PATENT ABSTRACT
An adaptive receiver to provide reliable estimates of the symbols for the high-order MQAM and MPSK modulated signals received in the presence of amplitude fading and phase dynamics induced due to time-varying atmospheric or terrestrial multipath fading encountered in wireless communication channels. The adaptive receiver encompasses an adaptive tracking loop comprised of adaptive channel fade envelope estimator derived from the high-order modulated signal, a novel phase detector to separately estimate the phase due to the fading channel and the reference oscillator from that due to the data modulation present in the received signal without the need of any pilot symbol or pilot carrier, and a Kalman filter, a fixed-lag smoother and a smoothed symbol detector.

PATENT DESCRIPTION
BACKGROUND 
       [0001]    The present invention relates to an adaptive receiver that acquires and tracks the carrier frequency and phase of QAM and MPSK signals under fading channel conditions, and provides reliable data estimates under such fading conditions encountered in wireless communication. The adaptive receiver utilizes an adaptive filter, a phase detector, a Kalman filter, a fixed lag smoother, and a smoothed symbol detector to provide an accurate estimate of the channel fade envelope, to compensate for the phase dynamics induced due to time-varying atmospheric or terrestrial multipath fading, and to provide reliable estimates of the symbols of high-order modulated signals. 
         [0002]    Wireless communication systems are currently in a rapid evolutionary phase in terms of development of various technologies, development of various applications, deployment of various services, and generation of many important standards in the field. Although there are many factors to be considered in the design of these systems, important factors include the bandwidth utilization efficiency due to the limited bandwidth allocation, flexibility in operation and robustness of the communication link in the presence of various disturbances such as fading while achieving the specified performance. 
         [0003]    The bandwidth and power efficient communication systems require coherent multilevel modulation techniques such as MQAM with M equal to 16 or higher, and MPSK signals. While both MQAM and MPSK signals have the same bandwidth efficiency for any value of M, the MQAM signals have a much higher power efficiency compared to MPSK. However, the detection of MQAM signals for M higher than 4 requires an accurate knowledge of the signal amplitude which is relatively difficult to obtain under time-varying random variations in the channel gain as occurs, for example, in Rayleigh fading channels in wireless communication. In the absence of a technique for a fast and accurate tracking of the fade envelope of the channel, the application of the MQAM techniques for M greater than 4 has serious limitations in terms of their applications to wireless channels. 
         [0004]    Additionally, both MQAM and MPSK techniques require an accurate estimation of the carrier frequency and phase which are corrupted by the phase noise introduced by the fading channel, in addition to any oscillator noise that is present in non fading channels. However, the magnitude of the phase noise due to the channel is much higher compared to that due to the oscillator. The required accuracy of the carrier phase increases with order M for both the MQAM and MPSK signals. Thus, successful applications of high order QAM techniques to wireless fading communication channels require both an accurate carrier phase acquisition and tracking, and a fast and accurate tracking of the channel fade envelope. 
         [0005]    In order to achieve high power efficiency, the carrier reference is obtained by some processing of the data modulated signal itself rather than transmitting a pilot carrier signal which results in power loss. Among such techniques is the loop involving a fourth power circuit followed by a narrow band phase lock loop that tracks four times the carrier frequency. This is suitable for QPSK modulation, which is the same as MQAM or MPSK with M equal to 4. See “Using Times-Four Carrier Recovery in M-QAM Digital Radio Receivers,” by A. J. Rustako, et. al., IEEE Journal on Selected Areas in Communications, vol. SAC-5, No. 3, pp. 524-533, April 1987. In a fading environment, a limiter precedes the fourth power circuit to eliminate the gain variations in the loop. See “A Limiter Aided 4th Multiplying PLL Carrier Recovery Technique for 16-QAM Signal,” IEEE, 1997. However, the process of limiting amplifies the noise. Also, the method is vulnerable to phase jitter induced by random data patterns. Such techniques involving limiting and taking power M of the signal are also applicable to higher order MPSK signals, albeit with the same disadvantages and do not extend to MQAM with M&gt;4. Decision-directed methods known as polarity-type Costas loop for the MPSK signals involve slicing the inphase and quadrature components of the received signal. See,” A Generalized Polarity-Type Costas Loop for Tracking MPSK Signals,” IEEE Transactions on Communications, Vol. 30, N0. 10, pp. 2289-2296, October 1982. This requires the knowledge of the signal amplitude and thus does not apply to MQAM signals with M&gt;4 under fading conditions. In the absence of accurate channel gain estimate in the fading environment, the application of the decision-directed methods to MQAM is not feasible and results in significant degradation in performance in the data estimates along with frequent loss of lock and long acquisition times. An earlier proposed solution to this problem by Dobrica, Carrier Synchronization Unit, U.S. Pat. No. 5,875,215, Feb. 23, 1999, used pilot symbols to estimate the channel gain. However, the use of pilot symbols result in significant reduction in capacity of the channel and requires interpolation of the amplitude and phase estimates from the pilot symbols to the subsequent data symbols. This leads to significant errors in amplitude and phase estimates during the data symbol detection, and imposes logistic difficulties in terms of maintaining the pilot symbol sequences, the need for frame synchronization, etc. Kumar, U.S. Pat. No. 6,693,979, Feb. 17, 2004 teaches a receiver for improved phase estimation using fixed-lag smoothing by estimating the fading channel amplitude. However, in the earlier teaching of Kumar there is no data modulation considered and thus does not solve the problem of reliable detection of high-order modulated signal over fading channels without the need for any pilot signals. These and other disadvantages are solved or reduced by the receiver of the present invention. 
       SUMMARY OF THE INVENTION 
       [0006]    The adaptive receiver architecture of this invention acquires and accurately tracks the carrier frequency and phase of a high-order MQAM modulation even in the presence of severe Rayleigh fading distortion without incurring any significant penalty compared to the case of known carrier amplitude and phase. The adaptive receiver also provides an accurate filtered estimate of the instantaneous channel fade envelope. The receiver of the present invention preferably employs an adaptive filter to estimate the channel fade envelope along with a phase detector, a Kalman filter, a fixed lag smoother, and a smoothed symbol detector to accurately track the carrier frequency and phase and provide reliable estimates of the symbols for high-order MQAM and MPSK modulated signals received in the presence of amplitude and phase dynamics induced due to time-varying atmospheric or terrestrial multipath fading. 
         [0007]    An object of this invention is to provide reliable estimates of the symbols for the high-order MQAM and MPSK modulated signals received in the presence of amplitude and phase dynamics induced due to time-varying atmospheric or terrestrial multipath fading. 
         [0008]    Another object of the invention is to provide real-time and accurate estimation of the channel fade envelope derived from the high-order MQAM and MPSK modulated signals. 
         [0009]    Another object of the invention is to provide an adaptive smoother to provide improved carrier phase estimation in the presence of amplitude and phase dynamics of the fading channel. 
         [0010]    Another object of the invention is to provide a phase detector to provide the prediction error to the Kalman filter and the inphase and quadrature components of the detected symbol. 
         [0011]    Yet another object of the invention is to provide a phase detector for more general modulation formats, including the MQAM and MPSK modulation. 
         [0012]    Yet another object of the invention is to provide improved estimates of the symbols of high-order MQAM and MPSK modulated signals by using the improved carrier phase provided by the adaptive smoother. 
         [0013]    Still another object of the invention is to provide a phase detector to provide an estimate of the carrier phase on the basis of the signals provided by an adaptive signal processor, the adaptive signal processor for a real-time estimation of the fade envelope, and to provide non fading normalized sampled baseband signals to the phase detector, a Kalman filter based phase locked loop for carrier phase estimation, to provide sampled baseband signals to the adaptive signal processor, and in combination with a fixed lag smoother and a smoothed symbol detector to provide improved phase estimates and improved data symbol estimation. 
         [0014]    The present invention is an adaptive receiver for solving the problems of degradation in the performance of a receiver for high order MQAM and MPSK signals in the presence channel fading conditions in wireless communication. The detection of MQAM signals for M higher than 4 requires an accurate knowledge of the signal amplitude, which is relatively difficult to obtain under time-varying random variations in the channel gain as occurs, for example, in Rayleigh fading channels. In the absence of a technique for a fast and accurate tracking of the fade envelope of the channel, the application of the MQAM techniques for M greater than 4 has serious limitations in terms of their applications to wireless channels. Additionally, both MQAM and MPSK techniques require an accurate estimation of the carrier frequency and phase, which is corrupted by the phase noise introduced by the fading channel in addition to any oscillator noise which is present in non fading channels. In the absence of an accurate channel gain estimate in the fading environment, the application of the decision-directed methods to MQAM results in significant degradation in performance in the data estimates along with frequent loss of lock and long acquisition times. This invention solves these and other problems by providing an adaptive signal processor that accurately estimates the channel fade envelope in real-time from the high-order MQAM or MPSK modulated signals. The adaptive receiver provides a phase detector to derive a prediction error from the non fading normalized sampled baseband signals provided by the adaptive signal processor. 
         [0015]    In the adaptive receiver of the invention for the high-order MQAM or MPSK modulated signals, the received RF signal after down conversion to an intermediate frequency (IF) signal is the input to a complex mixer along with the numerically controlled oscillator (NCO) output. The NCO output contains a sine and cosine carrier waveforms at IF frequency with the NCO phase determined by the phase tracking loop. The complex mixer outputs after being filtered by square-root raised cosine filters are digitized by analog-to-digital converters (ADC). The outputs of the ADCs provide the sampled baseband signals to the adaptive signal processor. 
         [0016]    The complex mixer, the ADCs, square-root raised cosine filters, adaptive signal processor, phase detector, Kalman filter and an NCO (numerically controlled oscillator) comprise a phase tracking loop that is made adaptive with the adaptive fade envelope filter that is part of the adaptive signal processor. The tracking loop provides an adaptive filtered estimate of the phase noise dynamics induced both due to the fading channel and the phase noise of the NCO, and adjusts the NCO instantaneous frequency such that the NCO phase is equal to the filtered phase estimate. In a preferred embodiment of the invention, the tracking loop phase detector, using a pair of threshold circuits and simple trigonometric identities, segregates the phase component due to channel induced phase and the oscillator phase noise from the phase of the data symbol thus providing a robust adaptive carrier phase tracking. Thus, the tracking loop adaptively provides an accurate estimate of the phase induced by the channel and the oscillator phase noise, and adjusts the NCO phase in such a manner that there is only a small degradation in the data symbol detection due to any phase tracking error even in the presence of channel fading. In addition, the adaptive tracking loop phase detector also provides the detected inphase and quadrature components of the data symbols based on the filtered phase and the fade envelope estimates. 
         [0017]    The phase estimate provided by the Kalman filter based adaptive tracking loop is optimum when no delay is included in the estimate. However reduction in any residual phase error due to the adaptive tracking loop is further reduced by a fixed-lag smoother providing a better estimate of the carrier phase with a delay of a few symbols. The normalized sampled baseband signals at the output of the adaptive signal processor are input to the smoothed symbol detector which, on the basis of phase corrections provided by the fixed-lag smoother, provides smoothed detected symbol output such that the probability of symbol error with the use of smoothed detected symbol output is smaller than the probability of symbol error using the detected symbol output provided by the phase detector of the adaptive phase tracking loop. The fixed-lag smoother and smoothed symbol detector achieve improved phase estimation and symbol detection by introducing a fixed delay in the estimation and detection process. This delay is similar to the delay introduced by the physical propagation channel and is only a very small fraction thereof, and thus does not adversely affect the performance of the adaptive receiver in any way. Thus the combination of the Kalman filter based adaptive phase tracking loop, the fixed lag smoother, and smoother symbol detector provide better performance than the adaptive tracking loop alone. Furthermore the adaptive tracking loop solves the difficulty of detection of high-order digitally modulated signals received over fading channels wherein the traditional non adaptive receivers will have relatively poor performance. The most significant benefits of a method of this invention are accurate and fast carrier phase and frequency acquisition and tracking, better probability of symbol error, and operational reliability over diverse communication channels including the fading channels encompassing both amplitude fading and severe phase noise distortion as are encountered in wireless communication. These and other advantages will become more apparent from the following detailed description of the preferred embodiment. 
     
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         [0018]    In the succeeding section, the invention will be described in detail with reference made to the accompanying drawings, in which 
           [0019]      FIG. 1  is a block diagram for a first preferred embodiment of the present invention,; 
           [0020]      FIG. 2  is a graph depicting the normalized signal constellation of a MQAM signal for M=16; 
           [0021]      FIG. 3  is a block diagram of a preferred embodiment of the adaptive signal processor block of  FIG. 1 ; 
           [0022]      FIG. 4  is a block diagram of a preferred embodiment of the adaptive fade envelope estimator in connection with the adaptive signal processor block of  FIG. 3 . 
           [0023]      FIG. 5  is a block diagram of a preferred embodiment of a phase detector;. 
           [0024]      FIG. 6  is a block diagram of a preferred embodiment of the Kalman Filter/Fixed-Lag Smoother of  FIG. 1 ; 
           [0025]      FIG. 7  is a block diagram of a preferred embodiment of a smoothed symbol detector; 
           [0026]      FIG. 8  is a block diagram of a second preferred embodiment of a phase detector; 
           [0027]      FIG. 9  is a block diagram of a third preferred embodiment of a phase detector; 
           [0028]      FIG. 10  is a block diagram of a first preferred embodiment of a complex symbol detector of the phase detector of  FIG. 9 ; 
           [0029]      FIG. 11  is a graph depicting a signal constellation diagram for an example MPSK modulation for the case of M=8; 
           [0030]      FIG. 12A  is a block diagram of a first preferred mapper for the case of M=8; and 
           [0031]      FIG. 12B  is a block diagram of a first preferred embodiment of the MUX 1 block of  FIG. 12A . 
       
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
       [0032]    An embodiment of the present invention as depicted in  FIG. 1  is disclosed and explained in detail with the aid of reference designators in the former drawing. More particularly, the present invention concerns a novel adaptive receiver architecture and carrier recovery loop propounded for, inter alia, M-QAM modulated signals. The algorithm performs an intermediate estimate of the amplitude envelope for normalization. The cosine and sine of the phase modulation term are further reckoned and removed through simple trigonometric identities, rendering the phase error estimate applied to the loop filter input. This technique offers several advantages over conventional carrier recovery structures, such as low cost, robustness and low complexity without infusing phase jitter due to random data patterns nor long acquisition times. 
         [0033]    An embodiment of the invention is described with reference to the drawings using reference numbers shown in the drawings. The adaptive receiver of this invention is preferably used in a communication system that uses high-order digital modulation techniques operating over fading communication channels. Referring to the drawing of the adaptive receiver shown in  FIG. 1 , the antenna  1  receives high-order modulated communication signal which is filtered and amplified by the RF front end  2 . The amplified and filtered RF signal  3  is downconverted to an intermediate frequency (IF) by a down converter  4 . 
         [0034]    The resulting IF signal  5  denoted by r(t) is input to the complex mixer  10  and may be expressed in the AM-PM form shown below 
         [0000]        r ( t )=α( t ) A   0   A   df ( t )cos [2π f   IF   t+φ   df ( t )+θ i ( t )]+ n ( t )   (1) 
         [0000]    where f IF  denotes the intermediate frequency, A 0  is a constant that includes the mean channel gain, α(t) is the fading envelope, A df (t) and φ df (t) denote respectively the filtered data amplitude and phase modulation waveforms, θ i (t) is the carrier phase induced by the fading channel, and n(t) denotes the receiver additive noise with two-sided power spectral density equal to N 0 /2. The amplitude and phase modulation terms A df (t) and φ df (t) may equivalently be also expressed in terms of the normalized inphase and quadrature modulation terms x If (t) and x Qf (t) by (2) below 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0000]    Or equivalently in term of the complex baseband envelope g(t) given by 
         [0000]        g ( t )= x   If ( t )+ jx   Qf ( t );  j ≡√{square root over (−1)}  (2c) 
         [0035]    In most cases in practice the filtered inphase and quadrature modulation terms x If (t) and x Qf (t) are obtained by filtering the corresponding unfiltered signals x I (t) and y I (t) by square-root raised cosine filters. However, in some applications, there may be no such band-limiting filtering involved. The receiver architecture described herein applies to all such cases. Over the k th  modulation symbol period of duration T s , the unfiltered signals x I (t) and y I (t) are constant equal to the real and imaginary parts of the k th  modulation symbol s(k) and denoted by i(k) and q(k) respectively for all integers k≧0. 
         [0036]    The inphase and quadrature modulation terms x I (t) and x Q (t) take values over a fixed set of values independent of the channel gain A 0  and the fading envelope α(t). For example, for the case of QPSK modulation both are equal to ±1. The fading envelope is assumed to have its second moment equal to 1, i.e., E└α 2 (t)┘=1 where E denotes the expected value operator. This is in view of the fact that the mean channel gain is absorbed in the factor A 0 . For the special case of non-fading additive white Gaussian noise (AWGN) channels α(t)≡1. 
         [0037]    The additive white Gaussian noise n(t) in (1) can similarly be expressed in terms of I-Q representation as 
         [0000]        n ( t )= v   I ( t )cos(2π f   IF   t +θ( t ))− v   Q ( t )sin(2π f   IF   t +θ( t ))   (3) 
         [0000]    where the inphase and quadrature noise terms v I (t) and v Q (t) are statistically independent Gaussian processes each with two-sided power spectral density ratio equal to N 0 . 
         [0038]    Referring to  FIG. 1  the IF signal  5  is input to a complex mixer  10 . The complex mixer comprises of a pair of real mixers  6   a,    6   b  and a π/2 phase shift circuit  11 . The first input to the complex mixer is the IF signal r(t). The other input to the complex mixer is the v LO  signal provided by the NCO (numerically controlled oscillator)  30  given by 
         [0000]        v   LO ( t )=2 cos [2π f   IF   t+θ   LO ( t )]=2 cos └2π f   IF   t+θ   pn ( t )+{circumflex over (θ)} P ( t )┘  (4) 
         [0039]    In Eqn. (4), the NCO phase θ LO (t) is the sum of the oscillator phase noise θ pn (t) and the phase {circumflex over (θ)} P (t) due to the signal  26  at the NCO input obtained from the Filter/Fixed Lag Smoother  25 . The 2f IF  frequency terms generated in the complex mixers  6   a,    6   b  are filtered out by the following low pass filters and are therefore ignored. The mixer outputs are filtered by the square-root raised cosine (SRRC) filters  8   a,    8   b  which are the matched filters for the case when the data modulation involves band limiting by SRRC filtering. For the case of no band-limiting, the SRRC filter is replaced by an integrate and dump (I&amp;D) filter. The outputs of both the matched filters, which may be either SRRC or I&amp;D filters depending upon whether the modulated data is band limited or not band limited respectively, are sampled by the ADC (analog-to-digital converters)  9   a  and  9   b  with the sampling rate selected equal to the modulation symbol rate. The sampled signals  10   a  and  10   b  at the outputs of ADCs  9   a  and  9   b  and denoted by Y I (k) and Y Q (t) are input to the adaptive signal processing block  15 . The sampled matched filtered baseband signals may be expressed in terms of various parameters of interest by (5) and (6) below 
         [0000]        Y   I ( k )=α( k ) A   0   A   d ( k )cos └φ d ( k )+{tilde over (θ)}( k )┘+ v   i ( k )cos └{tilde over (θ)}( k )┘  (5) 
         [0000]        Y   Q ( k )=α( k ) A   0   A   d ( k )cos └φ d ( k )+{tilde over (θ)}( k )┘+ v   q ( k )cos └{tilde over (θ)}( k )┘  (6) 
         [0040]    In equations (5) and (6), k denotes the sample time index and {tilde over (θ)}(k)=θ(k)−{circumflex over (θ)} P (k) denotes the phase tracking error with θ(k)=θ i (k)−θ pn (k). In an alternative embodiment of the invention with an all digital implementation, the IF signal r(t) is sampled and converted into digital form and the NCO  30  is replaced by a digital oscillator, such that all the operations including those of the complex mixer and SRRC can be performed in digital domain. 
         [0041]    For the M-QAM modulation signal in the absence of fading, the ideal signal constellation coordinates of the signal points are given by {±(2k+1)A o ,±(2k+1)A o } for k=0, 1, . . . ,((K/2)−1) with K=√{square root over (M)} where A 0  denotes the unknown channel gain. In this case the signal constellation is primarily a function of the parameter A 0 . The normalized signal constellation is one for which A 0 =1.  FIG. 2  depicts the normalized signal constellation for the example 16-QAM signal. As part of the phase estimation procedure, the adaptive signal processor  15  detailed in  FIG. 3  estimates the channel gain parameter from the sampled matched filtered baseband signals Y I (k) and Y Q (k) or from the sampled complex baseband signal Y(k)≡Y I (k)+√{square root over (−1)}Y Q (k) by the Â 0  estimate block  111 . 
         [0042]    The average power of the sampled complex signal Y(k) is related to the parameter A 0  by equations (7a) and (7b). 
         [0000]    
       
         
           
             
               
                 
                   
                     
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         [0000]    The Â 0  estimate block  111  first estimates the A rms  from the sampled baseband signals Y I (k) and Y Q (k) by equation ( 8 ) 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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         [0000]    The estimate for A 0  is then obtained in the Â 0  estimate block  111  by equation (9) 
         [0000]        Â   0   =Â   rms /√{square root over (β)}  (9) 
         [0043]    In Eqn. (8) the last term under the square root sign represents the E[v I   2 (k)+v Q   2 (k)] with E denoting the expected value and the estimation window size N is selected such that the estimation error is relatively small when there is no fading. For the case of fading channels, Â rms   2  from (8) is given by Eqn. (10) 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
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         [0000]    where the noise term in (10) denotes all terms that are dependent upon the additive noise terms v I (k) and v Q (k). For N sufficiently large, the noise term will become relatively small and the average over α 2 (n) will approach E[α 2 (n)] which is equal to 1. Thus the estimate Â rms  will approach the value A 0 √{square root over (β)} or A rms  as desired. In an alternative embodiment of the invention the rectangular averaging window in (8) can be replaced by an exponentially data weighed window. For the case of fading channels, N is selected to be much higher compared to fading time constants to obtain the desired results. 
         [0044]    Referring to  FIG. 3  the sampled baseband signals Y I (k) and Y Q (k) are input to the 1/Â 0 (k) blocks  112   a  and  112   b.  The 1/Â 0  blocks  112   a  and  112   b  blocks are also input with the output of the Â 0 (k) estimator block  111  and generate the normalized sampled baseband signals y I (k) and y Q (k) at their outputs. The normalized sampled baseband signals y I (k) and y Q (k) are given by Eqns. (11) and (12). 
         [0000]    
       
         
           
             
               
                 
                   
                     
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                   12 
                   ) 
                 
               
             
           
         
       
     
         [0045]    The normalized sampled baseband signals y I (k) and y Q (k) are related to various parameters such as the channel gain A 0 , amplitude and phase modulation A d (k) and φ d (k), and the phase error {tilde over (θ)}(k) by Eqns. (13) and (14). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       y 
                       I 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         α 
                          
                         
                           ( 
                           k 
                           ) 
                         
                       
                        
                       
                         
                           A 
                           0 
                         
                         
                           
                             
                               A 
                               ^ 
                             
                             0 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                        
                       
                         
                           A 
                           d 
                         
                          
                         
                           ( 
                           k 
                           ) 
                         
                       
                        
                       
                         cos 
                          
                         
                           [ 
                           
                             
                               
                                 φ 
                                 d 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                             + 
                             
                               
                                 θ 
                                 ~ 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           
                             v 
                             I 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                          
                         
                           cos 
                            
                           
                             [ 
                             
                               
                                 θ 
                                 ~ 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                             ] 
                           
                         
                       
                       
                         
                           
                             A 
                             ^ 
                           
                           0 
                         
                          
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       y 
                       Q 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         α 
                          
                         
                           ( 
                           k 
                           ) 
                         
                       
                        
                       
                         
                           A 
                           0 
                         
                         
                           
                             
                               A 
                               ^ 
                             
                             0 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                        
                       
                         
                           A 
                           d 
                         
                          
                         
                           ( 
                           k 
                           ) 
                         
                       
                        
                       
                         sin 
                          
                         
                           [ 
                           
                             
                               
                                 φ 
                                 d 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                             + 
                             
                               
                                 θ 
                                 ~ 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           
                             v 
                             Q 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                          
                         
                           cos 
                            
                           
                             [ 
                             
                               
                                 θ 
                                 ~ 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                             ] 
                           
                         
                       
                       
                         
                           
                             A 
                             ^ 
                           
                           0 
                         
                          
                         
                           ( 
                           k 
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
         [0046]    If the phase estimation error {tilde over (θ)}(k) and the channel gain estimation error └A 0 −Â 0 (k)┘ both are relatively small, then both y I (k) and y Q (k) are respectively equal to α(k) times the inphase and quadrature modulation terms plus noise. The normalized sampled baseband signals y I (k) and y Q (k) are input to 1/{circumflex over (α)}(k) blocks  114   a  and  114   b.  The normalized sampled baseband signals y I (k) and y Q (k) are also input to the adaptive fade envelope estimator  115 . The adaptive fade envelope estimator  115  processes the normalized sampled baseband signals y I (k) and y Q (k) so as to obtain an instantaneous estimate {circumflex over (α)}(k) of the fade envelope α(k) at the output. The fade envelope estimate {circumflex over (α)}(k) is input to the 1/{circumflex over (α)}(k) blocks  114   a  and  114   b.  The 1/{circumflex over (α)}(k) blocks  114   a  and  114   b  normalize the normalized sampled baseband signals y I (k) and y Q (k) by the fade envelope estimate {circumflex over (α)}(k) to produce non fading normalized sampled baseband signals y In (k) and y Qn (k) at the output as per Eqns. (15) and (16): 
         [0000]        y   In ( k )= y   I ( k )/{circumflex over (α)}( k )   (15) 
         [0000]        y   Qn ( k )= y   Q ( k )/{circumflex over (α)}( k )   (16) 
         [0047]    Referring to  FIG. 4 , the adaptive fade envelope estimator  115  is input with the normalized sampled baseband signals y I (k) and y Q (k) and the parameter 2σ 2   v  with σ 2   v  denoting the variance of v I (k) and v Q (k) normalized by Â 0 . Inside the fade envelope estimator  115  a preliminary sample estimate {circumflex over (α)} S   2 (k) of the square of the fade envelope α 2 (k) is obtained first by blocks  150   a,    150   b,    180   a,  and  184  according to Eqns. (17) and (18). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         α 
                         ^ 
                       
                       S 
                       2 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       1 
                       
                         β 
                         2 
                       
                     
                      
                     
                       [ 
                       
                         
                           
                             y 
                             I 
                             2 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                         + 
                         
                           
                             y 
                             Q 
                             2 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                         - 
                         
                           2 
                            
                           
                             σ 
                             v 
                             2 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       σ 
                       v 
                       2 
                     
                     = 
                     
                       
                         β 
                         m 
                       
                        
                       
                         
                           ( 
                           
                             
                               2 
                                
                               
                                 E 
                                 b 
                               
                             
                             
                               N 
                               0 
                             
                           
                           ) 
                         
                         
                           - 
                           1 
                         
                       
                     
                   
                   ; 
                   
                     m 
                     = 
                     
                       
                         log 
                         2 
                       
                        
                       
                         ( 
                         M 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where (E b /N 0 ) denotes the received bit energy to noise power spectral density ratio in the absence of fading and m is the number of bits per QAM symbol. The estimate given by Eqn. (17) is an asymptotically unbiased estimate of the square of the fade envelope α 2 (k) in that with increasing time index k, the expected value of the estimate approaches α 2 (k). However, it is effected by noise. The preliminary sample estimate {circumflex over (α)} S   2 (k) is input to the Type II filter  165  in the adaptive fade envelope estimator  115 . The Type II filter  165  is comprised of an adder  180   b,  an accumulator  182  with transfer function [z/(z−1)] and a filter  181  with transfer function F(z). The output of the type II filter is the final estimate {circumflex over (α)} 2 (k) of the square of the fade envelope α 2 (k). 
         [0048]    The final estimate {circumflex over (α)} 2 (k) with reduced noise is related to the preliminary sample estimate {circumflex over (α)} S   2 (k) by the recursive Eqn. (19). 
         [0000]      {circumflex over (α)} 2 ( k )= F ( z )(1− z   −1 )[{circumflex over (α)} S   2 ( k )−{circumflex over (α)} 2 ( k− 1)]  (19) 
         [0049]    In the type II filter  165  and the Eqn. (19) z denotes the Z-transform and F(z) is the filtering operator on the prediction error [{circumflex over (α)} S   2 (k)−{circumflex over (α)} 2 (k−1)] and in the preferred embodiment is given by Eqns. (20)-(22). 
         [0000]    
       
         
           
             
               
                 
                   
                     F 
                      
                     
                       ( 
                       z 
                       ) 
                     
                   
                   = 
                   
                     
                       γ 
                        
                       
                           
                       
                        
                       d 
                     
                     + 
                     
                       
                         γ 
                          
                         
                             
                         
                          
                         
                           d 
                           2 
                         
                       
                       
                         1 
                         - 
                         
                           z 
                           
                             - 
                             1 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
             
               
                 
                   γ 
                   = 
                   
                     4 
                      
                     
                       ζ 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
             
               
                 
                   d 
                   = 
                   
                     
                       4 
                        
                       
                         B 
                         A 
                       
                        
                       
                         T 
                         s 
                       
                     
                     
                       γ 
                       + 
                       1 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
         [0050]    In Eqns. (20)-(22), T s  is equal to the sampling period which is equal to the modulation symbol period, ζ is the damping coefficient, and B A  is the desired loop bandwidth for the estimation of α(k). In an alternative embodiment of the invention, the type II filter  165  may be replaced by a first order filter with its output {circumflex over (α)} 2 (k) related to the preliminary sample estimate {circumflex over (α)} S   2 (k) by the recursive Eqn. (23). 
         [0000]      {circumflex over (α)} 2 ( k )=λ s {circumflex over (α)} 2 ( k− 1)+(1−λ s ){circumflex over (α)} S   2 ( k )   (23) 
         [0051]    In Eqn. (23) λ s  is a constant between 0 and 1 and determines the filter averaging period which is approximately equal to [1/(1−λ S )] samples. In general for relatively small fade envelope bandwidths it is preferred to use the first order filter whereas for the case of relatively large fade envelope bandwidths the use of type II filter is preferred. 
         [0052]    Referring to  FIG. 4 , the final fade amplitude estimate {circumflex over (α)}(k)  190  is the output of the square-root processor  185  and is related to the final estimate {circumflex over (α)} 2 (k) by Eq. (24): 
         [0000]      {circumflex over (α)}( k )=√{square root over ({circumflex over (α)} 2 ( k ))}  (24) 
         [0053]    Referring to  FIG. 1 , the non fading normalized sampled baseband signals y In (k) and y Qn (k) at the output of the adaptive signal processor  15  are input to the phase detector  20  of the adaptive phase tracking loop. The adaptive phase tracking loop is comprised of the phase detector  20 , the Kalman filter  25 , and the NCO  30  (numerically controlled oscillator) in addition to the complex mixer  10 , SRRC filters  8   a,    8   b,  ADCs  9   a,    9   b  and the adaptive signal processor  15 . Referring to  FIG. 5  depicting the block diagram of the phase detector, the non fading normalized sampled baseband signals y In (k) and y Qn (k) are input to slicers  221   a  and  221   b  respectively. For the example QAM modulation scheme, the slicer  221   a  compares the input y In (k) against (K−1) number of thresholds V T,j =2j; j=−(K h −1), . . . , −1, 0, 1, . . . (K h −1) with K h =K/2, and determines the inphase detected signal i d (k) at the output according to Eqn. 25. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       i 
                       d 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               0.5 
                                
                               
                                 ( 
                                 
                                   
                                     V 
                                     
                                       T 
                                       , 
                                       j 
                                     
                                   
                                   + 
                                   
                                     V 
                                     
                                       T 
                                       , 
                                       
                                         j 
                                         + 
                                         1 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             ; 
                             
                               
                                 V 
                                 
                                   T 
                                   , 
                                   j 
                                 
                               
                               ≤ 
                               
                                 
                                   y 
                                   
                                     I 
                                     , 
                                     n 
                                   
                                 
                                  
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               &lt; 
                               
                                 V 
                                 
                                   T 
                                   , 
                                   
                                     j 
                                     + 
                                     1 
                                   
                                 
                               
                             
                             ; 
                           
                         
                         
                           
                             j 
                             = 
                             
                               - 
                               
                                 ( 
                                 
                                   
                                     K 
                                     h 
                                   
                                   , 
                                   … 
                                    
                                   
                                       
                                   
                                   , 
                                   
                                     - 
                                     1 
                                   
                                   , 
                                   0 
                                   , 
                                   1 
                                   , 
                                   … 
                                    
                                   
                                       
                                   
                                   , 
                                   
                                     K 
                                     h 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               - 
                               
                                 ( 
                                 
                                   K 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                             ; 
                           
                         
                         
                           
                             
                               
                                 y 
                                 
                                   I 
                                   , 
                                   n 
                                 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                             &lt; 
                             
                               V 
                               
                                 T 
                                 , 
                                 
                                   - 
                                   
                                     ( 
                                     
                                       
                                         K 
                                         h 
                                       
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 K 
                                 - 
                                 1 
                               
                               ) 
                             
                             ; 
                           
                         
                         
                           
                             
                               V 
                               
                                 T 
                                 , 
                                 
                                   ( 
                                   
                                     
                                       K 
                                       h 
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             ≤ 
                             
                               
                                 y 
                                 
                                   I 
                                   , 
                                   n 
                                 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
         [0054]    The operation of slicer  221   b  is similar to that of  221   a  and compares the input y Qn (k) against (K−1) number of thresholds V T,j =2j ; j=−(K h −1), . . . , −1, 0, 1, . . . (K h −1) with K h =K/2, and determines the quadrature detected signal q d (k) at the output according to Eqn. 25. The inphase and quadrature detected signals i d (k) and q d (k) are combined into the complex detected symbol s d (k) in the data combiner  35 . The data combiner  35  provides the detected symbol s d (k) to the detected symbol output  40 . 
         [0055]    Referring to  FIG. 5 , the normalized sampled baseband signal y I (k) is input to multipliers  223   a  and  223   c.  The normalized sampled baseband signals y Q (k) is input to multipliers  223   b  and  223   d.  The second input of multipliers  223   a  and  223   d  is q d (k) while the multipliers  223   b  and  223   c  have i d (k) as their second input. The output of multiplier  223   a  is subtracted from the output of multiplier  223   b  in the adder  224   a  to generate the first error signal Z I (k) at the output of the adder  224   a.  Similarly the output of multiplier  223   c  is added to the output of multiplier  223   d  in the adder  224   b  to generate the second error signal Z Q (k) at the output of the adder  224   b.  The error signals Z I (k) and Z Q (k) are related to the normalized sampled baseband signals y I (k) and y Q (k) and the detected signals i d (k) and q d (k) by Eqns. (26) and (27). 
         [0000]        Z   I ( k )= y   Q ( k ) i   d ( k )− y   I ( k ) q   d ( k )   (26) 
         [0000]        Z   Q ( k )= y   I ( k ) i   d ( k )+ y   Q ( k ) q   d ( k )   (27) 
         [0056]    In the absence of errors made in the generation of the detected signals i d (k) and i q (k), the error signals Z I (k) and Z Q (k) are related to the phase tracking error {tilde over (θ)}(k)=θ(k)−{circumflex over (θ)}(k) by equations (28) and (29). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Z 
                       I 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             α 
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                            
                           
                             A 
                             0 
                           
                            
                           
                             
                               A 
                               d 
                               2 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         
                           
                             
                               A 
                               ^ 
                             
                             0 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                        
                       
                         sin 
                          
                         
                           [ 
                           
                             
                               θ 
                               ~ 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           v 
                           ~ 
                         
                         I 
                       
                        
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       Z 
                       Q 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             α 
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                            
                           
                             A 
                             0 
                           
                            
                           
                             
                               A 
                               d 
                               2 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         
                           
                             
                               A 
                               ^ 
                             
                             0 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                        
                       
                         cos 
                          
                         
                           [ 
                           
                             
                               θ 
                               ~ 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           v 
                           ~ 
                         
                         Q 
                       
                        
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
         [0057]    In Eqns. (28) and (29) {tilde over (v)} I (k) and {tilde over (v)} Q (k) denoting the noise dependent terms. In the absence of errors made in the generation of the detected signals i d (k) and i q (k) and in the absence of fading, the noise terms {tilde over (v)} I (k) and {tilde over (v)} Q (k) are statistically independent and have variance approximately equal to σ {tilde over (v)}     1     2  which is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     σ 
                     
                       
                         v 
                         ~ 
                       
                       I 
                     
                     2 
                   
                   = 
                   
                     
                       
                         β 
                         2 
                       
                       m 
                     
                      
                     
                       
                         ( 
                         
                           
                             2 
                              
                             
                               E 
                               b 
                             
                           
                           
                             N 
                             0 
                           
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
         [0058]    The error signals Z I (k) and Z Q (k) are input to the inverse tangent block  226  which computes the prediction error signal η(k) as the four quadrant inverse tangent function from the inphase and quadrature error signals Z I (k) and Z Q (k). For relatively small phase error the prediction error η(k) has an approximate representation given by Eqn. (31) wherein {tilde over (v)}(k) is the noise term. 
         [0000]      η( k )=tan({tilde over (θ)}( k ))+{tilde over (v)}( k )≅{tilde over (θ)}( k )+{tilde over (v)}( k )   (31) 
         [0000]    The variance of {tilde over (v)}(k) is approximately equal to R given by (32): 
         [0000]    
       
         
           
             
               
                 
                   R 
                   = 
                   
                     
                       1 
                       m 
                     
                      
                     
                       
                         ( 
                         
                           
                             2 
                              
                             
                               E 
                               b 
                             
                           
                           
                             N 
                             0 
                           
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   32 
                   ) 
                 
               
             
           
         
       
     
         [0059]    In order to derive the estimate of the phase noise process θ(k) which may arise from the NCO or induced by the fading communication channel or both, the phase noise process θ(k) is modeled in terms of a state space model. A Kalman filter and a fixed lag smoother are used in the invention to provide two different estimates of θ(k). For this purpose the phase noise process is modeled by the second-order state space model described by Eqns. (33)-(37). 
         [0000]      θ( k+ 1)= H   T ( k+ 1) x ( k+ 1)   (33) 
         [0000]        x ( k+ 1)=Φ x ( k )+ w ( k )   (34) 
         [0000]    where T denotes the matrix transpose, 
         [0000]        H   T ( k+ 1)= a ( k+ 1)[1 0]  (35a) 
         [0000]    and for any integers k and j 
         [0000]    
       
         
           
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         w 
                          
                         
                           ( 
                           k 
                           ) 
                         
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   
                     35 
                      
                     b 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         
                           w 
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                          
                         
                           
                             v 
                             ~ 
                           
                            
                           
                             ( 
                             j 
                             ) 
                           
                         
                       
                       ] 
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   
                     35 
                      
                     c 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         
                           w 
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                          
                         
                           
                             w 
                             T 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       ] 
                     
                   
                   = 
                   Q 
                 
               
               
                 
                   ( 
                   
                     35 
                      
                     d 
                   
                   ) 
                 
               
             
             
               
                 
                   Q 
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 T 
                                 s 
                                 2 
                               
                               / 
                               3 
                             
                           
                           
                             
                               
                                 T 
                                 s 
                               
                               / 
                               2 
                             
                           
                         
                         
                           
                             
                               
                                 T 
                                 s 
                               
                               / 
                               2 
                             
                           
                           
                             1 
                           
                         
                       
                       ] 
                     
                      
                     
                       σ 
                       a 
                       2 
                     
                      
                     
                       T 
                       s 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   
                     35 
                      
                     e 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     E 
                      
                     
                       [ 
                       
                         
                           
                             v 
                             ~ 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                          
                         
                           
                             
                               v 
                               ~ 
                             
                             T 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                       
                       ] 
                     
                   
                   = 
                   R 
                 
               
               
                 
                   ( 
                   
                     35 
                      
                     f 
                   
                   ) 
                 
               
             
           
         
       
     
         [0000]    The state transition matrix Φ in (33) is given by 
         [0000]    
       
         
           
             
               
                 
                   Φ 
                   = 
                   
                     [ 
                     
                       
                         
                           1 
                         
                         
                           
                             T 
                             S 
                           
                         
                       
                       
                         
                           0 
                         
                         
                           1 
                         
                       
                     
                     ] 
                   
                 
               
               
                 
                   ( 
                   36 
                   ) 
                 
               
             
           
         
       
     
         [0060]    In Eqn. (35) T s  is the sampling period. The noise term {tilde over (v)}(k) in Eqn. (35) is the noise appearing in the prediction error η(k) given by (31). It has been shown above that in the absence of fading the variance R of {tilde over (v)}(k) is approximately equal to (2mE b /N 0 ) −1 . The ratio (σ a   2 /σ v   2 ) determines the effective loop bandwidth of the carrier tracking loop. In the first preferred embodiment, a(k+1) in (35a) is set equal to 1 for all integers k. In various possible modifications of the invention, state space models of order higher than 2 may be used and such modifications remain within the scope of this invention. In the state space model described by Eqns. (33)-(36), the state vector x(k) is of dimension 2 and has the phase θ(k) and the derivative of phase at time instance k as its components. The Kalman filter state and covariance matrix update equations for k=0, 1, . . . are given by Eqns. (37a)-(37g). The update equations (37) are initialized with some appropriate initial estimates {circumflex over (x)}(0/0) and P(0/0). 
         [0000]        {circumflex over (x)} ( k/k )= {circumflex over (x)} ( k/k− 1)+ K   0 ( k )η( k )   (37a) 
         [0000]        {circumflex over (x)} ( k/k− 1)=Φ {circumflex over (x)} ( k− 1/ k− 1)   (37b) 
         [0000]        P ( k/k− 1)=Φ P ( k− 1/ k− 1)Φ+ Q    (37c) 
         [0000]        S ( k )= H   T ( k ) P ( k/k− 1) H ( k )+ R    (37d) 
         [0000]        P ( k/k )= P ( k/k− 1)− P ( k/k− 1) H ( k ) S   −1 ( k ) H   T ( k ) P ( k/k− 1)   (37e) 
         [0000]        K   0 ( k )= P ( k/k− 1) H ( k ) S   −1 ( k )   (37f) 
         [0061]    In Eqn. (37) {circumflex over (x)}(k−1/k−1) and {circumflex over (x)}(k/k−1) respectively denote the filtered and predicted state estimates, P(k−1/k−1) and P(k/k−1) are the respective error covariance matrices corresponding to these two state estimates, and K 0 (k) denotes the Kalman gain vector. Referring to  FIG. 6  the Kalman filter/smoother gain processor  320  generates the Kalman gain vector K 0 (k)  330   a,  the prediction error is multiplied by the Kalman gain vector K 0 (k) in  327   a  to provide a correction to the predicted state {circumflex over (x)}(k/k−1) in the adder  331   a  to generate the filtered state {circumflex over (x)}(k/k) at the output of the adder  331   a.  The filtered state {circumflex over (x)}(k/k) delayed by the delay  329   a  and multiplied by the state transition matrix Φ in block  328  provides the predicted state {circumflex over (x)}(k/k−1) to the adder  331   a  for generation of the filtered state {circumflex over (x)}(k/k). 
         [0062]    The Kalman filter described by Eqns. (37) may also be replaced by an exponentially weighted Kalman filter. The predicted phase estimate {circumflex over (θ)} P (k) which is the estimate of the phase θ(k) derived from the sampled matched filter output signals Y I (j) and Y Q (j) for j=0, 1, . . . , k−1 is given in terms of the predicted state estimate {circumflex over (x)}(k+1/k) by Eqn. (38): 
         [0000]      {circumflex over (θ)} P ( k )={circumflex over (θ)}( k/k− 1)= H   T ( k ) {circumflex over (x)} ( k/k− 1)   (38) 
         [0063]    The predicted phase estimate {circumflex over (θ)} P (k) as computed in the Kalman filter/Smoother  25  is input to NCO  30 . The NCO  30  transfers the phase to its output  32  such that the phase {circumflex over (θ)} P (t) in Eqn. (4) at time t=kT s  is equal to {circumflex over (θ)} P (k). The NCO output  32  is the continuous time signal v LO (t) given by Eqn. (4). Referring to  FIG. 1 , the NCO output  32  is input to the complex mixer  10  for the generation of mixer output signals  7   a  and  7   b.    
         [0064]    In order to achieve a smaller phase tracking error than possible with the Kalman filter, the invention includes a fixed lag smoother. The fixed lag smoother further reduces the phase tracking error by basing the estimate of phase θ(k) on not only the sampled matched filter output signals Y I (j) and Y Q (j) for j=0, 1, . . . ,k, but also on some future sampled matched filter output signals Y I (j) and Y Q (j) for j=k+1, k+2, . . . , k+L for some positive integer L known as smoother lag for any time k. The fixed lag smoother involves a delay of L in the estimate of the phase. The fixed lag smoother operates in conjunction with the Kalman filter and is described by Eqns. (37) and (39) for i=1, 2, . . . L and k=1, 2, . . . . 
         [0000]        {circumflex over (x)}   i ( k/k )= {circumflex over (x)}   i−1 ( k− 1 /k− 1)+ K   i ( k )η( k )   (39a) 
         [0000]        K   i ( k )= P   i0 ( k/k− 1) H ( k ) S   −1 ( k )   (39b) 
         [0000]        P   i0 ( k/k )= P   i0 ( k/k− 1)− P   i0 ( k/k− 1) H ( k ) S   −1 ( k ) H   T ( k ) P   00 ( k/k− 1)   (39c) 
         [0000]        P   ii ( k/k )= P   ii ( k/k− 1)− P   i0 ( k/k− 1) H ( k ) S   −1 ( k ) H   T ( k ) P   i0   T ( k/k− 1)   (39d) 
         [0000]        P   ii ( k/k− 1)= P   i−1,i−1 ( k− 1/ k− 1)   (39e) 
         [0065]    In Eqn. (39a) {circumflex over (x)} i (k/k) denotes the estimate of x(k) with a lag i and is thus equal to {circumflex over (x)}(k−i/k) for i=1, 2, . . . , L with a similar definition for {circumflex over (x)} i (k/k−1). Also, in Eqn. (39a) {circumflex over (x)} 0 (k/k) is equal to the Kalman filter estimate obtained from Eqn. (37a). In Eqns. (39) P ii (k) denotes the error covariance matrix associated with the estimate {circumflex over (x)} i (k/k) with P 00 (k) equal to the filter error covariance matrix P obtained from Eqn. (37e). In Eqn. (39b), K i (k) denotes the smoother gain vectors for i=1, 2, . . . , L. Referring to  FIG. 6 , the Kalman filter/smoother gain processor generates the smoother gain vectors K i (k) for i=1, 2, . . . , L from Eqns. (39b)-(39e). The fixed delay smoother state equation (39a) is implemented by smoother gain multipliers  327   b, c, . . . , g,  smoother vector summers  331   b, c, . . . , g,  and the smoother delays  329   b, c, . . . , g− 1 where g is equal to smoother fixed delay L. The signals {circumflex over (x)}(k−i/k) at the output of the smoother vectors summers  331   b, c, . . . , g  are the smoothed state estimates with delay 1, 2, . . . , L with the output of the smoother vectors summer  331   g  equal to the final smoothed state estimate. 
         [0066]    The smoothed phase estimate {circumflex over (θ)} S (k−L) at the output of Kalman Filter/Fixed lag smoother  25  is given by Eqn. (40): 
         [0000]      {circumflex over (θ)} S ( k−L )= H   T ( k−L ) {circumflex over (x)}   L ( k/k )= H   T ( k−L ) {circumflex over (x)} ( k−L/k )   (40) 
         [0000]    where the smoothed state estimate {circumflex over (x)} L (k/k) is given by recursions (39a). The vector multiplier  333  in  FIG. 6  implements Eqn. (40) generating the smoothed phase estimate {circumflex over (θ)} S (k−L) at the output. 
         [0067]    The smoothed phase estimate {circumflex over (θ)} S (k) is used to obtain more accurate estimate of the real and imaginary parts i dS (k) and q dS (k) of the symbol s(k) by the smoothed symbol detector  45 . Referring to  FIG. 1 , the smoothed phase estimate {circumflex over (θ)} S (k−L), the predicted phase estimate with delay {circumflex over (θ)} P (k−L), and the delayed non fading normalized sampled baseband signals y In (k−L) and y Qn (k−L) are input to smoothed symbol detector  45  from the Kalman Filter/Fixed Lag Smoother  25 . The smoothed symbol detector  45  outputs the smoothed detected symbol s d,S (k) to the smoothed detected symbol output  50 . Referring to  FIG. 7 , the smoothed phase estimate {circumflex over (θ)} S (k−L) and the delayed predicted phase estimate {circumflex over (θ)} P (k−L) are input to adder  401   a  to generate the phase difference signal θ e (k−L) at the output. The phase difference signal θ e (k) is input to a cosine function block  402   a  and a sine function block  403 . The output of the cosine function block  402   a  is input to the multipliers  404   a  and  404   d.  The output of the sine function block  403  is input to the multipliers  404   b  and  404   c.  The other input to the multipliers  404   a  and  404   c  is the delayed non fading normalized sampled baseband inphase signal y In (k−L). The other input to the multipliers  404   b  and  404   d  is the delayed nonfading normalized sampled baseband quadrature signal y Qn (k−L). The output of multiplier  404   b  is subtracted from the output of multiplier  404   a  to provide the smoothed inphase signal y IS (k−L) given by Eqn. 41. Similarly the outputs of multiplier  404   c  is added to the output of multiplier  404   d  to provide the smoothed quadrature signal y QS (k−L) given by Eqn. 42. 
         [0000]        y   IS ( k−L )= y   In ( k−L )cos(θ e ( k−L ))− y   Qn ( k−L )sin(θ e ( k−L ))   (41) 
         [0000]        y   QS ( k−L )= y   In ( k−L )sin(θ e ( k−L ))+ y   Qn ( k−L )cos(θ e ( k−L ))   (42) 
         [0068]    The smoothed inphase signal y IS (k−L) is input to a slicer  406   a  similar to the slicers  221   a  and  221   b  and described by Eqn. 25 to provide smoothed detected real component i d,S (k−L) at the output of the slicer  406   a.  Similarly the smoothed quadrature signal y QS (k−L) is input to a slicer  406   b  to provide smoothed detected imaginary component q d,S (k−L) at the output of  406   b.  The smoothed inphase and quadrature components y IS (k−L) and y QS (k−L) are input to smoothed data combiner  407 , which combines the smoothed inphase and quadrature components into the smoothed detected symbol s dS (k−L). Referring to  FIG. 1 , the smoothed detected symbol s dS (k−L) is outputted to the smoothed detected symbol output block  50 . 
         [0069]    In an alternative embodiment of this invention, the phase detector  20  in  FIG. 1  is replaced by phase detector II shown in  FIG. 8 . Referring to  FIG. 8 , the non fading normalized sampled baseband signals y In (k) and y Qn (k) are input to slicers  621   a  and  621   b  respectively. The slicer  621   a  compares the input y In (k) against (K−1) number of thresholds V T,j =2j; j=−(K h −1), . . . , −1, 0, 1, . . . (K h −1) with K h =K/2, and determines the inphase detected signal i d (k) at the output according to Eqn. 43. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       i 
                       d 
                     
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     { 
                     
                       
                         
                           
                             
                               0.5 
                                
                               
                                 ( 
                                 
                                   
                                     V 
                                     
                                       T 
                                       , 
                                       j 
                                     
                                   
                                   + 
                                   
                                     V 
                                     
                                       T 
                                       , 
                                       
                                         j 
                                         + 
                                         1 
                                       
                                     
                                   
                                 
                                 ) 
                               
                             
                             ; 
                             
                               
                                 V 
                                 
                                   T 
                                   , 
                                   j 
                                 
                               
                               ≤ 
                               
                                 
                                   y 
                                   
                                     I 
                                     , 
                                     n 
                                   
                                 
                                  
                                 
                                   ( 
                                   k 
                                   ) 
                                 
                               
                               &lt; 
                               
                                 V 
                                 
                                   T 
                                   , 
                                   
                                     j 
                                     + 
                                     1 
                                   
                                 
                               
                             
                             ; 
                           
                         
                         
                           
                             j 
                             = 
                             
                               - 
                               
                                 ( 
                                 
                                   
                                     K 
                                     h 
                                   
                                   , 
                                   … 
                                    
                                   
                                       
                                   
                                   , 
                                   
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                                   , 
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                                   , 
                                   
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                                     h 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               - 
                               
                                 ( 
                                 
                                   K 
                                   - 
                                   1 
                                 
                                 ) 
                               
                             
                             ; 
                           
                         
                         
                           
                             
                               
                                 y 
                                 
                                   I 
                                   , 
                                   n 
                                 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                             &lt; 
                             
                               V 
                               
                                 T 
                                 , 
                                 
                                   - 
                                   
                                     ( 
                                     
                                       
                                         K 
                                         h 
                                       
                                       - 
                                       1 
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               ( 
                               
                                 K 
                                 - 
                                 1 
                               
                               ) 
                             
                             ; 
                           
                         
                         
                           
                             
                               V 
                               
                                 T 
                                 , 
                                 
                                   ( 
                                   
                                     
                                       K 
                                       h 
                                     
                                     - 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                             ≤ 
                             
                               
                                 y 
                                 
                                   I 
                                   , 
                                   n 
                                 
                               
                                
                               
                                 ( 
                                 k 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   43 
                   ) 
                 
               
             
           
         
       
     
         [0070]    The operation of slicer  621   b  is similar to that of  621   a  and compares the input y Qn (k) against (K−1) number of thresholds V T,j =2j; j=−(K h −1), . . . , −1, 0, 1, . . . (K h −1) with K h =K/2, and determines the quadrature detected signal q d (k) at the output according to Eqn. 43. 
         [0071]    Referring to  FIG. 8  the normalized sampled baseband signal y I (k) is input to multiplier  623   a  and the non fading normalized sampled baseband signal y Q (k) is input to multiplier  623   b.  The second inputs of multipliers  623   a  and  623   b  are q d (k) and i d (k), respectively. The output of multiplier  623   a  is subtracted from the output of multiplier  623   b  to generate the error signal Z I (k) at the output of the adder  624 . The inphase and quadrature detected signals i d (k) and q d (k) are input to the squaring blocks  630   a  and  630   b  respectively. The outputs of the squaring blocks  630   a  and  630   b  are added by the adder  635  to generate the estimate Â d   2  of the square of the data amplitude A d  at the output of the adder  635 . The estimate Â d   2  is input to the divider  640  with the other input of the divider  640  connected to the output Z I (k) of the adder  624 . The output of the divider  640  is the prediction error η(k) equal to Z I (k)/Â d   2 . The error signal Z I (k) is related to the non fading normalized sampled baseband signals y I (k) and y Q (k) and the detected signals i d (k) and q d (k) by Eqn. (26), and the prediction error η(k) is given by 
         [0000]    
       
         
           
             
               
                 
                   
                     η 
                      
                     
                       ( 
                       k 
                       ) 
                     
                   
                   = 
                   
                     
                       
                         
                           
                             α 
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                            
                           
                             A 
                             0 
                           
                            
                           
                             
                               A 
                               d 
                               2 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                         
                           
                             
                               
                                 A 
                                 ^ 
                               
                               0 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
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                                 A 
                                 ^ 
                               
                               d 
                               2 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                         
                       
                        
                       
                         sin 
                          
                         
                           [ 
                           
                             
                               θ 
                               ~ 
                             
                              
                             
                               ( 
                               k 
                               ) 
                             
                           
                           ] 
                         
                       
                     
                     + 
                     
                       
                         
                           v 
                           ~ 
                         
                         In 
                       
                        
                       
                         ( 
                         k 
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   44 
                   ) 
                 
               
             
           
         
       
     
         [0000]    The variance of the noise {tilde over (v)} In (k) is approximately equal to σ {tilde over (v)}     In     2  given by 
         [0000]    
       
         
           
             
               
                 
                   
                     σ 
                     
                       
                         v 
                         ~ 
                       
                       In 
                     
                     2 
                   
                   = 
                   
                     
                       1 
                       m 
                     
                      
                     
                       
                         ( 
                         
                           
                             2 
                              
                             
                               E 
                               b 
                             
                           
                           
                             N 
                             0 
                           
                         
                         ) 
                       
                       
                         - 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   45 
                   ) 
                 
               
             
           
         
       
     
         [0072]    For relatively small phase error the prediction error η(k) in (44) has an approximate representation given by Eqn. (46) 
         [0000]      η( k )=α( k )sin [{tilde over (θ)}( k )]+ {tilde over (v)}   In ( k )   (46) 
         [0073]    In this alternative embodiment of the invention, the smoother is an adaptive smoother described by Eqns. (37)-(39) with a(k+1) made equal to the fade envelope {circumflex over (α)}(k+1) in the definition of the vector H(k+1) in (35a). Both the embodiments of  FIGS. 1 and 8  can also be applied to an MPSK modulated signal with the slicers  621   a  and  621   b  appropriately modified for the MPSK signal. Such a modification can be made by a person knowledgeable in the art of the field of this invention. 
         [0074]    In a third alternative embodiment of this invention, the phase detector  20  in  FIG. 1  is replaced by a more general phase detector III  70  shown in  FIG. 9 . Referring to  FIG. 9 , the non fading normalized sampled baseband signals y In (k) and y Qn (k) are input to the complex symbol detector  710 . The complex valued output c d (k) of the complex symbol detector  710  is input to Re( ) block  720  to generate the real part i d (k) of c d (k) at the output of  720 . The Im( ) block  725  provides the imaginary part q d (k) of c d (k) at the output. 
         [0075]    Referring to  FIG. 9  the normalized sampled baseband signal y I (k) is input to multiplier  740   a  and the normalized sampled baseband signal y Q (k) is input to multiplier  740   b.  The second inputs of multipliers  740   a  and  740   b  are q d (k) and i d (k) respectively. The output of multiplier  740   a  is subtracted from the output of multiplier  740   b  to generate the error signal Z I (k) at the output of the adder  745 . The inphase and quadrature detected signals i d (k) and q d (k) are input to the squaring blocks  730   a  and  730   b  respectively. The outputs of the squaring blocks  730   a  and  730   b  are added by the adder  735  to generate the estimate Â d   2  of the square of the data amplitude A d  at the output of the adder  735 . The estimate Â d   2  is input to the divider  750  with the other input of the divider  750  connected to the output Z I (k) of the adder  745 . The output of the divider  750  is the prediction error Θ(k) equal to Z I (k)/Â d   2 . The prediction error η(k) is input to the Kalman Filter/Fixed Lag Smoother  25 . In a third alternative embodiment of the invention, the smoother is an adaptive smoother described by Eqns. (37)-(39) with a(k+1) made equal to the fade envelope {circumflex over (α)}(k+1) in the definition of the vector H(k+1) in (35a) as is true for the second alternative embodiment. The inphase and quadrature detected signals i d (k) and q d (k) at the output of phase detector  70  are input to the data combiner  35  of  FIG. 1 . 
         [0076]    For the example application of the third alternative embodiment of the invention to the MPSK signal, the complex symbol detector  710  is symbol detector for MPSK signal. Referring to  FIG. 10 , the complex detector for MPSK signal is comprised of inverse tangent block  720 , phase slicer  725 , and mapper  730 . For MPSK signals, the symbol phase φ d (k) in Eqn. (5) and (6) has possible values (2j+1)π/M; j=0, 1, . . . , (M−1) where M is the number of phases of the MPSK signal.  FIG. 11  shows the normalized signal constellation of the MPSK signal for the example M=8 case also depicting the indices of the M symbols. 
         [0077]    The output {circumflex over (φ)} d (k) of the inverse tangent block  720  is the estimate of φ d (k). The phase slicer  725  compares {circumflex over (φ)} d (k) with a set of thresholds equal to 2πj/M, j=0, 1, . . . , (M−1) and outputs the symbol index i s (k) according to Eqn. (47). 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                           i 
                           s 
                         
                          
                         
                           ( 
                           k 
                           ) 
                         
                       
                       = 
                       j 
                     
                     ; 
                     
                       
                         
                           
                             
                               2 
                                
                               π 
                             
                             M 
                           
                            
                           j 
                         
                         ≤ 
                         
                           
                             
                               φ 
                               ^ 
                             
                             d 
                           
                            
                           
                             ( 
                             k 
                             ) 
                           
                         
                         &lt; 
                         
                           
                             
                               2 
                                
                               π 
                             
                             M 
                           
                            
                           
                             ( 
                             
                               j 
                               + 
                               1 
                             
                             ) 
                           
                            
                           
                               
                           
                            
                           for 
                            
                           
                               
                           
                            
                           j 
                         
                       
                       = 
                       0 
                     
                   
                   , 
                   1 
                   , 
                   … 
                    
                   
                       
                   
                   , 
                   
                     ( 
                     
                       M 
                       - 
                       1 
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   47 
                   ) 
                 
               
             
           
         
       
     
         [0078]    The mapper  730  outputs the symbol c d (k) corresponding to the index i s (k) using the normalized signal constellation. As an example of the mapper,  FIG. 12A  shows implementation of the mapper  730  for M=8. Referring to  FIG. 12A  the symbol index i s (k) is represented in terms of BCD (binary coded decimal) form b 1 b 2 b 3 . The binary variables b 1 , b 2 , and b 3  take values 0 or 1 and are input to logic circuit blocks Logic 1 and Logic 2 which generate logic variables L 1 , . . . , L 8  at their outputs as shown in  FIG. 12A . From the normalized signal constellation in  FIG. 11  one obtains the following table 1 depicting the mapping required between i s (k) and c d (k). In the table δ 1 =cos(π/8) and δ 2 =sin(π/8) and j=√{square root over (−1)}. 
         [0000]    
       
         
               
             
               
               
               
               
               
             
           
               
                 TABLE 1 
               
             
             
               
                   
               
               
                 Mapping from symbol index to symbol 
               
             
          
           
               
                 i s   
                 b 1   
                 b 2   
                 b 3   
                 c d (k) 
               
               
                   
               
               
                 0 
                 0 
                 0 
                 0 
                   δ 1  + jδ 2   
               
               
                 1 
                 0 
                 0 
                 1 
                   δ 2  + jδ 1   
               
               
                 2 
                 0 
                 1 
                 0 
                 −δ 2  + jδ 1   
               
               
                 3 
                 0 
                 1 
                 1 
                 −δ 1  + jδ 2   
               
               
                 4 
                 1 
                 0 
                 0 
                 −δ 1  − jδ 2   
               
               
                 5 
                 1 
                 0 
                 1 
                 −δ 2  − jδ 1   
               
               
                 6 
                 1 
                 1 
                 0 
                   δ 2  − jδ 1   
               
               
                 7 
                 1 
                 1 
                 1 
                   δ 1  − jδ 2   
               
               
                   
               
             
          
         
       
     
         [0079]    Denoting by L 1 , L 2 , L 3  and L 4  the logic variables which take value 1 only when the real part of c d (k) takes values δ 1 , −δ 1 , δ 2 , and −δ 2  respectively, otherwise they take value 0. From Table 1, these logic variables have the following Boolean expressions. 
         [0000]        L   1   =b   1   b   2   b   3   +  b     1     b     2     b     3    (48a) 
         [0000]        L   2   =  b     1   b   2   b   3   +b   1     b     2     b     3    (48b) 
         [0000]        L   3   =  b     1     b     2   b   3   +b   1   b   2   b   3    (48c) 
         [0000]        L   4   =  b     1   b   2     b     3   +b   1     b     2     b     3    (48d) 
         [0080]    In Eqns.(48)  b   i  denotes logic inverse of b i  for i between 1 and 4. From Table 1, similar expressions can be derived for the logic variables L 5  to L 8 . Referring to  FIG. 12A , the logic variables L 1 , L 2 , L 3  and L 4  are input to a multiplexer MUX 1  734   a  which is also input with constants δ 1 , −δ 1 , δ 2 , and −δ 2 . The output of MUX 1  734   a  i d (k) is equal to δ 1 , −δ 1 , δ 2 , or −δ 2  depending upon which of the 4 logic variables L 1 , L 2 , L 3  and L 4  is equal to 1 in that order. The operation of MUX 2  734   b  is very similar to that of MUX 1. The inputs of MUX 2 are the logic variables L 5 , L 6 , L 7  and L 8  with the output q d (k) equal to δ 1 , −δ 1 , δ 2 , or −δ 2  depending upon which of the 4 logic variables L 5 , L 6 , L 7  and L 8  is equal to 1 in that order. The outputs of MUX 1  734   a  and MUX 1  734   b  are input to data combiner block  735  which generates the detected symbol c d (k) at the output. Referring to  FIG. 12B  the constants δ 1 , −δ 1 , δ 2 , and −δ 2  are stored in their signed binary forms in shift registers  755   a  to  755   d.  The contents of shift register  755   a  are input to AND gates  760   a  through  760   m.  The other input of the AND gates  760   a  through  760   m  is L 1 . The outputs of the AND gates  760   a  through  760   m  are input to the four input OR gates  780   a  through  780   m  respectively. The other inputs of the four input OR gates  780   a  through  780   m  are generated as a result of logic signals L 2 , L 3  and L 4  gating the shift registers containing −δ 1 , δ 2 , and −δ 2  respectively in their binary forms. The outputs of the four input OR gates  780   a  through  780   m  are input to shift register  785  which holds the real part i d (k) of the detected symbol c d (k). The MUX 2  734   b  in the likewise manner generates the imaginary part q d (k) of the detected symbol c d (k) based on the logic variables L 5 , L 6 , L 7  and L 8 . The advantage of the implementation of  FIG. 9  is that it does not involve any mathematical computations to obtain c d (k) from {circumflex over (φ)} d (k). Alternatively i d (k) and q d (k) can be obtained as cos({circumflex over (φ)} d (k)) and sin({circumflex over (φ)} d (k)) which requires computation of trigonometric functions. 
         [0081]    It will be understood that the embodiments described herein are merely exemplary and that a person skilled in the art may make many variations and modifications without departing from the spirit and scope of this invention. For example, a square grid for the normalized signal constellation is disclosed as an example. With appropriate changes to the slicers in the phase detector, the invention applies to rectangular and hexagonal grids as well as non-uniform grids. Another variation is the use of higher than type II filter in the adaptive fade envelope estimator which may be appropriate for channels experiencing higher order fade dynamics. Yet another variation is the use of a smoothed fade envelope estimate in the fixed lag smoother. All such variations and modifications are intended to be included within the scope of the invention as defined in the appended claims.