Multi-Relay Selection Design and Analysis for Multi-Stream Cooperative Communications
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Multi-Relay Selection Design and Analysis forMulti-Stream Cooperative Communications
Shunqing Zhang, Vincent K. N. Lau,
Senior Member, IEEE
Abstract
In this paper, we consider the problem of multi-relay selection for multi-stream cooperative MIMO systems with M relaynodes. Traditionally, relay selection approaches are primarily focused on selecting one relay node to improve the transmissionreliability given a single-antenna destination node. As such, in the cooperative phase whereby both the source and the selectedrelay nodes transmit to the destination node, it is only feasible to exploit cooperative spatial diversity (for example by meansof distributed space time coding). For wireless systems with a multi-antenna destination node, in the cooperative phase it ispossible to opportunistically transmit multiple data streams to the destination node by utilizing multiple relay nodes. Therefore,we propose a low overhead multi-relay selection protocol to support multi-stream cooperative communications. In addition, wederive the asymptotic performance results at high SNR for the proposed scheme and discuss the diversity-multiplexing tradeoff aswell as the throughput-reliability tradeoff. From these results, we show that the proposed multi-stream cooperative communicationscheme achieves lower outage probability compared to existing baseline schemes. I. I
NTRODUCTION
Cooperative communications for wireless systems has recently attracted enormous attention. By utilizing cooperation amongdifferent users, spatial diversity can be created and this is referred to as cooperative diversity [1]–[3]. In [1], the authorsconsidered the case where multiple relay nodes are available to assist the communication between the source and the destinationnodes using the decode-and-forward (DF) protocol, and they showed that a diversity gain of M (1 − r ) can be achieved with M relay nodes and a multiplexing gain of r . However, the advantages of utilizing more relay nodes is coupled with theconsumption of additional system resources and power, so it is impractical (or even infeasible) to activate many relay nodes inresource- or power-constrained systems. As a result, various relay selection protocols have been considered in the literature. Forexample, in [4]–[7] the authors considered cooperative MIMO systems with a single-antenna destination node and the selectionof one relay node in the cooperative phase. In [4] the authors proposed an opportunistic relaying protocol and showed that thisscheme can achieve the same diversity-multiplexing tradeoff (DMT) as systems that activate all the relay nodes to performdistributed space-time coding. In [5] the authors applied fountain code to facilitate exploiting spatial diversity. In [6], [7] theauthors proposed a dynamic decode-and-forward (DDF) protocol which allows the selected relay node to start transmitting assoon as it successfully decodes the source message. On the other hand, there are a number of works [8]–[10] that studiedthe capacity bounds and asymptotic performance (e.g. DMT relation) for single-stream cooperative MIMO systems with asingle-antenna destination node. In [8] the authors derived the DMT relation with multiple full-duplex relays and showed thatthe DMT relation is the same as the DMT upper bound for point-to-point MISO channels. In [9] the authors studied thenetwork scaling law based on the amplify-and-forward (AF) relaying protocol. In all the above works, the destination nodeis assumed to have single receive antenna and hence, only one data stream is involved in the cooperative phase. When thedestination node has multiple receive antennas, the system could support multiple data streams in the cooperative phase andthis could lead to a higher spectral efficiency.In this paper, we design a relay selection scheme for multi-stream cooperative systems and analyze the resultant systemperformance. In order to effectively implement multi-stream cooperative systems, there are several technical challenges thatrequire further investigations. • How to select multiple relay nodes to support multi-stream cooperation in the cooperative phase?
Most of the existingrelay selection schemes are designed with respect to having a single-antenna destination node and supporting cooperativespatial diversity. For example, in [11] the authors considered a rateless-coded system and proposed to select relay nodes forsupporting cooperative spatial diversity based on the criterion of maximizing the received SNR. In order to support multi-stream cooperation, the relay selection metric should represent the holistic channel condition between all the selected relaynodes and the destination node, but this property cannot be addressed by the existing relay selection schemes. • How much additional benefit can multi-stream cooperation achieve?
The spectral efficiency of the cooperative phasecan be substantially increased with multi-relay multi-stream cooperation compared to conventional schemes. However, the
This paper is funded by RGC 615609. The results in this paper were presented in part at the IEEE International Conference on Communications, May2008.S. Zhang was with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong. He is nowwith Huawei Technologies, Co. Ltd., China (e-mail: [email protected]).V. K. N. Lau is with the Department of Electronic and Computer Engineering, Hong Kong University of Science and Technology, Hong Kong (e-mail:[email protected]).
O APPEAR IN IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2010 2 system performance may be bottlenecked by the source-relay links. Therefore, we characterize the advantage of multi-streamcooperation in terms of the end-to-end performance gain over conventional schemes that are based on cooperative spatialdiversity.We propose a multi-stream cooperative relay protocol (DF-MSC-opt) for cooperative systems with a multi-antenna destinationnode. We consider optimized node selection in which a set of relay nodes is selected for multi-stream cooperation . Basedon the optimal relay selection criterion, we compare the outage capacity, the diversity-multiplexing tradeoff (DMT), and thethroughput-reliability tradeoff (TRT) of the proposed DF-MSC-opt scheme against traditional reference baselines.
Notation : In the sequel, we adopt the following notations. C M × N denotes the set of complex M × N matrices; Z denotesthe set of integers; upper and lower case letters denote matrices and vectors, respectively; ( · ) T denotes matrix transpose; Tr ( · ) denotes matrix trace; diag ( x , . . . , x L ) is a diagonal matrix with entries x , . . . , x L ; I ( · ) denotes the indicator function; Pr ( X ) denotes the probability of event X ; I N denotes the N × N identity matrix; ⌈·⌉ and ⌊·⌋ denote the ceiling and floor operations,respectively; ( · ) + = max( · , ; . = , ˙ ≤ and ˙ ≥ denote exponential equality and inequalities where, for example, f ( ρ ) . = ρ b if lim ρ →∞ log( f ( ρ ))log( ρ ) = b ; E Y [ · ] denotes expectation over Y ; and F ( x, k ) denotes the χ cumulative distribution function (CDF)for value x and degrees-of-freedom k . II. R ELAY C HANNEL M ODEL
We consider a system consisting of a single-antenna source node, M half-duplex single-antenna relay nodes, and a destinationnode with N r antennas. For notational convenience, we denote the source node as the th node and the M relay nodes asthe { , , . . . , M } -th node. We focus on block fading channels such that the channel coefficients for all links remain constantthroughout the transmission of a source message.We divide the transmission of a source message that requires N channel uses into two phases, namely the listening phase and cooperative phase . In the listening phase, all the relay nodes listen to the signals transmitted by the source node until K outof the M relay nodes can decode the source message . In the cooperative phase, the destination node chooses N r nodes fromamongst the source node and the successfully decoding relay nodes to transmit multiple data streams to the destination node.Specifically, let x = [ x (1) , x (2) , . . . , x ( N )] T ∈ C N × denote the signals transmitted by the source node over N channel uses.Similarly, let x m = [ x m (1) , x m (2) , . . . , x m ( N )] T ∈ C N × , m = 1 , . . . , M , denote the signals transmitted by the m th relaynode over N channel uses. The signals received by the m th relay node is given by y m = [ y m (1) , y m (2) , . . . , y m ( N )] T ∈ C N × , m = 1 , . . . , M , where y m ( n ) = h SR,m x ( n ) + z m ( n ) , (1) h SR,m is the fading channel coefficient between the source node and m th relay node, H SR = [ h SR, , h SR, , . . . , h SR,M ] T ∈ C M × is a vector containing the channel coefficients between the source and the M relay nodes, and z m ( n ) is the additive noisewith power normalized to unity. Each relay node attempts to decode the source message with each received signal observationuntil it can successfully decode the message. The listening phase ends and the cooperative phase begins after K relay nodessuccessfully decodes the source message. In the cooperative phase, let ˜ x k ( n ) , k = 1 , . . . , K , denote the signal relayed bythe k th successfully decoding relay node and the aggregate signal transmitted in the cooperative phase can be expressed as x D ( n ) = [ x ( n ) , ˜ x ( n ) , ˜ x ( n ) , . . . , ˜ x K ( n )] T ∈ C ( K +1) × . Accordingly, the received signals at the destination node are givenby Y = [ y (1) , y (2) , . . . , y ( N )] T ∈ C N × N r , where y ( n ) = (cid:26) H SD x ( n ) + z ( n ) for the listening phase H D ( D ) Vx D ( n ) + z ( n ) for the cooperative phase H SD ∈ C N r × represents the fading channel coefficients between the source and the destination nodes, D is the setrepresenting the K successfully decoding relay nodes, H D ( D ) = [ H SD , ˜ H RD, , ˜ H RD, , . . . , ˜ H RD,K ] ∈ C N r × ( K +1) representsthe aggregate channel in the cooperative phase with ˜ H RD,k ∈ C N r × being the fading channel coefficients between k th successfully decoding relay and the destination node, z ( n ) ∈ C N r × is the additive noise with power normalized to unity, and V is the node selection matrix . The selection matrix is defined as V = diag ( v , v , v , . . . , v K ) , where v k = 1 if the k th node( k ∈ { , . . . , K } ) is selected to transmit in the cooperative phase and v k = 0 if the node is not selected.The following assumptions are made throughout the rest of the paper. Assumption 1 (Half-duplex relay model):
The half-duplex relay nodes can either transmit or receive during a given timeinterval but not both.
Assumption 2 (Fading model):
We assume block fading channels such that the channel coefficients H SR and H D ( D ) remain unchanged within a fading block (i.e., N channel uses). Moreover, we assume the fading channel coefficients ofthe source-to-relay (S-R) links, relay-to-destination (R-D) links, and source to destination (S-D) links are independent andidentically distributed (i.i.d.) complex symmetric random Gaussian variables with zero-mean and variance σ SR , σ RD and σ SD ,respectively. The relay system cannot enter the cooperative phase if less than K relay nodes can decode the source message within N channel uses. O APPEAR IN IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2010 3
Assumption 3 (CSI model):
Each relay node has perfect channel state information (CSI) of the link between the source nodeand itself. The destination node has perfect CSI of the S-D link and all R-D links. For notational convenience, we denote theaggregate CSI as H = (cid:0) H SR , H D ( D ) (cid:1) . Assumption 4 (Transmit power constraints):
The transmit power of the source node is limited to ρ S . The transmit power ofthe k th relay node is limited to ρ k . III. P ROBLEM F ORMULATION
In this section, we first present the encoding-decoding scheme and transmission protocol of the proposed DF-MSC-optscheme. Based on that, we formulate the multi-relay selection problem as a combinatorial optimization problem.
A. Encoding and Decoding Scheme
The proposed multi-stream cooperation system is facilitated by random coding and maximum-likelihood (ML) decoding[12]. At the source node, an R -bit message W, drawn from the index set { , , . . . , R } , is encoded through an encodingfunction X N : { , , . . . R } → X N . The encoding function at the source node can be characterized by a vector codebook C = { X N (1) , X N (2) , . . . , X N (2 R ) } ∈ C N r × N . The m th codeword of codebook C is defined as X N ( m ) = x ( m )1 (1) . . . x ( m )1 ( N ) ... x ( m ) N r (1) . . . x ( m ) N r ( N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) x ( m )1 ( N +1) . . . x ( m )1 ( N ) ... x ( m ) N r ( N +1) . . . x ( m ) N r ( N ) , m = 1 , . . . , R , (2)which consists of N vector symbols of dimension N r × , and x ( m ) k ( n ) is the symbol to be transmitted by the k th antennaduring the n th channel use. The vector codebook C is known to all the M relay nodes (for decoding and re-encoding) as well asthe destination node for decoding. At the receiver side (relay node or destination node), the receiver decodes the R-bit messagebased on the observations, the CSI, and a decoding function Y N : ( Y N × H ) → { , , . . . , R } . We assume ML detection inthe decoding process. The detailed operation of the source node and the relay nodes in the listening and cooperative phasesare elaborated in the next subsection. B. Transmission Protocol for Multi-Stream Cooperation
The proposed transmission protocol is illustrated in Fig. 1, and the flow charts for the processing by the source, relay, anddestination nodes are shown in Fig. 2. Specifically, the N -symbol source message codeword (cf. (2)) is transmitted to thedestination node over two phases; the listening phase that spans the first N channel uses and the cooperative phase that spansthe remainder N = N − N channel uses.
1) Listening phase:
In the listening phase, the single-antenna source node transmits the first row of the message codeword(i.e. [ x ( m )1 (1) . . . x ( m )1 ( N )] as per (2)) to the relay and destination nodes. Each relay node attempts to decode the sourcemessage with each received signal observation. Although the source node transmits only the first row of the source messagecodeword, the relay nodes can still detect the source message using standard random codebook and ML decoding argument.Effectively, we can visualize a virtual system with a multi-antenna source node as shown in Fig. 3, and the missing rows inthe message codeword transmitted by the source node is equivalent to channel erasure in a virtual MISO source-relay channel.Once a relay node successfully decodes the source message, it sends an acknowledgement ACK RD to the destination nodethrough a dedicated zero-delay error free feedback link.
2) Control phase and signaling scheme:
Without loss of generality, we assume that K relay nodes can successfully decodethe source message with N received signal observations. Upon receiving the acknowledgement from the K successfullydecoding relay nodes, the destination node enters the control phase and selects N r nodes to participate in the multi-streamcooperation phase (cf. Section III-C). Specifically, the destination node indicates to the source and relay nodes the transition tothe cooperative phase as well as the node selection decisions via an ( M + 1) -bit feedback pattern. The first bit of the feedbackpattern is used to index the th node (the source node) and the last M bits are used to index the M relay nodes. The feedbackpattern contains N r bits that are set to 1; the m th bit of the feedback pattern is set to 1 if the corresponding node is selectedto participate in the cooperative phase, whereas the bit is set to 0 if the node is not selected. Note that the total number offeedback bits required by the proposed multi-stream cooperation scheme is K ACK RD plus one feedback pattern with M + 1 bits, which is less than 2 bits per relay node. The source node has a single transmit antenna and hence, could only transmit one row of the vector codeword X N ( m ) during the listening phase. O APPEAR IN IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2010 4
3) Cooperative phase:
In the cooperative phase, the N r selected nodes cooperate to transmit the ( N + 1) th to the N th columns of the source message codeword (cf. (2)) to the destination node to assist it with decoding the source message.Specifically, for k = 1 , . . . , N r , the node corresponding to the k th bit that is set to 1 in the feedback pattern transmits the k th row of the message codeword (i.e. [ x ( m ) k ( N + 1) . . . x ( m ) k ( N )] as per (2)) to the destination node.To better illustrate the proposed transmission protocol, we show in Fig. 1 an example of the proposed system with adestination node with N r = 2 antennas. Suppose the feedback pattern is given as follows: sourcenode |← M relay nodes →| Feedback Pattern 0 1 0 0 · · · |← M + 1 bits →| This corresponds to selecting the first relay node ( R ) and the M th relay node ( R M ) to participate in the cooperative phase. Inthe listening phase, the source node transmits the first row of the codeword X N ( m ) given by [ x ( m )1 (1) . . . x ( m )1 ( N )] . In thecooperative phase, R and R M transmit [ x ( m )1 ( N + 1) . . . x ( m )1 ( N )] and [ x ( m )2 ( N + 1) . . . x ( m )2 ( N )] , respectively. Therefore,the effective transmitted codeword can be expressed as X N ( m ) = " Transmitted by sourcenode in listening phase z }| { x ( m )1 (1) . . . x ( m )1 ( N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Transmitted by R in cooperative phase z }| { x ( m )1 ( N + 1) . . . x ( m )1 ( N ) x ( m )2 ( N + 1) . . . x ( m )2 ( N ) | {z } Transmitted by R M in cooperative phase . C. Problem Formulation for Multi-stream Cooperation
We focus on node selection by the destination node for optimizing outage performance (cf. Section III-B2), where thedestination node only has CSI of the S-D and R-D links H D ( D ) (cf. Assumption 3). The proposed Multi-stream CooperationScheme with Optimized node selection is called DF-MSC-opt and we define the outage event as follows: Definition 1 (Outage):
Outage refers to the event when the total instantaneous mutual information between the source andthe destination nodes is less than the target transmission rate R . Mathematically, outage can be expressed as I ( I DF-MSC-opt ( H , V ) < R ) , (3)where H = ( H SR , H D ( D )) is the aggregate channel realization and V is the node selection action given H D ( D ) .The total instantaneous mutual information of the multi-stream cooperative system in (3) is given by the following theorem. Theorem 1: (Instantaneous Mutual Information of Multi-Stream Cooperative System)
Given the aggregate channel realization H = ( H SR , H D ( D )) , the instantaneous mutual information I DF-MSC-opt ( H , V ) (bits/second/channel use) between the sourceand destination nodes for multi-stream cooperative system can be expressed as I DF-MSC-opt ( H , V )= N (cid:8) N log(1+ ρ S | H SD | )+ N log det( I N r + H D ( D ) VΓV H H D ( D ) H ) (cid:9) , (4)where N and N are the numbers of channel uses of the listening phase and the cooperative phase, respectively, and Γ = diag ( ρ S , ρ , ρ , . . . , ρ M ) is the transmit power matrix of the source node and the M relay nodes. Note that N and N arerandom variables that depend on the realization of the S-R links H SR .The proof of Theorem 1 can be extended from [12], [13] by applying random Gaussian codebook argument. Note that the firstterm in the mutual information in (4) corresponds to the contribution from the source transmission . By virtue of the proposedDF-MSC-opt scheme, multiple data streams are cooperatively transmitted in the cooperative phase (unlike traditional schemeswherein only a single stream is transmitted). Hence, we can fully exploit the spatial channels created from the multi-antennadestination node and therefore achieving higher mutual information in the second term of (4).By Definition 1, the average outage probability is given by P out ( V ) = E H (cid:2) I (cid:0) I DF-MSC-opt ( H , V ) < R (cid:1)(cid:3) , which is a function of the node selection policy V . It follows that, given the channel realization H , the optimal node selectionpolicy is given by V ⋆ = arg min V ∈ Ω I (cid:0) I DF-MSC-opt ( H , V ) < R (cid:1) , where the set of all feasible node selection actions is defined as Ω = { Λ ∈ { , } ( M +1) × ( M +1) | Λ is diagonal and Tr ( Λ ) = N r } . (5)Equivalently, for any transmission rate R , the optimal node selection policy is given by V ⋆ = arg max V ∈ Ω I DF-MSC-opt ( H , V ) . (6) The mutual information contributed by the source node can also be obtained from the mutual information of the virtual multi-antenna source model inFig. 3.
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IV. O
UTAGE A NALYSIS FOR THE
DF-MSC-
OPT S CHEME
In this section, we derive the asymptotic outage probability of the DF-MSC-opt scheme. For simplicity, we assume σ SD = σ RD = σ D and the power constraints of all the nodes are scaled up in the same order , i.e., lim ρ S →∞ ρ m ρ S = 1 for all m ∈ { , . . . , M } . The outage probability can be expressed as P ⋆out = E H (cid:2) I (cid:0) I DF-MSC-opt ( H , V ⋆ ) < R (cid:1)(cid:3) ( a ) = E H SR (cid:2) Pr (cid:0) I DF-MSC-opt ( H , V ⋆ ) < R | H SR (cid:1)(cid:3) , (7)where in (a) the randomness of I DF-MSC-opt ( H , V ⋆ ) is introduced by the aggregate channel gain in the cooperative phase H D ( D ) and the optimal node selection policy V ⋆ . Let H N denote the collection of all realizations of the S-R links H SR such that K out of M relay nodes can successfully decode the source message, and the listening phase ends at the ( N ) th channel use. We define the outage probability given H SR and the duration of the listening phase N as P SR ( H SR , N ) = Pr (cid:0) I DF-MSC-opt ( H , V ⋆ ) < R | H SR ∈ H N (cid:1) , which includes the following cases. • Case 1: The listening phase ends within N channel uses. In this case, N < N , N = N − N , and the outage probability P SR ( H SR , N ) can be expressed as P Case 1 SR ( H SR , N ) = Pr (cid:2)(cid:0) N N log(1 + ρ S | H SD | ) + N − N N g ( H , V ⋆ ) (cid:1) < R (cid:3) , (8)where g ( H , V ⋆ ) = max V ∈ Ω log det (cid:0) I N r + ρ S H D ( D ) VV H H D ( D ) H (cid:1) for K ≥ N r , and g ( H , V ⋆ ) = log det (cid:0) I N r + ρ S H D ( D ) H D ( D ) H (cid:1) for K < N r . • Case 2: The listening phase lasts for more than N channel uses. In this case, the outage event is only contributed by thedirect transmission between the source and the destination nodes. The outage probability P SR ( H SR , N ) can be expressedas P Case 2 SR ( H SR , N ) = Pr (cid:0) log(1 + ρ S | H SD | ) < R (cid:1) . (9)Substituting (8) and (9) into (7), the outage probability is given by P ⋆out = P Nl =1 P Case 1 SR ( H SR , l ) Pr ( H SR ∈ H l ) + P l>N P Case 2 SR ( H SR , l ) Pr ( H SR ∈ H l ) (10) = P Nl =1 P Case 1 SR ( H SR , l ) Pr ( H SR ∈ H l )+ Pr (cid:0) log(1+ ρ S | H SD | ) < R (cid:1) Pr ( H SR ∈ ∪ l>N H l ) , where Pr ( H SR ∈ H l ) is the probability that K out of M relay nodes can successfully decode the message at the l th channeluse. To evaluate the outage probability, we calculate each term in equation (10) as follows. First, it can be shown thatPr ( H SR ∈ H N ) = Pr ( H SR ∈ ∪ l>N − H l ) − Pr ( H SR ∈ ∪ l>N H l ) = Φ N − − Φ N (11)where Φ l = P K − i =0 (cid:0) Mi (cid:1)(cid:0) − exp( − NR/l − ρ S σ SR ) (cid:1) M − i (cid:0) exp( − NR/l − ρ S σ SR ) (cid:1) i , l = 1 , . . . , N, (12)denotes the probability that less than K relay nodes can successfully decode the source message at the l th channel use. Second, P Case 1 SR ( H SR , l ) can be upper-bounded as shown in the following lemma. Lemma 1:
The outage probability for the DF-MSC-opt scheme given that K relay nodes can successfully decode the sourcemessage at the l th channel use can be upper bounded as P Case 1 SR ( H SR , l ) ≤ Pr ( f ( α l , H , V ⋆ ) < R ) , l = 1 , . . . , N, (13)where α l = l/N , f ( α l , H , V ⋆ ) = α l log(1 + ρ S | H SD | ) + (1 − α l ) P L T i =1 log (cid:0) ρ S σ D κ ( i ) (cid:1) with L T = min( N r , K + 1) denoting the number of transmitted streams in the cooperative phase, and κ ( i ) , i = 1 , . . . , L T , are ( L T out of K + 1 ) ordered χ -distributed variables with N r degrees of freedom. Proof:
Refer to Appendix A for the proof.Third, when fewer than K relay nodes can successfully decode the source message within N channel uses, the sourcenode transmits to the destination node with the direct link only; it follows that Pr ( H SR ∈ ∪ l>N H l ) = Φ N and Pr (cid:0) log(1 + ρ S | H SD | ) < R (cid:1) = F (cid:16) R − ρ S σ D ; N r (cid:17) . Therefore, the outage probability (cf. (10)) is given by P ⋆out = Φ F (cid:16) R − ρ S σ D ; N r (cid:17) + P α l =1 /N (Φ l − − Φ l ) P Case 1 SR ( H SR , l ) ≤ Φ F (cid:16) R − ρ S σ D ; N r (cid:17) + P α l =1 /N (Φ l − − Φ l ) Pr ( f ( α l , H , V ⋆ ) < R )= Φ F (cid:16) R − ρ S σ D ; N r (cid:17) − Φ ′ l ′ Pr ( f ( α ′ l , H , V ⋆ ) < R ) , (14) As such, there is no loss of generality to assume
Γ = ρ S I M +1 when studying high SNR analysis. It is non-trivial to evaluate P Case 1 SR ( H SR , l ) exactly due to the dynamics of the optimal nodes selection. O APPEAR IN IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2010 6 where Φ ′ l and Φ ′ l ′ denote the first order derivative of Φ l with respect to α l and evaluated at l = α l N and l ′ = α ′ l N , respectively.Note that in the last step of (14) we apply the Mean Value Theorem [14] which guarantees the existence of the point α ′ l ∈ (0 , .Moreover, the outage probability upper bound of the DF-MSC-opt scheme is given by the following theorem. Theorem 2:
For sufficiently large N , the outage probability upper bound of the DF-MSC-opt scheme is given by P ⋆out ≤ Φ F (cid:16) R − ρ S σ D ; N r (cid:17)| {z } Direct Transmission − Φ ′ l ′ F (cid:18) Rα ′ l − ρ S σ D ; N r (cid:19) F (cid:18) R − α ′ l − ρ S σ D ; N r (cid:19) K +1 | {z } Relay-Assisted Transmission . (15) Proof:
Refer to Appendix B for the proof.For systems without relays, the outage probability is given by P out,pp = F (cid:16) R − N r ρ S σ D ; N r (cid:17) . Note that the DF-MSC-optscheme can achieve lower outage probability by taking advantage of multi-stream cooperative transmission. We can interpret − Φ ′ l ′ as the cooperative level , which quantifies the probability that the relay nodes can assist the direct transmission. Forexample, − Φ ′ l ′ = 1 implies that the DF-MSC-opt scheme can be fully utilized, whereas − Φ ′ l ′ = 0 implies that the source nodetransmits to the destination node with the direct link only. As per (14), the outage probability P ⋆out is a decreasing functionwith respect to − Φ ′ l ′ , whereas − Φ ′ l ′ is a decreasing function with respect to the strength of the S-R links σ SR . As shown inFig. 4, when the strength of the S-R links increases from 30dB to 40dB, the cooperative level increases and hence the outageprobability decreases. On the other hand, as shown in (15) and as illustrated in Fig. 5, the outage probability decreases withincreasing number of receive antennas at the destination node.V. DMT AND
TRT A
NALYSES FOR THE
DF-MSC-
OPT S CHEME
In this section, we focus on characterizing the performance of the DF-MSC-opt scheme in the high SNR regime. Specifically,we perform DMT and TRT analyses based on the outage probability P ⋆out as shown in (14). There are two fundamentalreasons for the performance advantage of the proposed DF-MSC-opt scheme, namely multi-stream cooperation and optimalnode selection V ⋆ (cf. (6)). To illustrate the contribution of the first factor, we shall compare the performance of DF-MSC-optwith the following baselines: • DF-SDiv (Baseline 1): The DF relay protocol for cooperative spatial diversity.
The listening phase and the cooperative phasehave fixed durations, i.e., each phase consists of N/ channel uses. The source node and all the successfully decoding relaynodes cooperate using distributed space-time coding. At the destination node, maximum ratio combining (MRC) is used tocombine the observations from the different receive antennas. • AF-SDiv (Baseline 2): The AF relay protocol for cooperative spatial diversity.
All the relay nodes transmit a scaled versionof their soft observations in the cooperative phase. At the destination node, MRC is used to combine the observations fromdifferent receive antennas. • DDF (Baseline 3):
The dynamic DF protocol [6].
Once a relay node successfully decodes the source message, it immediatelyjoins the transmission using distributed space-time coding. At the destination node, MRC is used to combine the observationsfrom different receive antennas.On the other hand, to illustrate the contribution of optimal node selection, we shall compare the performance of the DF-MSC-opt scheme with the DF-MSC-rand scheme (Baseline 4), which corresponds to a similar multi-stream cooperation scheme butwith randomized node selection V randomly generated from the node selection space Ω (cf. (5)). Table I summarizes the majordifferences among the DF-MSC-opt scheme and the baseline schemes. We illustrate in Fig. 6 the outage capacity versus SNR(with P out = 0 . ) of the cooperative systems with N r = 3 , K = 3 , M = 15 . Note that the proposed DF-MSC-opt schemecan achieve a gain of more than bit/channel use over the four baseline schemes. A. DMT Analysis
In order to analyze the DMT relation of the DF-MSC-opt scheme, we first derive the relation between the outage probabilityand the multiplexing gain. In the outage probability expression (14), for a sufficiently large ρ S , the asymptotic expression ofthe first term is given by Φ F (cid:16) R − ρ S σ D ; N r (cid:17) ˙= P K − i =0 (cid:0) Mi (cid:1) ρ − ( M − i )(1 − r ) + S ρ − N r (1 − r ) + S . = ρ − ( M − K +1+ N r )(1 − r ) + S , (16)and the asymptotic expression of Φ ′ l ′ is given by Φ ′ l ′ = P K − i =0 (cid:0) Mi (cid:1) (1 − φ l ′ ) M − i − ( φ l ′ ) i − (( M − i )(1 − φ l ′ ) ′ φ l ′ + i (1 − φ l ′ ) ( φ l ′ ) ′ )˙= P K − i =0 (cid:0) Mi (cid:1) ρ − ( M − i ) (cid:0) − rα ′ l (cid:1) + S ˙ ≤ ρ − min i ( M − i ) (cid:0) − rα ′ l (cid:1) + S = ρ − ( M − K +1) (cid:0) − rα ′ l (cid:1) + S (17) For a sufficiently large SNR ρ S , F (cid:18) Rα ′ l − ρ S σ D ; N r (cid:19) F (cid:18) RLT (1 − α ′ l ) − ρ S σ D ; N r (cid:19) K +1 is much smaller than F (cid:18) R − N r ρ S σ D ; N r (cid:19) . O APPEAR IN IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2010 7 where φ l ′ = exp (cid:16) − r log ρSαl ′ − ρ S σ SR (cid:17) . Moreover, we can express the term Pr ( f ( α ′ l , H , V ) < R ) in (14) asPr ( f ( α ′ l , H , V ) < R ) . = Pr (cid:0) ρ α ′ l (1 − δ ) + S Q L T i =1 ρ (1 − α ′ l )(1 − β i ) + S < ρ rS (cid:1) = Pr (cid:0) α ′ l (1 − δ ) + + P L T i =1 (1 − α ′ l )(1 − β i ) + < r (cid:1) . = RR B p ( δ, β ) d δ d β, (18)where − δ is the exponential order of σ SD , − β i is the exponential order of γ ( i ) , B = { δ, β : α ′ l (1 − δ ) + +(1 − α ′ l ) P i (1 − β i ) + The DMT relation for the DF-MSC-opt scheme can be expressed as d ( r, K ) DF-MSC-opt = min( d , d + d ) , (20)where d = ( M − K + 1 + N r )(1 − r ) + , d = ( M − K + 1)( − r − r ) + , and d = N r + ( N r − K ≤ r < / (cid:16) N r +( N r − θ )( K +1 − θ ) − ( N r + K − θ )( r − r − θ ) , ( N r − θ )( K +1 − θ )+ N r θ ( − rr ) (cid:17) θθ +1 ≤ r < θ +1 θ +2 N r L T ( − rr ) L T L T +1 ≤ r ≤ (21)with θ = 1 , , . . . , L T and L T = min( N r , K + 1) . Proof: Refer to Appendix C for the proof.We could further optimize the parameter K in the DF-MSC-opt scheme and the resulting DMT relation is given by d ( r ) ⋆ DF-MSC-opt = max K ∈ [1 , ,...,M ] d ( r, K ) DF-MSC-opt =max (cid:0) d ( r, ⌊ K ⋆ ⌋ ) DF-MSC-opt , d ( r, ⌈ K ⋆ ⌉ ) DF-MSC-opt (cid:1) , where K ⋆ = ( ( M +1)(2 − r + r )+ N r (2 − r + r )( N r − − r )+ r ≤ r ≤ / ( M +1)(1 − r ) − N r (3 r − r − (1 − r )2 θ ) − θ (3 θ (1 − r ) − − r ) N r (1 − r ) − r +(1 − r ) θθ +1 < r ≤ θ +1 θ +2 (22)is the optimal number of successfully decoding relay nodes to wait for in the listening phase.In the following, we show that the DF-MSC-opt scheme achieves superior DMT performance than the traditional cooperativediversity schemes. Specifically, as per [15], the AF-SDiv and DF-SDiv schemes have identical DMT relations given by d ( r ) AF/DF-SDiv = M (1 − r ) + + N r (1 − r ) + for ≤ r ≤ . For the DDF protocol, the DMT relation is given by thefollowing lemma. Lemma 2: The DMT relation for the DDF protocol with N r receive antennas at the destination node can be expressed as d ( r ) DDF = ( M + N r )(1 − r ) 0 ≤ r ≤ N r M + N r N r + M ( − r − r ) + N r M + N r ≤ r ≤ N r ( − rr ) ≤ r ≤ (23) Proof: Refer to Appendix D for the proof.In Fig. 7, we compare the DMT relations for the DF-MSC-opt scheme and the baseline schemes. For a system with M = 15 relay nodes and a destination node with N r = 3 antennas. Note that since the DF-MSC-opt scheme (as well as the DF-MSC-rand scheme) exploits multi-stream cooperation, it can achieve high diversity gain than the traditional cooperative diversityschemes (i.e. AF-SDiv, DF-SDiv, and DDF) that exploit single-stream cooperation. Moreover, since the DF-MSC-opt schemeoptimizes the node selection policy and the number of decoding relay nodes K , it can achieve better diversity gain than theDF-MSC-rand scheme. B. TRT Analysis The DMT analysis alone cannot completely characterize the fundamental tradeoff relation in the high SNR regime. Specifi-cally, the multiplexing gain gives the asymptotic growth rate of the transmission rate R at high SNR ρ and is only applicableto scenarios where R scales linearly with log ρ . Hence, there are many unique transmission rates that correspond to the samemultiplexing gain, and the DMT analysis only gives a first order comparison of the performance tradeoff at high SNR whenwe have different multiplexing gains. O APPEAR IN IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2010 8 In order to have a clearer picture on the tradeoff relations, in the following we study the SNR shift in the outage probability P out as we increase the transmission rate R by ∆ R . We quantify the more detailed relations among the three parameters ( R, log ρ, P out ( R, ρ )) by analyzing the TRT relation [16].Consider the outage probability expression for the DF-MSC-opt scheme (14). In order to facilitate studying the TRT, wechoose the parameter K such that the outage events are dominated by relay-assisted transmissions, i.e. K ≥ ⌈ K ⋆ ⌉ (where K ⋆ is given by (22)). The following theorem characterizes the asymptotic relationships among R , ρ S , and P ⋆out for the r > case . Theorem 4: The TRT relation for the DF-MSC-opt scheme under K ≥ ⌈ K ⋆ ⌉ and r > is given by lim ρ S →∞ , ( R,ρ S ) ∈R ( z ) log P ⋆out − c DF-MSC-opt ( z ) R log ρ S = − g DF-MSC-opt ( z ) (24)where R ( z ) , (cid:8) ( R, ρ S ) | z + 1 > R (1 − r ) log ρ S > z (cid:9) for z ∈ Z , ≤ z < L T , (25)denotes the z th operating region, c DF-MSC-opt ( z ) , K + 1 + N r − (2 z + 1) , and g DF-MSC-opt ( z ) , ( K + 1) N r − z ( z + 1) . Notethat g DF-MSC-opt ( z ) is defined as the reliability gain coefficient and t DF-MSC-opt ( z ) , g DF-MSC-opt ( z ) /c DF-MSC-opt ( z ) is defined asthe throughput gain coefficient. Proof: Refer to Appendix E for the proof.By applying Theorem 4, the SNR shift between two outage curves with a ∆ R rate difference is R/t ( z ) dB. Fig. 8 showsthe outage curves corresponding to ∆ R = 2 bits/channel use for N r = 3 , K = 3 and M = 15 . This scenario corresponds to theregion R (1) and the TRT relation for the DF-MSC-opt scheme can be expressed as g DF-MSC-opt (1) = 10 and t DF-MSC-opt (1) = .As we can see from the simulation results, the SNR shift is . dB for 2 bits/channel use increase in the transmission rate,which matches with our analysis. VI. C ONCLUSION In this paper, we proposed a multi-stream cooperative scheme (DF-MSC-opt) for multi-relay network. Optimal multi-relayselection is considered and we derived the associated outage capacity as well as the DMT and TRT relations. The proposeddesign has significant gains in both the outage capacity as well as the DMT relation due to (1) multi-stream transmissions inthe cooperative phase and (2) optimized relay selection.A PPENDIX AP ROOF OF L EMMA l , we characterize the function of g ( H , V ⋆ ) under different conditions. • Condition 1 K < N r : Under this condition, we allow all the successfully decoding relays to transmit in the cooperativephase. Thus, the communication links can be regarded as a conditional MIMO link [17] and g ( H , V ) = log det (cid:0) I N r + ρ S H D ( D ) H D ( D ) H (cid:1) ≤ P K +1 i =1 log (1 + ρ S σ D κ ( i )) , where κ ( i ) , i = 1 , . . . , K + 1 , are χ -distributed variables with N r degrees of freedom. • Condition 2 K ≥ N r : Under this condition, we select N r nodes out of the source and the successfully decodingrelay nodes from the node selection space Ω to participate in cooperative transmission. The analytical solution for theterm g ( H , V ) is in general not trivial [18]. Since in the proposed DF-MSC-opt scheme we choose the best N r out of K + 1 transmit nodes, the upper bound can be obtained similar to [18, (6)]. Thus, the capacity bound with transmit nodeselection is g ( H , V ) = max V ∈ Ω log det (cid:0) I N r + ρ S H D ( D ) VV H H D ( D ) H (cid:1) ≤ P N r i =1 log (1 + ρ S σ D κ ( i )) , where the κ ( i ) , i = 1 , . . . , N r , are ordered χ -distributed variables with N r degrees of freedom.Combining the two conditions above, we obtain the general form as shown in Lemma 1.A PPENDIX BP ROOF OF T HEOREM (cid:0) f ( α ′ l , H , V ) < R (cid:1) = Pr (cid:2) α ′ l log(1 + ρ S | H SD | ) + (1 − α ′ l ) P L T i =1 log (1 + ρ S σ D κ ( i )) < R (cid:3) , TRT analysis allows for investigating more general scenarios where the transmission rate R does not scale linearly with log ρ and helps to study the SNRgain of the outage probability vs SNR curve when we increase the transmission rate R by ∆ R . For the r ≤ case, the MIMO channel formed by the multiple relay nodes to the destination node can only operate with a multiplexing gain of 1 andthe TRT relation is trivial. O APPEAR IN IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2010 9 which can be relaxed asPr (cid:2) α ′ l log(1 + ρ S | H SD | ) + (1 − α ′ l ) P L T i =1 log (1 + ρ S σ D κ ( i )) < R (cid:3) ≤ Pr ( α ′ l log(1 + ρ S | H SD | ) < R ) Pr (cid:0) (1 − α ′ l ) P N r i =1 log (1 + ρ S σ D κ ( i )) < R (cid:1) = F (cid:16) Rα ′ l − ρ S σ D ; N r (cid:17) Pr (cid:16) P L T i =1 log (cid:0) ρ S σ D κ ( i ) (cid:1) < R − α ′ l (cid:17) . Moreover, the following relation holds for α ′ l > Pr (cid:16) P L T i =1 log (cid:0) ρ S σ D κ ( i ) (cid:1) < R − α ′ l (cid:17) ≤ Pr (cid:16) log (cid:0) ρ S σ D κ (1) (cid:1) < R − α ′ l (cid:17) = Pr (cid:16) κ (1) < R − α ′ l − ρ S σ D (cid:17) = F (cid:16) R − α ′ l − ρ S σ D ; N r (cid:17) K +1 (26)where in the last step we applied the results of order statistics [19]. Substituting (26) into (14), we have Theorem 2.A PPENDIX CP ROOF OF T HEOREM P ⋆out is given by (19) with B = { δ, β : α ′ l (1 − δ ) + +(1 − α ′ l ) P i (1 − β i ) + Define v k,j and u j,i as the exponential orders of g k,i and h j,i . We have the following relation, P ≤ ρ − P Mj =1 n j ( − min { P Nrk =1 v k, ,..., P Nrk =1 v k,j } ) + = ρ −{ P Mj =1 n j (1 − N r min { v ,...,v j } ) + , where v i = P Nrk =1 v k,i N r . Choose the rate R = r log ρ and the PEP bound for rate R is given by P ≤ ρ − P Mj =1 n j ( − min { P Nrk =1 v k, ,..., P Nrk =1 v k,j } ) + = ρ − N [ { P Mj =1 njN (1 − N r min { v ,...,v j } ) + − r ] . Hence, the set of channel realizations that satisfy { P Mj =1 n j N (1 − N r min { v , . . . , v j } ) + ≤ r results in the outage event.Define ¯ v j = min { v , . . . , v j } and we can separate the discussion in the following three cases. (Naturally, we have ¯ v ≥ ¯ v ≥ . . . ≥ ¯ v M ). • Case 1: ¯ v ≤ . In this case, we can follow the discussion from (62) to (71) in [6] and show that the diversity order d is d ≥ ( N r + M )(1 − r ) , N r N r + M ≥ r ≥ N r + M ( − r − r ) , ≥ r > N r N r + M N r ( − rr ) , ≥ r > . • Case 2: ¯ v M ≥ . It’s trivial and d = M . • Case 3: ¯ v i > ≥ ¯ v i +1 . Follow the same discussion from (72) to (82) in [6] , we conclude that d ≥ (cid:26) N r + M ( − r − r ) , ≥ r ≥ N r ( − rr ) , ≥ r > .Combining the above cases, we have Lemma 2. A PPENDIX EP ROOF OF T HEOREM r > and K ≥ ⌈ K ⋆ ⌉ , (14) can be simplified as P ⋆out ˙ ≤ − Φ ′ l ′ Pr ( f ( α ′ l , H , V ⋆ ) < R ) . Substituting the expression of f ( α ′ l , H , V ) , we have P ⋆out ≤ Pr ( α ′ l log(1 + ρ S | H SD | ) + (1 − α ′ l ) P L T i =1 log (1 + ρ S σ D γ ( i )) < R ) ≤ Pr ((1 − r ) P L T i =1 log (1 + ρ S σ D γ ( i )) < R ) . (29)Based on the above results and following the same lines as the proof of [16, Theorem 2], we have the result of Theorem 4.R EFERENCES[1] J. N. Laneman and G. W. 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O APPEAR IN IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2010 11 Dr. Shunqing Zhang obtained B.Eng from Fudan University and Ph.D. from Hong Kong University of Science and Technology(HKUST) in 2005 and 2009, respectively. He joined Huawei Technologies Co., Ltd in 2009, where he is now the system engineer ofGreen Radio Excellence in Architecture and Technology (GREAT) team. His current research interests include the energy consumptionmodeling of the wireless system, the energy efficient wireless transmissions as well as the energy efficient network architecture andprotocol design. Vincent K. N. Lau obtained B.Eng (Distinction 1st Hons) from the University of Hong Kong in 1992 and Ph.D. from CambridgeUniversity in 1997. He was with PCCW as system engineer from 1992-1995 and Bell Labs - Lucent Technologies as member oftechnical staff from 1997-2003. He then joined the Department of Electronic and Computer Engineering, Hong Kong Universityof Science and Technology as Professor. His current research interests include robust and delay-sensitive cross-layer schedulingof MIMO/OFDM wireless systems with imperfect channel state information, cooperative and cognitive communications as well asstochastic approximation and Markov Decision Process. O APPEAR IN IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2010 12 TABLE IC OMPARISONS AMONG THE DF-MSC- OPT SCHEME AND THE BASELINE SCHEMES . Relay protocol AF-SDiv DF-SDiv DDF DF-MSC-rand DF-MSC-optNo. of transmitted streams inthe cooperative phase N r N r No. of relay nodes in the coop-erative phase M depends onS-R links depends onS-R links N r N r chosenfrom K Receiver structure MRC MRC MRC ML ML O APPEAR IN IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2010 13 Fig. 1. Illustration of timing diagram and an example of feedback pattern design. In this example, two relay nodes R and R M are selected to transmitin the cooperative phase. The selected relay nodes will re-encode the same codeword (codeword X N ( m ) in this example) selected from the same commonvector codebook C and the R -bit message received in the listening phase. R will send out the “first” row of codeword X N ( m ) and R M will send out the“second” row. O APPEAR IN IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2010 14 StartEncode a new codewordand set n = 1Continue to transmit the current codeword n > N ?Yes K out of M relays can decode ?No No Perform multi-stream cooperation based on the feedback from the destinationYesn = n + 1 Source Node Flow StartReceive a new codewordand set n = 1Continue to receive the current codeword n > N ?Yes Can it be decoded now?No No Send an acknowledgement to the destinationYesn = n + 1 Receive the feedback from the destination node?Acknowledgement sent to the destination node before? YesNoPerform multi-stream cooperation or remain silent based on the feedbackNo Yes Relay Node Flow StartReceive a new codewordand set n = 1Continue to receive the current codeword using multi-stream cooperation schemen > N ?Yesn = n + 1 Received K acknowledgements out of the M relays?Does the feedback pattern sent to source and relay nodes?Send the feedback pattern to source and relay nodesContinue to receive the current codeword transmitted by the source node No YesYesNo No Destination Node Flow Fig. 2. Flow chart of the protocols at the source node, the relay node and the destination node.Fig. 3. The single-antenna source-relay channel is equivalent to a multi-antenna virtual MISO channel. O APPEAR IN IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2010 15 −6 −5 −4 −3 −2 −1 SNR (dB) O u t age P r obab ili t y Outage Probability vs. SNR ( Rate = 18.0 ) Monte−Carlo, σ SR2 = 50 dBClosed−Form Eq. (15), σ SR2 = 50 dBMonte−Carlo, σ SR2 = 40 dBClosed−Form Eq. (15), σ SR2 = 40 dB σ SR2 = 40 dB σ SR2 = 50 dB Fig. 4. Outage probability vs. SNR of the DF-MSC-opt scheme for different σ SR under N r = 3 , K = 6 and M = 15 . −5 −4 −3 −2 −1 Total Transmit Power (dB) O u t age P r obab ili t y Outage Probability vs. Total Transmit Power (Rate = 9.0) Monte−Carlo, N r = 3Closed−Form Eq. (15), N r = 3No Relay, N r = 3Monte−Carlo, N r = 2Closed−Form Eq. (15), N r = 2No Relay, N r = 2N r = 3 N r = 2 Fig. 5. Outage probability vs. SNR of the DF-MSC-opt scheme for different N r under σ SR = 10 dB , K = 6 and M = 15 . O u t age C apa c i t y Outage Capacity vs. Total Transmit Power AF−SDiv (Baseline 1)DF−SDiv (Baseline 2)DDF (Baseline 3)DF−MSC−rand (Baseline 4)DF−MSC−opt Fig. 6. Outage capacity comparison of different schemes for N r = 3 , K = 3 and M = 15 . The channel variances of the S-R, R-D, S-D links are normalizedto unity. O APPEAR IN IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, 2010 16 D i v e r s i t y G a i n d (r) Diversity Multiplexing Tradeoff for Different Relay ProtocolsSIMO BoundDDF (Baseline 3)DF−MSC−rand (Baseline 4)AF/DF−SDiv (Baseline 1 and 2)RepetitionNon CooperationDF−MSC−opt with K = N r DF−MSC−opt (with optimized K) Fig. 7. Diversity-multiplexing tradeoff comparison of different relay protocols. −4 −3 −2 −1 Total Transmit Power (dB) O u t age P r obab ili t y Outage Probability vs. Total Transmit Power DF−MSC−opt R = 12 DF−MSC−opt R = 14 DF−MSC−rand R = 12 DF−MSC−rand R = 142.4 dB 2.8 dB Fig. 8. Monte-Carlo simulation results for outage curves corresponding to ∆ R = 2 bits/channel use for N r = 3 , K = 3 and M = 15 case with normalizedS-R, R-D, S-D channel variances.case with normalizedS-R, R-D, S-D channel variances.