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Precoded Differential OFDM for Relay Networks

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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. Note that with optimal subcarrier grouping, the DD-




                                                                                                                SER
                                                                                                                        -2
                                                                                                                      10
           LCP-OFDM STBC and DD-GLCP-OFDM STBC both
           achieve the same diversity gain. We have further extended the
           analysis to multiple relay scenarios. We have presented the
           comprehensive Monte-Carlo simulations to corroborate the                                                   10
                                                                                                                        -3


           theoretical presentation.


       0
                                      4PSK, N=16 Subcarriers
      10                                                                                                                -4
                                                                                                                      10
                                                                      DD OFDM-STBC                                           0   2     4       6      8        10        12      14       16      18       20
                                                                      DD GLCP-OFDM-STBC                                                                   ESR2/N0 [dB]
                                                                      DD LCP-OFDM-STBC

                                                                                                          Fig.5. SER performance of DD-LCP-OFDM STBC system with three relays.
       -1
      10

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                                                                                66