PATENT ABSTRACT
Methods and systems are proposed for transmitting data from a source ( 110 ) to a destination ( 130 ) via a relay station ( 120 ) having multiple antennae ( 122, 124 ). The relay station ( 120 ) receives from the source a message containing the data and a first cyclic prefix. It does this using each of its antennae ( 122, 124 ), so producing multiple respective received signals. In certain embodiments, the relay station ( 120 ) removes the first cyclic prefix from the received signals, replacing it with a new one. In other embodiments, the relay station ( 120 ) removes only a portion of the first cyclic prefix. In either case, the relay station ( 120 ) may apply space-time coding to generate second signals, which it transmits to the destination ( 130 ), which extracts the data. Methods are also proposed for estimating parameters of the channel, to enable the destination ( 130 ) to decode the data.

PATENT DESCRIPTION
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    The present application is one of two patent applications claiming priority from U.S. provisional patent application No. 61/089,617. The present patent application relates to the usage of cyclic prefixes, whereas the other patent application relates to techniques for space-time coding. The techniques described in the respective two patent applications are independent since either can be used without the other, although it is equally possible to construct systems which combine techniques from the respective two patent applications. 
     
    
     FIELD OF THE INVENTION 
       [0002]    This invention relates to methods and systems for implementing cyclic prefixes in a wireless communication channel comprising a relay station capable of performing information coding, particularly but not exclusively, analog space-time coding. 
       BACKGROUND OF THE INVENTION 
       [0003]    Wireless relays have shown potential for extending communication range and providing good quality of experience. In a typical wireless communication system utilizing relays, the relay station would be time shared between different sources and destination. In such a scenario, the relay station needs to estimate the Angular Carrier Frequency Offset (ACFO) present between each pair of source and relay antennae (denoted ACFO φ r ). The ACFO that is present between each pair of relay and destination antennae (denoted ACFO φ d ) will be compensated for at the destination. 
         [0004]    Joint compensation of the source to relay and relay to destination ACFO is not possible in the prior art. Compensation for the source to relay ACFO φ r  can thus only be performed at the relay station. The relay station of the prior art will need to estimate and compensate the ACFO for each source-and-relay pair and this complicates the relay station hardware and increases the power required by the relay station. 
         [0005]    Implementations of analog space-time coding (ASTC) in the prior art also have the limitation that the transmission channel has to be flat fading. This is because inter-symbol interference (ISI) results when the transmission channel has frequency selective fading. The insertion of a cyclic prefix (CP) can be used to mitigate against ISI. The cyclic prefix inserted should preferably be of a length greater than that of the channel impulse response (CIR). 
         [0006]    Typical implementations of ASTC also require that the samples within a signal be re-ordered when performing the coding. 
         [0007]    It is an object of the present invention to provide a method for implementing cyclic prefixes and compensating for the channel effects of a wireless communication channel, which addresses at least one of the problems of the prior art and/or to provide the public with a useful choice. 
       SUMMARY OF THE INVENTION 
       [0008]    In general terms, the present invention relates to transmitting data though a wireless communication channel from a source to a destination via a relay station having multiple antennae. The relay station receives a message from the source containing the data and a first cyclic prefix. It does this using each of its antennae, so producing multiple respective received signals. In a first aspect of the invention, the relay station removes the first cyclic prefix from the received signals, replacing it with a new one. In a second aspect of the invention, the relay station removes only a portion of the first cyclic prefix. In either case, the relay station may apply space-time coding to generate second signals, which it transmits to the destination, which extracts the data. A further aspect of the invention relates to estimating parameters of the channel, to enable the destination to decode the data. 
         [0009]    The invention may be expressed alternatively as an integrated circuit configured to enable the above mentioned aspects of the invention. 
         [0010]    The cyclic prefix schemes of the present invention do not require that the transmission channels be flat fading. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0011]    By way of example only, embodiments will be described with reference to the accompanying drawings, in which: 
           [0012]      FIG. 1  is a schematic drawing of a communication channel having a source station S, relay station R and destination station D, according to an embodiment of the invention; 
           [0013]      FIG. 2  is a flow diagram of a scheme for inserting cyclic prefixes according to a Cyclic Prefix Scheme 1 of the example embodiment; 
           [0014]      FIG. 3  is a flow diagram of a scheme for inserting cyclic prefixes according to a Cyclic Prefix Scheme 2 of the example embodiment; 
           [0015]      FIG. 4  is a flow diagram of a method for performing channel estimation according to the example embodiment; 
       
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0016]      FIG. 1  shows a communication channel according to an example embodiment. This channel consists of a source station/node  110  (S), a relay station  120  (R) and a destination station/node  130  (D). The source and the destination each have only one antenna, and the relay station  120  has two antennae  122 , 124 . The antennae at the source station/node  110 , the relay station  120  and the destination station/node  130  can be configured to both transmit and receive. The source  110  sends signals to the relay station  120 , and the relay station  120  sends signals to the destination  130 . The signals sent from the source  110  to the relay station  120 , and from the relay station  120  to the destination  130  may comprise multiple orthogonal carrier frequencies, such as in the case where OFDM modulation is used. 
         [0017]    The present embodiment includes only one relay station  120 . An ASTC relaying scheme is applied using orthogonal frequency division multiplexing (OFDM) modulation at the source  110 . ASTC encoding is applied to individual carriers over two consecutive OFDM symbol periods. The relay station  120  receives the OFDM signal using the two antennae  122 ,  124 . It does simple analog processing (such as the sampling and storage of discrete time signals) of the received signal and performs coding. This coding can for example be 2×2 Alamouti coding. The coded signal is then transmitted to the destination node/station  130 . 
         [0018]    To avoid interference, the data transmission from the source node/station  110  to destination node/station  130  takes place over two phases. The transmission cycle thus consists of two phases. In the first phase, the source node/station  110  to relay station  120  link is activated and the destination node/station  130  remains silent. In the second phase, the relay station  120  to destination node/station  130  link is activated and the source node/station  110  remains silent. 
         [0019]    Simulations of the present embodiment were conducted using 16 carriers in each OFDM symbol. This however does not limit the number of carriers to 16 and a skilled reader will understand that the number of carriers can be varied. 
         [0020]    Alternative embodiments may also apply the ASTC relaying schemes to single-carrier cyclic prefix (SC-CP) systems where the ASTC encoding is also applied to individual symbols over two consecutive SC-CP blocks. 
         [0021]    Alternative embodiments may also have more than two antennae at the relay station  120 , in which case antenna selection can be adopted. Only two antennas are selected to implement the proposed schemes based on pre-defined selection criteria, e.g., best product channel SNR, etc. 
         [0022]    Alternative embodiments may also have multiple relay stations where relay selection can be performed in which one relay station is selected to implement the proposed schemes based on a pre-determined selection criteria, e.g., best product channel SNR, etc. 
         [0023]    Alternative embodiments may also have multiple relay stations where Coordinated Delay ASTC is implemented at each relay station, and the coordinated delays are applied at different relay station. In this case, the delay durations applied at the different relay station is a design parameter obtained from a central control. This embodiment may have the advantage that the carrier frequencies do not have to be orthogonal and thus signal collisions will not occur. 
         [0024]    Alternative embodiments may also have only one antenna at the relay station  120 . In this case, a solution is to implement cooperation using at least two relay sub-stations, where each relay sub-station is a relay station capable of performing information passing with other relay sub-stations. Information passing is implemented between the at least two relay sub-stations and the at least two relay sub-stations can then participate in the ASTC transmission. 
         [0025]    Angular Carrier Frequency Offset (ACFO) 
         [0026]    Let the number of carriers be N. The source  110  to relay station  120  and relay station  120  to destination  130  channels are assumed to be frequency selective multipath fading channels. Let the vectors h S,1  and h S,2  respectively denote the channel impulse response (CIR) from the source  110  to the first and second antennae  122 ,  124  of the relay station  120 . h S,1  and h S,2  both have the size of (L 1×1 ). Similarly, let the vectors h 1,D  and h 2,D  respectively denote the CIR from the first and second antennae  122 ,  124  of the relay station  120  to the destination  130 . h 1,D  and h 2,D  both have the size of (L 2×1 ). Let f 0 , f r  and f d  be the carrier frequency of the local oscillators at the source  110 , relay station  120  and destination  130 , respectively. Then Δf r =f 0 −f r  and Δf d =f 0 −f d  respectively denote the CFO at the relay  120  and destination  130 . The source  110 , relay station  120  and destination  130  can have independent local oscillators and thus their oscillating frequency need not be the same. This results in independent CFOs in the signal received at relay station  120  and the destination  130  i.e. Δf r ≠Δf d . 
         [0027]    The angular CFO (ACFO) is defined as 
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         [0000]    where φ r  denotes the ACFO that is present at the channels between the source  110  and either antenna  122 ,  124  of the relay station  120 , and φ d  denotes the ACFO that is present at the channels between either antenna  122 ,  124  of the relay station  120  and the destination  130 . T denotes the OFDM symbol duration. Scalars ε r =(Δf r )T and ε d =(Δf d )T are referred to as normalized CFOs and their magnitudes are bounded by |ε r |≦0.5 and |ε d |≦0.5. 
         [0028]    The ACFO φ r  and/or φ d  can be estimated during channel estimation and are used to perform compensation either at the relay station  120 , or at the destination  130 . 
         [0029]    Joint ACFO compensation can also be performed at the destination  130 , in which case no ACFO compensation needs to be done at the relay station  120 . What this means is that compensation for (φ r +φ d ) is done at the destination  130 . 
         [0030]    If the CFOs present in the signal received at the relay station  120  and destination  130  are not properly compensated, then the gains due to the relaying and usage of OFDM modulation cannot be realized. 
         [0031]    Joint ACFO compensation when done at the destination  130  is advantageous as it keeps the relay carrier frequency unaffected for any source-and-destination pair. 
         [0032]    Joint ACFO compensation when done at the destination  130  is also advantageous as it simplifies the implementation of the relay station  120  and reduces the computational complexity at the relay station  120 . 
         [0033]    Cyclic Prefix Schemes with Carrier Frequency Offset (CFO) Compensation 
         [0034]    The embodiment employs two cyclic prefix schemes which are described below with their associated methods of performing carrier frequency offset (CFO) compensation. The Cyclic Prefix Schemes 1 and 2 of the present invention permits the implementation of space-time codes with minimum signal processing complexity at the relay station. 
         [0035]    It is understood that the term “CP Scheme 1” is also used in this document to refer to the Cyclic Prefix Scheme 1 and “CP Scheme 2” is also used to refer to the Cyclic Prefix Scheme 2. 
         [0036]    Cyclic Prefix Scheme 1 
         [0037]      FIG. 2  shows a scheme for inserting cyclic prefixes according to the example embodiment. Let x j  be a (N×1) vector containing the N data symbols to be transmitted during a j-th OFDM symbol duration. 
         [0038]    While this example embodiment of the invention has been described using 2 OFDM symbols i.e. x 2n , and x 2n+1 , it is clear that the use of a different number of OFDM symbols is possible within the scope of the invention, as will be clear to a skilled reader. 
         [0039]    In step  202 , the source derives a first cyclic prefix (denoted CP j,1 ) with length P 1 =L 1 −1 from x j . L 1  is the length of the channel impulse response (CIR) represented by h S,1  and h S,2  respectively for the channel between the source and the first antenna of the relay station, and the channel between the source and the second antenna of the relay station. 
         [0040]    In step  204 , the source inserts CP j,1  in front of x j  and then transmits the resultant symbol sequence. 
         [0041]    Let the symbol sequence x j  consisting of N symbols be denoted by x n  where n=0 . . . N− 1 , i.e. 
         [0000]      x j =[x 0  . . . x n  . . . x N-1]   
         [0042]    The cyclic prefix CP j,1  of length P 1  is created by copying the last P 1  symbols of x j    
         [0000]      CP j,1 =└x N-P     1    . . . x N-2  . . . x N-1┘ 
 
         [0043]    The cyclic prefix CP j,1  is inserted in front of x j , producing the resultant symbol sequence └CP j,1  x j ┘ for transmission. 
         [0044]    The transmission from the source to relay station takes place in the first phase of the transmission cycle. 
         [0045]    In step  206 , the relay station receives └CP j,1  x j ┘ at both antennae of the station. 
         [0046]    In step  208 , after performing time synchronization, the CP j,1  portion of length P 1  in each OFDM symbol is removed leaving x j . Should ACFO be present, the signal vectors received over the span of 2 OFDM symbol durations i.e. for j={2n, 2n+1} are given after the removal of CP j,1  as 
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         [0000]    r R,i,j  denotes the received signal vector obtained after the removal of CP j,1 , with i and j respectively denoting the antenna index and OFDM symbol duration. r R,i,j  has the size of (N×1). n denotes the data transmission cycle index. The (N×N) matrix W N   H  is the inverse discrete Fourier transform (IDFT) matrix. The first column vector of the (N×N) circulant matrices h S,1  and H S,2  are [h S,1   T , 0 1×(N-L) ] T  and [h S,2   T , 0 1×(N-L) ] T , respectively. 
         [0047]    Vector v R,i,j  contains N samples of additive white Gaussian noise (AWGN) that distorts the signal received by the i-th relay antenna during j-th OFDM symbol duration. If 2d OFDM symbols are transmitted in each cycle with d being an integer value, then the matrix 
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                           1 
                         
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
         [0000]    will be of dimension (2N×2d) with OFDM symbol indexes of 2dn, 2dn+1, 2dn+2, . . . , 2dn+2d−1. 
         [0048]    It is assumed that the covariance of v R,i,j,    
         [0000]        E[v   R,i,j   v   R,i,j   H ]=σ R   2   I   N  for  i= 1,2 and  j= 2 n, 2 n+ 1 for  d&gt; 1.
 
         [0049]    Optionally, in the event that ACFO is present, compensation of ACFO φ r  can be performed at the relay station as step  230 . 
         [0050]    Assuming the complete removal of ACFO or if ACFO was absent, over the span of 2 OFDM symbol durations i.e. for j={2n, 2n+1}, the received signal vectors obtained after the removal of CP j,1  at the relay are given as follows: 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       r 
                       
                         R 
                         , 
                         
                             
                         
                          
                         1 
                         , 
                         
                             
                         
                          
                         
                           2 
                            
                           
                               
                           
                            
                           n 
                         
                       
                     
                   
                   
                     
                       r 
                       
                         R 
                         , 
                         
                             
                         
                          
                         1 
                         , 
                         
                             
                         
                          
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             n 
                           
                            
                           
                               
                           
                           + 
                           
                               
                           
                            
                           1 
                         
                       
                     
                   
                 
                 
                   
                     
                       r 
                       
                         R 
                         , 
                         
                             
                         
                          
                         2 
                         , 
                         
                             
                         
                          
                         
                           2 
                            
                           
                               
                           
                            
                           n 
                         
                       
                     
                   
                   
                     
                       r 
                       
                         R 
                         , 
                         
                             
                         
                          
                         2 
                         , 
                         
                             
                         
                          
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             n 
                           
                            
                           
                               
                           
                           + 
                           
                               
                           
                            
                           1 
                         
                       
                     
                   
                 
               
               ] 
             
             = 
             
               
                 [ 
                 
                     
                 
                  
                 
                   
                     
                       
                         
                           H 
                           
                             S 
                             , 
                             1 
                           
                         
                          
                         
                           W 
                           N 
                           H 
                         
                          
                         
                           x 
                           
                             2 
                              
                             n 
                           
                         
                       
                     
                     
                       
                         
                           H 
                           
                             S 
                             , 
                             1 
                           
                         
                          
                         
                           W 
                           N 
                           H 
                         
                          
                         
                           x 
                           
                             
                               2 
                                
                               n 
                             
                             + 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           H 
                           
                             S 
                             , 
                             2 
                           
                         
                          
                         
                           W 
                           N 
                           H 
                         
                          
                         
                           x 
                           
                             2 
                              
                             n 
                           
                         
                       
                     
                     
                       
                         
                           H 
                           
                             S 
                             , 
                             2 
                           
                         
                          
                         
                           W 
                           N 
                           H 
                         
                          
                         
                           x 
                           
                             
                               2 
                                
                               n 
                             
                             + 
                             1 
                           
                         
                       
                     
                   
                 
                 ] 
               
               + 
               
                   
                 
                   
                     [ 
                     
                       
                         
                           
                             v 
                             
                               R 
                               , 
                               1 
                               , 
                               
                                 2 
                                  
                                 n 
                               
                             
                           
                         
                         
                           
                             v 
                             
                               R 
                               , 
                               1 
                               , 
                               
                                 
                                   2 
                                    
                                   n 
                                 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           
                             v 
                             
                               R 
                               , 
                               2 
                               , 
                               
                                 2 
                                  
                                 n 
                               
                             
                           
                         
                         
                           
                             v 
                             
                               R 
                               , 
                               2 
                               , 
                               
                                 
                                   2 
                                    
                                   n 
                                 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                   , 
                 
               
             
           
         
       
     
         [0051]    Step  209  is then performed at the relay station. In step  209 , Simple processing is performed in order to implement space-time coding at the carrier level. The matrix Y R  is computed and arranged: 
         [0000]    
       
         
           
             
               Y 
               R 
             
             = 
             
               
                 [ 
                 
                   y 
                   
                     ij 
                     , 
                     R 
                   
                 
                 ] 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           y 
                           
                             11 
                             , 
                             R 
                           
                         
                       
                       
                         
                           y 
                           
                             12 
                             , 
                             R 
                           
                         
                       
                     
                     
                       
                         
                           y 
                           
                             21 
                             , 
                             R 
                           
                         
                       
                       
                         
                           y 
                           
                             22 
                             , 
                             R 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   [ 
                   
                     
                       
                         
                           r 
                           
                             R 
                             , 
                             1 
                             , 
                             
                               2 
                                
                               n 
                             
                           
                         
                       
                       
                         
                           r 
                           
                             R 
                             , 
                             1 
                             , 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                     
                     
                       
                         
                           ζ 
                            
                           
                             ( 
                             
                               r 
                               
                                 R 
                                 , 
                                 2 
                                 , 
                                 
                                   
                                     2 
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                               
                               * 
                             
                             ) 
                           
                         
                       
                       
                         
                           - 
                           
                             ζ 
                              
                             
                               ( 
                               
                                 r 
                                 
                                   R 
                                   , 
                                   2 
                                   , 
                                   
                                     2 
                                      
                                     n 
                                   
                                 
                                 * 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0052]    The space-time code implemented can for example be the Alamouti code. 
         [0000]    Given that 
         [0000]        H   S,1   W   N   H   x   j   =W   N   H Λ S,1   x   j  for  j= 2 n,  2 n+ 1 and  i= 1, 2,
 
         [0000]    where Λ S,i  is a (N×N) sized diagonal matrix whose diagonal elements are given by the N-point DFT of CIR h S,i  for i=1, 2, i.e., Λ S,i =√{square root over (N)}diag (W N [h S,i   T , 0 1×(N-L     i     ) ] T ), it can be shown that 
         [0000]    
       
         
           
             
               
                 Y 
                 R 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           r 
                           
                             R 
                             , 
                             1 
                             , 
                             
                               2 
                                
                               n 
                             
                           
                         
                       
                       
                         
                           r 
                           
                             R 
                             , 
                             1 
                             , 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                     
                     
                       
                         
                           ζ 
                            
                           
                             ( 
                             
                               r 
                               
                                 R 
                                 , 
                                 2 
                                 , 
                                 
                                   
                                     2 
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                               
                               * 
                             
                             ) 
                           
                         
                       
                       
                         
                           - 
                           
                             ζ 
                              
                             
                               ( 
                               
                                 r 
                                 
                                   R 
                                   , 
                                   2 
                                   , 
                                   
                                     2 
                                      
                                     n 
                                   
                                 
                                 * 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     
                       1 
                       
                         2 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               
                                 W 
                                 N 
                                 H 
                               
                                
                               
                                 Λ 
                                 
                                   S 
                                   , 
                                   1 
                                 
                               
                                
                               
                                 x 
                                 
                                   2 
                                    
                                   n 
                                 
                               
                             
                           
                           
                             
                               
                                 W 
                                 N 
                                 H 
                               
                                
                               
                                 Λ 
                                 
                                   S 
                                   , 
                                   1 
                                 
                               
                                
                               
                                 x 
                                 
                                   
                                     2 
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 W 
                                 N 
                                 H 
                               
                                
                               
                                 
                                   Z 
                                   N 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       π 
                                     
                                     N 
                                   
                                   ) 
                                 
                               
                                
                               
                                 Λ 
                                 
                                   S 
                                   , 
                                   2 
                                 
                                 * 
                               
                                
                               
                                 x 
                                 
                                   
                                     2 
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                                 * 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   W 
                                   N 
                                   H 
                                 
                               
                                
                               
                                 
                                   Z 
                                   N 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       π 
                                     
                                     N 
                                   
                                   ) 
                                 
                               
                                
                               
                                 Λ 
                                 
                                   S 
                                   , 
                                   2 
                                 
                                 * 
                               
                                
                               
                                 x 
                                 
                                   2 
                                    
                                   n 
                                 
                                 * 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                   + 
                   
                     
                       V 
                       ~ 
                     
                     
                       R 
                       , 
                       n 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where the (2N×2N) sized matrix {tilde over (V)} R,n  is given by 
         [0000]    
       
         
           
             
               
                 V 
                 ~ 
               
               
                 R 
                 , 
                 n 
               
             
             = 
             
               
                 [ 
                 
                   
                     
                       
                         v 
                         
                           v 
                           , 
                           1 
                           , 
                           
                             2 
                              
                             n 
                           
                         
                       
                     
                     
                       
                         v 
                         
                           R 
                           , 
                           1 
                           , 
                           
                             
                               2 
                                
                               n 
                             
                             + 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         ζ 
                          
                         
                           ( 
                           
                             v 
                             
                               R 
                               , 
                               2 
                               , 
                               
                                 
                                   2 
                                    
                                   n 
                                 
                                 + 
                                 1 
                               
                             
                             * 
                           
                           ) 
                         
                       
                     
                     
                       
                         - 
                         
                           ζ 
                            
                           
                             ( 
                             
                               v 
                               
                                 R 
                                 , 
                                 2 
                                 , 
                                 
                                   2 
                                    
                                   n 
                                 
                               
                               * 
                             
                             ) 
                           
                         
                       
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
         [0053]    Step  209  further comprises steps  210  and  212 . 
         [0054]    In step  210 , for those elements of Y R  where no signal conjugation is needed e.g.: for y 11,R  and y 12,R , nothing needs to be done. 
         [0055]    In step  212 , for those elements of Y R  where signal conjugation is needed e.g.: for y 21,R  and y 22,R , the OFDM symbol sequence is reordered using a mapping function ζ(.): 
         [0000]      ζ( a )=[ a ( N− 1),  a ( N− 2), . . . , a (0)] T ,
 
         [0000]    the (N×1) sized input vector a=[a(0), a(1), . . . , a(N−1)] T . This function can be easily implemented in hardware by storing the signal samples in a shift register and then reading the register in reverse order. This represents a significant advantage because reordering operations that are more complex than mere reversal, such as the insertion, deletions and swapping of signal samples do not have to be done. 
         [0056]    It can be shown that 
         [0000]    
       
         
           
             
               
                 
                   
                     ζ 
                      
                     
                       ( 
                       
                         
                           [ 
                           
                             
                               H 
                               
                                 S 
                                 , 
                                 i 
                               
                             
                              
                             
                               W 
                               N 
                               H 
                             
                              
                             
                               x 
                               j 
                             
                           
                           ] 
                         
                         * 
                       
                       ) 
                     
                   
                   = 
                     
                    
                   
                     ζ 
                      
                     
                       ( 
                       
                         
                           
                             H 
                             
                               S 
                               , 
                               i 
                             
                             * 
                           
                            
                           
                             ( 
                             
                               
                                 W 
                                 N 
                                 H 
                               
                                
                               
                                 x 
                                 j 
                               
                             
                             ) 
                           
                         
                         * 
                       
                       ) 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       ζ 
                        
                       
                         ( 
                         
                           H 
                           
                             S 
                             , 
                             i 
                           
                           * 
                         
                         ) 
                       
                     
                      
                     
                       
                         ( 
                         
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             j 
                           
                         
                         ) 
                       
                       * 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       W 
                       N 
                       H 
                     
                      
                     
                       
                         Z 
                         N 
                       
                        
                       
                         ( 
                         
                           
                             2 
                              
                             π 
                           
                           N 
                         
                         ) 
                       
                     
                      
                     
                       Λ 
                       
                         S 
                         , 
                         i 
                       
                       * 
                     
                      
                     
                       TW 
                       N 
                     
                      
                     
                       W 
                       N 
                       H 
                     
                      
                     
                       Tx 
                       j 
                       * 
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       W 
                       N 
                       H 
                     
                      
                     
                       
                         Z 
                         N 
                       
                        
                       
                         ( 
                         
                           
                             2 
                              
                             π 
                           
                           N 
                         
                         ) 
                       
                     
                      
                     
                       Λ 
                       
                         S 
                         , 
                         i 
                       
                       * 
                     
                      
                     
                       TTx 
                       j 
                       * 
                     
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       
                         W 
                         N 
                         H 
                       
                        
                       
                         
                           Z 
                           N 
                         
                          
                         
                           ( 
                           
                             
                               2 
                                
                               π 
                             
                             N 
                           
                           ) 
                         
                       
                        
                       
                         Λ 
                         
                           S 
                           , 
                           i 
                         
                         * 
                       
                        
                       
                         x 
                         j 
                         * 
                       
                     
                   
                    
                   
                       
                   
                 
               
             
           
         
       
       
         
           
             
                 
             
              
             
               
                 
                   for 
                    
                   
                       
                   
                    
                   j 
                 
                 = 
                 
                   2 
                    
                   n 
                 
               
               , 
               
                 
                   
                     2 
                      
                     n 
                   
                   + 
                   
                     1 
                      
                     
                         
                     
                      
                     and 
                      
                     
                         
                     
                      
                     i 
                   
                 
                 = 
                 1 
               
               , 
               2 
               , 
             
           
         
       
     
         [0000]    where matrix Z N (φ) denotes a (N×N) diagonal matrix given by 
         [0000]        Z   N (φ)=diag([1 ,e   jφ   ,e   j2φ   , . . . ,e   j(N-1)φ ] T )
 
         [0000]    and T is a (N×N) permutation matrix such that 
         [0000]        T =[ζ( I   N )] (1)  and  TT=I   N .
 
         [0057]    I N  is an identity matrix of size N. The operator [•]* represents a complex conjugation. 
         [0058]    The notation [a] (l)  denotes the circular shifting of vector a by l elements, where 
         [0000]      [ a]   (2)   =[a ( N− 2), a ( N− 1), a (0), a (1), . . . , a ( N− 3)] T . 
         [0059]    For a given matrix X, matrix [X] (l)  is obtained by circular shifting each column vector of X by l elements. 
         [0060]    In step  213 , a second cyclic prefix (denoted CP ij,2 ) with length P 2  where P 2 =L 2 −1 is obtained for each y ij,R . L 2  is the length of the CIRs denoted by the vectors h 1,D  and h 2,D  respectively for the channel between the first antenna of the relay station and the destination, and the channel between the second antenna of the relay station and the destination. Each of the cyclic prefixes CP ij,2  is derived from the corresponding y ij,R.    
         [0061]    The cyclic prefixes CP ij,2  are then inserted in front of y ij,R  to produce the symbol sequences └CP ij,2  y ij,R  ┘. It is to be noted that y ij,R  is a (N×1) sized vector. 
         [0062]    In step  214 , the other necessary steps for the ASTC implementation at the relay station are performed. The symbol sequences └CP ij,2  y ij,R  ┘ are then transmitted by the i-th relay antenna during the j-th OFDM symbol duration, where i=1, 2 and j={2n, 2n+1}. The transmission from the relay station to destination takes place in the second phase of the transmission cycle where the source remains silent. 
         [0063]    In step  216 , after frame synchronization, the destination removes the cyclic prefix of length P 2  from each OFDM symbol received. The OFDM symbol with the cyclic prefix removed can be denoted by the (N×1) signal vector r D,j  for each j-th OFDM symbol duration. 
         [0064]    Assuming that ACFO is present and ACFO compensation was done at the relay, for the j={2n, 2n+1}-th OFDM symbol duration, the (N×1) received signal vectors at the destination in the presence of ACFO φ d  are given by
       CP Scheme 1:       
 
         [0000]    
       
         
           
             
               r 
               
                 D 
                 , 
                 
                   2 
                    
                   n 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     
                       Z 
                       N 
                     
                      
                     
                       ( 
                       
                         φ 
                         d 
                       
                       ) 
                     
                   
                    
                   
                     [ 
                     
                       
                         
                           H 
                           
                             1 
                             , 
                             D 
                           
                         
                          
                         
                           H 
                           
                             S 
                             , 
                             1 
                           
                         
                          
                         
                           W 
                           N 
                           H 
                         
                          
                         
                           x 
                           
                             2 
                              
                             n 
                           
                         
                       
                       + 
                       
                         
                           H 
                           
                             2 
                             , 
                             D 
                           
                         
                          
                         
                           ζ 
                            
                           
                             ( 
                             
                               H 
                               
                                 S 
                                 , 
                                 2 
                               
                               * 
                             
                             ) 
                           
                         
                          
                         
                           W 
                           N 
                           H 
                         
                          
                         
                           Tx 
                           
                             sn 
                             + 
                             1 
                           
                           * 
                         
                       
                     
                     ] 
                   
                 
               
               + 
               
                 p 
                 
                   D 
                   , 
                   
                     2 
                      
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 r 
                 
                   D 
                   , 
                   
                     
                       2 
                        
                       n 
                     
                     + 
                     1 
                   
                 
               
               = 
               
                 
                   
                     
                        
                       
                         
                           j 
                            
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       n 
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 P 
                                 2 
                               
                             
                             ) 
                           
                         
                          
                         
                           φ 
                           d 
                         
                       
                     
                     
                       2 
                     
                   
                    
                   
                     
                       
                         Z 
                         N 
                       
                        
                       
                         ( 
                         
                           φ 
                           d 
                         
                         ) 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             H 
                             
                               1 
                               , 
                               D 
                             
                           
                            
                           
                             H 
                             
                               S 
                               , 
                               1 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                         - 
                         
                           
                             H 
                             
                               2 
                               , 
                               D 
                             
                           
                            
                           
                             ζ 
                              
                             
                               ( 
                               
                                 H 
                                 
                                   S 
                                   , 
                                   2 
                                 
                                 * 
                               
                               ) 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             Tx 
                             sn 
                             * 
                           
                         
                       
                       ] 
                     
                   
                 
                 + 
                 
                   p 
                   
                     D 
                     , 
                     
                       
                         2 
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where the noise vectors are given by 
         [0000]    
       
         
           
             
               p 
               
                 D 
                 , 
                 
                   2 
                    
                   
                       
                   
                    
                   n 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     
                       Z 
                       N 
                     
                      
                     
                       ( 
                       
                         φ 
                         d 
                       
                       ) 
                     
                   
                    
                   
                     [ 
                     
                       
                         
                           H 
                           
                             1 
                             , 
                             D 
                           
                         
                          
                         
                           v 
                           
                             R 
                             , 
                             1 
                             , 
                             
                               2 
                                
                               n 
                             
                           
                         
                       
                       + 
                       
                         
                           H 
                           
                             2 
                             , 
                             D 
                           
                         
                          
                         
                           ζ 
                            
                           
                             ( 
                             
                               v 
                               
                                 R 
                                 , 
                                 2 
                                 , 
                                 
                                   
                                     2 
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                               
                               * 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                 
               
               + 
               
                 v 
                 
                   D 
                   , 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               p 
               
                 D 
                 , 
                 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                   + 
                   1 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   N 
                                   + 
                                   
                                     P 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     
                       Z 
                       N 
                     
                      
                     
                       ( 
                       
                         φ 
                         d 
                       
                       ) 
                     
                   
                    
                   
                     [ 
                     
                       
                         
                           H 
                           
                             1 
                             , 
                             D 
                           
                         
                          
                         
                           v 
                           
                             R 
                             , 
                             1 
                             , 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                       - 
                       
                         
                           H 
                           
                             2 
                             , 
                             D 
                           
                         
                          
                         
                           ζ 
                            
                           
                             ( 
                             
                               v 
                               
                                 R 
                                 , 
                                 2 
                                 , 
                                 
                                   2 
                                    
                                   n 
                                 
                               
                               * 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                 
               
               + 
               
                 
                   v 
                   
                     D 
                     , 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
                 . 
               
             
           
         
       
     
         [0066]    Optionally, ACFO φ d  compensation can be performed at the destination as step  240 . 
         [0067]    Assuming that ACFO is present and no ACFO compensation was done at the relay, for the j={2n, 2n+1}-th OFDM symbol duration, 
         [0000]    
       
         
           
             
               r 
               
                 D 
                 , 
                 
                   2 
                    
                   
                       
                   
                    
                   n 
                 
               
             
             = 
             
               
                  
                 
                   
                     j 
                      
                     
                       ( 
                       
                         
                           2 
                            
                           
                               
                           
                            
                           
                             n 
                              
                             
                               ( 
                               
                                 N 
                                 + 
                                 
                                   P 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                         + 
                         
                           P 
                           2 
                         
                       
                       ) 
                     
                   
                    
                   
                     φ 
                     d 
                   
                 
               
                
               
                 
                   Z 
                   N 
                 
                  
                 
                   ( 
                   
                     φ 
                     d 
                   
                   ) 
                 
               
                
               
                   
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                    
                                   
                                     
                                       j 
                                        
                                       
                                         ( 
                                         
                                           
                                             2 
                                              
                                             
                                                 
                                             
                                              
                                             
                                               n 
                                                
                                               
                                                 ( 
                                                 
                                                   N 
                                                   + 
                                                   
                                                     P 
                                                     1 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           + 
                                           
                                             P 
                                             1 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       φ 
                                       r 
                                     
                                   
                                 
                                  
                                 
                                   H 
                                   
                                     1 
                                     , 
                                     D 
                                   
                                 
                                  
                                 
                                   
                                     Z 
                                     N 
                                   
                                    
                                   
                                     ( 
                                     
                                       φ 
                                       r 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   W 
                                   N 
                                   H 
                                 
                                  
                                 
                                   Λ 
                                   
                                     S 
                                     , 
                                     1 
                                   
                                 
                                  
                                 
                                   x 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     n 
                                   
                                 
                               
                               + 
                             
                           
                         
                         
                           
                             
                               
                                  
                                 
                                   
                                     - 
                                     
                                       j 
                                        
                                       
                                         ( 
                                         
                                           
                                             
                                               ( 
                                               
                                                 
                                                   2 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   n 
                                                 
                                                 + 
                                                 1 
                                               
                                               ) 
                                             
                                              
                                             
                                               ( 
                                               
                                                 N 
                                                 + 
                                                 
                                                   P 
                                                   1 
                                                 
                                               
                                               ) 
                                             
                                           
                                           + 
                                           
                                             P 
                                             1 
                                           
                                           + 
                                           N 
                                           - 
                                           1 
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                   
                                     φ 
                                     r 
                                   
                                 
                               
                                
                               
                                 H 
                                 
                                   2 
                                   , 
                                   D 
                                 
                               
                                
                               
                                 
                                   Z 
                                   N 
                                 
                                  
                                 
                                   ( 
                                   
                                     φ 
                                     r 
                                   
                                   ) 
                                 
                               
                                
                               
                                 W 
                                 N 
                                 H 
                               
                                
                               
                                 
                                   Z 
                                   N 
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       π 
                                     
                                     N 
                                   
                                   ) 
                                 
                               
                                
                               
                                 Λ 
                                 
                                   S 
                                   , 
                                   2 
                                 
                                 * 
                               
                                
                               
                                 x 
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                                 * 
                               
                             
                           
                         
                       
                       ] 
                     
                     + 
                     
                       
                         p 
                         
                           D 
                           , 
                           
                             2 
                              
                             
                                 
                             
                              
                             n 
                           
                         
                       
                        
                       
                         
 
                       
                        
                       
                         r 
                         
                           D 
                           , 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               n 
                             
                             + 
                             1 
                           
                         
                       
                     
                   
                   = 
                   
                     
                        
                       
                         
                           j 
                            
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       n 
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 P 
                                 2 
                               
                             
                             ) 
                           
                         
                          
                         
                           φ 
                           d 
                         
                       
                     
                      
                     
                       
                         Z 
                         N 
                       
                       ( 
                       
                           
                       
                        
                       
                         φ 
                         d 
                       
                       ) 
                     
                      
                     
                         
                       
                         
                           [ 
                           
                               
                           
                            
                           
                             
                               
                                 
                                   
                                     
                                        
                                       
                                         
                                           j 
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     
                                                       2 
                                                        
                                                       
                                                           
                                                       
                                                        
                                                       n 
                                                     
                                                     + 
                                                     1 
                                                   
                                                   ) 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     N 
                                                     + 
                                                     
                                                       P 
                                                       1 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               + 
                                               
                                                 P 
                                                 1 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                         
                                           φ 
                                           r 
                                         
                                       
                                     
                                      
                                     
                                       H 
                                       
                                         1 
                                         , 
                                         D 
                                       
                                     
                                      
                                     
                                       
                                         Z 
                                         N 
                                       
                                        
                                       
                                         ( 
                                         
                                           φ 
                                           r 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       W 
                                       N 
                                       H 
                                     
                                      
                                     
                                       Λ 
                                       
                                         S 
                                         , 
                                         1 
                                       
                                     
                                      
                                     
                                       x 
                                       
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           n 
                                         
                                         + 
                                         1 
                                       
                                     
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   
                                      
                                     
                                       
                                         - 
                                         
                                           j 
                                            
                                           
                                             ( 
                                             
                                               
                                                 2 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 
                                                   n 
                                                    
                                                   
                                                     ( 
                                                     
                                                       N 
                                                       + 
                                                       
                                                         P 
                                                         1 
                                                       
                                                     
                                                     ) 
                                                   
                                                 
                                               
                                               + 
                                               
                                                 P 
                                                 1 
                                               
                                               + 
                                               N 
                                               - 
                                               1 
                                             
                                             ) 
                                           
                                         
                                       
                                        
                                       
                                         φ 
                                         r 
                                       
                                     
                                   
                                    
                                   
                                     H 
                                     
                                       2 
                                       , 
                                       D 
                                     
                                   
                                    
                                   
                                     
                                       Z 
                                       N 
                                     
                                      
                                     
                                       ( 
                                       
                                         φ 
                                         r 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     W 
                                     N 
                                     H 
                                   
                                    
                                   
                                     
                                       Z 
                                       N 
                                     
                                      
                                     
                                       ( 
                                       
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           π 
                                         
                                         N 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     Λ 
                                     
                                       S 
                                       , 
                                       2 
                                     
                                     * 
                                   
                                    
                                   
                                     x 
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       n 
                                     
                                     * 
                                   
                                 
                               
                             
                           
                           ] 
                         
                         + 
                         
                             
                         
                          
                         
                           
                             p 
                             
                               D 
                               , 
                               
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   n 
                                 
                                 + 
                                 1 
                               
                             
                           
                            
                           
                               
                           
                           . 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0000]    H 1,D  and H 2,D  are (N×N) circulant matrices respectively with [h 1,D   T , 0 1×(N-L) ] T  and [h 2,D   T , 0 1×(N-L) ] T  as their first column vectors. The (N×1) vector P D,2n  represent additive noise. 
         [0068]    Assuming that ACFO compensation was performed, or if ACFO is absent, for the j=(2n)-th OFDM symbol duration 
         [0000]    
       
         
           
             
               
                 r 
                 
                   D 
                   , 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                 
               
               = 
               
                 
                   
                     H 
                     
                       1 
                       , 
                       D 
                     
                   
                    
                   
                     W 
                     N 
                     H 
                   
                    
                   
                     Λ 
                     
                       S 
                       , 
                       1 
                     
                   
                    
                   
                     x 
                     
                       2 
                        
                       
                           
                       
                        
                       n 
                     
                   
                 
                 + 
                 
                   
                     H 
                     
                       2 
                       , 
                       D 
                     
                   
                    
                   
                     W 
                     N 
                     H 
                   
                    
                   
                     
                       Z 
                       N 
                     
                      
                     
                       ( 
                       
                         
                           2 
                            
                           
                               
                           
                            
                           π 
                         
                         N 
                       
                       ) 
                     
                   
                    
                   
                     Λ 
                     
                       S 
                       , 
                       2 
                     
                     * 
                   
                    
                   
                     x 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         n 
                       
                       + 
                       1 
                     
                     * 
                   
                 
                 + 
                 
                   p 
                   
                     D 
                     , 
                     
                       2 
                        
                       
                           
                       
                        
                       n 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0069]    The (N×1) vector p D,2n  representing the additive noise is given by 
         [0000]        P   D,2n   =H   1,D   v   R,1,2n   +H   2,D ζ( v*   R,2,2n+1 )+ V   D,2n,  
 
         [0000]    where the (N×1) vector V D,2n  contains the N samples of AWGN at destination with covariance matrix E[v D,2n v D,2n   H ]=σ D   2 I N . 
         [0070]    Similarly, for the j=(2n+1)-th OFDM symbol duration, 
         [0000]    
       
         
           
             
               r 
               
                 D 
                 , 
                 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                   + 
                   1 
                 
               
             
             = 
             
               
                 
                   H 
                   
                     1 
                     , 
                     D 
                   
                 
                  
                 
                   W 
                   N 
                   H 
                 
                  
                 
                   Λ 
                   
                     S 
                     , 
                     1 
                   
                 
                  
                 
                   x 
                   
                     
                       2 
                        
                       
                           
                       
                        
                       n 
                     
                     + 
                     1 
                   
                 
               
               - 
               
                 
                   H 
                   
                     2 
                     , 
                     D 
                   
                 
                  
                 
                   W 
                   N 
                   H 
                 
                  
                 
                   
                     Z 
                     N 
                   
                    
                   
                     ( 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         π 
                       
                       N 
                     
                     ) 
                   
                 
                  
                 
                   Λ 
                   
                     S 
                     , 
                     2 
                   
                   * 
                 
                  
                 
                   x 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                   * 
                 
               
               + 
               
                 p 
                 
                   D 
                   , 
                   
                     
                       2 
                        
                       
                           
                       
                        
                       n 
                     
                     + 
                     1 
                   
                 
               
             
           
         
       
       
         
           
             
               p 
               
                 D 
                 , 
                 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                   + 
                   1 
                 
               
             
             = 
             
               
                 
                   H 
                   
                     1 
                     , 
                     D 
                   
                 
                  
                 
                   v 
                   
                     R 
                     , 
                     1 
                     , 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
               
               - 
               
                 
                   H 
                   
                     2 
                     , 
                     D 
                   
                 
                  
                 
                   ζ 
                    
                   
                     ( 
                     
                       v 
                       
                         R 
                         , 
                         2 
                         , 
                         
                           2 
                            
                           
                               
                           
                            
                           n 
                         
                       
                       * 
                     
                     ) 
                   
                 
               
               + 
               
                 
                   v 
                   
                     D 
                     , 
                     
                       
                         2 
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
                 . 
               
             
           
         
       
     
         [0071]    In step  218 , the destination performs ASTC decoding in the frequency domain using the channel state information (CSI) parameters that were obtained during training. N-point DFTs are performed on the signal vector r D,j  for the symbol durations j={2n, 2n+1}. Assuming that ACFO is absent, the N-point DFT of r D,2n  is 
         [0000]    
       
         
           
             
               
                 W 
                 N 
               
                
               
                 r 
                 
                   D 
                   , 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                 
               
             
             = 
             
               
                 
                   Λ 
                   
                     1 
                     , 
                     D 
                   
                 
                  
                 
                   Λ 
                   
                     S 
                     , 
                     1 
                   
                 
                  
                 
                   x 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                 
               
               + 
               
                 
                   Λ 
                   
                     2 
                     , 
                     D 
                   
                 
                  
                 
                   
                     Z 
                     N 
                   
                    
                   
                     ( 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         π 
                       
                       N 
                     
                     ) 
                   
                 
                  
                 
                   Λ 
                   
                     S 
                     , 
                     2 
                   
                   * 
                 
                  
                 
                   x 
                   
                     
                       2 
                        
                       
                           
                       
                        
                       n 
                     
                     + 
                     1 
                   
                   * 
                 
               
               + 
               
                 
                   W 
                   N 
                 
                  
                 
                   
                     p 
                     
                       D 
                       , 
                       
                         2 
                          
                         
                             
                         
                          
                         n 
                       
                     
                   
                   . 
                 
               
             
           
         
       
     
         [0072]    Similarly, the N-point DFT of the signal vector received by the destination during the (2n+1)-th OFDM symbol duration (i.e. r D,2n+1 ) is given by 
         [0000]    
       
         
           
             
               
                 W 
                 N 
               
                
               
                 r 
                 
                   D 
                   , 
                   
                     
                       2 
                        
                       
                           
                       
                        
                       n 
                     
                     + 
                     1 
                   
                 
               
             
             = 
             
               
                 
                   Λ 
                   
                     1 
                     , 
                     D 
                   
                 
                  
                 
                   Λ 
                   
                     S 
                     , 
                     1 
                   
                 
                  
                 
                   x 
                   
                     
                       2 
                        
                       
                           
                       
                        
                       n 
                     
                     + 
                     1 
                   
                 
               
               - 
               
                 
                   Λ 
                   
                     2 
                     , 
                     D 
                   
                 
                  
                 
                   
                     Z 
                     N 
                   
                    
                   
                     ( 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         π 
                       
                       N 
                     
                     ) 
                   
                 
                  
                 
                   Λ 
                   
                     S 
                     , 
                     2 
                   
                   * 
                 
                  
                 
                   x 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                   * 
                 
               
               + 
               
                 
                   W 
                   N 
                 
                  
                 
                   p 
                   
                     D 
                     , 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
               
             
           
         
       
     
         [0073]    A (2N×1) vector y D,n  is constructed as 
         [0000]    
       
         
           
             
               
                 
                   
                     y 
                     
                       D 
                       , 
                       n 
                     
                   
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             ( 
                             
                               
                                 W 
                                 N 
                               
                                
                               
                                 r 
                                 
                                   D 
                                   , 
                                   
                                     2 
                                      
                                     n 
                                   
                                 
                               
                             
                             ) 
                           
                           T 
                         
                         , 
                         
                           
                             ( 
                             
                               
                                 W 
                                 N 
                               
                                
                               
                                 r 
                                 
                                   D 
                                   , 
                                   
                                     
                                       2 
                                        
                                       n 
                                     
                                     + 
                                     1 
                                   
                                 
                               
                             
                             ) 
                           
                           H 
                         
                       
                       ] 
                     
                     T 
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       
                         
                           H 
                           P 
                         
                          
                         
                           [ 
                           
                             
                               
                                 
                                   x 
                                   
                                     2 
                                      
                                     n 
                                   
                                 
                               
                             
                             
                               
                                 
                                   x 
                                   
                                     
                                       2 
                                        
                                       n 
                                     
                                     + 
                                     1 
                                   
                                   * 
                                 
                               
                             
                           
                           ] 
                         
                       
                       + 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   W 
                                   N 
                                 
                                  
                                 
                                   p 
                                   
                                     D 
                                     , 
                                     
                                       2 
                                        
                                       n 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                       W 
                                       N 
                                     
                                      
                                     
                                       p 
                                       
                                         D 
                                         , 
                                         
                                           
                                             2 
                                              
                                             n 
                                           
                                           + 
                                           1 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                 * 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   , 
                 
               
             
           
         
       
     
         [0074]    Estimates of x 2n  and x 2n+1  (denoted {tilde over (x)} 2n  {tilde over (x)} 2n+1 ) can be obtained using the conjugate transpose of H P  i.e. H P   H   
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               x 
                               ~ 
                             
                             
                               2 
                                
                               n 
                             
                           
                         
                       
                       
                         
                           
                             
                               x 
                               ~ 
                             
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                             * 
                           
                         
                       
                     
                     ] 
                   
                   = 
                     
                    
                   
                     
                       H 
                       P 
                       H 
                     
                      
                     
                       y 
                       
                         D 
                         , 
                         n 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       H 
                       P 
                       N 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               
                                 W 
                                 N 
                               
                                
                               
                                 r 
                                 
                                   D 
                                   , 
                                   
                                     2 
                                      
                                     n 
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                     W 
                                     N 
                                   
                                    
                                   
                                     r 
                                     
                                       D 
                                       , 
                                       
                                         
                                           2 
                                            
                                           n 
                                         
                                         + 
                                         1 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               * 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
         [0075]    H P  is the (2N×2N) sized product channel matrix and is given by 
         [0000]    
       
         
           
             
               H 
               P 
             
             = 
             
               
                 [ 
                 
                   
                     
                       
                         
                           Λ 
                           
                             1 
                             , 
                             D 
                           
                         
                          
                         
                           Λ 
                           
                             S 
                             , 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           
                             Z 
                             N 
                           
                            
                           
                             ( 
                             
                               
                                 2 
                                  
                                 π 
                               
                               N 
                             
                             ) 
                           
                         
                          
                         
                           Λ 
                           
                             2 
                             , 
                             D 
                           
                         
                          
                         
                           Λ 
                           
                             S 
                             , 
                             2 
                           
                           * 
                         
                       
                     
                   
                   
                     
                       
                         
                           - 
                           
                             
                               Z 
                               N 
                             
                              
                             
                               ( 
                               
                                 - 
                                 
                                   
                                     2 
                                      
                                     π 
                                   
                                   N 
                                 
                               
                               ) 
                             
                           
                         
                          
                         
                           Λ 
                           
                             2 
                             , 
                             D 
                           
                           * 
                         
                          
                         
                           Λ 
                           
                             S 
                             , 
                             2 
                           
                         
                       
                     
                     
                       
                         
                           Λ 
                           
                             1 
                             , 
                             D 
                           
                           * 
                         
                          
                         
                           Λ 
                           
                             S 
                             , 
                             1 
                           
                           * 
                         
                       
                     
                   
                 
                 ] 
               
               . 
             
           
         
       
     
         [0076]    The product channel matrix H P  is a known parameter obtained during the product channel matrix estimation and forms part of the channel state information (CSI). H P  allows for a simple linear decoding at destination, since 
         [0000]        H   P   H   H   P   =I   2           (|Λ 1,D | 2 |Λ S,1   2 |+|Λ 2,D | 2 |Λ S,2 | 2 )
 
         [0000]    where           denotes performing a Kronecker product. 
         [0077]    In step  230 , the relay station optionally performs ACFO φ r  compensation. The signal vectors r R,i,j  received over the span of 2 OFDM symbol durations j={2n, 2n+1} for antennae i=1, 2 are: 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       r 
                       
                         R 
                         , 
                         1 
                         , 
                         
                           2 
                            
                           n 
                         
                       
                     
                   
                   
                     
                       r 
                       
                         R 
                         , 
                         1 
                         , 
                         
                           
                             2 
                              
                             n 
                           
                           + 
                           1 
                         
                       
                     
                   
                 
                 
                   
                     
                       r 
                       
                         R 
                         , 
                         2 
                         , 
                         
                           2 
                            
                           n 
                         
                       
                     
                   
                   
                     
                       r 
                       
                         R 
                         , 
                         2 
                         , 
                         
                           
                             2 
                              
                             n 
                           
                           + 
                           1 
                         
                       
                     
                   
                 
               
               ] 
             
             = 
             
                 
               
                 
                   [ 
                   
                       
                   
                    
                   
                     
                       
                         
                           
                              
                             
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                         n 
                                          
                                         
                                           ( 
                                           
                                             N 
                                             + 
                                             
                                               P 
                                               1 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       P 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 φ 
                                 r 
                               
                             
                           
                            
                           
                             
                               Z 
                               N 
                             
                              
                             
                               ( 
                               
                                 φ 
                                 r 
                               
                               ) 
                             
                           
                            
                           
                             H 
                             
                               S 
                               , 
                               1 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               2 
                                
                               n 
                             
                           
                         
                       
                       
                         
                           
                              
                             
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             2 
                                              
                                             n 
                                           
                                           + 
                                           1 
                                         
                                         ) 
                                       
                                        
                                       
                                         ( 
                                         
                                           N 
                                           + 
                                           
                                             P 
                                             1 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       P 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 φ 
                                 r 
                               
                             
                           
                            
                           
                             
                               Z 
                               N 
                             
                              
                             
                               ( 
                               
                                 φ 
                                 r 
                               
                               ) 
                             
                           
                            
                           
                             H 
                             
                               S 
                               , 
                               1 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                              
                             
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                         n 
                                          
                                         
                                           ( 
                                           
                                             N 
                                             + 
                                             
                                               P 
                                               1 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       P 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 φ 
                                 r 
                               
                             
                           
                            
                           
                             
                               Z 
                               N 
                             
                              
                             
                               ( 
                               
                                 φ 
                                 r 
                               
                               ) 
                             
                           
                            
                           
                             H 
                             
                               S 
                               , 
                               2 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               2 
                                
                               n 
                             
                           
                         
                       
                       
                         
                           
                              
                             
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             2 
                                              
                                             n 
                                           
                                           + 
                                           1 
                                         
                                         ) 
                                       
                                        
                                       
                                         ( 
                                         
                                           N 
                                           + 
                                           
                                             P 
                                             1 
                                           
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       P 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 φ 
                                 r 
                               
                             
                           
                            
                           
                             
                               Z 
                               N 
                             
                              
                             
                               ( 
                               
                                 φ 
                                 r 
                               
                               ) 
                             
                           
                            
                           
                             H 
                             
                               S 
                               , 
                               2 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                     
                   
                   ] 
                 
                 + 
                 
                     
                   
                     
                       [ 
                       
                         
                           
                             
                               v 
                               
                                 R 
                                 , 
                                 1 
                                 , 
                                 
                                   2 
                                    
                                   n 
                                 
                               
                             
                           
                           
                             
                               v 
                               
                                 R 
                                 , 
                                 1 
                                 , 
                                 
                                   
                                     2 
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               v 
                               
                                 R 
                                 , 
                                 2 
                                 , 
                                 
                                   2 
                                    
                                   n 
                                 
                               
                             
                           
                           
                             
                               v 
                               
                                 R 
                                 , 
                                 2 
                                 , 
                                 
                                   
                                     2 
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0078]    The ACFO φ r  is a known parameter obtained at the relay station during the ACFO {circumflex over (φ)} r , estimation step  410  and forms part of the channel state information (CSI). 
         [0079]    The effect of the ACFO φ r  can be removed for r R,1,2n  and r R,2,2n  by multiplying r R,1,2n  and r R,2,2n  with the conjugate of e j(2n(N+P     1     )+P     1     )φ     r   Z N (φ r ). The effect of the ACFO φ r  can be removed for r R,1,2n+1  and r R,2,2n+1  by multiplying r R,1,2n+1  and r R,2,2n+1  with the conjugate of e j((2n(N+P     1     )+P     1     )φ     r   Z N (φ r ). 
         [0080]    In step  240 , the destination optionally performs ACFO φ d  compensation. The (N×1) received signal vectors at the destination in the presence of ACFO φ d  are given by
       CP Scheme 1:       
 
         [0000]    
       
         
           
             
               r 
               
                 D 
                 , 
                 
                   2 
                    
                   n 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     
                       Z 
                       N 
                     
                      
                     
                       ( 
                       
                         φ 
                         d 
                       
                       ) 
                     
                   
                    
                   
                     [ 
                     
                       
                         
                           H 
                           
                             1 
                             , 
                             D 
                           
                         
                          
                         
                           H 
                           
                             S 
                             , 
                             1 
                           
                         
                          
                         
                           W 
                           N 
                           H 
                         
                          
                         
                           x 
                           
                             2 
                              
                             n 
                           
                         
                       
                       + 
                       
                         
                           H 
                           
                             2 
                             , 
                             D 
                           
                         
                          
                         
                           ζ 
                            
                           
                             ( 
                             
                               H 
                               
                                 S 
                                 , 
                                 2 
                               
                               * 
                             
                             ) 
                           
                         
                          
                         
                           W 
                           N 
                           H 
                         
                          
                         
                           Tx 
                           
                             
                               2 
                                
                               n 
                             
                             + 
                             1 
                           
                           * 
                         
                       
                     
                     ] 
                   
                 
               
               + 
               
                 p 
                 
                   D 
                   , 
                   
                     2 
                      
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 r 
                 
                   D 
                   , 
                   
                     
                       2 
                        
                       n 
                     
                     + 
                     1 
                   
                 
               
               = 
               
                 
                   
                     
                        
                       
                         
                           j 
                            
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       n 
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 P 
                                 2 
                               
                             
                             ) 
                           
                         
                          
                         
                           φ 
                           d 
                         
                       
                     
                     
                       2 
                     
                   
                    
                   
                     
                       
                         Z 
                         N 
                       
                        
                       
                         ( 
                         
                           φ 
                           d 
                         
                         ) 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             H 
                             
                               1 
                               , 
                               D 
                             
                           
                            
                           
                             H 
                             
                               S 
                               , 
                               1 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                         - 
                         
                           
                             H 
                             
                               2 
                               , 
                               D 
                             
                           
                            
                           
                             ζ 
                              
                             
                               ( 
                               
                                 H 
                                 
                                   S 
                                   , 
                                   2 
                                 
                                 * 
                               
                               ) 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             Tx 
                             
                               2 
                                
                               n 
                             
                             * 
                           
                         
                       
                       ] 
                     
                   
                 
                 + 
                 
                   p 
                   
                     D 
                     , 
                     
                       
                         2 
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where the noise vectors are given byp D,2n  and p D,2n+1 . The ACFO φ d  is a known parameter obtained at the destination during the ACFO {circumflex over (φ)} d     —   estimation step  420  and forms part of the channel state information (CSI). 
         [0082]    The effect of the ACFO φ d  can be removed by multiplying r D,2n  and r D,2n+1  respectively with the conjugate of e j(2n(N+P     2     )+P     2     )φ     d   Z N (φ d ) and e j((2n+1)(N+P     2     )+P     2     )φ     d   Z N (φ d ). 
         [0083]    The Cyclic Prefix Scheme 1 may have the advantage that only analog domain processing has to be done at the relay, and only linear processing has to be done for maximum likelihood decoding at the destination. 
         [0084]    Cyclic Prefix Scheme 2 
         [0085]      FIG. 3  shows an alternative scheme for inserting cyclic prefixes according to the example embodiment. Let x j  be a (N×1) vector containing the N data symbols to be transmitted during a j-th OFDM symbol duration. 
         [0086]    While this example embodiment of the invention has been described using 2 OFDM symbols i.e. x 2n  and x 2n+1 , it is clear that the use of a different number of OFDM symbols is possible within the scope of the invention, as will be clear to a skilled reader. 
         [0087]    In step  302 , the source derives a first cyclic prefix (denoted CP j,1 ) with length P from x j , where P=P 1 +P 2 =L 1 +L 2 −2. L 1  is the length of the channel impulse response (CIR) represented by h S,1  and h S,2  respectively for the channel between the source  110  and the first antenna  122  of the relay station  120 , and the channel between the source  110  and the second antenna  124  of the relay station  120 . L 2  is the length of the CIRs denoted by the vectors h 1,D  and h 2,D  respectively for the channel between the first antenna  122  of the relay station  120  and the destination  130 , and the channel between the second antenna  124  of the relay station  120  and the destination  130 . 
         [0088]    In step  304 , the source inserts CP j,1  in front of x j  and then transmits the resultant symbol sequence. 
         [0089]    Let the symbol sequence x j  consist of N symbols be denoted by x n  where n=0 . . . N−1, i.e. 
         [0000]      x j =[x 0  . . . x n  . . . x N−1]   
         [0090]    The cyclic prefix CP j,1  of length P 1  is created by copying the last Ncp words of x j    
         [0000]      CP j,1 =└x N-P     1    . . . x N-2  . . . x N-1┘ 
 
         [0091]    The cyclic prefix CP j,1  is inserted in front of x j , producing the resultant symbol sequence └CP j,1  x j  ┘ for transmission. 
         [0092]    The transmission from the source  110  to relay station  120  takes place in the first phase of the transmission cycle. 
         [0093]    In step  306 ; the relay station receives └CP j,1  x j ┘ at both antennae of the station. 
         [0094]    In step  308 , after performing time synchronization, the first P 1  samples of the CP j,1  portion of each OFDM symbol is removed, leaving a sequence represented by └CP j,2  x j  ┘. The first P 1  samples of each OFDM symbol are distorted by the ISI induced by the frequency selective fading occurring at the source to relay channels. The resulting OFDM symbol will then have (N+P 2 ) samples. Should ACFO be present, the signal vectors received over the span of 2 OFDM symbol durations i.e. for j={2n, 2n+1} are given after the removal of CP j,1  as 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       
                         r 
                         ~ 
                       
                       
                         R 
                         , 
                         1 
                         , 
                         
                           2 
                            
                           n 
                         
                       
                     
                   
                   
                     
                       
                         r 
                         ~ 
                       
                       
                         R 
                         , 
                         1 
                         , 
                         
                           
                             2 
                              
                             n 
                           
                           + 
                           1 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         r 
                         ~ 
                       
                       
                         R 
                         , 
                         2 
                         , 
                         
                           2 
                            
                           n 
                         
                       
                     
                   
                   
                     
                       
                         r 
                         ~ 
                       
                       
                         R 
                         , 
                         2 
                         , 
                         
                           
                             2 
                              
                             n 
                           
                           + 
                           1 
                         
                       
                     
                   
                 
               
               ] 
             
             = 
             
                 
               
                 
                   [ 
                   
                       
                   
                    
                   
                     
                       
                         
                           
                              
                             
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                         n 
                                          
                                         
                                           ( 
                                           
                                             N 
                                             + 
                                             P 
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       P 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 φ 
                                 r 
                               
                             
                           
                            
                           
                             
                               Z 
                               
                                 N 
                                 + 
                                 
                                   P 
                                   2 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 φ 
                                 r 
                               
                               ) 
                             
                           
                            
                           
                             
                               H 
                               ~ 
                             
                             
                               S 
                               , 
                               1 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               2 
                                
                               n 
                             
                           
                         
                       
                       
                         
                           
                              
                             
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             2 
                                              
                                             n 
                                           
                                           + 
                                           1 
                                         
                                         ) 
                                       
                                        
                                       
                                         ( 
                                         
                                           N 
                                           + 
                                           P 
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       P 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 φ 
                                 r 
                               
                             
                           
                            
                           
                             
                               Z 
                               
                                 N 
                                 + 
                                 
                                   P 
                                   2 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 φ 
                                 r 
                               
                               ) 
                             
                           
                            
                           
                             
                               H 
                               ~ 
                             
                             
                               S 
                               , 
                               1 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                              
                             
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                         n 
                                          
                                         
                                           ( 
                                           
                                             N 
                                             + 
                                             P 
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       P 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 φ 
                                 r 
                               
                             
                           
                            
                           
                             
                               Z 
                               
                                 N 
                                 + 
                                 
                                   P 
                                   2 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 φ 
                                 r 
                               
                               ) 
                             
                           
                            
                           
                             
                               H 
                               ~ 
                             
                             
                               S 
                               , 
                               2 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               2 
                                
                               n 
                             
                           
                         
                       
                       
                         
                           
                              
                             
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             2 
                                              
                                             n 
                                           
                                           + 
                                           1 
                                         
                                         ) 
                                       
                                        
                                       
                                         ( 
                                         
                                           N 
                                           + 
                                           P 
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       P 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 φ 
                                 r 
                               
                             
                           
                            
                           
                             
                               Z 
                               
                                 N 
                                 + 
                                 
                                   P 
                                   2 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 φ 
                                 r 
                               
                               ) 
                             
                           
                            
                           
                             
                               H 
                               ~ 
                             
                             
                               S 
                               , 
                               2 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                     
                   
                   ] 
                 
                 + 
                 
                     
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 v 
                                 ~ 
                               
                               
                                 R 
                                 , 
                                 1 
                                 , 
                                 
                                   2 
                                    
                                   n 
                                 
                               
                             
                           
                           
                             
                               
                                 v 
                                 ~ 
                               
                               
                                 R 
                                 , 
                                 1 
                                 , 
                                 
                                   
                                     2 
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 v 
                                 ~ 
                               
                               
                                 R 
                                 , 
                                 2 
                                 , 
                                 
                                   2 
                                    
                                   n 
                                 
                               
                             
                           
                           
                             
                               
                                 v 
                                 ~ 
                               
                               
                                 R 
                                 , 
                                 2 
                                 , 
                                 
                                   
                                     2 
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                               
                             
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0000]    {tilde over (r)} R,i,j  denotes the ((N+P 2 )×1) received signal vector obtained after the removal of P 1  samples, with i and j respectively denoting the antenna index and OFDM symbol duration. n denotes the data transmission cycle index. The (N×N) matrix W N   H  is the inverse discrete Fourier transform (IDFT) matrix. Matrix {tilde over (H)} S,1  for i=1, 2 is a ((N+P 2 )×N) matrix given by 
         [0000]    
       
         
           
             
               
                 
                   H 
                   ~ 
                 
                 
                   S 
                   , 
                   i 
                 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         
                           H 
                           
                             S 
                             , 
                             i 
                           
                         
                          
                         
                           ( 
                           
                             
                               
                                 N 
                                 - 
                                 
                                   P 
                                   2 
                                 
                                 + 
                                 1 
                               
                               : 
                               N 
                             
                             , 
                             : 
                           
                           ) 
                         
                       
                     
                   
                   
                     
                       
                         H 
                         
                           S 
                           , 
                           i 
                         
                       
                     
                   
                 
                 ] 
               
             
             , 
           
         
       
     
         [0000]    with H S,1  being a (N×N) circulant matrix with [h S,1   T , 0 1×(N-L     1     ) ] T  as its first column vector and H S,1 (N−P 2 +1: N,:) denotes the last P 2  row vectors of H S,1.    
         [0095]    The vector {tilde over (v)} R,i,j  of size ((N+P 2 )×1) contains the AWGN affecting the signal {tilde over (r)} R,i,j . The covariance matrix of the noise vector {tilde over (v)} R,i,j  at the relay station is given by 
         [0000]      E{{tilde over (v)} R,i,j {tilde over (v)} R,i,j   H }=σ R   2 I N+P     2    
 
         [0096]    Optionally, in the event that ACFO is present, ACFO φ r  compensation can be performed at the relay station as Step  330 . 
         [0097]    Assuming the complete removal of ACFO or if ACFO was absent, over the span of 2 OFDM symbol durations i.e. for j={2n, 2n+1}, the received signal vectors obtained after the removal of the first P 1  samples of the CP j,1  portion of each OFDM symbol are given as follows: 
         [0000]    
       
         
           
             
               
                 [ 
                 
                   
                     
                       
                         
                           r 
                           ~ 
                         
                         
                           R 
                           , 
                           1 
                           , 
                           
                             2 
                              
                             n 
                           
                         
                       
                     
                     
                       
                         
                           r 
                           ~ 
                         
                         
                           R 
                           , 
                           1 
                           , 
                           
                             
                               2 
                                
                               n 
                             
                             + 
                             1 
                           
                         
                       
                     
                   
                   
                     
                       
                         
                           r 
                           ~ 
                         
                         
                           R 
                           , 
                           2 
                           , 
                           
                             2 
                              
                             n 
                           
                         
                       
                     
                     
                       
                         
                           r 
                           ~ 
                         
                         
                           R 
                           , 
                           2 
                           , 
                           
                             
                               2 
                                
                               n 
                             
                             + 
                             1 
                           
                         
                       
                     
                   
                 
                 ] 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           
                             
                               H 
                               ~ 
                             
                             
                               S 
                               , 
                               1 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               2 
                                
                               
                                   
                               
                                
                               n 
                             
                           
                         
                       
                       
                         
                           
                             
                               H 
                               ~ 
                             
                             
                               S 
                               , 
                               1 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               H 
                               ~ 
                             
                             
                               S 
                               , 
                               2 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               2 
                                
                               
                                   
                               
                                
                               n 
                             
                           
                         
                       
                       
                         
                           
                             
                               H 
                               ~ 
                             
                             
                               S 
                               , 
                               2 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                     
                   
                   ] 
                 
                 + 
                 
                   [ 
                   
                     
                       
                         
                           
                             v 
                             ~ 
                           
                           
                             R 
                             , 
                             1 
                             , 
                             
                               2 
                                
                               
                                   
                               
                                
                               n 
                             
                           
                         
                       
                       
                         
                           
                             v 
                             ~ 
                           
                           
                             R 
                             , 
                             1 
                             , 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             v 
                             ~ 
                           
                           
                             R 
                             , 
                             2 
                             , 
                             
                               2 
                                
                               
                                   
                               
                                
                               n 
                             
                           
                         
                       
                       
                         
                           
                             v 
                             ~ 
                           
                           
                             R 
                             , 
                             2 
                             , 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                     
                   
                   ] 
                 
               
             
             , 
           
         
       
     
         [0098]    Step  309  is then performed at the relay station. In step  309 , processing is performed in order to implement space-time coding at the carrier level. The matrix Y R  is computed and arranged: 
         [0000]    
       
         
           
             
               Y 
               R 
             
             = 
             
               
                 [ 
                 
                   y 
                   
                     ij 
                     , 
                     R 
                   
                 
                 ] 
               
               = 
               
                 
                   [ 
                   
                     
                       
                         
                           y 
                           
                             11 
                             , 
                             R 
                           
                         
                       
                       
                         
                           y 
                           
                             12 
                             , 
                             R 
                           
                         
                       
                     
                     
                       
                         
                           y 
                           
                             21 
                             , 
                             R 
                           
                         
                       
                       
                         
                           y 
                           
                             22 
                             , 
                             R 
                           
                         
                       
                     
                   
                   ] 
                 
                 = 
                 
                   
                     1 
                     
                       2 
                     
                   
                    
                   
                     [ 
                     
                       
                         
                           
                             
                               r 
                               ~ 
                             
                             
                               R 
                               , 
                               1 
                               , 
                               
                                 2 
                                  
                                 n 
                               
                             
                           
                         
                         
                           
                             
                               r 
                               ~ 
                             
                             
                               R 
                               , 
                               1 
                               , 
                               
                                 
                                   2 
                                    
                                   n 
                                 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           
                             ζ 
                              
                             
                               ( 
                               
                                 
                                   r 
                                   ~ 
                                 
                                 
                                   R 
                                   , 
                                   2 
                                   , 
                                   
                                     
                                       2 
                                        
                                       n 
                                     
                                     + 
                                     1 
                                   
                                 
                                 * 
                               
                               ) 
                             
                           
                         
                         
                           
                             - 
                             
                               ζ 
                                
                               
                                 ( 
                                 
                                   
                                     r 
                                     ~ 
                                   
                                   
                                     R 
                                     , 
                                     2 
                                     , 
                                     
                                       2 
                                        
                                       n 
                                     
                                   
                                   * 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
               
             
           
         
       
     
         [0099]    The operator [•]* represents a complex conjugation. The space-time code implemented can for example be the Alamouti code. 
         [0100]    Step  309  further comprises steps  310  and  312 . 
         [0101]    In step  310 , for those elements of Y R  where no signal conjugation is needed e.g.: for y 11,R  and y 12,R , nothing needs to be done. 
         [0102]    In step  312 , for those elements of Y R  where signal conjugation is needed e.g.: for y 21,R  and y 22,R , the OFDM symbol sequence from the respective {tilde over (r)} R,i,j  is reordered using the mapping function ζ(.): 
         [0000]      ζ( a )=[ a ( N− 1), a ( N− 2), . . . , a (0)] T ,
 
         [0000]    the (N×1) input vector a=[a(0), a(1), . . . , a(N−1)] T . This function can be easily implemented in hardware by storing the signal samples in a shift register and then reading the register in reverse order. 
         [0103]    The mapping function ζ(.) can thus be re-expressed as 
         [0000]      ζ( a ( n ))= a ( N−n− 1)
 
         [0000]    where a(n) denotes the n-th symbol of the OFDM symbol sequence a. A time domain signal conjugation is then performed on each reordered symbol ζ(a(n)), for n=0 . . . N−1. 
         [0104]    Let b=ζ(a). It can be shown that after signal conjugation is done, taking B={B(0) . . . B(k) . . . B(N−1)} to be the corresponding frequency domain sequence of b 
         [0000]        B *( k ) W   N   k(N-P     2     −1)   =B *( k ) W   N   −k(P     2     +1)    
         [0000]    i.e., the phase-shifted conjugate sequence. 
         [0105]    The following Discrete Fourier Transform (DFT) properties are present:
       Linearity: ax(n)+by(n)+by(n)         aX(k)+bY(k);   Cyclic Shift x((n+m) N )         W N   −km X(k);   Symmetry x*((−n) N )         X*(k)
 
x(n) and y(n) to denote the time domain sequence, X(k) and Y(k) their corresponding frequency domain sequence, N the DFT size, and
       
 
         [0000]    
       
         
           
             
               W 
               N 
             
             = 
             
               
                 exp 
                  
                 
                   ( 
                   
                     
                       - 
                       j 
                     
                      
                     
                       
                         2 
                          
                         
                             
                         
                          
                         π 
                       
                       N 
                     
                   
                   ) 
                 
               
               . 
             
           
         
       
     
         [0109]    In step  314 , the other necessary steps for the ASTC implementation at the relay station are performed. The symbol sequences contained in 
         [0000]    
       
         
           
             
               Y 
               R 
             
             = 
             
               
                 [ 
                 
                   y 
                   
                     ij 
                     , 
                     R 
                   
                 
                 ] 
               
               = 
               
                 [ 
                 
                   
                     
                       
                         y 
                         
                           11 
                           , 
                           R 
                         
                       
                     
                     
                       
                         y 
                         
                           12 
                           , 
                           R 
                         
                       
                     
                   
                   
                     
                       
                         y 
                         
                           21 
                           , 
                           R 
                         
                       
                     
                     
                       
                         y 
                         
                           22 
                           , 
                           R 
                         
                       
                     
                   
                 
                 ] 
               
             
           
         
       
     
         [0000]    are then transmitted by the i-th relay antenna during the j-th OFDM symbol duration, where i=1,2 and j={2n, 2n+1}. The transmission from the relay station to destination takes place in the second phase of the transmission cycle where the source remains silent. 
         [0110]    In step  316 , after frame synchronization, the destination removes the cyclic prefix of length P 2  from each OFDM symbol received. Each OFDM symbol after cyclic prefix removal will be N samples long and can be denoted by the signal vector r D,j  for each j-th OFDM symbol duration. 
         [0111]    Assuming that ACFO is present and ACFO compensation was done at the relay, for the j={2n, 2n+1}-th OFDM symbol duration, the (N×1) received signal vectors at the destination in the presence of ACFO φ d  are given by
       CP Scheme 2:       
 
         [0000]    
       
         
           
             
               r 
               
                 D 
                 , 
                 
                   2 
                    
                   n 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     
                       Z 
                       N 
                     
                      
                     
                       ( 
                       
                         φ 
                         d 
                       
                       ) 
                     
                   
                    
                   
                     [ 
                     
                       
                         
                           H 
                           
                             1 
                             , 
                             D 
                           
                         
                          
                         
                           H 
                           
                             S 
                             , 
                             1 
                           
                         
                          
                         
                           W 
                           N 
                           H 
                         
                          
                         
                           x 
                           
                             2 
                              
                             n 
                           
                         
                       
                       + 
                       
                         
                           H 
                           
                             2 
                             , 
                             D 
                           
                         
                          
                         
                           G 
                           
                             S 
                             , 
                             2 
                           
                         
                          
                         
                           W 
                           N 
                           H 
                         
                          
                         
                           Tx 
                           
                             
                               2 
                                
                               n 
                             
                             + 
                             1 
                           
                           * 
                         
                       
                     
                     ] 
                   
                 
               
               + 
               
                 
                   p 
                   ~ 
                 
                 
                   D 
                   , 
                   
                     2 
                      
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 r 
                 
                   D 
                   , 
                   
                     
                       2 
                        
                       n 
                     
                     + 
                     1 
                   
                 
               
               = 
               
                 
                   
                     
                        
                       
                         
                           j 
                            
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       n 
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 P 
                                 2 
                               
                             
                             ) 
                           
                         
                          
                         
                           φ 
                           d 
                         
                       
                     
                     
                       2 
                     
                   
                    
                   
                     
                       
                         Z 
                         N 
                       
                        
                       
                         ( 
                         
                           φ 
                           d 
                         
                         ) 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             H 
                             
                               1 
                               , 
                               D 
                             
                           
                            
                           
                             H 
                             
                               S 
                               , 
                               1 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                         - 
                         
                           
                             H 
                             
                               2 
                               , 
                               D 
                             
                           
                            
                           
                             G 
                             
                               S 
                               , 
                               2 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             Tx 
                             
                               2 
                                
                               n 
                             
                             * 
                           
                         
                       
                       ] 
                     
                   
                 
                 + 
                 
                   
                     p 
                     ~ 
                   
                   
                     D 
                     , 
                     
                       
                         2 
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where the noise vectors are given by 
         [0000]    
       
         
           
             
               
                 p 
                 ~ 
               
               
                 D 
                 , 
                 
                   2 
                    
                   n 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     Z 
                     N 
                   
                    
                   
                     ( 
                     
                       φ 
                       d 
                     
                     ) 
                   
                 
                  
                 
                   
                     R 
                     cp 
                   
                    
                   
                     [ 
                     
                       
                         
                           U 
                           
                             1 
                             , 
                             D 
                           
                         
                          
                         
                           
                             v 
                             ~ 
                           
                           
                             R 
                             , 
                             1 
                             , 
                             
                               2 
                                
                               n 
                             
                           
                         
                       
                       + 
                       
                         
                           U 
                           
                             2 
                             , 
                             D 
                           
                         
                          
                         
                           ζ 
                            
                           
                             ( 
                             
                               
                                 v 
                                 ~ 
                               
                               
                                 R 
                                 , 
                                 2 
                                 , 
                                 
                                   
                                     2 
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                               
                               * 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                 
               
               + 
               
                 v 
                 
                   D 
                   , 
                   
                     2 
                      
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 p 
                 ~ 
               
               
                 D 
                 , 
                 
                   
                     2 
                      
                     n 
                   
                   + 
                   1 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   N 
                                   + 
                                   
                                     P 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     Z 
                     N 
                   
                    
                   
                     ( 
                     
                       φ 
                       d 
                     
                     ) 
                   
                 
                  
                 
                   
                     R 
                     cp 
                   
                    
                   
                     [ 
                     
                       
                         
                           U 
                           
                             1 
                             , 
                             D 
                           
                         
                          
                         
                           
                             v 
                             ~ 
                           
                           
                             R 
                             , 
                             1 
                             , 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                       - 
                       
                         
                           H 
                           
                             2 
                             , 
                             D 
                           
                         
                          
                         
                           ( 
                           
                             
                               v 
                               ~ 
                             
                             
                               R 
                               , 
                               2 
                               , 
                               
                                 2 
                                  
                                 n 
                               
                             
                             * 
                           
                           ) 
                         
                       
                     
                     ] 
                   
                 
               
               + 
               
                 
                   v 
                   
                     D 
                     , 
                     
                       
                         2 
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
                 . 
               
             
           
         
       
     
         [0113]    Optionally, ACFO φ d  compensation can be performed at the destination as step  340 . 
         [0114]    Assuming that ACFO is present and no ACFO compensation was done at the relay, for the j={2n, 2n+1}-th OFDM symbol duration, 
         [0000]        r   D,2n   =e   j(2n(N+P     2     )+P     2     )(φ     d     +φ     r     )   Z   N (φ d +φ r )[α n   {hacek over (H)}   1,D   H   S,1   W   N   H   x   2n γ n   {hacek over (H)}   2,D   G   S,2 ( W   N   H   x   2n+ )*]+ {tilde over (p)}   D,2n  
 
         [0000]        r   D,2n+1   =e   j((2n+1)(N+P     2     )+P     2     )(φ     d     +φ     r     )   Z   N (φ d +φ r )[α n   e   jP     1     φ     r     {hacek over (H)}   1,D   H   S,1   W   N   H   x   2n+1 −γ n   e   jP     1     φ     r     {hacek over (H)}   2,D   G   S,2 ( W   N   H   x   2n )*]+ {tilde over (p)}   D,2n+1,  
 
         [0000]    where {hacek over (H)} 1,D  and {hacek over (H)} 2,D  are (N×N) circulant matrix with Z N (−φ r ) [h 1,D   T , 0 1×(N-L     2)   ] T and Z   N (−φ r ) [h 2,D   T , 0 1×(N-L     2)   ] T  as their respective first column vectors and scalars α n  and γ n  are defined as follows: 
         [0000]      α n =e j(2n+1)P     1     φ     r    
 
         [0000]      γ n =e −j((4n+2)(N+P)−2nP     1     -1)φ     r.    
 
         [0115]    Optionally, joint ACFO compensation can be performed at the destination as step  350  to compensate for the ACFO φ r  and φ d . 
         [0116]    Assuming that ACFO compensation was performed, or if ACFO is absent, for the j=(2n)-th OFDM symbol duration 
         [0000]        r   D,2n   =H   1,D   H   S,1   W   N   H   x   2n   +H   2,D   G   S,2 ( W   N   H   x   2n+1 )*+ {tilde over (p)}   D,2n,    
         [0117]    The (N×N) matrix G S,2  is given by 
         [0000]        G   S,2 =ζ( {tilde over (H)}*   S,2 (1 :N ,:)).
 
         [0118]    The mapping ζ(.) for a 2-dimensional matrix A is computed as follows: 
         [0000]    Given that A has a size of (N 1 ×N 2 ) and A=[a 1 , a 2 , . . . , a N     2   ] where a i  is (N 1×1 ) vector for i=1, 2, . . . , N 2 , then ζ(A)=[ζ(a 1 ), ζ(a 2 ), . . . , ζ(a N     2   )]. 
         [0119]    The (N×1) vector {tilde over (p)} D,2n  is given by 
         [0000]        {tilde over (p)}   D,2n   =R   cp   [U   1,D   {tilde over (v)}   R,1,2n   +U   2,D ζ( {tilde over (v)}   R,2,2n+1 )] v   D,2n,  
 
         [0000]    where vector v D,2n  contains the AWGN received at the destination for i=1, 2, the matrix U 1,D  is a ((N+P 2 )×(N+P 2 )) Toeplitz matrix with [h 1,D   T , 0 1×(N+P     2     −L     2     ) ] T  as its first column vector and [h 1,D (1, 1), 0 1×(N+P     2     −1) ] as its first row vector. Matrix R cp  is a (N×(N+P 2 )) matrix defined as 
         [0000]      R cp =[0 N×P     2   ,I N ]. 
         [0120]    Similarly, the (N×1) sized signal vector received at the destination during (2n+1)th OFDM symbol duration is obtained to be 
         [0000]        r   D,2n+1   =H   1,D   H   S,1   W   N   H   x   2n+1   −H   2,D   G   S,2 ( W   N   H   x   2n )*+ {tilde over (p)}   D,2n+1,    
         [0000]        {tilde over (p)}   D,2n+1   =R   cp   [U   1,D   {tilde over (v)}R, 1 , 2 n+ 1   U 2,D ζ( {tilde over (v)}*   R,2,2n )]+ v   D,2n+.  
 
         [0121]    In step  318 , the destination performs the ASTC decoding in the frequency domain using the channel state information (CSI) parameters that were obtained during training. N-point DFTs are performed on the signal vector r D,j  for the symbol durations j={2n, 2n+1}, resulting in the vectors W N r D,2n  and W N r D,2n+1    
         [0122]    A (2N×1) vector y D,n  is constructed as 
         [0000]    
       
         
           
             
               
                 
                   
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                       , 
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         [0123]    The signals r D,2n  and r D,2n+1  can be represented as 
         [0000]        r   D,2n   =H   1,D   H   S,1   W   N   H   x   2n   +H   2,D   G   S,2 ( W   N   H   x   2n+1 )*+ {tilde over (p)}   D,2n,    
         [0000]        r   D,2n+1   =H   1,D   H   S,1   W   N   H   x   2n+1   −H   2,D   G   S,2 ( W   N   H   x   2n )* {tilde over (p)}   D,2n+1,    
         [0124]    The matrix G S,2  is defined to be a (N×N) matrix of the form: 
         [0000]        G   S,2 =ζ( {tilde over (H)}*   S,2 (1 :N ,:)).
 
         [0125]    It can be observed that matrix G S,2  can be obtained by permutating the column vectors of a (N×N) circulant matrix C S,2 =W N   H Λ *   S,2 W N . Using this knowledge, it can be shown that 
         [0000]    
       
         
           
             
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         [0126]    Substituting G S,2  in the (N×1)-sized vector H 2,D G S,2 (W N   H x 2n+1 )*, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
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                     . 
                   
                 
               
             
           
         
       
     
         [0127]    Similarly, the (N×1) vector H 2,D G S,2 (W N   H x 2n )* can be expressed as 
         [0000]    
       
         
           
             
               
                 H 
                 
                   2 
                   , 
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         [0128]    Substituting the expressions of H 2,D G S,2 (W N   H x 2n+1 )* and H 2,D G S,2 (W N   H x 2n )* into y D,n , 
         [0000]    
       
         
           
             
               
                 
                   
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                       ] 
                     
                     . 
                   
                 
               
             
           
         
       
     
         [0129]    Estimates of x 2n  and x 2n+1  (denoted {tilde over (x)} 2n  and {tilde over (x)}* 2n+1 ) can be obtained using the conjugate transpose of H P  i.e. H P   H   
         [0000]    
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               x 
                               ~ 
                             
                             
                               2 
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                               n 
                             
                           
                         
                       
                       
                         
                           
                             
                               x 
                               ~ 
                             
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                             * 
                           
                         
                       
                     
                     ] 
                   
                   = 
                     
                    
                   
                     
                       H 
                       P 
                       H 
                     
                      
                     
                       y 
                       
                         D 
                         , 
                         n 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       H 
                       P 
                       N 
                     
                      
                     
                       [ 
                       
                         
                           
                             
                               
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                                   , 
                                   
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                                         + 
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                                 ) 
                               
                               * 
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
             
           
         
       
     
         [0130]    H P  is the (2N×2N) sized product channel matrix and is given by 
         [0000]    
       
         
           
             
               H 
               P 
             
             = 
             
               [ 
               
                 
                   
                     
                       
                         Λ 
                         
                           1 
                           , 
                           D 
                         
                       
                        
                       
                         Λ 
                         
                           S 
                           , 
                           1 
                         
                       
                     
                   
                   
                     
                       
                         
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                         - 
                         
                           
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                             2 
                           
                         
                       
                        
                       
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                           2 
                           , 
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                           S 
                           , 
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                           1 
                           , 
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                         * 
                       
                        
                       
                         Λ 
                         
                           S 
                           , 
                           1 
                         
                         * 
                       
                     
                   
                 
               
               ] 
             
           
         
       
     
         [0131]    The product channel matrix H P  is a known parameter obtained during the product channel matrix estimation and forms part of the channel state information (CSI). H P  allows for a simple linear decoding at destination, since 
         [0000]        H   P   H   H   P   =I   2           (|Λ 1,D | 2 |Λ S,1 | 2 |Λ 2,D | 2 |Λ S,2 | 2 ).
 
         [0000]    where           denotes performing a Kronecker product. 
         [0132]    It can be seen that the present embodiment has an advantage in that there is no need for any sample reordering to be done at the destination. 
         [0133]    In the event that ACFO is present and joint ACFO compensation was done in step  316 , the ASTC decoding in the frequency domain can be performed using step  320  instead of step  318 . 
         [0134]    In step  320 , the (2N×1) vector y D,n  will be 
         [0000]    
       
         
           
             
               
                 
                   
                     y 
                     
                       D 
                       , 
                       n 
                     
                   
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 
                                   ( 
                                   
                                     
                                        
                                       
                                         
                                           - 
                                           
                                             j 
                                              
                                             
                                               ( 
                                               
                                                 
                                                   2 
                                                    
                                                   
                                                     n 
                                                      
                                                     
                                                       ( 
                                                       
                                                         N 
                                                         + 
                                                         
                                                           P 
                                                           2 
                                                         
                                                       
                                                       ) 
                                                     
                                                   
                                                 
                                                 + 
                                                 
                                                   P 
                                                   2 
                                                 
                                               
                                               ) 
                                             
                                           
                                         
                                          
                                         
                                           ( 
                                           
                                             φ 
                                             f 
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                     
                                       W 
                                       N 
                                     
                                      
                                     
                                       
                                         Z 
                                         N 
                                         H 
                                       
                                        
                                       
                                         ( 
                                         
                                           φ 
                                           f 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       r 
                                       
                                         D 
                                         , 
                                         
                                           2 
                                            
                                           n 
                                         
                                       
                                     
                                   
                                   ) 
                                 
                                 T 
                               
                               , 
                             
                           
                         
                         
                           
                             
                               
                                 ( 
                                 
                                   
                                      
                                     
                                       
                                         - 
                                         
                                           j 
                                            
                                           
                                             ( 
                                             
                                               
                                                 
                                                   ( 
                                                   
                                                     
                                                       2 
                                                        
                                                       n 
                                                     
                                                     + 
                                                     1 
                                                   
                                                   ) 
                                                 
                                                  
                                                 
                                                   ( 
                                                   
                                                     N 
                                                     + 
                                                     
                                                       P 
                                                       2 
                                                     
                                                   
                                                   ) 
                                                 
                                               
                                               + 
                                               
                                                 P 
                                                 2 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                        
                                       
                                         ( 
                                         
                                           φ 
                                           f 
                                         
                                         ) 
                                       
                                     
                                   
                                    
                                   
                                     W 
                                     N 
                                   
                                    
                                   
                                     
                                       Z 
                                       N 
                                       H 
                                     
                                      
                                     
                                       ( 
                                       
                                         φ 
                                         f 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     r 
                                     
                                       D 
                                       , 
                                       
                                         
                                           2 
                                            
                                           n 
                                         
                                         + 
                                         1 
                                       
                                     
                                   
                                 
                                 ) 
                               
                               H 
                             
                           
                         
                       
                       ] 
                     
                     T 
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         
                           
                               
                              
                             
                               [ 
                               
                                 
                                   
                                     
                                       
                                         α 
                                         n 
                                       
                                        
                                       
                                         
                                           Λ 
                                           ⋓ 
                                         
                                         
                                           1 
                                           , 
                                           D 
                                         
                                       
                                        
                                       
                                         Λ 
                                         
                                           S 
                                           , 
                                           1 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         
                                           
                                             γ 
                                             n 
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 Z 
                                                 N 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   
                                                     2 
                                                      
                                                     π 
                                                   
                                                   N 
                                                 
                                                 ) 
                                               
                                             
                                             ) 
                                           
                                         
                                         
                                           L 
                                           2 
                                         
                                       
                                        
                                       
                                         
                                           Λ 
                                           ⋓ 
                                         
                                         
                                           2 
                                           , 
                                           D 
                                         
                                       
                                        
                                       
                                         Λ 
                                         
                                           S 
                                           , 
                                           2 
                                         
                                         * 
                                       
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         - 
                                         
                                           γ 
                                           n 
                                           * 
                                         
                                       
                                        
                                       
                                         
                                           
                                              
                                             
                                               
                                                 - 
                                                 j 
                                               
                                                
                                               
                                                   
                                               
                                                
                                               
                                                 P 
                                                 1 
                                               
                                                
                                               
                                                 φ 
                                                 r 
                                               
                                             
                                           
                                            
                                           
                                             ( 
                                             
                                               
                                                 Z 
                                                 N 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   - 
                                                   
                                                     
                                                       2 
                                                        
                                                       π 
                                                     
                                                     N 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                             ) 
                                           
                                         
                                         
                                           L 
                                           2 
                                         
                                       
                                        
                                       
                                         
                                           Λ 
                                           ⋓ 
                                         
                                         
                                           2 
                                           , 
                                           D 
                                         
                                         * 
                                       
                                        
                                       
                                         Λ 
                                         
                                           S 
                                           , 
                                           2 
                                         
                                       
                                     
                                   
                                   
                                     
                                       
                                         α 
                                         n 
                                         * 
                                       
                                        
                                       
                                          
                                         
                                           
                                             - 
                                             j 
                                           
                                            
                                           
                                               
                                           
                                            
                                           
                                             P 
                                             1 
                                           
                                            
                                           
                                             φ 
                                             r 
                                           
                                         
                                       
                                        
                                       
                                         
                                           Λ 
                                           ⋓ 
                                         
                                         
                                           1 
                                           , 
                                           D 
                                         
                                         * 
                                       
                                        
                                       
                                         Λ 
                                         
                                           S 
                                           , 
                                           1 
                                         
                                         * 
                                       
                                     
                                   
                                 
                               
                               ] 
                             
                           
                            
                         
                         
                           
                             H 
                             ~ 
                           
                           F 
                         
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 x 
                                 
                                   2 
                                    
                                   n 
                                 
                               
                             
                           
                           
                             
                               
                                 x 
                                 
                                   
                                     2 
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                                 * 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                   
                 
               
             
             
               
                 
                     
                    
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 
                                    
                                   
                                     
                                       j 
                                        
                                       
                                         ( 
                                         
                                           
                                             2 
                                              
                                             
                                               n 
                                                
                                               
                                                 ( 
                                                 
                                                   N 
                                                   + 
                                                   
                                                     P 
                                                     2 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                           
                                           + 
                                           
                                             P 
                                             2 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         φ 
                                         f 
                                       
                                       ) 
                                     
                                   
                                 
                                  
                                 
                                   W 
                                   N 
                                 
                                  
                                 
                                   
                                     Z 
                                     N 
                                     H 
                                   
                                    
                                   
                                     ( 
                                     
                                       φ 
                                       f 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     p 
                                     ~ 
                                   
                                   
                                     D 
                                     , 
                                     
                                       2 
                                        
                                       n 
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                    
                                   
                                     
                                       - 
                                       
                                         j 
                                          
                                         
                                           ( 
                                           
                                             
                                               
                                                 ( 
                                                 
                                                   
                                                     2 
                                                      
                                                     n 
                                                   
                                                   + 
                                                   1 
                                                 
                                                 ) 
                                               
                                                
                                               
                                                 ( 
                                                 
                                                   N 
                                                   + 
                                                   
                                                     P 
                                                     2 
                                                   
                                                 
                                                 ) 
                                               
                                             
                                             + 
                                             
                                               P 
                                               2 
                                             
                                           
                                           ) 
                                         
                                       
                                     
                                      
                                     
                                       ( 
                                       
                                         φ 
                                         f 
                                       
                                       ) 
                                     
                                   
                                 
                                  
                                 
                                   W 
                                   N 
                                 
                                  
                                 
                                   
                                     Z 
                                     N 
                                     H 
                                   
                                    
                                   
                                     ( 
                                     
                                       φ 
                                       f 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     p 
                                     ~ 
                                   
                                   
                                     D 
                                     , 
                                     
                                       
                                         2 
                                          
                                         n 
                                       
                                       + 
                                       1 
                                     
                                   
                                 
                               
                             
                           
                         
                         ] 
                       
                        
                     
                     
                       a 
                       n 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               y 
               
                 D 
                 , 
                 n 
               
             
             = 
             
               
                 
                   
                     H 
                     ~ 
                   
                   F 
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           x 
                           
                             2 
                              
                             n 
                           
                         
                       
                     
                     
                       
                         
                           x 
                           
                             
                               2 
                                
                               n 
                             
                             + 
                             1 
                           
                           * 
                         
                       
                     
                   
                   ] 
                 
               
               + 
               
                 a 
                 n 
               
             
           
         
       
     
         [0135]    The scalars α n  and γ n  are defined as: 
         [0000]      α n =e j(2n+1)P     1     φ     r    
 
         [0000]      γ n =e −j((4n+2)(N+P)−2nP     1     −1)φ     r.    
 
         [0000]    where φ r  represents the ACFO φ r . The value of φ r  can be estimated at the relay using the ACFO {circumflex over (φ)} r  estimation step  410  and then forwarded to the destination. 
         [0136]    Estimates of x 2n  and x 2n+1  (denoted {tilde over (x)} 2n  and {tilde over (x)}* 2n+1 ) can thus be obtained using {tilde over (H)} F  such that 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       
                         x 
                         ~ 
                       
                       
                         2 
                          
                         n 
                       
                     
                   
                 
                 
                   
                     
                       
                         x 
                         ~ 
                       
                       
                         
                           2 
                            
                           n 
                         
                         + 
                         1 
                       
                       * 
                     
                   
                 
               
               ] 
             
             = 
             
               
                 
                   H 
                   ~ 
                 
                 F 
                 H 
               
                
               
                 
                   y 
                   
                     D 
                     , 
                     n 
                   
                 
                 . 
               
             
           
         
       
     
         [0137]    {tilde over (H)} F   H  represents the conjugate transpose of {tilde over (H)} F . {tilde over (H)} F  has the property that 
         [0000]        {tilde over (H)}   F   H   {tilde over (H)}   F   =I   2           (|{hacek over (Λ)} 1,D | 2 |Λ S,1 | 2 +|{hacek over (Λ)} 2,D | 2 |Λ S,2 | 2 ).
 
         [0000]    where           denotes performing a Kronecker product. {tilde over (H)} F  thus allows for a simple linear decoding at the destination. 
         [0138]    In step  330 , the relay station optionally performs ACFO φ r  compensation. The signal vectors {tilde over (r)} R,i,j  received over the span of 2 OFDM symbol durations j={2n,2n+ 1 } for antennae i=1, 2 are: 
         [0000]    
       
         
           
             
               [ 
               
                 
                   
                     
                       
                         r 
                         ~ 
                       
                       
                         R 
                         , 
                         1 
                         , 
                         
                           2 
                            
                           n 
                         
                       
                     
                   
                   
                     
                       
                         r 
                         ~ 
                       
                       
                         R 
                         , 
                         1 
                         , 
                         
                           
                             2 
                              
                             n 
                           
                           + 
                           1 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         r 
                         ~ 
                       
                       
                         R 
                         , 
                         2 
                         , 
                         
                           2 
                            
                           n 
                         
                       
                     
                   
                   
                     
                       
                         r 
                         ~ 
                       
                       
                         R 
                         , 
                         2 
                         , 
                         
                           
                             2 
                              
                             n 
                           
                           + 
                           1 
                         
                       
                     
                   
                 
               
               ] 
             
             = 
             
                 
               
                 
                   [ 
                   
                       
                   
                    
                   
                     
                       
                         
                           
                              
                             
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                         n 
                                          
                                         
                                           ( 
                                           
                                             N 
                                             + 
                                             P 
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       P 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 φ 
                                 r 
                               
                             
                           
                            
                           
                             
                               Z 
                               
                                 N 
                                 + 
                                 
                                   P 
                                   2 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 φ 
                                 r 
                               
                               ) 
                             
                           
                            
                           
                             
                               H 
                               ~ 
                             
                             
                               S 
                               , 
                               1 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               2 
                                
                               n 
                             
                           
                         
                       
                       
                         
                           
                              
                             
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             2 
                                              
                                             n 
                                           
                                           + 
                                           1 
                                         
                                         ) 
                                       
                                        
                                       
                                         ( 
                                         
                                           N 
                                           + 
                                           P 
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       P 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 φ 
                                 r 
                               
                             
                           
                            
                           
                             
                               Z 
                               
                                 N 
                                 + 
                                 
                                   P 
                                   2 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 φ 
                                 r 
                               
                               ) 
                             
                           
                            
                           
                             
                               H 
                               ~ 
                             
                             
                               S 
                               , 
                               1 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                              
                             
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                         n 
                                          
                                         
                                           ( 
                                           
                                             N 
                                             + 
                                             P 
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                     
                                       P 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 φ 
                                 r 
                               
                             
                           
                            
                           
                             
                               Z 
                               
                                 N 
                                 + 
                                 
                                   P 
                                   2 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 φ 
                                 r 
                               
                               ) 
                             
                           
                            
                           
                             
                               H 
                               ~ 
                             
                             
                               S 
                               , 
                               2 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               2 
                                
                               n 
                             
                           
                         
                       
                       
                         
                           
                              
                             
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             2 
                                              
                                             n 
                                           
                                           + 
                                           1 
                                         
                                         ) 
                                       
                                        
                                       
                                         ( 
                                         
                                           N 
                                           + 
                                           P 
                                         
                                         ) 
                                       
                                     
                                     + 
                                     
                                       P 
                                       1 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 φ 
                                 r 
                               
                             
                           
                            
                           
                             
                               Z 
                               
                                 N 
                                 + 
                                 
                                   P 
                                   2 
                                 
                               
                             
                              
                             
                               ( 
                               
                                 φ 
                                 r 
                               
                               ) 
                             
                           
                            
                           
                             
                               H 
                               ~ 
                             
                             
                               S 
                               , 
                               2 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                     
                   
                   ] 
                 
                 + 
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               v 
                               ~ 
                             
                             
                               R 
                               , 
                               1 
                               , 
                               
                                 2 
                                  
                                 n 
                               
                             
                           
                         
                         
                           
                             
                               v 
                               ~ 
                             
                             
                               R 
                               , 
                               1 
                               , 
                               
                                 
                                   2 
                                    
                                   n 
                                 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               v 
                               ~ 
                             
                             
                               R 
                               , 
                               2 
                               , 
                               
                                 2 
                                  
                                 n 
                               
                             
                           
                         
                         
                           
                             
                               v 
                               ~ 
                             
                             
                               R 
                               , 
                               2 
                               , 
                               
                                 
                                   2 
                                    
                                   n 
                                 
                                 + 
                                 1 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                   . 
                 
               
             
           
         
       
     
         [0139]    The ACFO φ r  is a known parameter obtained at the relay station during the ACFO {circumflex over (φ)} r  estimation step  410  and forms part of the channel state information (CSI). 
         [0140]    The effect of the ACFO φ r  can be removed for {tilde over (r)} R,1,2n  and {tilde over (r)} R,2,2n  by multiplying with the conjugate of e j(2n(N+P     1     )+P     1     )φ     r   Z N (φ r ). The effect of the ACFO φ r  can be removed for {tilde over (r)} R,1,2n+1  and {tilde over (r)} R,2,2n+1  by multiplying with the conjugate of e j((2n+1)(N+P     1     )+P     1     )φ     r   Z N (φ r ). 
         [0141]    In step  340 , the destination optionally performs ACFO φ d  compensation. The (N×1) received signal vectors at the destination in the presence of ACFO φ d  are given by
       CP Scheme 2:       
 
         [0000]    
       
         
           
             
               r 
               
                 D 
                 , 
                 
                   2 
                    
                   n 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     
                       Z 
                       N 
                     
                      
                     
                       ( 
                       
                         φ 
                         d 
                       
                       ) 
                     
                   
                    
                   
                     [ 
                     
                       
                         
                           H 
                           
                             1 
                             , 
                             D 
                           
                         
                          
                         
                           H 
                           
                             S 
                             , 
                             1 
                           
                         
                          
                         
                           W 
                           N 
                           H 
                         
                          
                         
                           x 
                           
                             2 
                              
                             n 
                           
                         
                       
                       + 
                       
                         
                           H 
                           
                             2 
                             , 
                             D 
                           
                         
                          
                         
                           G 
                           
                             S 
                             , 
                             2 
                           
                         
                          
                         
                           W 
                           N 
                           H 
                         
                          
                         
                           Tx 
                           
                             
                               2 
                                
                               n 
                             
                             + 
                             1 
                           
                           * 
                         
                       
                     
                     ] 
                   
                 
               
               + 
               
                 
                   p 
                   ~ 
                 
                 
                   D 
                   , 
                   
                     2 
                      
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 r 
                 
                   D 
                   , 
                   
                     
                       2 
                        
                       n 
                     
                     + 
                     1 
                   
                 
               
               = 
               
                 
                   
                     
                        
                       
                         
                           j 
                            
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       n 
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 P 
                                 2 
                               
                             
                             ) 
                           
                         
                          
                         
                           φ 
                           d 
                         
                       
                     
                     
                       2 
                     
                   
                    
                   
                     
                       
                         Z 
                         N 
                       
                        
                       
                         ( 
                         
                           φ 
                           d 
                         
                         ) 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             H 
                             
                               1 
                               , 
                               D 
                             
                           
                            
                           
                             H 
                             
                               S 
                               , 
                               1 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                         - 
                         
                           
                             H 
                             
                               2 
                               , 
                               D 
                             
                           
                            
                           
                             G 
                             
                               S 
                               , 
                               2 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             Tx 
                             
                               2 
                                
                               n 
                             
                             * 
                           
                         
                       
                       ] 
                     
                   
                 
                 + 
                 
                   
                     p 
                     ~ 
                   
                   
                     D 
                     , 
                     
                       
                         2 
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where the noise vectors are given by {tilde over (p)} D,2n  and {tilde over (p)} D,2n+1 . The ACFO φ d  is a known parameter obtained during the ACFO {circumflex over (φ)} d  estimation step  420  and forms part of the channel state information (CSI). 
         [0143]    The effect of the ACFO φ d  can be removed by multiplying r D,2n  and r D,2n+1  respectively with the conjugate of e j(2n(N+P     2     )+P     2     )φ     d   Z N (φ d ) and e j((2n+1)(N+P     2     )+P     2     )φ     d   Z N (φ d ). 
         [0144]    In step  350 , the destination optionally performs joint ACFO compensation. The (N×1) signal vectors received at the destination for the j={2n, 2n+1}-th OFDM symbol duration are given by 
         [0000]        r   D,2n   =e   j(2n(N+P     2     )+P     2     )(φ     d     +φ     r     )   Z   N (φ d +φ r )[α n   {hacek over (H)}   1,D   H   S,1   W   N   H   x   2n +γ n   {hacek over (H)}   2,D   G   S,2 ( W   N   H   x   2n+1 )*]
 
         [0000]        r   D,2n+1   =e   j((2n+1)(N+P     2     )+P     2     )(φ     d     +φ     r     )   Z   N (φ d +φ r )[α n   e   jP     1     φ     r     {hacek over (H)}   1,D   H   S,1   W   N   H   x   2n+1 −γ n   e   jP     1     φ     r     {hacek over (H)}   2,D   G   S,2 ( W   N   H   x   2n )*]+ {tilde over (p)}   D,2n+1,  
 
         [0000]    where the noise vectors are given by {tilde over (p)} D,2n  and {tilde over (p)} D,2n+1.    
         [0145]    The scalars α n  and γ n  are defined as follows: 
         [0000]      α n =e j(2n+1)P     1     φ     r    
 
         [0000]      γ n =e −j((4n+2)(N+P)−2nP     1     -1)φ     r.    
 
         [0146]    It can be seen that ACFO φ r  and ACFO φ d  are present in the signal vectors r D,2n  and r D,2n+1 . ACFO φ r  and ACFO φ d  can be added together and be represented by ACFO φ f  i.e. φ f =φ d +φ r . The ACFO φ f  is a known parameter obtained during the joint ACFO (φ r +φ d ) estimation step  430  and forms part of the channel state information (CSI). 
         [0147]    The effect of the ACFO φ f  can be removed by multiplying r D,2n  and r D,2n+1  respectively with the conjugates of e j(2n(N+P     2     )+P     2     )(φ     d     +φ     r     ) Z N (φ d +φ r ) and e j((2n+1)(N+P     2     )+P     2     )(φ     d     +φ     r     ) Z N (φ d +φ r ). 
         [0148]    The Cyclic Prefix Scheme 2 may have the advantage that only analog domain processing has to be done at the relay, and only linear processing has to be done for maximum likelihood decoding at the destination for each carrier. 
         [0149]    The Cyclic Prefix Scheme 2 may also be advantageous over the prior art because the relay station does not have to perform time sharing between different sources and destinations. In the prior art, the relay station will need to estimate and compensate the ACFO for each source-and-destination pair as joint ACFO compensation at the destination is not done. This can complicate the relay station hardware and could increase the computational complexity required at the relay station. 
         [0150]    Channel Estimation 
         [0151]    Channel estimation is performed before actual information is transmitted according to the method shown in the flow-chart of  FIG. 2  or  FIG. 3 . The purpose of channel estimation is to obtain a set of Channel State Information (CSI) parameters for use in decoding the signals received at the destination and optionally to compensate for Angular Carrier Frequency Offset (ACFO) at either the relay station or the destination. 
         [0152]    Channel estimation according to the example embodiment will be described next with reference to  FIG. 4 . 
         [0153]    In step  402 , a simple pair of pilot symbols (or training symbols) is used to estimate the product channel. The pair of pilot symbols comprising of 
         [0000]      [x 2n x 2n+1]   
         [0000]    is transmitted over two symbol intervals from the source. The pilot symbols are defined as 
         [0000]      x 2n =a 
         [0000]    
       
      
       X 
       2n+1 
       =−a  
      
     
         [0000]    where a is a (N×1) vector such that |a(i)|=1 for i=0, 1, . . . , N−1 and a has low peak to average power ratio (PAPR). n denotes the data transmission cycle index. 
         [0154]    ACFO {circumflex over (φ)} r  Estimation at the Relay Station 
         [0155]    In step  410 , ACFO {circumflex over (φ)} r  estimation is performed at the relay station for either of Cyclic Prefix Schemes 1 or 2. Given the known values of the pilot symbols [x 2n  x 2n+1 ], the ACFO at the relay {circumflex over (φ)} r  can be easily estimated as 
         [0000]    
       
         
           
             CP 
              
             
                 
             
              
             Scheme 
              
             
                 
             
              
             1 
              
             
               : 
             
           
         
       
       
         
           
             
               
                 φ 
                 ^ 
               
               r 
             
             = 
             
               
                 
                   ∠ 
                    
                   
                     ( 
                     
                       
                         - 
                         
                           r 
                           
                             R 
                             , 
                             1 
                             , 
                             
                               2 
                                
                               n 
                             
                           
                           H 
                         
                       
                        
                       
                         r 
                         
                           R 
                           , 
                           1 
                           , 
                           
                             
                               2 
                                
                               n 
                             
                             + 
                             1 
                           
                         
                       
                     
                     ) 
                   
                 
                 + 
                 
                   ∠ 
                    
                   
                     ( 
                     
                       
                         - 
                         
                           r 
                           
                             R 
                             , 
                             2 
                             , 
                             
                               2 
                                
                               n 
                             
                           
                           H 
                         
                       
                        
                       
                         r 
                         
                           R 
                           , 
                           2 
                           , 
                           
                             
                               2 
                                
                               n 
                             
                             + 
                             1 
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 2 
                  
                 
                   ( 
                   
                     N 
                     + 
                     P 
                   
                   ) 
                 
               
             
           
         
       
       
         
           
             CP 
              
             
                 
             
              
             Scheme 
              
             
                 
             
              
             2 
              
             
               : 
             
           
         
       
       
         
           
             
               
                 φ 
                 ^ 
               
               r 
             
             = 
             
               
                 
                   
                     ∠ 
                      
                     
                       ( 
                       
                         
                           - 
                           
                             
                               r 
                               ~ 
                             
                             
                               R 
                               , 
                               1 
                               , 
                               
                                 2 
                                  
                                 n 
                               
                             
                             H 
                           
                         
                          
                         
                           
                             r 
                             ~ 
                           
                           
                             R 
                             , 
                             1 
                             , 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                       ) 
                     
                   
                   + 
                   
                     ∠ 
                      
                     
                       ( 
                       
                         
                           - 
                           
                             
                               r 
                               ~ 
                             
                             
                               R 
                               , 
                               2 
                               , 
                               
                                 2 
                                  
                                 n 
                               
                             
                             H 
                           
                         
                          
                         
                           
                             r 
                             ~ 
                           
                           
                             R 
                             , 
                             2 
                             , 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                       ) 
                     
                   
                 
                 
                   2 
                    
                   
                     ( 
                     
                       N 
                       + 
                       P 
                     
                     ) 
                   
                 
               
               . 
             
           
         
       
     
         [0156]    The operator ∠(.) returns the phase of (.) and hence the returned scalar is within the interval [−π,π). It is to be noted that 
         [0000]    
       
         
           
             
               
                 φ 
                 ^ 
               
               r 
             
             = 
             
               
                 2 
                  
                 
                   πε 
                   r 
                 
               
               N 
             
           
         
       
     
         [0000]    and hence the maximum normalized CFO that can be estimated is given by the bounds 
         [0000]    
       
         
           
             
                
               
                 
                   ε 
                   ^ 
                 
                 r 
               
                
             
             ≤ 
             
               0.5 
                
               
                 N 
                 
                   N 
                   + 
                   P 
                 
               
             
             &lt; 
             0.5 
           
         
       
     
         [0157]    The above limitation can be overcome by designing suitable pilot symbols. The estimated ACFO value {circumflex over (φ)} r  can be used by the relay to do the compensation during the transmission of actual information. 
         [0158]    Cyclic Prefix Scheme 2 may have the advantage that ACFO compensation at relay is not necessary because joint ACFO compensation can be performed at the destination. 
         [0159]    ACFO {circumflex over (φ)} d  Estimation at the Destination 
         [0160]    In step  420 , ACFO {circumflex over (φ)} d  estimation is performed at the destination station for either of Cyclic Prefix Schemes 1 or 2. 
         [0161]    For Cyclic Prefix Scheme 1, the (N×1) received signal vectors at the destination in the presence of ACFO φ d  are given by
       CP Scheme 1       
 
         [0000]    
       
         
           
             
               r 
               
                 D 
                 , 
                 
                   2 
                    
                   
                       
                   
                    
                   n 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     
                       Z 
                       N 
                     
                      
                     
                       ( 
                       
                         φ 
                         d 
                       
                       ) 
                     
                   
                    
                   
                     [ 
                     
                       
                         
                           H 
                           
                             1 
                             , 
                             D 
                           
                         
                          
                         
                           H 
                           
                             S 
                             , 
                             1 
                           
                         
                          
                         
                           W 
                           N 
                           H 
                         
                          
                         
                           x 
                           
                             2 
                              
                             
                                 
                             
                              
                             n 
                           
                         
                       
                       + 
                       
                         
                           H 
                           
                             2 
                             , 
                             D 
                           
                         
                          
                         
                           ζ 
                            
                           
                             ( 
                             
                               H 
                               
                                 S 
                                 , 
                                 2 
                               
                               * 
                             
                             ) 
                           
                         
                          
                         
                           W 
                           N 
                           H 
                         
                          
                         
                           Tx 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               n 
                             
                             + 
                             1 
                           
                           * 
                         
                       
                     
                     ] 
                   
                 
               
               + 
               
                 p 
                 
                   D 
                   , 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 r 
                 
                   D 
                   , 
                   
                     
                       2 
                        
                       
                           
                       
                        
                       n 
                     
                     + 
                     1 
                   
                 
               
               = 
               
                 
                   
                     
                        
                       
                         
                           j 
                            
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       n 
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 P 
                                 2 
                               
                             
                             ) 
                           
                         
                          
                         
                           φ 
                           d 
                         
                       
                     
                     
                       2 
                     
                   
                    
                   
                     
                       
                         Z 
                         N 
                       
                        
                       
                         ( 
                         
                           φ 
                           d 
                         
                         ) 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             H 
                             
                               1 
                               , 
                               D 
                             
                           
                            
                           
                             H 
                             
                               S 
                               , 
                               1 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                         - 
                         
                           
                             H 
                             
                               2 
                               , 
                               D 
                             
                           
                            
                           
                             ζ 
                              
                             
                               ( 
                               
                                 H 
                                 
                                   S 
                                   , 
                                   2 
                                 
                                 * 
                               
                               ) 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             Tx 
                             
                               2 
                                
                               
                                   
                               
                                
                               n 
                             
                             * 
                           
                         
                       
                       ] 
                     
                   
                 
                 + 
                 
                   p 
                   
                     D 
                     , 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where the noise vectors are given by 
         [0000]    
       
         
           
             
               p 
               
                 D 
                 , 
                 
                   2 
                    
                   
                       
                   
                    
                   n 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     
                       Z 
                       N 
                     
                      
                     
                       ( 
                       
                         φ 
                         d 
                       
                       ) 
                     
                   
                    
                   
                     [ 
                     
                       
                         
                           H 
                           
                             1 
                             , 
                             D 
                           
                         
                          
                         
                           v 
                           
                             R 
                             , 
                             1 
                             , 
                             
                               2 
                                
                               
                                   
                               
                                
                               n 
                             
                           
                         
                       
                       + 
                       
                         
                           H 
                           
                             2 
                             , 
                             D 
                           
                         
                          
                         
                           ζ 
                            
                           
                             ( 
                             
                               v 
                               
                                 R 
                                 , 
                                 2 
                                 , 
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                               
                               * 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                 
               
               + 
               
                 v 
                 
                   D 
                   , 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               p 
               
                 D 
                 , 
                 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                   + 
                   1 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   N 
                                   + 
                                   
                                     P 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     
                       Z 
                       N 
                     
                      
                     
                       ( 
                       
                         φ 
                         d 
                       
                       ) 
                     
                   
                    
                   
                     [ 
                     
                       
                         
                           H 
                           
                             1 
                             , 
                             D 
                           
                         
                          
                         
                           v 
                           
                             R 
                             , 
                             1 
                             , 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                       - 
                       
                         
                           H 
                           
                             2 
                             , 
                             D 
                           
                         
                          
                         
                           ζ 
                            
                           
                             ( 
                             
                               v 
                               
                                 R 
                                 , 
                                 2 
                                 , 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   n 
                                 
                               
                               * 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                 
               
               + 
               
                 
                   v 
                   
                     D 
                     , 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
                 . 
               
             
           
         
       
     
         [0163]    Similarly, the (N×1) received signal vectors at the destination for CP Scheme 2 are given by
       CP Scheme 2:       
 
         [0000]    
       
         
           
             
               r 
               
                 D 
                 , 
                 
                   2 
                    
                   
                       
                   
                    
                   n 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     
                       Z 
                       N 
                     
                      
                     
                       ( 
                       
                         φ 
                         d 
                       
                       ) 
                     
                   
                    
                   
                     [ 
                     
                       
                         
                           H 
                           
                             1 
                             , 
                             D 
                           
                         
                          
                         
                           H 
                           
                             S 
                             , 
                             1 
                           
                         
                          
                         
                           W 
                           N 
                           H 
                         
                          
                         
                           x 
                           
                             2 
                              
                             
                                 
                             
                              
                             n 
                           
                         
                       
                       + 
                       
                         
                           H 
                           
                             2 
                             , 
                             D 
                           
                         
                          
                         
                           G 
                           
                             S 
                             , 
                             2 
                           
                         
                          
                         
                           W 
                           N 
                           H 
                         
                          
                         
                           Tx 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               n 
                             
                             + 
                             1 
                           
                           * 
                         
                       
                     
                     ] 
                   
                 
               
               + 
               
                 p 
                 
                   D 
                   , 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 r 
                 
                   D 
                   , 
                   
                     
                       2 
                        
                       
                           
                       
                        
                       n 
                     
                     + 
                     1 
                   
                 
               
               = 
               
                 
                   
                     
                        
                       
                         
                           j 
                            
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       n 
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 P 
                                 2 
                               
                             
                             ) 
                           
                         
                          
                         
                           φ 
                           d 
                         
                       
                     
                     
                       2 
                     
                   
                    
                   
                     
                       
                         Z 
                         N 
                       
                        
                       
                         ( 
                         
                           φ 
                           d 
                         
                         ) 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             H 
                             
                               1 
                               , 
                               D 
                             
                           
                            
                           
                             H 
                             
                               S 
                               , 
                               1 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             x 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                         - 
                         
                           
                             H 
                             
                               2 
                               , 
                               D 
                             
                           
                            
                           
                             G 
                             
                               S 
                               , 
                               2 
                             
                           
                            
                           
                             W 
                             N 
                             H 
                           
                            
                           
                             Tx 
                             
                               2 
                                
                               
                                   
                               
                                
                               n 
                             
                             * 
                           
                         
                       
                       ] 
                     
                   
                 
                 + 
                 
                   p 
                   
                     D 
                     , 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where the noise vectors are given by 
         [0000]    
       
         
           
             
               
                 p 
                 ~ 
               
               
                 D 
                 , 
                 
                   2 
                    
                   
                       
                   
                    
                   n 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     Z 
                     N 
                   
                    
                   
                     ( 
                     
                       φ 
                       d 
                     
                     ) 
                   
                 
                  
                 
                   
                     R 
                     cp 
                   
                    
                   
                     [ 
                     
                       
                         
                           U 
                           
                             1 
                             , 
                             D 
                           
                         
                          
                         
                           
                             v 
                             ~ 
                           
                           
                             R 
                             , 
                             1 
                             , 
                             
                               2 
                                
                               
                                   
                               
                                
                               n 
                             
                           
                         
                       
                       + 
                       
                         
                           U 
                           
                             2 
                             , 
                             D 
                           
                         
                          
                         
                           ζ 
                            
                           
                             ( 
                             
                               
                                 v 
                                 ~ 
                               
                               
                                 R 
                                 , 
                                 2 
                                 , 
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                               
                               * 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                 
               
               + 
               
                 v 
                 
                   D 
                   , 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 p 
                 ~ 
               
               
                 D 
                 , 
                 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                   + 
                   1 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   N 
                                   + 
                                   
                                     P 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     Z 
                     N 
                   
                    
                   
                     ( 
                     
                       φ 
                       d 
                     
                     ) 
                   
                 
                  
                 
                   
                     R 
                     cp 
                   
                    
                   
                     [ 
                     
                       
                         
                           U 
                           
                             1 
                             , 
                             D 
                           
                         
                          
                         
                           
                             v 
                             ~ 
                           
                           
                             R 
                             , 
                             1 
                             , 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                         
                       
                       - 
                       
                         
                           H 
                           
                             2 
                             , 
                             D 
                           
                         
                          
                         
                           ζ 
                            
                           
                             ( 
                             
                               
                                 v 
                                 ~ 
                               
                               
                                 R 
                                 , 
                                 2 
                                 , 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   n 
                                 
                               
                               * 
                             
                             ) 
                           
                         
                       
                     
                     ] 
                   
                 
               
               + 
               
                 
                   v 
                   
                     D 
                     , 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
                 . 
               
             
           
         
       
     
         [0165]    The following estimation applies for either of Cyclic Prefix Schemes 1 or 2. Using the pilot symbols of x 2n =a and x 2n =−a that were transmitted by the source, it is assumed that W N   H a=b. The (N×N) matrix W N   H  is the inverse discrete Fourier transform (IDFT) matrix. Therefore, vector H 1,D H S,1 W N   H x 2n  can be expressed as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       H 
                       
                         1 
                         , 
                         D 
                       
                     
                      
                     
                       H 
                       
                         S 
                         , 
                         1 
                       
                     
                      
                     
                       W 
                       N 
                       H 
                     
                      
                     
                       x 
                       
                         2 
                          
                         n 
                       
                     
                   
                   = 
                     
                    
                   
                     
                       H 
                       
                         1 
                         , 
                         D 
                       
                     
                      
                     
                       H 
                       
                         S 
                         , 
                         1 
                       
                     
                      
                     b 
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       A 
                        
                       
                           
                       
                        
                       
                         h 
                         P 
                       
                     
                   
                   , 
                 
               
             
           
         
       
     
         [0000]    where the (N×(P+1)) Toeplitz matrix A is given by 
         [0000]      A=[[b ] (0) ,[b] (1) , . . . ,[b] (P)]   
         [0000]    and the ((P+1)×1) vector hp is 
         [0000]    
       
      
       h 
       P 
       =h 
       1,D 
       *h 
       S,1  
      
     
         [0000]    with * denoting convolution operation. Let us assume that d=W N   H Ta*. 
         [0166]    Vector H 2,D  ζ(H* S,2 )d can be written as 
         [0000]        H   2,D ζ( H*   S,2 ) d=B{tilde over (h)}   P ,
 
         [0000]    where the (N×(P+1)) Toeplitz matrix B is given by 
         [0000]      B=[[f] (−L     1     +1) ,[f](−L   1     +2) , . . . ,[f] (−L     1     +P+1)] 
 
         [0000]    with f=ζ(d) and the ((P+1)×1) vector {tilde over (h)} P  is 
         [0000]        {tilde over (h)}   P   =h   2,D *ζ( h*   S,2 ).
 
         [0167]    Similarly, vector H 2,D G S,2 d can be expressed as 
         [0000]      H 2,D G S,2 d=C{tilde over (h)} P , 
         [0000]    where the (N×(P+1)) Toeplitz matrix C is given by 
         [0000]      c=[[f] (−2L     2     +1) ,[f] (−2L     2     +2) , . . . ,[f] (−2L     2     +P+1)].    
         [0168]    Substituting the above results into r D,2n  and r D,2n+1  and bearing in mind that r D,2n  and r D,2n+1  are the received signal vectors of the pilot symbols at the destination, r D,2n  and r D,2n+1  can be rewritten as
       CP Scheme 1:       
 
         [0000]    
       
         
           
             
               r 
               
                 D 
                 , 
                 
                   2 
                    
                   
                       
                   
                    
                   n 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     
                       Z 
                       N 
                     
                      
                     
                       ( 
                       
                         φ 
                         d 
                       
                       ) 
                     
                   
                    
                   
                     [ 
                     
                       A 
                       , 
                       B 
                     
                     ] 
                   
                 
                  
                 
                   h 
                   F 
                 
               
               + 
               
                 p 
                 
                   D 
                   , 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               r 
               
                 D 
                 , 
                 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                   + 
                   1 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               
                                 ( 
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     n 
                                   
                                   + 
                                   1 
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   N 
                                   + 
                                   
                                     P 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     
                       Z 
                       N 
                     
                      
                     
                       ( 
                       
                         φ 
                         d 
                       
                       ) 
                     
                   
                    
                   
                     [ 
                     
                       
                         - 
                         A 
                       
                       , 
                       
                         - 
                         B 
                       
                     
                     ] 
                   
                 
                  
                 
                   h 
                   F 
                 
               
               + 
               
                 p 
                 
                   D 
                   , 
                   
                     
                       2 
                        
                       
                           
                       
                        
                       n 
                     
                     + 
                     1 
                   
                 
               
             
           
         
       
       
         
           
             CP Scheme 2: 
           
         
       
     
         [0000]    
       
         
           
             
               r 
               
                 D 
                 , 
                 
                   2 
                    
                   
                       
                   
                    
                   n 
                 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   
                     
                       Z 
                       N 
                     
                      
                     
                       ( 
                       
                         φ 
                         d 
                       
                       ) 
                     
                   
                    
                   
                     [ 
                     
                       A 
                       , 
                       C 
                     
                     ] 
                   
                 
                  
                 
                   h 
                   F 
                 
               
               + 
               
                 
                   p 
                   ~ 
                 
                 
                   D 
                   , 
                   
                     2 
                      
                     
                         
                     
                      
                     n 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 r 
                 
                   D 
                   , 
                   
                     
                       2 
                        
                       
                           
                       
                        
                       n 
                     
                     + 
                     1 
                   
                 
               
               = 
               
                 
                   
                     
                        
                       
                         
                           j 
                            
                           
                             ( 
                             
                               
                                 
                                   ( 
                                   
                                     
                                       2 
                                        
                                       
                                           
                                       
                                        
                                       n 
                                     
                                     + 
                                     1 
                                   
                                   ) 
                                 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                               + 
                               
                                 P 
                                 2 
                               
                             
                             ) 
                           
                         
                          
                         
                           φ 
                           d 
                         
                       
                     
                     
                       2 
                     
                   
                    
                   
                     
                       
                         Z 
                         N 
                       
                        
                       
                         ( 
                         
                           φ 
                           d 
                         
                         ) 
                       
                     
                      
                     
                       [ 
                       
                         
                           - 
                           A 
                         
                         , 
                         
                           - 
                           C 
                         
                       
                       ] 
                     
                   
                    
                   
                     h 
                     F 
                   
                 
                 + 
                 
                   
                     p 
                     ~ 
                   
                   
                     D 
                     , 
                     
                       
                         2 
                          
                         
                             
                         
                          
                         n 
                       
                       + 
                       1 
                     
                   
                 
               
             
             , 
           
         
       
     
         [0000]    where the (2(P+1)×1) vector h F    
         [0000]      h F =[h P   T , {tilde over (h)} P   T ] T . 
         [0171]    Let q n  be a (2N×1) vector constructed as 
         [0000]    
       
         
           
             
               
                 
                   
                     q 
                     n 
                   
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           r 
                           
                             D 
                             , 
                             
                               2 
                                
                               n 
                             
                           
                           T 
                         
                         , 
                         
                           r 
                           
                             D 
                             , 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                           T 
                         
                       
                       ] 
                     
                     T 
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       
                         
                           F 
                            
                           
                             ( 
                             
                               φ 
                               d 
                             
                             ) 
                           
                         
                          
                         
                           h 
                           F 
                         
                       
                       + 
                       
                         e 
                         n 
                       
                     
                   
                   , 
                 
               
             
           
         
       
     
         [0000]    where e n =[p D,2n   T , p D,2n+1   T ] T  for CP Scheme 1 and e n =[{tilde over (p)} D,2n   T , {tilde over (p)} D,2n+1   T ] T  for CP Scheme 2. The (2N×2(P+1)) matrix F(φ d ) is given by
       CP Scheme 1       
 
         [0000]    
       
         
           
             
               F 
                
               
                 ( 
                 
                   φ 
                   d 
                 
                 ) 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         d 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           
                             Z 
                             N 
                           
                            
                           
                             ( 
                             
                               φ 
                               d 
                             
                             ) 
                           
                         
                       
                       
                         
                           0 
                           
                             N 
                             × 
                             N 
                           
                         
                       
                     
                     
                       
                         
                           0 
                           
                             N 
                             × 
                             N 
                           
                         
                       
                       
                         
                           
                              
                             
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 φ 
                                 d 
                               
                             
                           
                            
                           
                             
                               Z 
                               N 
                             
                              
                             
                               ( 
                               
                                 φ 
                                 d 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ] 
                 
               
                
               
                 [ 
                 
                   
                     
                       A 
                     
                     
                       
                         - 
                         B 
                       
                     
                   
                   
                     
                       
                         - 
                         A 
                       
                     
                     
                       
                         - 
                         B 
                       
                     
                   
                 
                 ] 
               
             
           
         
       
       
         
           
             CP Scheme 2 
           
         
       
     
         [0000]    
       
         
           
             
               F 
                
               
                 ( 
                 
                   φ 
                   d 
                 
                 ) 
               
             
             = 
             
               
                 
                   
                     
                        
                       
                         
                           j 
                            
                           
                             ( 
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   n 
                                    
                                   
                                     ( 
                                     
                                       N 
                                       + 
                                       
                                         P 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                               
                               + 
                               
                                 P 
                                 2 
                               
                             
                             ) 
                           
                         
                          
                         
                           φ 
                           d 
                         
                       
                     
                     
                       2 
                     
                   
                    
                   
                     [ 
                     
                       
                         
                           
                             
                               Z 
                               N 
                             
                              
                             
                               ( 
                               
                                 φ 
                                 d 
                               
                               ) 
                             
                           
                         
                         
                           
                             0 
                             
                               N 
                               × 
                               N 
                             
                           
                         
                       
                       
                         
                           
                             0 
                             
                               N 
                               × 
                               N 
                             
                           
                         
                         
                           
                             
                                
                               
                                 
                                   j 
                                    
                                   
                                     ( 
                                     
                                       N 
                                       + 
                                       
                                         P 
                                         2 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   φ 
                                   d 
                                 
                               
                             
                              
                             
                               
                                 Z 
                                 N 
                               
                                
                               
                                 ( 
                                 
                                   φ 
                                   d 
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     ] 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         A 
                       
                       
                         
                           - 
                           C 
                         
                       
                     
                     
                       
                         
                           - 
                           A 
                         
                       
                       
                         
                           - 
                           C 
                         
                       
                     
                   
                   ] 
                 
               
               . 
             
           
         
       
     
         [0174]    Using q n , a maximum likelihood (ML) based ACFO and CIR estimator can be formulated. The ACFO is estimated as 
         [0000]    
       
         
           
             
               
                 φ 
                 ^ 
               
               d 
             
             = 
             
               arg 
                
               
                 
                   min 
                   φ 
                 
                  
                 
                   
                     ( 
                     
                       
                         
                           q 
                           n 
                           H 
                         
                          
                         
                           [ 
                           
                             
                               I 
                               
                                 2 
                                  
                                 N 
                               
                             
                             - 
                             
                               
                                 F 
                                  
                                 
                                   ( 
                                   φ 
                                   ) 
                                 
                               
                                
                               
                                 
                                   ( 
                                   
                                     
                                       
                                         F 
                                         H 
                                       
                                        
                                       
                                         ( 
                                         φ 
                                         ) 
                                       
                                     
                                      
                                     
                                       F 
                                        
                                       
                                         ( 
                                         φ 
                                         ) 
                                       
                                     
                                   
                                   ) 
                                 
                                 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   F 
                                   H 
                                 
                                  
                                 
                                   ( 
                                   φ 
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         q 
                         n 
                       
                     
                     ) 
                   
                   . 
                 
               
             
           
         
       
     
         [0000]    {circumflex over (φ)} d  can be estimated using a grid search. 
         [0175]    It is to be noted that, for the pilot symbols used in step  402 , the matrix F H (φ)F(φ) is given by 
         [0000]        F   H (φ) F (φ)= NI   2(P+1).  
 
         [0176]    This implies that the inverse of F H (φ)F(φ) need not be computed for every search point. 
         [0177]    Hence the ACFO {circumflex over (φ)} d  can be estimated as 
         [0000]    
       
         
           
             
               
                 φ 
                 ^ 
               
               d 
             
             = 
             
               arg 
                
               
                 
                   min 
                   φ 
                 
                  
                 
                   
                     ( 
                     
                       
                         
                           q 
                           n 
                           H 
                         
                          
                         
                           [ 
                           
                             
                               I 
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 N 
                               
                             
                             - 
                             
                               
                                 1 
                                 N 
                               
                                
                               
                                 F 
                                  
                                 
                                   ( 
                                   φ 
                                   ) 
                                 
                               
                                
                               
                                 
                                   F 
                                   H 
                                 
                                  
                                 
                                   ( 
                                   φ 
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                       
                        
                       
                         q 
                         n 
                       
                     
                     ) 
                   
                   . 
                 
               
             
           
         
       
     
         [0178]    It can be seen that the ACFO {circumflex over (φ)} d  estimation at the destination is not computationally demanding as no inverse matrix computation needs to be done for each search point. 
         [0179]    Once the ACFO {circumflex over (φ)} d  is estimated, vector h F  can be estimated as 
         [0000]    
       
         
           
             
               
                 h 
                 ^ 
               
               F 
             
             = 
             
               
                 1 
                 N 
               
                
               
                 F 
                  
                 
                   ( 
                   
                     
                       φ 
                       ^ 
                     
                     d 
                   
                   ) 
                 
               
                
               
                 
                   F 
                   H 
                 
                  
                 
                   ( 
                   
                     
                       φ 
                       ^ 
                     
                     d 
                   
                   ) 
                 
               
                
               
                 
                   q 
                   n 
                 
                 . 
               
             
           
         
       
     
         [0180]    Joint ACFO (φ r +φ d ) Estimation at the Destination 
         [0181]    In step  430 , joint ACFO (φ r +φ d ) estimation is performed at the destination for the Cyclic Prefix Scheme 2. The (N×1) received signal vectors at destination (without ACFO compensation at relay) can be written as follows: 
         [0000]        r   D,2n   =e   j(2n(N+P     2     )+P     2     )(φ     d     +φ     r     )   Z   n (φ d +φ r )[α n   {hacek over (H)}   1,D   H   S,1   W   N   H   x   2n +γ n   {hacek over (H)}   2,D   G   S,2 ( W   N   H   x   2n+1 )*]
 
         [0000]        r   D,2n+1   =e   j((2n+1)(N+P     2     )+P     2     )(φ     d     +φ     r     )   Z   N (φ d +φ r )[α n   e   jP     1     φ     r     {hacek over (H)}   1,D   H   S,1   W   N   H   x   2n+1 −γ n   e   jP     1     φ     r     {hacek over (H)}   2,D   G   S,2 ( W   N   H   x   2n )*] {tilde over (p)}   D,2n+1,  
 
         [0000]    where {hacek over (H)} 1,D  and {hacek over (H)} 2,D  respectively are (N×N) circulant matrix with Z N (−φ r )[h 1,D   T ,0 1×(N-L     2     ) ] T  and Z N (−φ r )[h 2,D   T , 0 1×(N-L     2     )]   T  as their first column vectors, and scalars α n  and γ n  are defined as follows: 
         [0000]      α n =e j(2n+1)P     1     φ     r    
 
         [0000]      γ n =e j((4n+2)(N+P)−2nP     1     −1)φ     r.    
 
         [0182]    It is defined that φ f =φ d +φ r . We have 
         [0000]    
       
         
           
             
               
                 
                   
                     q 
                     n 
                   
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                           r 
                           
                             D 
                             , 
                             
                               2 
                                
                               n 
                             
                           
                           T 
                         
                         , 
                         
                           r 
                           
                             D 
                             , 
                             
                               
                                 2 
                                  
                                 n 
                               
                               + 
                               1 
                             
                           
                           T 
                         
                       
                       ] 
                     
                     T 
                   
                 
               
             
             
               
                 
                   
                     = 
                       
                      
                     
                       
                         
                           
                             F 
                             ~ 
                           
                            
                           
                             ( 
                             
                               
                                 φ 
                                 f 
                               
                               , 
                               
                                 φ 
                                 r 
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             h 
                             ~ 
                           
                           F 
                         
                       
                       + 
                       
                         e 
                         n 
                       
                     
                   
                   , 
                 
               
             
           
         
       
     
         [0000]    where the (2N×2(P+1)) matrix {tilde over (F)}(φ f , φ r ) is given by 
         [0000]    
       
         
           
             
               
                 F 
                 ~ 
               
                
               
                 ( 
                 
                   
                     φ 
                     f 
                   
                   , 
                   
                     φ 
                     r 
                   
                 
                 ) 
               
             
             = 
             
               
                 
                   
                      
                     
                       
                         j 
                          
                         
                           ( 
                           
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 n 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                             
                             + 
                             
                               P 
                               2 
                             
                           
                           ) 
                         
                       
                        
                       
                         φ 
                         f 
                       
                     
                   
                   
                     2 
                   
                 
                  
                 
                   [ 
                   
                     
                       
                         
                           
                             Z 
                             N 
                           
                            
                           
                             ( 
                             
                               φ 
                               f 
                             
                             ) 
                           
                         
                       
                       
                         
                           0 
                           
                             N 
                             × 
                             N 
                           
                         
                       
                     
                     
                       
                         
                           0 
                           
                             N 
                             × 
                             N 
                           
                         
                       
                       
                         
                           
                              
                             
                               
                                 j 
                                  
                                 
                                   ( 
                                   
                                     N 
                                     + 
                                     
                                       P 
                                       2 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                 φ 
                                 f 
                               
                             
                           
                            
                           
                             
                               Z 
                               N 
                             
                              
                             
                               ( 
                               
                                 φ 
                                 f 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ] 
                 
               
                
               
                   
                 
                   [ 
                   
                     
                       
                         
                           
                             α 
                             n 
                           
                            
                           A 
                         
                       
                       
                         
                           
                             - 
                             
                               γ 
                               n 
                             
                           
                            
                           C 
                         
                       
                     
                     
                       
                         
                           
                             - 
                             
                               α 
                               n 
                             
                           
                            
                           
                              
                             
                               j 
                                
                               
                                   
                               
                                
                               
                                 P 
                                 1 
                               
                                
                               
                                 φ 
                                 r 
                               
                             
                           
                            
                           A 
                         
                       
                       
                         
                           
                             - 
                             
                               γ 
                               n 
                             
                           
                            
                           
                              
                             
                               j 
                                
                               
                                   
                               
                                
                               
                                 P 
                                 1 
                               
                                
                               
                                 φ 
                                 r 
                               
                             
                           
                            
                           C 
                         
                       
                     
                   
                   ] 
                 
               
             
           
         
       
     
         [0000]    and {tilde over (h)} F  is a (2(P+1)×1) vector constructed as 
         [0000]        {tilde over (h)}   F =[( {hacek over (h)}   1,D   *h   S,1 ) T ,( {hacek over (h)}   2,D *ζ( h*   S,2 )) T ] T  
 
         [0000]    with {hacek over (h)} 1,D =Z L     2   (−φ r )h 1,D  and {hacek over (h)} 2,D =Z L     2   (−φ r )h 2,D . It can be shown that 
         [0000]        {tilde over (F)}   H (φ f ,φ r )  {tilde over (F)}(φ   f ,φ r )= NI   2(P+1).  
 
         [0183]    The Maximum Likelihood estimate for φ f  is obtained as 
         [0000]    
       
         
           
             
               
                 φ 
                 ^ 
               
               f 
             
             = 
             
               arg 
                
               
                 
                   min 
                   φ 
                 
                  
                 
                   ( 
                   
                     
                       
                         q 
                         n 
                         H 
                       
                        
                       
                         [ 
                         
                           
                             I 
                             
                               2 
                                
                               
                                   
                               
                                
                               N 
                             
                           
                           - 
                           
                             
                               1 
                               N 
                             
                              
                             
                               
                                 F 
                                 ~ 
                               
                                
                               
                                 ( 
                                 
                                   φ 
                                   , 
                                   
                                     φ 
                                     r 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                               
                                 
                                   F 
                                   ~ 
                                 
                                 H 
                               
                                
                               
                                 ( 
                                 
                                   φ 
                                   , 
                                   
                                     φ 
                                     r 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                         ] 
                       
                     
                      
                     
                       q 
                       n 
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    and {tilde over (h)} F  is estimated as 
         [0000]    
       
         
           
             
               
                 
                   h 
                   ~ 
                 
                 ^ 
               
               F 
             
             = 
             
               
                 1 
                 N 
               
                
               
                 
                   F 
                   ~ 
                 
                  
                 
                   ( 
                   
                     
                       
                         φ 
                         ^ 
                       
                       f 
                     
                     , 
                     
                       φ 
                       r 
                     
                   
                   ) 
                 
               
                
               
                 
                   
                     F 
                     ~ 
                   
                   H 
                 
                  
                 
                   ( 
                   
                     
                       
                         φ 
                         ^ 
                       
                       f 
                     
                     , 
                     
                       φ 
                       r 
                     
                   
                   ) 
                 
               
                
               
                 
                   q 
                   n 
                 
                 . 
               
             
           
         
       
     
         [0184]    While example embodiments of the invention have been described in detail, many variations are possible within the scope of the invention as will be clear to a skilled reader. Variations can be made, for example to parameters such as the number of carriers, the cyclic prefix lengths P, P 1  and P 2 , the coding technique used at the relay station, or the way in which the product channel is estimated. Variations can also be made to the design of the pilot symbols used. Further, the CP schemes need not necessary use space-time coding and other forms of coding may be used.