Abstract:
A receiver for processing frequency division multiplexing (FDM) signals, the receiver includes a processor configured to: convert the FDM signals from at least two transmitters into frequency domain signals; determine a first component of the frequency domain signals, the first component of the frequency domain signals comprising a channel noise and a composite residual inter-carrier interference (ICI) contributed by the at least two transmitters; calculate a set of correlation values corresponding to the first component of the frequency domain signals; and process the first component of the frequency domain signals based on the set of correlation values.

Description:
RELATED APPLICATION 
       [0001]    This application claims priority to U.S. Provisional Application No. 61/761,611, filed Feb. 6, 2013, which is hereby incorporated by reference in its entirety. 
     
    
     BACKGROUND 
       [0002]    The present disclosure relates generally to a frequency-division multiplexing (FDM) communication. For example, the present disclosure relates to a device and a method for processing orthogonal frequency-division multiplexing (OFDM) signals in a cooperative communication system. 
         [0003]    A cooperative communication system may achieve spatial diversity gains by employing distributed multi-transmitters. Normally, every single distributed transmitter in the cooperative communication system may rarely have accurately aligned carrier frequency. Accordingly, multiple carrier frequency offsets (MCFOs) may occur due to a receiver may constantly have high relative velocity with respect to the distributed multi-transmitters. Moreover, Doppler shifts or Doppler spread in channel response, as well as uncorrected CFOs, may result in inter-carrier interference (ICI). The MCFOs and ICI may severely deteriorate the performance of a cooperative communication system using an orthogonal frequency-division multiplexing (OFDM) scheme. 
         [0004]    It may therefore be desirable to have a device and a method to mitigate the MCFOs and ICI in the cooperative OFDM communication system. 
       BRIEF SUMMARY 
       [0005]    A simplified summary is provided herein to help enable a basic or general understanding of various aspects of non-limiting embodiments that follow in the more detailed description and the accompanying drawings. This summary is not intended, however, as an extensive or exhaustive overview. Instead, the sole purpose of this summary is to present some concepts related to some exemplary non-limiting embodiments in a simplified form as a prelude to the more detailed description of the various embodiments that follow. 
         [0006]    Example embodiments may provide a receiver for processing frequency division multiplexing (FDM) signals, the receiver includes a processor configured to: convert the FDM signals from at least two transmitters into frequency domain signals; determine a first component of the frequency domain signals, the first component of the frequency domain signals comprising a channel noise and a composite residual inter-carrier interference (ICI) contributed by the at least two transmitters; calculate a set of correlation values corresponding to the first component of the frequency domain signals; and process the first component of the frequency domain signals based on the set of correlation values. 
         [0007]    Some example embodiments may provide a method for processing frequency-division multiplexing (FDM) signals, the method includes the steps of: receiving the FDM signals from at least two transmitters; converting the FDM signals to frequency domain signals; determining a first component of the frequency domain signals, the first component of the frequency domain signals comprising a channel noise and a composite residual inter-carrier interference (ICI) contributed by the at least two transmitters; calculating a set of correlation values corresponding to the first component of the frequency domain signals; and processing the first component of the frequency domain signals based on the set of correlation values. 
         [0008]    It is to be understood that both the foregoing general description and the following detailed description are exemplary and explanatory only and are not restrictive. Therefore, the disclosed subject matter should not be limited to any single embodiment, or group of embodiments described herein, but rather should be construed in breadth and scope in accordance with the appended claims. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0009]    The foregoing summary, as well as the following detailed description of the various embodiments, will be better understood when read in conjunction with the appended drawings. For the purpose of illustrating the various embodiments, there are shown in the drawings various examples. It should be understood, however, that the various embodiments are not limited to the precise arrangements and instrumentalities shown and that other similar embodiments can be used or modifications and additions can be made to the described embodiments for performing the same, similar, alternative, or substitute function of the disclosed subject matter without deviating therefrom. 
           [0010]    Numerous aspects, embodiments, objects and advantages of the present invention will be apparent upon consideration of the following detailed description, taken in conjunction with the accompanying drawings, in which like reference characters refer to like parts throughout, and in which: 
           [0011]      FIG. 1  is a block diagram of the baseband part of a cooperative orthogonal frequency-division multiplexing (OFDM) communication system in accordance with an example embodiment; 
           [0012]      FIG. 2  illustrates a channel matrix of a channel in the cooperative OFDM communication system illustrated in  FIG. 1  in accordance with an example embodiment; 
           [0013]      FIG. 3A  is a block diagram of the baseband part of a cooperative OFDM communication system in accordance with another example embodiment; 
           [0014]      FIG. 3B  illustrates channel matrices of the channels in the cooperative OFDM communication system illustrated in  FIG. 3A  in accordance with another example embodiment; 
           [0015]      FIG. 4A  is a block diagram of a device for performing the blockwise whitening process and a signal detector in the cooperative OFDM communication system illustrated in  FIG. 3A  in accordance with another example embodiment; 
           [0016]      FIG. 4B  illustrates the sub-vectors in the cooperative OFDM communication system illustrated in  FIG. 3A  in accordance with another example embodiment; 
           [0017]      FIG. 4C  illustrates a channel sounding method performed by the channel estimators illustrated in  FIG. 4A  in accordance with another example embodiment; 
           [0018]      FIG. 4D  illustrates the channel matrices of the channels in the cooperative OFDM communication system illustrated in  FIG. 3A  in accordance with another example embodiment; 
           [0019]      FIG. 4E  illustrates an operation for calculating composite residual ICI plus channel noise in the blockwise whitening process in accordance with another example embodiment; 
           [0020]      FIG. 4F  is a block diagram of a device for performing the blockwise whitening process and the signal detection in accordance with yet another example embodiment; 
           [0021]      FIG. 4G  is a block diagram of a device for performing the blockwise whitening process and a device for performing the signal detection in accordance with still another example embodiment; 
           [0022]      FIG. 5A  illustrates an Alamouti-type coding for the cooperative OFDM communication system illustrated in  FIG. 3A  in accordance with another example embodiment; 
           [0023]      FIG. 5B  illustrates carrier frequency offsets (CFOs) in the cooperative OFDM communication system illustrated in  FIG. 3A  in accordance with another example embodiment; 
           [0024]      FIG. 5C  illustrates sub-matrices of the channels as well as corresponding sub-vectors in the cooperative OFDM communication system illustrated in  FIG. 3A  in accordance with another example embodiment; 
           [0025]      FIG. 5D  illustrates CFOs in a cooperative OFDM communication system in accordance with still another example embodiment; 
           [0026]      FIG. 5E  illustrates the channel matrices of the channels in the cooperative OFDM communication system illustrated in  FIG. 5D  in accordance with still another example embodiment; and 
           [0027]      FIG. 5F  illustrates the sub-matrices of the channels as well as corresponding sub-vectors in the cooperative OFDM communication system illustrated in  FIG. 5D  in accordance with still another example embodiment. 
       
    
    
     DETAILED DESCRIPTION 
       [0028]    Reference will now be made in detail to the present examples of the various embodiments, examples of which are illustrated in the accompanying drawings. Wherever possible, the same reference numbers will be used throughout the drawings to refer to the same or like parts. 
         [0029]      FIG. 1  is a block diagram of the baseband part of a cooperative orthogonal frequency-division multiplexing (OFDM) communication system  1  in accordance with an example embodiment. Referring to  FIG. 1 , the cooperative OFDM communication system  1  may include a plurality of transmitters  10  and a receiver  20 . In this example embodiment, the number of the plurality of transmitters  10  may be denoted as N t , and the N t  transmitters  10  may be communicatively coupled to the receiver  20  through a plurality of channels  30  respectively. That is, a channel  30   n   t  of the channels  30  may correspond to the (n t -th) transmitter  10   n   t  of the N t  transmitters  10  (wherein 1≦n t ≦N t ), and the transmitter  10   n   t  may be communicatively coupled to the receiver  20  through the channel  30   n   t . 
         [0030]    The transmitters  10  may be configured to transmit signals to the receiver  20  through the channels  30  respectively. Specifically, the signal transmitted by the transmitter  10   n   t  may be denoted as x n   n     t    with discrete time index “n” and the signal x n   n     t    may be transmitted to the receiver  20  through the channel  30   n   t . The channel  30   n   t  may include a time-varying multipath fading channel, which may be characterized by a set of discrete-time complex gains denoted as {h n,l   n     t   } with “n” denoting the discrete time index and “l” denoting the channel path index. That is, h n,l   n     t    may direct to a complex gain of the l-th channel path at time n that corresponds to the transmitter  10   n   t . In one example embodiment of the present invention each of the channels  30  in the cooperative OFDM communication system  1  may be wide-sense stationary uncorrelated scattering (WSSUS) as characterized by the following equation: 
         [0000]        E[h   n,l   n     th     n−q,l−m   n     t     * ]=σ l,n     t     2 γ l   n     t   ( q )δ( m )  eq. (1)
 
         [0031]    In equation (1), the terms σ l,n     t     2 , γ l   n     t   (q) and δ(m) may be defined as the following: 
         [0032]    σ l,n     t     2  may denote the variance of the tap gain h l   n     t    of the l-th channel path of the channel  30   n   t , 
         [0033]    γ l   n     t   (q) may denote the normalized autocorrelation function of the tap gain h l   n     t    of the l-th channel path of the channel  30   n   t  with γ l   n     t   (0)=1, and 
         [0034]    δ(m) may denote the Kronecker delta function. 
         [0035]    Furthermore, the operation E[.] may denote expectation, and the superscript “*” may denote complex conjugation. 
         [0036]    Moreover, the l-th channel path of the channel  30   n   t  may have a normalized Doppler power spectral density (PSD) P l,n     t   (f), and the mentioned normalized autocorrelation function γ l   n     t   (q) may be expressed by the following equation: 
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         [0037]    In equation (2), the terms f d  may denote the peak Doppler frequency of all the channels  30 . 
         [0038]    In this example embodiment, different channel paths of each of the channels  30  may have arbitrary and different fading, thus each different channel path of each of the channels  30  may have a different normalized Doppler PSD P l,n     t   (f). In addition, the normalized Doppler PSD P l,n     t   (f) of each channel path of each of the channels  30  may be asymmetric about the zero frequency (e.g., f=0). 
         [0039]    On the other hand, regarding the receiver side, the receiver  20  may be configured to receive signals from all of the transmitters  10 . The signal received by the receiver  20  may be denoted as y n  with “n” denoting the discrete time index. The received signal y n  may include contributions from all the transmitted signals {x n   n     t}|     ∀n     t     ∈(1,2, . . . ,N     t     )  by all the transmitters  10 . Accordingly, the received signal y n  may be also defined as “composite received signal” (being composite of contributions from all the transmitted signals {x n   n     t}|     ∀n     t     ∈(1,2, . . . ,N     t     )  and noise) and expressed by the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0040]    In equation (3), “L” denotes the number of multipaths of each of the channels  30 , and w n  denotes a complex additive noise at time n. 
         [0041]    In this example embodiment, the cyclic prefix (CP) may be capable of covering the maximum possible length of channel impulse response of each of the channels  30  (wherein, the maximum possible length of the channel impulse response may be denoted as “LT sa ” with “T sa ” denoting the sampling period for the transmitted signal x n   n     t    and the received signal y n ). Moreover, in this example embodiment, each of the transmitters  10  and the receiver  20  of the cooperative OFDM communication system  1  may be configured to operate with a discrete Fourier transform (DFT) size of “N”. In order not to over-burden the mathematical notation, hereinafter, all integer indexes to frequency-domain quantities are to be understood as modulo-N. For example, l means l%N when indexing a frequency-domain quantity and (m−k) means (m−k)%N when indexing a frequency-domain quantity, where “%” denotes modulo operation in the sense that “a%N” for any integer a means taking the nonnegative remainder of integer division of a by N, that is, a%N=a−└a/N┘N where “└ ┘” is the floor operation that outputs the largest integer equal to or smaller than its argument. Accordingly, as expressed in the DFT domain, the composite received signal y n  may be expressed by the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0042]    In equation (4), the terms Y m , X k   n     t   , W m     t    and H l,n     t     (m−k)  may be defined as the following: 
         [0043]    Y m  with a subcarrier index “m” may denote the DFT of the received signal y n  (e.g., Y m =DFT(y n )), 
         [0044]    X k   n     t    with a subcarrier index “k” may denote the DFT of the transmitted signal x n   n     t    from the transmitter  10   n   t  (e.g., X k   n     t   =DFT (x n   n     t   )), 
         [0045]    W m  with the subcarrier index “m” may denote the DFT of the complex additive noise w n  (e.g., W m =DFT(w n )), and 
         [0046]    H l,n     t     (m−k)  with the subcarrier indexes “k” and “m” may denote frequency spreading function of the l-th channel path of the channel  30   n   t  which corresponds to the transmitter  10   n   t . 
         [0047]    Furthermore, the frequency spreading function H l,n     t     (m−k)  may be expressed by the following equation, given that the subcarrier index “(m−k)” is replaced by the subcarrier index “k”: 
         [0000]    
       
         
           
             
               
                 
                   
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         [0048]    Moreover, expanding the subcarrier index “m” in equation (4) to “[0,N−1]”, a set of received signal {Y m | m∈[0,N−1] } in the DFT domain (also defined as a set of “frequency domain received signals {Y m }”) may be expressed in matrix-vector form as the following equation: 
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         [0049]    In equation (6), the terms Y, H n     t   , X n     t    and W may be defined as the following: 
         [0050]    Y=[Y 0 , Y 1 , . . . , Y N−1 ]′, which denote a vector form of the set of frequency domain received signals {Y m } corresponding to the subcarrier indexed “0” up to the subcarrier indexed “N−1,” 
         [0051]    X n     t   =[X 0   n     t   , X 1   n     t   , . . . , X N−1   n     t   ]′, which denote a vector form of a set of frequency domain transmitted signals {X k   n     t   } from the transmitter  10   n   t , which correspond to the subcarrier indexed “0” up to the subcarrier indexed “N−1,” and 
         [0052]    W=[W 0 , W 1 , . . . , W N−1 ]′, which denote a vector form of a set of frequency domain complex additive noise {W m } corresponding to the subcarrier indexed “0” up to the subcarrier indexed “N−1.” 
         [0053]    In the vector forms of Y, X n     t    and W defined as the above, the symbol “/” denotes the matrix-vector transpose. Furthermore, H n     t    may be defined as a “channel matrix” of the channel  30   n   t  corresponding to the transmitter  10   n   t , which may have a size of N×N and expressed as the following: 
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         [0054]    In equation (7), each of the entities {a m,k   n     t   } of the channel matrix H n     t    may be defined as a “channel coefficient.” The channel coefficient a m,k   n     t    may direct to a coefficient associated with a contribution on the frequency domain received signal Y m  corresponding to the subcarrier indexed “m”, which is induced by the frequency domain transmitted signal X k   n     t    corresponding to the subcarrier indexed “k” from the transmitter  10   n   t . The contribution of X k   n     t    in Y m  through a m,k   n     t    for k≠m is commonly considered as ICI. Such “ICI contributions” may be caused by uncorrected CFOs and Doppler shifts or Doppler spread due to time-variation of the channels  30 . The channel coefficient a m,k   n     t    may be described by the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
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                            
                           2 
                            
                           
                               
                           
                            
                           π 
                            
                           
                             kl 
                             N 
                           
                         
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     8 
                     ) 
                   
                 
               
             
           
         
       
     
         [0055]    Provided the channel coefficients {a m,k   n     t   }, the frequency domain received signal Y m  corresponding to the subcarrier indexed “m” may be alternatively expressed in terms of the channel coefficients {a m,k   n     t   } and the frequency domain transmitted signals {X k   n     t   }, as the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     Y 
                     m 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           
                             n 
                             t 
                           
                           = 
                           1 
                         
                         
                           N 
                           t 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             a 
                             
                               m 
                               , 
                               k 
                             
                             
                               n 
                               t 
                             
                           
                            
                           
                             X 
                             k 
                             
                               n 
                               t 
                             
                           
                         
                       
                     
                     + 
                     
                       W 
                       m 
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     9 
                     ) 
                   
                 
               
             
           
         
       
     
         [0056]    In this example embodiment, the receiver  20  may be configured to perform a receiver-based and frequency-domain signal processing to mitigate the effects of MCFO and ICI induced on the frequency domain received signals {Y m }. The mentioned “receiver-based” processing may direct to a non-closed-loop MCFO controlling scheme in which the transmitters  10  may not be requested by the receiver  20  to adjust carrier frequencies thereof. Furthermore, in order to reduce the computation complexity for the receiver  20 , the mentioned receiver-based and frequency-domain signal processing may be executed under a condition that the receiver  20  may not have full space-frequency channel state information (CSI) of the channels  30 . That is, the receiver  20  may not need to estimate all entities {a m,k   n     t}|     n     t     ∈[1,N     1     ]  of the channel matrix H n     t|     n     t     ∈[1,N     1     ]  for all the channels  30  corresponding to all the transmitters  10 . Instead, the receiver  20  may have only “partial” CSI of each of the channels  30 , wherein only selected entities {a m,k   n     t   } need to be estimated by the receiver  20 , as will be discussed in the following paragraphs by reference to  FIG. 2 . 
         [0057]      FIG. 2  illustrates the channel matrix H n     t    of the channel  30   n   t  in the cooperative OFDM communication system  1  illustrated in  FIG. 1  in accordance with an example embodiment. Referring to  FIG. 2 , a “band approximation” with a bandwidth “K” may be applied to the channel matrix H n     t   , and selected entities {a m,k   n     t}|     k∈[m−K,M+K]  residing within such a “band” may be defined as “in-band coefficients.” In this example embodiment, the channel matrix H n     t    may be band-approximated with bandwidth K=1, and the in-band coefficients may thus include entities {a m,k   n     t}|     k∈[m−1,m+1]  residing on and within the band defined by the dotted-lines A-A′ and B-B′, circularly in an end-around fashion along each row of the channel matrix as illustrated in  FIG. 2 . On the other hand, the remaining entities {a m,k   n     t}|     k∉[m−1,m+1]  of the channel matrix H n     t    other than the in-band coefficients may be defined as “out-of-band ICI coefficients”. Given the above definitions for the in-band coefficients and out-of-band ICI coefficients, the frequency domain received signal Y m  obtained by equation (9) may be separated into an “in-band portion” and an “out-of-band portion” as following: 
         [0000]    
       
         
           
             
               
                 
                   
                     Y 
                     m 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           
                             n 
                             t 
                           
                           = 
                           1 
                         
                         
                           N 
                           t 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             
                               m 
                               - 
                               K 
                             
                           
                           
                             m 
                             + 
                             K 
                           
                         
                          
                         
                             
                         
                          
                         
                           
                             a 
                             
                               m 
                               , 
                               k 
                             
                             
                               n 
                               t 
                             
                           
                            
                           
                             X 
                             k 
                             
                               n 
                               t 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           
                             n 
                             t 
                           
                           = 
                           
                             
                               1 
                                
                               k 
                             
                             ∉ 
                             
                               [ 
                               
                                 
                                   m 
                                   - 
                                   K 
                                 
                                 , 
                                 
                                   m 
                                   + 
                                   K 
                                 
                               
                               ] 
                             
                           
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           a 
                           
                             m 
                             , 
                             k 
                           
                           
                             n 
                             t 
                           
                         
                          
                         
                           X 
                           k 
                           
                             n 
                             t 
                           
                         
                       
                     
                     + 
                     
                       W 
                       m 
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     10 
                     ) 
                   
                 
               
             
           
         
       
     
         [0058]    In equation (10), the in-band portion 
         [0000]    
       
         
           
             
               ∑ 
               
                 
                     
                 
                  
                 
                   
                     n 
                     t 
                   
                   = 
                   1 
                 
               
               
                 N 
                 t 
               
             
              
             
                 
             
              
             
               
                 ∑ 
                 
                   k 
                   = 
                   
                     m 
                     - 
                     K 
                   
                 
                 
                   m 
                   + 
                   K 
                 
               
                
               
                   
               
                
               
                 
                   a 
                   
                     m 
                     , 
                     k 
                   
                   
                     n 
                     t 
                   
                 
                  
                 
                   X 
                   k 
                   
                     n 
                     t 
                   
                 
               
             
           
         
       
     
         [0000]    may direct to in-band contributions on the received signal Y m  corresponding to the subcarrier indexed “m”, which are contributed by all the transmitted signals X m−K   n     t   , X m−K+1   n     t   , X m−K+2   n     t   , . . . , X m+K   n     t    from all the transmitters  10 . Therefore, the in-band portion 
         [0000]    
       
         
           
             
               ∑ 
               
                 
                     
                 
                  
                 
                   
                     n 
                     t 
                   
                   = 
                   1 
                 
               
               
                 N 
                 t 
               
             
              
             
                 
             
              
             
               
                 ∑ 
                 
                   k 
                   = 
                   
                     m 
                     - 
                     K 
                   
                 
                 
                   m 
                   + 
                   K 
                 
               
                
               
                   
               
                
               
                 
                   a 
                   
                     m 
                     , 
                     k 
                   
                   
                     n 
                     t 
                   
                 
                  
                 
                   X 
                   k 
                   
                     n 
                     t 
                   
                 
               
             
           
         
       
     
         [0000]    may be also defined as “composite in-band signal,” which may be composite of all contributions from the transmitted signals X m−K   n     t   , X m−K+1   n     t   , X m−K+2   n     t   , . . . , X m+K   n     t    by all of the transmitters  10 . On the other hand, the out-of-band portion 
         [0000]    
       
         
           
             
               ∑ 
               
                 
                   n 
                   t 
                 
                 = 
                 1 
               
               
                 N 
                 t 
               
             
              
             
                 
             
              
             
               
                 ∑ 
                 
                   k 
                   ∉ 
                   
                     [ 
                     
                       
                         m 
                         - 
                         K 
                       
                       , 
                       
                         m 
                         + 
                         K 
                       
                     
                     ] 
                   
                 
               
                
               
                   
               
                
               
                 
                   a 
                   
                     m 
                     , 
                     k 
                   
                   
                     n 
                     t 
                   
                 
                  
                 
                   X 
                   k 
                   
                     n 
                     t 
                   
                 
               
             
           
         
       
     
         [0000]    may be defined as “composite residual ICI,” which may be composite of all contributions from the transmitted signals X 0   n     t   , X 1   n     t   , . . . , X m−K−1   n     t   , X m+K+1   n     t   , . . . , X N−1   n     t   , X N   n     t    by all of the transmitters  10 . 
         [0059]    In one example, the receiver  20  may be configured to perform channel estimation to estimate the in-band coefficients {a m,k   n     t}|     k∈[m−K,m+K] . Furthermore, based on the estimated in-band coefficients, the receiver  20  may be configured to perform frequency-domain equalizing on the composite in-band signal of the frequency domain received signal Y m  and leave the composite residual ICI causing performance floors. In another example, signal detection may be performed on the frequency domain received signal Y m  regarding only the composite in-band signal, wherein performance floors may be caused by the composite residual ICI as well. 
         [0060]    Thanks to the statistical property of the composite residual ICI in the cooperative OFDM communication system  1  (that is, the normalized autocorrelation of the composite residual ICI may be substantially invariant with respect to various system settings and channel conditions, and the first few lags of the normalized autocorrelation function of the composite residual ICI may have relatively high values given that {X n     t|     n     t     ∈[1,N     t     ] } are equal or independent), the composite residual ICI may be performed by a “whitening process” substantially independent to the properties of the channels  30  and system settings of the cooperative OFDM communication system  1 . Such a whitening process may lower the performance floors caused by the composite residual ICI. 
         [0061]    In operation, the whitening process may be performed on the frequency domain received signal Y m . Thereby, the whitening process may be also performed on the sum of the composite residual ICI 
         [0000]    
       
         
           
             
               ∑ 
               
                 
                   n 
                   t 
                 
                 = 
                 1 
               
               
                 N 
                 t 
               
             
              
             
                 
             
              
             
               
                 ∑ 
                 
                   k 
                   ∉ 
                   
                     [ 
                     
                       
                         m 
                         - 
                         K 
                       
                       , 
                       
                         m 
                         + 
                         K 
                       
                     
                     ] 
                   
                 
               
                
               
                   
               
                
               
                 
                   a 
                   
                     m 
                     , 
                     k 
                   
                   
                     n 
                     t 
                   
                 
                  
                 
                   X 
                   k 
                   
                     n 
                     t 
                   
                 
               
             
           
         
       
     
         [0000]    and the channel noise W m  within the frequency domain received signal Y m . Wherein, the whitened received signal may be denoted as “Y m ”. Furthermore, subsequent to the whitening process, the receiver  20  may be configured to perform signal detection on the whitened received signal {tilde over (Y)} m  so as to detect data (e.g., bit information) conveyed in the transmitted signals X m   n     t    by all the transmitters  10 . Detailed operation of the whitening process will be discussed with the aid of an example embodiment described in the following paragraphs by reference to  FIGS. 3A and 3B . In the following example embodiment, for simplicity, a cooperative OFDM communication system including two transmitters (N t =2) is considered. 
         [0062]      FIG. 3A  is a block diagram of the baseband part of a cooperative OFDM communication system  2  in accordance with another example embodiment, and  FIG. 3B  illustrates channel matrices H 1  and H 2  of the channels  30   a - 1  and  30   a - 2  in the cooperative OFDM communication system  2  illustrated in  FIG. 3A  in accordance with another example embodiment. Referring to  FIG. 3A , the cooperative OFDM communication system  2  may be similar to the cooperative OFDM communication system  1  as illustrated in  FIG. 1  except that, the cooperative OFDM communication system  2  may include but not limited to two transmitters  10   a - 1  and  10   a - 2 . Furthermore, the cooperative OFDM communication system  2  may operate with but not limited to a DFT size of 128, which corresponds to 128 subcarriers (e.g., subcarrier indexed “0” up to subcarrier indexed “127”). 
         [0063]    The transmitters  10   a - 1  and  10   a - 2  may be communicatively coupled to a receiver  20   a  through channels  30   a - 1  and  30   a - 2  respectively, and the transmitters  10   a - 1  and  10   a - 2  may be configured to transmit signals to the receiver  20   a  through the channels  30   a - 1  and  30   a - 2  respectively. Specifically, the signal transmitted by the transmitter  10   a - 1  may be denoted as X 1 , while the signal transmitted by the transmitter  10   a - 2  may be denoted as x n   2 . Furthermore, the channel  30   a - 1  may be characterized by a set of discrete-time complex gains {h n,l   1 }, while the channel  30   a - 2  may be characterized by a set of discrete-time complex gains {h n,l   2 }. In this example embodiment, each of the channels  30   a - 1  and  30   a - 2  may have but not limited to six channel paths. The signals x n   1  and x n   2  which may be convolved with the complex gains {h n,l   1 } and {h n,l   2 } respectively, may then be received by the receiver  20   a . The received signal at the receiver  20   a  may be denoted as y n , and the received signal y n  may be expressed by the following equation (wherein channel noise w n  may be included): 
         [0000]    
       
         
           
             
               
                 
                   
                     y 
                     n 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           0 
                         
                         5 
                       
                        
                       
                           
                       
                        
                       
                         
                           h 
                           
                             n 
                             , 
                             l 
                           
                           1 
                         
                          
                         
                           x 
                           
                             n 
                             - 
                             l 
                           
                           1 
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           0 
                         
                         5 
                       
                        
                       
                           
                       
                        
                       
                         
                           h 
                           
                             n 
                             , 
                             l 
                           
                           2 
                         
                          
                         
                           x 
                           
                             n 
                             - 
                             l 
                           
                           2 
                         
                       
                     
                     + 
                     
                       w 
                       n 
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     11 
                     ) 
                   
                 
               
             
           
         
       
     
         [0064]    Moreover, being transformed to the DFT domain, the frequency domain received signal Y m  which corresponds to subcarrier indexed “m,” may be expressed in terms of frequency domain transmitted signals “X k   1 ” and “X k   2 ”, frequency domain complex additive noise “W m ” and frequency spreading functions “H l,1   (m−k) ” (and “H l,2   (m−k) ” of the l-th channel path of the channels  30   a - 1  and  30   a - 2 , as the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     Y 
                     m 
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         127 
                       
                        
                       
                           
                       
                        
                       
                         
                           ∑ 
                           
                             l 
                             = 
                             0 
                           
                           5 
                         
                          
                         
                             
                         
                          
                         
                           
                             X 
                             k 
                             1 
                           
                            
                           
                             H 
                             
                               l 
                               , 
                               1 
                             
                             
                               ( 
                               
                                 m 
                                 - 
                                 k 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 
                                   - 
                                   j 
                                 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 lk 
                               
                               128 
                             
                           
                         
                       
                     
                     + 
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         127 
                       
                        
                       
                           
                       
                        
                       
                         
                           ∑ 
                           
                             l 
                             = 
                             0 
                           
                           5 
                         
                          
                         
                             
                         
                          
                         
                           
                             X 
                             k 
                             2 
                           
                            
                           
                             H 
                             
                               l 
                               , 
                               2 
                             
                             
                               ( 
                               
                                 m 
                                 - 
                                 k 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 
                                   - 
                                   j 
                                 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 lk 
                               
                               128 
                             
                           
                         
                       
                     
                     + 
                     
                       W 
                       m 
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     12 
                     ) 
                   
                 
               
             
           
         
       
     
         [0065]    In addition, to be expressed in matrix-vector forms, equation (12) may be expressed as the following: 
         [0000]        Y=H   1   X   1   +H   2   X   2   +W   eq. (13)
 
         [0066]    In equation (13), the set of frequency domain received signals {Y m }| m∈[0,127]  which correspond to the subcarrier indexed “0” up to the subcarrier indexed “127,” may be expressed in a vector form of Y=[Y 0 , Y 1 , . . . , Y 127 ]′. Furthermore, the set of frequency domain transmitted signals {X k   1 }| k∈[0,127]  from the transmitter  10   a - 1  that correspond to the subcarrier indexed “0” up to the subcarrier indexed “127,” may be expressed in a vector form of X 1 =[X 0   1 , X 1   1 , . . . , X 127   1 ]′. Likewise, the set of frequency domain transmitted signals {X k   2 }| k∈[0,127]  from the transmitter  10   a - 2  that correspond to the subcarrier indexed “0” up to the subcarrier indexed “127,” may be expressed in a vector form of X 2 =[X 0   2 , X 1   2 , . . . , X 127   2 ]′. In the same manner, the set of frequency domain complex additive noise {W m }| m∈[0,127]  that correspond to the subcarrier indexed “0” up to the subcarrier indexed “127,” may be expressed in a vector form of W=[W 0 , W 1 , . . . , W 127 ]′. 
         [0067]    On the other hand, the channel matrices H 1  and H 2  in equation (13) may have a size of 128×128 with channel coefficients {a m,k   1 }| m,k∈[0,127]  and {a m,k   2 }| m,k∈[0,127]  as their entities. The channel coefficients {a m,k   1 }| m,k∈[0,127]  and {a m,k   2 }| m,k∈[0,127]  may be described using the following equations: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       a 
                       
                         m 
                         , 
                         k 
                       
                       1 
                     
                     = 
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           0 
                         
                         5 
                       
                        
                       
                           
                       
                        
                       
                         
                           H 
                           
                             l 
                             , 
                             1 
                           
                           
                             ( 
                             
                               m 
                               - 
                               k 
                             
                             ) 
                           
                         
                          
                         
                            
                           
                             
                               - 
                               j 
                             
                              
                             
                                 
                             
                              
                             2 
                              
                             π 
                              
                             
                               kl 
                               128 
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     14 
                     ) 
                   
                 
               
             
             
               
                 
                   
                     a 
                     
                       m 
                       , 
                       k 
                     
                     2 
                   
                   = 
                   
                     
                       ∑ 
                       
                         l 
                         = 
                         0 
                       
                       5 
                     
                      
                     
                         
                     
                      
                     
                       
                         H 
                         
                           l 
                           , 
                           2 
                         
                         
                           ( 
                           
                             m 
                             - 
                             k 
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                             j 
                           
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           π 
                            
                           
                             kl 
                             128 
                           
                         
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     15 
                     ) 
                   
                 
               
             
           
         
       
     
         [0068]    In this example embodiment, the cooperative OFDM communication system  2  may have a bandwidth K=1 (e.g., the channel matrices H 1  and H 2  may thus be band-approximated with bandwidth K=1), hence, the entities {a m,k   1 }| m,k∈[0,127]  and {a m,k   2 }| m,k∈[0,127]  of the channel matrices H 1  and H 2  may be categorized as the in-band coefficients and the out-of-band coefficients as shown in  FIG. 3B . Based on the above categorization, the receiver  20   a  may be configured to perform whitening process on the composite residual ICI 
         [0000]    
       
         
           
             
               ∑ 
               
                 k 
                 ∉ 
                 
                   [ 
                   
                     
                       m 
                       - 
                       1 
                     
                     , 
                     
                       m 
                       + 
                       1 
                     
                   
                   ] 
                 
               
             
              
             
                 
             
              
             
               ( 
               
                 
                   
                     a 
                     
                       m 
                       , 
                       k 
                     
                     1 
                   
                    
                   
                     X 
                     k 
                     1 
                   
                 
                 + 
                 
                   
                     a 
                     
                       m 
                       , 
                       k 
                     
                     2 
                   
                    
                   
                     X 
                     k 
                     2 
                   
                 
               
               ) 
             
           
         
       
     
         [0000]    contributed from the transmitters  10   a - 1  and  10   a - 2 . Meanwhile, such a whitening process may be also performed on the channel noise W m . 
         [0069]    In order to reduce computation complexity, in this example embodiment, the whitening process may be performed block-by-block (thus defined as “blockwise whitening process”) with each block corresponding to several selected subcarriers, instead of whole sequence corresponding to all the 128 subcarriers. The receiver  20   a  may include a device to perform such a blockwise whitening process. An exemplary hardware structure of such a device and exemplary operations thereof will be discussed in the following paragraphs by reference to  FIGS. 4A to 4E . 
         [0070]      FIG. 4A  is a block diagram of a device  40  for performing the blockwise whitening process and a signal detector  46  in the cooperative OFDM communication system  2  illustrated in  FIG. 3A  in accordance with another example embodiment, and  FIG. 4B  illustrates the sub-vectors Ys m , Xs m   1  and Xs m   2  in the cooperative OFDM communication system  2  illustrated in  FIG. 3A  in accordance with another example embodiment. Referring to  FIG. 4A , the device  40  which may be configured to perform the blockwise whitening process, may include a truncator  41 , at least two channel estimators  42  and  43 , a processor  44  and a filter  45 . 
         [0071]    The truncator  41  may be configured to receive the set of frequency domain received signals {Y m }| m∈[0,127]  in series, and truncate the set of frequency domain received signals {Y m }| m∈[0,127]  into sub-blocks (denoted as “sub-vectors {Ys m }”). The sub-vector Ys m  may have a length “Q” and center at the subcarrier indexed “m.” That is, the sub-vector Ys m  may include a subset of the frequency domain received signals 
         [0000]    
       
         
           
             
               { 
               
                 
                   Y 
                   
                     m 
                     - 
                     
                       ⌊ 
                       
                         
                           Q 
                           - 
                           1 
                         
                         2 
                       
                       ⌋ 
                     
                   
                 
                 , 
                 … 
                  
                 
                     
                 
                 , 
                 
                   Y 
                   
                     m 
                     - 
                     1 
                   
                 
                 , 
                 
                   Y 
                   m 
                 
                 , 
                 
                   Y 
                   
                     m 
                     + 
                     1 
                   
                 
                 , 
                 … 
                  
                 
                     
                 
                 , 
                 
                   Y 
                   
                     m 
                     + 
                     
                       ⌈ 
                       
                         
                           Q 
                           - 
                           1 
                         
                         2 
                       
                       ⌉ 
                     
                   
                 
               
               } 
             
              
             
                 
             
           
         
       
     
         [0000]    near the subcarrier indexed “m” where “┌ ┐” denotes the ceiling operation that outputs the smallest integer equal to or greater than its argument. In this example embodiment, the sub-vector Ys m  having a length Q=3 and centering at the subcarrier indexed “5” may be expressed in vector form of [Y 4 , Y 5 , Y 6 ]′ as shown in  FIG. 4B . 
         [0072]    Likewise, the set of frequency domain transmitted signals {X m   1 }| m∈[0,127]  from the transmitter  10   a - 1  and the set of frequency domain transmitted signals {X m   2 }| m∈[0,127]  from the transmitter  10   a - 2  may be also truncated into sub-blocks (denoted as “sub-vectors {Xs m   1 } and {Xs m   2 }”) respectively. Each of the sub-vectors Xs m   1  and Xs m   2  may have a length “P 1 ” and “P 2 ” respectively and center at the subcarrier indexed “m”. As shown in  FIG. 4B , the sub-vectors Xs m   1  and Xs m   2  having a length P 1 =P 2 =3 and centering at the subcarrier indexed “5” may be expressed in vector form of [X 4   1 , X 5   1 , X 6   1 ]′ and [X 4   2 , X 5   2 ,X 6   2 ]′ respectively. Furthermore, in this example embodiment, the sub-vectors Ys m , Xs m   1  and Xs m   2  may not be limited to have equal length. 
         [0073]    Referring back to  FIG. 4A , the channel estimators  42  and  43  may be configured to estimate channel state information of the channels  30   a - 1  and  30   a - 2  respectively. Based on the estimated channel state information, channel coefficients corresponding to the channels  30   a - 1  and  30   a - 2  may be obtained. Thereafter, the channel matrices H 1  and H 2  which correspond to the channels  30   a - 1  and  30   a - 2  respectively may be constructed using the obtained channel coefficients as their entities. In one example embodiment, the channel state information of the channels  30   a - 1  and  30   a - 2  may be estimated by the channel estimators  42  and  43  exploiting a channel sounding method. 
         [0074]      FIG. 4C  illustrates the channel sounding method performed by the channel estimators  42  and  43  illustrated in  FIG. 4A  in accordance with another example embodiment. Referring to  FIG. 4C , the transmitter  10   a - 1  may be configured to transmit a sounding signal (which may alternatively be referred to as a pilot signal) S 1  through the channel  30   a - 1 . The sounding signal S 1  may pass through the channel  30   a - 1  and thereafter received by the receiver  20   a . The received sounding signal at the receiver  20   a  may be denoted as S R   1 , and channel state information of channel  30   a - 1  may be derived from the received sounding signal S R   1 . Likewise, the transmitter  10   a - 2  may be configured to transmit a sounding signal S 2  through the channel  30   a - 2 , and channel state information of the channel  30   a - 2  may be derived from the received sounding signal S R   2  at the receiver  20   a . Referring back to  FIG. 4A , the processor  44  may include computing units  441 ,  442  and  443 . The computing unit  441  may be configured to decompose the channel matrices H 1  and H 2  into a plurality of sub-matrices {H m,m   1 } and {H m,m   2 }, as will be discussed in the following paragraphs by reference to  FIG. 4D . 
         [0075]      FIG. 4D  illustrates the channel matrices H 1  and H 2  of the channels  30   a - 1  and  30   a - 2  in the cooperative OFDM communication system  2  illustrated in  FIG. 3A  in accordance with an example embodiment, and  FIG. 4E  illustrates an operation for calculating composite residual ICI plus channel noise z m  in the blockwise whitening process in accordance with an example embodiment. Referring to  FIG. 4D , to fit the length Q of the sub-vector Ys m  and the lengths P 1  and P 2  of the sub-vectors Xs m   1  and Xs m   2 , the sub-matrices {H m,m   1 } and {H m,m   2 } may have sizes Q×P 1  and Q×P 2 , respectively. Furthermore, the sub-matrices H m,m   1  and H m,m   2  may correspond to the subcarrier indexed “m” and include channel coefficients residing near entities a m,m   1  and a m,m   2  respectively. For example, the sub-matrix H 5,5   1  which may have a size of 3×3 and correspond to the subcarrier indexed “5,” may include channel coefficients {a 4,4   1 , a 5,4   1 , a 6,4   1 , a 4,5   1 , a 5,5   1 , a 6,5   1 , a 4,6   1 , a 5,6   1 , a 6,6   1 } as its entities. Likewise, the sub-matrix H 5,5   2  which may also have a size of 3×3 and correspond to the subcarrier indexed “5,” may include channel coefficients {a 4,4   2 ,a 5,4   2 ,a 6,4   2 ,a 4,5   2 ,a 5,5   2 ,a 6,5   2 ,a 4,6   2 ,a 5,6   2 ,a 6,6   2 } as its entities. 
         [0076]    Providing the mentioned sub-vectors Xs m   1  and Xs m   2  and the mentioned sub-matrices H m,m   1  and H m,m   2 , the sub-vector Ys m  may be expressed by the following equation: 
         [0000]        Ys   m   =H   m,m   1   Xs   m   1   +H   m,m   2   Xs   m   2   +z   m   eq. (16)
 
         [0077]    In equation (16), the portion H m,m   1 Xs m   1 +H m,m   2 Xs m   2  may include composite in-band contributions on the frequency domain received signals 
         [0000]    
       
         
           
             
               Y 
               
                 m 
                 - 
                 
                   ⌊ 
                   
                     
                       Q 
                       - 
                       1 
                     
                     2 
                   
                   ⌋ 
                 
               
             
             , 
             … 
              
             
                 
             
             , 
             
               Y 
               
                 m 
                 - 
                 1 
               
             
             , 
             
               Y 
               m 
             
             , 
             
               Y 
               
                 m 
                 + 
                 1 
               
             
             , 
             … 
              
             
                 
             
             , 
             
               and 
                
               
                   
               
                
               
                 Y 
                 
                   m 
                   + 
                   
                     ⌈ 
                     
                       
                         Q 
                         - 
                         1 
                       
                       2 
                     
                     ⌉ 
                   
                 
               
             
             , 
           
         
       
     
         [0000]    which are contributed by the frequency domain transmitted signals 
         [0000]    
       
         
           
             
               X 
               
                 m 
                 - 
                 
                   ⌊ 
                   
                     
                       
                         P 
                         1 
                       
                       - 
                       1 
                     
                     2 
                   
                   ⌋ 
                 
               
               1 
             
             , 
             … 
              
             
                 
             
             , 
             
               X 
               
                 m 
                 - 
                 1 
               
               1 
             
             , 
             
               X 
               m 
               1 
             
             , 
             
               X 
               
                 m 
                 + 
                 1 
               
               1 
             
             , 
             … 
              
             
                 
             
             , 
             
               and 
                
               
                   
               
                
               
                 X 
                 
                   m 
                   + 
                   
                     ⌈ 
                     
                       
                         
                           P 
                           1 
                         
                         - 
                         1 
                       
                       2 
                     
                     ⌉ 
                   
                 
                 1 
               
             
           
         
       
     
         [0000]    from the transmitter  10   a - 1  and the frequency domain transmitted signals 
         [0000]    
       
         
           
             
               X 
               
                 m 
                 - 
                 
                   ⌊ 
                   
                     
                       
                         P 
                         2 
                       
                       - 
                       1 
                     
                     2 
                   
                   ⌋ 
                 
               
               2 
             
             , 
             … 
              
             
                 
             
             , 
             
               X 
               
                 m 
                 - 
                 1 
               
               2 
             
             , 
             
               X 
               m 
               2 
             
             , 
             
               X 
               
                 m 
                 + 
                 1 
               
               2 
             
             , 
             … 
              
             
                 
             
             , 
             
               and 
                
               
                   
               
                
               
                 X 
                 
                   m 
                   + 
                   
                     ⌈ 
                     
                       
                         
                           P 
                           2 
                         
                         - 
                         1 
                       
                       2 
                     
                     ⌉ 
                   
                 
                 2 
               
             
           
         
       
     
         [0000]    from the transmitter  10   a - 2 . On the other hand, the portion “z m ” may include the channel noise and the composite residual ICI contributed by the transmitters  10   a - 1  and  10   a - 2  corresponding to the channels  30   a - 1  and  30   a - 2 . More particularly, the portion z m  may include all the remaining terms for the sub-vector Ys m  in the right-hand-side (RHS) of equation (13), which are left out of the portion H m,m   1 Xs m   1 +H m,m   2 Xs m   2 . Accordingly, the portion z m  may be obtained by subtracting the portion H m,m   1 Xs m   1 +H m,m   2 Xs m   2  from the sub-vector Ys m , as illustrated in  FIG. 4E . The operation shown in  FIG. 4E  may be executed by the computing unit  442  of the processor  44 . 
         [0078]    Furthermore, thanks to the statistical property of the composite residual ICI within the portion z m , the portion “z m ” can be whitened in a nearly channel-independent manner. In this example embodiment, the portion “z m ” may be whitened by performing the blockwise whitening process thereon. To perform the mentioned blockwise whitening process, covariance matrix (denoted as “K z ”) of the portion “z m ” needs to be calculated in advance. In this example embodiment, the computing unit  443  of the processor  44  may be configured to execute an operation to calculate the covariance matrix K z  as the following: 
         [0000]        K   z   =E[z   m   z   m   H ]  eq. (17)
 
         [0079]    By the independence between the composite residual ICI and the channel noise, K z =K l +K w  where K w  is the Q×Q covariance matrix of the channel noise component in z m , and K l  is the Q×Q covariance matrix of the composite residual ICI component in z m . In one embodiment of this invention, K w  may be calculated by estimating the variance of the channel noise and letting K w  be a diagonal matrix with its diagonal terms equal to the variance of the channel noise, and K l  may be calculated by estimating the variance of the composite residual ICI and employing the statistical property of the composite residual ICI. 
         [0080]    Moreover, referring back to  FIG. 4A , the covariance matrix K z  may be provided to the filter  45 , and the filter  45  (also denoted as “whitening filter”) may be configured to perform blockwise whitening process on the sub-vector Ys m  and in turn the portion “z m ”, using the following operation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Y 
                       ~ 
                     
                      
                     
                       s 
                       m 
                     
                   
                   = 
                   
                     
                       K 
                       z 
                       
                         - 
                         
                           1 
                           2 
                         
                       
                     
                      
                     
                       Ys 
                       m 
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     18 
                     ) 
                   
                 
               
             
           
         
       
     
         [0081]    In equation (18), the term {tilde over (Y)}s m  denotes the whitened received signal. The whitened received signal {tilde over (Y)}s m  may be further expanded as the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Y 
                       ~ 
                     
                      
                     
                       s 
                       m 
                     
                   
                   = 
                   
                     
                       
                         
                           K 
                           z 
                           
                             - 
                             
                               1 
                               2 
                             
                           
                         
                          
                         
                           H 
                           
                             m 
                             , 
                             m 
                           
                           1 
                         
                          
                         
                           Xs 
                           m 
                           1 
                         
                       
                       + 
                       
                         
                           K 
                           z 
                           
                             - 
                             
                               1 
                               2 
                             
                           
                         
                          
                         
                           H 
                           
                             m 
                             , 
                             m 
                           
                           2 
                         
                          
                         
                           Xs 
                           m 
                           2 
                         
                       
                       + 
                       
                         
                           K 
                           z 
                           
                             - 
                             
                               1 
                               2 
                             
                           
                         
                          
                         
                           z 
                           m 
                         
                       
                     
                     = 
                     
                       
                         
                           
                             H 
                             ~ 
                           
                           
                             m 
                             , 
                             m 
                           
                           1 
                         
                          
                         
                           Xs 
                           m 
                           1 
                         
                       
                       + 
                       
                         
                           
                             H 
                             ~ 
                           
                           
                             m 
                             , 
                             m 
                           
                           2 
                         
                          
                         
                           Xs 
                           m 
                           2 
                         
                       
                       + 
                       
                         
                           z 
                           ~ 
                         
                         m 
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     19 
                     ) 
                   
                 
               
             
           
         
       
     
         [0082]    In equation (19), the portion “{tilde over (z)} m ”denotes the whitened composite residual ICI plus channel noise. 
         [0083]    Subsequent to the blockwise whitening process, the whitened received signal {tilde over (Y)}s m  may be sent to a signal detector  46 , and the signal detector  46  may be configured to detect the whitened received signal {tilde over (Y)}s m  by various detection methods. In this example embodiment, the whitened received signal {tilde over (Y)}s m  may be detected by a maximum-likelihood sequence estimation (MLSE)-based detection. 
         [0084]    Regarding the above-mentioned MLSE-based detection performed on the whitened received signal {tilde over (Y)}s m , specifically, given that the whitened composite residual ICI plus channel noise {tilde over (z)} m  for all the subcarriers indexed “0” to “127” (e.g., 0≦m≦127) are mutually independent, the joint likelihood function of the whitened received signal {tilde over (Y)}s m  for all the subcarriers indexed “0” to “127” (e.g., 0≦m≦127) may take a form of the following: 
         [0000]    
       
         
           
             
               
                 
                   
                     f 
                      
                     
                       ( 
                       
                         
                           
                             Y 
                             ~ 
                           
                            
                           
                             s 
                             0 
                           
                         
                         , 
                         
                           
                             Y 
                             ~ 
                           
                            
                           
                             s 
                             1 
                           
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           
                             
                               
                                 Y 
                                 ~ 
                               
                                
                               
                                 s 
                                 127 
                               
                             
                             | 
                             
                               Xs 
                               m 
                               
                                 n 
                                 t 
                               
                             
                           
                           ; 
                           
                             0 
                             ≤ 
                             m 
                             ≤ 
                             127 
                           
                         
                         , 
                         
                           
                             n 
                             t 
                           
                           ∈ 
                           
                             [ 
                             
                               1 
                               , 
                               2 
                             
                             ] 
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       f 
                        
                       
                         ( 
                         
                           
                             
                               z 
                               ~ 
                             
                             0 
                           
                           , 
                           
                             
                               z 
                               ~ 
                             
                             1 
                           
                           , 
                           … 
                            
                           
                               
                           
                           , 
                           
                             
                               z 
                               ~ 
                             
                             127 
                           
                         
                         ) 
                       
                     
                     = 
                     
                       
                         ∏ 
                         
                           n 
                           = 
                           0 
                         
                         127 
                       
                        
                       
                           
                       
                        
                       
                         f 
                          
                         
                           ( 
                           
                             
                               z 
                               ~ 
                             
                             n 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   eq 
                   . 
                   
                       
                   
                    
                   
                     ( 
                     20 
                     ) 
                   
                 
               
             
           
         
       
     
         [0085]    In case the above set of {tilde over (z)} m  are not mutually independent, equation (20) may still be used as a possibly approximate mathematical model to deal with {tilde over (z)} m . 
         [0086]    Accordingly, the log-likelihood functions Λ m  may be defined as the following: 
         [0000]      Λ m ≡log  f ( {tilde over (z)}   0   ,{tilde over (z)}   1   , . . . ,{tilde over (z)}   m ) for 0 ≦m≦ 127  eq. (21)
 
         [0087]    Furthermore, the above log-likelihood functions Λ m  may have a recursive relation as the following: 
         [0000]      Λ m =Λ m−1 +log  f ( {tilde over (Y)}s   m   −{tilde over (H)}   m,m   1   Xs   m   1   −{tilde over (H)}   m,m   2   Xs   m   2 ) for  m≧ 1  eq. (22)
 
         [0088]    With the above recursive relation, trellis structure for Viterbi algorithm may be formed and applied to the signal detector  46  of the receiver  20   a  in this example embodiment. 
         [0089]    In yet another example embodiment, the device  40  for performing the blockwise whitening process and the signal detector  46  for performing the signal detection may be integrated into a single device, as will be discussed in the following paragraphs by reference to  FIG. 4F . 
         [0090]      FIG. 4F  is a block diagram of a device  50  for performing the blockwise whitening process and the signal detection in accordance with yet another example embodiment. Referring to  FIG. 4F , the device  50  may include a processor or a micro control unit (MCU) which may be configured to execute computer-based instructions to perform the blockwise whitening process and the signal detection. 
         [0091]    In this example embodiment, the device  50  may include computing units  51  to  58 . The computing units  51  to  58  may correspond to the truncator  41 , the channel estimators  42  and  43 , the computing units  441 ,  442  and  443 , the filter  45  and the signal detector  46  illustrated by  FIG. 4A  respectively. Specifically, the computing unit  51  may be configured to truncate the frequency domain received signals {Y m } into subvectors Ys m . Furthermore, the computing units  52  and  53  may be configured to estimate channel state information of the channels  30   a - 1  and  30   a - 2  and generate channel matrices H 1  and H 2 . Moreover, the computing unit  54  may be configured to decompose the channel matrices H 1  and H 2  into sub-matrices H m,m   1  and H m,m   2 . In addition, based on the subvectors Ys m  and the sub-matrices H m,m   1  and H m,m   2 , the computing unit  55  may be configured to calculate the portion z m  which includes the composite residual ICI and the channel noise, and the computing unit  56  may be configured to calculate the covariance matrix K z  of the portion z m . Based on the covariance matrix K z , the computing unit  57  may be configured to perform whitening process on the subvectors Ys m  to obtain whitened received signal {tilde over (Y)}s m . Thereafter, the computing unit  58  may be configured to perform signal detection on the whitened received signal {tilde over (Y)}s m  using a MLSE-based detection. 
         [0092]      FIG. 4G  is a block diagram of a device  40   a  for performing the blockwise whitening process and a device  46   a  for performing the signal detection in accordance with still another example embodiment. Referring to  FIG. 4G , the device  40   a  may be similar to the device  40  illustrated in  FIG. 4A  except that, the computing unit  441   a  of the device  40   a  may be configured to decompose the channel matrices H 1  and H 2  into a plurality of sub-matrices {H m,m     1     1 } and {H m,m     2     2 }. 
         [0093]    More particularly, the sub-matrices {H m,m     1     1 } and {H m,m     2     2 } may be similar to the sub-matrices {H m,m   1 } and {H m,m   2 } expressed in equation (16) and illustrated by  FIG. 4A  except that H m,m     1     1  is defined as a Q×P 1  sub-matrix of H 1  consisting of the intersection of the 
         [0000]    
       
         
           
             
               ( 
               
                 m 
                 - 
                 
                   ⌊ 
                   
                     
                       Q 
                       - 
                       1 
                     
                     2 
                   
                   ⌋ 
                 
               
               ) 
             
             th 
           
         
       
     
         [0000]    to the 
         [0000]    
       
         
           
             
               ( 
               
                 m 
                 + 
                 
                   ⌈ 
                   
                     
                       Q 
                       - 
                       1 
                     
                     2 
                   
                   ⌉ 
                 
               
               ) 
             
             th 
           
         
       
     
         [0000]    rows of H 1  and the 
         [0000]    
       
         
           
             
               ( 
               
                 
                   m 
                   1 
                 
                 - 
                 
                   ⌊ 
                   
                     
                       
                         P 
                         1 
                       
                       - 
                       1 
                     
                     2 
                   
                   ⌋ 
                 
               
               ) 
             
             th 
           
         
       
     
         [0000]    to the 
         [0000]    
       
         
           
             
               ( 
               
                 
                   m 
                   1 
                 
                 + 
                 
                   ⌈ 
                   
                     
                       
                         P 
                         1 
                       
                       - 
                       1 
                     
                     2 
                   
                   ⌉ 
                 
               
               ) 
             
             th 
           
         
       
     
         [0000]    columns of H 1  but may have some elements therein set to zero and, on the other hand, H m,m     2     2  is defined as a Q×P 2  sub-matrix of H 2  consisting of the intersection of the 
         [0000]    
       
         
           
             
               ( 
               
                 m 
                 - 
                 
                   ⌊ 
                   
                     
                       Q 
                       - 
                       1 
                     
                     2 
                   
                   ⌋ 
                 
               
               ) 
             
             th 
           
         
       
     
         [0000]    to the 
         [0000]    
       
         
           
             
               ( 
               
                 m 
                 + 
                 
                   ⌈ 
                   
                     
                       Q 
                       - 
                       1 
                     
                     2 
                   
                   ⌉ 
                 
               
               ) 
             
             th 
           
         
       
     
         [0000]    rows of H 2  and the 
         [0000]    
       
         
           
             
               ( 
               
                 
                   m 
                   2 
                 
                 - 
                 
                   ⌊ 
                   
                     
                       
                         P 
                         2 
                       
                       - 
                       1 
                     
                     2 
                   
                   ⌋ 
                 
               
               ) 
             
             th 
           
         
       
     
         [0000]    to the 
         [0000]    
       
         
           
             
               ( 
               
                 
                   m 
                   2 
                 
                 + 
                 
                   ⌈ 
                   
                     
                       
                         P 
                         2 
                       
                       - 
                       1 
                     
                     2 
                   
                   ⌉ 
                 
               
               ) 
             
             th 
           
         
       
     
         [0000]    columns of H 2  but may have some elements therein set to zero. Given the above definitions of H m,m     1     1  and H m,m     2     2 , equation (16) may be more generally organized into the following form: 
         [0000]        Ys   m   =H   m,m     1     1   Xs   m     1     1   +H   m,m     2     2   Xs   m     2     2   +z   m   eq. (23)
 
         [0094]    In equation (23), Xs m     1     1  is defined similarly to Xs m   l  of equation (16) except that the subscript m thereof is substituted by m 1 , and Xs m     2     2  is defined similarly to Xs m   2  of equation (16) except that the subscript m thereof is substituted by m 2 . Furthermore, z m  includes all the remaining terms for the sub-vector Ys m  in the RHS of equation (13) which are left out of the portion H m,m     1     1 Xs m     1     1 +H m,m     2     2  Xs m     2     2 . Accordingly, in this example embodiment of the present invention, the computing unit  442   a  may be configured to calculate the portion z m  by subtracting the portion H m,m     1     1 Xs m     1     1 +H m,m     2     2 Xs m     2     2  from the sub-vector Ys m . 
         [0095]    Moreover, the detector  46   a  may be configured to perform signal detection (for example, MLSE detection) on the whitened received signal {tilde over (Y)}s m  with the aid of sub-matrices H m,m   1  and H m,m     2     2 . 
         [0096]    To operate with the receiver  20   a  which uses the MLSE-based detection performed by either the signal detector  46  illustrated by  FIG. 4A , the computing unit  58  illustrated by  FIG. 4F  or the signal detector  46   a  illustrated by  FIG. 4G , the transmitters  10   a - 1  and  10   a - 2  may be configured to operate with an Alamouti-type coding. Detail operation of such transmitters  10   a - 1  and  10   a - 2  will be discussed in the following paragraphs by reference to  FIGS. 5A to 5F . 
         [0097]      FIG. 5A  illustrates an Alamouti-type coding for the cooperative OFDM communication system  2  illustrated in  FIG. 3A  in accordance with another example embodiment. Referring to  FIG. 5A , data denoted as X 0  and X 1  may be two successive data from a data source (not shown) associated with the transmitters  10   a - 1  and  10   a - 2 . Furthermore, subcarriers indexed “1,0” and “1,1” corresponding to the transmitter  10   a - 1  (which may be also denoted as f 1,0  and f 1,1 ), may be two successive subcarriers in an OFDM symbol. Likewise, subcarriers indexed “2,0” and “2,1” corresponding to the transmitter  10   a - 2  (which may be also denoted as f 2,0  and f 2,1 ), may be two successive subcarriers in an OFDM symbol. With the Alamouti-type coding, the transmitter  10   a - 1  may be configured to transmit data “−X 1 *” over the subcarrier f 1,1 , while the transmitter  10   a - 2  may be configured to transmit data “X 0 *” over the subcarrier f 2,1 , with the superscript “*” denoting complex conjugation. 
         [0098]      FIG. 5B  illustrates carrier frequency offsets (CFOs) in the cooperative OFDM communication system  2  illustrated in  FIG. 3A  in accordance with an example embodiment. Referring to  FIG. 5B , each of the transmitters  10   a - 1  and  10   a - 2  may have a CFO with respect to the receiver  20   a . The carrier frequency of the transmitter  10   a - 1  may be denoted as f c1 , while the carrier frequency of the transmitter  10   a - 2  may be denoted as f c2 . On the other hand, the frequency of a sinusoidal signal generated by a local oscillator (not shown) of the receiver  20   a  may be denoted as f LO . The difference between f c1  and f LO  may be defined as the CFO between the transmitter  10   a - 1  and the receiver  20   a . Likewise, the difference between f c2  and f LO  may be defined as the CFO between the transmitter  10   a - 2  and the receiver  20   a . In this example embodiment, the CFOs for the transmitters  10   a - 1  and  10   a - 2  may be normalized with respect to the subcarrier spacing Δf. Such a normalized CFO for the transmitter  10   a - 1  may be denoted as ∈ 1 , while the normalized CFO for the transmitter  10   a - 2  may be denoted as ∈ 2 . Furthermore, a difference between ∈ i  and ∈ 2  may be denoted as Δ∈. 
         [0099]    In this example embodiment, the receiver  20   a  may be synchronized to the transmitter  10   a - 1 . Therefore, the normalized CFO ∈ 1  may be equal to zero, and the normalized CFO ∈ 2  may thus be equal to Δ∈. Furthermore, the cooperative OFDM communication system  2  may have a MCFO span less than one subcarrier spacing, such as, Δ∈=0.5. Moreover, each of the channels (not shown) between the transmitters  10   a - 1 ,  10   a - 2  and the receiver  20   a  may have a Doppler spread with a nonzero peak Doppler frequency f d =0.5 Hz. 
         [0100]    Regarding such a fractional MCFO span in relation to the subcarrier spacing not exceeding 0.5 in value and such a small Doppler spread, the receiver  20   a  may be configured to perform the blockwise whitening process based on relatively small lengths Q, P 1  and P 2  for the sub-vectors Ys m , Xs m   1  and Xs m   2  and relatively small size for the sub-matrices H m,m   1  and H m,m   2 . For example, each of the sub-vectors Ys m , Xs m   1  and Xs m   2  may have a length of 2, and each of the sub-matrices H m,m   1  and H m,m   2  may have a size of 2×2. In addition, the MLSE-based detection, which may be executed subsequent to the blockwise whitening process, may be performed based on trellis structure formed according to Xs m   1 , Xs m   2 , and the sub-matrices H m,m   1  and H m,m   2 . 
         [0101]      FIG. 5C  illustrates sub-matrices H 5,5   1  and H 5,5   2  of the channels  30   a - 1  and  30   a - 2  in the cooperative OFDM communication system  2  illustrated in  FIG. 3A  in accordance with another example embodiment, as well as the corresponding sub-vectors Ys 5 , Xs 5   1  and Xs 5   2 . Referring to  FIG. 5C  and taking the subcarrier indexed “5” as an example, the trellis structure may be formed according to the center diagonals a 5,5   1  and a 6,6   1  of the sub-matrix H 5,5   1  together with the center diagonals a 5,5   2  and a 6,6   2  of the sub-matrix H 5,5   2 . 
         [0102]      FIG. 5D  illustrates CFOs in a cooperative OFDM communication system  3  in accordance with still another example embodiment, and  FIG. 5E  illustrates the channel matrices H 1  and H 2  of the channels  30   b - 1  and  30   b - 2  in the cooperative OFDM communication system  3  illustrated in  FIG. 5D  in accordance with still another example embodiment. Referring to  FIG. 5D , the cooperative OFDM communication system  3  may be similar to the cooperative OFDM communication system  2  illustrated in  FIGS. 5A and 5B  except that, the cooperative OFDM communication system  3  may have a MCFO span greater than one subcarrier spacing, such as, Δ∈=1.5. Due to such a relatively large MCFO span, the main signal and ICI power associated with the in-band portion of channel matrix H 2  may have a shift with respect to the diagonal, as shown in  FIG. 5E . To cover such a MCFO span and hence the shift of the main signal and ICI power, the receiver  20   b  of the cooperative OFDM communication system  3  may be configured to perform blockwise whitening process and the subsequent MLSE-based detection based on the sub-vectors Ys m  and Xs m   1 , the sub-matrix H m,m   1 , and a shifted sub-vector Xs m−1   2  and a shifted sub-matrix H m,m−1   2 . In this example embodiment, each of the sub-vectors Ys m , Xs m   1  and Xs m−1   2  may have a length of 3, and each of the sub-matrices H m,m   1  and H m,m−1   2  may have a size of 3×3. In addition, the MLSE-based detection, which may be executed subsequent to the blockwise whitening process, may be performed based on trellis structure formed according to Xs m   1 , Xs m−1   2 , and the sub-matrices H m,m   1 , and H m,m−1   2 . 
         [0103]      FIG. 5F  illustrates the sub-matrices H 5,5   1  and H 5,4   2  of the channels  30   b - 1  and  30   b - 2 , together with the corresponding sub-vectors Ys 5 , Xs 5   1  and Xs 4   2 , in the cooperative OFDM communication system  3  illustrated in  FIG. 5D  in accordance with another example embodiment. Referring to  FIG. 5F  and taking the subcarrier indexed “5” as an example, in the sub-matrix H 5,4   2 , main ICI power may have a shift and thus reside on the first sub-diagonal element a 5,4   2 . Accordingly, in this example embodiment, the trellis structure at the subcarrier indexed “5” for the MLSE-based detection may be formed according to the sub-vectors Xs 5   1  and Xs 4   2  and the sub-matrices H 5,5   1  and H 5,4   2 . 
         [0104]    It will be appreciated by those skilled in the art that changes could be made to the examples described above without departing from the broad inventive concept thereof. It is understood, therefore, that the various embodiments are not limited to the particular examples disclosed, but it is intended to cover modifications within the spirit and scope of the various embodiments and as defined by the appended claims. 
         [0105]    Further, in describing representative examples of the various embodiments, the specification may have presented the method and/or process as a particular sequence of steps. However, to the extent that the method or process does not rely on the particular order of steps set forth herein, the method or process should not be limited to the particular sequence of steps described. As one of ordinary skill in the art would appreciate, other sequences of steps may be possible. Therefore, the particular order of the steps set forth in the specification should not be construed as limitations on the claims. In addition, the claims directed to the method and/or process of the various embodiments should not be limited to the performance of their steps in the order written, and one skilled in the art can readily appreciate that the sequences may be varied and still remain within the spirit and scope of the various embodiments.