Abstract:
A method and apparatus are disclosed for canceling cross-talk in a frequency-division multiplexed communication system. The disclosed frequency-division multiplexed communication system employs multiple carriers having overlapping channels and provides an improved cross-talk cancellation mechanism to address the resulting interference. Bandwidth compression is achieved using n level amplitude modulation in each frequency band. An FDM receiver is also disclosed that decomposes the received broadband signal into each of its respective frequency bands and returns the signal to baseband in the analog domain. Analog requirements are relaxed by removing cross-talk from adjacent RF channels, from image bands, and minimizing the performance degradation caused by In-phase and Quadrature-phase (I/Q) phase and gain mismatches in modulators and demodulators. The disclosed transmitter or receiver (or both) can be fabricated on a single integrated circuit.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a divisional of U.S. patent application Ser. No. 12/020,722, filed Jan. 28, 2008, which is continuation of U.S. patent application Ser. No. 10/219,906, filed Aug. 15, 2002, which claims the benefit of U.S. Provisional Application Ser. No. 60/366,923, filed Mar. 22, 2002, each incorporated by reference herein. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The present invention relates generally to cross-talk cancellation techniques, and more particularly, to methods and apparatus for reducing cross-talk in frequency division multiplexed (FDM) communication systems. 
       BACKGROUND OF THE INVENTION 
       [0003]    The explosive growth of digital communications technology has resulted in an ever-increasing demand for bandwidth for communicating digital information, such as data, audio and/or video information. To keep pace with the increasing bandwidth demands, new or improved network components and technologies must constantly be developed to perform effectively at the ever-increasing data rates. In optical communication systems, however, the cost of deploying improved optical components becomes prohibitively expensive at such higher data rates. For example, it is estimated that the cost of deploying a 40 Gbps optical communication system would exceed the cost of existing 10 Gbps optical communication systems by a factor of ten. Meanwhile, the achievable throughput increases only by a factor of four. 
         [0004]    Thus, much of the research in the area of optical communications has attempted to obtain higher throughput from existing optical technologies. For example, a number of techniques have been proposed or suggested to employ multi-carrier transmission techniques over fiber channels. Conventional multi-carrier transmission techniques, however, space the multiple carriers so that they do not interfere with one another. The required carrier spacing, however, leads to poor spectral efficiency and thus limits the throughput that can be achieved within the available frequencies. A further proposal has suggested the use of orthogonal carrier frequencies to minimize interference. A system employing orthogonal carrier frequencies, however, will require an all-digital implementation that is particularly challenging with existing analog-to-digital and digital-to-analog converters at optical rates (10 Gbps and higher). 
         [0005]    A need therefore exists for a multi-carrier transmission technique that provides improved spectral efficiency and allows for an analog implementation. Among other benefits, improved spectral efficiency will allow greater tolerance to dispersion and the use of generic and available optical technologies. 
       SUMMARY OF THE INVENTION 
       [0006]    Generally, a method and apparatus are disclosed for canceling cross-talk in a frequency-division multiplexed communication system. The disclosed frequency-division multiplexing communication system employs multiple carriers having overlapping channels and provides an improved cross-talk cancellation mechanism to address the resulting interference. 
         [0007]    Generally, the carrier spacing of the multiple carriers can approach the Nyquist limit. Another feature of the invention achieves bandwidth compression using n level amplitude modulation in each frequency band. The multi-level signaling improves spectral efficiency by a factor of log 2 n, at the expense of a higher signal-to-noise ratio (SNR) requirement. 
         [0008]    According to another feature of the invention, an FDM receiver is disclosed where the received broadband signal is decomposed into each of its respective frequency bands and returned to baseband all in the analog domain. Duncdcrto relax the analog requirements, a digital cross-talk canceller is also disclosed that removes cross-talk from adjacent RF chaimels, from image bands, and minimizes the performance degradation caused by In-phase and Quadrature-phase (I/Q) phase and gain mismatches in modulators and demodulators. The disclosed transmitter or receiver (or both) can be fabricated on a single integrated circuit. 
         [0009]    A more complete understanding of the present invention, as well as further features and advantages of the present invention, will be obtained by reference to the following detailed description and drawings. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0010]      FIG. 1  illustrates a frequency-division multiplexing (FDM) system in which the present invention can operate; 
           [0011]      FIG. 2  illustrates a desired band and image bands at the frequency ω k  and t−ω k , respectively, that result from the modulation of a symbol s n,k ; 
           [0012]      FIG. 3  illustrates the return of a carrier frequency, ω j , to baseband and the resulting image signal at 2ω j ; 
           [0013]      FIG. 4  illustrates an exemplary matched filter p(t) that may be employed in the transmitter of  FIG. 1 ; 
           [0014]      FIG. 5  illustrates three adjacent carrier signals and their corresponding image bands following filtering by a pulse shaping filter p(t) embodied as a rectangular function of duration T′; 
           [0015]      FIG. 6  illustrates three adjacent carrier signals following filtering by a pulse shaping filter p(t) embodied as a square root raised cosine filter of period T; 
           [0016]      FIG. 7  illustrates a particular example of a pulse shaping filter p(t) embodied as a square root raised cosine filter of period T; 
           [0017]      FIG. 8  is a schematic block diagram of a receiver  800  incorporating a cross-talk cancellation scheme in accordance with the present invention; 
           [0018]      FIG. 9  is a schematic block diagram illustrating the cross-talk cancellation scheme of  FIG. 8  in further detail; 
           [0019]      FIG. 10  illustrates a rule that may be employed to simplify the structure of  FIG. 9 ; 
           [0020]      FIG. 11  employs the rule of  FIG. 10  to illustrate two equivalent representations of the filter structures of  FIG. 9 ; 
           [0021]      FIG. 12  is a schematic block diagram of one variation of the cross-talk cancellation scheme of  FIG. 9 ; 
           [0022]      FIG. 13  is a schematic block diagram of another variation of the cross-talk cancellation scheme of  FIG. 9 ; and 
           [0023]      FIG. 14  is a schematic block diagram of a receiver incorporating a generalized cross-talk cancellation/equalization scheme in accordance with the present invention. 
       
    
    
     DETAILED DESCRIPTION 
       [0024]      FIG. 1  illustrates a frequency-division multiplexing (FDM) system  100  in which the present invention can operate. According to one aspect of the invention, the exemplary frequency-division multiplexing (FDM) system  100  employs multiple carriers with arbitrary carrier frequencies having carrier spacings that are close to the Nyquist limit. The present invention also achieves bandwidth compression by making use of n point (constellation) amplitude modulated transmission in each frequency band. The multi-level signaling improves spectral efficiency by a factor of log 2 n, at the expense of a higher signal-to-noise ratio (SNR) requirement. 
         [0025]    In one exemplary implementation of a 40 Gbps system, there are 16 quadrature amplitude modulated (QAM) constellations (four levels each for in-phase and quadrature-phase bands) having 2.5 Gbps per carrier frequency. In this system, each channel requires a bandwidth greater than 622 MHz (the baud rate per channel is 622 Mbaud). Therefore, the total bandwidth must be slightly greater than 10 GHz (16×622 MHz). Thus, the multi-channel QAM transmission scheme can provide a 40 Gbps throughput using only 10 Gbps optical components. The present invention recognizes that while the close spacing of each carrier exhibits excellent spectral efficiency, cross-talk will be introduced in the received signal. Thus, another aspect of the invention provides an improved cross-talk cancellation mechanism. 
         [0026]    As shown in  FIG. 1 , K input broadband RF signals, S n,0  through S n,k−1 , each corresponding to one of K frequency bands, are received at an input of a transmitter  110 . Each of the K input signals are applied to pulse shaping filters p(t)  115 - 0  through  115 -K−1 and then converted to the corresponding carrier frequency using an array of input multipliers  120 - 0  through  120 -K−1. The K carrier frequency signals are then added by an adder  125  and typically only the real component is transmitted (please note that for single sideband (SSB) signals, both real and imaginary signals can be used) before being filtered to extract the real components at stage  130 . The signal is then transmitted over a channel  140  that is represented as a band-limited filter  142  and an adder  144  that introduces noise, N(t). 
         [0027]    The transmitted signal is received by a receiver  150 . As shown in  FIG. 1 , the broadband RF signal is received at an input of the receiver  150 . The input broadband signal is then decomposed into K frequency bands and returned to baseband using an array of input multipliers  160 - 0  through  160 -K−1. Thereafter, each of the K baseband signals is filtered by a corresponding matched pulse shaping filter p(−t)  170 - 0  through  170 -K−1 and sampled by a corresponding switch  180 - 0  through  180 -K−1 at a sampling period, T, to produce the sampled signal elements, rx n,0  through rx n,K− 1. A Fourier transform is applied to the sampled signal elements by a signal processor  190  to produce the digitized signal y n,0  through y n,K− 1 (K samples at a time). 
         [0028]    Let tx(t) be the transmit signal: 
         [0000]    
       
         
           
             
               tx 
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               2 
                
               
                 Re 
                 ( 
                 
                   
                     ∑ 
                     n 
                   
                    
                   
                     
                       p 
                        
                       
                         ( 
                         
                           t 
                           - 
                           nT 
                         
                         ) 
                       
                     
                     · 
                     
                       
                         ∑ 
                         k 
                       
                        
                       
                         
                           s 
                           
                             n 
                             , 
                             k 
                           
                         
                          
                         
                            
                           
                             
                               ω 
                               k 
                             
                              
                             t 
                           
                         
                       
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               tx 
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 n 
               
                
               
                 ( 
                 
                   
                     p 
                      
                     
                       ( 
                       
                         t 
                         - 
                         nT 
                       
                       ) 
                     
                   
                   · 
                   
                     
                       ∑ 
                       k 
                     
                      
                     
                       ( 
                       
                         
                           
                             s 
                             
                               n 
                               , 
                               k 
                             
                           
                            
                           
                              
                             
                               
                                 ω 
                                 k 
                               
                                
                               t 
                             
                           
                         
                         + 
                         
                           
                             s 
                             
                               n 
                               , 
                               k 
                             
                             * 
                           
                            
                           
                              
                             
                               
                                 - 
                                 
                                   ω 
                                   k 
                                 
                               
                                
                               t 
                             
                           
                         
                       
                       ) 
                     
                   
                 
                 ) 
               
             
           
         
       
     
         [0000]    where s n,k  is the nth symbol transmitted on the carrier k: 
         [0029]    Thus, as shown in  FIG. 2 , when the symbol s n,k  is modulated, you get the desired main signal at the carrier frequency W k  and the corresponding image band at the frequency—ω k . The received signal rx j (t) on carrier j can be written (for the case of a perfect channel  140 ): 
         [0000]        rx   j ( t )=∫ tx ( u )· e   −iw     j     u   ·p (−( t−u ))· du  
 
         [0030]      FIG. 3  illustrates the return of a carrier frequency, ω j , to a baseband signal and the resulting image signal at 2 ω j . The resulting image band signal is generally filtered using an analog filter. The received signal rx j (t) can be expressed as follows: 
         [0000]    
       
         
           
             
               
                 rx 
                 j 
               
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 n 
               
                
               
                 
                   ∑ 
                   k 
                 
                  
                 
                   ⌊ 
                   
                     
                       
                         s 
                         
                           n 
                           , 
                           k 
                         
                       
                        
                       
                         ∫ 
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 u 
                                 - 
                                 nT 
                               
                               ) 
                             
                           
                            
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   - 
                                   t 
                                 
                                 + 
                                 u 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       
                                         ω 
                                         j 
                                       
                                       - 
                                       
                                         ω 
                                         k 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                               u 
                             
                           
                            
                           
                              
                             u 
                           
                         
                       
                     
                     + 
                     
                       
                         s 
                         
                           n 
                           , 
                           k 
                         
                         * 
                       
                        
                       
                         ∫ 
                         
                           
                             p 
                              
                             
                               ( 
                               
                                 u 
                                 - 
                                 nT 
                               
                               ) 
                             
                           
                            
                           
                             p 
                              
                             
                               ( 
                               
                                 
                                   - 
                                   t 
                                 
                                 + 
                                 u 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       
                                         ω 
                                         j 
                                       
                                       + 
                                       
                                         ω 
                                         k 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                               u 
                             
                           
                            
                           
                              
                             u 
                           
                         
                       
                     
                   
                   ⌋ 
                 
               
             
           
         
       
       
         
           
             
               
                 rx 
                 j 
               
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 n 
               
                
               
                 
                   ∑ 
                   k 
                 
                  
                 
                   ⌊ 
                   
                     
                       
                         s 
                         
                           n 
                           , 
                           k 
                         
                       
                        
                       
                          
                         
                           
                              
                              
                             
                               ( 
                               
                                 
                                   ω 
                                   j 
                                 
                                 - 
                                 
                                   ω 
                                   k 
                                 
                               
                               ) 
                             
                           
                            
                           nT 
                         
                       
                        
                       
                         ∫ 
                         
                           
                             p 
                              
                             
                               ( 
                               v 
                               ) 
                             
                           
                            
                           
                             p 
                              
                             
                               ( 
                               
                                 nT 
                                 - 
                                 t 
                                 + 
                                 v 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       
                                         ω 
                                         j 
                                       
                                       - 
                                       
                                         ω 
                                         k 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                               v 
                             
                           
                            
                           
                              
                             v 
                           
                         
                       
                     
                     + 
                     
                       
                         s 
                         
                           n 
                           , 
                           k 
                         
                         * 
                       
                        
                       
                          
                         
                           
                              
                              
                             
                               ( 
                               
                                 
                                   ω 
                                   j 
                                 
                                 + 
                                 
                                   ω 
                                   k 
                                 
                               
                               ) 
                             
                           
                            
                           nT 
                         
                       
                        
                       
                         ∫ 
                         
                           
                             p 
                              
                             
                               ( 
                               v 
                               ) 
                             
                           
                            
                           
                             p 
                              
                             
                               ( 
                               
                                 nT 
                                 - 
                                 t 
                                 + 
                                 v 
                               
                               ) 
                             
                           
                            
                           
                              
                             
                               
                                 - 
                                 
                                    
                                    
                                   
                                     ( 
                                     
                                       
                                         ω 
                                         j 
                                       
                                       + 
                                       
                                         ω 
                                         k 
                                       
                                     
                                     ) 
                                   
                                 
                               
                                
                               v 
                             
                           
                            
                           
                              
                             v 
                           
                         
                       
                     
                   
                   ⌋ 
                 
               
             
           
         
       
     
         [0031]    The values a(t) and b(t) are defined as follows: 
         [0000]        a   j,k ( t )=∫ p ( v ) p ( v−t ) e   −i(ω     j     −ω   k   )v   dv  
 
         [0000]        b   j,k ( t )=∫ p ( v ) p ( v−t ) e   −i(ω     j     −ω   k   )v   dv  
 
         [0032]    The receive signal rx j (t) is then written in a more compact form: 
         [0000]    
       
         
           
             
               
                 rx 
                 j 
               
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 n 
               
                
               
                 
                   ∑ 
                   k 
                 
                  
                 
                   ⌊ 
                   
                     
                       
                         
                           
                             a 
                             
                               j 
                               , 
                               k 
                             
                           
                            
                           
                             ( 
                             
                               t 
                               - 
                               nT 
                             
                             ) 
                           
                         
                         · 
                         
                            
                           
                             
                                
                                
                               
                                 ( 
                                 
                                   
                                     ω 
                                     j 
                                   
                                   - 
                                   
                                     ω 
                                     k 
                                   
                                 
                                 ) 
                               
                             
                              
                             nT 
                           
                         
                       
                        
                       
                         s 
                         
                           n 
                           , 
                           k 
                         
                       
                     
                     + 
                     
                       
                         
                           
                             b 
                             
                               j 
                               , 
                               k 
                             
                           
                            
                           
                             ( 
                             
                               t 
                               - 
                               nT 
                             
                             ) 
                           
                         
                         · 
                         
                            
                           
                             
                                
                                
                               
                                 ( 
                                 
                                   
                                     ω 
                                     j 
                                   
                                   + 
                                   
                                     ω 
                                     k 
                                   
                                 
                                 ) 
                               
                             
                              
                             nT 
                           
                         
                       
                        
                       
                         s 
                         
                           n 
                           , 
                           k 
                         
                         * 
                       
                     
                   
                   ⌋ 
                 
               
             
           
         
       
     
         [0000]    where the first term represents the main signal and cross-talk with other carriers and the second term represents the image bands after demodulation. The second term may be negligible if receive filters are very selective, or for particular choices of the carrier frequencies coj. 
         [0033]    The sampled receive signal rx j (mT) is given by: 
         [0000]    
       
         
           
             
               
                 rx 
                 j 
               
                
               
                 ( 
                 mT 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 n 
               
                
               
                 
                   ∑ 
                   k 
                 
                  
                 
                   ⌊ 
                   
                     
                       
                         
                           
                             a 
                             
                               j 
                               , 
                               k 
                             
                           
                            
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   m 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               T 
                             
                             ) 
                           
                         
                         · 
                         
                            
                           
                             
                                
                                
                               
                                 ( 
                                 
                                   
                                     ω 
                                     j 
                                   
                                   - 
                                   
                                     ω 
                                     k 
                                   
                                 
                                 ) 
                               
                             
                              
                             nT 
                           
                         
                       
                        
                       
                         s 
                         
                           n 
                           , 
                           k 
                         
                       
                     
                     + 
                     
                       
                         
                           
                             b 
                             
                               j 
                               , 
                               k 
                             
                           
                            
                           
                             ( 
                             
                               
                                 ( 
                                 
                                   m 
                                   - 
                                   n 
                                 
                                 ) 
                               
                                
                               T 
                             
                             ) 
                           
                         
                         · 
                         
                            
                           
                             
                                
                                
                               
                                 ( 
                                 
                                   
                                     ω 
                                     j 
                                   
                                   + 
                                   
                                     ω 
                                     k 
                                   
                                 
                                 ) 
                               
                             
                              
                             nT 
                           
                         
                       
                        
                       
                         s 
                         
                           n 
                           , 
                           k 
                         
                         * 
                       
                     
                   
                   ⌋ 
                 
               
             
           
         
       
     
         [0034]    A convenient simplification is obtained when assuming that the shaping filter&#39;s impulse response p(t) has a finite duration T (T=baud-period): 
         [0000]        a   j,k (( m−n ) T )=0 if  m≠n    
         [0000]        a   j,k (( m−n ) T )= a   j,k  if  m=n    
         [0000]        b   j,k (( m−n ) T )=0 if  m≠n    
         [0000]        b   j,k (( m−n ) T )= b   j,k  if  m=n    
         [0000]    rx j (mT) is therefore expressed in the simplified form: 
         [0000]    
       
         
           
             
               
                 rx 
                 j 
               
                
               
                 ( 
                 mT 
                 ) 
               
             
             = 
             
               
                 ∑ 
                 k 
               
                
               
                 ⌊ 
                 
                   
                     
                       
                         a 
                         
                           j 
                           , 
                           k 
                         
                       
                       · 
                       
                          
                         
                           
                              
                              
                             
                               ( 
                               
                                 
                                   ω 
                                   j 
                                 
                                 - 
                                 
                                   ω 
                                   k 
                                 
                               
                               ) 
                             
                           
                            
                           mT 
                         
                       
                     
                      
                     
                       s 
                       
                         m 
                         , 
                         k 
                       
                     
                   
                   + 
                   
                     
                       
                         b 
                         
                           j 
                           , 
                           k 
                         
                       
                       · 
                       
                          
                         
                           
                              
                              
                             
                               ( 
                               
                                 
                                   ω 
                                   j 
                                 
                                 + 
                                 
                                   ω 
                                   k 
                                 
                               
                               ) 
                             
                           
                            
                           mT 
                         
                       
                     
                      
                     
                       s 
                       
                         m 
                         , 
                         k 
                       
                       * 
                     
                   
                 
                 ⌋ 
               
             
           
         
       
     
         [0000]    where: 
         [0000]        a   j,k=∫p   2 ( v ) e   −i(ω     j     −ω     k     )v   dv    
         [0000]        b   j,k   =∫p   2 ( v ) e   −i(ω     j     −ω     k     )v   dv    
         [0035]    The pulse shaping filters p(t) in the transmitter  110  and the corresponding matched filters p(−t) in the receiver may be embodied as rectangular functions of duration T′ (where T′≦T) or as square root raised cosine filters of period T, and excess bandwidth α. When the pulse shaping filters p(t) and corresponding matched filters p(−t) are embodied as rectangular functions of duration T′ (where T′≦T), in an exemplary return-to-zero (RZ) transmitter and an “integrate and dump” receiver, the filter p(t), shown in  FIG. 4 , can be expressed as follows (assuming ISI free): 
         [0000]        p ( t )=Π T′ ( t ) T′≦T  
 
         [0000]    where: 
         [0000]    
       
         
           
             
               
                 Π 
                 
                   T 
                   ′ 
                 
               
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     1 
                   
                   
                     
                       
                         if 
                         - 
                         
                           
                             T 
                             ′ 
                           
                           / 
                           2 
                         
                       
                       ≤ 
                       t 
                       ≤ 
                       
                         
                           + 
                           
                             T 
                             ′ 
                           
                         
                         / 
                         2 
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     otherwise 
                   
                 
               
             
           
         
       
     
         [0036]    The value of a can be computed (and similarly b) as follows: 
         [0000]        a   j,k ( mT )=∫Π T′ ( v )Π T′ ( v−mT ) e   −i(ω     j     −ω     k     )v   dv  
 
         [0000]    If a is 0 and m is non zero, the following is true: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       a 
                       
                         j 
                         , 
                         k 
                       
                     
                      
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                     
                    
                   
                     ∫ 
                     
                       
                         
                           Π 
                           
                             T 
                             ′ 
                           
                         
                          
                         
                           ( 
                           v 
                           ) 
                         
                       
                        
                       
                         
                           Π 
                           
                             T 
                             ′ 
                           
                         
                          
                         
                           ( 
                           v 
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                             
                                
                                
                               
                                 ( 
                                 
                                   
                                     ω 
                                     j 
                                   
                                   - 
                                   
                                     ω 
                                     k 
                                   
                                 
                                 ) 
                               
                             
                           
                            
                           v 
                         
                       
                        
                       
                          
                         v 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       ∫ 
                       
                         
                           - 
                           
                             T 
                             ′ 
                           
                         
                         / 
                         2 
                       
                       
                         
                           + 
                           
                             T 
                             ′ 
                           
                         
                         / 
                         2 
                       
                     
                      
                     
                       
                          
                         
                           
                             - 
                             
                                
                                
                               
                                 ( 
                                 
                                   
                                     ω 
                                     j 
                                   
                                   - 
                                   
                                     ω 
                                     k 
                                   
                                 
                                 ) 
                               
                             
                           
                            
                           v 
                         
                       
                        
                       
                          
                         v 
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       [ 
                       
                         
                            
                           
                             
                               - 
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       ω 
                                       j 
                                     
                                     - 
                                     
                                       ω 
                                       k 
                                     
                                   
                                   ) 
                                 
                               
                             
                              
                             v 
                           
                         
                         
                           - 
                           
                              
                              
                             
                               ( 
                               
                                 
                                   ω 
                                   j 
                                 
                                 - 
                                 
                                   ω 
                                   k 
                                 
                               
                               ) 
                             
                           
                         
                       
                       ] 
                     
                     
                       v 
                       = 
                       
                         
                           - 
                           
                             T 
                             ′ 
                           
                         
                         / 
                         2 
                       
                     
                     
                       v 
                       = 
                       
                         
                           + 
                           
                             T 
                             ′ 
                           
                         
                         / 
                         2 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   
                     
                       a 
                       
                         j 
                         , 
                         k 
                       
                     
                      
                     
                       ( 
                       0 
                       ) 
                     
                   
                   = 
                     
                    
                   
                     
                       T 
                       ′ 
                     
                     · 
                     
                       
                         
                            
                           
                             
                               + 
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       ω 
                                       j 
                                     
                                     - 
                                     
                                       ω 
                                       k 
                                     
                                   
                                   ) 
                                 
                               
                             
                              
                             
                               
                                 T 
                                 ′ 
                               
                               / 
                               2 
                             
                           
                         
                         - 
                         
                            
                           
                             
                               - 
                               
                                  
                                  
                                 
                                   ( 
                                   
                                     
                                       ω 
                                       j 
                                     
                                     - 
                                     
                                       ω 
                                       k 
                                     
                                   
                                   ) 
                                 
                               
                             
                              
                             
                               
                                 T 
                                 ′ 
                               
                               / 
                               2 
                             
                           
                         
                       
                       
                         2 
                          
                         
                           
                             i 
                              
                             
                               ( 
                               
                                 
                                   ω 
                                   j 
                                 
                                 - 
                                 
                                   ω 
                                   k 
                                 
                               
                               ) 
                             
                           
                           · 
                           
                             
                               T 
                               ′ 
                             
                             / 
                             2 
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       T 
                       ′ 
                     
                     · 
                     
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               ( 
                               
                                 
                                   ω 
                                   j 
                                 
                                 - 
                                 
                                   ω 
                                   k 
                                 
                               
                               ) 
                             
                              
                             
                               
                                 T 
                                 ′ 
                               
                               / 
                               2 
                             
                           
                           ) 
                         
                       
                       
                         
                           ( 
                           
                             
                               ω 
                               j 
                             
                             - 
                             
                               ω 
                               k 
                             
                           
                           ) 
                         
                          
                         
                           
                             T 
                             ′ 
                           
                           / 
                           2 
                         
                       
                     
                   
                 
               
             
             
               
                 
                   = 
                     
                    
                   
                     
                       
                         T 
                         ′ 
                       
                       · 
                       sin 
                     
                      
                     
                         
                     
                      
                     
                       c 
                        
                       
                         ( 
                         
                           
                             ( 
                             
                               
                                 ω 
                                 j 
                               
                               - 
                               
                                 ω 
                                 k 
                               
                             
                             ) 
                           
                            
                           
                             
                               T 
                               ′ 
                             
                             / 
                             2 
                           
                         
                         ) 
                       
                     
                   
                 
               
             
           
         
       
     
         [0037]    Thus, if ω k  equals kω T′  (where ω T′ =2π/T′), then: 
         [0000]        a   j,k ( nT )=δ j,k ·δ m  
 
         [0000]        b   j,k ( mT )=0 
         [0038]    In this case, the basis functions are orthogonal, and the transmission is referred to as orthogonal frequency division multiplexing (OFDM). Please note that this is due to the particular choice of T′ and ω k , the quantity ω T′ /ω T  can be considered as “excess bandwidth” of the shaping filter p(t): 
         [0000]      ω k+1 −ω k =ω T′ =(1+α)ω T  
 
         [0000]      FIG. 5  illustrates three adjacent carrier signals, ω, ω k , ω k+1 , and their corresponding image bands following filtering by the rectangular function pulse shaping filter p(t). In the above analysis, orthogonality is achieved for a specific sampling instant, however in practice other considerations make the use of this approach impractical. Since transmit and receive filters are not very selective, the considerable cross-talk energy due to adjacent channels tends to considerably reduce the well-known eye opening at the receiver side (except around the correct sampling time), and hence causes very little tolerance to high frequency or random timing jitter. 
         [0039]    It was observed that coarse filters, such as the transmit pulse shaping filter p(t) being embodied as a continuous-time third order Butterworth filter, and the receive matched filter being embodied as a continuous-time fifth order Butterworth, demonstrated degraded performance at the correct sampling time due to intersymbol interference, but the greater selectivity of the coarse filters provided a much wider eye opening making the receiver  150  more tolerant to random timing jitter. 
         [0040]    When the pulse shaping filters p(t) and the corresponding matched filters p(−t) are embodied as square root raised cosine filters of period T, and excess bandwidth α, each carrier frequency is better isolated, as shown in  FIG. 6  for three adjacent carrier signals, ω, ω k , ω k+1 . As indicated above, 
         [0000]      α j,k ( t )=∫ p ( v ) p ( v−t ) e   −i(ω     j     −ω     k     )v   dv  
 
         [0041]    By setting f(u) equal to p(u) and g(u) equal to p(u-t), the following is obtained: 
         [0000]        a   j,k ( t )=∫ f ( v ) g ( v ) e   −i(ω     j     −ω     k     )v   dv  
 
         [0042]    In the above expression, the Fourier transform of the product function (f·g) is recognized. This Fourier transform can also be expressed as the convolution of the individual Fourier transforms of F and G, respectively: 
         [0000]        a   j,k ( t )=∫ F (ω−Ω) G (Ω) dΩ 
 
         [0000]      where F and G are given by: 
         [0000]        F (ω−Ω)= P (ω−Ω)
 
         [0000]        G (Ω)=∫ p ( v−t ) e   −iΩv   dv=e   −iΩt   ·∫p ( u ) e   −iΩu   du=e   −iΩt   ·P (Ω)
 
         [0000]      Therefore, 
         [0000]        a   j,k ( t )=∫ P ((ω j −ω k )−Ω) P (Ω) e   −iΩt   dΩ 
 
         [0043]    The raised-cosine pulse is defined as follows: 
         [0000]    
       
         
           
             
               c 
                
               
                 ( 
                 t 
                 ) 
               
             
             = 
             
               
                 ( 
                 
                   
                     sin 
                      
                     
                       ( 
                       
                         π 
                         · 
                         
                           t 
                           / 
                           T 
                         
                       
                       ) 
                     
                   
                   
                     t 
                     / 
                     T 
                   
                 
                 ) 
               
                
               
                 ( 
                 
                   
                     cos 
                      
                     
                       ( 
                       
                         απ 
                         · 
                         
                           t 
                           / 
                           T 
                         
                       
                       ) 
                     
                   
                   
                     1 
                     - 
                     
                       
                         ( 
                         
                           2 
                            
                           
                             α 
                             · 
                             
                               t 
                               / 
                               T 
                             
                           
                         
                         ) 
                       
                       2 
                     
                   
                 
                 ) 
               
             
           
         
       
     
         [0044]    and its Fourier transform is expressed as: 
         [0000]    
       
         
           
             
               C 
                
               
                 ( 
                 ω 
                 ) 
               
             
             = 
             
               { 
               
                 
                   
                     T 
                   
                   
                     
                       0 
                       ≤ 
                       
                          
                         ω 
                          
                       
                       ≤ 
                       
                         
                           ( 
                           
                             1 
                             - 
                             α 
                           
                           ) 
                         
                          
                         
                           π 
                           / 
                           T 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         T 
                         2 
                       
                        
                       
                         [ 
                         
                           1 
                           - 
                           
                             sin 
                              
                             
                               ( 
                               
                                 
                                   T 
                                   
                                     2 
                                      
                                     α 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     
                                        
                                       ω 
                                        
                                     
                                     - 
                                     
                                       π 
                                       T 
                                     
                                   
                                   ) 
                                 
                               
                               ) 
                             
                           
                         
                         ] 
                       
                     
                   
                   
                     
                       
                         
                           ( 
                           
                             1 
                             - 
                             α 
                           
                           ) 
                         
                          
                         
                           π 
                           / 
                           T 
                         
                       
                       ≤ 
                       
                          
                         ω 
                          
                       
                       ≤ 
                       
                         
                           ( 
                           
                             1 
                             + 
                             α 
                           
                           ) 
                         
                          
                         
                           π 
                           / 
                           T 
                         
                       
                     
                   
                 
                 
                   
                     0 
                   
                   
                     
                       
                          
                         ω 
                          
                       
                       &gt; 
                       
                         
                           ( 
                           
                             1 
                             + 
                             α 
                           
                           ) 
                         
                          
                         
                           π 
                           / 
                           T 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0045]    If p(t) is embodied as the square root raised cosine then: 
         [0000]        P (ω)= √{square root over (C(ω))} 
 
         [0046]    If the following is assumed for all j and k: 
         [0000]      |ω j −ω k |&gt;2(1+α)π/T, if j≠k
 
         [0000]    then using the above frequency domain expression of a, the following is obtained: 
         [0000]        a   j,k ( t )=∫ P ((ω j −ω k )−Ω) P (Ω) e   −iΩt   dΩ= 0 if ω j ≠ω k  
 
         [0000]    Similarly, (assuming that all carrier frequencies are non-zero): 
         [0000]      ∀( j,k ) b   j,k ( t )=0
 
         [0000]      and: 
         [0000]        a   j,j ( t )=∫ P   2 (Ω) e   −iΩt   dΩ=∫C (Ω) e   −iΩt   dΩ=∫C (Ω) e   +iΩt   dΩ=c ( t )
 
         [0000]      and therefore: 
         [0000]        a   j,j ( mT )= c ( mT ) 
         [0047]    It is noted that: 
         [0000]        a   j,j ( mT )=0 if  m≠ 0 
         [0000]    Again, under these assumptions of the pulse shaping filter, p(t), the transmission is orthogonal (OFDM). However in practice, the pulse shape is realized in the sampled domain, and due to speed limitations, it is desirable to limit oversampling to 2-4×. For instance, if the baudrate equals 666 MBaud, the minimum D/A sampling rate required with T/2 fractional spacing would be 1.3 GS/s which is already quite high. The example shown in  FIG. 7  illustrates the case where a equals 30%, with T/2 fractional spacing. It is noted that the cross-talk energy from image bands due to the D/A sampling rate is not negligible, and unless image bands are totally suppressed, in practice, the FDM transmission is not cross-talk free. Image bands, however, can be considerably reduced by either T/4 or T/8 fractional spacing and simple RC continuous-time filtering or T/2 fractional spacing combined with more selective continuous-time filters. 
         [0048]      FIG. 8  is a schematic block diagram of a receiver  800  incorporating a cross-talk cancellation scheme in accordance with the present invention. As shown in  FIG. 8 , the receiver  800  receives the broadband RF signal rx(t). The input broadband signal is then decomposed into K frequency bands and returned to baseband using an array of input multipliers  810 - 0  through  810 -K−1. Thereafter, each of the K baseband signals is filtered by a corresponding pulse shaping filter p(t)  820 - 0  through  820 -K−1 and sampled by a corresponding switch  830 - 0  through  830 -K−1 at a sampling period. T, to produce the sampled signal elements, rx n,0  through rx n,K−1 . 
         [0049]    A rotator  840  in the receiver  800  removes the rotation introduced in the sampled signal elements, rx n,0  through rx n,K−1 , by the multipliers  810  in the receiver  800 . A time-frequency analyzer  850  cancels the cross-talk and equalizes the channel  140 , in a manner discussed further below in conjunction with  FIG. 9 . A rotator  870  in the receiver  800  removes the rotation introduced in the signal by the multipliers  120  in the transmitter  110 , to produce the digitized signal y n,0  through y n,K−1  (K samples at a time). It is noted that the following notation is used in  FIG. 8  and elsewhere in the present application: 
         [0000]      Θ n,j,k =δ j,k   −iω     j     nT  
 
         [0050]      FIG. 9  is a schematic block diagram illustrating the cross-talk cancellation scheme of  FIG. 8  in further detail. As shown in  FIG. 9 , the rotator  840  of  FIG. 8  is embodied as an array of multipliers  910 - 0  through  910 -K−1, to remove the rotation introduced in the sampled signal elements, rx n,0  through rx n,K−1 , by the multipliers  810  in the receiver  800 . The time-frequency analyzer  850  of  FIG. 8  is embodied as an array  920  of parallel time domain filters comprised of K filters for each of the K frequency bands to cancel the cross-talk and equalizes the channel  140 . The rotator  840  of  FIG. 8  is embodied as an array of multipliers  930 - 0  through  930 -K−1, to remove the rotation introduced signal by the multipliers  120  in the transmitter  110 , to produce the digitized signal y n,0  through y n,K−1  (K samples at a time). 
         [0051]    Each time domain filter in the array  920 , represented as Λ j,k,1:N , is an N-tap filter with the following z-transform: 
         [0000]      Λ j,k,1:N ( z )=λ j,k,0   ·z   0 +δ j,k,l   ·z   −1 + . . . +λ j,k,N−1   ·z   −(N−1)  
 
         [0052]    In a discrete-time system, the modulator and Λ can be permutated following the rule shown in  FIG. 10 , where: 
         [0000]      Λ( z )=λ 0   ·z   0 +λ 1   ·z   −1 + . . . +λ N−1   ·z   −(N−1)  
 
         [0000]      Thus, 
         [0000]        y   n =λ 0   ·e   iωnT   x   n +λ 1   ·e   iω(n−1)T   x   n−1 + . . . +λ N−1   ·e   iω(n−N+1)T   x   n−N+1  
 
         [0000]        y   n   =e   iωnT ·(λ 0   ·e   iω0T   x   n +λ 1   ·e   −iω1·T   x   n−1 + . . . +λ N−1   ·e   −iω(N−1)T   x   n− 
 
         [0053]      FIG. 11  employs the rule of  FIG. 10  to illustrate two equivalent representations of the filter structures, where: 
         [0000]      Λ′( z )=λ 0   ·e   −iω0·T   ·z   0 +λ 1   ·e   −iω1·T   ·z   −1 + . . . +λ N−1   ·e   −ω(N−1)·T   ·z   −(   N− 
 
         [0000]    Equivalent forms of the structure of  FIG. 9  can be obtained by permuting the order of filters and rotators or de-rotators. For instance, an equivalent form is obtained by exchanging the order of the filters and the output de-rotators.  FIG. 12  is a schematic block diagram of one variation of the cross-talk cancellation scheme of  FIG. 9 . As shown in  FIG. 12 , the multipliers  910  and  930  for each frequency band have been integrated into a single multiplier  1210  associated with each of the K filters for each of the K frequency bands. In addition, each filter in the array  920  from  FIG. 9  has been replaced with an equivalent filter in an array  1220 , represented as Λ′ j,k,1:N , an N-tap filter with the following z-transform: 
         [0000]      Λ′ j,k,1:N ( z )=λ j,k,0   ·e   i     ωk     0·T   ·z   0 +λ j,k,1   ·e   i     ωκ     1·T   ·z   −1 + . . . +λ j,k,N−1   ·e   i     ωκ     (N−1)·T   ·z   −(N−1)  
 
         [0054]      FIG. 13  is a schematic block diagram of another variation of the cross-talk cancellation scheme of  FIG. 9  that exchanges the order of the filters and the input rotators. As shown in  FIG. 13 , the multipliers  910  and  930  for each frequency band have been integrated into a single multiplier  1330  (following the filters) associated with each of the K filters for each of the K frequency bands. In addition, each filter in the array  920  from  FIG. 9  has been replaced with an equivalent filter in an array  1320 , represented as Λ″ j,k,1:N , an N-tap filter with the following z-transform: 
         [0000]      Λ″ j,k,1:N ( z )=λ j,k,0   ·e   +i     ωκ     0·T   ·z   0 +λ j,k,1   ·e   i     ωκ     1·   ·z   −1 + . . . +λ j,k,N−1   ·e   i     ωκ     (N−1)·T   ·z   −(N−1) =Λ′ j,k,1:N ( z )
 
         [0055]      FIG. 14  is a schematic block diagram of a receiver  1400  incorporating a generalized cross-talk cancellation/equalization scheme in accordance with the present invention. Generally, the receiver  1400  has a similar structure to the receiver  800  of  FIG. 8 , with an added lower section to remove the undesired image bands. As shown in  FIG. 14 , the receiver  1400  receives the broadband RF signal rx(t). The input broadband signal is then decomposed into K frequency bands and returned to baseband using an array of input multipliers  1410 - 0  through  1410 -K−1. Thereafter, each of the K baseband signals is filtered by a corresponding pulse shaping filter p(t)  1420 - 0  through  1420 -K−1 and sampled by a corresponding switch  1430 - 0  through  1430 -K−1 at a sampling period, T, to produce the sampled signal elements, rx n,0  through rx n,K−1 . 
         [0056]    A rotator  1440  in the receiver  1400  removes the rotation introduced in the sampled signal elements, rx n,0  through rx n,K−1 , by the multipliers  1410  in the receiver  1400 . A time-frequency equalizer  1450  cancels the cross-talk and equalizes the channel  140 , in a similar manner to the time-frequency equalizer  850  discussed above in conjunction with  FIG. 8 . A de-rotator  1470  in the receiver  1400  removes the rotation introduced in the signal by the multipliers  120  in the transmitter  110 , to produce the digitized signal y n,0  through y n,K−1  (K samples at a time). It is noted that the following notation is used in  FIG. 14  and elsewhere in the present application: 
         [0000]      Θ n,j,k =δ j,k   e   −iω     j     nT  
 
         [0057]    As shown in  FIG. 14 , the generalized receiver  1400  also includes a second time-frequency equalizer  1455  cancels the undesired image bands and is applied to the complex conjugate ( 1445 ) of the rotated signal. It is noted that the second time-frequency equalizer  1455  has the same structure as the time-frequency equalizer  1450  with different coefficients. 
         [0058]    It is noted that the architecture of  FIG. 14  seem to have double complexity. However, using the following property the complexity is generally not increased in terms of the number of required real multiplications, and that the update part has a negligible complexity increase if the baud-rate update is not used. 
         [0000]      Λ( x+jy )+Γ( x−jy )=(Λ+Γ) x+j (Λ−Γ) y=Λ   t   x+jΓ   t   y  
 
         [0059]    It is noted that the right side of the above expression comprises two multiplications of complex and real numbers, thus equaling four real multiplications. 
         [0060]    The least mean square (LMS) update algorithm for the generalized adaptive filters of  FIG. 14  can therefore be written as follows: 
         [0000]      Λ n+1   t =Λ t   n   +μe   n real( X   n )
 
         [0000]      Γ n+1   t =δ n   t   +μe   n imag( X   n )
 
         [0061]    In a more detailed form: 
         [0000]      real(Λ n+1   t )=real(Λ n   t )+μ·real( e   n )·real( x   n )
 
         [0000]      imag(Λ n+1   t )=imag(Λ n   t )+μ·imag( e   n )·real( X   n )
 
         [0000]      real(Γ n+1   t )=real(Γ n   t )+μ·real( e   n )·imag( X   n )
 
         [0000]      imag(Γ n+1   t )=imag(Γ n   t )+μ·imag( e   n )·imag( x   n )
 
         [0062]    This modified LMS update algorithm has the added feature that it provides compensation for a fairly large I/Q phase and gain mismatch in the analog component. This is possible because of the added two degrees of freedom. 
         [0063]    It is to be understood that the embodiments and variations shown and described herein are merely illustrative of the principles of this invention and that various modifications may be implemented by those skilled in the art without departing from the scope and spirit of the invention. For example, additional filters can be employed in the transmitter to provide additional resilience to cross-talk, in a known manner. In addition, while the exemplary embodiment shown in  FIG. 9  employs both input rotation and output rotation, it is noted that the present invention can be employed with only one of the input rotation and output rotation, as would be apparent to a person of ordinary skill in the art (as illustrated in  FIGS. 11-13 ).