Abstract:
A method implemented by a user equipment includes selecting a first estimate of a signal-to-noise (SNR) ratio, calculating a first amplitude and first noise variance, calculating a second amplitude and a second noise variance, calculating a second SNR, calculating a resolution value, adjusting the first SNR, and performing estimation iterations until the resolution value is equal to a predetermined value.

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application is a continuation of U.S. patent application Ser. No. 12/060,541, filed Apr. 1, 2008, which is a continuation of U.S. patent application Ser. No. 10/867,508, filed Jun. 14, 2004 and issued on May 20, 2008 as U.S. Pat. No. 7,376,178, which is a continuation of U.S. patent application Ser. No. 10/269,606, filed Oct. 11, 2002 and issued on Jul. 6, 2004 as U.S. Pat. No. 6,760,370, which claims the benefit of U.S. Provisional Patent Application Ser. No. 60/369,655 filed Apr. 3, 2002, which are incorporated by reference as if fully set forth. 
     
    
     FIELD OF THE INVENTION 
       [0002]    The invention relates to processing of communications signals. More particularly, the invention relates to the estimation of the communication signal power in terms of signal-to-noise ratio. 
       BACKGROUND OF THE INVENTION 
       [0003]    In the field of communications, various types of systems use algorithms that depend on a signal-to-noise ratio (SNR) estimate for proper operation. Code division multiple access (CDMA) systems, such as time division duplex CDMA (TDD/CDMA) and time division synchronous CDMA (TDSCDMA) and frequency division duplex CDMA (FDD/CDMA) and CDMA 2000, use SNR estimation for power control to maintain the required link quality while using the minimum transmitted power. An asymmetric digital subscriber loop (ADSL) system uses SNR for the bit allocation algorithm to select the maximum transmission data rate. In turbo decoders, both the determined signal power and noise power are required. Rate adaptive transmission systems often use SNR to dynamically adapt the modulation scheme or the coding rate. 
         [0004]    Several algorithms are known for performing SNR estimation. One such algorithm, the received data-aided (RXDA) estimation, is based on the following equation: 
         [0000]    
       
         
           
             
               
                 
                   SNR 
                   = 
                   
                     
                       
                         ( 
                         
                           
                             1 
                             N 
                           
                            
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                                
                               
                                 r 
                                 k 
                               
                                
                             
                           
                         
                         ) 
                       
                       2 
                     
                     
                       
                         
                           1 
                           N 
                         
                          
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             r 
                             k 
                             2 
                           
                         
                       
                       - 
                       
                         
                           ( 
                           
                             
                               1 
                               N 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                  
                                 
                                   r 
                                   k 
                                 
                                  
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   1 
                 
               
             
           
         
       
     
         [0000]    where r is the received signal vector and N is the number of sample points read by the receiver for the vector r. 
         [0005]    Another known algorithm is the transmitted data-aided (TXDA) algorithm, which is represented by the equation: 
         [0000]    
       
         
           
             
               
                 
                   SNR 
                   = 
                   
                     
                       
                         ( 
                         
                           
                             1 
                             N 
                           
                            
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                               
                                 r 
                                 k 
                               
                                
                               
                                 a 
                                 k 
                               
                             
                           
                         
                         ) 
                       
                       2 
                     
                     
                       
                         
                           1 
                           
                             N 
                             - 
                             3 
                           
                         
                          
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             r 
                             k 
                             2 
                           
                         
                       
                       - 
                       
                         
                           1 
                           
                             N 
                              
                             
                               ( 
                               
                                 N 
                                 - 
                                 3 
                               
                               ) 
                             
                           
                         
                          
                         
                           
                             ( 
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                 
                                   r 
                                   k 
                                 
                                  
                                 
                                   a 
                                   k 
                                 
                               
                             
                             ) 
                           
                           2 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   2 
                 
               
             
           
         
       
     
         [0006]    A third known algorithm for SNR estimation is represented as: 
         [0000]    
       
         
           
             
               
                 
                   SNR 
                   = 
                   
                     
                       N 
                       2 
                     
                      
                     
                       
                         ( 
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               1 
                             
                             
                               N 
                               / 
                               2 
                             
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   
                                      
                                     
                                       r 
                                       
                                         
                                           2 
                                            
                                           
                                               
                                           
                                            
                                           k 
                                         
                                         - 
                                         1 
                                       
                                     
                                      
                                   
                                   - 
                                   
                                      
                                     
                                       r 
                                       
                                         2 
                                          
                                         
                                             
                                         
                                          
                                         k 
                                       
                                     
                                      
                                   
                                 
                                 ) 
                               
                               2 
                             
                             
                               
                                 r 
                                 
                                   
                                     2 
                                      
                                     
                                         
                                     
                                      
                                     k 
                                   
                                   - 
                                   1 
                                 
                                 2 
                               
                               + 
                               
                                 r 
                                 
                                   2 
                                    
                                   
                                       
                                   
                                    
                                   k 
                                 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   3 
                 
               
             
           
         
       
     
         [0007]    The algorithms for Equations 1 and 3 are performed blind without any pilot signal. In contrast, the TDXA algorithm uses a pilot signal with known training sequences, which provides enhanced performance. The drawback of TDXA is that additional equipment is required to process the training sequence. Although the RXDA and Equation 3 algorithms work well when the SNR is high, their performance suffers at low and negative SNRs, where they are known to have a high bias. This is problematic for various communication systems. For example, turbo code applications are known to experience negative ratios of symbol energy to noise density. In CDMA systems, the chip energy to noise density is often negative. Hence, there is a need to develop a blind SNR estimation method that works well at low and negative values without the benefit of a training sequence. 
       SUMMARY 
       [0008]    An apparatus and method for low bias estimation of small or negative signal-to-noise ratio (SNR) for a communication signal is presented. The estimation is iterative and comprises choosing an initial minimum and maximum estimate of the signal amplitude and determining the mean thereof. Associated minimum and maximum noise variances are calculated based on the amplitude values. Using probability density, maximum likelihood estimates of the minimum, maximum and mean amplitudes are derived. Based on whether the mean amplitude estimate increases or decreases, the initial minimum or maximum estimate is set equal to the maximum likelihood mean amplitude, and the resolution between the new minimum and maximum estimates is determined. These steps are repeated until the resolution is within the acceptable limit, at which point the SNR is calculated from the mean amplitude. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0009]      FIG. 1  shows a process flow diagram of method  100  for SNR estimation. 
           [0010]      FIG. 2  shows a process flow diagram of method  200  for SNR estimation. 
           [0011]      FIG. 3  shows a graph of calculated SNRs plotted against assumed SNRs using method  100 . 
           [0012]      FIG. 4  shows a comparison of mean SNR estimates of a BPSK signal performed by method  200 , RXDA, TXDA and a third other algorithm, with sampling N=1024. 
           [0013]      FIG. 5  shows a comparison of mean square errors (MSE) normalized to SNR of a BPSK signal, performed by method  200 , RXDA, TXDA, a third algorithm and the Cramer-Rao (CR) bound, with sampling N=1024. 
           [0014]      FIG. 6  shows a comparison of mean SNR estimates of an 8PSK signal performed by method  200 , a decision directed algorithm, RXDA and TXDA, with sampling N=100. 
           [0015]      FIG. 7  shows a comparison of mean square errors (MSE) normalized to SNR of an 8PSK signal performed by method  200 , a decision directed algorithm, RXDA, TXDA and the Cramer-Rao (CR) bound, with sampling N=100. 
           [0016]      FIG. 8  shows a comparison of mean SNR estimates of an 8PSK signal performed by method  200 , a decision directed algorithm, RXDA and TXDA, with sampling N=1024. 
           [0017]      FIG. 9  shows a comparison of mean square errors (MSE) normalized to SNR of an 8PSK signal performed by method  200 , a decision directed algorithm, RXDA, TXDA and the Cramer-Rao (CR) bound, with sampling N=1024. 
           [0018]      FIG. 10  shows a comparison of mean SNR estimates of a 16PSK signal performed by method  200 , a decision directed algorithm, RXDA and TXDA, with sampling N=100. 
           [0019]      FIG. 11  shows a comparison of mean square errors (MSE) normalized to SNR of a 16PSK signal performed by method  200 , a decision directed algorithm, RXDA, TXDA and the Cramer-Rao (CR) bound, with sampling N=100. 
           [0020]      FIG. 12  shows a comparison of mean SNR estimates of a 16PSK signal performed by method  200 , a decision directed algorithm, RXDA and TXDA, with sampling N=1024. 
           [0021]      FIG. 13  shows a comparison of mean square errors (MSE) normalized to SNR of a 16PSK signal performed by method  200 , a decision directed algorithm, RXDA, TXDA and the Cramer-Rao (CR) bound, with sampling N=1024. 
           [0022]      FIG. 14  shows a convergence of estimation iterations for several trials of method  200 . 
           [0023]      FIG. 15  shows a system for communications between a base station and user equipments employing SNR estimation methods  100  and  200 . 
       
    
    
     DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0024]    For a BPSK modulated signal, the time and carrier phase synchronization can be obtained so the received samples can be expressed as: 
         [0000]        r   k   =s   k   +n   k ,   Equation 4 
         [0000]    where s k  is the transmitted signal taking amplitude values from {−A, A} with equal probability and n k  is real additive white Gaussian noise with variance of σ 2 . In order to determine the unknown value A, a probability density function is a preferred technique. The probability density function of r k  can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     f 
                      
                     
                       ( 
                       
                         r 
                         k 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       2 
                     
                      
                     
                       { 
                       
                         
                           
                             f 
                             + 
                           
                            
                           
                             ( 
                             
                               r 
                               k 
                             
                             ) 
                           
                         
                         + 
                         
                           
                             f 
                             - 
                           
                            
                           
                             ( 
                             
                               r 
                               k 
                             
                             ) 
                           
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   5 
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       f 
                       + 
                     
                      
                     
                       ( 
                       
                         r 
                         k 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             π 
                           
                         
                          
                         σ 
                       
                     
                      
                     
                        
                       
                         - 
                         
                           
                             
                               ( 
                               
                                 
                                   r 
                                   k 
                                 
                                 - 
                                 A 
                               
                               ) 
                             
                             2 
                           
                           
                             2 
                              
                             
                                 
                             
                              
                             
                               σ 
                               2 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   6 
                 
               
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       f 
                       - 
                     
                      
                     
                       ( 
                       
                         r 
                         k 
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         
                           
                             2 
                              
                             
                                 
                             
                              
                             π 
                           
                         
                          
                         σ 
                       
                     
                      
                     
                       
                          
                         
                           - 
                           
                             
                               
                                 ( 
                                 
                                   
                                     r 
                                     k 
                                   
                                   + 
                                   A 
                                 
                                 ) 
                               
                               2 
                             
                             
                               2 
                                
                               
                                   
                               
                                
                               
                                 σ 
                                 2 
                               
                             
                           
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   7 
                 
               
             
           
         
       
     
         [0025]    For a received sample of consecutive symbols of length N (r 1 ,r 2 , . . . ,r N ), the probability density function can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       f 
                       N 
                     
                      
                     
                       ( 
                       
                         
                           r 
                           1 
                         
                         , 
                         
                           r 
                           2 
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           r 
                           N 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∏ 
                       
                         k 
                         = 
                         1 
                       
                       N 
                     
                      
                     
                         
                     
                      
                     
                       f 
                        
                       
                         ( 
                         
                           r 
                           k 
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   8 
                 
               
             
           
         
       
     
         [0026]    An equation for amplitude A which maximizes the probability function can be determined by taking the partial derivative of Equation 8 with respect to amplitude A, and setting the partial derivative equal to zero: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∂ 
                       
                         
                           f 
                           N 
                         
                          
                         
                           ( 
                           
                             
                               r 
                               1 
                             
                             , 
                             
                               r 
                               2 
                             
                             , 
                             … 
                              
                             
                                 
                             
                             , 
                             
                               r 
                               N 
                             
                           
                           ) 
                         
                       
                     
                     
                       ∂ 
                       A 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   9 
                 
               
             
           
         
       
     
         [0000]    The determination of a maximum likelihood estimate of A is then the solution to Equation 10: 
         [0000]    
       
         
           
             
               
                 
                   A 
                   = 
                   
                     
                       1 
                       N 
                     
                      
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                         
                           r 
                           k 
                         
                          
                         
                           th 
                            
                           
                             ( 
                             
                               
                                 Ar 
                                 k 
                               
                               
                                 σ 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   10 
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]    
       
         
           
             
               
                 
                   
                     th 
                      
                     
                       ( 
                       x 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                            
                           x 
                         
                         - 
                         
                            
                           
                             - 
                             x 
                           
                         
                       
                       
                         
                            
                           x 
                         
                         + 
                         
                            
                           
                             - 
                             x 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   11 
                 
               
             
           
         
       
     
         [0027]    Since the SNR is unknown, it may possibly be high or low. If the SNR is high, an acceptable approximation for value th can be made as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     th 
                      
                     
                       ( 
                       
                         
                           Ar 
                           k 
                         
                         
                           σ 
                           2 
                         
                       
                       ) 
                     
                   
                   ≅ 
                   
                     { 
                     
                       
                         
                           
                             
                               + 
                               1 
                             
                             , 
                             
                               
                                 r 
                                 k 
                               
                               &gt; 
                               0 
                             
                           
                         
                       
                       
                         
                           
                             
                               - 
                               1 
                             
                             , 
                             
                               
                                 r 
                                 k 
                               
                               &lt; 
                               0 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   12 
                 
               
             
           
         
       
     
         [0028]    The decision-directed amplitude estimate is then: 
         [0000]    
       
         
           
             
               
                 
                   
                     A 
                     ^ 
                   
                   = 
                   
                     
                       1 
                       N 
                     
                      
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                           
                       
                        
                       
                          
                         
                           r 
                           k 
                         
                          
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   13 
                 
               
             
           
         
       
     
         [0029]    The noise power can be estimated as total power minus the signal power, and the SNR can therefore be estimated as: 
         [0000]    
       
         
           
             
               
                 
                   
                     S 
                      
                     
                         
                     
                      
                     N 
                      
                     
                         
                     
                      
                     R 
                   
                   = 
                   
                     
                       
                         ( 
                         
                           
                             1 
                             N 
                           
                            
                           
                             
                               ∑ 
                               
                                 k 
                                 = 
                                 1 
                               
                               N 
                             
                              
                             
                                 
                             
                              
                             
                                
                               
                                 r 
                                 k 
                               
                                
                             
                           
                         
                         ) 
                       
                       2 
                     
                     
                       
                         
                           1 
                           N 
                         
                          
                         
                           
                             ∑ 
                             
                               k 
                               = 
                               1 
                             
                             N 
                           
                            
                           
                             r 
                             k 
                             2 
                           
                         
                       
                       - 
                       
                         
                           ( 
                           
                             
                               1 
                               N 
                             
                              
                             
                               
                                 ∑ 
                                 
                                   k 
                                   = 
                                   1 
                                 
                                 N 
                               
                                
                               
                                   
                               
                                
                               
                                  
                                 
                                   r 
                                   k 
                                 
                                  
                               
                             
                           
                           ) 
                         
                         2 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   14 
                 
               
             
           
         
       
     
         [0030]    In an alternative embodiment, for a signal in which the time synchronization and the carrier phase synchronization have been obtained for MPSK modulation, the value s k  of Equation 4 is the transmitted M-ary PSK signal, represented as: 
         [0000]        Ae   j2πk/M   ,k= 0,1, . . . , M− 1   Equation 15 
         [0000]    with equal probability of 1/M, and A as the amplitude of MPSK signal s k . Value n k  from Equation 4 is the complex additive white Gaussian noise with variance of 2σ 2 . The probability density function of r k , where 
         [0000]        r   k   =x   k   +jy   k    Equation 16 
         [0000]    can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     f 
                      
                     
                       ( 
                       
                         
                           x 
                           k 
                         
                         , 
                         
                           y 
                           k 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       M 
                     
                      
                     
                       
                         ∑ 
                         
                           l 
                           = 
                           0 
                         
                         
                           M 
                           - 
                           1 
                         
                       
                        
                       
                           
                       
                        
                       
                         
                           1 
                           
                             
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 π 
                               
                             
                              
                             σ 
                           
                         
                          
                         exp 
                          
                         
                           { 
                           
                             - 
                             
                               
                                 
                                   
                                     
                                       
                                         
                                           ( 
                                           
                                             
                                               x 
                                               k 
                                             
                                             - 
                                             
                                               
                                                 X 
                                                 l 
                                               
                                                
                                               A 
                                             
                                           
                                           ) 
                                         
                                         2 
                                       
                                       + 
                                     
                                   
                                 
                                 
                                   
                                     
                                       
                                         ( 
                                         
                                           
                                             y 
                                             k 
                                           
                                           - 
                                           
                                             
                                               Y 
                                               l 
                                             
                                              
                                             A 
                                           
                                         
                                         ) 
                                       
                                       2 
                                     
                                   
                                 
                               
                               
                                 2 
                                  
                                 
                                     
                                 
                                  
                                 
                                   σ 
                                   2 
                                 
                               
                             
                           
                           } 
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   17 
                 
               
             
           
         
       
     
         [0000]    where 
         [0000]        X   l   +jY   l   =e   j2πl/M    Equation 18 
         [0000]    and j=√{square root over (−1)}. For a received sample of consecutive MPSK symbols of length N (r 1 ,r 2 , . . . ,r N ), the probability density function can be expressed as: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       f 
                       N 
                     
                      
                     
                       ( 
                       
                         
                           r 
                           1 
                         
                         , 
                         
                           r 
                           2 
                         
                         , 
                         … 
                          
                         
                             
                         
                         , 
                         
                           r 
                           N 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       ∏ 
                       
                         k 
                         = 
                         1 
                       
                       N 
                     
                      
                     
                         
                     
                      
                     
                       f 
                        
                       
                         ( 
                         
                           
                             x 
                             k 
                           
                           , 
                           
                             y 
                             k 
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   19 
                 
               
             
           
         
       
     
         [0000]    Using Equation 9, the partial derivative of Equation 19 with respect to amplitude A is performed and set to zero, resulting in the following equation: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ∑ 
                       
                         k 
                         = 
                         1 
                       
                       N 
                     
                      
                     
                         
                     
                      
                     
                       
                         
                           ∂ 
                           
                             f 
                              
                             
                               ( 
                               
                                 
                                   x 
                                   k 
                                 
                                 , 
                                 
                                   y 
                                   k 
                                 
                               
                               ) 
                             
                           
                         
                         
                           ∂ 
                           A 
                         
                       
                       
                         f 
                          
                         
                           ( 
                           
                             
                               x 
                               k 
                             
                             , 
                             
                               y 
                               k 
                             
                           
                           ) 
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   20 
                 
               
             
           
         
       
     
         [0000]    According to Equation 20, the equation for amplitude A which maximizes the probability function is derived and expressed as follows: 
         [0000]    
       
         
           
             
               
                 
                   A 
                   = 
                   
                     
                       1 
                       N 
                     
                      
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               M 
                               - 
                               1 
                             
                           
                            
                           
                             
                               [ 
                               
                                 
                                   
                                     x 
                                     k 
                                   
                                    
                                   
                                     X 
                                     l 
                                   
                                 
                                 + 
                                 
                                   
                                     y 
                                     k 
                                   
                                    
                                   
                                     Y 
                                     l 
                                   
                                 
                               
                               ] 
                             
                              
                             exp 
                              
                             
                               { 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           x 
                                           k 
                                         
                                          
                                         
                                           X 
                                           l 
                                         
                                       
                                       + 
                                       
                                         
                                           y 
                                           k 
                                         
                                          
                                         
                                           Y 
                                           l 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   A 
                                 
                                 
                                   σ 
                                   2 
                                 
                               
                               } 
                             
                           
                         
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               M 
                               - 
                               1 
                             
                           
                            
                           
                               
                           
                            
                           
                             exp 
                              
                             
                               { 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           x 
                                           k 
                                         
                                          
                                         
                                           X 
                                           l 
                                         
                                       
                                       + 
                                       
                                         
                                           y 
                                           k 
                                         
                                          
                                         
                                           Y 
                                           l 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   A 
                                 
                                 
                                   σ 
                                   2 
                                 
                               
                               } 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   21 
                 
               
             
           
         
       
     
         [0000]    If the actual SNR is high, an acceptable decision-directed amplitude estimation is then: 
         [0000]    
       
         
           
             
               
                 
                   
                     A 
                     ^ 
                   
                   ≈ 
                   
                     
                       1 
                       N 
                     
                      
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                           
                       
                        
                       
                         [ 
                         
                           
                             
                               x 
                               k 
                             
                              
                             
                               
                                 X 
                                 ^ 
                               
                               k 
                             
                           
                           + 
                           
                             
                               y 
                               k 
                             
                              
                             
                               
                                 Y 
                                 ^ 
                               
                               k 
                             
                           
                         
                         ] 
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   22 
                 
               
             
           
         
       
     
         [0000]    where ({circumflex over (X)} k , Ŷ k ) is the estimated signal that maximizes X l  and Y l : 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           X 
                           ^ 
                         
                         k 
                       
                       , 
                       
                         
                           Y 
                           ^ 
                         
                         k 
                       
                     
                     ) 
                   
                   = 
                   
                     arg 
                      
                     
                       { 
                       
                         
                           max 
                           
                             
                               X 
                               l 
                             
                             , 
                             
                               Y 
                               l 
                             
                           
                         
                          
                         
                           { 
                           
                             
                               
                                 
                                   x 
                                   k 
                                 
                                  
                                 
                                   X 
                                   l 
                                 
                               
                               + 
                               
                                 
                                   y 
                                   k 
                                 
                                  
                                 
                                   Y 
                                   l 
                                 
                               
                             
                             , 
                             
                               l 
                               = 
                               0 
                             
                             , 
                             1 
                             , 
                             … 
                              
                             
                                 
                             
                             , 
                             
                               M 
                               - 
                               1 
                             
                           
                           } 
                         
                       
                       } 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   23 
                 
               
             
           
         
       
     
         [0031]    A method  100  for an iterative SNR estimation for a BPSK signal using Equation 10 is shown in  FIG. 1 . Given an amplitude estimate A 0  and a noise variance estimate σ 0   2 , a new amplitude estimate A 1  is calculated by Equation 24, which is based on Equation 10: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       A 
                       1 
                     
                     = 
                     
                       
                         1 
                         N 
                       
                        
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             1 
                           
                           N 
                         
                          
                         
                             
                         
                          
                         
                           
                             r 
                             k 
                           
                            
                           
                             th 
                              
                             
                               ( 
                               
                                 
                                   
                                     A 
                                     0 
                                   
                                    
                                   
                                     r 
                                     k 
                                   
                                 
                                 
                                   σ 
                                   0 
                                   2 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   24 
                 
               
             
           
         
       
     
         [0000]    and a new noise variance estimate σ 1   2  by: 
         [0000]    
       
         
           
             
               
                 
                   
                     σ 
                     1 
                     2 
                   
                   = 
                   
                     
                       
                         1 
                         N 
                       
                        
                       
                         
                           ∑ 
                           
                             k 
                             = 
                             1 
                           
                           N 
                         
                          
                         
                             
                         
                          
                         
                           r 
                           k 
                           2 
                         
                       
                     
                     - 
                     
                       A 
                       1 
                       2 
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   25 
                 
               
             
           
         
       
     
         [0032]    As the method is updated, A 0   2 /σ 0   2  converges to A 1   2 /σ 1   2 . Since the SNR to be estimated is unknown, an initial SNR is assumed (step  101 ), denoted as: 
         [0000]      SNR 0   =A   0   2 /σ 0   2    Equation 26 
         [0033]    In step  102 , corresponding values for A 0  and σ 2  are calculated as: 
         [0000]    
       
         
           
             
               
                 
                   
                     A 
                     0 
                   
                   = 
                   
                     
                       
                         S 
                          
                         
                             
                         
                          
                         N 
                          
                         
                             
                         
                          
                         
                           R 
                           0 
                         
                       
                       
                         1 
                         + 
                         
                           S 
                            
                           
                               
                           
                            
                           N 
                            
                           
                               
                           
                            
                           
                             R 
                             0 
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   27 
                 
               
             
           
         
       
     
         [0000]    and 
         [0000]    
       
         
           
             
               
                 
                   
                     σ 
                     0 
                     2 
                   
                   = 
                   
                     
                       1 
                       
                         1 
                         + 
                         
                           S 
                            
                           
                               
                           
                            
                           N 
                            
                           
                               
                           
                            
                           
                             R 
                             0 
                           
                         
                       
                     
                     . 
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   28 
                 
               
             
           
         
       
     
         [0034]    Next in step  103 , Equations 24 and 25 are used to calculate A 1 , σ 1   2 , and SNR 1  is calculated in step  104  by Equation 29: 
         [0000]      SNR 1   =A   1   2 /σ 1   2    Equation 29 
         [0035]    Step  105  performs a decision as to whether estimate SNR 0  is within a predetermined acceptable resolution compared to the calculated SNR 1 . If the resolution is acceptable, then SNR 0  can be accepted as the final estimate (step  107 ). Otherwise, SNR 0  is adjusted (step  106 ) and the process repeats starting at step  102 . As an example with a predetermined acceptable resolution of 0.1 dB as the benchmark, steps  102  through  106  are repeated until the difference between calculated SNR 1  and estimate SNR 0  is less than or equal to 0.1 dB. Alternatively, steps  102  through  106  are repeated for a predetermined number of times before bringing an end to the estimation process (step  107 ), and accepting the resulting estimate value, regardless of the intermediate resolutions. 
         [0036]    A similar method for MPSK signals can be performed by replacing Equation 24 in step  103  with Equation 30, which is based on Equation 21, to calculate amplitude A 1 : 
         [0000]    
       
         
           
             
               
                 
                   
                     A 
                     1 
                   
                   = 
                   
                     
                       1 
                       N 
                     
                      
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                           
                       
                        
                       
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               M 
                               - 
                               1 
                             
                           
                            
                           
                             
                               [ 
                               
                                 
                                   
                                     x 
                                     k 
                                   
                                    
                                   
                                     X 
                                     l 
                                   
                                 
                                 + 
                                 
                                   
                                     y 
                                     k 
                                   
                                    
                                   
                                     Y 
                                     l 
                                   
                                 
                               
                               ] 
                             
                              
                             exp 
                              
                             
                               { 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           x 
                                           k 
                                         
                                          
                                         
                                           X 
                                           l 
                                         
                                       
                                       + 
                                       
                                         
                                           y 
                                           k 
                                         
                                          
                                         
                                           Y 
                                           l 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     A 
                                     0 
                                   
                                 
                                 
                                   σ 
                                   0 
                                   2 
                                 
                               
                               } 
                             
                           
                         
                         
                           
                             ∑ 
                             
                               l 
                               = 
                               0 
                             
                             
                               M 
                               - 
                               1 
                             
                           
                            
                           
                               
                           
                            
                           
                             exp 
                              
                             
                               { 
                               
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           x 
                                           k 
                                         
                                          
                                         
                                           X 
                                           l 
                                         
                                       
                                       + 
                                       
                                         
                                           y 
                                           k 
                                         
                                          
                                         
                                           Y 
                                           l 
                                         
                                       
                                     
                                     ) 
                                   
                                    
                                   
                                     A 
                                     0 
                                   
                                 
                                 
                                   σ 
                                   0 
                                   2 
                                 
                               
                               } 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   30 
                 
               
             
           
         
       
     
         [0037]      FIG. 3  shows curves of calculated SNRs versus assumed SNRs for twenty 1024-point sample vectors each with a real SNR of 3 dB. Each curve crosses the straight line “calculated SNR=assumed SNR” at one point. The crossing point is the estimated SNR for a converged method. It is noteworthy that the crossing points are concentrated around the true SNR of 3 dB. Variations among the 20 curves are due to the random nature of the noise component during each trial. The calculated values vary approximately between −1 dB and +0.5 dB. When the assumed SNR value is greater than the actual SNR, the calculated SNR value is less than the assumed value. This relationship is useful for quick convergence as each successive assumed SNR value can properly be increased or reduced accordingly. 
         [0038]    An alternative method is to iteratively solve for amplitude A, then to compute the SNR estimate upon convergence, as shown by flow diagram of method  200  in  FIG. 2 . In Step  201 , the received vector is normalized such that: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       1 
                       N 
                     
                      
                     
                       
                         ∑ 
                         
                           k 
                           = 
                           1 
                         
                         N 
                       
                        
                       
                           
                       
                        
                       
                         r 
                         k 
                         2 
                       
                     
                   
                   = 
                   1 
                 
               
               
                 
                   Equation 
                    
                   
                       
                   
                    
                   31 
                 
               
             
           
         
       
     
         [0000]    Assumed minimum and maximum amplitudes of interest A min  and A max  are selected, and a predetermined resolution Δ is selected. Values A 0  and A 1  are initialized as follows: A 0 =A min  and A 1 =A max . 
         [0039]    In steps  202  and  203 , the mean of A 0  and A 1  is calculated by: 
         [0000]        A   m =( A   0   +A   1 )/2   Equation 32 
         [0000]    and the corresponding noise variances are determined by: 
         [0000]      σ 0   2 =1− A   0   2    Equation 33 
         [0000]      σ 1   2 =1− A   1   2    Equation 34 
         [0000]      σ m   2 =1− A   m   2    Equation 35 
         [0040]    In step  204 , three estimated amplitude values, A′ 0 , A′ 1  and A′ m  are calculated using Equation 24 by substituting the initial amplitude values A 0 , A 1  and A m , for A 0  in Equation 24 and initial noise variances σ 0 , σ 1  and σ m , respectively, for σ 0  in Equation 24. 
         [0041]    For step  205 , if A m &gt;A′ m , then the maximum amplitude A 1  is updated as follows: A 1 =A′ m . Otherwise, the minimum amplitude A 0  is updated: A 0 =A′ m . In an alternative embodiment for step  205 , if A m &gt;A′ m , then amplitude A 1  can be updated so that A 1 =A m ; otherwise the minimum amplitude A 0  is updated: A 0 =A m . 
         [0042]    For step  206 , the resolution Δ is evaluated. If A 1 −A 0 &lt;Δ, then the estimated amplitude is the updated value A OUT =(A 0 +A 1 )/2 with either A 0  or A 1  as updated amplitude values from step  205 . The final estimated signal-to-noise ratio SNR OUT  is calculated from the estimated amplitude value A OUT  as follows: SNR OUT =A OUT   2 /(1−A OUT   2 ). Otherwise the process is repeated by returning to step  202  and repeating the steps through step  206  until an acceptable resolution Δ is achieved. 
         [0043]    As with method  100 , method  200  can be modified to accommodate an MPSK signal. This is achieved by calculating amplitude estimates A′ 0 , A′ 1  and A′ m  using Equation 30 instead of Equation 24 in step  204 . 
         [0044]    The lower bias of method  200  can be seen in  FIGS. 4-13  in which the mean SNR and normalized mean square error (MSE) results are compared against various SNR algorithms. Simulation results for the iterative SNR estimation method  200  are graphically compared to the RXDA, the TXDA, and the Equation 3 SNR estimation algorithms as shown in  FIGS. 4-5 . As aforementioned, the TXDA algorithm is based on exact knowledge of the received data, which is only applicable to known training sequences. The TXDA curve is therefore shown as a baseline for comparison purposes. 
         [0045]      FIG. 4  shows means of the various SNR estimations generated using a received vector of 1024 samples (N=1024) versus the actual SNR. The iterative SNR estimation method  200  has a lower bias (excepting the known data case) and the useful range extends down to about −5 dB. For comparison, the useful range in each case for RXDA and Equation 3 algorithms only extends down to about 8 dB. 
         [0046]      FIG. 5  shows the normalized mean square error (MSE) of the SNR estimations where N=1024 and also shows the Cramer-Rao (CR) bound that is lower bounded by 
         [0000]    
       
         
           
             
               C 
                
               
                   
               
                
               R 
             
             ≥ 
             
               2 
                
               
                 
                   { 
                   
                     
                       2 
                       
                         
                           A 
                           2 
                         
                          
                         N 
                       
                     
                     + 
                     
                       1 
                       N 
                     
                   
                   } 
                 
                 . 
               
             
           
         
       
     
         [0000]    The estimation by method  200  produces results having a lower normalized MSE than that for RXDA and Equation 3. 
         [0047]      FIGS. 6-9  show mean and MSE results of method  200  compared with RXDA, TXDA, and decision-directed for an 8PSK signal. Comparison of  FIGS. 6 ,  7  to  FIGS. 8 ,  9  show the improvement in mean and MSE versus SNR by method  200  when the sample length is increased from N=100 to N=1024, respectively. It should be noted that improvements are evident for method  200  whereas those for Equations 1 and 3 show no improvement. 
         [0048]    Similarly,  FIGS. 10-11  show mean and MSE results for a 16PSK signal for N=100 and  FIGS. 12 ,  13  show mean and MSE results for a 16PSK signal for N=1024 with similar results. 
         [0049]      FIG. 14  shows several trajectories of convergence within 9 iterations for method  200 . In general, the number of iterations depends on A min , A max  and the resolution Δ. In this example, A min =0.001, A max =0.999 and Δ=0.0002. As shown in  FIG. 14 , the estimated SNR stabilizes after 7 iterations and by the 9 th  iteration, A 1 −A 0 &lt;Δ, and the estimation is finished. 
         [0050]      FIG. 15  shows an embodiment for using methods  100  and  200 , comprising a system for wireless communications, such as CDMA, with base station  301  and user equipments (UEs)  302 - 305 . Base station  301  and (UEs)  302 - 305  each include an SNR estimator which performs the low bias SNR estimation method  200 . Improved SNR estimation provides several advantages for base station and UE performance. For instance, improved SNR estimation for a UE is enhanced power control from having more accurate assessment of the required uplink power. At the base station, improved channel selection results from better SNR estimation.