A Novel Distributed Space-Time Trellis Code for Asynchronous Cooperative Communications under Frequency-Selective Channels Zhimeng Zhong†, Shihua Zhu†, and Arumugam Nallanathan‡ †Department of Electronics and Information Engineering, Xi’an Jiaotong University, Xi’an 710049 P.R. China Email: {zmzhong, szhu}@mail.xjtu.edu.cn ‡Division of Engineering, King’s College London, London WC2R 2LS. U.K Email: arumugam.nallanathan@kcl.ac.uk I. I NTRODUCTION Fading severely affects wireless communications performance, causing large variations in signal strength as a function of the user position. Diversity is a powerful technique against fading, used into spatial, temporal, and frequency domains [1]. To exploit the spatial diversity, multiple antennas can be equipped at the transmitter and/or the receiver. However, it is difficult to place multiple antennas on a mobile terminal or a sensor node due to the size limit and the hardware complexity. It is shown recently that the spatial diversity can be exploited if cooperation is adopted among users [2][3]. The resulted transmission scheme is referred to as the cooperative communications [3]. In [3], [4], different cooperative protocols are derived. The distributed space-time code (DSTC) protocol is proposed in [4], where space-time code (STC) is applied at each relay nodes to pass the data from source to destination simultaneously. Most of the cooperative diversity works assume perfect synchronization among users. Recently, asynchronous cooperative diversity has been discussed in [5]-[12]. Note that, the DSTC in [9]-[11] are designed under the assumption that the channels between relays and destination are flat fading. In this paper, we use the stack construction method from [13], [14] to build a family of DSTTC with the minimum memory order for asynchronous cooperative communications where the channels are considered frequency-selective. We derive sufficient conditions on the code design such that the full diversity can be achieved. Simulations demonstrate that R1 Phase II Phase I R2 T S …… Abstract— In most cooperative communications works, perfect synchronization among relay nodes is assumed in order to achieve cooperative diversity. However, this assumption is not realistic due to the distributed nature of each relay node. In this paper, we propose a family of distributed space-time trellis code (DSTTC) that does not require the synchronization assumption. It is shown that the proposed DSTTC can achieve full cooperative and full multipath diversities for the asynchronous transmission by utilizing a specifically designed generator matrix. We derive sufficient conditions on the DSTTC with the minimum memory order. Finally, various numerical examples are provided to verify the analytical studies. The newly proposed codes exhibit good properties, e.g. high energy efficiency and low synchronization cost, and can be applied to distributed wireless networks. RM 1 RM Fig. 1. Transmission Protocol. the newly proposed DSTTC can achieve both full cooperative diversity and full multipath diversity for asynchronous communications. This paper is organized as follows. In Section II, the system model is presented and the problem is formulated. In Section III, the DSTTC is developed. We derive sufficient conditions on the code design such that the full diversity can be achieved in this section. Finally, simulations are provided in section IV and conclusions are drawn in Section V. Notations: Vectors and matrices are boldface small and capital letters; the transpose, complex conjugate, Hermitian of the matrix A are denoted by AT , A∗ , AH , respectively; Tr(A) and AF are the trace and the Frobenius norm of A; (A)i,j is the (i, j)th entry of A and diag{a} denotes a diagonal matrix with the diagonal element constructed from a; I is the identity matrix. II. S YSTEM M ODEL AND P ROBLEM F ORMULATION Consider a system with M + 2 nodes that communicate cooperatively as shown in Fig. 1. We assume that there is one source node S, one destination node T , and M relays Ri , i = 1, 2, . . . , M . We also assume that there is no direct connection between the source and the destination (for example due to shadowing or too large separation) and that all terminals operate in half-duplex fashion [3]. We consider the decodeand-forward (DF) transmission protocol that consists of two phases. During phase I, S broadcasts its information to all the relays. During phase II, each relay firstly checks whether 978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings. the decoding is successful according to Cyclic Redundancy Check (CRC) bits that was inserted by the source, then, if the decoding is successful, the relays will encode the information and forward the encoded data to the destination. We assume that CRC is able to detect all the packet errors [5], [10]. Define Rs as the set of potential relays that decode successfully, where Ms = |Rs | is the cardinality of Rs . Clearly, Rs is determined by the channel quality between the source and the relay nodes, and Ms can be considered as a random variable. Since a space-time code designed to M relays has full diversity property, also has full diversity if M − Ms relays are deleted [12]. Hence, without loss of generality, we assume that M relays are all enrolled in phase II, and T receives y (n) = L M   hi (l)si (n − l − τi ) + z(n) , (1) i=1 l=0 where hi (l) is the lth path gain from Ri to T which is a circularly complex Gaussian random variable with variance σi2 (l), and L is the length of the channel impulse Lresponse (CIR). The channel gains are normalized such that l=0 σi2 (l) = 1. Since the terminals are separated by sufficiently large distances, different hi (l) can be reasonably assumed independent from each other. We consider the quasi-static fading process where hi (l) is constant over one packet of N symbols but may vary independently from packet to packet. Moreover, CIR is assumed perfectly known at the destination. si (n) is the symbol transmitted by Ri which is encoded based on the information decoded by Ri , and z(n) is the additive white Gaussian noise whose variance is N0 . In addition, τi is the timing delay for relay Ri . We denote τ = max(τ1 , . . . , τM ). We assume that τi is integer multiples of the symbol duration [10], [11]. Although we assume the delays are integer multiples of the symbol duration, following [15], we should note that the proposed code can still achieve full diversity when the delays are fractional symbol duration by borrowing the notions of DS-CDMA spread spectrum system. We also assume that these relative timing errors are known at the receiver but not known at the transmitter. Equation (1) can be rewritten in the matrix form: a y = hS + z , (2) where y is the received row vector of length (N + L + τ ), z is the noise vector of length (N + L + τ ), h is the 1 × M (L + 1) vector with the form h  [h1 (0), . . . , h1 (L), h2 (0), . . . , h2 (L), . . . , hM (L)]. Define T  S = ST1 , ST2 , . . . , STM , ⎡ si (1) si (2) ⎢ ⋆ si (1) ⎢ Si = ⎢ . .. ⎣ .. . ⋆ ⋆ ... ... .. . si (N ) si (N − 1) .. . ... ... .. . ... si (N − L) . . . ⋆ ⋆ .. . ⎤ ⎥ ⎥ ⎥. ⎦ si (N ) Considering the timing errors, Sa of dimension M (L + 1) × (N + L + τ ) can be expressed as ⎡ ⎤ ⋆(L+1)×τ1 S1 ⋆(L+1)×(τ −τ1 ) ⎢ ⋆(L+1)×τ2 S2 ⋆(L+1)×(τ −τ2 ) ⎥ ⎢ ⎥ Sa = ⎢ (3) ⎥. .. .. .. ⎣ ⎦ . . . ⋆(L+1)×τM SM ⋆(L+1)×(τ −τM ) where ⋆m×n is the m × n all-dumb-symbol matrix. In the above asynchronous cooperative communications, although the symbol synchronization is not required, to eliminate inter-packet interference, we assume that each packet in different enrolled relays is preceded by the dumb signals ⋆ as preamble, and the length of preamble is not less than Le + L, where Le is the upper bound of the timing errors (one may view this upper bound as a system parameter set by the physical layer design). In order to achieve full diversity order without the synchronous transmission assumption among relays, in the following, we will design S. III. C ODE C ONSTRUCTION A. Generator Matrix Model for DSTTC Before proceeding to discuss the code construction, we first clarify some definitions which will be used for the rest of this paper. Definition 1: Basic vector is defined as the vector v over the binary field F2  {0, 1} whose most left 1 corresponds to the first column. For example, v = 1011 is a basic vector. Definition 2: The length l(v) of a binary row vector v is defined as the number of components between the most left and the most right 1’s in v, including the two 1’s themselves. In particular, let l(0) = 0 and the length of a vector with only one nonzero component is defined as 1. For example, the length l(v) of v = 1011 is 4. We define V to be the set of all binary row vectors with finite lengths. Definition 3: For any vector v ∈ V, v(j) denotes the row vector resulted from v with each component shifted j bits to the right and zeros are padded to its two ends if needed. For example, the 3 bits right-shift of the binary row vector v = 1011 is v(3) = 0001011. To get a matrix as T G = vT , (v(j) )T , . . . , we need to add zeros to the right end of each row if necessary. Considering the above example, G = vT , (v(3) )T T can be expressed as  1 0 1 1 0 0 0 . G= 0 0 0 1 0 1 1  Definition 4: For two vectors v, u ∈ V, v ◦ u denotes their convolution and v+u is the component-wise addition over the binary field F2 . If the source information bits are correctly detected by a relay Ri , they will be sent to a linear shift register with tapped coefficients (gi,0 , gi,1 , . . . , gi,v ), where gi,d ∈ F2 for d = 0, 1, . . . , v, and v is the maximal memory order. Define gi (D)  gi,0 + gi,1 D + . . . + gi,v Dv , where D represents the symbol delay. The proposed DSTTC is based 978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings. on the idea of “virtual transmit antennas”. Namely, each Ri with L + 1 resolvable paths is equivalent to a relay with L + 1 antennas whose transfer function matrix is gi (D)  [gi0 (D), gi1 (D), . . . , giL (D)] and gij (D) = Dj gi (D) for each path [13]. Denote GM (D)  [g1 (D), g2 (D), . . . , gM (D)]. The coefficient matrix of gi (D) is the matrix of dimension (L + 1) × (v + L + 1): ⎡ ⎤ ⎡ ⎤ gi gi0 ⎢ gi1 ⎥ ⎢ gi(1) ⎥ ⎢ ⎥ ⎢ ⎥ i G = ⎢ . ⎥ = ⎢ . ⎥. (4) ⎣ .. ⎦ ⎣ .. ⎦ (L) giL gi The generator matrix under frequency-selective channels is then equivalent to (5) GM = [(G1 )T , (G2 )T , . . . , (GM )T ]T . If the binary source information bits detected in the relays in one packet are ū ∈ F21×Lu , then the binary output of all the paths belongs to the set M (L+1)×(v+Lu +L) C = {C(ū) ∈ F2 | (c10 (ū)T , c11 (ū)T , . . . , cM L (ū) ) , ū = (u1 , . . . , uLu ) ∈ F21×Lu } T T (6) where cij (ū) is binary output vector for the jth path of s1i : cij (ū) = ū ◦ gij (7) for i = 1, 2, . . . , M , j = 0, 1, . . . , L C(ū) generated by (7) can be rewritten as C(ū) = [(ū ◦ g10 )T , · · · , (ū ◦ g1L )T , · · · , (ū ◦ gM L )T ]T  ū ◦ GM . (8) In turn, the DSTTC generated by GM belongs to the set S ={S ∈ CM (L+1)×(v+Lu +L) |(S)m,n = (−1)(C(ū))m,n , C(ū) ∈ C}. (9) In this code structure, if the length of information bits in one packet is Lu , the rate of the space-time trellis code S generated by GM is Lu /(Lu + v + L + Le ) bits/s/Hz. For long data packet, the rate approaches 1 bit/s/Hz. In the following, we investigate conditions on the generator matrix GM to achieve full diversity. The timing delay of relay Ri is τi , i.e., τi dumb symbols ⋆ are padded to the left of (i − 1)L + jth row in the signal matrix Sa , j = 1, . . . , L + 1, as shown in (3). If dumb symbol ⋆ = 1, then it is equivalent to that τi zeros are padded to the left of the (i − 1)L + jth row of binary matrix C(ū), j = 1, . . . , L. These matrices can be generated by [Dτ1 g1 (D), Dτ2 g2 (D), . . . , DτM gM (D)]. To ensure the full diversity in the asynchronous cooperative communications, there are requirements on the tapped coefficients gi,d , i = 1, 2, . . . , M, d = 0, 1, . . . , v, stated in the following theorem. Theorem 1: Define the asynchronous version of the generator matrix GM as GaM = [(G1a )T , (G2a )T , · · · , (GM a )T ]T  T T T T   (τ ) (τ ) (τ ) , · · · , g1L1 , . . . , gMM . = g101 L The STC generated by g1 (D), g2 (D), . . . , gM (D) achieves full diversity in the asynchronous cooperative communication if and only if GaM has full rank in the binary field F2 for arbitrary τ1 , τ2 , . . . , τM . Proof: The proof follows the same argument of the stacking construction in flat-fading channels [10]. In Theorem 1, GaM is a submatrix-shifted version of GM and for each submatrix Gi , the shifted amount τi is arbitrary. The importance of Theorem 1 lies in that, we only need to construct gi (D) such that any submatrix-shifted version GaM of the generator matrix GM has full rank. Remark 1: The main difference between Theorem 1 and [10, Theorem 1] is that, gi (D) here is constructed in a way to ensure that any sub-matrix-shifted version GaM of the generator matrix GM has full rank. If L = 0, i.e. flat fading channels, each sub-matrix Gi in GM is degraded to one row only. Remark 2: In [11], a family of the generator matrices, called shift-full-rank (SFR) matrices, is constructed such that they have full row rank no matter how their rows are shifted. However, the SFR matrices are not shift-full-rank in frequencyselective channels, since each relay has L + 1 paths and the tapped coefficients of each path are equivalent to the right shifted versions of each relay’s tapped coefficients, as shown in (4). Besides, GM consists of the tapped coefficients of the relays and the right shifted versions of each relay’s tapped coefficients. If GM is an SFR matrix, we cannot ensure the matrix ḠM obtained by adding some rows (the nonzero shifted versions of the rows in GM ) to GM is an SFR matrix. An example is given here. Example 1: If there are two relays and channels are flat fading, the generator matrix   1 0 0 . G2 = 0 1 1 is an SFR matrix and it has full row rank no matter how its two rows are shifted. However in frequency-selective channels, if each relay has two paths. Then, according to (4), (5), Ḡ2 obtained by G2 is written as ⎡ ⎤ 1 0 0 0 ⎢0 1 0 0⎥ ⎥ Ḡ2 = ⎢ ⎣0 1 1 0⎦ . 0 0 1 1 It can be easily checked that the asynchronous version Ḡa2 of Ḡ2 may not have full rank. In the following, we give sufficient conditions under which the asynchronous versions GaM of GM always have full rank. The discussion is divided into two cases. B. Generator Matrix Construction for DSTTC Theorem 2: A necessary condition to ensure the full rank property of GM is that the maximum memory order v of the generating polynomial gi (D), i = 1, . . . , M should not be less than (M − 1)(L + 1). 978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings. TABLE I THE BINARY VECTORS IN OCTAL FOR DIFFERENT M UNDER FREQUENCY- SELECTIVE CHANNELS M L 2 3 4 1 2 3 1 2 1 2 3 4 5 v1 15 35 41 21 111 401 101 2017 403 v2 17 25 61 31 141 601 143 1403 601 v3 v4 v5 33 143 711 151 1501 641 177 777 743 326 Proof: Since GM is a binary matrix of dimension M (L+ 1) × (v + L + 1), v + L + 1 must not be less than M (L + 1), i.e., v ≥ (M − 1)(L + 1) in order to achieve full row rank M (L + 1). Theorem 3: For any two different basic vectors v1 , v2 with l(v1 )  =  l(v2 ) ≥ L + 2, if the generator matrix    (1) G2 = v1T , v1 T (L) , . . . , v1 T (1) , v2T , v2 T (L) T T , . . . , v2 has full row rank in the binary field F2 , its asynchronous versions Ga2 will also have full rank in the binary field F2 . Proof: see in [16]. Theorem 3 indicates that when there are two relays and the generator matrix G2 is constructed by two different basic vectors v1 , v2 with l(v1 ) = l(v2 ) = L + 2, the asynchronous matrix Ga2 has full row rank if G2 is full row rank. We should note that L + 2 is the minimum length of basic vectors for M = 2 according to both Theorem 2 and Theorem 3. Hence we only need to construct the full row rank binary matrix G2 from two different basic vectors that have the minimum length L + 2. The following theorem gives a sufficient condition to ensure that G2 is of full row rank. Theorem 4: For any two different basic vectors v1 , v2 with l(v1 ) = l(v2 ) = L+2, where L is the maximum length of the (L) CIR for two relays, if l((v1 + v2 ) + (v1 + v2 ) ) < L + 2, then G2 is of full row rank. Proof: see in [16]. Example 2: If L = 1, the basic vectors v1 , v2 are 101 and 111 according to Theorem 3 and Theorem 4. Then, generator matrix G2 can be written as ⎡ ⎤ 1 0 1 0 ⎢0 1 0 1 ⎥ ⎥ G2 = ⎢ ⎣1 1 1 0 ⎦ . 0 1 1 1 We also list the basic vectors for the cases that L = 2, 3, 4 in Table I. Normally when there is only one antenna in the destination receiver, the performance gain achieved by more than four transmit antennas is marginal compared with that achieved by four transmit antennas [17], [18]. In addition, if M is too large, the resulted decoding complexity is prohibitively high. Hence, we only consider the case that M ≤ 5 in the following discussions. For M = 3, we have the following theorem. Theorem 5: For any three different basic vectors v1 , v2 and v3 with l(v1 ) = l(v2 ) = 2L + 3, if the generator  matrix G = l(v3 ) ≥ T T T 3  T (L) (L) (L) T T T v1 , . . . , v1 , v2 , . . . , v2 , v3 , . . . , v3 has full row rank in the binary field F2 , its asynchronous versions Ga3 will also have full rank in the binary field F2 . Proof: see in [16]. For scenario of M > 3, it is intractable to find a sufficient condition for GM to ensure full diversity in asynchronous cooperative communications. However, according to Theorem 1 and Theorem 2, we can find the binary vectors that make all the asynchronous versions of the generator matrix full rank for arbitrary M by using computer search. Hence, we give the code search results in Table I for different length L and M = 4, 5 as well as those for M = 2, 3 which are found from Theorem 3,4 and 5. It should be mentioned that the above results are obtained for BPSK signals only. For the other constellations, a general result can also be obtained based on the unified construction method [14]. The process is straightforward, and is omitted. IV. S IMULATION R ESULTS In this section, we evaluate the performance of our DSTTC through various numerical examples. In the examples, we assume that a frame contains 60 information bits and the channels are quasi-static Rayleigh fading. Furthermore, we assume that no errors occur in phase I transmission. We also assume that there is only one antenna in every node, and the random delays are uniformly selected from the set {0, 1, . . . , Le }, Le = 2 unless otherwise stated. In addition, BPSK modulation is used. We first compare the frame error rate (FER) performances for delay diversity (DD) method in [18], the space-time trellis code generated by SFR matrix in example 1 and our DSTTC in example 2. Two relays with L1 = L2 = 1 are assumed. Both the asynchronous and synchronous transmissions are considered. The FER performances versus SNR of these three codes are shown in Fig. 2. We see that our DSTTC outperforms the DD method in both the asynchronous and synchronous cases. That is to say, our DSTTC can achieve full diversity order of 4, which agrees well with the theoretical studies. In addition, the performance of the DD method degrades significantly for asynchronous case since its code error matrix does not have full rank and the achievable diversity order for DD in asynchronous case is 2. It is also seen that the performance of the code generated by SFR matrix degrades for asynchronous case, because it can only achieve the diversity order of 3 under frequency-selective channels. On the other hand, our proposed DSTTC has the same complexity with the DD method and the space-time trellis code generated by SFR matrix due to the same number of states used. Next we show the performance of the proposed DSTTC in example 2 under the synchronous case and the asynchronous case with Le = 2 and Le = 4 in Fig. 3. We see that the performance of our proposed DSTTC improves when the relative timing errors range increases. This is because that 978-1-4244-2324-8/08/$25.00 © 2008 IEEE. This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings. 0 0 10 10 Delay Diversity, synchronous DSTTC, asynchronous −1 −1 10 FER FER 10 −2 10 −2 10 DSTTC,G12,asynchronous DSTTC,G12,synchronous −3 10 −4 10 −3 10 SFR code, asynchronous SFR code, synchronous delaydiversity, asynchronous delaydiversity,synchronous 0 5 −4 10 10 15 3 SNR(dB) Fig. 2. 6 9 12 SNR(dB) Performance Comparison for DD, SFR code and DSTTC. Fig. 4. Comparison of FER for DSTTC generated by [21,31,33] and DD. 0 10 M=2,L=1,Le=0 M=2,L=1,Le=2 M=2,L=1,Le=4 −1 FER 10 −2 10 −3 10 −4 10 3 Fig. 3. 6 9 SNR(dB) 12 15 Performance Comparison for different Le . the case with asynchronous transmission or larger Le has a larger memory order. According to Eq.(8-2-36) in [1], the hamming free distance is upper bounded by the memory order, so the upper bound of the Hamming free distance in the asynchronous case is higher than that of the synchronous case. Thus the asynchronous case or larger Le case will have bigger coding gain than the synchronous case or smaller Le . Finally, in Fig.4 we set M = 3, L = 1 and compare the FER of DSTTC generated by the generator polynomials in Table I in the asynchronous case and the DD method in the synchronous case. From Fig.4, we can see that our DSTTC in the asynchronous case outperforms the DD method in the synchronous case by about 2dB at FER=10−2 . This is because our DSTTC can achieve full diversity over frequency-selective channels and has larger hamming free distance than the delay diversity, which agrees well with the theoretical studies. V. C ONCLUSIONS In this paper we propose a novel family of DSTTC that achieves full cooperative and multipath diversities for asynchronous cooperative communications. We give sufficient conditions to construct such family of DSTTC with minimum memory order. The proposed new technique can be applied to distributed wireless networks to enhance transmission energy efficiency and reduce synchronization cost. R EFERENCES [1] J. G.Proakis, Digital Communications, 4th ed. Hill, 2001. New York: McGraw- [2] A. Sendonaris, E. Erkip, and B. Aazhang, “User cooperation diversity— part I: System description,” IEEE Trans. Commun., vol. 51, no. 11, pp. 1927–1938, Nov. 2003. [3] J. N. Laneman, D. N. C. Tse, and G. W. Wornell, “Cooperative diversity in wireless networks: efficient protocols and outage behavior,” IEEE Trans. Inf. Theory, vol. 50, no. 12, pp. 3062– 3080, Dec. 2004. [4] J. N. Laneman and G. W. 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This full text paper was peer reviewed at the direction of IEEE Communications Society subject matter experts for publication in the IEEE "GLOBECOM" 2008 proceedings.