+ (14)

where ()

, ()

el, ()

, and ()

el can be

attained using the same procedure as above.

3. AUB Analysis for AF Cooperative Systems

With the above assumption and the estimated channel

gains, maximum ratio combining (MRC) [16] can be

applied at the destination terminal to minimize the SER

of the system. Deﬁne MB 1/1= − and 1)3/(=

−

MKQ.

The accurate SER expression for the considered system

with M-QAM is [14]

=()

PPl

−

−∑ (15)

where

=)(lP

/2 1

14()

(, ,)

2sin

() 2sin

lx dxd

πθ

αθ

∫∫

/4 1

14()

(, ,)

2sin

() 2sin

lx dxd

πθ

αθ

−

∫∫

[]

3() ()

(,,)=(,,)(,,)

llx

lxsexpalxs blxs

αβ

⎛⎞

−+

⎜⎟

−

⎝⎠

223

()

(,, )=()(2()())(,, )

al xslsllx lxs

αααυ

+− ++

(,, )=(2()())lxssll xsx

υαα

−+−+

()

223

(,, )=

() [()()]

bl xs

ls llxsx

ααα

+− +−

123 3

()=()()()[() ()]llllexpll

αα ααβ

[()]

()= [()]

SeSD

ShSD

PlN

lPl

ασ

0,0

ˆ,

()()

()= ()

ReRD

RhRD

PN lN

lPl

ασ

ˆ,

()

()= ()

SeSR

ShSR

PlN

lPl

ασ

222 2

,,, 0

,00,0

()() (1())

()=(())[()()]

SReSReRDeRD

SeSRR eRD

PPlllN

lPlNPN lN

σσ σ

βσσ

++ +

Note that although the numerical evaluation of the

above expression of the SER is straightforward, it is not

insightful in terms of its dependence on the system pa-

rameters like the pilot spacing or power allocation be-

tween the source terminal and relay terminal. To opti-

mize the system parameters using (15) seems to be in-

tractable. Therefore, a simple and insightful AUB of the

SER is of special interest.

As derived in [14], we know that the instantaneous

SNR of the output signal from the MRC detector is the

sum of two terms: the first term is determined by the

direct signal from the source terminal and the second

term is determined by the relay signal from the relay

terminal. Using the result in [14], the instantaneous SNR

determined by the relay signal can be rewritten as

() ()

()= ()() ()

xlx l

lxll

(16)

where

|()|

()= () ()

hRD

xl ll

ασ

|()|

()= () ()

hSR

xl ll

ασ

From the definition of )(l

in (15), it can be found

that 0>)(l

. Therefore, if we set 0=)(l

in (16),

we get an upper bound of the )(

. With this observa-

tion, we obtain an upper bound of the SER by simply

setting 0=)(l

in (15). After some manipulations, we

obtain the AUB of the SER

12 3

=()()()

Plll

ααα

−

⎡⎤

⎣⎦

−∑ (17)

Note that 0/R

NP tends to zero in high SNR regions.

Y. WU ET AL. 19

Therefore, the AUB of the SER can be further simplified as

[]

12 3

=()()()

Plll

ααα

−

′+

−∑ (18)

where

ˆ,

()

()= ()

()

ReRD

RhRD

PlN

αα

′≈

As shown in Section 5, the AUB of the SER in (18) is

very close to the exact SER, especially in high SNR regions.

4. Parameter Optimization

As can be seen from (18), the AUB of the SER is deter-

mined by the function

12 3

( ,,,)=()()()

LNP Plll

ααα

−

′+

⎡⎤

⎣⎦

∑ (19)

It should be pointed out that 123

(), (),()lll

αα

′, for

=1,2, ,1lL−L are related to the parameters ,, ,

LNP

and R

P. This can be deduced from their definition in

(15). As a result, we establish the relation between the

AUB of the SER and the parameters which need to be

optimized. Using the above metric as an optimality crite-

rion, we can now study the parameter optimization prob-

lem of the considered system. In principle, we should

optimize the four parameters ,, ,

LNP and R

P jointly

to get the optimum system performance. However, the

joint optimization problem is difficult to solve due to the

form of the metric in (19). Therefore, we propose to op-

timize the parameters of the PSAM and the power allo-

cation separately as shown below. Although this method

is suboptimal, our simulation results show that this

method provides a satisfactory performance.

4.1. PSAM Parameter Optimization

For the PSAM scheme, there exists a tradeoff between

the system performance, pilot overhead, and receiver

complexity. While a smaller pilot spacing L leads to a

better channel estimation, the overhead imposed by the

pilot symbols reduces the effective SNR and transmis-

sion efficiency. A similar conflict also exists for the

choice of the Wiener ﬁlter length N. A larger value of

N is required to improve the channel estimation, but

this will increase the receiver complexity. Therefore, the

parameters L and N should be accordingly chosen

by taking all these factors into account. We will use the

metric in (19) as the optimality criterion for determining

appropriate values of L and N. In particular, we will

set /2== PPP RS , where

is the total transmitted

power, and try to minimize the metric (, ,,)

LNP P

which characterizes (asymptotically) the performance of

the considered system. Since there is no closed-form

solution to this minimization problem, the suitable values

of L and N can only be obtained by examining the met-

ric ),,,( RSPPNLM , which is presented in the next section.

4.2. Power Allocation Optimization

Now, we will study the power allocation problem for the

considered system. We assume that the parameters L,

N are fixed and the total transmitted power is

RS PPP

=. Under these constraints, we are going to

optimize S

P and R

P so that the SER performance of

the system is minimized. Since the metric

),,,( RS PPNLM characterizes (asymptotically) the SER

performance of the considered system, we can state the

power allocation problem as follows.

Problem Statement: Given positive integers L, N,

find a pair of real numbers S

P and R

P such that the

metric function ),,,( RS PPNLM is minimized under

the power constraint of the transmitted power which is

fixed to P, i.e.,

{, }=(,, ,)

SR SR

PPP

PParg minMLNPP

(20)

Note that the derivatives of the metric

),,,( RS PPNLM with respect to S

P and R

P will be

expressed as the sum of several high-order polynomials.

This prevents us from finding a closed-form solution for

P and R

P. Therefore, we propose to find the optimum

power allocation by means of a gradient search over a con-

tinuous range.

5. Numerical Results

In this section, we will first verify by simulations the

correctness of the derived expression found for the AUB

of the SER. We will then present some examples illus-

trating the parameter optimization procedure. We con-

sider an AF cooperative communication system with

4-QAM modulation formats using the PSAM scheme for

the channel estimation. Unless stated otherwise, the fol-

lowing parameters are used in the numerical work. We

set RS PP = and assume that the variance of the noise

was chosen to be 1=

N. We also assume that the com-

plex channel gains are described by the autocorrelation

functions 0max

()=()=()=(2)

SD SR RDs

RR JfT

κκπκ

where )(

0xJ is the zeroth order Bessel function of the

first kind, max

f is the maximum Doppler frequency,

and s

T is the symbol duration. Note that the variances

of the complex channel gains are normalized to unity.

We further assume that a pilot spacing of 6=L is used

20 Y. WU ET AL.

in the PSAM scheme and the LMMSE with 6=N is

used for the channel estimation. Note that the power loss

resulting from the pilots is accounted for all curves.

Figure 2 shows the theoretical AUB and the Monte

Carlo simulation results of the SER for the AF coopera-

tive communication system with 4-QAM. The results are

presented for two different levels of the normalized

maximum Doppler frequency, i.e 0.01=

max s

Tf and

max =0.05

fT . From Figure 2, we observe that the AUB

ﬁts very well with the simulated SER for both cases in

high SNR regions.

Assuming =

PP, Figure 3 plots the metric

(, ,,)

LNP P as a function of the pilot spacing L

and the Wiener ﬁlter length N at SNR=20 dB with a

normalized maximum Doppler frequency of

max =0.05

fT . We observe that for a given N, the

metric ),,,( RS PPNLM decreases rapidly with L for

4L≤. This is because the energy spent by pilot sym-

bols decreases rapidly with L for 4L≤. As a result,

the energy assigned to each data symbol increases, and

this leads to a fast decrease in the SER. On the other

hand, we also find that the metric (, ,,)

LNP P

increases with L for 7>L. This is easy to under-

stand since large L will increase the channel estima-

tion error, and thus increase the SER. By taking all these

factors into account, we suggest to choose =6L. Now

let us consider the choice of N. From Figure 3, we

observe that for a given L, the metric (, ,,)

LNP P

decreases rapidly with N for 6N≤. However, the

decrease in ),,,( RS PPNLM obtained by increasing

N beyond 6 is minor. Since large N leads to a high

receiver complexity, we suggest to choose =6N for

this particular case.

Figure 2. Comparison of the theoretical AUB and simula-

tion results of the SER for the AF cooperative communica-

tion systems with various values of the normalized maxi-

mum Doppler frequency max

Now, we turn our attention to the power allocation

strategies. As discussed earlier, we use a constrained

gradient-search algorithm to find the power tradeoff be-

tween the source terminal and the relay terminal. For

example, in case of 222

== =1

SD SR RD

σσσ

, and

0.01=

max s

Tf we find the optimum power allocation is

/=0.66

PP , and 0.34=/PPR. The performance

comparison of the equal power scheme and the optimum

power allocation scheme is presented in Figure 4. This

figure illustrates that the performance of the system with

optimum power allocation is better than that of the sys-

tem with equal power allocation. We can see that a

greater performance improvement can be achieved from

Figure 3. The metric ()

M L,N,P,P at SNR=20 dB with

a normalized maximum Doppler frequency of

max0.05

fT= .

Figure 4. SER performance of the AF cooperative commu-

nication systems assuming equal power allocation and op-

timum power allocation.

Y. WU ET AL. 21

Figure 5. SER performance of the AF cooperative commu-

nication systems assuming equal power allocation and op-

timum power allocation.

the optimum power allocation scheme if the ratio

SR RD

σ σ

decreases. For example, in case of

1== 22

SRSD

σσ

, 2=10

and =0.01

max s

fT , we find

the optimum power allocation is 0.83=/PPS, and

0.17=/PPR. The corresponding performance compari-

son is plotted in Figure 5. As can be seen from this figure,

the optimum power allocation scheme leads to an im-

provement of 1.5 dB over the equal power scheme. This

further demonstrates the effectiveness of the power allo-

cation optimization.

6. Conclusions

We dealt with the problem of parameter optimization of

AF cooperative communication systems with a PSAM-

based LMMSE scheme used for the channel estimation.

A tight and easy to-evaluate AUB of the SER was de-

rived for the considered system with QAM constella-

tions. Using the derived AUB, we proposed a criterion

for the choice of parameters in the PSAM scheme, i.e.,

pilot spacing and Wiener filter length. We also formu-

lated an optimum power allocation problem for the con-

sidered system. The optimum power allocation was

found by means of a gradient search over a continuous

range. Some illustrative examples for the parameter op-

timization were presented. The benefits of parameter

optimization were demonstrated by the numerical re-

sults.

7. References

[1] G. Kramer, M. Gastpar, and P. Gupta, “Cooperative

strategies and capacity theorems for relay networks,”

IEEE Transactions on Information Theory, Vol. 51, No. 9,

pp. 3037-3063, September 2005.

[2] A. Reznik, M. R. Kulkarni, and S. Verdu, “Degraded

Gaussian multirelay channel: Capacity and optimal power

allocation,” IEEE Transactions on Information Theory,

Vol. 50, No. 12, pp. 3037-3046, December 2004.

[3] A. Host-Madsen and J. Zhang, “Capacity bounds and

power allocation for wireless relay channels,” IEEE

Transactions on Information Theory, Vol. 51, No. 6, pp.

2020-2040, June 2005.

[4] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Coop-

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