Precoder Design for Correlated Multi-Antenna Cooperative Systems with Partial Decode and Forward Protocol and MMSE-SIC Receivers †
Shunqing Zhang†,∗ , Eddy Chiu† and Vincent K. N. Lau† Dept. of ECE, Hong Kong University of Science and Technology ∗ Huawei Technologies, Co. Ltd. Email: {eezsq,eechiua}@ust.hk,
[email protected]
Abstract—Cooperative communication is an important technology in next generation wireless networks. Partial decode-andforward provides an alternative solution for the conventional amplify-and-forward as well as decode-and-forward relay protocols. However, there are several important issues to be addressed regarding the application of partial decode-and-forward protocol. In this paper, we address the practical issues by proposing a joint MIMO precoder design for the multi-antenna cooperative system with correlated MIMO fading and partial decode-and-forward relay protocol. In addition, the MIMO precoders at both the source and the relay are matched to the MMSE-SIC receivers at the destination. We find that under similar system settings, joint MIMO precoder design with partial decode-and-forward relay protocol and MMSE-SIC receivers achieves substantial performance enhancement and has important practical significance. Index Terms—Correlated MIMO Systems, Cooperative Communication, Successive Interference Cancelation
capacity bound of PDF relay protocol. However, there are several important practical issues to be addressed regarding the application of PDF protocol in multi-antenna cooperative systems. •
I. I NTRODUCTION Cooperative communication is an important technology in next generation wireless networks. Due to the broadcast nature of wireless transmission, cooperation among different users can be simply utilized and significantly increases reliability as well as the achievable rates. Taking into consideration of the duplexing constraint, various relaying protocols have been proposed in the current literature. Amplify-and-forward (AF) [1] is a simple scheme where the relay forwards the received signal to the destination. One disadvantage of AF protocol is the noise amplification in the process of repeating the received signal. On the other hand, in the decode-andforward (DF) [1] protocol, the relay forward a clean copy of the decoded message from the source to the destination. However, in the DF scheme, the relay can participate only if it can decode the source signal reliably and hence, it does not utilize all the available resource in the system, resulting in some performance loss. To overcome the above issues, various relaying protocols have been proposed in the literature. For example, Yuksel and Erkip proposed a partial decode-and-forward (PDF) relaying protocol in the Gaussian broadcast channel [2], where a 2level superposition coding scheme is assumed at the source node and the relay node forwards partial information to the destination node based on the decoding ability. The information theoretic study of the relaying protocols characterizes the
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Joint Precoder Design for Cooperative PDF Systems: Precoding has been shown to be very important to boost the performance of multi-antenna cooperative systems. Yet, most of the works considered precoder design at the relay only [3]. As we shall illustrate in this paper, joint design of precoder at both the relay and the source is very important to exploit the full benefit of MIMO precoder. Furthermore, most of the existing works on MIMO precoder design for cooperative system have focused on AF/DF relaying protocol only. It is quite challenging to jointly design the precoders at both the source and the relay for the PDF relaying protocol. Precoder Design for Practical MMSE-SIC Receiver: Precoder design is tightly coupled with the receiver structure at the destination. Traditional precoder design for multi-antenna cooperative systems either assumed ML receiver or simple LMMSE receiver [4]. The ML receiver assumption allows simpler formulation and could serve as a performance reference but it is difficult to realize. On the other hand, the performance of LMMSE receiver is usually not good enough and there is a large performance gap compared with the ML design. As a result, MMSE-SIC is an important practical receiver which could bridge the gap between performance and complexity. As far as we are aware, there is no work addressing the precoding design matching the MMSESIC receiver. One key challenge involved is the issue of error propagation in the SIC process.
In this paper, we shall address the above issues by proposing a joint MIMO precoder design for the multi-antenna cooperative system with correlated MIMO fading and PDF relay protocol. In addition, the MIMO precoders at the source and the relay are matched to the MMSE-SIC receivers at the destination and hence, the work is of practical significance. Specifically, the precoders at the multi-antenna source and relay are jointly designed to minimize the average end-toend outage probability with MMSE-SIC receivers and PDF
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
U tRD
U rSR
RD H ind
Relay
H
SR ind
U rRD SD H ind
U tSR
U tSD Source
U rSD
Listening Phase
Forwarding Phase
Source
Transmit
Idle/Quiet
Relay
Receive
Transmit
Dest.
Receive
Receive
Dest.
Fig. 1. The correlated channel model of the cooperative relay systems and cooperative transmission relations.
relay protocol. We find that under similar system settings, joint MIMO precoder design with PDF relay protocol and MMSESIC receivers achieves substantial performance enhancement compared with joint MIMO precoder design for conventional AF/DF relay protocol and ML receivers, and has important practical significance. II. S YSTEM M ODEL A. Cooperative Transmission Signal Model Consider a cooperative system with a single BS (source), a single relay node and a mobile terminal (destination) as shown in Fig. 1. The source node communicates to the destination node with the help of a half-duplex relay. Let nS , nR , nD be the number of transmit/receive antennas equipped at the source, relay and destination nodes respectively. We define the listening phase to be the transmission period for the BS to transmit and define the forwarding phase to be the period for the transmission of the relay node as shown in Fig. 1. Denote T to be the transmission time intervals1 and X ∈ CnS ×T to be the transmitted signals by the source node. Signals transmitted by the relay node are denoted by XR ∈ CnR ×T . The variances of X and XR are normalized to unity. Received signals at the relay node and the mobile in the listening phase are denoted by Yl1 ∈ CnR ×T and Yl ∈ CnD ×T , respectively. Received signals at the mobile in the forwarding phase are denoted by Yf ∈ CnD ×T . The following equations show the relations among these signals. Yl = HSD PS X + Zl for the listening phase YlR = HSR PS X + Z1l Yf = HRD PR XR + Zf
for the forwarding phase
where HSD ∈ CnD ×nS , HSR ∈ CnR ×nS , HRD ∈ CnD ×nR are the fading coefficients of Source-Destination (SD), SourceRelay (SR) and Relay-Destination (RD) channels respectively. PS ∈ CnS ×nS and PR ∈ CnR ×nR denote the precoding matrices at the source and the relay nodes. Zl , Zf ∈ CnD ×T and Z1l ∈ CnR ×T are the additive white Gaussian noises (AWGN) with zero mean and normalized variances. Notice 1 The transmission time interval is defined to be the time duration where the channel fading coefficients in the cooperative networks remains unchanged.
that the signal transmitted by the relay node, XR , depends on the relay protocol adopted in the cooperative systems. For example, we have the following relations for the AF/DF relaying protocols. X for the DF relaying protocol (1) XR = R R Yl /Yl for the AF relaying protocol where · denotes the 2 norm of the corresponding vector. B. Correlated MIMO Channel Model The channel coefficients are modeled by Rayleigh fading with the second order statistics described via a general, mathematically tractable canonical decomposition of the channel along the transmit and receive covariance bases [5]. In this SD SD,H , where HSD model, HSD = USD r Hind Ut ind has independent, but not necessarily identically distributed entries. USD t and USD are the unitary discrete Fourier transform (DFT) r matrices at the transmitter and the receiver, which transforms the angular domain domain the virtual propagation spread into SD,H SD SD SD,H SD and Λ = E H H = E H [6]. ΛSD r t ind ind ind Hind are the nD × nD and nS × nS diagonal matrices representing the transmit and the receive covariances in the respectively. HSR and HRD can be modeled in the similar manner, i.e. HSR
=
SR SR,H USR r Hind Ut
(2)
RD RD,H URD r Hind Ut
= (3) H SR,H SR with ΛSR , ΛSR = E HSR = E HSR,H r t ind Hind ind Hind , RD,H RD ΛRD and ΛRD = E HRD = E HRD,H H r t ind Hind ind ind denoting the corresponding transmit and receive covariance matrices. The following assumptions are made through the rest of the paper. Firstly, the receiver at the relay node has perfect CSI knowledge of the SR link only and the receiver at the destination node has perfect CSI knowledge of the SD and RD links but no CSI knowledge of the SR link. Secondly, the source and the relay nodes only has statistical knowledge of the MIMO channels in the relay networks (e.g. the antenna correlation matrix). Moreover, we consider block fading channels where the aggregate CSI H = (HSD , HSR , HRD ) remains quasistatic within a fading block (i.e. transmission time interval T ) but varies between different fading blocks. RD
III. PDF R ELAY P ROTOCOL AND P ROBLEM F ORMULATION A. PDF Relay Protocol Consider two information streams (denoted as X1 and X2 ) to be transmitted over the multi-antenna cooperative systems. Without loss of generality, we assume at the source node, the two information streams X1 and X2 are delivered through n1S and n2S transmit antennas respectively with n1S + n2S = nS . To exploit the possible diversity provided by the transmit antenna arrays, orthogonal space-time block code (OSTBC) [7] scheme is applied for the i-th information stream, which ˜ 1, X ˜ 2 to be the spans over the entire niS antennas. Denote X OSTBC encoded symbols for the information streams X1 , X2
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
and the transmitted symbols at the source node are given by ˜ 2 ]. ˜1 X X = [X The transmitter at the relay node is delivering an amplified version of its observation when AF relay protocol is applied. For the DF relay protocol, the transmitted symbols at the relay node is a clean copy of the transmitted signals at the source node, i.e. XR = X. At the destination side, the receiver is suggested to decode the information streams according to the received signal strengths (e.g. the post-processing SNR). Both AF and DF schemes transmit full information streams at the relay node and it provides a solution for the optimal detection algorithms such as ML equalizer. However, when we consider the practical receiving structure (e.g. ZF/MMSE SIC receiver) with the transmitted information together, the relay may not be always desirable to transmit the full information streams to the destination. In our proposed PDF relay protocol, the signal processing procedure at the relay node can be described as follows. In the listening phase, the relay node shall try to decode the transmitted information streams from the source node through MIMO detection schemes (e.g. MMSE-SIC receiver). If the first information stream X1 can be decoded correctly, the relay node shall map the information stream X1 into nR transmit antennas using OSTBC scheme and deliver the corresponding ˜ in the RD link. If the information stream X1 symbols X 1 cannot decode correctly and the information stream X2 can be successfully decoded, the relay shall follow the same procedure except that the transmitted symbols are OSTBC encoded version of the second information stream X2 , which ˜ . Otherwise, the relay node shall remain is denoted as X 2 quiet since there is no reliable information stream available for transmission in the cooperative phase. Mathematically, the transmitted signals at the relay node for the proposed PDF relay protocol are given by ⎧ ˜ ⎪ ⎨X1 X1 is decoded successfully. R ˜ X2 is decoded successfully. (4) X = X 2 ⎪ ⎩ 0 X1 , X2 cannot be decoded. For easy illustration, we define the information stream transmitted by the relay node as the cooperative stream and define the other stream as the regular payload stream. At the destination node, the receiver shall try to decode the cooperative stream first which is more reliable. If the cooperative stream can be successfully decoded, the decoder can subtract the interference generated by the cooperative stream and perform the traditional SIC processing as suggested in V-BLAST [8] to decode the regular payload stream. Comparing with the conventional relay transmission protocols (e.g. AF and DF relay protocols as illustrated in the previous part), the proposed PDF relay protocol has the following advantages. • Error Performance. In the SIC process, decoding the first stream is extremely important since the error in the first stream will strongly affect the performance of the other stream. In the conventional AF/DF scheme,
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the diversity orders for the two information streams at the destination side are identical, e.g. every transmitted symbol will receive the same amount of “copies” at the destination side. Moreover, since all the streams interacts with each other, the decoding performance of the first steam is limited, especially in the high SNR regime. In the proposed PDF scheme, however, it provides the nonuniform diversity protections for different information streams. In addition, since the transmitted stream in the RD link has no interference from the payload stream, the decoding performance at the destination node shall no longer be limited by the inter-stream interference, especially for high SNR regime. As a result, the decoding performance of the cooperative stream is much better than other relay protocols and the proposed PDF scheme can be regarded as an efficient solution for the V-BLAST relay transmission with SIC receivers. Cooperative Level. In the relay protocol, we refer the cooperative level to be the probability that the forward procedure becomes active. In the traditional AF scheme, the cooperative level equals to 1 because the relay always forwards the information under all the cases. Since the traditional AF scheme introduces additional noises in the forwarding process, the performance may be limited especially when the SR link is weak. In the traditional DF scheme, the cooperative level is limited since it has potentially low probability that all the streams can be decoded at the relay side. PDF scheme provides a better tradeoff solution, due to the following two reasons. (1) Since the PDF requires to decode the information stream before the forwarding process, it provides a more reliable result than the traditional AF scheme. (2) Since the PDF scheme requires to decode only one of the information stream, it has potentially larger chance to help forward the message compared with traditional DF scheme.
B. Problem Formulation In this part, we consider a practical system scenario where in each fading block, the source node sends to two packets (each of L bits information) to the destination node with the help of the relay using PDF relay protocol. MMSE-SIC is performed at the relay as well as the destination node. Moreover, strong coding such as Convolutional Turbo Code (CTC) / LowDensity Parity-Check (LDPC) Code is used to encode the data and as a result, packet error is mainly contributed by channel outage events. Specifically, we first have the following definition on end-to-end outage. Definition 1 (End-to-End Outage): The term end-to-end outage is defined as the event when the instantaneous mutual information (for a given channel realization H) between the transmitted information stream Xi for i = 1, 2 and the received symbols Y, is smaller than the transmission rate L/T . Mathematically, the outage event for i-th information stream can be described through the following equation
(5) I IPi DF (H, PS , PR ) < L/T
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
where I(A) is the indicate function, taking value 1 when A is true and 0 otherwise. By the definition of the outage event represented in (5), the average end-to-end outage probability (over all the transmitted information streams) is a function of the preS R S R coding matrices P , P and given by P out (P , P ) = 2 1 i S R . Our objective i=1 EH I IP DF (H, P , P ) < L/T 2 is to minimize the average end-to-end outage probability based on the joint design of the precoding matrices at the source and the relay nodes. Hence, the optimal precoding matrices (PS, , PR, ) can be found by solving the following optimization problem arg min
(PS ,PR )
subject to
Pout (PS , PR )
Tr PS PS,H ≤ αS , Tr PR PR,H ≤ αR αS + αR ≤ P0 , αS , αR ≥ 0 (6)
where Tr(·) denotes the matrix trace operation, αS , αR denote the power consumed at the source and the relay nodes respectively and the last inequality stands for a sum-power constraint in the relay network. In general, the above optimization problem (6) is non-trivial because of the following reasons. Firstly, deriving a closed-form expression for the average end-to-end outage probability is not straight-forward due to the effects of error propagation in the MMSE-SIC receiver structure. Secondly, since the source and the relay nodes only have the statistical knowledge of the channel matrices, the traditional channel diagonalization with optimal power waterfilling schemes [9] cannot be directly applied here. IV. J OINT MIMO P RECODER D ESIGN FOR PDF P ROTOCOL AND MMSE-SIC R ECEIVER In this section, our target is to jointly design the optimal MIMO precoder at both the source and the relay nodes based on solving the optimization problem given by (6). To simplify the analysis of the average end-to-end outage probability, we shall describe the behavior of the cooperative systems with PDF relay protocol and derive the average end-to-end outage probability under different cases. Case 1 (A1 ): The relay node is able to decode the first information stream X1 in the listening phase. In this case, X1 is the cooperative stream, X2 is the regular payload stream and the average end-to-end outage probability is given by the following lemma. Lemma 1: Under MMSE-SIC receivers at the relay and the destination node, the average end-to-end outage probability in case 1 and the probability that case 1 happens are given RD P (λSD 2dB 5
10
15
20
SNR (dB)
Fig. 2. PER versus SNR with the proposed PDF relay protocol, without relay, and with conventional DF and AF relay protocols.
with respect to αS and αR respectively. Moreover, αS and αR are related by αS + αR ≤ P0 , which is a linear constraint. Hence, the optimization problem (11) belongs to the quasiconvex optimization problem and we solve the above problem via convex feasibility problems [10] (e.g. bisection search algorithms). V. S IMULATION R ESULTS Consider the following practical multi-antenna cooperation system with MMSE-SIC based receivers. We assume that uniform linear antenna arrays are used [11], where the source node has nS = 4 antennas, the relay node has nR = 2 antennas, and the destination node has nD = 2 antennas. At the source node, information streams X1 and X2 are transmitted through n1S = n2S = 2 antennas. We assume the SR and RD links are stronger than the SD link, where the average channel gain per transmit antenna for the SD link SR is Tr(ΛSD t )/nS = 1, for the SR link is Tr(Λt )/nS = 2, RD and for the RD link is Tr(Λt )/nR = 4. We apply the CTC code specified in [11, Section 8.4.9.2.3], where each data packet consists of 12 data bytes, coded at rate 1/2, and modulated using QPSK. Finally, we define the SNR w.r.t. the total transmit power P0 and the noise power at the relay and the destination nodes. In Fig. 2, we compare the PER with the proposed PDF relay protocol, without relay (baseline 1), and with conventional DF (baseline 2) and AF (baseline 3) relay protocols. It can be seen that at PER of 10−3 the PDF protocol has SNR gain in excess of 7dB compared to without relay, and has SNR gains of over 2dB compared to DF and AF relay protocols. The superior error performance by applying the PDF protocol is manifested from the mitigation of inter-stream interference at the destination node (compared to DF and AF) as well as enhanced cooperative level (compared to DF). To demonstrate the effectiveness of the proposed precoding structure, we illustrate in Fig. 3 the PER versus SNR for the following systems: PDF with the proposed joint MMSE-SICbased precoding structure; (baseline 1) PDF without precoding; (baseline 2) PDF with disjoint precoding between the source node and the relay node; and (baseline 3) PDF with
978-1-4244-4148-8/09/$25.00 ©2009 This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2009 proceedings.
0
10
PDF: without precoding PDF: disjoint precoding PDF: naive precoding PDF: proposed precoding structure −1
10
PER
Baseline 1: PDF without precoding Baseline 2: PDF disjoint precoding
−2
10
Baseline 3: PDF naive precoding PDF proposed precoding structure 1.5dB
−3
2dB
10
6
8
10 SNR (dB)
12
14
16
Fig. 3. PER versus SNR with different precoding structures. Naive precoding (baseline 3) refers to a precoding design assuming ML detection at the relay and destination nodes. In all cases, MMSE-SIC receivers are applied at the RS and the destination nodes.
naive precoding (precoding design assuming ML detection at the RS and the destination nodes). In all cases, the receivers at the RS and destination nodes are MMSE-SIC. It can be seen that the proposed precoding structure yields better error performance than all the baseline systems for all SNR. In particular, the proposed MMSE-SIC based precoding structure is indeed more effective than naive precoding which cannot accommodate the impact of inter-stream interference for SICbased receivers.
variance, and the distribution of hRD follows the multivariate normal distribution with its probability density function RD,H −1 RD Σ h , where Σ = given by p(hRD ) = (2π)41det Σ e−h R,H RD RD RD,H R RD Ut Λt Ut P ⊗Λr . By changing of variables, P the p.d.f. of the random variable λRD = hRD,H hRD can 4 δmax −δmin λRD be upper bounded by p(λRD ) ≤ (2π) where 4e δmin / δmax is the minimum / maximum eigenvalue of the matrix Σ−1 and we have substituted Σ−1 ≥ δmin I into the expression p(hRD ). To minimize the outage probability P (λRD < 2L/T − 1), we can minimize the p.d.f. with respect to λRD or equivalently, minimize δmax . Hence, the optimization problem (7) is equivalent to the following problem. min
PR ,δmax
subject to
δmax Σ−1 ≤ δmax I,
Tr(PR PR,H ) ≤ αR
RD RD,H R Σ = PR,H URD P ⊗ ΛRD t Λt Ut r
The above optimization problem belongs to the Schur-concave optimization and the optimal solution is to diagonal the R channel Σ. Hence, the optimal design of PR = αR URD t Σ with Tr(ΣR,2 ) = 1. Substituting into (12), the optimization R,2 ⊗ ΛRD problem becomes maxTr(ΣR,2 )=1 λmin (ΛRD t Σ r ) where λmin (A) denotes the minimum eigenvalue of the matrix A. The optimal value of ΣR,2, is chosen to be /Tr(ΛRD,−1 ) and the physical meaning behind is to ΛRD,−1 t t equalize the covariance channel at the transmit side. R EFERENCES
VI. C ONCLUSION In this paper, we consider the joint precoder design at the source and the relay nodes for the correlated multi-antenna cooperative systems with PDF relay protocol and MMSESIC receivers. By formulating the precoder design problem as an optimization problem, we have derived the closed-form expression of the optimal precoder. The optimal precoder matches with the MMSE-SIC receivers as well as the channel statistics of the cooperative systems. The performance of the optimal precoding schemes is compared with several baselines. As suggested by the empirical results, the proposed PDF protocol can achieve significant performance gain compared to those baseline systems. A PPENDIX A P ROOF OF T HEOREM 1 Denote vec(·) to be the vectorization operation which stacks the columns of the matrix into a long column vector. The random variable λRD can be represented as λRD = HRD PR 2F = vec(HRD PR )H vec(HRD PR ). RD,1/2 RD RD,1/2 Hiid Λt , where HRD Since HRD ind = Λr iid contains independent identical distributed (i.i.d.) entries, we can RD,1/2 RD RD,1/2 RD,H R Hiid Λt Ut P . rewrite HRD PR = URD r Λr Due to the property of the vectorization operation, we RD,1/2 RD,H R RD R P ) = Λt Ut P ⊗ have hRD = vec(H
RD,1/2 RD URD , where ⊗ denotes the Kronecker Λ · vec H r r iid product operation. Each entry in HRD iid follows the complex Gaussian distribution with zero mean and normalized
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