PATENT ABSTRACT
Process for computing an estimation of the frequency offset in a receiver for a UMTS communication network said receiver receives the signal transmitted by two antennas and including two Common Pilot CHannels (CPICH), said process involving the steps of separating the two signals by means of computation and computing the frequency offset on the base of the two separated signals

PATENT DESCRIPTION
TECHNICAL FIELD 
       [0001]    The invention relates to the field of wireless communication and more particularly to a process for estimating the frequency offset in third generation wireless communication systems, particularly for UMTS. 
       BACKGROUND ART 
       [0002]    Universal Mobile Telecommunications Systems (UMTS) is based on the W-CDMA technology and is in the heart of the 3G communication networks providing packet based transmission of text, digitized voice, and multimedia data at rates up to 2 Megabits per second. 
         [0003]    One particular case of UMTS is defined in the 3G specification, where one base station of such a communication systems may comprise two emitting antennas each transmitting one signal comprising one corresponding Common Pilot CHannel (CPICH). The transmitted sequence of pilots from each antenna is constructed by applying a sign pattern given by  FIG. 1  to the symbol(1+j), where j is to the imaginary complex such as j 2 =−1. 
         [0004]      FIG. 1 : This figure represents the pattern of the transmitted CPICH pilots&#39; sequence. 
         [0005]    Those two pilot signals which are transmitted by the base station are used for the determination of the channel characteristics but also for estimating the frequency offset between the base station reference clock and the internal clock of the User Equipment (UE). The received signal corresponding to the transmission of the k th  pilot sample in the absence of frequency offset is given by: 
         [0000]    
       
         
           
             
               r 
                
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 
                   ( 
                   
                     1 
                     + 
                     j 
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         h 
                         1 
                       
                        
                       
                         [ 
                         k 
                         ] 
                       
                     
                     + 
                     
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         
                           ⌊ 
                           
                             
                               k 
                               + 
                               1 
                             
                             2 
                           
                           ⌋ 
                         
                       
                        
                       
                         
                           h 
                           2 
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                   
                   ) 
                 
               
               + 
               
                 n 
                  
                 
                   [ 
                   k 
                   ] 
                 
               
             
           
         
       
     
         [0000]    while when the UE is subject to a frequency offset with respect to the base station, the received signal is then 
         [0000]    
       
         
           
             
               r 
                
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 
                   ( 
                   
                     1 
                     + 
                     j 
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         h 
                         1 
                       
                        
                       
                         [ 
                         k 
                         ] 
                       
                     
                     + 
                     
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         
                           ⌊ 
                           
                             
                               k 
                               + 
                               1 
                             
                             2 
                           
                           ⌋ 
                         
                       
                        
                       
                         
                           h 
                           2 
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                   
                   ) 
                 
                  
                 
                   exp 
                    
                   
                     ( 
                     
                       
                         j 
                          
                         
                             
                         
                          
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         k 
                          
                         
                             
                         
                          
                         Δ 
                          
                         
                             
                         
                          
                         f 
                          
                         
                             
                         
                          
                         T 
                       
                       + 
                       
                         ϕ 
                         0 
                       
                     
                     ) 
                   
                 
               
               + 
               
                 n 
                  
                 
                   [ 
                   k 
                   ] 
                 
               
             
           
         
       
     
         [0006]    Where h m [k] corresponds to the channel response from antenna m (m=1, 2) and at the time instant of transmission of the k th  pilot, Δf is the frequency offset, T is the time duration between two consecutive samples and φ 0  is an initial phase rotation. 
         [0007]    Some techniques are already known for computing an estimation of the frequency offset based on the computation of phase discriminator. 
         [0008]    However, the techniques which are known do not achieve both accuracy in the estimation of the offset and still operate in a relatively wide range of frequency offsets. 
         [0009]    It is particularly desirable to be able to estimate an offset in the maximum range allowed by an estimation based on the CPICH, while providing accuracy in the estimation. This maximum range is given by 
         [0000]    
       
         
           
             
               
                  
                 
                   Δ 
                    
                   
                       
                   
                    
                   f 
                 
                  
               
               ≤ 
               
                 1 
                 
                   2 
                    
                   T 
                 
               
             
             , 
           
         
       
     
         [0000]    where T is the time between two consecutive CPICH pilots. In UMTS 
         [0000]    
       
         
           
             T 
             = 
             
               
                 1 
                 15 
               
                
               
                 10 
                 
                   - 
                   3 
                 
               
             
           
         
       
     
         [0000]    seconds, which corresponds to covering a range of frequency offsets from minus to plus 7500 Hz. 
         [0010]    Such is the goal of the present invention. 
       SUMMARY OF THE INVENTION 
       [0011]    It is an object of the present invention to provide an efficient frequency offset estimation method which is suitable for direct-sequence (DS) spread-spectrum systems and particularly for the 3GPP UMTS standard. 
         [0012]    It is a further object of the present invention to provide a method which provides both accuracy and wide range of estimation of the frequency offset. 
         [0013]    It is another object of the present invention to provide a receiver for a UMST communication system which incorporates improved frequency offset estimation mechanisms. 
         [0014]    These and other objects of the invention are achieved by means of a process for computing an estimation of the frequency offset which involves a pre-processing based on the separation at the receiver of the two pilot signals transmitted from the two antennas by means of the computation of 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                       NoTxd 
                     
                      
                     
                       [ 
                       k 
                       ] 
                     
                   
                   = 
                     
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                       r 
                        
                       
                         [ 
                         k 
                         ] 
                       
                     
                     + 
                     
                       r 
                        
                       
                         [ 
                         
                           k 
                           + 
                           2 
                         
                         ] 
                       
                     
                   
                 
               
             
             
               
                 
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                         ( 
                         
                           1 
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                        
                       
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                                   - 
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                                 2 
                               
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                                 ] 
                               
                             
                           
                         
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                         [ 
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                       Txd 
                     
                      
                     
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                         ( 
                         
                           - 
                           1 
                         
                         ) 
                       
                       
                         ⌊ 
                         
                           
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                             + 
                             1 
                           
                           2 
                         
                         ⌋ 
                       
                     
                      
                     
                       ( 
                       
                         
                           r 
                            
                           
                             [ 
                             k 
                             ] 
                           
                         
                         - 
                         
                           r 
                            
                           
                             [ 
                             
                               k 
                               + 
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                             ] 
                           
                         
                       
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                   = 
                     
                    
                   
                     
                       
                         ( 
                         
                           1 
                           + 
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                         ) 
                       
                        
                       
                         ( 
                         
                           
                             
                               S 
                               2 
                             
                              
                             
                               [ 
                               k 
                               ] 
                             
                           
                           + 
                           
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 ⌊ 
                                 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   2 
                                 
                                 ⌋ 
                               
                             
                              
                             
                               
                                 D 
                                 1 
                               
                                
                               
                                 [ 
                                 k 
                                 ] 
                               
                             
                           
                         
                         ) 
                       
                     
                     + 
                     
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         
                           ⌊ 
                           
                             
                               k 
                               + 
                               1 
                             
                             2 
                           
                           ⌋ 
                         
                       
                        
                       
                         
                           N 
                           d 
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0015]    Where 
         [0016]    r[k] is the received signal at the instant k; 
         [0017]    Where r NoTxd  and r Txd  are the received pilot signals from antenna 1 and from antenna 2 respectively, assuming that the channel does not change (i.e. h 1 [k]=h 1  and h 2 [k]=h 2 ) and where for m=1,2: 
         [0000]        S   m   [k]=h   m  exp( jφ   0 )(exp( jk Φ)+exp( j ( k+ 2)Φ)),
 
         [0000]        D   m [2 k]=h   m  exp( jφ   0 )(exp( jk Φ)−exp( j ( k+ 2)Φ))
 
         [0000]        N   s   [k]=n[k]+n[k+ 2] 
         [0000]        N   d   [k]=n[k]−n[k+ 2] 
         [0018]    and Φ=2 π Δf T 
         [0019]    we then compute 
         [0000]        R   r     NoTxd     r     NoTxd     [k]=r   NoTxd   [k ]*conj( r   NoTxd   [k+ 1]) 
         [0000]        R   r     Txd     r     Txd     [k]=r   Txd   [k ]*conj( r   Txd   [k+ 1]) 
         [0020]    where conj(.) is the complex conjugate operator. 
         [0021]    With the assumption that the channel does not change (i.e. h 1 [k]=h 1  and h 2 [k]=h 2 ) and after simplification, we have 
         [0000]    
       
         
           
             
               
                 R 
                 
                   
                     r 
                     NoTxd 
                   
                    
                   
                     r 
                     NoTxd 
                   
                 
               
                
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 2 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             R 
                             
                               
                                 S 
                                 1 
                               
                                
                               
                                 S 
                                 1 
                               
                             
                           
                           + 
                           
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 ⌊ 
                                 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   2 
                                 
                                 ⌋ 
                               
                             
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                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 ⌊ 
                                 
                                   
                                     k 
                                     + 
                                     2 
                                   
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                                 ⌋ 
                               
                             
                              
                             
                               R 
                               
                                 
                                   D 
                                   2 
                                 
                                  
                                 
                                   D 
                                   2 
                                 
                               
                             
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
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                               R 
                               
                                 
                                   S 
                                   1 
                                 
                                  
                                 
                                   D 
                                   2 
                                 
                               
                             
                           
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                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 ⌊ 
                                 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   2 
                                 
                                 ⌋ 
                               
                             
                              
                             
                               R 
                               
                                 
                                   D 
                                   2 
                                 
                                  
                                 
                                   S 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                   
                   ) 
                 
               
               + 
               
                 
                   R 
                   
                     
                       N 
                       s 
                     
                      
                     
                       N 
                       s 
                     
                   
                 
                  
                 
                   [ 
                   k 
                   ] 
                 
               
             
           
         
       
       
         
           
             
               
                 R 
                 
                   
                     r 
                     Txd 
                   
                    
                   
                     r 
                     Txd 
                   
                 
               
                
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 2 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             R 
                             
                               
                                 S 
                                 2 
                               
                                
                               
                                 S 
                                 2 
                               
                             
                           
                           + 
                           
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 ⌊ 
                                 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   2 
                                 
                                 ⌋ 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 ⌊ 
                                 
                                   
                                     k 
                                     + 
                                     2 
                                   
                                   2 
                                 
                                 ⌋ 
                               
                             
                              
                             
                               R 
                               
                                 
                                   D 
                                   1 
                                 
                                  
                                 
                                   D 
                                   1 
                                 
                               
                             
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 ⌊ 
                                 
                                   
                                     k 
                                     + 
                                     2 
                                   
                                   2 
                                 
                                 ⌋ 
                               
                             
                              
                             
                               R 
                               
                                 
                                   S 
                                   2 
                                 
                                  
                                 
                                   D 
                                   1 
                                 
                               
                             
                           
                           + 
                           
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 ⌊ 
                                 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   2 
                                 
                                 ⌋ 
                               
                             
                              
                             
                               R 
                               
                                 
                                   D 
                                   1 
                                 
                                  
                                 
                                   S 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                   ) 
                 
               
               + 
               
                 
                   
                     ( 
                     
                       - 
                       1 
                     
                     ) 
                   
                   
                     ⌊ 
                     
                       
                         k 
                         + 
                         1 
                       
                       2 
                     
                     ⌋ 
                   
                 
                  
                 
                   
                     ( 
                     
                       - 
                       1 
                     
                     ) 
                   
                   
                     ⌊ 
                     
                       
                         k 
                         + 
                         2 
                       
                       2 
                     
                     ⌋ 
                   
                 
                  
                 
                   
                     R 
                     
                       
                         N 
                         d 
                       
                        
                       
                         N 
                         d 
                       
                     
                   
                    
                   
                     [ 
                     k 
                     ] 
                   
                 
               
             
           
         
       
     
         [0000]    where, for m=1,2 and n=1,2 
         [0000]        R   S     m     S     m     =S   m   [k ].conj( S   m   [k+ 1])=2| h   m | 2 (exp( j Φ)(1+cos(2Φ)))
 
         [0000]        R   D     m     D     m     =D   m   [k ].conj( D   m   [k+ 1])=2| h   m | 2 (exp( j Φ)(1−cos(2Φ)))
 
         [0000]        R   S     m     D     n     =S   m   [k ].conj( D   n   [k+ 1])=−2 jh   m   h   n *exp( j Φ)sin(2Φ)
 
         [0000]        R   N     s     N     s     =N   s   [k ].conj( N   s   [k+ 1]) 
         [0000]        R   N     d     N     d     =N   d   [k ].conj( N   d   [k+ 1]) 
         [0022]    In one embodiment, the estimation is based on one estimator R even, avr  computed in accordance with the following formulas: 
         [0000]    
       
         
           
             
               R 
               
                 even 
                 , 
                 avr 
               
             
             = 
             
               
                 1 
                 
                   
                     N 
                     2 
                   
                   - 
                   
                     N 
                     1 
                   
                 
               
                
               
                 
                   ∑ 
                   
                     
                       k 
                       = 
                       
                         N 
                         1 
                       
                     
                     , 
                     
                       k 
                        
                       
                           
                       
                        
                       even 
                     
                   
                   
                     N 
                     2 
                   
                 
                  
                 
                   ( 
                   
                     
                       
                         R 
                         
                           
                             r 
                             Txd 
                           
                            
                           
                             r 
                             Txd 
                           
                         
                       
                        
                       
                         [ 
                         k 
                         ] 
                       
                     
                     + 
                     
                       
                         R 
                         
                           
                             r 
                             NoTxd 
                           
                            
                           
                             r 
                             NoTxd 
                           
                         
                       
                        
                       
                         [ 
                         k 
                         ] 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    which simplifies to: 
         [0000]    
       
         
           
             
               R 
               
                 even 
                 , 
                 avr 
               
             
             = 
             
               
                 4 
                  
                 
                     
                 
                  
                 
                   exp 
                    
                   
                     ( 
                     
                       j 
                        
                       
                           
                       
                        
                       Φ 
                     
                     ) 
                   
                 
                  
                 
                   ( 
                   
                     
                       
                          
                         
                           h 
                           1 
                         
                          
                       
                       2 
                     
                     + 
                     
                       
                          
                         
                           h 
                           2 
                         
                          
                       
                       2 
                     
                   
                   ) 
                 
                  
                 
                   cos 
                    
                   
                     ( 
                     
                       2 
                        
                       Φ 
                     
                     ) 
                   
                 
               
               + 
               
                 
                   1 
                   
                     
                       N 
                       2 
                     
                     - 
                     
                       N 
                       1 
                     
                   
                 
                  
                 
                   
                     ∑ 
                     
                       
                         k 
                         = 
                         
                           N 
                           1 
                         
                       
                       , 
                       
                         k 
                          
                         
                             
                         
                          
                         even 
                       
                     
                     
                       N 
                       2 
                     
                   
                    
                   
                     ( 
                     
                       
                         
                           R 
                           
                             
                               N 
                               s 
                             
                              
                             
                               N 
                               s 
                             
                           
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                       - 
                       
                         
                           R 
                           
                             
                               N 
                               d 
                             
                              
                             
                               N 
                               d 
                             
                           
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where N 1  and N 2  are respectively the indices of the first and the last CPICH symbols used for the frequency offset estimation. The use of the average tends to attenuate the effect of noise by averaging over several symbols. 
         [0023]    The frequency offset estimate being provided by the formula: 
         [0000]    
       
         
           
             
               Δ 
                
               
                   
               
                
               f 
             
             = 
             
               
                 1 
                 
                   2 
                    
                   π 
                    
                   
                       
                   
                    
                   T 
                 
               
                
               
                 arctan 
                  
                 
                   ( 
                   
                     
                        
                        
                       
                           
                       
                        
                       
                         m 
                          
                         
                           ( 
                           
                             R 
                             
                               even 
                               , 
                               avr 
                             
                           
                           ) 
                         
                       
                     
                     
                       ℜ 
                        
                       
                           
                       
                        
                       
                         e 
                          
                         
                           ( 
                           
                             R 
                             
                               even 
                               , 
                               avr 
                             
                           
                           ) 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    where          (.) and ℑm(.) are respectively the imaginary part and real part operators. Alternatively, the frequency offset estimation is based on one estimator R odd, avr  computed in accordance with the following formulas: 
         [0000]    
       
         
           
             
               R 
               
                 odd 
                 , 
                 avr 
               
             
             = 
             
               
                 1 
                 
                   
                     N 
                     2 
                   
                   - 
                   
                     N 
                     1 
                   
                 
               
                
               
                 
                   ∑ 
                   
                     
                       k 
                       = 
                       
                         N 
                         1 
                       
                     
                     , 
                     
                       k 
                        
                       
                           
                       
                        
                       odd 
                     
                   
                   
                     N 
                     2 
                   
                 
                  
                 
                   ( 
                   
                     
                       
                         R 
                         
                           
                             r 
                             Txd 
                           
                            
                           
                             r 
                             Txd 
                           
                         
                       
                        
                       
                         [ 
                         k 
                         ] 
                       
                     
                     + 
                     
                       
                         R 
                         
                           
                             r 
                             NoTxd 
                           
                            
                           
                             r 
                             NoTxd 
                           
                         
                       
                        
                       
                         [ 
                         k 
                         ] 
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    which simplifies to 
         [0000]    
       
         
           
             
               R 
               
                 odd 
                 , 
                 avr 
               
             
             = 
             
               
                 4 
                  
                 
                     
                 
                  
                 
                   exp 
                    
                   
                     ( 
                     
                       j 
                        
                       
                           
                       
                        
                       Φ 
                     
                     ) 
                   
                 
                  
                 
                   ( 
                   
                     
                       
                          
                         
                           h 
                           1 
                         
                          
                       
                       2 
                     
                     + 
                     
                       
                          
                         
                           h 
                           2 
                         
                          
                       
                       2 
                     
                   
                   ) 
                 
               
               + 
               
                 
                   1 
                   
                     
                       N 
                       2 
                     
                     - 
                     
                       N 
                       1 
                     
                   
                 
                  
                 
                   
                     ∑ 
                     
                       
                         k 
                         = 
                         
                           N 
                           1 
                         
                       
                       , 
                       
                         k 
                          
                         
                             
                         
                          
                         odd 
                       
                     
                     
                       N 
                       2 
                     
                   
                    
                   
                     ( 
                     
                       
                         
                           R 
                           
                             
                               N 
                               s 
                             
                              
                             
                               N 
                               s 
                             
                           
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                       + 
                       
                         
                           R 
                           
                             
                               N 
                               d 
                             
                              
                             
                               N 
                               d 
                             
                           
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                     ) 
                   
                 
               
             
           
         
       
     
         [0000]    where N 1  and N 2  are respectively the indices of the first and the last pilot symbols used for the frequency offset estimation. The use of the average tends to attenuate the effect of noise by averaging over several symbols. 
         [0024]    The frequency offset estimate being provided by the formula: 
         [0000]    
       
         
           
             
               Δ 
                
               
                   
               
                
               f 
             
             = 
             
               
                 1 
                 
                   2 
                    
                   π 
                    
                   
                       
                   
                    
                   T 
                 
               
                
               
                 arctan 
                  
                 
                   ( 
                   
                     
                        
                        
                       
                           
                       
                        
                       
                         m 
                          
                         
                           ( 
                           
                             R 
                             
                               odd 
                               , 
                               avr 
                             
                           
                           ) 
                         
                       
                     
                     
                       ℜ 
                        
                       
                           
                       
                        
                       
                         e 
                          
                         
                           ( 
                           
                             R 
                             
                               odd 
                               , 
                               avr 
                             
                           
                           ) 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    In one particular, embodiment, the estimation is computed on the basis of both R odd, avr  and R even, avr  in accordance with the formula: 
         [0000]      If          ( R   odd, avr )|&gt;|ℑ m ( R   odd, avr )|
 
         [0000]    
       
      
       R=R 
       odd, avr 
       +R 
       even, avr  
      
     
         [0000]      else 
         [0000]    
       
      
       R=R 
       odd, avr 
       −R 
       even, avr  
      
     
         [0000]    and the frequency offset estimate being provided by the formula: 
         [0000]    
       
         
           
             
               Δ 
                
               
                   
               
                
               f 
             
             = 
             
               
                 1 
                 
                   2 
                    
                   π 
                    
                   
                       
                   
                    
                   T 
                 
               
                
               
                 arctan 
                  
                 
                   ( 
                   
                     
                        
                        
                       
                           
                       
                        
                       
                         m 
                          
                         
                           ( 
                           R 
                           ) 
                         
                       
                     
                     
                       ℜ 
                        
                       
                           
                       
                        
                       
                         e 
                          
                         
                           ( 
                           R 
                           ) 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    The invention also provides a process for processing the CPICH pilot signals r(i) received from the two antennas of a base station which involves the steps: 
         [0025]    extracting the received CPICH signal included in the CPICH channel; 
         [0026]    performing a first processing branch (left) comprising the steps of:
       computing a first intermediate value x(i) in accordance with the formula:       
 
         [0000]        x ( i )= r ( i )+ r ( i+ 2)       computing a second intermediate value y(i) derived from the first intermediate value x(i) in accordance with the formula:         
         [0000]        y ( i )= x ( i ).  x *( i+ 1)       where x*(i+1) is the complex conjugate of x(i+1)         
         [0030]    performing a second processing branch (right) comprising the steps of:
       computing a third intermediate value x′(i) in accordance with the formula:       
 
         [0000]        x ′( i )= r ( i )− r ( i+ 2)
       computing a fourth intermediate value t(i) derived from the third intermediate value x′(i) in accordance with the formula:       
 
         [0000]        t ′(2 k )= x ′(2 k ) and
 
         [0000]        t ′(2 k+ 1)=− x ′(2 k+ 1)
       computing a fifth intermediate value y′(i) derived from the fourth intermediate value t′(i) in accordance with the formula:       
 
         [0000]        y ′( i )= t ′( i ).  t ′*( i+ 1)
 
         [0034]    computing a sixth and seventh intermediate values Z and Z′ in accordance with the formulas: 
         [0000]        Z (2 k )= y (2 k )+ y ′(2 k )
 
         [0000]        Z ′(2 k+ 1)= y (2 k+ 1)+ y ′(2 k+ 1)
 
         [0035]    respectively computing an eight and a ninth values S and S′ in accordance with the following formulation: 
         [0000]        S= 1/ N Σ Z ( k ) for  k= 1 to  N    
         [0000]        S′= 1/ N Σ Z ′( k ) for  k= 1 to  N  
 
         [0036]    performing a test to determine whether the absolute value of the real part of S′ is superior to the absolute value of the imaginary part of S′ and, in that case, computing (30) a tenth intermediate value R as follows: 
         [0000]    
       
      
       R=S′+S  
      
     
         [0000]    And, conversely if the absolute value of the real part of S′ is inferior to the absolute value of the imaginary part of S′, computing R in accordance with the formula: 
         [0000]    
       
      
       R=S′−S  
      
     
         [0000]    The offset estimation is then simply computed as follows: 
         [0000]    
       
         
           
             
               Δ 
                
               
                   
               
                
               f 
             
             = 
             
               
                 1 
                 
                   2 
                    
                   π 
                    
                   
                       
                   
                    
                   T 
                 
               
                
               
                 arctan 
                  
                 
                   ( 
                   
                     
                        
                        
                       
                           
                       
                        
                       
                         m 
                          
                         
                           ( 
                           R 
                           ) 
                         
                       
                     
                     
                       ℜ 
                        
                       
                           
                       
                        
                       
                         e 
                          
                         
                           ( 
                           R 
                           ) 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0037]    where          (R) and ℑm(R) are the real and imaginary parts, respectively of the tenth intermediate value R. 
         [0038]    The invention is particularly suitable for the achievement of a UMTS receiver of a User Equipment (UE) such as a mobile telephone or a Portable Document Assistant. 
     
    
     
       DESCRIPTION OF THE DRAWINGS 
         [0039]    Other features of one or more embodiments of the invention will best be understood by reference to the following detailed description when read in conjunction with the accompanying drawings. 
           [0040]      FIG. 1  illustrates the structure of the sign patter (applied to (1+j)) of the two CPICH pilots sequences which are used in the UMTS using transmit diversity (TxD) 
           [0041]      FIG. 2  illustrates one embodiment of the method for computing the offset frequency in accordance with the present invention. 
           [0042]      FIG. 3  illustrates the principle of the separation of the two signals transmitted through the two emitting antennas. 
           [0043]      FIG. 4  illustrates the Root Mean Square Error (RMSE) of the frequency offset estimation (FOE) as a function of the frequency offset (FO) for an AWGN environment and SNR=0 dB 
           [0044]      FIG. 5  illustrates the RMSE of the frequency offset estimation as a function of lor/loc, in an AWGN environment and for FO=0 Hz 
           [0045]      FIG. 6  illustrates the RMSE of the frequency offset as a function of lor/loc, in case 1 environment (defined in the 3GPP specs) and for FO=0 Hz. 
       
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENT 
       [0046]    The invention can be implemented in digital electronic circuitry forming a mobile telephone or a Portable Digital Assistant (PDA) including hardware circuits with a combination of firmware and software. 
         [0047]    The method which is proposed is based on the computation of the phase discriminator, that is to say the sum of the autocorrelation of the samples sequence. 
         [0048]    To simplify the mathematical treatment, we use a symbol-rate model for the CPICH symbols transmitted from Tx antenna 1 and Tx antenna 2 in symbol time instants k=0,1, . . . , 149 . . . , during one 10-ms radio frame, which reads: 
         [0000]    
       
         
           
             
               
                 p 
                 1 
               
                
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               ( 
               
                 1 
                 + 
                 j 
               
               ) 
             
           
         
       
       
         
           
             
               
                 p 
                 2 
               
                
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 ( 
                 
                   1 
                   + 
                   j 
                 
                 ) 
               
                
               
                 
                   ( 
                   
                     - 
                     1 
                   
                   ) 
                 
                 
                   ⌊ 
                   
                     
                       k 
                       + 
                       1 
                     
                     2 
                   
                   ⌋ 
                 
               
             
           
         
       
     
         [0049]      FIG. 1  illustrates the sign pattern (applied to (1+j)) resulting from those formulas. 
         [0050]    These Tx symbols are transmitted via time-varying channel coefficients to obtain the effective received symbol (on one single Rx antenna) in time instants, described by k=0,1, . . . , 149. 
         [0000]    
       
         
           
             
               r 
                
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 
                   ( 
                   
                     
                       
                         
                           h 
                           1 
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                        
                       
                         
                           p 
                           1 
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           h 
                           2 
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                        
                       
                         
                           p 
                           2 
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                   
                   ) 
                 
                  
                 
                   exp 
                    
                   
                     ( 
                     
                       ( 
                       
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           k 
                            
                           
                               
                           
                            
                           Δ 
                            
                           
                               
                           
                            
                           f 
                            
                           
                               
                           
                            
                           T 
                         
                         + 
                         
                           ϕ 
                           0 
                         
                       
                       ) 
                     
                     ) 
                   
                 
               
               + 
               
                 n 
                  
                 
                   [ 
                   k 
                   ] 
                 
               
             
           
         
       
       
         
           
             
               r 
                
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 
                   ( 
                   
                     1 
                     + 
                     j 
                   
                   ) 
                 
                  
                 
                   ( 
                   
                     
                       
                         h 
                         1 
                       
                        
                       
                         [ 
                         k 
                         ] 
                       
                     
                     + 
                     
                       
                         
                           ( 
                           
                             - 
                             1 
                           
                           ) 
                         
                         
                           ⌊ 
                           
                             
                               k 
                               + 
                               1 
                             
                             2 
                           
                           ⌋ 
                         
                       
                        
                       
                         
                           h 
                           2 
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                     
                   
                   ) 
                 
                  
                 
                   exp 
                    
                   
                     ( 
                     
                       ( 
                       
                         
                           j 
                            
                           
                               
                           
                            
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           k 
                            
                           
                               
                           
                            
                           Δ 
                            
                           
                               
                           
                            
                           f 
                            
                           
                               
                           
                            
                           T 
                         
                         + 
                         
                           ϕ 
                           0 
                         
                       
                       ) 
                     
                     ) 
                   
                 
               
               + 
               
                 n 
                  
                 
                   [ 
                   k 
                   ] 
                 
               
             
           
         
       
     
         [0051]    where h m [k] is the channel gain from antenna m at time instant k and n[k] is the additive Gaussian noise with zero mean and variance σ 2 . 
         [0052]    Δf corresponds to the frequency offset and assumed to be identical for both antennas. 
         [0053]    The method is based on a separation of the received pilot signals from the two transmit antennas—summarized in FIG.  3 —in order to keep the sampling period of the resulting samples equal to T, which is a condition for keeping a relatively wide estimation of frequency offsets. 
         [0054]    Now considering r NoTxd  and r Txd [k] being the computed signals respectively containing only the pilot transmitted from antenna 1 and antenna 2, computed write the following formula: 
         [0000]    
       
         
           
             
                 
             
              
             
               
                 
                   r 
                   NoTxd 
                 
                  
                 
                   [ 
                   k 
                   ] 
                 
               
               = 
               
                 
                   
                     r 
                      
                     
                       [ 
                       k 
                       ] 
                     
                   
                   + 
                   
                     r 
                      
                     
                       [ 
                       
                         k 
                         + 
                         2 
                       
                       ] 
                     
                   
                 
                 = 
                 
                   
                     
                       ( 
                       
                         1 
                         + 
                         j 
                       
                       ) 
                     
                      
                     
                       ( 
                       
                         
                           
                             S 
                             1 
                           
                            
                           
                             [ 
                             k 
                             ] 
                           
                         
                         + 
                         
                           
                             
                               ( 
                               
                                 - 
                                 1 
                               
                               ) 
                             
                             
                               ⌊ 
                               
                                 
                                   k 
                                   + 
                                   1 
                                 
                                 2 
                               
                               ⌋ 
                             
                           
                            
                           
                             
                               D 
                               2 
                             
                              
                             
                               [ 
                               k 
                               ] 
                             
                           
                         
                       
                       ) 
                     
                   
                   + 
                   
                     
                       N 
                       2 
                     
                      
                     
                       [ 
                       k 
                       ] 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               
                 r 
                 Txd 
               
                
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 
                   
                     ( 
                     
                       - 
                       1 
                     
                     ) 
                   
                   
                     ⌊ 
                     
                       
                         k 
                         + 
                         1 
                       
                       2 
                     
                     ⌋ 
                   
                 
                  
                 
                   ( 
                   
                     
                       r 
                        
                       
                         [ 
                         k 
                         ] 
                       
                     
                     - 
                     
                       r 
                        
                       
                         [ 
                         
                           k 
                           + 
                           2 
                         
                         ] 
                       
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     ( 
                     
                       1 
                       + 
                       j 
                     
                     ) 
                   
                    
                   
                     ( 
                     
                       
                         
                           S 
                           2 
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                       + 
                       
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           
                             ⌊ 
                             
                               
                                 k 
                                 + 
                                 1 
                               
                               2 
                             
                             ⌋ 
                           
                         
                          
                         
                           
                             D 
                             1 
                           
                            
                           
                             [ 
                             k 
                             ] 
                           
                         
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     
                       ( 
                       
                         - 
                         1 
                       
                       ) 
                     
                     
                       ⌊ 
                       
                         
                           k 
                           + 
                           1 
                         
                         2 
                       
                       ⌋ 
                     
                   
                    
                   
                     
                       N 
                       d 
                     
                      
                     
                       [ 
                       k 
                       ] 
                     
                   
                 
               
             
           
         
       
     
         [0055]    Where for m=1,2 
         [0000]        S   m   [k]=h   m  exp( jφ   0 )(exp( jk Φ)+exp( j ( k+ 2)Φ))
 
         [0000]        D   m [2 k]=h   m  exp( jφ   0 )(exp( jk Φ)−exp( j ( k+ 2)Φ))
 
         [0000]        N   s   [k]=n[k]+n[k+ 2] 
         [0000]        N   d   [k]=n[k]−n[k+ 2] 
         [0056]    h m  corresponding to the channel response for antenna m (m=1, 2), 
         [0057]    j corresponding to the imaginary complex such as j 2 =−1 and Φ=2 π Δf T 
         [0058]    It can be seen that the sampling period of r NoTxd  and r Txd [k] sequences is T, which will allow a possible estimation range of 
         [0000]    
       
         
           
             
                
               
                 Δ 
                  
                 
                     
                 
                  
                 f 
               
                
             
             ≤ 
             
               1 
               
                 2 
                  
                 T 
               
             
           
         
       
     
         [0059]    We then compute R NoTxd [k] and R Txd [k] from the separated signal r NoTxd  and r Txd [k] as 
         [0000]        R   r     NoTxd     r     NoTxd     [k]=r   NoTxd   [k ]*conj( r   NoTxd   [k+ 1]) 
         [0000]        R   r     Txd     r     Txd     [k]=r   Txd   [k ]*conj( r   Txd   [k+ 1]) 
         [0060]    which simplifies, with the assumption that the channel is constant (i.e. h 1 [k]=h 1  and h 2 [k]=h 2 ), to 
         [0000]    
       
         
           
             
               
                 R 
                 
                   
                     r 
                     NoTxd 
                   
                    
                   
                     r 
                     NoTxd 
                   
                 
               
                
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 2 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             R 
                             
                               
                                 S 
                                 1 
                               
                                
                               
                                 S 
                                 1 
                               
                             
                           
                           + 
                           
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 ⌊ 
                                 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   2 
                                 
                                 ⌋ 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 ⌊ 
                                 
                                   
                                     k 
                                     + 
                                     2 
                                   
                                   2 
                                 
                                 ⌋ 
                               
                             
                              
                             
                               R 
                               
                                 
                                   D 
                                   2 
                                 
                                  
                                 
                                   D 
                                   2 
                                 
                               
                             
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 ⌊ 
                                 
                                   
                                     k 
                                     + 
                                     2 
                                   
                                   2 
                                 
                                 ⌋ 
                               
                             
                              
                             
                               R 
                               
                                 
                                   S 
                                   1 
                                 
                                  
                                 
                                   D 
                                   2 
                                 
                               
                             
                           
                           + 
                           
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 ⌊ 
                                 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   2 
                                 
                                 ⌋ 
                               
                             
                              
                             
                               R 
                               
                                 
                                   D 
                                   2 
                                 
                                  
                                 
                                   S 
                                   1 
                                 
                               
                             
                           
                         
                       
                     
                   
                   ) 
                 
               
               + 
               
                 
                   R 
                   
                     
                       N 
                       s 
                     
                      
                     
                       N 
                       s 
                     
                   
                 
                  
                 
                   [ 
                   k 
                   ] 
                 
               
             
           
         
       
       
         
           
             
               
                 R 
                 
                   
                     r 
                     Txd 
                   
                    
                   
                     r 
                     Txd 
                   
                 
               
                
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 2 
                  
                 
                   ( 
                   
                     
                       
                         
                           
                             R 
                             
                               
                                 S 
                                 2 
                               
                                
                               
                                 S 
                                 2 
                               
                             
                           
                           + 
                           
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 ⌊ 
                                 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   2 
                                 
                                 ⌋ 
                               
                             
                              
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 ⌊ 
                                 
                                   
                                     k 
                                     + 
                                     2 
                                   
                                   2 
                                 
                                 ⌋ 
                               
                             
                              
                             
                               R 
                               
                                 
                                   D 
                                   1 
                                 
                                  
                                 
                                   D 
                                   1 
                                 
                               
                             
                           
                           + 
                         
                       
                     
                     
                       
                         
                           
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 ⌊ 
                                 
                                   
                                     k 
                                     + 
                                     2 
                                   
                                   2 
                                 
                                 ⌋ 
                               
                             
                              
                             
                               R 
                               
                                 
                                   S 
                                   2 
                                 
                                  
                                 
                                   D 
                                   1 
                                 
                               
                             
                           
                           + 
                           
                             
                               
                                 ( 
                                 
                                   - 
                                   1 
                                 
                                 ) 
                               
                               
                                 ⌊ 
                                 
                                   
                                     k 
                                     + 
                                     1 
                                   
                                   2 
                                 
                                 ⌋ 
                               
                             
                              
                             
                               R 
                               
                                 
                                   D 
                                   1 
                                 
                                  
                                 
                                   S 
                                   2 
                                 
                               
                             
                           
                         
                       
                     
                   
                   ) 
                 
               
               + 
               
                 
                   
                     ( 
                     
                       - 
                       1 
                     
                     ) 
                   
                   
                     ⌊ 
                     
                       
                         k 
                         + 
                         1 
                       
                       2 
                     
                     ⌋ 
                   
                 
                  
                 
                   
                     ( 
                     
                       - 
                       1 
                     
                     ) 
                   
                   
                     ⌊ 
                     
                       
                         k 
                         + 
                         2 
                       
                       2 
                     
                     ⌋ 
                   
                 
                  
                 
                   
                     R 
                     
                       
                         N 
                         d 
                       
                        
                       
                         N 
                         d 
                       
                     
                   
                    
                   
                     [ 
                     k 
                     ] 
                   
                 
               
             
           
         
       
     
         [0000]    where, for m=1,2 and n=1,2 
         [0000]        R   S     m     S     m     =S   m   [k ].conj( S   m   [k+ 1])=2| h   m | 2 (exp( j Φ)(1+cos(2Φ)))
 
         [0000]        R   D     m     D     m     =D   m   [k ].conj( D   m   [k+ 1])=2| h   m | 2 (exp( j Φ)(1−cos(2Φ)))
 
         [0000]        R   S     m     D     n     =S   m   [k ].conj( D   n   [k+ 1])=−2 jh   m   h   n *exp( j Φ)sin(2Φ)
 
         [0000]        R   N     s     N     s     =N   s   [k ].conj( N   s   [k+ 1]) 
         [0000]        R   N     d     N     d     =N   d   [k ].conj( N   d   [k+ 1]) 
         [0061]    If k is even, one may demonstrate that: 
         [0000]        R   even   [k]=R   r     Txd     r     Txd     [k]+R   r     NoTxd     r     NoTxd     [k]= 4 exp( j Φ)(| h   1 | 2   +|h   2 | 2 )cos(2Φ)+ R   N     s     N     s     [k]−R   N     d     N     d     [k] 
 
         [0062]    And if k is odd, 
         [0000]        R   odd   [k]=R   r     Txd     r     Txd     [k]+R   r     NoTxd     r     NoTxd     [k]= 4 exp( j Φ)(| h   1 | 2   +|h   2 | 2 )+( R   N     s     N     s     [k]+R   N     d     N     d     [k ])
 
         [0063]    In order to attenuate the noise effect, we compute an average over several samples of R even [k] and R odd [k] as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           R 
                           
                             even 
                             , 
                             avr 
                           
                         
                         = 
                           
                          
                         
                           
                             1 
                             
                               
                                 N 
                                 2 
                               
                               - 
                               
                                 N 
                                 1 
                               
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 
                                   k 
                                   = 
                                   
                                     N 
                                     1 
                                   
                                 
                                 , 
                                 
                                   k 
                                    
                                   
                                       
                                   
                                    
                                   even 
                                 
                               
                               
                                 N 
                                 2 
                               
                             
                              
                             
                               
                                 R 
                                 even 
                               
                                
                               
                                 [ 
                                 k 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             4 
                              
                             
                                 
                             
                              
                             
                               exp 
                                
                               
                                 ( 
                                 jΦ 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                      
                                     
                                       h 
                                       1 
                                     
                                      
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                      
                                     
                                       h 
                                       2 
                                     
                                      
                                   
                                   2 
                                 
                               
                               ) 
                             
                              
                             
                               cos 
                                
                               
                                 ( 
                                 
                                   2 
                                    
                                   Φ 
                                 
                                 ) 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             1 
                             
                               
                                 N 
                                 2 
                               
                               - 
                               
                                 N 
                                 1 
                               
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 
                                   k 
                                   = 
                                   
                                     N 
                                     1 
                                   
                                 
                                 , 
                                 
                                   k 
                                    
                                   
                                       
                                   
                                    
                                   even 
                                 
                               
                               
                                 N 
                                 2 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                     R 
                                     
                                       
                                         N 
                                         s 
                                       
                                       , 
                                       
                                         N 
                                         s 
                                       
                                     
                                   
                                    
                                   
                                     [ 
                                     k 
                                     ] 
                                   
                                 
                                 - 
                                 
                                   
                                     R 
                                     
                                       
                                         N 
                                         d 
                                       
                                        
                                       
                                         N 
                                         d 
                                       
                                     
                                   
                                    
                                   
                                     [ 
                                     k 
                                     ] 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
             
             
               
                 
                   
                     
                       
                         
                           R 
                           
                             odd 
                             , 
                             avr 
                           
                         
                         = 
                           
                          
                         
                           
                             1 
                             
                               
                                 N 
                                 2 
                               
                               - 
                               
                                 N 
                                 1 
                               
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 
                                   k 
                                   = 
                                   
                                     N 
                                     1 
                                   
                                 
                                 , 
                                 
                                   k 
                                    
                                   
                                       
                                   
                                    
                                   odd 
                                 
                               
                               
                                 N 
                                 2 
                               
                             
                              
                             
                               
                                 R 
                                 odd 
                               
                                
                               
                                 [ 
                                 k 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             4 
                              
                             
                                 
                             
                              
                             
                               exp 
                                
                               
                                 ( 
                                 jΦ 
                                 ) 
                               
                             
                              
                             
                               ( 
                               
                                 
                                   
                                      
                                     
                                       h 
                                       1 
                                     
                                      
                                   
                                   2 
                                 
                                 + 
                                 
                                   
                                      
                                     
                                       h 
                                       2 
                                     
                                      
                                   
                                   2 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               1 
                               
                                 
                                   N 
                                   2 
                                 
                                 + 
                                 
                                   N 
                                   1 
                                 
                               
                             
                              
                             
                               ∑ 
                               
                                 
                                   k 
                                   = 
                                   
                                     N 
                                     1 
                                   
                                 
                                 , 
                                 
                                   k 
                                    
                                   
                                       
                                   
                                    
                                   odd 
                                 
                               
                               
                                 N 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           ( 
                           
                             
                               
                                 R 
                                 
                                   
                                     N 
                                     s 
                                   
                                    
                                   
                                     N 
                                     s 
                                   
                                 
                               
                                
                               
                                 [ 
                                 k 
                                 ] 
                               
                             
                             + 
                             
                               
                                 R 
                                 
                                   
                                     N 
                                     d 
                                   
                                    
                                   
                                     N 
                                     d 
                                   
                                 
                               
                                
                               
                                 [ 
                                 k 
                                 ] 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
             
           
         
       
     
         [0064]    Both of these two sums R even, avr  and R odd, avr  can be considered as independent estimators and the frequency offset estimation is given respectively by 
         [0000]    
       
         
           
             
               Δ 
                
               
                   
               
                
               f 
             
             = 
             
               
                 1 
                 
                   2 
                    
                   π 
                    
                   
                       
                   
                    
                   T 
                 
               
                
               
                 arctan 
                  
                 
                   ( 
                   
                     
                        
                        
                       
                           
                       
                        
                       
                         m 
                          
                         
                           ( 
                           
                             R 
                             
                               even 
                               , 
                               
                                   
                               
                                
                               avr 
                             
                           
                           ) 
                         
                       
                     
                     
                        
                        
                       
                           
                       
                        
                       
                         e 
                          
                         
                           ( 
                           
                             R 
                             
                               even 
                               , 
                               
                                   
                               
                                
                               avr 
                             
                           
                           ) 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               Δ 
                
               
                   
               
                
               f 
             
             = 
             
               
                 1 
                 
                   2 
                    
                   π 
                    
                   
                       
                   
                    
                   T 
                 
               
                
               
                 arctan 
                  
                 
                   ( 
                   
                     
                        
                        
                       
                           
                       
                        
                       
                         m 
                          
                         
                           ( 
                           
                             R 
                             
                               odd 
                               , 
                               
                                   
                               
                                
                               avr 
                             
                           
                           ) 
                         
                       
                     
                     
                        
                        
                       
                           
                       
                        
                       
                         e 
                          
                         
                           ( 
                           
                             R 
                             
                               odd 
                               , 
                               
                                   
                               
                                
                               avr 
                             
                           
                           ) 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
         [0065]    The one based on R even, avr  is dependent on the FO due to the cos(2Φ) and is expected to show poor results around 
         [0000]    
       
         
           
             
                
               Φ 
                
             
             = 
             
               
                 
                   π 
                   4 
                 
                  
                 
                     
                 
                  
                 or 
                  
                 
                     
                 
                  
                 
                    
                   Φ 
                    
                 
               
               = 
               
                 
                   3 
                    
                   
                       
                   
                    
                   π 
                 
                 4 
               
             
           
         
       
       
         
           
             ( 
             
               
                 i 
                 . 
                 e 
                 . 
                 
                     
                 
                  
                 FO 
               
               = 
               
                 
                   
                     ± 
                     1875 
                   
                    
                   
                       
                   
                    
                   or 
                    
                   
                       
                   
                    
                   FO 
                 
                 = 
                 
                   ± 
                   5625 
                 
               
             
             ) 
           
         
       
     
         [0066]    Also, it should be noticed that this cos(2Φ) introduces a sign inversion for 
         [0000]    
       
         
           
             
                
               Φ 
                
             
             ∈ 
             
               [ 
               
                 
                   π 
                   4 
                 
                 , 
                 
                   
                     3 
                      
                     
                         
                     
                      
                     π 
                   
                   4 
                 
               
               ] 
             
           
         
       
     
         [0067]    Thus, in one particular advantageous embodiment, the two discrimators values R even, avr  and R odd, avr  are accumulated as follows: 
         [0000]      If          ( R   odd, avr )|&gt;|ℑ m ( R   odd, avr )|
 
         [0000]    
       
      
       R=R 
       odd, avr 
       +R 
       even, avr  
      
     
         [0000]      else 
         [0000]    
       
      
       R=R 
       odd, avr 
       −R 
       even, avr  
      
     
         [0068]    The correlation R is passed to the arc tan function in order to extract the is frequency offset estimate: 
         [0000]    
       
         
           
             
               
                 Δ 
                  
                 
                     
                 
                  
                 f 
               
               = 
               
                 
                   1 
                   
                     2 
                      
                     π 
                      
                     
                         
                     
                      
                     T 
                   
                 
                  
                 
                   arctan 
                    
                   
                     ( 
                     
                       
                          
                          
                         
                             
                         
                          
                         
                           m 
                            
                           
                             ( 
                             R 
                             ) 
                           
                         
                       
                       
                          
                          
                         
                             
                         
                          
                         
                           e 
                            
                           
                             ( 
                             R 
                             ) 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
              
             
                 
             
           
         
       
     
         [0069]    With respect to  FIG. 2 , there is now described one particular embodiment of a method for computing the frequency offset Δf which provides both accuracy and a relatively wide range of frequency use. 
         [0070]    In a step  21 , the process extracts the received CPICH signal included in the CPICH channel. As known in the art, it is assumed that the signal received at the antenna of the user equipment is properly equalized, de-spreaded and descrambled in accordance with the W-CDMA standard. Such operations are well known to a skilled man and, therefore, do not need further development. It suffices to recall that the received despreaded signal is modulated with the appropriate code in order to extract the predetermined CPICH channel. Such channel generates the sum of the two pilot signals transmitted by the base station. 
         [0071]    From step  21 , two parallel sequences are performed in order to separate the two pilot channels, with a left branch based on steps  22 - 23  and a right branch based on steps  24 - 25 - 26 . 
         [0072]    Considering the left branch, one sees that the process proceeds to a step  22 , the process proceeds with the computation of a first intermediate value x(i) in accordance with the formula: 
         [0000]        x ( i )= r ( i )+ r ( i+ 2) 
         [0073]    It can be seen that step  21  achieves the computation of r NoTxd [k] which was mentioned above: 
         [0000]    
       
         
           
             
               
                 r 
                 NoTxd 
               
                
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 
                   r 
                    
                   
                     [ 
                     k 
                     ] 
                   
                 
                 + 
                 
                   r 
                    
                   
                     [ 
                     
                       k 
                       + 
                       2 
                     
                     ] 
                   
                 
               
               = 
               
                 
                   
                     ( 
                     
                       1 
                       + 
                       j 
                     
                     ) 
                   
                    
                   
                     ( 
                     
                       
                         
                           S 
                           1 
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                       + 
                       
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           
                             ⌊ 
                             
                               
                                 k 
                                 + 
                                 1 
                               
                               2 
                             
                             ⌋ 
                           
                         
                          
                         
                           
                             D 
                             2 
                           
                            
                           
                             [ 
                             k 
                             ] 
                           
                         
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     N 
                     s 
                   
                    
                   
                     [ 
                     k 
                     ] 
                   
                 
               
             
           
         
       
     
         [0074]    Then, in a step  23 , the process proceeds with the computation of a second intermediate value y(i) derived from the first intermediate value x(i) in accordance with the formula: 
         [0000]        y ( i )= x ( i ).  x *( i+ 1) 
         [0075]    with x*(i+1) being the conjugate value of x(i+1). 
         [0076]    Such second intermediate value corresponds to the computation of R r     NoTxd     r     NoTxd   [k] mentioned above. 
         [0077]    Considering the right branch, one sees that, after completion of step  21 , the process proceeds to a step  24 , the process proceeds with the computation of a third intermediate value x′(i) in accordance with the formula: 
         [0000]        x ′( i )= r ( i )− r ( i+ 2)
 
         [0078]    Then, in a step  25 , the process proceeds with the computation of a fourth intermediate value t(i) derived from the third intermediate value x′(i) in accordance with the formula: 
         [0000]        t ′(2 k )= x ′(2 k ) and
 
         [0000]        t ′(2 k+ 1)=− x ′(2 k+ 1)
 
         [0079]    It can be seen that steps  24 - 25  achieves the computation of the value of r txd [k] which was mentioned above: 
         [0000]    
       
         
           
             
               
                 r 
                 Txd 
               
                
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 
                   
                     ( 
                     
                       - 
                       1 
                     
                     ) 
                   
                   
                     ⌊ 
                     
                       
                         k 
                         + 
                         1 
                       
                       2 
                     
                     ⌋ 
                   
                 
                  
                 
                   ( 
                   
                     
                       r 
                        
                       
                         [ 
                         k 
                         ] 
                       
                     
                     - 
                     
                       r 
                        
                       
                         [ 
                         
                           k 
                           + 
                           2 
                         
                         ] 
                       
                     
                   
                   ) 
                 
               
               = 
               
                 
                   
                     ( 
                     
                       1 
                       + 
                       j 
                     
                     ) 
                   
                    
                   
                     ( 
                     
                       
                         
                           S 
                           2 
                         
                          
                         
                           [ 
                           k 
                           ] 
                         
                       
                       + 
                       
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           
                             ⌊ 
                             
                               
                                 k 
                                 + 
                                 1 
                               
                               2 
                             
                             ⌋ 
                           
                         
                          
                         
                           
                             D 
                             1 
                           
                            
                           
                             [ 
                             k 
                             ] 
                           
                         
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     
                       ( 
                       
                         - 
                         1 
                       
                       ) 
                     
                     
                       ⌊ 
                       
                         
                           k 
                           + 
                           1 
                         
                         2 
                       
                       ⌋ 
                     
                   
                    
                   
                     
                       N 
                       d 
                     
                      
                     
                       [ 
                       k 
                       ] 
                     
                   
                 
               
             
           
         
       
     
         [0080]    Then, in a step  26 , the process proceeds with the computation of a fifth intermediate value y′(i) derived from the fourth intermediate value t′(i) in accordance with the formula: 
         [0000]        y ′( i )= t ′( i ).  t ′*( i+ 1)
 
         [0081]    where t′*(i+1) corresponds to the conjugate of the complex t′(i+1). 
         [0082]    Step  26  achieves the computation of R r     Txd     r     Txd   [k] mentioned above 
         [0083]    Then, in a step  27 , the process proceeds with the computation of a sixth and seventh intermediate values, respectively Z and Z′ which are computed as follows: 
         [0000]        Z (2 k )= y (2 k )+ y ′(2 k )
 
         [0000]        Z ′(2 k+ 1)= y (2 k+ 1)+ y ′(2 k+ 1)
 
         [0084]    Then, in a step  28 , the process proceeds with the average of the Z and Z′ value over a period of N samples in order to respectively generate an eight and a ninth values S and S′ in accordance with the following formulation: 
         [0000]        S= 1/ N Σ Z ( k ) for  k= 1 to  N    
         [0000]        S′= 1/ N Σ Z ′( k ) for  k= 1 to  N  
 
         [0085]    The process then proceeds with a step  29  which is a test for determining whether the absolute value of the real part of S′ is superior to the absolute value of the imaginary part of S′, in which case, the process proceeds with a step  30  where the two values S and S′ are added in order to generate a tenth intermediate value R as follows: 
         [0000]    
       
      
       R=S′+S  
      
     
         [0086]    The process then proceeds to a step  32 . 
         [0087]    If the absolute value of the real part of S′ is inferior to the absolute value of the imaginary part of S′ in step  29 , the process proceeds to a step  31  where the value of S is subtracted from that of S′ in order to compute R in accordance with the formula: 
         [0000]    
       
      
       R=S′−S  
      
     
         [0088]    After completion of steps  30  and  31 , the process proceeds to a step  32  where the frequency offset estimation is computed as follows: 
         [0000]    
       
         
           
             
               
                 Δ 
                  
                 
                     
                 
                  
                 f 
               
               = 
               
                 
                   1 
                   
                     2 
                      
                     π 
                      
                     
                         
                     
                      
                     T 
                   
                 
                  
                 
                   arctan 
                    
                   
                     ( 
                     
                       
                          
                          
                         
                             
                         
                          
                         
                           m 
                            
                           
                             ( 
                             R 
                             ) 
                           
                         
                       
                       
                          
                          
                         
                             
                         
                          
                         
                           e 
                            
                           
                             ( 
                             R 
                             ) 
                           
                         
                       
                     
                     ) 
                   
                 
               
             
              
             
                 
             
           
         
       
     
         [0089]    where          (R) and ℑm(R) are the real and imaginary parts, respectively of the tenth intermediate value R. 
         [0090]    The method which was described above shows a sampling period of T, which allows coverage of possible estimation range for a value of 
         [0000]    
       
         
           
             
                
               
                 Δ 
                  
                 
                     
                 
                  
                 f 
               
                
             
             ≤ 
             
               1 
               
                 2 
                  
                 T 
               
             
           
         
       
     
         [0000]      FIGS. 4 ,  5  and  6  are comparative flow charts allowing comparison of the process of the invention which was described above (and referred to as scheme n°3) with respect of two prior art method, namely scheme 1 and 2. 
         [0091]      FIG. 4  represents the Root Mean Squared Error (RMSE) of the frequency offset estimate (FOE) as a function of the frequency offset (FO) for AWGN and a SNR=0 dB. It should be noticed that a prior art method—referred to as scheme 2—only have limited performances to FO&lt;3500 Hz since this estimator cannot estimate larger FOs. The curve which is represented highlights the expected behaviors of the different estimators. As expected, estimator 3 (which is the one of the invention) shows degraded performances around 1875 Hz and 5625 Hz which is equivalent to the degradation shown by scheme 2 around 3.5 KHz. However, our proposed estimator has performance very close to those of estimator 2 for small FOs, which corresponds to the working regime after the AFC loop convergence. Estimator 2 has the advantage of covering a much larger interval. 
         [0092]      FIGS. 5 and 6  plot the FOE&#39;s RMSE as a function of lor/loc for FO=0 Hz (for AWGN and case 1 propagation scenarios respectively). 
         [0093]    The considered SPW simulations setups are as follows: 
         [0094]    For AWGN: 
         [0095]    Measurement Reference Channel: 12.2 Kbps 
         [0096]    lorx/loc=−i dB 
         [0097]    DPCH_Ec/lorx=−16.6 dB 
         [0098]    loc=−60 dBm 
         [0099]    For Fading Multi-Path Case 1: 
         [0100]    Measurement Reference Channel: 12.2 Kbps lorx/loc=9 dB DPCH_Ec/lorx=−15 dB loc=−60 dBm 
         [0101]    Multi-path propagation: 2 paths
       Powers: 0 dB, −10 dB   Delays: 0 ns, 976 ns