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

Description:
TECHNICAL FIELD 
     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 
     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. 
     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. 
       FIG. 1 : This figure represents the pattern of the transmitted CPICH pilots&#39; sequence. 
     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: 
               r   ⁡     [   k   ]       =         (     1   +   j     )     ⁢     (         h   1     ⁡     [   k   ]       +         (     -   1     )       ⌊       k   +   1     2     ⌋       ⁢       h   2     ⁡     [   k   ]           )       +     n   ⁡     [   k   ]               
while when the UE is subject to a frequency offset with respect to the base station, the received signal is then
 
               r   ⁡     [   k   ]       =         (     1   +   j     )     ⁢     (         h   1     ⁡     [   k   ]       +         (     -   1     )       ⌊       k   +   1     2     ⌋       ⁢       h   2     ⁡     [   k   ]           )     ⁢     exp   ⁡     (       j   ⁢           ⁢   2   ⁢           ⁢   π   ⁢           ⁢   k   ⁢           ⁢   Δ   ⁢           ⁢   fT     +     φ   0       )         +     n   ⁡     [   k   ]               
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.
 
     Some techniques are already known for computing an estimation of the frequency offset based on the computation of phase discriminator. 
     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. 
     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 
                      Δ   ⁢           ⁢   f          ≤     1     2   ⁢           ⁢   T         ,         
where T is the time between two consecutive CPICH pilots. In UMTS
 
             T   =       1   15     ⁢     10     -   3               
seconds, which corresponds to covering a range of frequency offsets from minus to plus 7500 Hz.
 
     Such is the goal of the present invention. 
     SUMMARY OF THE INVENTION 
     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. 
     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. 
     It is another object of the present invention to provide a receiver for a UMST communication system which incorporates improved frequency offset estimation mechanisms. 
     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 
     
       
         
           
             
               
                 r 
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                       N 
                       d 
                     
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                       k 
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     Where 
     r[k] is the received signal at the instant k; 
     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:
 
 S   m   [k]=h   m exp( jφ   0 )(exp( jkΦ )+exp( j ( k+ 2)Φ)),
 
 D   m [2 k]=h   m exp( jφ   0 )(exp( jkΦ )−exp( j ( k+ 2)Φ))
 
 N   s   [k]=n[k]+n[k+ 2]
 
 N   d   [k]=n[k]−n[k+ 2]
 
     and Φ=2π Δf T 
     we then compute
 
 R   r     NoTxd     r     NoTxd     [k]=r   NoTxd   [k] *conj( r   NoTxd   [k+ 1])
 
 R   r     Txd     r     Txd     [k]=r   Txd   [k]* conj( r   Txd   [k+ 1])
 
     where conj(.) is the complex conjugate operator. 
     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 
                 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   ]                 
where, for m=1,2 and n=1,2
 
 R   S     m     S     m     =S   m   [k]. conj( S   m   [k+ 1])=2| h   m | 2 (exp( jΦ )(1+cos(2Φ)))
 
 R   D     m     D     m     =D   m   [k]. conj( D   m   [k+ 1])=2| h   m | 2 (exp( jΦ )(1−cos(2Φ)))
 
 R   S     m     D     n     =S   m   [k]. conj( D   n   [k+ 1])=−2 jh   m   h   n *exp( jΦ )sin(2Φ)
 
 R   N     s     N     s     =N   s   [k]. conj( N   s   [k+ 1])
 
 R   N     d     N     d     =N   d   [k]. conj( N   d   [k+ 1])
 
     In one embodiment, the estimation is based on one estimator R even,avr  computed in accordance with the following formulas: 
               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   ]         )               
which simplifies to:
 
               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   ]         )                 
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. The frequency offset estimate being provided by the formula:
 
               Δ   ⁢           ⁢   f     =       1     2   ⁢           ⁢   π   ⁢           ⁢   T       ⁢   arc   ⁢           ⁢     tan   ⁡     (       ⁢     m   ⁡     (     R     even   ,   avr       )           ⁢     e   ⁡     (     R     even   ,   avr       )           )               
where            e(.) and          m(.) 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:

               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   ]         )               
which simplifies to
 
               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   ]         )                 
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. The frequency offset estimate being provided by the formula:
 
               Δ   ⁢           ⁢   f     =       1     2   ⁢           ⁢   π   ⁢           ⁢   T       ⁢   arc   ⁢           ⁢     tan   ⁡     (       ⁢     m   ⁡     (     R     odd   ,   avr       )           ⁢     e   ⁡     (     R     odd   ,   avr       )           )               
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:
 
If              e ( R   odd,avr )|&gt;           m ( R   odd,avr )|
 
 R=R   odd,avr   +R   even,avr  

     else
 
 R=R   odd,avr   −R   even,avr  
 
and the frequency offset estimate being provided by the formula:
 
               Δ   ⁢           ⁢   f     =       1     2   ⁢           ⁢   π   ⁢           ⁢   T       ⁢   arc   ⁢           ⁢     tan   ⁡     (       ⁢     m   ⁡     (   R   )           ⁢     e   ⁡     (   R   )           )               
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:
         extracting the received CPICH signal included in the CPICH channel;   performing a first processing branch (left) comprising the steps of:
           computing a first intermediate value x(i) in accordance with the formula:
 
 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:
 
 y ( i )= x ( i ).  x* ( i+ 1)
   where x* (i+1) is the complex conjugate of x(i+1)   
           performing a second processing branch (right)) comprising the steps of:
           computing a third intermediate value x′(i) in accordance with the formula:
 
 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:
 
 t′ (2 k )= x′ (2 k ) and
 
 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:
 
 y′ ( i )= t′ ( i ).  t′* ( i+ 1)
   
           computing a sixth and seventh intermediate values Z and Z′ in accordance with the formulas:
 
 Z (2 k )= y (2 k )+ y′ (2 k )
 
 Z′ (2 k+ 1)= y (2 k+ 1)+ y′ (2 k+ 1)
   respectively computing an eight and a ninth values S and S′ in accordance with the following formulation:
 
 S= 1/ N Σ Z ( k ) for  k= 1 to  N  
 
 S′= 1/ N Σ Z′ ( k ) for  k= 1 to  N  
   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:
 
 R=S′+S  
 
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:
 
 R=S′−S  
 
The offset estimation is then simply computed as follows:
       

     
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               f 
             
             = 
             
               
                 1 
                 
                   2 
                   ⁢ 
                   
                       
                   
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                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   T 
                 
               
               ⁢ 
               arc 
               ⁢ 
               
                   
               
               ⁢ 
               
                 tan 
                 ⁡ 
                 
                   ( 
                   
                     
                       ⁢ 
                       
                         m 
                         ⁡ 
                         
                           ( 
                           R 
                           ) 
                         
                       
                     
                     
                       ⁢ 
                       
                         e 
                         ⁡ 
                         
                           ( 
                           R 
                           ) 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     where            e(R)and          m(R) are the real and imaginary parts, respectively of the tenth intermediate value R.
     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 
       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. 
         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) 
         FIG. 2  illustrates one embodiment of the method for computing the offset frequency in accordance with the present invention. 
         FIG. 3  illustrates the principle of the separation of the two signals transmitted through the two emitting antennas. 
         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 
         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 
         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 
     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. 
     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. 
     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: 
     
       
         
           
             
               
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                 1 
               
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       FIG. 1  illustrates the sign pattern (applied to (1+j)) resulting from those formulas. 
     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. 
     
       
         
           
             
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     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 . 
     Δf corresponds to the frequency offset and assumed to be identical for both antennas. 
     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. 
     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: 
     
       
         
           
             
               
                 r 
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                             ] 
                           
                         
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     
                       ( 
                       
                         - 
                         1 
                       
                       ) 
                     
                     
                       ⌊ 
                       
                         
                           k 
                           + 
                           1 
                         
                         2 
                       
                       ⌋ 
                     
                   
                   ⁢ 
                   
                     
                       N 
                       d 
                     
                     ⁡ 
                     
                       [ 
                       k 
                       ] 
                     
                   
                 
               
             
           
         
       
     
     Where for m=1,2
 
 S   m   [k]=h   m exp( jφ   0 )(exp( jkΦ )+exp( j ( k+ 2)Φ))
 
 D   m [2 k]=h   m exp( jφ   0 )(exp( jkΦ )−exp( j ( k+ 2)Φ))
 
 N   s   [k]=n[k]+n[k+ 2]
 
 N   d   [k]=n[k]−n[k+ 2]
 
     h m  corresponding to the channel response for antenna m (m=1, 2), 
     j corresponding to the imaginary complex such as j 2 =−1 and Φ=2π Δf T 
     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 
     
       
         
           
             
                
               
                 Δ 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 f 
               
                
             
             ≤ 
             
               1 
               
                 2 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 T 
               
             
           
         
       
     
     We then compute R NoTxd [k] and R Txd [k] from the separated signal r NoTxd  and r Txd [k] as
 
 R   r     NoTxd     r     NoTxd     [k]=r   NoTxd   [k]* conj( r   NoTxd   [k+ 1])
 
 R   r     Txd     r     Txd     [k]=r   Txd   [k]* conj( r   Txd   [k+ 1])
 
     which simplifies, with the assumption that the channel is constant (i.e. h 1 [k]=h 1  and h 2 [k]=h 2 ), to 
                 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   ]                 
where, for m=1,2 and n=1,2
 
 R   S     m     S     m     =S   m   [k] .conj( S   m   [k+ 1])=2| h   m | 2 (exp( jΦ )(1+cos(2Φ)))
 
 R   D     m     D     m     =D   m   [k] .conj( D   m   [k+ 1])=2| h   m | 2 (exp( jΦ )(1−cos(2Φ)))
 
 R   S     m     D     n     =S   m   [k] .conj( D   n   [k+ 1])=−2 jh   m   h   n *exp( jΦ )sin(2Φ)
 
 R   N     s     N     s     =N   s   [k] .conj( N   s   [k+ 1])
 
 R   N     d     N     d     =N   d   [k] .conj( N   d   [k+ 1])
 
     If k is even, one may demonstrate that:
 
 R   even   [k]=R   r     Txd     r     Txd     [k]+R   r     NoTxd     r     NoTxd     [k]= 4exp( jΦ )(| h   1 | 2   +|h   2 | 2 )cos(2Φ)+ R   N     s     N     s     [k]−R   N     d     N     d     [k] 
 
     And if k is odd,
 
 R   odd   [k]=R   r     Txd     r     Txd     [k]+R   r     NoTxd     r     NoTxd     [k]= 4exp( jΦ )(| h   1 | 2   +|h   2 | 2 )+( R   N     s     N     s     [k]+R   N     d     N     d     [k] )
 
     In order to attenuate the noise effect, we compute an average over several samples of R even [k] and R odd [k] as 
     
       
         
           
             
               R 
               
                 even 
                 , 
                 avr 
               
             
             = 
             
               
                 
                   1 
                   
                     
                       N 
                       2 
                     
                     - 
                     
                       N 
                       1 
                     
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       
                         k 
                         = 
                         
                           N 
                           1 
                         
                       
                       , 
                       
                         k 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         even 
                       
                     
                     
                       N 
                       2 
                     
                   
                   ⁢ 
                   
                     
                       R 
                       even 
                     
                     ⁡ 
                     
                       [ 
                       k 
                       ] 
                     
                   
                 
               
               = 
               
                 
                   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 
                             ] 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               R 
               
                 odd 
                 , 
                 avr 
               
             
             = 
             
               
                 
                   1 
                   
                     
                       N 
                       2 
                     
                     - 
                     
                       N 
                       1 
                     
                   
                 
                 ⁢ 
                 
                   
                     ∑ 
                     
                       
                         k 
                         = 
                         
                           N 
                           1 
                         
                       
                       , 
                       
                         k 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         odd 
                       
                     
                     
                       N 
                       2 
                     
                   
                   ⁢ 
                   
                     
                       R 
                       odd 
                     
                     ⁡ 
                     
                       [ 
                       k 
                       ] 
                     
                   
                 
               
               = 
               
                 
                   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 
                             ] 
                           
                         
                       
                       ) 
                     
                   
                 
               
             
           
         
       
     
     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 
     
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               f 
             
             = 
             
               
                 1 
                 
                   2 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   T 
                 
               
               ⁢ 
               arc 
               ⁢ 
               
                   
               
               ⁢ 
               
                 tan 
                 ⁡ 
                 
                   ( 
                   
                     
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         m 
                         ⁡ 
                         
                           ( 
                           
                             R 
                             
                               even 
                               , 
                               avr 
                             
                           
                           ) 
                         
                       
                     
                     
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         e 
                         ⁡ 
                         
                           ( 
                           
                             R 
                             
                               even 
                               , 
                               avr 
                             
                           
                           ) 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     and 
     
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               f 
             
             = 
             
               
                 1 
                 
                   2 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   T 
                 
               
               ⁢ 
               arc 
               ⁢ 
               
                   
               
               ⁢ 
               
                 tan 
                 ⁡ 
                 
                   ( 
                   
                     
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         m 
                         ⁡ 
                         
                           ( 
                           
                             R 
                             
                               odd 
                               , 
                               avr 
                             
                           
                           ) 
                         
                       
                     
                     
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         e 
                         ⁡ 
                         
                           ( 
                           
                             R 
                             
                               odd 
                               , 
                               avr 
                             
                           
                           ) 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     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 
                  Φ        =         π   4     ⁢           ⁢   or   ⁢           ⁢        Φ          =       3   ⁢           ⁢   π     4             (i.e. FO=±1875 or FO=±5625)
 
     Also, it should be noticed that this cos(2Φ) introduces a sign inversion for 
     
       
         
           
             
                
               Φ 
                
             
             ∈ 
             
               [ 
               
                 
                   π 
                   4 
                 
                 , 
                 
                   
                     3 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     π 
                   
                   4 
                 
               
               ] 
             
           
         
       
     
     Thus, in one particular advantageous embodiment, the two discriminators values R even,avr  and R odd,avr  are accumulated as follows:
 
If              e ( R   odd,avr )|&gt;|           m ( R   odd,avr )|
 
 R=R   odd,avr   +R   even,avr  

     else
 
 R=R   odd,avr   −R   even,avr  
 
     The correlation R is passed to the arctan function in order to extract the frequency offset estimate: 
     
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               f 
             
             = 
             
               
                 1 
                 
                   2 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   T 
                 
               
               ⁢ 
               arc 
               ⁢ 
               
                   
               
               ⁢ 
               
                 tan 
                 ⁡ 
                 
                   ( 
                   
                     
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         m 
                         ⁡ 
                         
                           ( 
                           R 
                           ) 
                         
                       
                     
                     
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         e 
                         ⁡ 
                         
                           ( 
                           R 
                           ) 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     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. 
     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. 
     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 . 
     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:
 
 x ( i )= r ( i )+ r ( i+ 2)
 
     It can be seen that step  21  achieves the computation of r NoTxd [k] which was mentioned above: 
     
       
         
           
             
               
                 r 
                 NoTxd 
               
               ⁡ 
               
                 [ 
                 k 
                 ] 
               
             
             = 
             
               
                 
                   r 
                   ⁡ 
                   
                     [ 
                     k 
                     ] 
                   
                 
                 + 
                 
                   r 
                   ⁡ 
                   
                     [ 
                     
                       k 
                       + 
                       2 
                     
                     ] 
                   
                 
               
               = 
               
                 
                   
                     ( 
                     
                       1 
                       + 
                       j 
                     
                     ) 
                   
                   ⁢ 
                   
                     ( 
                     
                       
                         
                           S 
                           1 
                         
                         ⁡ 
                         
                           [ 
                           k 
                           ] 
                         
                       
                       + 
                       
                         
                           
                             ( 
                             
                               - 
                               1 
                             
                             ) 
                           
                           
                             ⌊ 
                             
                               
                                 k 
                                 + 
                                 1 
                               
                               2 
                             
                             ⌋ 
                           
                         
                         ⁢ 
                         
                           
                             D 
                             2 
                           
                           ⁡ 
                           
                             [ 
                             k 
                             ] 
                           
                         
                       
                     
                     ) 
                   
                 
                 + 
                 
                   
                     N 
                     s 
                   
                   ⁡ 
                   
                     [ 
                     k 
                     ] 
                   
                 
               
             
           
         
       
     
     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:
 
 y ( i )= x ( i ).  x* ( i+ 1)
 
     with x*(i+1) being the conjugate value of x(i+1) . 
     Such second intermediate value corresponds to the computation of R r     NoTxd     r     NoTxd   [k] mentioned above. 
     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:
 
 x′ ( i )= r ( i )− r ( i+ 2)
 
     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:
 
 t′ (2 k )= x′ (2 k ) and
 
 t′ (2 k+ 1)=− x′ (2 k+ 1)
 
     It can be seen that steps  24 - 25  achieves the computation of the value of r txd [k] which was mentioned above: 
     
       
         
           
             
               
                 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 
                       ] 
                     
                   
                 
               
             
           
         
       
     
     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:
 
 y′ ( i )= t′ ( i ).  t′* ( i+ 1)
 
     where t′*(i+1) corresponds to the conjugate of the complex t′(i+1). 
     Step  26  achieves the computation of R r     Txd     r     Txd   [k] mentioned above 
     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:
 
 Z (2 k )= y (2 k )+ y′ (2 k )
 
 Z′ (2 k+ 1)= y (2 k+ 1)+ y′ (2 k+ 1)
 
     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:
 
S=1/N Σ Z(k) for k=1 to N
 
S′=1/N Σ Z′(k) for k=1 to N
 
     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:
 
 R=S′+S  
 
     The process then proceeds to a step  32 . 
     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:
 
 R=S′−S  
 
     After completion of steps  30  and  31 , the process proceeds to a step  32  where the frequency offset estimation is computed as follows: 
     
       
         
           
             
               Δ 
               ⁢ 
               
                   
               
               ⁢ 
               f 
             
             = 
             
               
                 1 
                 
                   2 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   π 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   T 
                 
               
               ⁢ 
               arc 
               ⁢ 
               
                   
               
               ⁢ 
               
                 tan 
                 ⁡ 
                 
                   ( 
                   
                     
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         m 
                         ⁡ 
                         
                           ( 
                           R 
                           ) 
                         
                       
                     
                     
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         e 
                         ⁡ 
                         
                           ( 
                           R 
                           ) 
                         
                       
                     
                   
                   ) 
                 
               
             
           
         
       
     
     where            e(R) and          m(R) are the real and imaginary parts, respectively of the tenth intermediate value R.
     The method which was described above shows a sampling period of T, which allows coverage of possible estimation range for a value of 
                    Δ   ⁢           ⁢   f          ≤     1     2   ⁢   T               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 no 3) with respect of two prior art method, namely scheme 1 and 2.
 
       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. 
       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). 
     The considered SPW simulations setups are as follows: 
     For AWGN: 
     Measurement Reference Channel: 12.2 Kbps 
     lorx/loc=−i dB 
     DPCH_Ec/lorx=−16.6 dB 
     loc=−60 dBm 
     For fading multi-path case 1: 
     Measurement Reference Channel: 12.2 Kbps lorx/loc=9 dB DPCH_Ec/lorx=−15 dB loc=−60 dBm 
     Multi-path propagation: 2 paths
         Powers: 0 dB, −10 dB   Delays: 0 ns, 976 ns