Doubly Iterative Receiver for MIMOAmplify-and-Forward Relay Networks

The-Hanh Pham†, Hari Krishna Garg†, Ying-Chang Liang‡ and Arumugam Nallanathan§† National University of Singapore, Singapore 119260

({elepth,eleghk}@nus.edu.sg)

‡ Institute for Infocomm Research, 1 Fusionopolis Way, #21-01 Connexis, Singapore 138632(ycliang@i2r.a-star.edu.sg

)§ Division of Engineering, King’s College London, London WC2R 2LS, U.K.

(arumugam.nallanathan@kcl.ac.uk

)

Abstract—In this paper we consider a Multi-Input Multi-Output (MIMO) wireless relay network which comprises ofa source terminal, a destination terminal and a relay. Thetransmission from the source terminal to the destination isdivided into two phases. During Phase I, the source terminalsends its signals to the relay. The relay only amplifies andforwards its received signals to the destination terminal duringPhase II, i.e., the network works in the Amplify-and-Forward(AF) mode. In this mode, the noise elements at the relay atPhase I are also amplified and forwarded through a link (fromthe relay to the destination terminal). It makes the overallnoises across antennas of the destination terminal correlated.This correlation is characterized by a noise covariance matrix.Hence, for the purpose of data detection at the destinationterminal, we need to estimate channel coefficients as well as thenoise covariance matrix. In the considered network, an errorcorrecting code (ECC) is used. We propose a doubly iterativereceiver to estimate channel information and decode transmittedsignals. The estimation part is an application of the ExpectationConditional Maximization (ECM) algorithm. Turbo-equalizationis used to provide soft information of transmitted signals to theECM-based algorithm. Simulation results show that our doublyiterative receiver provides an excellent BER performance.

I. INTRODUCTION

Communications based on relays have drawn great atten-tions recently [1–3]. A typical relay-based communicationnetwork comprises of a source terminal, S, a destinationterminal, D, communicating with each other via a relay, R. Thesignal transmission from the source terminal to the destinationterminal is usually divided into two Phases. During PhaseI, the source terminal transmits its signals to the relay. AtPhase II, the relay amplifies and forwards its received signalsfrom the source terminal to the destination terminal, i.e., thenetwork operates in the amplify-and-forward (AF) mode. TheAF mode requires simpler implementations at the relay andno information about structure of signals from the sourceterminal.

When a network works in the AF mode, as pointed out in [4,5], the overall channel information from the source terminalto the destination terminal is estimated at the destinationonly. Note that in [4, 5], authors consider the case where thesource and destination terminals have only one antenna each.

This research was supported by NUS research grants, grant numbers areR-263-000-421-112 and R-263-000-436-112.

On the other hand, it is well-known that deploying multi-antenna transmission over multi-input multi-output (MIMO)channels is effective in increasing the channel capacity andcombating multipath fading [6, 7]. However, if we deploymultiple antennas at both the source and destination terminals,due to the noise propagation from the relay to the destinationterminal during Phase II, the overall noise elements at antennasof the destination terminal become correlated. The correlationof those elements are characterized by a noise covariancematrix. Hence, deploying multiple antennas at two terminalsforces us to estimate another set of parameters beside thechannel coefficients, i.e., the noise covariance matrix, at thedestination terminal.

In this paper we consider a relay network in which multipleantennas are deployed at the source, destination terminals andat the relay. An error correcting code (ECC) is used at thesource terminal. We design a doubly iterative receiver at thedestination terminal. The iterative receiver is to estimate thechannel information, i.e., the overall channel coefficients fromthe source terminal to the destination terminal and the noisecovariance matrix. It also decodes the transmitted signals fromthe source terminal. Information given by detected signalsat the destination terminal are used in the process of esti-mating the channel information. The estimation algorithm isan application of the Expectation Conditional Maximization(ECM) algorithm [8] which is an variation of the Expecta-tion Maximization (EM) algorithm [9–11]. The ECM-basedestimation algorithm requires soft information in terms of thelog-likelihood ratios (LLRs) of the transmitted symbols. Thisinformation is provided by the turbo-equalization [12].

The rest of this paper is structured as follows. Section IIdescribes the system model. The proposed doubly iterativereceiver is given in Section III. Simulation results are providedin Section IV. Finally, conclusions are given in Section V.

Notation: The capital bold letters denote matrices and thesmall bold letters denote row/column vectors. |a|, a∗, �{a}and �{a} denotes the absolute value, conjugate value, real partand imaginary part of a number a, respectively. Transpose,Hermitian transpose of a vector/matrix are denoted by (·)Tand (·)H. diag{a} denotes the diagonal matrix with diagonalelements that belong to a. tr{A} stands for the trace operation.Determinant of matrix A is det{A}. The (m,n)th element ofA is denoted by [A]m,n. In stands for the identity matrix ofsize n. 0n×m is a n × m size zero matrix. The hyperbolic

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978-1-4244-6398-5/10/$26.00 ©2010 IEEE

tangent is denoted by tanh{·}. E{·} denotes expectationoperator.

II. SYSTEM MODEL

We consider a relay network in which a source terminal, S,communicates with a destination terminal, D, via a relay, R.The S, R and D have Nt1 , Nr and Nt2 antennas, respectively.The channel coefficient between the kth antenna of S andthe lth antenna of R is denoted by hl,k. We assume thathl,k is a complex Gaussian random variable with zero meanand variance β2

1 . Meanwhile, the channel coefficient betweenthe kth antenna of D and the lth antenna of R is denotedby gl,k which is a complex zero mean Gaussian randomvariable with variance β2

2 . An EEC is deployed at S in whicha sequence of information bits m is encoded to producea sequence of coded bits. Those coded bits are fed to aninterleaver and then to a modulator which is a binary phaseshift keying (BPSK) modulator in this paper. The outputs ofthe modulator are delivered to transmit antennas of S througha serial-to-parallel (S/P) converter. More specifically, we havea sequence of symbols after the modulator

{s(n)

}TNt1−1

n=0.

Those symbols are S/P converted into T blocks of size Nt1 .The tth block, which is transmitted from S at time t, is denotedby s(t) = [s(t; 1) s(t; 2) · · · s(t;Nt1)]

T where s(t; k) =s(k + (t − 1)Nt1), k = 1, 2, . . . , Nt1 and t = 1, 2, . . . , T . Wefurther assume that |s(t, k)|2 = E1.

During Phase I, S transmits T signal blocks, {s(t)}Tt=1, to

R. The received signal vector at R at time t can be written as

y(t) = Hs(t) + n(t), t = 1, 2, . . . , T, (1)

where

y = [y1(t) y2(t) · · · yNr(t)]T ∈ C

Nr×1 (2)

H =

⎡⎢⎢⎢⎣

h1,1 h1,2 · · · h1,Nt1

h2,1 h2,2 · · · h2,Nt1...

.... . .

...hNr,1 hNr,2 · · · hNr,Nt1

⎤⎥⎥⎥⎦ ∈ C

Nr×Nt1 , (3)

n(t) = [n1(t) n2(t) · · · nNr(t)]T ∈ C

Nr×1, (4)

and n(t) ∼ CN (0Nr×1, σ2rINr

).The vector y(t), t = 1, 2, . . . , T , is then amplified at R

by a real coefficient α. The value of α is to guarantee thatthe average power transmitted from each antenna of R is Er.The resultant vector αy(t) is transmitted from R to D at time(t + T ) during Phase II.

The received signal vector at D at time (t + T ) is given by

z(t + T ) = αGHy(t) + v(t + T )= αGHs(t) + αGn(t) + v(t + T )= Ks(t) + w(t + T ), t = 1, 2, . . . , T, (5)

where

K �[k1 · · · kNt1

]= αGH ∈ C

Nt2×Nt1 , (6)

G =

⎡⎢⎢⎢⎣

g1,1 g1,2 · · · g1,Nr

g2,1 g2,2 · · · g2,Nr

......

. . ....

gNt2 ,1 gNt2 ,2 · · · gNt2 ,Nr

⎤⎥⎥⎥⎦ ∈ C

Nt2×Nr , (7)

v(t + T ) = [v1(t + T ) · · · vNt2(t + T )]T ∈ C

Nt2×1, (8)

w(t + T ) = αGn(t) + v(t + T ), (9)

and v(t + T ) ∼ CN (0Nt2×1, σ22INt2

). Note that the overallnoise vector w(t + T ) at D is a realization of a Gaussianrandom vector with zero mean and covariance matrix givenby

Σ � E{w(t + T )wH(t + T )

}= α2σ2

rGGH + σ22INt2

. (10)

If T vectors, {z(t+T )}Tt=1, are grouped into a matrix Z �[

z(T +1) z(T +2) · · · z(2T )] ∈ C

Nt2×T , (5) can be writtenin another form as follows

Z = KS + W , (11)

where

S =[s(1) s(2) · · · s(T )

], (12)

W =[w(T + 1) w(t + 2) · · · w(2T )

]. (13)

Our objective is to estimate the parameters {K,Σ} and todetect the information bits of S at D. The proposed doublyiterative receiver to do this task is presented in the nextSection.

III. PROPOSED DOUBLY ITERATIVE RECEIVER

In this section we describe our proposed doubly iterativereceiver which is illustrated in Fig. 1. The receiver has twomajor blocks: the first one is the turbo-equalization block andthe other is the ECM-based channel information estimationalgorithm block. The turbo-equalization block receives thechannel information, i.e., the estimated values of K and Σ.Then it provides the soft information of transmitted signals tothe ECM-based channel information estimation blocks. Thedetails of the two blocks are explained below.

A. Turbo-equalization block

Suppose that after the (e−1)th iteration of the ECM-based

algorithm we obtain K[e]

and Σ[e]

. The turbo-equalizationblock calculates the approximation of a posteriori probabilityof each element of the information sequence m in an iterativemanner. Turbo-equalization consists of one soft-input-soft-output (SISO) equalizer and one SISO channel decoder. In thisconnection, SISO equalizer ignores the structure of the ECC inuse; meanwhile, SISO channel decoder ignores the effects ofthe channel. The SISO equalizer can be any equalizer which

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accepts soft inputs and produces soft outputs. In this paper weadopt the effective Minimum Mean Square Error (MMSE)-with Soft Interference Cancelation (SIC) equalizer which isproposed in [13] for CDMA systems. The algorithm can beextended to our correlated noise scenario.

Suppose that after the (p − 1)th iteration of turbo-

equalization using K[e]

and Σ[e]

, the SISO decoder pro-vides to the SISO equalizer the LLR values of s(t; k) asλ

[e,p−1]d (t; k), t = 1, 2, . . . , T and k = 1, 2, . . . , Nt1 . The

mean and variance of s(t; k) are determined based on thisLLR value as

s[e,p−1](t; k) =√

E1tanh(λ

[e,p−1]d (t; k)

2

), (14)

s[e,p−1](t; k) = E1 − (s[e,p−1](t; k))2. (15)

The equalizer first performs soft interference cancelation fors(t; k) as follows

z[e,p−1]k (t + T ) � z(t + T ) − K

[e]

k s[e,p−1]t,k , (16)

where K[e]

k is the matrix K[e]

without its kth column ands[e,p−1]t,k = [s[e,p−1](t; 1), · · · , s[e,p−1](t; k − 1), s[e,p−1](t; k +

1), · · · , s[e,p−1](t;Nt1)]T . After that, the residual interference

is further reduced by using the MMSE filtering, i.e., to findq[e,p−1](t; k) according to the following criterion

q[e,p−1](t; k) = arg minq[e,p−1](t;k)

E{|s(t; k)

− (q[e,p−1](t; k))Hz[e,p−1]k (t + T )|2}. (17)

The vector q[e,p−1](t; k) is given by

q[e,p−1](t; k)

=(E1k

[e]

k (k[e]

k )H + K[e]

k A[e,p−1]t,k (K

[e]

k )H + Σ[e]

)−1

E1k[e]

k ,

(18)

where A[e,p−1]t,k = diag{s[e,p−1](t; 1), · · · , s[e,p−1](t; k −

1), s[e,p−1](t; k + 1), · · · , s[e,p−1](t;Nt1)} and k[e]

k is the kth

column of K[e]

.The output of the MMSE filtering is approximated as [13]

s[e,p−1](t; k) = (q[e,p−1](t; k))Hz[e,p−1]k (t + T )

≈ ν[e,p−1](t; k)s(t; k) + ε[e,p−1](t; k), (19)

where ν[e,p−1](t; k) = (q[e,p−1](t; k))Hk[e]

k and ε[e,p−1](t; k)is a zero mean complex Gaussian random variable. Based ons[e,p−1](t; k), the LLR value from the equalizer forwarded tothe decoder is determined by

λ[e,p]e (t; k) =

4�{s[e,p−1](t; k)}1 − ν[e,p−1](t; k)

. (20)

The decoder uses λ[e,p]e (t; k) and the code structure to refine

λ[e,p]d (t; k). The latter is then provided to the ECM-based

algorithm which is presented in the next subsection.

B. ECM-based estimation algorithm

The matrix Z in (11) is considered as the missing data spacefor the parameter {K,Σ}. We define {Z,S} as the completedata space for {K,Σ}. The eth iteration of the GEM-basedestimation algorithm consists of the following two steps.

1) E-step: In this step, we compute the following quantity

Q(K,Σ

∣∣K [e], Σ

[e]) � E{ln f

(Z,S

∣∣K,Σ)∣∣Z, K

[e], Σ

[e]}.

(21)

We have

f(Z,S

∣∣K,Σ)

= f(Z

∣∣S,K,Σ)f(S

∣∣K,Σ). (22)

where

f(Z

∣∣S,θ)

=1

(det{Σ})T

× exp{−tr

{(Z − KS)HΣ−1(Z − KS)

}}, (23)

and f(S

∣∣K,Σ)

is independent from {K,Σ}.Replacing (23) into (21), (21) can be written as

Q(K,Σ

∣∣K [e], Σ

[e])= C1 − T log det{Σ}

− tr{Σ−1

(ZZH − HS

[e,p]ZH − Z(S[e,p])T HH

+ HS[e,p]

HH)},(24)

where C1 is a constant that is not related to parameter {K,Σ}and

S[e,p] = E

{S

∣∣Z, θ[e]}

, (25)

S[e,p]

= E{SST ∣∣Z, θ

[e]}. (26)

The above two matrices are determined by

[S

[e,p]]k,t

=√

E1tanh(λ

[e,p]d (t; k)

2

), (27)

[S

[e,p]]k1,k2

=

{TE1 if k1 = k2∑T

t=1

[S

[e,p]]k1,t

[S

[e,p]]k2,t

if k1 �= k2,

k, k1, k2 = 1, 2, . . . , Nt1 , t = 1, 2, . . . , T. (28)

2) CM-step: In this step we need to determine

{K[e+1], Σ

[e+1]} to maximize (24). This maximizationis highly complicated due to high-dimensional {K,Σ}.Therefore, CM-step maximizes (24) with respect to (w.r.t) K(Σ) while Σ (K) is kept at its most recently updated value.Specifically, we have:

a) Step 1: In this step we determine K[e+1]

by thefollowing equation

K[e+1]

= arg maxK

Q(K, Σ

[e]∣∣K [e], Σ

[e])= arg max

K

{tr

{(Σ

[e])−1

(ZZH − KS

[e,p]ZH

− Z(S[e,p])T KH + KS[e,p]

KH)}}.(29)

Differentiating the function inside {·} in (29) w.r.t. K [14]and equating the result to 0, we obtain

K[e+1]

= Z(S

[e,p])T (S

[e,p])−1. (30)

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b) Step 2: In this step we determine Σ[e+1]

by

Σ[e+1]

= arg maxΣ

Q(K

[e+1],Σ

∣∣K [e], Σ

[e])= arg max

Σ

{−T log det{Σ}

− tr{(Σ

(ZZH − K

[e+1]S

[e,p]ZH − Z(S[e,p])T (K

[e+1])H

+ K[e+1]

S[e,p]

(K[e+1]

)H)}}

. (31)

Similar to Step 1, we differentiate the function inside {·} in(31) w.r.t. Σ and equate the result to 0. This leads to

Σ[e+1]

=1T

(ZZH − K

[e+1]S

[e,p]ZH

− Z(S[e,p])T (K[e+1]

)H + K[e+1]

S[e,p]

(K[e+1]

)H).(32)

IV. SIMULATION RESULTS

In this section we provide simulation results for a relaynetwork with Nt1 = Nr = Nt2 = 4. At the source terminal,a binary information sequence is encoded to code bits usinga recursive convolutional ECC with 1/2 rate, memory 2and generator [5 7]. The channel coefficients of both linksare realizations of independent zero mean Gaussian randomvariables with variance β2

1 = β22 = 1. We assume that those

coefficients are static over the transmission of 16 pilot vectorsand 256 data vectors from S. We choose E1 = Er = 1 andassume that the average signal-to-noise ratios (SNRs) of the

two links are equal. The pilot vectors are used to obtain K[0]

based on the least-square principle. The initial value of Σ isdetermined by

Σ[0]

=1T

T∑t=1

z(t + T )zH(t + T ). (33)

A. BER performance

Fig. 2 illustrates the BER performance of the above set-upin which the number of iterations of turbo-equalization is 3and the number of iterations of the ECM-based algorithm is4. The “Initial” curve in Fig. 2 is the BER performance of

the turbo-equalization using K[0]

and Σ[0]

. We observe thatonly one iteration of the ECM-based algorithm can greatlyimprove the BER performance at high SNR values. The morenumber of iterations of the ECM-based algorithm, the lessBER improvement. In Fig. 2 we also provide a benchmarkwhich is the BER performance of turbo-equalization withperfect K and Σ. At the BER level of 10−4, the gap betweenthe proposed receiver with 4 iterations of the ECM-basedalgorithm and the benchmark is less than 1 dB.

B. MSE performance

Fig. 3 provides the MSE performance of K. As compared

with the estimated K[0]

given by pilot vectors, our algorithmimproves the quality of estimated K greatly.

Fig. 4 illustrates the MSE performance of Σ. Similarly forthe case of K, the difference between estimated Σ and itstrue value becomes smaller as the number of iterations of theproposed receiver increases.

V. CONCLUSIONS

In this paper we consider an MIMO one-way relay networkconsisting of a source, a destination and a relay terminal.The transmission process from the source terminal to thedestination terminal are divided into two Phases. During PhaseI, the signals from the source terminal are sent to the relay.In Phase II, the received signals at the relay are amplifiedand forwarded to the destination terminal. Due to the AFworking mode, noise elements across all antennas of thedestination terminal become correlated. We propose a doublyiterative receiver to estimate the overall channel coefficientsfrom the source to the destination terminal as well as the noisecovariance matrix based on the ECM algorithm. Simulationresults show that our proposed iterative receiver provides BERclose to the ideal scenario.

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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the WCNC 2010 proceedings.

Turbo-equalization

ECM-based algorithm

[ ]1ˆ e+K [ ]1ˆ e+�

Z

[ ] ( ), ;e pd t kλ

Fig. 1. Block diagram of the proposed iterative receiver.

0 2 4 6 8 10 1210

−4

10−3

10−2

10−1

100

SNR (in dB)

BE

R

InitialECM − 1 iterationECM − 2 iterationsECM − 3 iterationsECM − 4 iterations

Turbo−equalizationprefect channel information

Fig. 2. BER performance.

0 2 4 6 8 10 1210

−2

10−1

100

101

SNR (in dB)

MSE

from pilot vectors

ECM − 1 iteration

ECM − 2 iterations

ECM − 3 iterations

ECM − 4 iterations

Fig. 3. MSE performance of K.

0 2 4 6 8 10 1210

−2

10−1

100

101

102

103

SNR (in dB)

MSE

Initial

ECM − 1 iteration

ECM − 2 iterations

ECM − 3 iterations

ECM − 4 iterations

Fig. 4. MSE performance of Σ.

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