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Cyber Journals: Multidisciplinary Journals in Science and Technology, Journal of Selected Areas in Telecommunications (JSAT), January Edition, 2011 Precoded Differential OFDM for Relay Networks Homa Eghbali and Sami Muhaidat investigated for OFDM systems. However, the proposed Abstract — We study the performance of differential space- system could not fully exploit the underlying multipath time codes with linear constellation precoding (LCP) for diversity. Motivated by this practical concern, we extend the orthogonal frequency division multiplexing (OFDM) cooperative work in [26] to linearly coded and grouped precoded OFDM networks over frequency selective channels. Through exploiting systems for cooperative communications. the unitary structure of the orthogonal STBCs, we design a low complexity differential STBC-LCP-OFDM receiver for LCP-OFDM was developed in [25] for multicarrier wireless cooperative networks. We assume the amplify-and-forward transmissions over frequency selective fading channels. Not protocol and consider both single relay and multi-relay scenarios. only LCP improves the uncoded OFDM performance, but also Under the assumption of perfect power control for the relay doesn't reduce the transmission rates of uncoded OFDM, and terminal and high signal-to-noise ratio for the underlying links, guarantees symbol detectability [24]. LCP-OFDM was our performance analysis demonstrates that the considered extended to GLCP-OFDM to exploit the correlation structure scheme is able to exploit fully the spatial diversity. of subchannels, such that, the set of correlated sunchannels are Index Terms — Linear constellation precoding, orthogonal split into subsets of less correlated subchannels. Within each frequency division multiplexing, differential detection, subset of subcarriers, a linear constellation precoder (LCP) is cooperative communications. designed to maximize diversity and coding gains. The LCPs are in general complex and could possibly be nonunitary. I. INTRODUCTION While greatly reducing the system complexity, subcarrier grouping maintains the maximum possible diversity and There has been a growing demand for high data rate coding gains [25]. GLCP-OFDM is a considerably flexible services for wireless multimedia and internet services. Spatial system that offers maximum multipath diversity, as well as diversity offers significant improvement in link reliability and large coding gains, and guaranteed symbol detectability with spectral efficiency through the use of multiple antennas at the low decoding complexity. transmitter and/or receiver side [1]-[4]. Recently, cooperative Related work and contributions: Although there have been communications have gained much attention due to the ability considerable research efforts on differential STBC to explore the inherent spatial diversity in relay channels [5]- (conventional and distributed) for frequency flat fading [9]. The idea behind cooperative diversity is that in a wireless channels (see for example [10]-[17]), only a few isolated environment, the signal transmitted by the source nodes is results have been reported on conventional differential STBC overheard by other nodes, which are also known as partners. for frequency-selective channels [18]-[20]. Distributed The source and its partners jointly process and transmit their differential STBC multi-carrier transmission for broadband information, creating a "virtual antenna array" although each cooperative networks was investigated in [26], yet was of them is equipped with only one antenna. suboptimal. Our contributions in this work are summarized as Most the current works on cooperative diversity consider follows: coherent detection and assume the availability of perfect 1. We propose a distributed differential linear channel state information (CSI) at the receiver. In fading constellation precoded OFDM (DD-LCP-OFDM) channels where the coherence time is large enough, the STBC scheme for broadband cooperative systems channel estimation can be carried out through the use of pilot with amplify-and-forward (AF) relaying. The symbols [10]. For fast fading channels where the phase carrier proposed scheme can be considered as an extension recovery is more difficult, differential detection provides a of the DD-OFDM-STBC scheme proposed in [26]. more practical solution. In [10-14], differential detection has Carefully exploiting the underlying orthogonality of been investigated for cooperative transmission scenarios. The distributed STBC, the proposed scheme is able to works in [10-14] assume an idealized transmission fully exploit the available underlying diversity. environment with an underlying frequency-flat fading channel. 2. To further reduce the decoding complexity while This assumption can be justified for narrowband cooperative preserving the performance, we extend the DD-LCP- scenarios with fixed infrastructure; however, it is impractical OFDM STBC scheme to grouped linear constellation if wideband cooperative networks are considered. In [26], the precoded (GLCP)-OFDM system. We observe that applicability of differential STBC to broadband cooperative the optimal performance of the DD-GLCP-OFDM transmission over frequency-selective channels was relies on the design of the GLCP precoder. 3. We present a comprehensive Monte Carlo simulation H. Eghbali and S. Muhaidat are with the School of Engineering Science, Simon Fraser University, Burnaby, B.C., Canada (e-mail: hea7@sfu.ca, study to confirm the analytical observations and give muhaidat@ieee.org). insight into system performance. We extend our work 59 on single-relay scenarios to multi-relay scenarios and j j j vectors h SR , h RD , and h SD , are assumed to be independent analyze system's performance via simulation results. zero-mean complex Gaussian with power delay profile vectors denoted by v SR = [σ SR (0),K , σ SR ( LSR )] 2 2 , The rest of the paper is organized as follows: In Section II, the transmission model is introduced. The differential scheme v RD = [σ RD (0),K , σ RD ( LRD )] 2 2 , and under consideration for distributed LCP-0FDM STBC is v SD = [σ 2 SD (0),K , σ 2 SD ( LSD )] that are normalized such that described in Sections III, and is extended to DD-GLCP- ∑ ( lSR ) = 1 ∑ σ RD ( lRD ) = 1 LSR LRD lSR = 0 σ 2 SR , lRD = 0 2 , and OFDM STBC in section IV. Section V extends our analysis on ∑ σ SD ( lSD ) = 1 . The CIRs are assumed to be constant LSD single relay scenarios to multiple relay scenarios. Numerical 2 lSD = 0 results are presented in Section VI and the paper is concluded in Section VII. over four consecutive blocks and vary independently every four blocks. Notation: (.)* , (.) T , and (.) H denote conjugate, transpose, We consider LCP-OFDM transformation [25] where the and Hermitian transpose operations, respectively. ⊗ denotes LCP is defined by the N × N matrix Φ that has entries over Kronecker product, . denotes the absolute value, . denotes the complex field. Φ should satisfy the transmit-power constraint tr ( ΦΦ H ) = N . Information symbols are parsed the Euclidean norm of a vector, [.]k ,l denotes the (k , l )th entry into frames, where each frame consists of two information of a matrix, [.]k denotes the k th entry of a vector, and blocks. Let the N ×1 vector y SRD represent the data vector that tr ( ·) denotes the matrix trace. I N denotes the identity matrix is transmitted to the relay terminal during the first block of of size N, and 0M × M denotes all-zero matrix of size M × M . each frame, whose entries are complex MPSK symbols that are generated through differential space-time (ST) encoding. For a vector a = [ a0 K a N −1 ]T , After linear constellation precoding, the N ×1 precoded [ P a]s = a ( ( N − s + q ) mod N ) q N where PN is a block y SRD = Φ y SRD is IFFT processed by the inverse FFT % N × N permutation matrix. Q represents the N × N FFT matrix Q H to yield the discrete time signal Q H Φ y SRD . To matrix whose (l , k )th element is given by further remove the IBI, a cyclic prefix of length l ≥ max ( LSR , LRD , LSD ) is inserted per transmitted OFDM Q(l , k ) = 1/ N exp(− j 2π lk / N ) where 0 ≤ l , k ≤ N − 1 . A block and is removed from the corresponding received block. circularly symmetric complex Gaussian random variable is a In the following, without loss of generality, we drop the index random variable Z = X + jY ~ CN (0, σ 2 ) , where X and Y are j for brevity. σ2 The signal received at the relay terminal during the first i.i.d. N (0, ) . Bold upper-case letters denote matrices and signalling interval (broadcasting phase) of each frame after the 2 bold lower-case letters denote vectors. CP removal is rR = E SR H SR Q H Φ y SRD + n R , , (1) II. SYSTEM MODEL Where E SR is the average signal energy over one symbol A single-relay assisted cooperative communication scenario period received at relay terminal, H SR is the N × N circulant is considered. All terminals are equipped with single transmit matrix with entries [ H SR ]m , n = h SR ( ( m − n ) mod N ) , and n R and receive antennas. We assume AF relaying and adopt the user cooperation protocol proposed by Nabar et. al. [9]. is the additive white Gaussian noise vector with each entry Specifically, the source terminal communicates with the relay having zero-mean and variance of N 0 2 per dimension. To terminal during the first signaling interval. There is no ensure that the power budget is not violated, the relay transmission from source-to-destination within this period. In terminals normalizes each entry of the respective received the second signaling interval, both the relay and source signal [rR ]n , n = 1,2,K ,N , by a factor of E (| [rR ]n |2 ) = ESR + N 0 to ensure unit average energy and terminals communicate with the destination terminal. The CIRs for S → R , S → D and R → D links for the j th retransmits the signal during the second signalling interval T transmission block are by h SR = hSR [0],K , hSR [ LSR ] j j j , (relaying phase) of each frame. After some mathematical T manipulations, the received signal at the destination terminal h SD = hSD [0],K , hSD [ LSD ] j j j , and during the relaying phase is given by T ERD ESR h RD = hRD [0],K , hRD [ LRD ] , respectively, where L SR , j j j rD = H RD H SR Q H Φy SRD ESR + N 0 (2) L SD , L RD denote the corresponding channel memory lengths. + ESD H SD Q Φy SD + n D , H All S → R , S → D , and R → D links are assumed to experience frequency selective Rayleigh fading. The random where E RD and E SD are the average signal energies over one 60 symbol period received at destination terminal, H RD and 4R 8 64744 H SD are the N × N circulant matrices with entries Qr ( t ) Qn ( t ) * = * + [ H RD ]m,n = h RD ( ( m − n ) mod N ) and Qr ( t + 1) Qn ( t + 1) (5) γ 1 Λ RD Λ SR Φ γ 2 Λ SD Φ y SRD ( t ) [ H SD ]m ,n = h SD ( ( m − n ) mod N ) , respectively, and n D is . γ 2 Λ SD Φ * − γ 1 Λ* Λ* Φ y SD ( t ) RD SR conditionally (conditioned on h RD ) complex Gaussian with Assuming without loss of generality the symbols to have ERD LRD 2 zero mean and variance σ n D = N 0 1 + 2 ∑0 h RD (m) . variance σ y = σ y 2 2 = 1 , applying the minimum mean square ESR + N 0 m = SRD SD error (MMSE) equalization we have The destination terminal further normalizes the received signal y SRD ( t ) ˆ by a factor of ρ , = E RD LRD y SD ( t ) ˆ ∑0 h RD ( m ) . Note that this (6) 2 where ρ = 1 + α = 1 + γ 1 Φ H Λ * Λ* A ESR + N 0 RD SR γ 2 Φ H Λ SD A 1444m = 4 24444 3 R, α γ 2 Φ H Λ* A SD − γ 1 Φ H Λ RD Λ SR A does not affect the SNR, but simplifies the ensuing where A = ( γ 1Λ RD Λ SR ΦΦ H Λ* Λ* + γ 2 Λ SD ΦΦ H Λ* ) . In the −1 presentation [21]. After normalization, we obtain SR RD SD r = γ 1 H RD H SR Q H Φy SRD + γ 2 H SD Q H Φy SD + n, (3) following, we will discuss the distributed differential (DD)- where n is C N (0, N 0 ) and the scaling coefficients γ 1 and LCP-OFDM system and the ST encoding and decoding procedures employed by it. a c γ 2 are defined as γ 1 = , and γ 2 = respectively, where b b III. DD-LCP-OFDM LRD a = SR E , b = 1+ E / N + E ∑ h RD ( m ) ERD / N0 , The data vectors di ( t ) = di0 ( t ) ,K , diN −1 ( t ) 2 T N 0 RD i = 1,2, SR 0 m=0 represent the OFDM-STBC symbols, where t is the time and c = 1 + SR E . Note that at the end of each frame, E N 0 SD index and complex symbols din ( t ) n = 1,K, N , are drawn the receiver is provided with the time-domain observation r in from an unit-energy MPSK constellation. We are encoding the (3). LCP-OFDM-STBC data vectors d1 ( t ) = Φd1 ( t ) and Inspired by Alamouti code [4], we can further extend (3) to d 2 ( t ) = Φd 2 ( t ) into their linear constellation precoded a distributed (D)STBC-LCP-OFDM scenario, by using the transmit diversity scheme y SRD ( t + 1) = −y * ( t ) and SD differentially encoded frequency domain counterparts y SRD ( m ) = y ( m ) , …, y ( m ) 0 N −1 T and y SD ( t + 1) = y* ( t ) , leading to SRD SRD SRD y SD ( m ) = ySD ( m ) , …, ySD−1 ( m ) m = 2t,2t+1 as following T r ( t ) = γ 1 H RD H SR Q H y SRD ( t ) 0 N + γ 2 H SD Q H Φy SD ( t ) + n ( t ) , Y n ( t ) = Dn ( t ) Y n ( t − 1) (7) (4) where 1 d1 ( t ) d 2n ( t ) n r ( t + 1) = − γ 1 H RD H SR Q Φy H * (t ) SD Dn ( t ) = *, (8) 2 − ( d 2n ( t ) ) ( d1n (t ) ) * + γ 2 H SD Q Φy SRD ( t ) + n ( t + 1) , H * Exploiting the circulant structure of the channel matrices and H RD , H SR , and H SD , we have H i = Q H Λ i Q , where Λ i , y n ( 2t ) y SD ( 2t ) n Y n ( t ) = n SRD , (9) y SRD ( 2t + 1) ySD ( 2t + 1) i denotes SR, RD, SD, is a diagonal matrix whose n ( n, n ) th element is equal to the nth DFT coefficient of hi . Thus, having transforming the received signal r ( 2t ) to the frequency ySRD ( 2t + 1) = − ( ySD ( 2t ) ) , n n * domain by multiplying it with Q matrix and further writing (10) ySD ( 2t + 1) = ( ySRD ( 2t ) ) . n n * the result in matrix form we have Note that (7) is relating 4 frames of information in a differential manner. Applying (10) at the sequence level to the OFDM blocks ( 2t ) and ( 2t + 1) in (9) and substituting them into (4), we obtain 61 r ( 2t ) = γ 1 H RD H SR Q H y SRD ( 2t ) n r n ( 2t ) U1n ( t ) % * = * + + γ 2 H SD Q H y SD ( 2t ) + n ( 2t ) , ( r n ( 2t + 1) ) (U 2 ( t ) ) n n % (15) (11) 1 r ( 2t − 2 ) % n r ( 2t − 1) d n ( t ) % n r ( 2t + 1) = − γ 1 H RD H SR Q ( y ( 2t ) ) * * 1 H n . ( r n ( 2t − 1) ) − ( r n ( 2t − 2 ) ) d 2 ( t ) * n SD 2 % % + γ 2 H SD Q H ( y SRD ( 2t ) ) + n ( 2t + 1) , n * To perform MMSE equalization, we can further cascade every which represents the two consecutive received OFDM frames nth elements to use the vector forms in (15) as follows ( 2t ) and ( 2t + 1) at the destination terminal. r ( 2t ) U1 ( t ) % %* = * + To recover the OFDM data vectors d1 ( t ) and d 2 ( t ) from r ( 2t + 1) U 2 ( t ) (16) the differentially encoded received signals in (12), we exploit 1 r ( 2t − 2 ) r ( 2t − 1) d1 ( t ) % % * . the circulant structure of the channel matrices H RD , H SR , and 2 r ( 2t − 1) −r ( 2t − 2 ) d 2 ( t ) % %* H SD , similar to (7) as following Thus, performing blind MMSE equalization with no access to Qr ( 2t ) = r ( 2t ) = γ 1 Λ RD Λ SR y % n SRD ( 2t ) CSI, we have d1 ( t ) + γ 2 Λ SD y n SD ( 2t ) + Qn ( 2t ) , 124 4 3 = n ( 2t ) % d 2 ( t ) (17) (12) γ 1 Φ H r H ( 2t − 2 ) B % γ 2 Φ H r ( 2t − 1) B % % R, Qr ( 2t + 1) = r ( 2t + 1) = − γ 1 Λ RD Λ SR ( y ( 2t ) ) γ 2 Φ H r H ( 2t − 1) B − γ 1 Φ H r ( 2t − 2 ) B * % n SD % % + γ 2 Λ SD ( y SRD ( 2t ) ) + Qn ( 2t + 1) , R = r ( 2t ) r * ( 2t + 1) T * % % % n where and 14 324 B = ( γ 1r ( 2t − 2 ) ΦΦ H r H ( 2t − 2 ) + γ 2 r ( 2t − 1) ΦΦ H r H ( 2t − 1) ) . n ( 2 t +1) % −1 % % % % Note that in here, we are performing the differential decoding upon the nth subchannel and nth subcarrier to recover d1n ( t ) and d 2 ( t ) . Considering the nth subchannel, writing n IV. DD-GLCP-OFDM (12) in the matrix form we have Due to high decoding complexity of LCP-OFDM, an R (t ) n n (t ) %n 64748 64748 optimal subcarrier grouping technique was proposed in [25] in r ( 2t ) n ( 2t ) % n % n which the decoder's complexity is reduced by dividing the set n = n + r ( 2t + 1) n ( 2t + 1) % % of all subcarriers into nonintersecting subsets of subcarriers, (13) called subcarrier groups. In this approach, every information 64444 Y (t ) n 4 744444 64 744 8 4Λ n 8 symbol is transmitted over subcarriers within only one of these ySRD ( 2t ) n ySD ( 2t ) γ 1 Λ n Λ n n subsets. Note that if the subcarrier grouping is properly done, * RD SR , not only the decoding complexity is reduced, but also systems − ( ySD ( 2t ) ) ( ySRD ( 2t ) ) γ 2 Λ n n * n SD performance is preserved [25]. where Λ SR , Λ RD , and Λ SD , stand for the nth diagonal n n n The GLCP matrix θ is designed such that the decoding complexity is reduced, while preserving the maximum elements of matrices Λ SR , Λ RD , and Λ SD . Thus, substituting diversity and coding gains. Assuming that N = KM , in DD- (7) into (13), the current input to the distributed STBC GLCP-OFDM STBC system, the information symbols differential detector, R n ( t ) , for each sub-channel is related to di ( t ) , i = 1, 2, are divided into M blocks, dim ( t ) = ψ m di ( t ) , the previous input, R n ( t − 1) , according to where ψ m is a K × N permutation matrix built from the rows R n ( t −1) − n n ( t −1) ( m − 1) K + 1 → mK % 64748 of I N ; and then precoded by the GLCP R n ( t ) = Dn ( t ) Y n ( t − 1) Λ n + n n ( t ) % (14) matrix θ . Thus, as an example, (12) can be rewritten as = D ( t ) R ( t − 1) + n ( t ) − D ( t ) n n ( t − 1) . n n % n n % 1444 24444 4 3 Un (t ) r ( 2t ) = γ 1 Λ m Λ m g SRD ( 2t ) % RD SR m We set Y ( 0 ) ; n = 0,K, N − 1 to I2 and linearly detect n + γ 2 Λ m g SD ( 2t ) + n ( 2t ) , SD m % D ( t ) from Y ( t ) using the orthogonal structure in (13). This n n (18) can be done by rewriting (14) as follows r ( 2t + 1) = − γ 1 Λ Λ ( g ( 2t ) ) m m m * % RD SR SD + γ 2 Λ m ( g SRD ( 2t ) ) + n ( 2t + 1) , m * SD % 62 where Λ im = ψ m Λ iψ m , i = SR,RD,SD , and T g im ( 2t ) , normalized received signal to the destination terminal, and the 4 8 12 16 20 24 28 32 destination receives rD , rD , rD , rD , rD , rD , rD , rD , where i = SRD,SD , are differentially encoded from the GLCP- t rD denotes the received signal at the destination terminal at OFDM symbols g im ( t ) = θ dim ( t ) following similar steps as in time slot t . Assuming that the Information symbols are first (7). parsed as four streams of N ×1 blocks xi , i = 1, 2, 3, 4 and Following [24-25], for any K , QAM, PAM, BPSK, and QPSK constellation, the optimal can be constructed through encoded to cti using the orthogonal space-time block code LCP-A, which can be generally written as a Vandermonde design C , (7) holds true with matrix as following x1n x2 x3 n n 1 α1 L α1K −1 − x2 x1 − x4 n n n 1 1 α 2 L α 2K −1 − x3 x4 x1n θ= n n , (19) β M M M M − x4 − x3 x12 n n n 1 α K L α K 1 K −1 ( x1 ) ( x2 ) ( x3 ) , * * * Dn ( t ) = n n n (20) where β is a constant such that tr (θθ H ) = K , and the 3 − ( x2 ) ( x1n ) − ( x4 ) n * * n * parameters {α k }k =1 are selected depending on K [25]. K − ( x n )* ( x n )* ( x n )* 3 4 1 V. EXTENSION TO MULTIPLE RELAY SCENARIOS − ( x4 ) − ( x3 ) ( x12 ) n * n * n * We consider a multiple-relay assisted cooperative wireless communication system with a single source S , R half-duplex and Y n ( t ) changes accordingly. Following this, R n ( t ) in (13) relay terminals Ri , i = 1, 2,K, R , and a single destination D . changes to The source, destination, and all relays are equipped with single R n ( t ) = rD T transmit and receive antennas. We adopt the transmission 4 8 rD 12 rD 16 rD 20 rD 24 rD 28 rD rD . 32 (21) protocol in [27] and consider non-regenrative relays. Note that unlike [27], we assume that there is no direct transmission VI. NUMERICAL RESULTS between the source and destination terminals due to the In this section, we present Monte-Carlo simulation results for presence of shadowing. the proposed receiver. The CIRs for S → Ri and Ri → D links for the i th relay T Fig.1. depicts the SER performance of the DD-LCP-OFDM terminal are given by h SRi = hSRi [0],..., hSR [ LSRi ] , STBC scheme assuming for the following three different T scenarios: h Ri D = hRi D [0],..., hRi D [ LRi D ] , respectively , where LSRi and 1) L SR =L RD =L SD = 0, L Ri D denote the corresponding channel memory lengths. All 2) L SR =L RD =L SD = 1, the S → Ri and Ri → D links are assumed to be frequency 3) L SR =5, L RD =2, L SD = 1. selective Rayleigh fading. The random vectors h SRi and h Ri D For LCP-OFDM to achieve maximum diversity order, are assumed to be independent zero-mean complex Gaussian maximum diversity encoders should be used. Two classes of with power delay profile vectors denoted by maximum achievable diversity order (MADO) enabling LCP v SRi = [σ SRi (0),..., σ SRi ( LSRi )] 2 2 and encoders are introduced in [24], namely: Vandermonde encoders and cosine encoders. In here, we are using v Ri D = [σ Ri D (0),..., σ Ri D ( LRi D )] that are normalized such that 2 2 Vandermonde encoders and we assume 4-PSK modulation. To further minimize the receiver complexity, we are applying the ∑ σ SR ( lSR ) = 1 and ∑ σ R D ( lR D ) = 1 . LSRi 2 LRi D 2 lSRi = 0 i i lRi D = 0 i i low-cost minimum mean-square error (MMSE) equalizer. We consider a complex space-time block code C with its Our simulation results indicate that with the optimal design of entries cti [1]. For demonstration purposes, we focus on the LCP encoder matrix, the DD-LCP-OFDM STBC system is able to achieve full spatial and multipath diversity, G3 code [3] with three relays in a cooperative DD-OFDM- min ( LSR , LRD ) + LSD + 1 and it has been confirmed. STBC scenario with AF relaying. The extension to other orthogonal space-time block codes and DD-LCP/GLCP- Following [25], we suggest an alternative low complexity OFDM STBC systems is straightforward. During every four implementation of the DD-LCP-OFDM STBC system, time slots, three time slots are devoted to signal transmission namely, DD grouped linear constellation precoded GLCP- from source to relays, one at a time, and during the last time OFDM STBC subsystems. The aim is to reduce system slot, all R relays retransmit the normalized received signals to complexity while preserving maximum possible diversity and the destination terminal. During the last signaling interval of coding gains. The proposed system's optimal performance every four time slots, all the three relays retransmit the relies on the design of the GLCP matrix [25]. 63 Fig. 2 depicts the SER performance of the DD-GLCP- subsets, respectively, where N = MK . As is illustrated in Fig. OFDM STBC scheme for the following combinations of the 7, the optimal subcarrier grouping improves performance, i.e. underlying channel memory lengths: by ~ 2 dB at BER= 10 −3 . 1) L SR =L RD =L SD = 0, 2) L SR =L RD =L SD = 1, 0 10 3) L SR =5, L RD =2, L SD = 1. The MADO GLCP encoders used for scenarios 1, 2, and 3 are θ 2 , θ 4 , and θ8 , respectively [25]. We analyze system's -1 10 performance for α = 1,10 , i.e. ESR N 0 = α ESD N 0 . To -2 minimize the receiver complexity, MMSE equalizer is 10 implemented at the receiver side. In the case of α = 1 , for all three scenarios where the S → R and S → D links are SER -3 10 balanced, the SER performance degrades compared to the corresponding scenario when α = 10 , while preserving the LSR=LRD=LSD=0, α=1, θ2 achieved diversity order. -4 10 LSR=LRD=LSD=0, α=10, θ2 LSR=LRD=LSD=1, α=1, θ4 0 10 LSR=LRD=LSD=1, α=10, θ4 LSR=LRD=LSD=0, α=10 -5 10 LSR=5,LRD=1,LSD=2, α=1, θ8 LSR=LRD=LSD=1, α=10 LSR=5,LRD=1,LSD=2, α=10, θ8 LSR=5, LRD=2, LSD=1, α=10 -1 10 -6 10 0 5 10 15 20 25 ESD/N0 -2 10 Fig.2. SER performances of DD-GLCP-OFDM STBC over frequency- selective S→R , R→D , and S→D links SER ( ESR / N 0 = α ESD / N 0 , α = 1,10 ) -3 10 Next, we extend our analysis to a multiple relay scenario, where we adopt the transmission protocol in [27] and consider -4 10 non-regenerative relays. Note that unlike [27], we assume that there is no direct transmission between the source and destination terminals due to the presence of shadowing. We assume that there are three relay nodes, where each node is 0 2 4 6 8 10 ESD/N0 12 14 16 18 20 equipped with one antenna. We set SNRSR1 = SNRSR3 = 25 dB, and ER1D = ER2 D = ER3 D = 5 dB, and the SER curve is plotted Fig.1. SER performances of DD-LCP-OFDM STBC over frequency- selective S→R , R→D , and S→D links against ESR2 N 0 . 4-PSK modulation and G3 code [28] are ( ESR / N 0 = α ESD / N 0 , α = 10 ) used. The scenarios with different combinations of channel memory lengths are considered for DD-LCP-OFDM-STBC In Fig.3 the SER performance of DD-OFDM-STBC is system: compared with that of DD-LCP-OFDM-STBC and DD- 1) LSR1 = LR1D = LSR2 = LR2 D = LSR3 = LR3 D = 0, GLCP-OFDM-STBC, for L SR = L RD = L SD = 1 , α = 10 , and 2) LSR1 = LR1D = LSR2 = LR2 D = LSR3 = LR3 D = 1, θ 4 [25]. As can be observed from Fig. 6, DD-LCP-OFDM- STBC outperforms both DD-GLCP-OFDM-STBC and DD- 3) LSR1 = LR1D = LSR2 = LR2 D = LSR3 = LR3 D = 2. OFDM-STBC. However, both DD-LCP-OFDM-STBC and As is illustrated in Fig. 5, as an example, at BER= 10 −2 , the DD-GLCP-OFDM-STBC achieve the same diversity gain. As third scenario outperforms the first and second scenarios by ~ an example, at BER= 10 −4 , the DD-LCP-OFDM-STBC system 3 dB and ~ 8 dB, respectively. The second scenario outperforms DD-GLCP-OFDM-STBC and DD-OFDM-STBC outperforms the first scenario by ~ 4 dB at BER= 10 −2 . systems by ~ 2 dB and ~ 4 dB, respectively. In Fig. 4 the SER performance of DD-GLCP-OFDM is VII. CONCLUSION provided with optimal and suboptimal subcarrier grouping, We have investigated distributed differential LCP-OFDM assuming L SR = L RD = L SD = 1 and 4-PSK modulation. The STBC and GLCP-OFDM for cooperative communications optimal and suboptimal grouping are specified with over frequency-selective fading channels. We have carefully I m ,opt = {m − 1, M + m − 1, …, ( K − 1) M + m − 1} and exploited the unitary structure of STBCs to design a low complexity distributed differential STBC receiver for I m , subopt = {( m − 1) K + 1, ( m − 1) K + 2, …, mK } m ∈ [1, M ] broadband cooperative networks. With the optimal design of 64 the LCP encoder matrix, the DD-LCP-OFDM STBC system is 0 able to achieve full diversity gain. An alternative low- 10 LSR1=LR1D=LSR2=LR2D=LSR3=LR3D=1 complexity implementation of the DD-LCP-OFDM STBC LSR1=LR1D=LSR2=LR2D=LSR3=LR3D=0 system, namely DD-GLCP-OFDM-STBC, reduces the LSR1=LR1D=LSR2=LR2D=LSR3=LR3D=2 complexity while preserving the maximum possible diversity -1 10 and coding gains by dividing the set of all subcarriers into non-intersecting subcarrier groups. The DD-GLCP-OFDM STBC system’s performance relies on the design of the GLCP matrix. 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