IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 4, APRIL 2013 689

Robust Relay Precoding Design for Two Way Relay Systems withDelayed and Limited Feedback

Ji Wang, Student Member, IEEE, Yifei Wang, Xin Gui, and Ping Zhang, Member, IEEE

Abstract—This letter addresses the robust linear precoding de-sign at the relay node in two way relay systems with independentsource-relay and relay-source channels. An antenna configurationin which both the two source nodes S1 and S2 and the relaynode R are equipped with multiple antennas, is assumed. Dueto delayed and finite-rate feedback, only outdated and quantizedchannel direction information (CDI) of the downlink channel isavailable at the transmitter of the relay during the broadcast(BC) phase. To tackle the performance degradation caused bythe joint effects of channel quantization error and feedback delay,we propose a robust precoding scheme based on minimizingthe expected weighted sum mean squared error (WS-MSE)conditioned on the induced channel uncertainties subject torelay power constraints. We also find that CDI-only feedback issufficient for calculating the robust precoding matrix. Numericalresults validate the robustness of the proposed scheme.

Index Terms—Two way relay, delayed limited feedback, meansquared error (MSE), robust precoding.

I. INTRODUCTION

THE two way relaying technology has attracted muchresearch interest in the past few years, because it can

alleviate the spectral efficiency loss brought by traditionalone way relaying [1]. When the relay node is equipped withmultiple antennas, additional spatial degrees of freedom canbe provided to enhance link reliability. Recently, several multi-antenna based relay precoding schemes have been proposedto exploit spatial diversity [1]-[3]. In [1], the authors derivedoptimal relay beamforming structure aiming to achieve thecapacity region of the two way relay system. On the otherhand, linear relay precoding designs based on the minimummean square error (MMSE) criterion were proposed in [2]-[3].

However, all of the aforementioned beamforming schemesare established by assuming that perfect channel state informa-tion (CSI) is available at both the relay and the sources, whichmay not be a feasible assumption in practice. Hence, severalrobust beamforming schemes have been proposed by takingchannel uncertainties into consideration [4]-[6]. However, themajority of the current research works in the literature assumereciprocal channels and no robust precoder has been designedfor the non-reciprocal feedback based channel model.

In this letter, we consider more general and practical casewhere the corresponding source-relay and relay-source chan-nels are independent, i.e., non-reciprocal, as in [5]. In this

Manuscript received December 23, 2012. The associate editor coordinatingthe review of this letter and approving it for publication was T. Oechtering.

This work was supported by NSFC-AF-60911130512, National S & TMajor Projects (2012ZX03003012-004), Creative Research Groups of China(61121001), PCSIRT No.IRT1049, project under No.201105, and the Post-graduate Innovation Fund of Zhongxun (CITC) & SICE, BUPT, 2011-2012.

The authors are with the Key Laboratory of Universal Wireless Communi-cations, Ministry of Education, Beijing University of Posts and Telecommu-nications, Beijing, P. R. China (e-mail: wangji.bupt@hotmail.com).

Digital Object Identifier 10.1109/LCOMM.2013.021913.122881

1[ ]nH

SourceS1

SourceS2

Relay

2[ ]nH

2[ ]nG1[ ]nG

1 RN

MA Phase BC Phase Feedback Channel

2 2ˆ [ ]n DG

1

M

1

M

1 1ˆ [ ]n DG

Fig. 1. Two way relay system with delayed and limited feedback.

scenario, a limited feedback channel with non-zero delay fromsource to relay is used to provide the relay with necessary CSIto perform precoding. However, the induced CSI imperfectionsdue to the inevitable quantization error and feedback delayseverely degrade the system performance [7]-[10]. Thereforein this letter, we investigate the characteristics of delayed andlimited feedback, and devise a closed-form robust beamformerat the multi-antenna relay to compensate the quantizationerror and delay effect. The proposed precoding is based onminimizing the expected weighted sum mean square error(WS-MSE) cost function conditioned on the outdated andquantized channel direction information (CDI). We also findthat the CDI-only feedback is sufficient for calculating the ro-bust precoding matrix. Simulations confirm that the proposedscheme is robust against the considered channel uncertaintiesand outperforms conventional precoding algorithms.

As for notations, we employ uppercase boldface lettersfor matrices and lowercase boldface for vectors. E(·), Tr(·),vec(·), Re(·), ‖ · ‖, (·)∗, (·)T and (·)H return the expectation,trace, vectorization, real part, Euclidean 2-norm, conjugate,transpose and conjugate transpose of the input, respectively.IM stands for an identity matrix of size M ×M . In addition,X⊗Y denotes the Kronecker product of matrices X and Y.

II. SYSTEM MODEL

As shown in Fig. 1, We consider a two way relay systemwhere two source nodes, denoted by S1 and S2, wish tocommunicate with each other assisted by an intermediate relaynode denoted by R. There exist no direct links between S1 andS2. It is assumed that S1 and S2 employ M antennas each,whereas R is equipped with NR antennas. The entries of theinvolved channels are independent and identically distributed(i.i.d.) complex Gaussian with zero mean and unit variance.Block fading is assumed where the channel realizations areconstant over one block but correlated across blocks.

1089-7798/13$31.00 c© 2013 IEEE

690 IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 4, APRIL 2013

Based on analogue network coding (ANC), one round ofinformation exchange can be accomplished in two consecutivetime slots, also known as the multiple access (MA) phase andbroadcast (BC) phase, respectively. During MA, both S1 andS2 transmit concurrently to R, and the transmitted complex-valued vector from source i at the n-th discrete time instantis denoted by si[n] ∈ CM×1 with E(si[n]s

Hi [n]) = IM (i =

1, 2). Then the received signal at R can be represented as

yR[n] = H1[n]s1[n] +H2[n]s2[n] + vR[n], (1)

where Hi[n] ∈ CNR×M denotes the uplink channel coefficientmatrix from source i to R at time instant n. Moreover, vR[n] ∼CN (0, σ2

RINR) stands for the additive noise at the relay.During the following BC phase, the relay node R mul-

tiplies the received superimposed signal by a beamformingmatrix W[n] ∈ CNR×NR and broadcasts the precoded signalxR[n] = W[n]yR[n] to S1 and S2. Note that the relaybeamforming matrix W[n] satisfies the power constraint PR =E(‖W[n]yR[n]‖2) = Tr(W[n]Q[n]WH [n]), where PR isthe total relay transmit power, and Q[n] = H1[n]H

H1 [n] +

H2[n]HH2 [n] + σ2

RINR is assumed for brevity.Next, the received signal at source i can be formulated as

yi[n] = GHi [n]W[n]Hi[n]si[n] +GH

i [n]W[n]Hi[n]si[n]

+GHi [n]W[n]vR[n] + vi[n] (i = 1, 2), (2)

where i = 2 if i = 1 and i = 1 if i = 2. The downlink channelfrom R to source i at time instant n is denoted by Gi[n] ∈C

NR×M and independent of Hi[n]. The additive noise atsource i follows the distribution vi[n] ∼ CN (0, σ2

i IM ).We assume that the sources can obtain the cascaded equiv-

alent channel gains required to perform self-interference can-celation (SIC) and recover the data of interest perfectly andtimely via training-based channel estimation [1], [5]. Thus, theback propagated self-interference term can be subtracted as

yi[n] = GHi [n]W[n]Hi[n]si[n] +GH

i [n]W[n]vR[n] +vi[n].(3)

Furthermore, it is straightforward to assume that all thethree nodes have perfect and instantaneous knowledge of thelocal channel state information at the receiver side (CSIR) [5].Thus, Hi[n](i = 1, 2) are known to R and Gi[n] is availableat source i. Unfortunately, the availability of accurate Gi[n]at R, which is required to perform precoding during BC, isdifficult to achieve. Therefore in this paper, since the uplinkand downlink channels are independent, we resort to the finite-rate feedback channels from the sources to relay which canprovide R with partial information of the downlink channelsand ensure a reasonable uplink overhead as well.

Specifically, at time instant n, source i quantizes the spa-tial channel direction Gi[n], i.e., subspace spanned by thecolumns of Gi[n] to a matrix isotropically distributed inCNR×M denoted by Gi[n]. In this letter, we write the SVD ofHermitian Gi[n]G

Hi [n] as Gi[n]G

Hi [n] = Gi[n]Λi[n]G

Hi [n]

[8]. Then we get Gi[n] = Gi[n]Λ1/2i [n]. The quantization

is chosen from a predesigned codebook composed of 2Bi

matrices Ci = {Ci1,Ci2, · · ·,Ci2Bi } as Gi[n] = Cik

according to the minimum chordal distance criterion k =argmin1�j�2Bid

2(Gi[n],CHij ), where Bi denotes the number

of feedback bits allocated to source i. Then the selected index

k is fed back to R via an error free limited feedback channelwith non-zero delay denoted by Di, indicating that Di symboldurations have been delayed owing to the quantization at thesource and the transmission in the feedback channel. Note thatthe codebooks of the two sources are generated independently.

III. ANALYSIS ON DELAYED AND LIMITED FEEDBACK

At time instant n, the relay is only aware of the knowledgeof the outdated and quantized CDI denoted by Gi[n−Di](i =1, 2). In this section, we characterize the relation betweenGi[n−Di] and the current channel Gi[n], which will be usedin the robust beamforming design in the next section.

Firstly, due to channel quantization, the channel directionGi[n] can be decomposed mathematically as [8]

Gi[n] = Gi[n]Xi[n]Yi[n] + Si[n]Zi[n], (4)

where Xi[n] ∈ CM×M is an unitary and uniformly distributedrandom matrix. Zi[n],Yi[n] ∈ CM×M are upper triangularwith positive diagonal elements indicating the accuracy ofquantization satisfying Tr(ZH

i [n]Zi[n]) = d2(Gi[n], Gi[n])and YH

i [n]Yi[n] = IM − ZHi [n]Zi[n]. Moreover, Si[n] ∈

CNR×M is an orthonormal basis for an isotropically dis-tributed M dimensional plane in the NR − M dimensionalleft null space of Gi[n]. And with random codebooks thequantization error can be computed by

δi � E{d2(Gi[n], Gi[n])} = E(Tr(ZHi [n]Zi[n]))

.=

Γ(1/T )

Tς−1/T 2−Bi/T , (5)

where T = M(NR −M) and ς = 1T !

∏Mi=1

(NR−i)!(M−i)! .

Secondly, we employ the following widely adopted Gauss-Markov block fading autoregressive model to represent thetemporal channel correlation [10]

Gi[n] = ρiGi[n−Di] +√1− ρ2iEi[n], (6)

where ρi stands for the normalized temporal correlation co-efficient. The error matrix Ei [n] is unitary and uncorrelatedwith Gi[n−Di]. For Jake’s fading spectrum, we have ρi =J0(2πDifdiTs), where fdi represents the Doppler frequencyassociated with source i, Ts denotes the symbol duration, andJ0(·) is the zero-th order Bessel function of the first kind.

Then according to the error models (4) and (6), the depen-dence of Gi[n] and Gi[n−Di] can be determined as

Gi[n] = ρiGi[n−Di]Ai+ρiSi[n−Di]Bi+√1− ρ2iEi[n],

(7)where we assume Ai = Xi[n−Di]Yi[n−Di]Λ

1/2i [n−Di]

and Bi = Zi[n−Di]Λ1/2i [n−Di] for notational brevity.

Lemma 1: The statistical characteristics of the indepen-dent random matrices can be calculated as E(Ai) � αIM ,E(AiA

Hi ) = NR(1 − δi

M )IM , E(BiBHi ) = NRδi

M IM , andE(Si[n]S

Hi [n]) = M

NR−M (INR − Gi[n]GHi [n]).

Proof: Please see Appendix A.

WANG et al.: ROBUST RELAY PRECODING DESIGN FOR TWO WAY RELAY SYSTEMS WITH DELAYED AND LIMITED FEEDBACK 691

IV. ROBUST MMSE BEAMFORMING DESIGN

In this section, the proposed robust beamforming scheme ispresented. We target at designing the robust MSE-based relaybeamformer in the presence of delayed and limited feedback.

The MSE for transmitted signal si can be calculated as

εi(W[n], β) = Esi[n],vR[n],vi[n]{‖si[n]− β−1yi[n]‖2}= Tr(β−2GH

i [n]W[n](Hi[n]HHi [n] + σ2

RINR)WH [n]Gi[n]

− 2β−1Re(GHi [n]W[n]Hi[n]) + β−2σ2

i IM + IM ). (8)

where β ∈ R is an automatic gain control scalar at thereceivers of the sources. Then the WS-MSE can be representedby ε(W[n], β) =

∑2i=1 ωiεi(W[n], β) at time instant n,

where ωi is the given positive MSE weight for si.To include the joint impacts of the quantization error and

feedback delay in the design, we employ the expected WS-MSE over Gi[n] conditioned on Gi[n−Di] as the objectivefunction, i.e., EGi[n]|Gi[n−Di]

{ε(W[n], β)} � μ(W[n], β).In order to get μ(W[n], β), we calculate E{εi(W[n], β)}

first which can be rewritten as

E{εi(W[n], β)} � μi(W[n], β)

= β−2E{‖GH

i [n]W[n]Hi[n]‖2}+ β−2σ2RE{‖GH

i [n]W[n]‖2}−2β−1

E{Tr(Re(GHi [n]W[n]Hi[n]))}+ (β−2σ2

i + 1)IM . (9)

By substituting (7) into (9) and using Lemma 1, theexpectations on right-hand side (RHS) of (9) can be furthercomputed. Specifically, as to the second expectation we have

E{‖GHi [n]W[n]‖2}

(a)= ρ2iEAi

{Tr(WH [n]Gi[n−Di]AiAHi GH

i [n−Di]W[n])}+ ρ2iEBi,Si

{Tr(WH [n]Si[n−Di]BiBHi SH

i [n−Di]W[n])}+ (1− ρ2i )EEi

{Tr(WH [n]Ei[n]EHi [n]W[n])}

(b)= Ψi||Gi[n−Di]W[n]||2 +Υi‖W[n]‖2, (10)

denoting Ψi = ρ2iNR(1 − δi

M − δiNR−M ) and Υi =

ρ2iNRδi

NR−M +

(1− ρ2i). Note that (a) and (b) hold because of the statistical

independence properties described in the previous sectionand the fact that E(Si[n]) = E(Ei[n]) = 0. Similarly,E{‖GH

i[n]W[n]Hi[n]‖2} can also be calculated.

Next, computing the last expectation in (9) resultsin E{Tr(Re(GH

i[n]W[n]Hi[n]))} = αρiTr(Re(GH

i[n −

Di]W[n]Hi[n])) accordingly. Then we can further obtain

μi(W[n], β)

= β−2(Ψi‖GHi [n−Di]W[n]Hi[n]‖2 −Υi‖W[n]Hi[n]‖2)

+ β−2σ2RΨi‖GH

i [n−Di]W[n]‖2 + β−2(σ2i + PRΥi) + 1

− 2β−1αρiTr(Re(GHi [n−Di]W[n]Hi[n])). (11)

where the relay power constraint is applied. Compared with(8), we find there formally exists an additional term Υi(PR−‖W[n]Hi[n]‖2) in (11) arising from both the quantizationerror Si[n] and the delay error Ei[n].

Mathematically, the proposed robust beamforming can beformulated as the convex problem

minW[n],β

μ(W[n], β) =∑2

i=1ωiμi(W[n], β)

s.t. Tr(W[n]Q[n]WH [n]) = PR. (12)

To solve (12), we construct the Lagrangian as

L = μ(W[n], β) + λ(Tr(W[n]Q[n]WH [n])− PR). (13)

Taking derivative of L(W[n], β, λ) with respect to W∗[n]and β respectively and equating them to zero, we have

∑2

i=1αβωiρiGi[n−Di]H

Hi [n]

=∑2

i=1ωiΨiGi[n−Di]G

Hi [n−Di]W[n]Hi[n]H

Hi [n]

+∑2

i=1ωiΨiσ

2RGi[n−Di]G

Hi [n−Di]W[n]

−∑2

i=1ωiΥiW[n]Hi[n]H

Hi [n] + λβ2W[n]Q[n], (14a)∑2

i=1Tr(Re(αβωiρiG

Hi [n−Di]W[n]Hi[n]))

=∑2

i=1ωiΨi‖GH

i [n−Di]W[n]hi[n]‖2

+∑2

i=1ωiΨiσ

2R‖GH

i [n−Di]W[n]‖2

−∑2

i=1ωi[Υi‖W[n]Hi[n]‖2 − (σ2

i + PRΥi)], (14b)

where ∂Tr(ZX0ZHX1)/∂Z

∗ = X1ZX0 is employed.Postmultiplying (14a) by WH [n] and taking traces on both

sides, we can find that the left-hand sides (LHS) of this newformula and (14b) are equal by recalling the property thatTr(Re(X)) = Tr(X) if Tr(X) = Tr(XH). Comparing theirRHSs, we can get λβ2 = P−1

R

∑2i=1 ωi(σ

2i+ PRΥi) � ξ in

closed form. Denote η = αβ and let W[n] = ηW[n]. Then itfollows that η = P

1/2R [Tr(W[n]Q[n]WH [n])]−1/2. Substitut-

ing ξ and W[n] into (14a), we find η can be eliminated and(14a) becomes an generalization form of Sylvester equationwith respect to W[n]. Then W[n] can be solved by

vec(W[n]) = Ω−1vec(Ξ), (15)

where we denote Ξ =∑2

i=1 ωiρiGi[n − Di]HHi [n] and

Ω = QT [n] ⊗ (ξINR) +∑2

i=1{(Hi[n]HHi [n] + σ2

RINR)T ⊗

(ωiΨiGi[n−Di]GHi[n−Di])−(Hi[n]H

Hi[n])T⊗(ωiΥiINR)}

which is nonsingular in practice [2]. Thus, we obtain theclosed form solution of (12) eventually with W[n] = ηW[n].

The following remarks can be made from (15). Firstly, thescalar α carrying channel magnitude information has beeneliminated together with η, which demonstrates that CDI-only feedback is sufficient for the robust precoding design.Accordingly, the considered different feedback bits, delaysas well as MSE weights between S1 and S2 make oursolution more adjustable. Furthermore, from (15) we can seeexplicitly how the CSI imperfections modify and impact thebeamforming design with coefficients Bi, Di, δi and ρi.Particularly, when Bi → ∞ and Di → 0, i.e., the inducedchannel uncertainties vanishes, we have δi = zi[n] = 0 andρi = 1. In this ideal case, Gi[n − Di] approaches Gi[n]and the robust MMSE beamforming matrix reduces to theconventional beamformer derived in [2]. Also, when M = 1,the analysis on feedback reduces to the single-stream casein [7] and the proposed scheme reduce to the single-streamcase accordingly. Finally, the proposed scheme has analogouscomplexity to conventional ones thanks to the similar closedform beamforming structure they have.

692 IEEE COMMUNICATIONS LETTERS, VOL. 17, NO. 4, APRIL 2013

V. SIMULATION RESULTS

In this section, we provide numerical results to evaluate therobustness of the proposed scheme. We assume Jake’s fadingspectrum and identical Doppler spread for S1 and S2. In theexamples, the normalized Doppler frequency fdiTs = 0.1,and we have ρi = 1, 0.9037 corresponding to Di = 0, 1.Random quantization codebooks are applied for simulation[8]. For convenience, we set NR = 4, M = 2, B1 = B2 = B,D1 = D2 = D, ω1 = ω2 = 1/2, PR = 1 and σ2

1 = σ22 =

σ2R = σ2. Thus, the signal-to-noise ratio (SNR) is defined by

SNR = 1/σ2. Moreover, QPSK constellations are used tomodulate the source information bits. All simulation resultsare averaged over 106 channel realizations.

Fig. 2 depicts the bit error rate (BER) performance compar-ison of the proposed design (R-MMSE) with the conventionalMMSE beamforming scheme (C-MMSE), which treat theimperfect CDI as the real channel when perform beamforming[2], [7]. The BER of the ideal case when accurate CSI isavailable at R is given for reference. From Fig. 2, we canobserve that R-MMSE compensates the CSI imperfections andR-MMSE outperforms C-MMSE in all the considered casesand in the whole SNR region. In low SNR regime where noisevariance dominates over the other coefficients in (11) and (15),the performance gap is negligible. And as SNR increases,the gain over C-MMSE becomes more and more significant.Furthermore, the BER curves under different delays illustratethat the achieved performance gain over C-MMSE becomesmore pronounced as D decreases. In addition, when theallocated feedback bits increase from B = 4 to B = 8, theBER performance of R-MMSE can be improved due to theenhanced accuracy of the channel quantization.

VI. CONCLUSION

In this letter, we design the robust precoding matrix atthe relay node in two way relay systems with delayed andlimited feedback. The proposed precoding algorithm is basedon minimizing the expected WS-MSE cost function condi-tioned on the outdated and quantized CDI. Numerical resultsdemonstrate the robustness of our scheme against the jointeffects of channel quantization error and feedback delay.

APPENDIX APROOF OF LEMMA 1

It is known that the channel direction Gi[n] and thediagonal eigenvalues matrix Λi[n] in the SVD of Gi[n]G

Hi [n]

are independent [8]. Then we can obtain E(Λi[n]) = NRIM .And according to (5), we have E(ZH

i [n]Zi[n]) = δiM IM ,

E(YHi [n]Yi[n]) = NR(1− δi

M )IMIM and E(XHi [n]Xi[n]) =

IM . Note that Ai, Bi, Si[n] and Ei[n] are all independent.Thus the expectations can be calculated as

E(AiAHi ) = E(Xi[n]Yi[n]Λi[n]Y

Hi [n]XH

i [n])

= NR(1− δiM

)IM ,

E(BiBHi ) = E(Zi[n−Di]Λi[n]Z

Hi [n−Di]) =

NRδiM

IM ,

E(Ai) = E(Xi[n]Yi[n]Λ1/2i [n])

= E{(IM − ZHi [n]Zi[n])

1/2}E(Λ1/2i [n]) � αIM ,

0 5 10 15 20 25 30 35 4010

−6

10−5

10−4

10−3

10−2

10−1

100

BER Performance Comparison

SNR(dB)

Unc

oded

BE

R

IdealR−MMSE, B=4, D=0C−MMSE, B=4, D=0R−MMSE, B=4, D=1C−MMSE, B=4, D=1R−MMSE, B=8, D=0C−MMSE, B=8, D=0

Fig. 2. Uncoded BER versus SNR for NR = 4, M = 2 and fdiTs = 0.1.

To get E(Si[n]SHi [n]), we extend Gi[n] to an orthonor-

mal basis U = (Gi[n],uM+1, ...,uNR) of the Grassmannmanifold ΩNR,M . Since Si[n] is an orthonormal basis for anisotropically distributed M dimensional plane in the NR−Mdimensional left null space of Gi[n], it can be expressedas Si[n] = U1V, where U1 = (uM+1, ...,uNR) and V isuniformly and isotropically distributed in ΩNR−M,M . Then

E(Si[n]SHi [n]) = E(U1[n]VVHUH

1 [n])

(a)=

M

NR −MU1[n]U

H1 [n]

(b)=

M

NR −M(INR − Gi[n]G

Hi [n]),

where (a) follows from E(VVH ) = MNR−M INR−M . To get

(b), note U is unitary and U1U1H = UUH − Gi[n]G

Hi [n].

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