Patent Publication Number: US-2012033754-A1

Title: Ofdm channel estimation method and apparatus

Description:
CROSS-REFERENCE TO RELATED APPLICATION 
     This application is a continuation of U.S. application Ser. No. 12/265,220, filed Nov. 5, 2008, now pending, the disclosure of which is incorporated herein by reference. 
    
    
     TECHNICAL FIELD 
     The present invention generally relates to channel estimation, and more particularly relates to channel estimation for OFDM systems. 
     BACKGROUND 
     The response of a wireless communication channel is estimated by a wireless receiver so that the receiver can coherently demodulate received data symbols. Channel estimation is typically performed using known symbols referred to as pilot symbols. Pilot symbols are transmitted in the time-frequency domain in OFDM (Orthogonal Frequency-Division Multiplexing) systems. The pilot symbols sample the time-frequency response of the OFDM channel which is a slow-varying, two-dimensional process. The response of an OFDM channel is conventionally estimated based on pilot symbols using minimum mean-square error estimation (MMSE), maximum-likelihood (ML) estimation or other approaches that are suboptimal such as interpolation. 
     MMSE channel estimation typically requires prior knowledge of channel statistics and tends to be highly complex because of the matrix inversion performed during the channel estimation process. The size of the matrix inversion performed by an MMSE estimator depends on the number of pilot symbol observations. In OFDM systems, pilot symbol density must be sufficiently large so that receiving devices can properly reconstruct an accurate representation of the time-frequency channel response. This in turn can dramatically increase the size of the matrix inversion performed by an MMSE channel estimator, significantly increasing MMSE estimation complexity. ML channel estimation also tends to be relatively complex, but not as complex as MMSE estimation. When pilot symbols are regularly spaced, ML estimation can be implemented using a two-dimensional FFT (fast-Fourier transform) process that yields the channel response in the delay-Doppler domain. 
     The complexity of both MMSE and ML estimators tends to increase significantly when pilot symbol spacing is irregular. Many types of OFDM systems have high pilot symbol density and irregular pilot symbol spacing. For example, the downlink specified in IEEE (the Institute of Electrical and Electronics Engineers) standard 802 16e (WiMAX) has 120 pilots for each OFDM symbol. Pilot spacing is also irregular, meaning that pilot symbols are transmitted at different time and frequency intervals. The Long Term Evolution (LTE) standard specified in Release 8 of 3GPP (3rd Generation Partnership Project) also calls for irregular pilot symbol spacing. The matrix inversion performed by a conventional MMSE estimator increases significantly when pilot symbol spacing is irregular. The complexity of a conventional ML estimator also increases when the estimator processes irregularly-spaced pilot symbols. An interpolation-based technique can be used to perform channel estimation when pilot symbol density is relatively high, but is sub-optimal and sacrifices accuracy for less complexity. 
     SUMMARY 
     Smaller patterns of regularly-spaced pilot symbols can be discerned from a larger pattern of irregularly-spaced pilot symbols transmitted in the time-frequency domain. Accordingly, the irregularly-spaced pilot symbols can be partitioned into at least two different groups of regularly-spaced pilot symbols in the time-frequency domain. Each group of regularly-spaced pilot symbols is individually processed with lower complexity and the results combined to generate an accurate time-frequency channel response estimate. 
     According to an embodiment, a set of irregularly-spaced pilot symbols is transmitted over a time-frequency window. Channel response is estimated based on the pilot symbols by grouping the pilot symbols into subsets of regularly-spaced pilot symbols. An intermediate quantity is generated for each subset of regularly-spaced pilot symbols as a function of the pilot symbols included in the subset. The channel response is estimated over the time-frequency window as a function of the intermediate quantities. 
     Of course, the present invention is not limited to the above features and advantages. Those skilled in the art will recognize additional features and advantages upon reading the following detailed description, and upon viewing the accompanying drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  illustrates a block diagram of an embodiment of a wireless receiver. 
         FIG. 2  illustrates an embodiment of a method for estimating channel response based on a set of irregularly-spaced pilot symbols transmitted over a time-frequency window. 
         FIG. 3  illustrates a plot diagram of an embodiment of irregularly-spaced WiMAX pilot symbols grouped into subsets of regularly-spaced pilot symbols. 
         FIG. 4  illustrates a plot diagram of an embodiment of irregularly-spaced LTE pilot symbols grouped into subsets of regularly-spaced pilot symbols. 
         FIG. 5  illustrates an embodiment of a method for estimating channel response based on subsets of regularly-spaced pilot symbols. 
         FIG. 6  illustrates another embodiment of a method for estimating channel response based on subsets of regularly-spaced pilot symbols. 
     
    
    
     DETAILED DESCRIPTION 
       FIG. 1  illustrates an embodiment of a wireless receiver  100  such as a mobile phone or a wireless Local Area Network (LAN) adapter. The receiver  100  processes OFDM signals transmitted in the time-frequency domain, e.g., in accordance with LTE, WiMAX, etc. Irregularly-spaced pilot symbols are transmitted in the time-frequency domain to aid in channel estimation at the receiver  100 . The receiver  100  includes one or more antennas  110 , front end circuitry  120 , a baseband processor  130  and a post processor  140 . Each antenna  110  receives an OFDM signal transmitted over a multipath fading channel. Received signals are down-converted to corresponding baseband signals by the front end circuitry  120  such as one or more amplifiers, filters, mixers, digitizers, etc. The baseband processor  130  demodulates received data symbols based on an estimate of the channel. A channel estimator  132  included in or associated with the baseband processor  130  generates the channel estimate. A pilot symbol partition unit  134  included in or associated with the baseband processor  130  partitions the irregularly-spaced pilot symbols into at least two different groups of regularly-spaced pilot symbols in the time-frequency domain. The channel estimator  132  processes the smaller subsets of regularly-spaced pilot symbols to estimate the channel instead of a single larger set of irregularly-spaced pilot symbols, reducing channel estimation complexity while maintaining sufficient accuracy. The baseband processor  130  uses the channel estimate to obtain soft bit values from the demodulated data symbols, e.g., by weighting and combining various components of the baseband signal. The baseband processor  130  provides the soft bit values to the post processor  140  for subsequent processing such as error correction, etc. 
       FIG. 2  illustrates an embodiment of program logic implemented by the baseband processor  130  for performing channel estimation. The program logic begins by the pilot symbol partition unit  134  grouping a set of irregularly-spaced pilot symbols into subsets of regularly-spaced pilot symbols in the time-frequency domain (Step  200 ). The channel estimator  132  generates an intermediate quantity for each subset of regularly-spaced pilot symbols as a function of the pilot symbols included in the subset (Step  210 ). The channel estimator  132  also computes an estimate of the channel response as a function of the different intermediate quantities (Step  220 ). The baseband processor  130  uses the composite channel estimate obtained based on the pilot symbol groupings to demodulate received data symbols as is well known in the art. According to one embodiment, the irregularly-spaced pilot symbols are grouped into subsets of regularly-spaced pilot symbols based on a priori pilot spacing information. 
       FIG. 3  shows an embodiment of how the pilot symbol partition unit  134  groups a set of irregularly-spaced pilot symbols based on a priori WiMAX pilot spacing information. Pilot symbols transmitted over a particular time-frequency window in accordance with WiMAX are irregularly spaced as shown in  FIG. 3 . The pilot symbol partition unit  134  partitions the irregularly-spaced pilot symbol pattern into at least four subsets of regularly-spaced WiMAX pilot patterns and identifies the pilot symbols associated with each regularly-spaced pattern. The first regularly-spaced pattern has an initial frequency offset (φ) of 0 and a time offset (ψ) of 4. The second regularly-spaced pattern has an initial frequency offset of 0 and a time offset of 8. The third regularly-spaced pattern has an initial frequency offset of 1 and a time offset of 0 and the fourth pattern has an initial frequency offset of 1 and a time offset of 12. The pilot symbols included in each subset are regularly-spaced apart in frequency (M=14 sub-carriers) and time (N=2). 
       FIG. 4  shows an embodiment of how the pilot symbol partition unit  134  groups a set of irregularly-spaced pilot symbols based on a priori LTE pilot spacing information. The pilot symbol partition unit  134  partitions the irregularly-spaced LTE pilot symbol pattern into at least two regularly-spaced pilot patterns over the time-frequency window. The first regularly-spaced pattern has an initial frequency offset (φ) of 0 and a time offset (ψ) of 0. The second regularly-spaced pattern has an initial frequency offset of 4 and a time offset of 3. The pilot symbols included in each subset are regularly-spaced apart in frequency (M=6 sub-carriers) and time (N=7). 
     In yet another embodiment, irregularly-spaced OFDM pilot symbols can be grouped into subsets of regularly-spaced pilot symbols by blindly detecting the communication mode used to transmit the pilot symbols. The baseband processor  130  and/or the pilot symbol partition unit  134  can determine if a pilot symbol pattern is irregular based on the communication mode, and if so, whether the irregular pattern can be partitioned into smaller, regularly-spaced patters. The communication mode may correspond to the transmission technology (e.g., WiMAX, LTE, etc.) or can be any other type of information relating to the communication of information over a wireless channel that can be used to determine pilot spacing. The pilot symbols are then grouped into subsets of regularly-spaced pilot symbols based on the communication mode. In each pilot grouping embodiment, the channel estimator  132  generates a composite channel estimate based on the smaller subsets of regularly-spaced pilot symbols instead of a single larger set of irregularly-spaced pilot symbols. 
       FIG. 5  illustrates one embodiment where the intermediate quantities derived by the channel estimator  132  for each subset of regularly-spaced pilot symbols are computed using an MMSE model. In more detail, a discrete frequency domain received OFDM sample can be expressed as: 
         X[t,f]=H[t,f]A[t,f]+Z[t,f]   (1)
 
     where the index [t,f] corresponds to the f th  frequency sub-carrier in the t th  OFDM symbol, H[t,f] is the time-frequency response of the channel at a particular point, Λ[t,f] is the transmitted symbol and Z[t,f] represents Additive White Gaussian Noise (AWGN). The baseband processor  130  estimates the channel response H[t,f] based on known pilot symbols and uses the channel response H[t,f] to coherently demodulate received data symbols. The pilot observations X[t,f] can be concisely expressed in matrix form by arranging the observations into a column vector as given by: 
         X   1 =Λ 1   H   1   +Z   1    (2)
 
     where Λ 1  is a diagonal matrix containing the pilot symbols as diagonal elements and X 1 , H 1  and Z 1  are column vectors of the same size corresponding to the pilot observation, channel and noise, respectively. 
     The observations from all regular pilots can be stacked into a column vector as given by: 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             X 
                             1 
                           
                         
                       
                       
                         
                           
                             X 
                             2 
                           
                         
                       
                       
                         
                           M 
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       
                         [ 
                         
                           
                             
                               
                                 Λ 
                                 1 
                               
                             
                             
                               0 
                             
                             
                               Λ 
                             
                           
                           
                             
                               0 
                             
                             
                               
                                 Λ 
                                 2 
                               
                             
                             
                               Λ 
                             
                           
                           
                             
                               M 
                             
                             
                               M 
                             
                             
                               O 
                             
                           
                         
                         ] 
                       
                        
                       
                         [ 
                         
                           
                             
                               
                                 H 
                                 1 
                               
                             
                           
                           
                             
                               
                                 H 
                                 2 
                               
                             
                           
                           
                             
                               M 
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       [ 
                       
                         
                           
                             
                               Z 
                               1 
                             
                           
                         
                         
                           
                             
                               Z 
                               2 
                             
                           
                         
                         
                           
                             M 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     The column-vector form of equation (3) can be converted to matrix-form by introducing the subscript t as given by: 
         X   t =Λ t   H   t   +Z   t    (4)
 
     In one embodiment, the channel H[t, f] can be modeled as a two-dimensional, zero-mean White Sense Stationary (WSS) Gaussian random process having the following correlation: 
       Γ[t 1 −t 2 ]≡E{H[t 1 ,f 1 ]H*[t 2 f 2 ]}  (5)
 
     The channel correlation represented by equation (5) is typically known or can be estimated from past observation. The MMSE channel estimation can be formulated based on the channel statistics and the matrix representation of equation (4) as explained next. 
     In more detail, let H denote the column vector containing the channel response at the locations to be estimated in the time-frequency domain. For example, this can be any part of the 16 OFDM symbols by 56 frequency sub-carriers shown in the window area of  FIG. 3 . The MMSE estimate of H is the mean of H conditioned on the observation X t  as given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             H 
                             ^ 
                           
                            
                           
                             ( 
                             
                               X 
                               t 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             E 
                              
                             
                               { 
                               
                                 H 
                                 | 
                                 
                                   X 
                                   t 
                                 
                               
                               } 
                             
                           
                           = 
                           
                             
                               Π 
                               HXt 
                             
                              
                             
                               Π 
                               Xt 
                               
                                 - 
                                 1 
                               
                             
                              
                             
                               X 
                               t 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             Π 
                             HHt 
                           
                            
                           
                             
                               
                                 Λ 
                                 t 
                                 H 
                               
                                
                               
                                 ( 
                                 
                                   
                                     
                                       Λ 
                                       t 
                                     
                                      
                                     
                                       Π 
                                       Ht 
                                     
                                      
                                     
                                       Λ 
                                       t 
                                       H 
                                     
                                   
                                   + 
                                   
                                     
                                       σ 
                                       Z 
                                       2 
                                     
                                      
                                     I 
                                   
                                 
                                 ) 
                               
                             
                             
                               - 
                               1 
                             
                           
                            
                           
                             X 
                             t 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     where Π Ht =E{H t H t   H } denotes the auto covariance matrix of H t , Π HHt =E{HH t   H } is the covariance matrix between H and H t  and H HXt  is the similarly defined covariance matrix between H and X t . The elements in the different covariance matrices can be derived using the channel correlation function of equation (5). Accordingly, the MMSE channel estimation process becomes a linear operation that transforms the pilot observation vector X t  into a vector containing the channel estimates at the desired locations. 
     The pilot symbol observations  500  corresponding to the i th  regular pilot symbol pattern over the time-frequency window limited by the symbol range 0≦t&lt;N SYM  and the frequency sub-carrier range 0≦f&lt;N SUB  are given by: 
     
       
         
           
             
               
                 
                   
                     
                       X 
                       i 
                     
                      
                     
                       [ 
                       
                         t 
                         , 
                         f 
                       
                       ] 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 
                                   X 
                                    
                                   
                                     [ 
                                     
                                       T 
                                       , 
                                       f 
                                     
                                     ] 
                                   
                                 
                                  
                                 
                                     
                                 
                                  
                                 for 
                                  
                                 
                                     
                                 
                                  
                                 t 
                               
                               = 
                               
                                 nN 
                                 + 
                                 
                                   ϕ 
                                   i 
                                 
                               
                             
                             , 
                           
                         
                         
                           
                             f 
                             = 
                             
                               mM 
                               + 
                               
                                 ψ 
                                 i 
                               
                             
                           
                         
                       
                       
                         
                           
                             0 
                             , 
                           
                         
                         
                           otherwise 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     where φ i  represents frequency offset, ψ i  represents time offset, 0≦n&lt;Q, 0≦m&lt;K, N SYM =NQ and N SUB =MK . A more concise matrix expression of equation (4) when considering i subsets of regularly-spaced pilot symbols is given by: 
         X   i;n,m =Λ i   H   i;n,m   +Z   i;n,m    (8)
 
     According to the MMSE estimation embodiment, the intermediate quantity derived for each subset of regularly-spaced pilot symbols is an intermediate channel response estimate Ĥ i [t, f]  510  computed using equations (6) and (8) where the index i corresponds to the i th  regularly-spaced pilot pattern  500  under consideration. Thus, each subset of regularly-spaced pilot symbols determined by the pilot symbol partition unit  134  is used to generate an intermediate MMSE channel estimate Ĥ i [t, f]  510 . A signal combiner  520  included in or associated with the channel estimator  132  combines the different intermediate MMSE channel estimates Ĥ i [t, f] to form a composite MMSE estimate of the channel as given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         H 
                         ^ 
                       
                       COMP 
                     
                      
                     
                       [ 
                       
                         t 
                         , 
                         f 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       ∑ 
                       i 
                     
                      
                     
                       
                         
                           H 
                           ^ 
                         
                         i 
                       
                        
                       
                         [ 
                         
                           t 
                           , 
                           f 
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
       FIG. 6  illustrates another embodiment where the intermediate quantities derived by the channel estimator  132  for each subset of regularly-spaced pilot symbols are computed using a delay-Doppler model. In more detail, an intermediate delay-Doppler image  610  of the channel defined over the delay range 0≦τ&lt;K and the Doppler range 0≦v&lt;Q and derived from the i th  regular pilot pattern  600  is given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             I 
                             i 
                           
                            
                           
                             [ 
                             
                               τ 
                               , 
                               v 
                             
                             ] 
                           
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               t 
                               = 
                               0 
                             
                             
                               
                                 N 
                                 SYM 
                               
                               - 
                               1 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 f 
                                 = 
                                 0 
                               
                               
                                 
                                   N 
                                   SUB 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 
                                   X 
                                   i 
                                 
                                  
                                 
                                   [ 
                                   
                                     t 
                                     , 
                                     f 
                                   
                                   ] 
                                 
                               
                                
                               
                                  
                                 
                                   
                                     
                                       j2π 
                                        
                                       
                                           
                                       
                                        
                                       f 
                                        
                                       
                                           
                                       
                                        
                                       τ 
                                     
                                     
                                       N 
                                       SUB 
                                     
                                   
                                   - 
                                   
                                     
                                       j2π 
                                        
                                       
                                           
                                       
                                        
                                       v 
                                        
                                       
                                           
                                       
                                        
                                       t 
                                     
                                     
                                       N 
                                       SYM 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               Q 
                               - 
                               1 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 m 
                                 = 
                                 0 
                               
                               
                                 K 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 X 
                                 
                                   
                                     i 
                                     ; 
                                     n 
                                   
                                   , 
                                   m 
                                 
                               
                                
                               
                                  
                                 
                                   
                                     j2π 
                                      
                                     
                                       ( 
                                       
                                         mM 
                                         + 
                                         
                                           ψ 
                                           i 
                                         
                                       
                                       ) 
                                     
                                   
                                   KM 
                                 
                               
                                
                               
                                  
                                 
                                   
                                     j2π 
                                      
                                     
                                         
                                     
                                      
                                     
                                       v 
                                        
                                       
                                         ( 
                                         
                                           nM 
                                           + 
                                           
                                             ϕ 
                                             i 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   NQ 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                              
                             
                               
                                 
                                   
                                     j2πψ 
                                     i 
                                   
                                    
                                   τ 
                                 
                                 KM 
                               
                               - 
                               
                                 
                                   j2π 
                                    
                                   
                                       
                                   
                                    
                                   v 
                                    
                                   
                                       
                                   
                                    
                                   
                                     ϕ 
                                     i 
                                   
                                 
                                 NQ 
                               
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 Q 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   m 
                                   = 
                                   0 
                                 
                                 
                                   K 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   X 
                                   
                                     
                                       i 
                                       ; 
                                       n 
                                     
                                     , 
                                     m 
                                   
                                 
                                  
                                 
                                    
                                   
                                     
                                       j 
                                        
                                       
                                           
                                       
                                        
                                       2 
                                        
                                       π 
                                        
                                       
                                           
                                       
                                        
                                       m 
                                        
                                       
                                           
                                       
                                        
                                       τ 
                                     
                                     K 
                                   
                                 
                                  
                                 
                                    
                                   
                                     - 
                                     
                                       
                                         j 
                                          
                                         
                                             
                                         
                                          
                                         2 
                                          
                                         π 
                                          
                                         
                                             
                                         
                                          
                                         v 
                                          
                                         
                                             
                                         
                                          
                                         n 
                                       
                                       Q 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     Each intermediate delay-Doppler image I i [τ, v]  610  represents the two-dimensional DFT (discrete Fourier transform) of the observation  600  of size (Q,K) corresponding to the i th regular pilot symbol pattern. Patterns of different offsets (φ i ,ψ i )are phase modulated by the leading term in equation (10). A signal combiner  620  included in or associated with the channel estimator  132  combines the different intermediate delay-Doppler images I i [τ,v]  610  to obtain a composite delay-Doppler image as given by: 
     
       
         
           
             
               
                 
                   
                     
                       I 
                       COMP 
                     
                      
                     
                       [ 
                       
                         τ 
                         , 
                         v 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       ∑ 
                       i 
                     
                      
                     
                       
                         I 
                         i 
                       
                        
                       
                         [ 
                         
                           τ 
                           , 
                           v 
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     The baseband processor  130  and/or the channel estimator  132  convert the composite delay-Doppler image I COMP [τ, v] from the delay-Doppler domain to the time-frequency domain to obtain a composite estimate Ĥ COMP  of the channel. In one embodiment, the composite delay-Doppler image is converted to the time-frequency domain by a two-dimensional inverse DFT (IDFT) function  630  of size(N SYM ,N SUB ). In one embodiment, the composite delay-Doppler image I COMP [τ,v] is filtered before conversion to the time-frequency domain. According to this embodiment, the two-dimensional IDFT function  630  is performed over the entire time-frequency window to yield the composite channel estimate Ĥ COMP  as given by: 
     
       
         
           
             
               
                 
                   
                     
                       
                         H 
                         ^ 
                       
                       COMP 
                     
                      
                     
                       [ 
                       
                         t 
                         , 
                         f 
                       
                       ] 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         τ 
                         = 
                         0 
                       
                       
                         
                           N 
                           SUB 
                         
                         - 
                         1 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           v 
                           = 
                           0 
                         
                         
                           
                             N 
                             SYM 
                           
                           - 
                           1 
                         
                       
                        
                       
                         
                           
                             I 
                             COMP 
                           
                            
                           
                             [ 
                             
                               τ 
                               , 
                               v 
                             
                             ] 
                           
                         
                          
                         
                           W 
                            
                           
                             [ 
                             
                               τ 
                               , 
                               v 
                             
                             ] 
                           
                         
                          
                         
                            
                           
                             
                               
                                 - 
                                 j2π 
                               
                                
                               
                                   
                               
                                
                               f 
                                
                               
                                   
                               
                                
                               t 
                             
                             
                               N 
                               SUB 
                             
                           
                         
                          
                         
                            
                           
                             
                               j2π 
                                
                               
                                   
                               
                                
                               v 
                                
                               
                                   
                               
                                
                               t 
                             
                             
                               N 
                               SYM 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     where W[τ,v] is a windowing function for removing sampling artifacts. In one embodiment, the windowing function W[τ,v] is a two-dimensional brick-wall filter that eliminates non-zero elements of the composite delay-Doppler image outside the maximum delay-Doppler spread as given by: 
     
       
         
           
             
               
                 
                   
                     W 
                      
                     
                       [ 
                       
                         τ 
                         , 
                         v 
                       
                       ] 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               
                                 1 
                                  
                                 
                                     
                                 
                                  
                                 for 
                                  
                                 
                                     
                                 
                                  
                                 0 
                               
                               ≤ 
                               
                                 τ 
                                 max 
                               
                             
                             , 
                           
                         
                         
                           
                             0 
                             ≤ 
                             v 
                             &lt; 
                             
                               v 
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     With the above range of variations and applications in mind, it should be understood that the present invention is not limited by the foregoing description, nor is it limited by the accompanying drawings. Instead, the present invention is limited only by the following claims, and their legal equivalents.