Abstract:
A method of wireless transmission for estimating the carrier frequency offset in a base station of a received transmission from a user equipment (UE) accessing a radio access network. The method time de-multiplexes selected symbols of a received sub-frame, computes the frequency-domain symbols received from each antenna through an FFT, de-maps the UEs selected sub-carriers for each antenna, computes metrics associated to a carrier frequency offset hypothesis spanning a searched frequency offset window, repeats these steps on subsequent received sub-frames from the UE over an estimation interval duration, non-coherently accumulates the computed metrics and selects the carrier frequency offset hypothesis with largest accumulated metric amplitude.

Description:
CLAIM OF PRIORITY 
       [0001]    This application claims priority under 35 U.S.C. 119(e)(1) to U.S. Provisional Application No. 61/079,933 filed Jul. 11, 2008, U.S. Provisional Application No. 61/095,405 filed Sep. 9, 2008 and U.S. Provisional Application No. 61/100,104 filed Sep. 25, 2008. 
     
    
     TECHNICAL FIELD OF THE INVENTION 
       [0002]    The technical field of this invention is wireless communication. 
       BACKGROUND OF THE INVENTION 
       [0003]      FIG. 1  shows an exemplary wireless telecommunications network  100 . The illustrative telecommunications network includes base stations  101 ,  102  and  103 , though in operation, a telecommunications network necessarily includes many more base stations. Each of base stations  101 ,  102  and  103  are operable over corresponding coverage areas  104 ,  105  and  106 . Each base station&#39;s coverage area is further divided into cells. In the illustrated network, each base station&#39;s coverage area is divided into three cells. Handset or other user equipment (UE)  109  is shown in Cell A  108 . Cell A  108  is within coverage area  104  of base station  101 . Base station  101  transmits to and receives transmissions from UE  109 . As UE  109  moves out of Cell A  108  and into Cell B  107 , UE  109  may be handed over to base station  102 . Because UE  109  is synchronized with base station  101 , UE  109  can employ non-synchronized random access to initiate handover to base station  102 . 
         [0004]    Non-synchronized UE  109  also employs non-synchronous random access to request allocation of up-link  111  time or frequency or code resources. If UE  109  has data ready for transmission, which may be traffic data, measurements report, tracking area update, UE  109  can transmit a random access signal on up-link  111 . The random access signal notifies base station  101  that UE  109  requires up-link resources to transmit the UEs data. Base station  101  responds by transmitting to UE  109  via down-link  110 , a message containing the parameters of the resources allocated for UE  109  up-link transmission along with a possible timing error correction. After receiving the resource allocation and a possible timing advance message transmitted on down-link  110  by base station  101 , UE  109  optionally adjusts its transmit timing and transmits the data on up-link  111  employing the allotted resources during the prescribed time interval. 
         [0005]    Long Term Evolution (LTE) wireless networks, also known as Evolved Universal Terrestrial Radio Access Network (E-UTRAN), are being standardized by the 3GPP working groups (WG). Orthogonal frequency division multiple access (OFDMA) and SC-FDMA (single carrier FDMA) access schemes were chosen for the down-link (DL) and up-link (UL) of E-UTRAN, respectively. User Equipments (UEs) are time and frequency multiplexed on a physical uplink shared channel (PUSCH) and time and frequency synchronization between UEs guarantees optimal intra-cell orthogonality. In UL, frequency offsets (FO) can be due to local oscillator (LO) drifts at both the UE and the Base Station, also referred to as eNodeB, but also to the UE speed translating into Doppler shift in line of sight (LOS) propagation conditions. If the large sub-carrier spacing of LTE (15 kHz) makes it robust with respect to orthogonality loss due to Doppler shift, since even high speed trains would not generate a Doppler shift exceeding one tenth of a sub-carrier, the frequency offset still has a negative impact on BLER due to fast channel variation within a sub-frame: 
         [0006]    Rayleigh TU channel: FO range of 0 to 300 Hz with the maximum frequency inaccuracies at eNB of 0.05 ppm and at UE of 0.1 ppm; 
         [0007]    AWGN (LOS) channel: FO range of 0 to 1600 Hz with the UE speed translating into Doppler shift. 
         [0008]      FIG. 2  shows the BLER performance degradation due to various frequency offsets with quadrature phase shift keying (QPSK) modulation and turbo coding rate of ⅓ with AWGN.  FIG. 3  similarly shows the BLER performance degradation with TU-6 fading channel. There is a need to estimate and remove the frequency offset before channel estimation and demodulation. This invention is a frequency estimation method which applies directly on the de-mapped frequency sub-carriers of a UEs symbol and compares its performance with other published methods. 
         [0009]    On top of these scenarios, the 3GPP Working Group #4 defined propagation channels specifically addressing the frequency offset estimation and compensation function which worst-case scenarios are expected to be encountered along High Speed Trains (HST) lines.  FIG. 4  illustrates the frequency offset time behavior of both channels.  FIG. 5  illustrates the resulting frequency variations observed within a 30 ms interval. 
         [0010]    In an additional channel scenario, Rician fading is considered where Rician factor K=10 dB is the ratio between the dominant signal power and the variant of the other weaker signals. 
         [0011]    It is clear from  FIGS. 4 and 5  that in order to track the abrupt frequency offset variations of the HST scenarios, the eNB must permanently estimate the frequency offset of each UE. This patent application assumes the following scheme for concurrent frequency offset estimation and compensation illustrated in  FIG. 6 . In  FIG. 6  frequency estimation is performed for a given UE during an estimation interval  610  and a frequency estimate is issued at the end of interval  610 . During a given estimation interval  620  of a given UE, the frequency estimate issued by the previous interval  610  is replaced. 
       SUMMARY OF THE INVENTION 
       [0012]    This patent application details the design choices for the LTE frequency offset estimation and compensation, from theoretical derivations and performance evaluations. It is shown that the state-of-the-art does not satisfactorily addresses the high-end performance requirements of LTE so that an alternate approach is needed. This invention is a maximum-likelihood based solution using the available frequency-domain interference-free symbols de-mapped (or de-multiplexed) at the output of the Fast Fourier Transform (FFT) of an OFDMA multi-user receiver. The estimator operates on frequency-domain OFDMA user symbols extracted from an OFDMA multiplex with sub-carrier modulation is known a-priori. Other solutions either rely on pilot repetition or on an interference-free cyclic prefix (CP) not used in OFDMA systems such as LTE. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0013]    These and other aspects of this invention are illustrated in the drawings, in which: 
           [0014]      FIG. 1  is a diagram of a communication system of the prior art related to this invention having three cells; 
           [0015]      FIG. 2  illustrates the BLER performance degradation due to various frequency offsets with quadrature phase shift keying (QPSK) modulation and turbo coding rate of ⅓ with AWGN; 
           [0016]      FIG. 3  is similar to  FIG. 2  and illustrates the BLER performance degradation with TU- 6  fading channel; 
           [0017]      FIG. 4  illustrates the frequency offset time behavior of two channel types; 
           [0018]      FIG. 5  illustrates the resulting frequency variations in these two channel types observed within a 30 ms interval; 
           [0019]      FIG. 6  illustrates the invention for concurrent frequency offset estimation and compensation; 
           [0020]      FIG. 7  illustrates the principles of the single carrier frequency division multiple access (SC-FDMA) UL transmitter and receiver of LTE; 
           [0021]      FIG. 8  illustrates two ways to remove the frequency offset once frequency offset compensation has been estimated; 
           [0022]      FIG. 9  illustrates the number of complex multiples required for various time and frequency-domain FO compensation methods for the full range of allocated RBs per UE; 
           [0023]      FIG. 10  illustrates a maximum-likelihood algorithm which tests a number of frequency offset hypothesis on the demodulation reference symbols (DMRS) located at the centre of each slot according to this invention; 
           [0024]      FIG. 11  illustrates computing the metric M(δf) by multiplying the received de-mapped FFT output samples R S (k) with the complex conjugate of frequency shifted replicas of the expected DMRS frequency-domain sequence S(k,δf i ); 
           [0025]      FIG. 12  illustrates computing the metric M(δf) by multiplying the frequency shifted replicas of the received de-mapped FFT output sample R S(k ,δf i ) with the complex conjugate of the expected DMRS sequence S(k); 
           [0026]      FIG. 13  illustrates the continuous frequency signal of a DMRS sequence (Drifted Symbol) and specifically an extended Zadoff-Chu sequence (EZC); 
           [0027]      FIG. 14  illustrates the corresponding bias of the FO estimator illustrated in  FIG. 13 ; 
           [0028]      FIG. 15  illustrates the residual bias of a compensated estimator; 
           [0029]      FIG. 16  illustrates that the frequency-bins with common channel estimation method can be mapped onto a parabola; 
           [0030]      FIG. 17  illustrates that the frequency-bins with per-bin channel estimation method can be mapped onto a parabola; 
           [0031]      FIG. 18  is a flow chart illustrating the frequency offset estimation and compensation algorithm of this invention; and 
           [0032]      FIG. 19  illustrates the time behavior of the frequency elementary estimates within an estimation interval obtained from simulation. 
       
    
    
     DETAILED DESCRIPTION OF PREFERRED EMBODIMENTS 
       [0033]      FIG. 7  illustrates the principles of the single carrier frequency division multiple access (SC-FDMA) UL transmitter and receiver of LTE. The SC-FDMA scheme is also referred to as discrete Fourier transform spread orthogonal frequency division multiplexing (DFT-SOFDM) and is an OFDMA system with subband carrier assignment scheme (CAS). For the UE transmitter coder  711  encodes received modulated symbols. These are transformed into the frequency domain by discrete Fourier transform (DFT)  712 . These are filtered by filter  713 . Block  714  combines the symbols with zeros to fill particular slots. Block  715  is an inverse Fast Fourier Transform (IFFT) which converts back into the time domain. Block  716  adds the cyclic prefix (CP) for transmission via the antenna (TxRF). For the base station (eNB) receiver block  721  removes the cyclic prefix. Block  722  is a Fast Fourier Transform (FFT). Filter  723  separates the channels corresponding to separate UEs. Processing continues with one channel for each active UE.  FIG. 7  illustrates two channels. The two channels include filters  734  and  754 . Blocks  735  and  755  perform channel estimation and compensation. Following filters  736  and  756 , blocks  737  and  757  return to the time domain by an inverse discrete Fourier transform (IDFT). Filters  738  and  758  drive respective demodulators  739  and  759  which recover the originally transmitted signals. 
         [0034]      FIG. 7  assumes no frequency offset so frequency offset compensation is not shown.  FIG. 8  illustrates two ways to remove the frequency offset once frequency offset compensation has been estimated at the base station receiver: time-domain frequency offset compensation  810 ; and frequency-domain frequency offset compensation  820 . In time-domain frequency offset compensation block  811  removes the cyclic prefix from the received signals. A bank of frequency compensating filters  812  for each active UE frequency compensates by e −j2πf(i)t . Block  813  converts this into the frequency domain via a Fast Fourier Transform (FFT). Filter  814  separates the signals corresponding to the individual UEs  815 . In frequency-domain frequency offset compensation block  821  removes the cyclic prefix. Block  722  converts into the frequency domain via a Fast Fourier Transform (FFT). Filter  823  separates the corresponding to the individual UEs. Each UE channel includes filters  834  and  835 , a frequency domain interpolation  835  and  855  to generate the individual UE signals  836  and  856 . 
         [0035]    The frequency-domain interpolation  FIG. 8   820  operates on de-mapped frequency sub-carriers of a given UE as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Z 
                       i 
                     
                      
                     
                       ( 
                       
                         k 
                         , 
                         ɛ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         N 
                         FFT 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           0 
                         
                         
                           N 
                           
                             sc 
                             - 
                             1 
                           
                         
                       
                        
                       
                         
                           Z 
                            
                           
                             ( 
                             l 
                             ) 
                           
                         
                          
                         
                           C 
                            
                           
                             ( 
                             
                               
                                 l 
                                 - 
                                 k 
                               
                               , 
                               ɛ 
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where: N FFT  is the FFT size; N SC  is the UE allocation size in sub-carriers; ε is the normalized frequency shift equal to δf/Δf sc ; C(l−k,ε) are the interpolator coefficients; Z i (k,ε) is the interpolated point from the Z(l) points. Further: 
         [0000]    
       
         
           
             
               
                 
                   
                     C 
                      
                     
                       ( 
                       
                         k 
                         , 
                         ɛ 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         sin 
                          
                         
                             
                         
                          
                         
                           π 
                            
                           
                             ( 
                             
                               k 
                               + 
                               ɛ 
                             
                             ) 
                           
                         
                       
                       
                         sin 
                          
                         
                           ( 
                           
                             
                               π 
                                
                               
                                 ( 
                                 
                                   k 
                                   + 
                                   ɛ 
                                 
                                 ) 
                               
                             
                             / 
                             
                               N 
                               FFT 
                             
                           
                           ) 
                         
                       
                     
                      
                     
                        
                       
                         
                           jπ 
                            
                           
                             ( 
                             
                               k 
                               + 
                               ɛ 
                             
                             ) 
                           
                         
                          
                         
                           
                             
                               N 
                               FFT 
                             
                             - 
                             1 
                           
                           
                             N 
                             FFT 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    According to the expression of Equation (1) this method is also called frequency offset compensation through circular convolution. This is derived as follows. Let Z(k), k=0 . . . N SC −1 be the frequency complex samples at the DFT output of the UE transmitter in  FIG. 7 . The Z(k) samples are mapped onto a wideband signal Z WB (k) as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Z 
                       WB 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               Z 
                                
                               
                                 ( 
                                 
                                   k 
                                   - 
                                   
                                     k 
                                     1 
                                   
                                 
                                 ) 
                               
                             
                             ; 
                           
                         
                         
                           
                             
                               k 
                               = 
                               
                                 k 
                                 1 
                               
                             
                             , 
                             
                               
                                 k 
                                 1 
                               
                               + 
                               1 
                             
                             , 
                             … 
                              
                             
                                 
                             
                             , 
                             
                               
                                 k 
                                 1 
                               
                               + 
                               
                                 N 
                                 SC 
                               
                               - 
                               1 
                             
                           
                         
                       
                       
                         
                           0 
                         
                         
                           elsewhere 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This signal is then converted into the time domain for transmission. The cyclic prefix plays no role in this derivation so it is omitted without loss of generality. This conversion is as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             z 
                             WB 
                           
                            
                           
                             ( 
                             n 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             1 
                             
                               
                                 N 
                                 FFT 
                               
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 0 
                               
                               
                                 
                                   N 
                                   FFT 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 
                                   Z 
                                   WB 
                                 
                                  
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                                
                               
                                  
                                 
                                   j2 
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     kn 
                                     / 
                                     
                                       N 
                                       FFT 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             
                               
                                 N 
                                 FFT 
                               
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
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                                   N 
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                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 Z 
                                  
                                 
                                   ( 
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                                   j2 
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                                    
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                                    
                                   
                                       
                                   
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                                     ( 
                                     
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                                       + 
                                       
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                                     ) 
                                   
                                    
                                   
                                     n 
                                     / 
                                     
                                       N 
                                       FFT 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
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                                  
                                 
                                     
                                 
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                                     nk 
                                     1 
                                   
                                   / 
                                   
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                                 N 
                                 FFT 
                               
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
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                                   N 
                                   SC 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 Z 
                                  
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                                
                               
                                  
                                 
                                   j2 
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     kn 
                                     / 
                                     
                                       N 
                                       FFT 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The normalized frequency offset 
         [0000]    
       
         
           
             
               ɛ 
               = 
               
                 
                   δ 
                    
                   
                       
                   
                    
                   f 
                 
                 
                   f 
                   sc 
                 
               
             
             , 
           
         
       
     
         [0000]    where δf is the frequency offset (Hz) and f sc  is the sub-carrier spacing, applies to the time domain signal as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             z 
                             
                               WB 
                               - 
                               FO 
                             
                           
                            
                           
                             ( 
                             
                               n 
                               , 
                               ɛ 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             
                               z 
                               WB 
                             
                              
                             
                               ( 
                               n 
                               ) 
                             
                           
                            
                           
                              
                             
                               j2π 
                                
                               
                                   
                               
                                
                               n 
                                
                               
                                   
                               
                                
                               
                                 ɛ 
                                 / 
                                 
                                   N 
                                   FFT 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                                
                               
                                 j2 
                                  
                                 
                                     
                                 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     nk 
                                     1 
                                   
                                   / 
                                   
                                     N 
                                     FFT 
                                   
                                 
                               
                             
                             
                               
                                 N 
                                 FFT 
                               
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 0 
                               
                               
                                 
                                   N 
                                   SC 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 Z 
                                  
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                                
                               
                                  
                                 
                                   j2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     π 
                                     ( 
                                     
                                         
                                     
                                      
                                     
                                       k 
                                       + 
                                       ɛ 
                                     
                                     ) 
                                   
                                    
                                   
                                     n 
                                     / 
                                     
                                       N 
                                       FFT 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    At the receiver, the frequency domain signal Z WB-FO  is obtained by applying an FFT to z WB-FO : 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             Z 
                             
                               WB 
                               - 
                               FO 
                             
                           
                            
                           
                             ( 
                             
                               k 
                               , 
                               ɛ 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             1 
                             
                               
                                 N 
                                 FFT 
                               
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 
                                   N 
                                   FFT 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 
                                   z 
                                   
                                     WB 
                                     - 
                                     FO 
                                   
                                 
                                  
                                 
                                   ( 
                                   
                                     n 
                                     , 
                                     ɛ 
                                   
                                   ) 
                                 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     j2 
                                   
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     kn 
                                     / 
                                     
                                       N 
                                       FFT 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             
                               N 
                               FFT 
                             
                           
                            
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               
                                 N 
                                 FFT 
                               
                               - 
                               1 
                             
                           
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             [ 
                             
                               
                                 ∑ 
                                 
                                   l 
                                   = 
                                   0 
                                 
                                 
                                   
                                     N 
                                     SC 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   Z 
                                    
                                   
                                     ( 
                                     l 
                                     ) 
                                   
                                 
                                  
                                 
                                    
                                   
                                     j2 
                                      
                                     
                                         
                                     
                                      
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                                      
                                     
                                         
                                     
                                      
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           k 
                                           1 
                                         
                                         + 
                                         ɛ 
                                       
                                       ) 
                                     
                                      
                                     
                                       n 
                                       / 
                                       
                                         N 
                                         FFT 
                                       
                                     
                                   
                                 
                               
                             
                             ] 
                           
                            
                           
                              
                             
                               
                                 - 
                                 j2π 
                               
                                
                               
                                   
                               
                                
                               
                                 kn 
                                 / 
                                 
                                   N 
                                   FFT 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             1 
                             
                               N 
                               FFT 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 l 
                                 = 
                                 0 
                               
                               
                                 
                                   N 
                                   SC 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 Z 
                                  
                                 
                                   ( 
                                   l 
                                   ) 
                                 
                               
                                
                               
                                 
                                   ∑ 
                                   
                                     n 
                                     = 
                                     0 
                                   
                                   
                                     
                                       N 
                                       FFT 
                                     
                                     - 
                                     1 
                                   
                                 
                                  
                                 
                                    
                                   
                                     j2 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     
                                       ( 
                                       
                                         l 
                                         + 
                                         
                                           k 
                                           1 
                                         
                                         - 
                                         k 
                                         + 
                                         ɛ 
                                       
                                       ) 
                                     
                                      
                                     
                                       n 
                                       / 
                                       
                                         N 
                                         FFT 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Finally, the de-mapped signal Z FO (s, ε) is extracted from Z WB-FO  as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               
                                 Z 
                                 FO 
                               
                                
                               
                                 ( 
                                 
                                   s 
                                   , 
                                   ɛ 
                                 
                                 ) 
                               
                             
                             = 
                               
                              
                             
                               
                                 Z 
                                 
                                   WB 
                                   - 
                                   FO 
                                 
                               
                                
                               
                                 ( 
                                 
                                   
                                     k 
                                     = 
                                     
                                       
                                         k 
                                         1 
                                       
                                       + 
                                       s 
                                     
                                   
                                   , 
                                   ɛ 
                                 
                                 ) 
                               
                             
                           
                           ; 
                           
                             s 
                             = 
                             
                               
                                 0 
                                  
                                 
                                     
                                 
                                  
                                 … 
                                  
                                 
                                     
                                 
                                  
                                 
                                   N 
                                   SC 
                                 
                               
                               - 
                               1 
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               1 
                               
                                 N 
                                 FFT 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   l 
                                   = 
                                   0 
                                 
                                 
                                   
                                     N 
                                     SC 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   Z 
                                    
                                   
                                     ( 
                                     l 
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     ∑ 
                                     
                                       n 
                                       = 
                                       0 
                                     
                                     
                                       
                                         N 
                                         FFT 
                                       
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                      
                                     
                                       
                                         j2π 
                                          
                                         
                                           ( 
                                           
                                             l 
                                             - 
                                             s 
                                             + 
                                             ɛ 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         n 
                                         / 
                                         
                                           N 
                                           FFT 
                                         
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                            
                           
                             
                               1 
                               
                                 N 
                                 FFT 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   l 
                                   = 
                                   0 
                                 
                                 
                                   
                                     N 
                                     sc 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   Z 
                                    
                                   
                                     ( 
                                     l 
                                     ) 
                                   
                                 
                                  
                                 
                                   C 
                                    
                                   
                                     ( 
                                     
                                       
                                         l 
                                         - 
                                         s 
                                       
                                       , 
                                       ɛ 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   
                     where 
                      
                     
                       : 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         
                           C 
                            
                           
                             ( 
                             
                               k 
                               , 
                               ɛ 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               
                                 N 
                                 FFT 
                               
                               - 
                               1 
                             
                           
                            
                           
                              
                             
                               j2π 
                                
                               
                                   
                               
                                
                               
                                 
                                   n 
                                    
                                   
                                     ( 
                                     
                                       k 
                                       + 
                                       ɛ 
                                     
                                     ) 
                                   
                                 
                                 / 
                                 
                                   N 
                                   FFT 
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               sin 
                                
                               
                                   
                               
                                
                               
                                 π 
                                  
                                 
                                   ( 
                                   
                                     k 
                                     + 
                                     ɛ 
                                   
                                   ) 
                                 
                               
                             
                             
                               sin 
                                
                               
                                 ( 
                                 
                                   
                                     π 
                                      
                                     
                                       ( 
                                       
                                         k 
                                         + 
                                         ɛ 
                                       
                                       ) 
                                     
                                   
                                   / 
                                   
                                     N 
                                     FFT 
                                   
                                 
                                 ) 
                               
                             
                           
                            
                           
                              
                             
                               
                                 jπ 
                                  
                                 
                                   ( 
                                   
                                     k 
                                     + 
                                     ɛ 
                                   
                                   ) 
                                 
                               
                                
                               
                                 
                                   
                                     N 
                                     FFT 
                                   
                                   - 
                                   1 
                                 
                                 
                                   N 
                                   FFT 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
         [0036]    The interpolation coefficients apply directly onto the demapped symbol sub-carriers and are not dependent on the UE location in the frequency multiplex, here modelled by the origin k 1  of the user&#39;s sub-carrier mapping. Thus once the frequency offset is known at the receiver, the same set of coefficients can be used for a given UE and allocation size irrespective of its sub-carrier mapping. The coefficient set for smaller number of sub-carriers allocation is a sub-set of the coefficient set of larger number of sub-carriers allocation. The set of interpolator coefficients for a given frequency offset can be computed once for the largest allocation size and stored in a look up table (LUT) to be used for smaller allocation sizes. For example for a 20 MHz LTE spectrum, the maximum allocation is 1200 sub-carriers so that the storage requirements for the interpolator of a given UE is 1200 complex coefficients, each of 2 by 16 bits for a total of 4.69 KBytes. 
         [0037]    Table 1 assumes a 20 MHz LTE spectrum, with the following numerology from 3GPP TS 36.211 v8.1.0 2007-11 standard. 
         [0000]    
       
         
               
               
               
             
           
               
                   
                 TABLE 1 
               
               
                   
                   
               
             
             
               
                   
                 Symbol duration 
                 66.67 μs 
               
               
                   
                 Spectrum 
                 20 MHz 
               
               
                   
                 BW efficiency 
                 90% 
               
               
                   
                 Sampling rate 
                 30.72 Msps 
               
               
                   
                 Sub-carrier 
                 15 kHz 
               
               
                   
                 size 
               
               
                   
                 FFT size 
                 2048 
               
               
                   
                 RB size 
                 12 subcarriers = 180 kHz 
               
               
                   
                 Number of RBs 
                 100 
               
               
                   
                   
               
             
          
         
       
     
         [0038]    Table 2 is an example of complexity comparison in the number of complex multiplications. Table 2 lists the number of complex multiples required for various time and frequency-domain FO compensation methods. Table 2 assumes 50 UEs each with two allocated resource blocks (RBs) frequency multiplexed in a 20 MHz symbol. The complexity of the radix-4 FFT is taken as 3*N FFT /4*(ln(N FFT )/ln(4)) and the complexity of the frequency domain FO compensation is taken as N SC   2 .  FIG. 9  illustrates this comparison for the full range of allocated RBs per UE. 
         [0000]    
       
         
               
               
               
               
               
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
               
               
               
               
               
               
             
           
               
                   
                 TABLE 2 
               
             
             
               
                   
                   
               
               
                   
                 Number of 
                 Time domain 
                   
                 Frequency domain 
                   
               
             
          
           
               
                   
                 complex 
                 TD freq 
                 radix-4 
                 radix-4 
                 FD freq 
               
               
                   
                 multiplies 
                 comp 
                 FFT 
                 FFT 
                 comp. 
               
               
                   
                   
               
             
          
           
               
                   
                 Common to  
                   
                   
                 8448 
                   
               
               
                   
                 all UEs 
               
               
                   
                 Per UE 
                 2048 
                 8448 
                   
                 576 
               
               
                   
                 Sub-total 
                 102400 
                 844800 
                 8448 
                 28800 
               
             
          
           
               
                   
                 Total 
                 947200 
                   
                 37248 
               
               
                   
                   
               
             
          
         
       
     
         [0039]    Table 2 and  FIG. 9  show frequency-domain FO compensation is more efficient than the time-domain FO compensation when the number of multiplexed UEs per symbol exceeds about 10% of the multiplexing capacity. 
         [0040]    In OFDMA and SC-FDMA systems such as LTE, the estimated UL frequency offset of each multiplexed UE in an OFDM or SC-FDMA symbol is removed in frequency domain after sub-carrier de-mapping by means of the frequency domain interpolator defined in Equations (1) and (2). 
         [0041]    The interpolation coefficients apply directly onto the demapped symbol sub-carriers. The are not dependent on the UE location in the frequency multiplex. Therefore the same set of coefficients can be used for a given UE and allocation size irrespective of its sub-carrier mapping. 
         [0042]    The coefficient set for smaller number of sub-carriers allocation is a sub-set of the coefficient set of larger number of sub-carriers allocation. Thus the set of interpolator coefficients for a given frequency offset can be computed once for the largest allocation size and stored in a LUT to be used for smaller allocation sizes. 
         [0043]    Most algorithms published on the topic of frequency offset estimation in OFDM systems are variations of the Correlation Based method that can be used both in time and frequency domains. Let z, be a sequence of time domain complex samples such that some of these samples are duplicated in time: 
         [0000]      z n+LN =z n   ; n= 0,1, . . . ,  N− 1; N,L integers  (9) 
         [0000]    This sequence is transmitted over the air and received distorted at the demodulator unit by AWGN samples w n  and a frequency offset δf: 
         [0000]        r   n   =z   n   e   j2πnδf/f     s     +w   n   (10) 
         [0000]    where: f s  is the sampling rate. The correlation p in time of the received repeated samples is: 
         [0000]    
       
         
           
             
               
                 
                   ρ 
                   = 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                      
                     
                       
                         r 
                         n 
                       
                        
                       
                         r 
                         
                           n 
                           + 
                           LN 
                         
                         * 
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with a statistical average of: 
         [0000]    
       
         
           
             
               
                 
                   
                     〈 
                     ρ 
                     〉 
                   
                   = 
                   
                     
                        
                       
                         
                           - 
                           j 
                         
                          
                         
                             
                         
                          
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         LN 
                          
                         
                             
                         
                          
                         δ 
                          
                         
                             
                         
                          
                         
                           f 
                           / 
                           
                             f 
                             s 
                           
                         
                       
                     
                      
                     
                       
                         ∑ 
                         
                           n 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                            
                           
                             z 
                             n 
                           
                            
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
         [0000]    A non-biased estimator for the normalized frequency offset ε=N δf/f s  is: 
         [0000]    
       
         
           
             
               
                 
                   
                     ɛ 
                     ^ 
                   
                   = 
                   
                     - 
                     
                       ∠ρ 
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         L 
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The same estimator can be derived in frequency domain as follows. The N-length FFT R k  of the received samples r n  is: 
         [0000]    
       
         
           
             
               
                 
                   
                     R 
                     k 
                   
                   = 
                   
                     
                       
                         1 
                         
                           N 
                         
                       
                        
                       
                         
                           ∑ 
                           
                             n 
                             = 
                             0 
                           
                           
                             N 
                             - 
                             1 
                           
                         
                          
                         
                           
                             r 
                             n 
                           
                            
                           
                              
                             
                               
                                 - 
                                 j 
                               
                                
                               
                                   
                               
                                
                               2 
                                
                               
                                   
                               
                                
                               π 
                                
                               
                                   
                               
                                
                               
                                 nk 
                                 / 
                                 N 
                               
                             
                           
                         
                       
                     
                     = 
                     
                       
                         
                           1 
                           
                             N 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               N 
                               - 
                               1 
                             
                           
                            
                           
                             
                               z 
                               n 
                             
                              
                             
                                
                               
                                 
                                   - 
                                   j 
                                 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 π 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     n 
                                      
                                     
                                       ( 
                                       
                                         k 
                                         - 
                                         ɛ 
                                       
                                       ) 
                                     
                                   
                                   / 
                                   N 
                                 
                               
                             
                           
                         
                       
                       + 
                       
                         w 
                         n 
                         ′ 
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where: w n ′ are AWGN samples with same statistical properties as w n . Similarly, the N-length FFT R k ′ of the received samples r n+LN  is: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           R 
                           k 
                           ′ 
                         
                         = 
                           
                          
                         
                           
                             1 
                             
                               N 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 N 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 r 
                                 
                                   n 
                                   + 
                                   LN 
                                 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     j 
                                   
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   
                                     nk 
                                     / 
                                     N 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                  
                                 
                                   j 
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   π 
                                    
                                   
                                       
                                   
                                    
                                   L 
                                    
                                   
                                       
                                   
                                    
                                   ɛ 
                                 
                               
                               
                                 N 
                               
                             
                              
                             
                               
                                 ∑ 
                                 
                                   n 
                                   = 
                                   0 
                                 
                                 
                                   N 
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   z 
                                   n 
                                 
                                  
                                 
                                    
                                   
                                     
                                       - 
                                       j 
                                     
                                      
                                     
                                         
                                     
                                      
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     π 
                                      
                                     
                                         
                                     
                                      
                                     
                                       
                                         n 
                                          
                                         
                                           ( 
                                           
                                             k 
                                             - 
                                             ɛ 
                                           
                                           ) 
                                         
                                       
                                       / 
                                       N 
                                     
                                   
                                 
                               
                             
                           
                           + 
                           
                             w 
                             
                               n 
                               + 
                               LN 
                             
                             ′ 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The correlation Γ in time of the received and FFTed repeated samples is: 
         [0000]    
       
         
           
             
               
                 
                   Γ 
                   = 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         0 
                       
                       
                         N 
                         - 
                         1 
                       
                     
                      
                     
                       
                         R 
                         k 
                       
                        
                       
                         R 
                         k 
                         
                           ′ 
                           * 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0000]    with a statistical average of: 
         [0000]    
       
         
           
             
               
                 
                   
                     〈 
                     Γ 
                     〉 
                   
                   = 
                   
                     
                        
                       
                         j2π 
                          
                         
                             
                         
                          
                         L 
                          
                         
                             
                         
                          
                         ɛ 
                       
                     
                      
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         
                           N 
                           - 
                           1 
                         
                       
                        
                       
                         
                            
                           
                             R 
                             k 
                           
                            
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0000]    so that a non-biased estimator for the normalized frequency offset ε is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     ɛ 
                     ^ 
                   
                   = 
                   
                     - 
                     
                       
                         ∠ 
                          
                         
                             
                         
                          
                         Γ 
                       
                       
                         2 
                          
                         π 
                          
                         
                             
                         
                          
                         L 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This derivation assumes the same (flat) channel over the two correlated symbols and any unpredictable phase variation in between would lead to a wrong estimate in equation (18). Therefore the time interval between the correlated symbols should be small enough to fulfill this requirement. 
         [0044]    This method is especially attractive for its low-complexity when applied to OFDM systems because it can make use of either the repeated reference symbols or the inherent duplication of the OFDM symbol tail in the cyclic prefix.  FIG. 10  illustrates the sub-frame structure defined in the LTE standard. The sub-frame includes two 0.5 ms slots (slot  1  and slot  2 ). Each slot includes six DFT-SOFDM (also denoted as SC-FDMA) data symbols and one central reference symbol. Both the cyclic prefix (CP) correlation of each symbol and the reference symbols correlation are used in a combined estimator. 
         [0045]    For each UE two de-mapped reference symbols of each sub-frame are correlated in the frequency domain. The LTE specification allows for both intra sub-frame frequency hopping and sequence hopping in either a base sequence, a cyclic shift or both. In intra sub-frame frequency hopping the sequence of the reference symbol changes from one slot to another. Therefore the symbol cannot be considered anymore as a repeated symbol. In sequence hopping the flat channel assumption of the estimator of equation (18) is violated and the method cannot be used. Therefore for most cases the reference symbol correlation method is impractical for LTE. 
         [0046]    Denote r(n) a received OFDM symbol including the CP illustrated in  FIG. 10  for samples n=0, 1, . . . , N FFT +N CP −1, where N FFT  is the FFT size and N CP  is the CP length. The frequency domain correlation method requires converting both the CP and the corresponding tail of the symbol in frequency domain to perform the correlation of equation 16. However this can be derived from the already available FFT of the complete symbol. This saves performing an additional FFT. 
         [0047]    Denote R CP (k) as the FFTs of the CP: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       R 
                       CP 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         0 
                       
                       
                         
                           N 
                           CP 
                         
                         - 
                         1 
                       
                     
                      
                     
                       
                         r 
                          
                         
                           ( 
                           n 
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                             j 
                           
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             
                               π 
                                
                               kn 
                             
                             / 
                             
                               N 
                               FFT 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Denote R Tail (k) as the tail of the OFDM symbol: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       R 
                       Tail 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       ∑ 
                       
                         n 
                         = 
                         0 
                       
                       
                         
                           N 
                           CP 
                         
                         - 
                         1 
                       
                     
                      
                     
                       
                         r 
                          
                         
                           ( 
                           
                             n 
                             + 
                             
                               N 
                               FFT 
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                             j 
                           
                            
                           
                               
                           
                            
                           2 
                            
                           
                               
                           
                            
                           
                             
                               π 
                                
                               kn 
                             
                             / 
                             
                               N 
                               FFT 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0000]    Denote R S (k) as the complete OFDM symbol: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             R 
                             S 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               0 
                             
                             
                               
                                 N 
                                 FFT 
                               
                               - 
                               1 
                             
                           
                            
                           
                             
                               r 
                                
                               
                                 ( 
                                 
                                   n 
                                   + 
                                   
                                     N 
                                     CP 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 
                                   - 
                                   j 
                                 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     π 
                                      
                                     kn 
                                   
                                   / 
                                   
                                     N 
                                     FFT 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               ∑ 
                               
                                 n 
                                 = 
                                 0 
                               
                               
                                 
                                   N 
                                   FFT 
                                 
                                 - 
                                 
                                   N 
                                   CP 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 r 
                                  
                                 
                                   ( 
                                   
                                     n 
                                     + 
                                     
                                       N 
                                       CP 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                  
                                 
                                   
                                     - 
                                     j 
                                   
                                    
                                   
                                       
                                   
                                    
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   
                                     
                                       π 
                                        
                                       kn 
                                     
                                     / 
                                     
                                       N 
                                       FFT 
                                     
                                   
                                 
                               
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                          
                         
                           
                             ∑ 
                             
                               n 
                               = 
                               
                                 
                                   N 
                                   FFT 
                                 
                                 - 
                                 
                                   N 
                                   CP 
                                 
                               
                             
                             
                               
                                 N 
                                 FFT 
                               
                               - 
                               1 
                             
                           
                            
                           
                             
                               r 
                                
                               
                                 ( 
                                 
                                   n 
                                   + 
                                   
                                     N 
                                     CP 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 
                                   - 
                                   j 
                                 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   
                                     π 
                                      
                                     kn 
                                   
                                   / 
                                   
                                     N 
                                     FFT 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The relevant term reflecting the OFDM tail in is the second term, which can be re-written as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             R 
                             
                               S 
                               - 
                               Tail 
                             
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             ∑ 
                             
                               m 
                               = 
                               0 
                             
                             
                               
                                 N 
                                 CP 
                               
                               - 
                               1 
                             
                           
                            
                           
                             
                               r 
                                
                               
                                 ( 
                                 
                                   m 
                                   + 
                                   
                                     N 
                                     FFT 
                                   
                                 
                                 ) 
                               
                             
                              
                             
                                
                               
                                 
                                   - 
                                   j 
                                 
                                  
                                 
                                     
                                 
                                  
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   π 
                                    
                                   
                                     ( 
                                     
                                       m 
                                       - 
                                       
                                         N 
                                         CP 
                                       
                                       + 
                                       
                                         N 
                                         FFT 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   k 
                                   / 
                                   
                                     N 
                                     FFT 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               R 
                               Tail 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                            
                           
                              
                             
                               j 
                                
                               
                                   
                               
                                
                               2 
                                
                               π 
                                
                               
                                   
                               
                                
                               k 
                                
                               
                                   
                               
                                
                               
                                 
                                   N 
                                   CP 
                                 
                                 / 
                                 
                                   N 
                                   FFT 
                                 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
         [0000]    To compensate for the phase difference between the desired R Tail (k) and the available R S , R CP (k) must be weighted by e j2πkN     CP     /N     FFT    before correlating with R S (k) and the correlation in equation (16) is computed as: 
         [0000]    
       
         
           
             
               
                 
                   Γ 
                   = 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         0 
                       
                       
                         
                           N 
                           SC 
                         
                         - 
                         1 
                       
                     
                      
                     
                       
                         
                           R 
                           S 
                         
                          
                         
                           ( 
                           
                             k 
                             + 
                             
                               k 
                               0 
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           R 
                           CP 
                           * 
                         
                          
                         
                           ( 
                           
                             k 
                             + 
                             
                               k 
                               0 
                             
                           
                           ) 
                         
                       
                        
                       
                          
                         
                           
                             - 
                             j 
                           
                            
                           
                               
                           
                            
                           2 
                            
                           π 
                            
                           
                               
                           
                            
                           k 
                            
                           
                               
                           
                            
                           
                             
                               N 
                               CP 
                             
                             / 
                             
                               N 
                               FFT 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   23 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where: N SC  is the number of allocated sub-carriers starting from sub-carrier k 0 . Simulations verify that this implementation optimization does not cause any performance degradation compared with using the FFT of only the tail of the OFDM symbol R Tail (k). 
         [0048]    This method was originally designed for OFDM systems where only one UE occupies the spectrum at a time. In that case both the CP and the OFDM symbol are free of interference. In OFDMA systems the CP is not interference-free because, unlike the OFDM symbol, its spectrum is not limited to the UEs allocated sub-carriers. The CP spectrums spills over adjacent sub-carriers where other UEs are expected to be scheduled. 
         [0049]    Simulation results show that this CP interference results in estimation errors proportional to the frequency offset. Thus the frequency estimates must be used to compensate the frequency offset before computing a new estimation. The only practical method for frequency compensation in OFDMA systems operates in frequency domain on the sub-carriers allocated to the UE. There are two options for the CP: remove the frequency offset on the de-mapped sub-carriers; or remove the frequency offset on the full bandwidth. Removing the frequency offset on the de-mapped sub-carriers is obviously the more attractive from a complexity standpoint. However this truncates the frequency-domain CP, thus resulting in erroneous frequency correction. Removing the frequency offset on the full bandwidth performs an exact frequency correction in absence of any interferer. However, this method suffers from interference as soon as other UEs are multiplexed in the same symbol. Frequency-domain frequency offset generation or compensation method provide exactitude when applied to the OFDM symbol because the Drifted and Interpolated symbol provide results similar to each other. 
         [0050]    Thus despite the low complexity of the CP-correlation method, a higher performance method is needed to estimate each UE&#39;s frequency offset in LTE. This invention uses the available frequency-domain interference-free symbols de-mapped (or de-multiplexed) at the output of the FFT of an OFDMA multi-user receiver. 
         [0051]    The criterion to design a frequency offset estimation algorithm for LTE is that it should circumvent the drawbacks of the state of the art methods. The new method should only operate on the de-mapped interference-free OFDM symbols. The new method should not rely on symbol repetition. 
         [0052]    This invention is a maximum-likelihood algorithm which tests a number of frequency offset hypothesis on the demodulation reference symbols (DMRS) located at the center of each slot as shown in  FIG. 10 . This is expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Δ 
                       ^ 
                     
                      
                     
                         
                     
                      
                     f 
                   
                   = 
                   
                     arg 
                      
                     
                         
                     
                      
                     
                       
                         max 
                         
                           δ 
                            
                           
                               
                           
                            
                           f 
                         
                       
                        
                       
                         { 
                         
                           M 
                            
                           
                             ( 
                             
                               δ 
                                
                               
                                   
                               
                                
                               f 
                             
                             ) 
                           
                         
                         } 
                       
                     
                   
                 
               
               
                 
                   ( 
                   24 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where: the metric M(δf) can be computed in a number of ways. A first M(δf) computation multiplies the received de-mapped FFT output samples R S (k) with the complex conjugate of frequency shifted replicas of the expected DMRS frequency-domain sequence S(k, δf i ): 
         [0000]    
       
         
           
             
               
                 
                   
                     M 
                      
                     
                       ( 
                       
                         δ 
                          
                         
                             
                         
                          
                         f 
                       
                       ) 
                     
                   
                   = 
                   
                      
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         
                           
                             N 
                             sc 
                           
                           - 
                           1 
                         
                       
                        
                       
                         
                           
                             R 
                             S 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                          
                         
                           
                             S 
                             * 
                           
                            
                           
                             ( 
                             
                               k 
                               , 
                               
                                 δ 
                                  
                                 
                                     
                                 
                                  
                                 f 
                               
                             
                             ) 
                           
                         
                       
                     
                      
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0000]    A second M(δf) computation multiplies the frequency shifted replicas of the received de-mapped FFT output sample R S(k , δf i ) with the complex conjugate of the expected DMRS sequence S(k): 
         [0000]    
       
         
           
             
               
                 
                   
                     M 
                      
                     
                       ( 
                       
                         δ 
                          
                         
                             
                         
                          
                         f 
                       
                       ) 
                     
                   
                   = 
                   
                      
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         
                           
                             N 
                             sc 
                           
                           - 
                           1 
                         
                       
                        
                       
                         
                           
                             R 
                             S 
                           
                            
                           
                             ( 
                             
                               k 
                               , 
                               
                                 δ 
                                  
                                 
                                     
                                 
                                  
                                 f 
                               
                             
                             ) 
                           
                         
                          
                         
                           
                             S 
                             * 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                     
                      
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
         [0000]      FIGS. 11 and 12  illustrate the principles of these two approaches. In  FIG. 11  the received sampled are divided into frequency bins in bin filter  1101 . The binned signals are converted to frequency domain via Fast Fourier Transform ( 1102 ). Filter  1103  separates the signals corresponding to individual active UEs in exemplary channel  1104 . Multiplier  1105  multiplies the channel signal by the processed DMRS sequence via block  1105 . Block  1107  performs the required summation and block  1108  finds the maximum.  FIG. 12  operates similarly except the DMRS signal S(k)  1201  is subject to the frequency bins in block  1202 . This is processed in block  1203  for multiplication by the corresponding separated channel signal in multiplier  1106 . 
         [0053]    Both approaches can be simplified using the frequency-domain frequency offset generation or compensation method to build the frequency bins. The two approaches are equivalent in performance-wise. This application describes the first approach. For DMRS sequence planning with hopping disabled, the frequency bins S*(k,δf) of a given UEs DMRS sequence can be pre-computed and stored in a LUT. Enabling sequence hopping might result in a reduced set of expected sequences per UE where the LUT approach is also beneficial. For the worst-case cell configuration where DMRS sequences change on every slot (known as base sequence group hopping) for a given slot S(k,δf) is the same for all UEs. 
         [0054]      FIG. 13  illustrates the continuous frequency signal of a DMRS sequence (Drifted Symbol) and specifically an extended Zadoff-Chu sequence (EZC).  FIG. 14  illustrates the corresponding bias of the FO estimator.  FIG. 13  shows the constant amplitude property of such sequence does not hold true between the sub-carrier centre positions located at integer multiples of 15 kHz. Thus  FIG. 14  does not illustrate a constant gain across the frequency range. Thus a frequency offset estimator is required. 
         [0055]    The metric gain is a function of the sequence including the EZC index and cyclic shift and the number of allocated sub-carriers G(S,NSC,δf): 
         [0000]    
       
         
           
             
               
                 
                   
                     G 
                      
                     
                       ( 
                       
                         S 
                         , 
                         
                           N 
                           SC 
                         
                         , 
                         
                           δ 
                            
                           
                               
                           
                            
                           f 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                       
                       
                         
                           N 
                           sc 
                         
                         - 
                         1 
                       
                     
                      
                     
                       
                         S 
                          
                         
                           ( 
                           
                             k 
                             , 
                             
                               δ 
                                
                               
                                   
                               
                                
                               f 
                             
                           
                           ) 
                         
                       
                        
                       
                         
                           S 
                           * 
                         
                          
                         
                           ( 
                           
                             k 
                             , 
                             
                               δ 
                                
                               
                                   
                               
                                
                               f 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
         [0000]    and can be compensated for in the estimator metric as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     M 
                      
                     
                       ( 
                       
                         δ 
                          
                         
                             
                         
                          
                         f 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         G 
                          
                         
                           ( 
                           
                             S 
                             , 
                             
                               N 
                               SC 
                             
                             , 
                             
                               δ 
                                
                               
                                   
                               
                                
                               f 
                             
                           
                           ) 
                         
                       
                     
                      
                     
                        
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             
                               N 
                               sc 
                             
                             - 
                             1 
                           
                         
                          
                         
                           
                             
                               R 
                               S 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                            
                           
                             
                               S 
                               * 
                             
                              
                             
                               ( 
                               
                                 k 
                                 , 
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   f 
                                 
                               
                               ) 
                             
                           
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
         [0000]    or as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     M 
                      
                     
                       ( 
                       
                         δ 
                          
                         
                             
                         
                          
                         f 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                          
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               
                                 N 
                                 sc 
                               
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                 R 
                                 S 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                              
                             
                               
                                 S 
                                 * 
                               
                                
                               
                                 ( 
                                 
                                   k 
                                   , 
                                   
                                     δ 
                                      
                                     
                                         
                                     
                                      
                                     f 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                          
                       
                       
                         
                           G 
                            
                           
                             ( 
                             
                               S 
                               , 
                               
                                 N 
                                 SC 
                               
                               , 
                               
                                 δ 
                                  
                                 
                                     
                                 
                                  
                                 f 
                               
                             
                             ) 
                           
                         
                       
                     
                     = 
                     
                       
                          
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               
                                 N 
                                 sc 
                               
                               - 
                               1 
                             
                           
                            
                           
                             
                               
                                 R 
                                 S 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                              
                             
                               
                                 S 
                                 * 
                               
                                
                               
                                 ( 
                                 
                                   k 
                                   , 
                                   
                                     δ 
                                      
                                     
                                         
                                     
                                      
                                     f 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                          
                       
                       
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               0 
                             
                             
                               
                                 N 
                                 sc 
                               
                               - 
                               1 
                             
                           
                            
                           
                              
                             
                               S 
                                
                               
                                 ( 
                                 
                                   k 
                                   , 
                                   
                                     δ 
                                      
                                     
                                         
                                     
                                      
                                     f 
                                   
                                 
                                 ) 
                               
                             
                              
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The resulting metric is no longer biased. This can be mathematically verified by checking that it is always maximum at the received frequency offset: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                        
                       
                         M 
                          
                         
                           ( 
                           
                             δ 
                              
                             
                                 
                             
                              
                             f 
                           
                           ) 
                         
                       
                     
                     
                       
                          
                         δ 
                       
                        
                       
                           
                       
                        
                       f 
                     
                   
                    
                   
                     
                        
                       
                         
                           δ 
                            
                           
                               
                           
                            
                           f 
                         
                         = 
                         
                           δ 
                            
                           
                               
                           
                            
                           
                             f 
                             u 
                           
                         
                       
                     
                      
                     
                       = 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where: δf u  is the frequency offset of the received sequence R S (k). 
         [0056]      FIG. 15  illustrates the resulting residual bias of the associated estimator, which is now only due to the quantization of the searched frequency space (frequency bins). In the case of no sequence hopping or if sequence hopping leads to a reduced set of expected sequences per UE, the gain compensation factor √{square root over (G)} in equation (27) can be incorporated in the frequency bins S*(k,δf) stored in LUT. 
         [0057]    The maximum likelihood metric assumes a flat channel across the allocated sub-carriers of a given UE. This is not the case under fading conditions especially for large allocation sizes. Equalizing the de-mapped sub-carriers R S (k) with the frequency-domain channel estimates Ĥ(k) before frequency estimation circumvents this issue as follows: 
         [0000]        R   S−eq ( k )= R   S ( k ) Ĥ ( k )  (31) 
         [0000]    Using the simplest channel estimation derived from weighting the received de-mapped samples with the complex conjugate of the expected frequency-domain DMRS sequence is: 
         [0000]        Ĥ ( k )= R   S ( k ) S *( k )  (32) 
         [0000]    As a result, the metric is now upgraded to apply to fading channels as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     M 
                      
                     
                       ( 
                       
                         δ 
                          
                         
                             
                         
                          
                         f 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         G 
                          
                         
                           ( 
                           
                             S 
                             , 
                             
                               N 
                               SC 
                             
                             , 
                             
                               δ 
                                
                               
                                   
                               
                                
                               f 
                             
                           
                           ) 
                         
                       
                     
                      
                     
                        
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             
                               N 
                               sc 
                             
                             - 
                             1 
                           
                         
                          
                         
                           
                             
                                
                               
                                 
                                   R 
                                   S 
                                 
                                  
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                                
                             
                             2 
                           
                            
                           
                             S 
                              
                             
                               ( 
                               
                                 k 
                                 , 
                                 0 
                               
                               ) 
                             
                           
                            
                           
                             
                               S 
                               * 
                             
                              
                             
                               ( 
                               
                                 k 
                                 , 
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   f 
                                 
                               
                               ) 
                             
                           
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   33 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The metric of equation (33) only operates on DMRS symbols and is accumulated over antennas, slots and sub-frames as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         M 
                          
                         
                           ( 
                           
                             
                               S 
                               f 
                             
                             , 
                             
                               δ 
                                
                               
                                   
                               
                                
                               f 
                             
                           
                           ) 
                         
                       
                       = 
                       
                         
                           ∑ 
                           
                             
                               s 
                               f 
                             
                             = 
                             1 
                           
                           
                             S 
                             f 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               a 
                               = 
                               1 
                             
                             
                               N 
                               a 
                             
                           
                            
                           
                             
                               ∑ 
                               
                                 
                                   s 
                                   l 
                                 
                                 = 
                                 1 
                               
                               2 
                             
                              
                             
                               
                                 m 
                                  
                                 
                                   ( 
                                   
                                     a 
                                     , 
                                     
                                       s 
                                       l 
                                     
                                     , 
                                     
                                       s 
                                       f 
                                     
                                     , 
                                     
                                       N 
                                       SC 
                                     
                                     , 
                                     
                                       δ 
                                        
                                       
                                           
                                       
                                        
                                       f 
                                     
                                   
                                   ) 
                                 
                               
                                
                               
                                   
                               
                                
                               with 
                             
                           
                         
                       
                     
                      
                     
                       
 
                     
                      
                     
                       
                         m 
                          
                         
                           ( 
                           
                             a 
                             , 
                             
                               s 
                               l 
                             
                             , 
                             
                               s 
                               f 
                             
                             , 
                             
                               N 
                               SC 
                             
                             , 
                             
                               δ 
                                
                               
                                   
                               
                                
                               f 
                             
                           
                           ) 
                         
                       
                       = 
                       
                         
                            
                           
                             
                               
                                 
                                   
                                     ∑ 
                                     
                                       k 
                                       = 
                                       0 
                                     
                                     
                                       
                                         N 
                                         sc 
                                       
                                       - 
                                       1 
                                     
                                   
                                    
                                   
                                     
                                        
                                       
                                         
                                           R 
                                           DMRS 
                                         
                                          
                                         
                                           ( 
                                           
                                             
                                               k 
                                               + 
                                               
                                                 k 
                                                 0 
                                               
                                             
                                             , 
                                             a 
                                             , 
                                             
                                               s 
                                               l 
                                             
                                             , 
                                             
                                               s 
                                               f 
                                             
                                           
                                           ) 
                                         
                                       
                                        
                                     
                                     2 
                                   
                                 
                               
                             
                             
                               
                                 
                                   S 
                                    
                                   
                                     ( 
                                     
                                       k 
                                       , 
                                       
                                         s 
                                         l 
                                       
                                       , 
                                       
                                         s 
                                         f 
                                       
                                       , 
                                       0 
                                     
                                     ) 
                                   
                                    
                                   
                                     
                                       S 
                                       * 
                                     
                                      
                                     
                                       ( 
                                       
                                         k 
                                         , 
                                         
                                           s 
                                           l 
                                         
                                         , 
                                         
                                           s 
                                           f 
                                         
                                         , 
                                         
                                           δ 
                                            
                                           
                                               
                                           
                                            
                                           f 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                            
                         
                          
                         
                           
                             G 
                              
                             
                               ( 
                               
                                 
                                   s 
                                   l 
                                 
                                 , 
                                 
                                   s 
                                   f 
                                 
                                 , 
                                 
                                   N 
                                   SC 
                                 
                                 , 
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   f 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   34 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where: δf is the frequency offset hypothesis; S f  is the estimation interval duration in sub-frames; k is the sub-carrier index; s l  is the slot index; a is the antenna index; s f  is the sub-frame index; S( . . . , δf) is the frequency shifted replica of the DMRS sequence expected in slot s l  of sub-frame s f , and computed according to equations (1) and (2); R DMRS ( . . . ) is the FFT samples of the DMRS symbol of slot s l  in sub-frame s f  from antenna a; G is the gain compensation factor of equation (27); and N SC  is the number of allocated sub-carriers starting from sub-carrier k 0 . 
         [0058]    The resulting frequency estimator for this estimation interval is: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Δ 
                       ^ 
                     
                      
                     
                         
                     
                      
                     
                       f 
                        
                       
                         ( 
                         
                           S 
                           f 
                         
                         ) 
                       
                     
                   
                   = 
                   
                     
                       
                         arg 
                          
                         
                             
                         
                          
                         max 
                       
                       
                         δ 
                          
                         
                             
                         
                          
                         f 
                       
                     
                      
                     
                       { 
                       
                         M 
                          
                         
                           ( 
                           
                             
                               S 
                               f 
                             
                             , 
                             
                               δ 
                                
                               
                                   
                               
                                
                               f 
                             
                           
                           ) 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       H 
                       ^ 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         R 
                         S 
                       
                        
                       
                         ( 
                         k 
                         ) 
                       
                     
                      
                     
                       
                         S 
                         * 
                       
                        
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
         [0000]    As a result, the metric is now upgraded to apply to fading channels as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           M 
                            
                           
                             ( 
                             
                               δ 
                                
                               
                                   
                               
                                
                               f 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                              
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                                 
                                   
                                     N 
                                     sc 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   
                                     R 
                                     S 
                                   
                                    
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           
                                             R 
                                             S 
                                           
                                            
                                           
                                             ( 
                                             k 
                                             ) 
                                           
                                         
                                          
                                         
                                           
                                             S 
                                             * 
                                           
                                            
                                           
                                             ( 
                                             
                                               k 
                                               , 
                                               0 
                                             
                                             ) 
                                           
                                         
                                       
                                       ] 
                                     
                                     * 
                                   
                                   
                                      
                                     
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   channel 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   estimates 
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   for 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   fading 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       
                                         
                                           channel 
                                         
                                       
                                     
                                   
                                 
                                  
                                 
                                   
                                     S 
                                     * 
                                   
                                    
                                   
                                     ( 
                                     
                                       k 
                                       , 
                                       
                                         δ 
                                          
                                         
                                             
                                         
                                          
                                         f 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                              
                           
                           
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                                 
                                   
                                     N 
                                     sc 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   S 
                                    
                                   
                                     ( 
                                     
                                       k 
                                       , 
                                       
                                         δ 
                                          
                                         
                                             
                                         
                                          
                                         f 
                                       
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     S 
                                     * 
                                   
                                    
                                   
                                     ( 
                                     
                                       k 
                                       , 
                                       
                                         δ 
                                          
                                         
                                             
                                         
                                          
                                         f 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                              
                             
                               
                                 
                                   ∑ 
                                   
                                     k 
                                     = 
                                     0 
                                   
                                 
                                 
                                   
                                     N 
                                     sc 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   
                                      
                                     
                                       
                                         R 
                                         S 
                                       
                                        
                                       
                                         ( 
                                         k 
                                         ) 
                                       
                                     
                                      
                                   
                                   2 
                                 
                                  
                                 
                                   S 
                                    
                                   
                                     ( 
                                     
                                       k 
                                       , 
                                       0 
                                     
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     S 
                                     * 
                                   
                                    
                                   
                                     ( 
                                     
                                       k 
                                       , 
                                       
                                         δ 
                                          
                                         
                                             
                                         
                                          
                                         f 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                              
                           
                           
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                                 
                                   
                                     N 
                                     sc 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                    
                                   
                                     S 
                                      
                                     
                                       ( 
                                       
                                         k 
                                         , 
                                         
                                           δ 
                                            
                                           
                                               
                                           
                                            
                                           f 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   37 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This is referred to as the frequency-bins with common channel estimation method. The method of equation (29) is referred to as the frequency-bins without channel estimation method. 
         [0059]    The estimator of equation (37) is biased even without fading because the introduced channel estimate term R S (k)S*(k,0) has a non-constant gain across the frequency offset range of the received sequence R S (k). This is fixed by estimating the channel specifically for each frequency hypothesis δf and normalizing the resulting metric as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           M 
                            
                           
                             ( 
                             
                               δ 
                                
                               
                                   
                               
                                
                               f 
                             
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                              
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                                 
                                   
                                     N 
                                     sc 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   
                                     R 
                                     S 
                                   
                                    
                                   
                                     ( 
                                     k 
                                     ) 
                                   
                                 
                                  
                                 
                                   
                                     
                                       [ 
                                       
                                         
                                           
                                             R 
                                             S 
                                           
                                            
                                           
                                             ( 
                                             k 
                                             ) 
                                           
                                         
                                          
                                         
                                           
                                             S 
                                             * 
                                           
                                            
                                           
                                             ( 
                                             
                                               k 
                                               , 
                                               
                                                 δ 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 f 
                                               
                                             
                                             ) 
                                           
                                         
                                       
                                       ] 
                                     
                                     * 
                                   
                                   
                                      
                                     
                                       
                                         
                                           
                                             
                                               
                                                 
                                                   frequency 
                                                    
                                                   
                                                     - 
                                                   
                                                    
                                                   specific 
                                                 
                                               
                                             
                                             
                                               
                                                 
                                                   channel 
                                                    
                                                   
                                                       
                                                   
                                                    
                                                   estimates 
                                                 
                                               
                                             
                                           
                                         
                                       
                                       
                                         
                                           
                                             for 
                                              
                                             
                                                 
                                             
                                              
                                             fading 
                                              
                                             
                                                 
                                             
                                              
                                             channel 
                                           
                                         
                                       
                                     
                                   
                                 
                                  
                                 
                                   
                                     S 
                                     * 
                                   
                                    
                                   
                                     ( 
                                     
                                       k 
                                       , 
                                       
                                         δ 
                                          
                                         
                                             
                                         
                                          
                                         f 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                              
                           
                           
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                                 
                                   
                                     N 
                                     sc 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                   [ 
                                   
                                     
                                       S 
                                        
                                       
                                         ( 
                                         
                                           k 
                                           , 
                                           
                                             δ 
                                              
                                             
                                                 
                                             
                                              
                                             f 
                                           
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       
                                         S 
                                         * 
                                       
                                        
                                       
                                         ( 
                                         
                                           k 
                                           , 
                                           
                                             δ 
                                              
                                             
                                                 
                                             
                                              
                                             f 
                                           
                                         
                                         ) 
                                       
                                     
                                   
                                   ] 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                               
                               
                                 
                                   N 
                                   sc 
                                 
                                 - 
                                 1 
                               
                             
                              
                             
                               
                                 
                                    
                                   
                                     
                                       R 
                                       S 
                                     
                                      
                                     
                                       ( 
                                       k 
                                       ) 
                                     
                                   
                                    
                                 
                                 2 
                               
                                
                               
                                 
                                    
                                   
                                     S 
                                      
                                     
                                       ( 
                                       
                                         k 
                                         , 
                                         
                                           δ 
                                            
                                           
                                               
                                           
                                            
                                           f 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                 
                                 2 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   0 
                                 
                                 
                                   
                                     N 
                                     sc 
                                   
                                   - 
                                   1 
                                 
                               
                                
                               
                                 
                                    
                                   
                                     S 
                                      
                                     
                                       ( 
                                       
                                         k 
                                         , 
                                         
                                           δ 
                                            
                                           
                                               
                                           
                                            
                                           f 
                                         
                                       
                                       ) 
                                     
                                   
                                    
                                 
                                 4 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   38 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This is referred to as the frequency-bins with per-bin channel estimation method. 
         [0060]    One concern with this frequency-bin approach is the complexity associated with the number of bins needed to cover a sufficient frequency range with an adequate granularity. This can be solved by restricting the number of bins to only three in the searched frequency window: [−f max , 0, f max ] and using a parabolic interpolation to locate the position of the metric maximum.  FIGS. 16 and 17  illustrate that both the frequency-bins with common channel estimation method for equation (37) or the frequency-bins with per-bin channel estimation method of equation (38) can be mapped onto a parabola. 
         [0061]    Three Cartesian points (x i , y i ) for i=1, 2, 3 where y i =ax i   2 +bx i +c, the abscissa x max  of the maximum y max  of the parabola is given by: 
         [0000]    
       
         
           
             
               
                 
                   
                     x 
                     max 
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       [ 
                       
                         
                           x 
                           1 
                         
                         + 
                         
                           x 
                           2 
                         
                         - 
                         
                           
                             
                               ( 
                               
                                 
                                   x 
                                   1 
                                 
                                 - 
                                 
                                   x 
                                   3 
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 
                                   x 
                                   3 
                                 
                                 - 
                                 
                                   x 
                                   2 
                                 
                               
                               ) 
                             
                              
                             
                               ( 
                               
                                 
                                   y 
                                   1 
                                 
                                 - 
                                 
                                   y 
                                   2 
                                 
                               
                               ) 
                             
                           
                           
                             
                               
                                 ( 
                                 
                                   
                                     y 
                                     1 
                                   
                                   - 
                                   
                                     y 
                                     3 
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     1 
                                   
                                   - 
                                   
                                     x 
                                     2 
                                   
                                 
                                 ) 
                               
                             
                             - 
                             
                               
                                 ( 
                                 
                                   
                                     y 
                                     1 
                                   
                                   - 
                                   
                                     y 
                                     2 
                                   
                                 
                                 ) 
                               
                                
                               
                                 ( 
                                 
                                   
                                     x 
                                     1 
                                   
                                   - 
                                   
                                     x 
                                     3 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This reduces significantly the complexity of the frequency-bin approach and makes it a viable alternative to correlation-based approaches. 
         [0062]    The frequency range covered by the frequency bins is constant during an estimation interval to allow for consistent accumulation of the metrics of equations (37) and (38). Dynamic control across estimation intervals narrows down, when possible, the scope of the searched frequency offset. Empirical simulations identified the following frequency window adaptation scheme: 
         [0000]    
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               f 
                               searched 
                             
                             ∈ 
                             
                               [ 
                               
                                 
                                   
                                     
                                       - 
                                       
                                         f 
                                         max 
                                       
                                     
                                   
                                   
                                     
                                       f 
                                       max 
                                     
                                   
                                 
                               
                               ] 
                             
                           
                            
                           
                               
                           
                         
                       
                     
                     
                       
                         
                           
                             with 
                              
                             
                                 
                             
                              
                             
                               f 
                               max 
                             
                           
                           = 
                           
                             min 
                              
                             
                               { 
                               
                                 
                                   200 
                                    
                                   
                                       
                                   
                                    
                                   Hz 
                                 
                                 ; 
                                 
                                   max 
                                    
                                   
                                     { 
                                     
                                       
                                         2000 
                                          
                                         
                                             
                                         
                                          
                                         Hz 
                                       
                                       ; 
                                       
                                         3 
                                          
                                         
                                           
                                             δ 
                                              
                                             
                                                 
                                             
                                              
                                             f 
                                           
                                           ^ 
                                         
                                       
                                     
                                     } 
                                   
                                 
                               
                               } 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where: δf is the residual frequency estimate of the previous estimation interval. 
         [0063]    The metrics of equations (37) and (38) involve the power of the received frequency domain samples. This results in a sub-optimal estimator since squaring increases the estimator variance. In another embodiment of this invention a novel metric expression for frequency offset estimation mitigates the estimator variance increase using absolute values instead of powers. A normalization function maintains a normalized gain providing a non-biased estimator as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     M 
                      
                     
                       ( 
                       
                         δ 
                          
                         
                             
                         
                          
                         f 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           0 
                         
                         
                           
                             N 
                             sc 
                           
                           - 
                           1 
                         
                       
                        
                       
                         
                            
                           
                             
                               R 
                               S 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                            
                         
                          
                         
                            
                           
                             S 
                              
                             
                               ( 
                               
                                 k 
                                 , 
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   f 
                                 
                               
                               ) 
                             
                           
                            
                         
                       
                     
                     
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             0 
                           
                           
                             
                               N 
                               sc 
                             
                             - 
                             1 
                           
                         
                          
                         
                           
                              
                             
                               S 
                                
                               
                                 ( 
                                 
                                   k 
                                   , 
                                   
                                     δ 
                                      
                                     
                                         
                                     
                                      
                                     f 
                                   
                                 
                                 ) 
                               
                             
                              
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   41 
                   ) 
                 
               
             
           
         
       
     
         [0000]    This method is referred to as frequency-bins with min-variance per-bin channel estimation. This approach also reduces the complexity as the absolute value |z| of z=x+j*y is accurately approximated as follows. Let a=max(|x|, |y|) and b=min(|x|, |y|). If b&gt;a/4, |z|=0.875*a+0.5*b. Otherwise, |z|=a. Thus computation of equation (41) requires 4 additions/subtractions in 2&#39;s complement arithmetic. 
         [0064]      FIG. 18  is a flow chart  1800  illustrating the complete frequency offset estimation and compensation algorithm. Flow chare  1800  begins with initialization of the data for all UEs and n slot  in step  1801 . Step  1801  computes and stores the value of γ(k, δf, n slot , N SC ) for all frequency bins. Step  1803  sets n symb  to zero. Step  1804  sets n ant  to zero. Step  1805  performs a Fast Fourier Transform of the next symbol. Step  1806  sets n symb  to zero. Step  1807  demaps the symbol of UE n UE . Step  1808  performs a frequency offset (FO) removal of the symbol producing R S (n UE ). Step  1809  tests to determine if the current symbol is a DMRS symbol. If this is true (Yes at step  1809 ), then step  1810  calculates Γ(n UE ,δf i ) for all frequency bins δf i . Flow then advances to step  1811 . If this is not true (No at step  1809 ), then flow advances to step  1811 . Step  1811  increments n UE . Step  1812  tests to determine if n UE  equals the number N b  of UEs. If not true (No at step  1812 ), flow loops back to step  1807 . If true (Yes at step  1812 ), the flow advances to step  1813 . Step  1813  increments n ant . Step  1814  tests to determine if n ant  equals the number N b  of antennas. If not true (No at step  1814 ), flow loops back to step  1805 . If true (Yes at step  1814 ), the flow advances to step  1815 . Step  1815  increments n symb . Step  1816  tests to determine if n symb  equals seven. If not true (No at step  1816 ), flow loops back to step  1804 . If true (Yes at step  1816 ), the flow advances to step  1817 . Step  1817  increments n slot . Step  1818  tests to determine if n slot  equals twice N subframes . If not true (No at step  1818 ), flow loops back to step  1802 . If true (Yes at step  1819 ), the flow advances to step  1819 . Step  1819  completes the calculation of the frequency offset for all UEs. 
         [0065]      FIG. 18  illustrates the scheduling of the functions involved with the frequency-bin based estimators. The difference between the various estimators is the metric used and the maximum argument algorithm. This maximum argument algorithm could be a basic maximum or a parabolic interpolation. More specifically, the set of coefficients γ(k,δf,n slot ,N sc ) is given by: 
         [0000]    
       
         
           
             
               
                 
                   { 
                   
                       
                     
                       
                         
                           
                             
                               
                                 γ 
                                  
                                 
                                   ( 
                                   
                                     k 
                                     , 
                                     
                                       δ 
                                        
                                       
                                           
                                       
                                        
                                       f 
                                     
                                     , 
                                     
                                       n 
                                       slot 
                                     
                                     , 
                                     
                                       N 
                                       sc 
                                     
                                   
                                   ) 
                                 
                               
                               = 
                               
                                 
                                   
                                     S 
                                      
                                     
                                       ( 
                                       
                                         k 
                                         , 
                                         0 
                                       
                                       ) 
                                     
                                   
                                    
                                   
                                     
                                       S 
                                       * 
                                     
                                      
                                     
                                       ( 
                                       
                                         k 
                                         , 
                                         
                                           δ 
                                            
                                           
                                               
                                           
                                            
                                           f 
                                         
                                       
                                       ) 
                                     
                                   
                                 
                                 
                                   
                                     
                                       ∑ 
                                       
                                         k 
                                         = 
                                         0 
                                       
                                       
                                         
                                           N 
                                           sc 
                                         
                                         - 
                                         1 
                                       
                                     
                                      
                                     
                                       
                                          
                                         
                                           S 
                                            
                                           
                                             ( 
                                             
                                               k 
                                               , 
                                               
                                                 δ 
                                                  
                                                 
                                                     
                                                 
                                                  
                                                 f 
                                               
                                             
                                             ) 
                                           
                                         
                                          
                                       
                                       2 
                                     
                                   
                                 
                               
                             
                             ; 
                             
                               common 
                                
                               
                                   
                               
                                
                               channel 
                                
                               
                                   
                               
                                
                               estimate 
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 γ 
                                  
                                 
                                   ( 
                                   
                                     k 
                                     , 
                                     
                                       δ 
                                        
                                       
                                           
                                       
                                        
                                       f 
                                     
                                     , 
                                     
                                       n 
                                       slot 
                                     
                                     , 
                                     
                                       N 
                                       sc 
                                     
                                   
                                   ) 
                                 
                               
                               = 
                               
                                 
                                   
                                      
                                     
                                       S 
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                             ; 
                             
                               per 
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                               channel 
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                               estimate 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
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                   42 
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         [0066]    The duration of each estimation interval is controlled dynamically via a stopping criterion. This stopping criterion is the standard deviation of the elementary estimates {circumflex over (Δ)}f(S f −L+1), {circumflex over (Δ)}f(S f −L+2), . . . , {circumflex over (Δ)}f(S f ) in an L-length sliding window ending on the last estimate {circumflex over (Δ)}f(S f ) of the current interval.  FIG. 19  illustrates the time behavior of the frequency elementary estimates within an estimation interval obtained from simulation. 
         [0067]    The value of the standard deviation threshold σ thresh  used as stopping criterion is optimized using simulations. A link-level simulator with ideal frequency offset compensation with only one UE, no timing errors and the a varying number of allocated RBs. The average durations in number of sub-frames of the estimation interval for various values of the stopping criterion σ thresh  Beyond σ thresh =50 Hz the estimation interval duration does not improve further and remains at about 30 sub-frames. 
         [0068]    The BLER performance of the PUSCH with 2 RB allocations for various values of the stopping criterion σ thresh  using QPSK and 16QAM modulation varies. Using the largest value of σ thresh  of 50 Hz does not degrade the BLER compared to smaller values. As a result σ thresh =50 Hz minimizes the estimation interval duration to 30 sub-frames while achieving high performance BLER for both QPSK and 16QAM modulations. 
         [0069]    The frequency range covered by the frequency bins is constant during an estimation interval to allow for consistent accumulation of the metric of equation (34). The frequency range is dynamically controlled across estimation intervals in order to narrow down when possible the scope of the searched frequency offset. For simplicity the frequency range of the frequency bins of this invention is preferably limited to four uncertainty windows: +/−2000 Hz; +/−1000 Hz; +/−500 Hz; and +/−200 Hz. After each estimation interval, the algorithm selects the smaller uncertainty window not exceeding three times the last estimated frequency offset. The preferred embodiment of this invention uses 100 frequency bins regardless the uncertainty window. Note that too fine a frequency granularity is overkill and this number may be reduced to provide about 20 Hz frequency granularity. 
         [0070]    The performance of both the CP-correlation and frequency-bins algorithms in a realistic multi-user SC-FDMA multiplex with ideal and real frequency offset compensation was evaluated via simulation. For the CP-correlation algorithm the estimated frequency offset was subtracted from the modeled frequency offset of the UE under test. For the frequency-bins algorithm the estimated frequency offset was compensated in the receiver. In both cases the simulator models a number of UEs with equal allocation size sharing the total bandwidth. The frequency mapping of the UEs was re-selected randomly every sub-frame and intra sub-frame frequency hopping was enabled. The DMRS sequence of the UEs was selected randomly for each new estimation interval but was kept the same within the estimation interval by allowing no sequence hopping. The simulator modeled timing errors of the UEs were chosen randomly within a maximum time uncertainty window. The other UEs other than the UE under test were given random power and frequency offsets within maximum uncertainty windows. The CP-correlation method was modeled using the same algorithm except that equation (34) was used as the accumulated metric and all symbols of the sub-frame were used in the accumulation. Table 3 lists all parameters of the simulation. 
         [0000]    
       
         
               
               
               
             
           
               
                   
                 TABLE 3 
               
               
                   
                   
               
               
                   
                 Parameter 
                 Value or range 
               
               
                   
                   
               
             
             
               
                   
                 System Bandwidth 
                 5 MHz 
               
               
                   
                 Number of users 
                 6 
               
               
                   
                 Number of scheduled RBs 
                 4 
               
               
                   
                 MCS 
                 QPSK ⅓-16QAM ½ 
               
               
                   
                 RS sequences 
                 EZC with random selection of ZC 
               
               
                   
                   
                 index and cyclic shift across freq 
               
               
                   
                   
                 estimation intervals 
               
               
                   
                 Scheduling scheme 
                 Frequency hopping, no sequence 
               
               
                   
                   
                 hopping within frequency 
               
               
                   
                   
                 estimation interval 
               
               
                   
                 Frequency estimation 
                 CP correlation on all symbols and 
               
               
                   
                 methods 
                 frequency bins on DMRS symbols 
               
               
                   
                   
                 only 
               
               
                   
                 Number of frequency bins 
                 100 
               
               
                   
                 Frequency search window 
                 Initial: +/−2 kHz, then 
               
               
                   
                 size 
                 dynamically adapted as a function 
               
               
                   
                   
                 of the estimated frequency 
               
               
                   
                 σ measurement interval 
                 5 sub-frames 
               
               
                   
                 size 
               
               
                   
                 σ thresh   
                 50 Hz 
               
               
                   
                 Frequency compensation 
                 Ideal, based on FO estimates, or 
               
               
                   
                   
                 real method at the receiver. In 
               
               
                   
                   
                 the latter case, CP FO 
               
               
                   
                   
                 compensation is implemented on 
               
               
                   
                   
                 either de-mapped CP or full 
               
               
                   
                   
                 bandwidth CP. 
               
               
                   
                 Max timing uncertainty 
                 +/−0.5 μs 
               
               
                   
                 window 
               
               
                   
                 Max frequency 
                 +/−500 Hz 
               
               
                   
                 uncertainty window 
               
               
                   
                 (other UEs) 
               
               
                   
                 Max power offset 
                 +/−3 dB 
               
               
                   
                 uncertainty (other UEs) 
               
               
                   
                 Channels 
                 AWGN, TU6-3 km/h, 
               
               
                   
                   
                 High Speed Train #3 
               
               
                   
                   
               
             
          
         
       
     
         [0071]    The following evaluation uses as performance criterions the statistics of the residual frequency offset error after estimation and compensation and the resulting Block Error Rate (BLER). The following evaluation provides the statistics of the mean, standard deviation and the sum of both so as to reflect the overall performance of the estimator/compensation scheme. 
         [0072]    This set of simulations allows assessing the impact of non-orthogonal CP in the CP correlation method. The frequency offsets are initialized to 800 Hz for AWGN channels and 300 Hz for TU channels. Then, every new frequency offset estimate is subtracted from the UEs current frequency offset. For both AWGN and TU channels the frequency bin method is insensitive to synchronization and power errors. This is not the case of the CP correlation method which outperforms the frequency bin method in ideal synchronization conditions but performs worse as soon as a realistic multiplex is modeled. In this case the frequency bin method outperforms the CP correlation method despite working on one instead of seven symbols per slot. 
         [0073]    This set of simulations allows assessing the impact of both the non-orthogonal CP method and the practical (non-ideal) CP frequency offset compensation method. The 3GPP RAN WG#4 specified High Speed Train (HST) scenarios with time-varying frequency offsets specifically addressing the high performance frequency tracking requirements of LTE receivers. These simulations use the third scenario out of the three scenarios. This provides the sharpest frequency variations. 
         [0074]    For the CP correlation method, simulations tested both wideband CP and de-mapped CP FO compensation methods. The CP correlation method produces erroneous average estimates at both ends of the Doppler shift and that compensating the FO on the wideband CP slightly outperforms the compensation on the de-mapped CP. Of the three tested methods, only the frequency bin method allows accurate tracking of the Doppler shift trajectory. All algorithms but the frequency bins have error floors at high SNR, where the frequency offset has larger impact on the BLER. 
         [0075]    The simulations assessed the performance of the various metrics elaborated for the frequency-bin approach including: frequency-bins without channel estimation; frequency-bins with common channel estimation; and frequency-bins with per-bin channel estimation. 
         [0076]    As expected, the metric of equation (29) designed to optimize the AWGN channel performance outperforms the other metrics as long as an ideal SC-FDMA multiplex is achieved. In a realistic multiplex where other UEs also experience some timing errors within the range tolerated by LTE, frequency errors and power differences, the performance of this metric becomes worse than the other metrics. Thus the frequency-bins without channel estimation method is not suitable in practice for real systems. The two remaining metrics including the common channel estimation and the per-bin channel estimation have very similar performance on the static AWGN channel and are insensitive to the interference resulting from non-ideal multiplexing in all channels. The metrics for the per-bin channel estimation shows better frequency offset tracking than the metric with common channel estimation. This is due to the FO-dependent bias upon AWGN of the former metric which is fixed in the latter metric thus accelerating the convergence when tracking FO variations. On the other hand, the metric with common channel estimation is more robust in fading channels. As a result, the preferred embodiment of this invention uses the metric involving per-bin channel estimation on AWGN channels and the metric involving common channel estimation on fading channels. 
         [0077]    The frequency-bins with per-bin channel estimation metric of equation (38) is used for the AWGN and HST#3 channels and the frequency-bins with common channel estimation metric of equation (37) is used for the TU 3 km/h channel. The parabolic interpolation method improves the performance with respect to the brute-force equal-spaced frequency bin approach. This is due to the resulting non-quantized frequency offset estimate. The parabolic interpolation scheme is quite robust with respect to a non-interference-free realistic multiplex. 
         [0078]    This patent application describes in details the design choices for the LTE frequency offset estimation and compensation, from theoretical derivations and performance evaluations. In particular, it is shown that the state-of-the-art does not satisfactorily addresses the high-end performance requirements of LTE so that an alternate approach is needed. A maximum-likelihood based solution is provided, taking profit of the available frequency-domain interference-free symbols de-mapped (or de-multiplexed) at the output of the FFT of an OFDMA multi-user receiver.