Cascade Orthogonal Space-Time Block Codes for Wireless Multi-Hop Relay | N 1, Papers of Health sciences

Download Cascade Orthogonal Space-Time Block Codes for Wireless Multi-Hop Relay | N 1 and more Papers Health sciences in PDF only on Docsity! 1 Cascaded Orthogonal Space-Time Block Codes for Wireless Multi-Hop Relay Networks Rahul Vaze and Robert W. Heath Jr. The University of Texas at Austin Department of Electrical and Computer Engineering Wireless Networking and Communications Group 1 University Station C0803 Austin, TX 78712-0240 email: vaze@ece.utexas.edu, rheath@ece.utexas.edu Abstract Distributed space-time block coding is a diversity technique to mitigate the effects of fading in multi-hop wireless networks, where multiple relay stages are used by a source to communicate with its destination. This paper proposes a new distributed space-time block code called the cascaded orthogonal space-time block code (COSTBC) for the case where the source and destination are equipped with multiple antennas and each relay stage has one or more single antenna relays. Each relay stage is assumed to have receive channel state information (CSI) for all the channels from the source to itself, while the destination is assumed to have receive CSI for all the channels. To construct the COSTBC, multiple orthogonal space-time block codes are used in cascade by the source and each relay stage. In the COSTBC, each relay stage separates the constellation symbols of the orthogonal space-time block code sent by the preceding relay stage using its CSI, and then transmits another orthogonal space- time block code to the next relay stage. COSTBCs are shown to achieve the maximum diversity gain This work was funded in part by Samsung Electronics and DARPA through IT-MANET grant no. W911NF-07-1-0028. 2 in a multi-hop wireless network with flat Rayleigh fading channels. Several explicit constructions of COSTBCs are also provided. It is also shown that COSTBCs require minimum decoding complexity thanks to the connection to orthogonal space-time block codes. I. INTRODUCTION Distributed space-time block coding (DSTBC) is a technique to improve reliability in a relay assisted communication, where one or more relays help the source to communicate with its destination. In DSTBCs, relay antennas are used together with the source antennas in a distributed manner to transmit a space time block code (STBC) [1] to the destination. By introducing redundancy in space and time, DSTBCs increase the reliability of the communication by increasing the diversity gain, defined as the negative of the exponent of the signal-to-noise ratio (SNR) in the pairwise error probability expression at high SNR [1]. Relay assisted communication is required in the deployment of large wireless networks, such as an ad-hoc network or a sensor network, where most often the destination is out of the source’s communication range and multiple relays/hops are required for the source signal to reach the destination. Relay assisted communication is also used in a cellular wireless networks to improve the performance of cell edge users, and has been incorporated in wireless standards such as IEEE 802.16j and IEEE 802.16j and IEEE 802.11s. Consequently, there is a need for construction of DSTBCs that can achieve maximum diversity gain in the presence of two or more hops between source and destination. Unfortunately, most prior DSTBC designs [2]–[13] are restricted to a two-hop network. Moreover, the decoding complexity of known DSTBC designs [2]–[12] is quite high and none of them allow simple decoding like the Alamouti code [14], except for [13]. By exploiting extended Clifford algebra properties, the DSTBC construction [13] has lower decoding complexity than other maximum diversity gain DSTBC constructions [2]–[12], however, it still requires decoding complexity more than that of Alamouti code decoding. In this paper we design maximum diversity gain achieving DSTBCs for multi-hop wireless networks that require minimum decoding complexity. Construction of DSTBCs with minimum 5 trace of matrix A are denoted by det(A) and tr(A). The field of real and complex numbers is denoted by R and C, respectively. The space of M × N matrices with complex entries is denoted by CM×N . The Euclidean norm of a vector a is denoted by |a|. An m × m identity matrix is denoted by Im and 0m is as an all zero m×m matrix. The superscripts T ,∗ ,† represent the transpose, transpose conjugate and element wise conjugate. The expectation of function f(x) with respect to x is denoted by Ef(x). A circularly symmetric complex Gaussian random variable x with zero mean and variance σ2 is denoted as x ∼ CN (0, σ). We use the symbol .= to represent exponential equality i.e., let f(x) be a function of x, then f(x) .= xa if limx→∞ log(f(x)) log x = a and similarly . ≤ and . ≥ denote the exponential less than or equal to and greater than or equal to relation, respectively. To define a variable we use the symbol :=. II. SYSTEM MODEL Consider a multi-hop wireless network where a source terminal with M0 antennas wants to communicate with a destination terminal with MN antennas via N − 1 relay stages as shown in Fig. 1. Each relay in any relay stage has a single antenna; Mn denotes the number of relays in the nth relay stage. It is assumed that the relays do not generate their own data and only operate in half-duplex mode. No assumption is made on the half-duplex or full-duplex functionality of the relays. In the half-duplex case the rate of transmission gets halved. Similar to the model considered in [18], we assume that any relay of relay stage n can only receive the signal from any relay of relay stage n−1, i.e. we consider a directed multi-hop wireless network. This assumption is valid for the case when successive relay stages appear in increasing order of distance from the source towards the destination and any two relay nodes are chosen to lie in adjacent relay stages if they have sufficiently good SNR between them. In any practical setting there will be interference received at any relay node of stage n because of the signals transmitted from relay nodes of relay stage 0, . . . , n−2 and n+2, . . . , N−1. Due to relatively large distances between non adjacent relay stages, however, this interference is quite small and we account for that in the additive noise term. Throughout this paper we refer to this multi-hop wireless network with N − 1 relay stages as an N -hop network. 6 As shown in Fig. 1, the channel between the source and the ith relay of the first stage of relays is denoted by hi = [h1i h2i . . . hM0i] T , i = 1, 2, . . . ,M1, between the jth relay of relay stage s and the kth relay of relay stage s + 1 by f sjk, s = 0, 1, . . . , N − 2, j = 1, 2, . . . ,Ms, k = 1, 2, . . . ,Ms+1 and the channel between the relay stage N − 1 and the `th antenna of the destination by g` = [g1` g2` . . . gMN−1`] T , ` = 1, 2, . . . ,MN . We assume that hi ∈ CM0×1, f sjk ∈ C1×1, gl ∈ CMN−1×1 with independent and identically distributed (i.i.d.) CN (0, 1) entries for all i, j, k, `, s. We assume that the mth relay of nth stage knows hi, f sjk, ∀ i, j, k, s = 1, 2, . . . , n− 2, fn−1jm ∀j and the destination knows hi, f sjk,gl, ∀ i, j, k, l, s. We further assume that all these channels are frequency flat and block fading, where the channel coefficients remain constant in a block of time duration Tc and change independently from block to block. We assume that the Tc is at least max{M0, M1, . . . ,MN−1}. A. Problem Formulation Definition 1: (STBC) [21] A rate-L/T T × Nt design D is a T × Nt matrix with entries that are complex linear combinations of L complex variables s1, s2, . . . , sL and their complex conjugates. A rate-L/T T×Nt STBC S is a set of T×Nt matrices that are obtained by allowing the L variables s1, s2, . . . , sL of the rate-L/T T×Nt design D to take values from a finite subset Cf of the complex field C. The cardinality of S = |Cf |L, where |Cf | is the cardinality of C. We refer to s1, s2, . . . , sL as the constituent symbols of the STBC. Definition 2: A DSTBC C for a N -hop network is a collection of codes {S0,S1, . . . ,SN−1}, where S0 is the STBC transmitted by the source and Sn = [f1n(Sn−1) . . . f Mn n (Sn−1)] is the STBC transmitted by relay stage n, where f jn(Sn−1) is the vector transmitted by the j th relay of stage n which is a function of Sn−1, j = 0, . . . ,Mn, n = 1, . . . , N − 1. An example of a DSTBC is illustrated in Fig. 2. Definition 3: The diversity gain [1], [3] of a DSTBC C is defined as dC = − lim E→∞ log Pe (E) log E , 7 Pe (E) is the pairwise error probability (PEP) using coding strategy C, and E is the sum of the transmit power used by each node in the network. The problem we consider in this paper is to design DSTBCs that achieve the maximum diversity gain in a N -hop network. To identify the limits on the maximum possible diversity gain in a N -hop network, an upper bound on the diversity gain achievable with any DSTBC is presented next. Theorem 1: The diversity gain dC of DSTBC C for an N -hop network is upper bounded by min {MnMn+1} n = 0, 1, . . . , N − 1. Proof: See Proposition 2.1 [17]. Theorem 1 implies that the maximum diversity gain achievable in a N -hop network is equal to the minimum of the maximum diversity gain achievable between any two relay stages, when all the relays in each relay stage are allowed to collaborate. In the next section we propose COSTBCs that are shown to achieve this upper bound on the diversity gain. III. CASCADED ORTHOGONAL SPACE-TIME CODE In this section we introduce the COSTBC design for a N -hop network. Before introducing COSTBCs, we need the following definitions. Definition 4: With T ≥ Nt, a rate L/T T × Nt STBC S is called full-rank or fully-diverse or is said to achieve maximum diversity gain if the difference of any two matrices M1,M2 ∈ S is full-rank, minM1 6=M2, M1,M2∈S rank(M1 −M2) = Nt. Definition 5: A rate-L/K K × K STBC S is called an orthogonal space-time block code (OSTBC) if the design D from which it is derived is orthogonal i.e. DD∗ = (|s1|2+. . .+|sL|2)IK . Definition 6: Let S be a rate-L/K K×K STBC. Then, using CSI, if each of the constituent symbols si, i = 1, . . . , L of S can be separated/decoded independently of sj ∀i 6= j i, j = 1, . . . , L with independent noise terms, then S is called a single symbol decodable STBC. Remark 1: OSTBCs are single symbol decodable STBCs [15]. With these definitions we are now ready to describe COSTBC for a N -hop network. 10 Theorem 3 is proved in Appendix II. Discussion: In this section we constructed COSTBCs by cascading OSTBCs at each relay stage. OSTBCs are cascaded at each relay stage by first separating each constellation symbol of the OSTBC transmitted from the preceding relay stage and then transmitting another OSTBC to the next relay stage. The proposed OSTBC cascading strategy is novel and different than other approaches that use Alamouti code or OSTBC in a distributed manner [12], [22]. In [22], for the case of two-hop network, with a single relay and single antenna at the source and the relay, the relay node decodes the received signal from the source when the mutual information is more than the rate of transmission and then transmits an Alamouti code together with the source to the destination. In [12], [22] for the case of two-hop network, with multiple relays and single antenna at the source and each relay, the relay nodes scale the received signals and transmit an OSTBC in the next time slot. The key difference between COSTBCs and [12], [22] is that in COSTBCs all constellation symbol are decoupled at each relay stage without any decoding, and transmitted using an OSTBC to the next relay stage, rather than just scaling by each relay [12], or decoding by the relay [22]. As a result, COSTBCs simplify the problem of construction of maximum diversity gain DSTBCs for a N -hop network to the problem of construction of N -OSTBCs for each relay stage, which is well known [15]. Moreover, COSTBCs can be constructed for any number of antennas at the source and relays, and for multiple hop networks. We showed that by cascading OSTBCs, the single symbol decodable property of OSTBCs is preserved. Therefore, COSTBCs require minimum decoding complexity, which is quite critical for practical implementations. We also proved that COSTBCs maximum diversity gain in a N - hop network. To obtain this result we used the single symbol decodable property of COSTBCs. Using the single symbol decodable property, the destination decouples the different constellation symbols of the OSTBC transmitted by the source, where the channel gain of each of the constellation symbols is shown to have at least minn=0,1,...,N−1{MnMn+1} channel coefficients. Therefore the diversity gain of COSTBC is equal to minn=0,1,...,N−1{MnMn+1} which meets the upper bound Theorem 1. 11 IV. EXPLICIT CODE CONSTRUCTIONS In this section, we explicitly construct COSTBCs that achieve maximum diversity gain in N -hop networks. We present examples of COSTBCs for N = 2, M0 = M1 = 2 using the Alamouti code [14], N = 2, M0 = M1 = 4 using the rate-3/4 4 antenna OSTBC [15] and N = 2, M0 = M1 = 4 using the rate-3/4 4 antenna OSTBC and the Alamouti code. Example 1: (Cascaded Alamouti Code) We consider N = 2, M0 = M1 = 2 case and let S0 be the Alamouti code given by: Sala =  s1 s2 −s∗2 s∗1  where s1 and s2 are constituent symbols of the Alamouti code. The 2× 1 received signal at relay m is r1m r2m  =√E0  s1 s2 −s∗2 s∗1  h1m h2m +  n1m n2m  for m = 1, 2. Transforming this in the usual way r1m −r∗2m  =√E0  h1m h2m −h∗2m h∗1m  ︸ ︷︷ ︸ H̃m  s1 s2 +  n1m −n∗2m  for m = 1, 2. We define h̃m := |h1m|2 + |h2m|2, η1m := (n1mh∗1m + n∗2mh2m), and η2m := (n1mh ∗ 2m − n∗2mh1m). Pre-multiplying by H̃∗m,[ r̂1m r̂ ∗ 2m ]T := H̃∗m [ r̂1m r̂ ∗ 2m ]T = √ E0 [ h̃ms1 h̃ms2 ]T + [ η1m η2m ]T for m = 1, 2. Now using A1 =  1 0 0 1  ,B1 = 02, A2 = 02,B2 =  0 −1 1 0  (7) the STBC S1 formed by the two relays is equal to STala which is also an OSTBC as required. Note that Ai,Bi i = 1, 2 satisfy the requirements of (5). We call this the cascaded Alamouti code. 12 Example 2: In this example we consider the case N = 2, M0 = 4, M1 = 4. We choose S0 to be the rate-3/4 OSTBC for 4 transmit antennas given by S0 =  s1 s2 s3 0 −s∗2 s∗1 0 s3 s∗3 0 −s∗1 s2 0 s∗3 −s∗2 −s1  (8) and use A1 =  1 0 0 0 0 0 0 0 0 0 0 0  ,A2 =  0 1 0 0 0 0 0 0 0 0 0 0  , A3 =  0 0 1 0 0 0 0 0 0 0 0 0  , A4 =  0 0 0 0 0 1 0 1 0 −1 0 0  and B1 =  0 0 0 0 −1 0 0 0 1 0 0 0  ,B2 =  0 0 0 1 0 0 0 0 0 0 0 1  , B3 =  0 0 0 0 0 0 −1 0 0 0 −1 0  , B4 =  0 0 0 0 0 0 0 0 0 0 0 0  . It is easy to verify that tr(A∗i Ai + B ∗ i Bi) = 3 and A ∗ i Bi = −B∗i Ai, i = 1, 2, 3, 4 as required. Then the STBC S1 = S0 using these Ai,Bi i = 1, 2, 3, 4, which is a rate-3/4 OSTBC as described above. In both the previous examples we constructed COSTBC for N = 2-hop case by repeatedly using the same OSTBC at both the source and the relay stage. Using a similar procedure, it is easy to see that when Mi = Mj ∀ i, j = 0, 1, . . . , N − 1, i 6= j we can construct COSTBCs by using particular OSTBC for M0 antennas at the source and each relay stage, e.g. if O is an OSTBC for M0 antennas, then by using Sn = O, n = 0, 1, . . . , N − 1 we obtain a maximum diversity gain achieving COSTBC. OSTBC constructions for different number of antennas can be found in [15]. In the next example we construct COSTBC for M0 = 4 and M1 = 2 by cascading the rate-3/4 4 antenna OSTBC with the Alamouti code. 15 implementation. We then gave an explicit construction of COSTBCs for various numbers of source, destination, and relay antennas. The only restriction that COSTBCs impose is that the source and all the relay stages have to use an OSTBC. It is well known that high rate OSTBC do not exist, therefore the COSTBCs have rate limitations. For future work it will be interesting to see whether the OSTBC requirement can be relaxed without sacrificing the maximum diversity gain and minimum decoding complexity of the COSTBCs. APPENDIX I In this section we prove that COSTBCs have the single symbol decodable property. We first show this for 2-hop networks and then generalize it to N -hop networks where N is any arbitrary integer. Let S0 be the transmitted OSTBC from the source and s = [s1, . . . , sL]T be the vector of the constituent symbols of S0. Then from (3), using CSI, the received signal r1k at the k th relay of relay stage 1 can be transformed into r̂1k where r̂ 1 k = √ E0H 1 2 s + n̂1k, H is defined in (2) and the entries of n̂k are independent and CN (0, 1) distributed. For N = 2, from (6) the received signal at the jth antenna of the destination can be written as yj = [t11 t 1 2 . . . t 1 M1 ]gj +zj for j = 1, 2, . . . M2, where t1k is the transmitted vector from relay k (4) of relay stage 1. The received signal yj can also be written as yj = √ E0E1M Lγ S1 [ √∑M0 m=1 |hm1|2g1j √∑M0 m=1 |hm2|2g2j . . . √∑M0 m=1 |hmM1|2gM1j ]T + √ E1M1 Lγ [A1n̂1 + B1n̂ † 1 A2n̂2 + B2n̂ † 2 . . . AM1n̂M1 + BM1n̂ † M1 ]gj + zj︸ ︷︷ ︸ wj where S1 = [A1s + B1s† A2s + B2s† . . . AM1s + BM1s †]. Since S1 is an OSTBC, invoking the single symbol decodable property of OSTBC (2) and using the fact that entries of wj are independent, it follows that, using CSI, the received signal yj can be transformed into ŷj , where ŷj = √ E0E1M1 Lγ  ∑M1 k=1 |gkj|2 (∑M0 m=1 |hmk|2 ) 0 0 0 . . . 0 0 0 ∑M1 k=1 |gkj|2 (∑M0 m=1 |hmk| )2  s + ŵj 16 and the entries of ŵj are independent. Thus, it is clear that all the constituent symbols s1, . . . , sL can be separated with independent noise terms and we conclude that COSTBCs have the single symbol decodable property for a 2-hop network. Now assume that the result is true for a k-hop network. Then by induction hypothesis, the received signal at the jth antenna of destination of the k-hop network can be written as r̂kj = pk−1H 1 2 jks+ n̂ k j , where pk−1 is a function of E0, . . . , Ek−1, Hjk is a diagonal matrix with each entry equal to the channel gain and the entries of n̂kj are independent and CN (0, 1) distributed. Now we extend the k-hop network to a k + 1-hop network by assuming that the actual destination to be one more hop away and using the destination of the k-hop case as the kth relay stage with Mk relays by separating the Mk antennas into Mk relays with single antenna each. With this extension, let the OSTBC transmitted by kth relay stage be Sk using received signal rkj , then the received signal at the i th antenna of the destination of k + 1-hop network is rk+1i = pkSk [ H 1 2 1k(1)g1i . . . H 1 2 Mkk (1)gM1i ]T + Ek[A k 1n̂ k 1 + B k 1n̂ k† 1 . . . A k M1 n̂kMk + B k Mk n̂k†Mk ]gj + z k+1 i︸ ︷︷ ︸ wk+1i , where Hjk(1), j = 1, 2, . . . ,Mk is the (1, 1) element of H1k, pk is a function of E0, . . . , Ek, Sk = [A k 1s+B k 1s † Ak2s+B k 2s † . . . AkMks+B k Mk s†] and zk+1i is CN (0, 1) added at the destination. Since Sk is an OSTBC, therefore, using CSI, the received signal at the destination of the k + 1-hop network can be transformed to r̂k+1i , where r̂k+1i = pk  ∑Mk j=1 |gji|2Hk(1) 0 0 0 . . . 0 0 0 ∑Mk j=1 |gji|2Hjk(1)  s + ŵk+1i and the entries of ŵk+1i are independent. Thus, we conclude that the COSTBCs have single symbol decodable property. APPENDIX II We prove Theorem 3 using induction. First we show that COSTBCs achieve maximum diversity gain for N = 2 and then extend the result for a k-hop network, where k is any 17 arbitrary natural number. The outage probability Pout(R) is defined as Pout(R) := P (I(s; r) ≤ R) , where s is the input and r is the output of the channel and I(s; r) is the mutual information between s and r [23]. Let dout(r) be the SNR exponent of Pout with rate of transmission R scaling as r log SNR, i.e. log Pout(r log SNR) . = SNR−dout(r). Then, if Pe(SNR) . = SNR−d(r), then from [19], and the compound channel argument [18], Pout(r log SNR) . = Pe(SNR), d(r) . = dout(r). Therefore to compute d(r), it is sufficient to compute dout(r). In the following we compute dout(r) for the COSTBC with a 2-hop network. For the 2-hop network, using the single symbol decodable property of COSTBCs (Appendix I), the received signal can be separated in terms of the individual constituent symbols of the OSTBC transmitted by the source. Therefore, the received signal can be written as rl = √ θE M2∑ j=1 M1∑ k=1 |gkj|2 ( M0∑ m=1 |hmk|2 ) sl + zl (9) where θ is the normalization constant so as to ensure the total power constraint of E in the network, sl is the lth, l = 1, 2, . . . , L symbol transmitted from the source and zl is the additive white Gaussian noise (AWGN) with variance σ2. Note that zl depends on the channel coefficients, however, as shown in Theorem 1.2 [18], zl is white in the scale of interest and without loss of generality zl can be modeled as CN (0, σ2) that is independent of channel coefficients in the outage analysis. Let SNR := θE σ2 , then Pout(r log SNR) = P ( log ( 1 + SNR M2∑ j=1 M1∑ k=1 |gkj|2 ( M0∑ m=1 |hmk|2 )) ≤ r log SNR ) . ≤ P  M1∑ k=1 min{M0,M2}∑ j=1 |gkj|2|hjk|2 ≤ SNR−(1−r)  ≤ P ( max {j=1,...,min{M0,M2}, k=1,...,M1} |gkj|2|hjk|2 ≤ SNR−(1−r) ) . Since |gkj|2|hjk|2 are i.i.d. for j = 1, . . . , min{M0, M2}, k = 1, . . . ,M1 and the total number of terms are min{M0M1, M1M2}, Pout(r log SNR) . = P ( |g11|2|h11|2 ≤ SNR−(1−r) )min{M0M1, M1M2} . 20 By induction hypothesis, the diversity gain of COSTBCs with ci is α, i.e., P ( ci ≤ SNR−(1−r) z ) = ∫ SNR−(1−r)/z 0 fci(y)dy ≤ k4 ( SNR−(1−r) z )α where k4 is a constant. Thus, Pout(r log SNR) ≤ P ( Z ≤ SNR−(1−r) ) + ∫ ∞ SNR−(1−r) k4SNR −α(1−r) ( 1 z )α fZ(z)dz. (13) Since Z is a gamma distributed random variable with PDF e −zzMk+1−1 Mk+1−1! , the first term in Pout(r log SNR) expression can be found in [19] and is given by P ( Z ≤ SNR−(1−r) ) . = SNR−Mk+1(1−r). Now we are left with computing the second term which can be done as follows.∫ ∞ SNR−(1−r) k4SNR −α(1−r) ( 1 z )α fZ(z)dz . = k4SNR −α(1−r) ∫ ∞ SNR−(1−r) z−α e−zzMk+1−1 Mk+1 − 1! dz . = SNR−α(1−r). Thus, from (13) it follows that Pout(r log SNR) . = SNR−Mk+1(1−r) + SNR−α(1−r) . = SNR−min{Mk+1,α}(1−r). 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Relay 2 M 0 Relay M 1 h 1








Relay 1 Relay j RelayRelay M s M s+1 Relay 1 f ij s Relay i Source Relay 1




M N Relay M N−1 Relay p g p Destination


























































































Stage 1 1 2 Stage s Stage s+1 1 2 Fig. 1. System Block Diagram of a N-hop Wireless Network S S S Relay M2Relay M1 function of S n 1 Source Destination Relay 1 2 Relay 1 Relay 1 S 0 f 1 1 f 1 M 1 f M2 2 2(S ) Stage 1 Stage 2 Relay MN−1 Stage N−1 N−1 f (S )01 1 f i j n(S ) is a f 1 N−1 (S ) 0 (S )1 f M N−1 N−1 (S )N−2 (S )N−2 Fig. 2. An Illustration Of The DSTBC Design Problem