Patent Publication Number: US-7715505-B2

Title: Adaptive synchronization enhancement technique for serial modulated waveforms

Description:
CROSS REFERENCE TO RELATED APPLICATIONS 
     This application claims priority of U.S. Provisional Patent Application Ser. No. 60/703,095, filed Jul. 28, 2005. 
    
    
     STATEMENT REGARDING FEDERALLY SPONSORED RESEARCH 
     This invention was made with Government Support Under Agreement No. DAAB07-03-9-K601 awarded by the United States Army. The Government has certain rights in the invention. 
    
    
     FIELD OF THE INVENTION 
     The present invention relates to the field of communication systems. More specifically, the present invention relates to methods and apparatus for synchronization detection of a serial offset quadrature pulse shaped (such as phase shift keyed (PSK)) signal in a receiver. 
     SUMMARY OF THE INVENTION 
     The present invention is embodied in a method for synchronization (SYNC) detection of a received serial offset quadrature pulse shaped waveform modulated by a predetermined SYNC sequence of symbols. The method serially demodulates the received waveform into a serial baseband signal. The serial baseband signal has the same predetermined SYNC sequence of symbols. The method also correlates the serial baseband signal with a plurality of segments of the SYNC sequence of symbols to form a plurality of parallel correlated output signals. The plurality of segments form the predetermined SYNC sequence of symbols. The method further combines the parallel correlated output signals to form a combined correlation signal and detects SYNC using the combined correlation signal. 
     The present invention is also embodied in a method for SYNC detection of a received serial offset quadrature pulse shaped waveform modulated by a predetermined SYNC sequence of symbols. The method serially demodulates the is received waveform into a serial baseband signal. The serial baseband signal has the same predetermined SYNC sequence of symbols. The method also correlates the serial baseband signal with a plurality of segments of the SYNC sequence of symbols to form a plurality of parallel correlated output signals. The plurality of segments form the predetermined SYNC sequence of symbols. The method further computes correlation strength estimates of each of the parallel correlated output signals. The method further combines the correlation strength estimates to form a combined correlation signal and detects SYNC using the combined correlation signal. 
     The present invention is further embodied in a receiver. The receiver includes a demodulator for serially demodulating a received waveform into a serial baseband signal. The serial baseband signal has a predetermined SYNC sequence of symbols. The receiver further includes a matched filter for correlating the serial baseband signal with a plurality of segments of the SYNC sequence of symbols to form a plurality of parallel correlated output signals. The plurality of segments form the predetermined SYNC sequence of symbols. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The invention is best understood from the following detailed description when read in connection with the accompanying drawing. It is emphasized that, according to common practice, the various features of the drawing are not to scale. On the contrary, the dimensions of the various features are arbitrarily expanded or reduced for clarity. Included in the drawing are the following figures: 
         FIG. 1  is a block diagram of a SQBL-MSK modulator, in accordance with an embodiment of the present invention; 
         FIG. 2  is a block diagram of a SQBL-MSK demodulator front end, in accordance with an embodiment of the present invention; 
         FIG. 3  is a plot of a QBL-MSK autocorrelation function, in accordance with an embodiment of the present invention; 
         FIG. 4  is a block diagram of a phase rotator, in accordance with an embodiment of the present invention; 
         FIG. 5  is a block diagram of another phase rotator, in accordance with an embodiment of the present invention; 
         FIG. 6  is a block diagram of an adaptive SYNC detector and SYNC threshold adaptor, in accordance with an embodiment of the present invention; 
         FIG. 7  is a block diagram of a 32 symbol digital matched filter (DMF) structure, in accordance with an embodiment of the present invention; 
         FIG. 8  depicts plots of QBL-MSK and squared QBL-MSK autocorrelation curves for use with the SYNC detector of  FIG. 6 ; 
         FIG. 9  is a block diagram of a DMF structure; 
         FIG. 10  is a block diagram of a SYNC combining structure, in accordance with an embodiment of the present invention; 
         FIG. 11  is a block diagram of an adaptive SYNC threshold adjustment module, in accordance with an embodiment of the present invention; 
         FIG. 12  is an illustrative example of an adaptive SYNC threshold control signal and a correlation strength estimation signal, in accordance with an embodiment of the present invention; 
         FIG. 13  is a block diagram of an alternative adaptive SYNC threshold adjustment module having a fast decay response, in accordance with an embodiment of the present invention; and 
         FIG. 14  is an illustrative example of an adaptive SYNC threshold control signal having a fast decay response and a correlation strength estimation signal, in accordance with an embodiment of the present invention. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     An embodiment of the present invention uses a serial quasi-bandlimited-minimum shift keyed (SQBL-MSK) signal as a modulation waveform. The SQBL-MSK signal enables a serial correlation structure to be used, with an in-phase (I) and quadrature (Q) correlator using the same synchronization (SYNC) sequence. Standard parallel QBL-MSK uses a four correlator structure based on the even and odd symbols on the I and Q sequence. The serial correlation structure used for synchronization (SYNC) detection and short segment correlations for frequency estimation advantageously provides a simplified binary phase shift keyed (BPSK) correlation operation versus a parallel structure. This reduces the complexity of the SYNC and frequency estimation operation. 
     Although QBL-MSK is selected as an exemplary modulation waveform, it is understood that other constant or near constant envelope modulation waveforms, such as Offset Quadrature Phase Shift Keying (OQPSK), Minimum Shift Keying (MSK), Gaussian MSK, Tamed Frequency Modulation (TFM), Intersymbol Jitter Free Offset Quadrature Phase Shift Keying (IJF-OQPSK), Raised Cosine Filtered Offset Quadrature Phase Shift Keying (RC-OQPSK), and bandwidth efficient Continuous Phase Modulation (CPM) methods may be used for the modulation. Non-constant envelope modulation waveforms, such as BPSK and filtered BPSK, may also be used for the modulation. 
     To simplify the SYNC and frequency estimation operation, it is desirable to implement serial formatting on these modulation waveforms. Serial formatting for quadrature pulse-shaped signals may be applied by adding a serial formatting term to the modulation waveform, which multiplies the nonreturn to zero (NRZ) symbol sequence with a repetitive 1, 1, −1, −1 sequence. For SQBL-MSK, the modulation waveform for the SYNC portion of the waveform can be written as follows: 
     
       
         
           
             
               
                 
                   
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                   ( 
                   
                     eqn 
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     For the data modulated SQBL-MSK waveform equation given above, T s  represents the symbol period, c i  represents the SYNC symbols at time iT s , 2M is the number of symbols in the SYNC sequence, p(t) is the QBL pulse-shaping function, f o  is the carrier center frequency, and the (−1) i  terms which multiplies the symbol value represents the serial formatting. For 128 symbol SYNC, for example, M is equal to 64. The SYNC symbols (c i ) take on either a +1 or −1 value. 
       FIG. 1  shows a block diagram of SQBL-MSK modulator  100  having I {x(t)} and Q {y(t)} as input signals. As shown in  FIG. 1 , the I and Q data signals are mixed with a carrier signal at mixers  102  and  106 . The outputs are then combined at summer  104 . The resulting signal s(t) is the modulation waveform. The equations for the I {x(t)} and Q {y(t)} signals modulating the carrier during the SYNC section of the waveform are obtained from equation 1 as follows: 
     
       
         
           
             
               
                 
                   
                     
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       FIG. 2  shows a block diagram for a SQBL-MSK demodulator front-end of receiver  200 , which down-converts the received signal to baseband I and Q signals, digitizes the I and Q signals, digitally filters the I and Q signals with the respective symbol matched filters and performs a phase rotator operation to generate the serial I and Q signals. As shown in  FIG. 2 , the received signal is mixed by mixers  202  and  210  with respective quadrature signals at the carrier frequency, resulting in the desired baseband I and Q signals (mixing difference term) and the undesired signal at twice the carrier frequency (mixing sum term). Lowpass filtering by LPF  204  and LPF  212  follows the down-converter function to remove the respective undesired mixing summation term. Baseband I and Q signals are digitized by the I and Q analog-to-digital converters (ADC)  206  and  214 , respectively. As shown in  FIG. 2 , the sampling rate is equal to twice the symbol rate. 
     Following digitization, the I and Q signals are filtered by respective symbol matched filters  208  and  216 , which maximize the signal-to-noise ratio (SNR). Phase rotator  218  converts the respective I and Q symbol matched filter outputs, from symbol matched filters  208  and  216 , into serial I and Q signals, which are sent to a SYNC detection operation and demodulation section (described later). 
     The QBL-MSK symbol matched filter coefficients are based on the QBL-MSK pulse-shaping function defined by: 
                     p   ⁡     (   t   )       =     {                 [       sin   ⁡     (       π   ⁡     [     t   -     2   ⁢           ⁢     T   s         ]         2   ⁢           ⁢     T   s         )         (       π   ⁡     [     t   -     2   ⁢           ⁢     T   s         ]         2   ⁢           ⁢     T   s         )       ]     3     ;           0   ≤   t   ≤     4   ⁢           ⁢     T   s                   0   ;           elsewhere   .           ;               (     eqn   ⁢           ⁢   5     )               
where T s  corresponds to the symbol period for the SYNC sequence.
 
     Because the QBL-MSK pulse-shaping function is non-zero over a four symbol period interval, the digital QBL-MSK symbol matched filter operating at twice the symbol rate may include 9 samples defined by the following equation: 
     
       
         
           
             
               
                 
                   
                     
                       
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                   ( 
                   
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                     6 
                   
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     Recognizing that the filter value for k equal to 0 and 8 is equal to zero, the digital QBL-MSK symbol matched filter response can be simplified to 7 samples defined by the following equation: 
     
       
         
           
             
               
                 
                   
                     
                       
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     Convolution of the QBL-MSK symbol pulse shape with the QBL-MSK symbol matched filter results in the QBL-MSK autocorrelation function {g(t)}.  FIG. 3  shows a plot of QBL-MSK autocorrelation function  300 , {g(t)}. As shown in this figure, autocorrelation function  300  is zero or near zero for time 2.5T s  from the desired optimum sampling point (time 0). 
     Using the QBL-MSK autocorrelation function {g(t)}, the I and Q signals outputted from the symbol matched filters are: 
                       x   2     ⁡     (     0.5   ⁢           ⁢     nT   s       )       =         [       ∑     i   =   0       M   -   1       ⁢         (     -   1     )     i     ⁢       c     2   ⁢           ⁢   i       ·     g   ⁡     (       0.5   ⁢           ⁢     nT   s       -     2   ⁢           ⁢     iT   s         )             ]     ⁢     cos   ⁡     (       π   ⁢           ⁢   n   ⁢           ⁢   Δ   ⁢           ⁢   f   ⁢           ⁢     T   s       +   ϕ     )         +             [       ∑     i   =   0       M   -   1       ⁢         (     -   1     )     i     ⁢       c       2   ⁢           ⁢   i     +   1       ·     g   ⁡     (       0.5   ⁢           ⁢     nT   s       -       [       2   ⁢           ⁢   i     +   1     ]     ⁢     T   s         )             ]     ⁢     sin   ⁡     (       π   ⁢           ⁢   n   ⁢           ⁢   Δ   ⁢           ⁢     fT   s       +   ϕ     )       ⁢     
     ⁢   and                 (     eqn   ⁢           ⁢   8     )                   y   2     ⁡     (     0.5   ⁢           ⁢     nT   s       )       =         -     [       ∑     i   =   0       M   -   1       ⁢         (     -   1     )     i     ⁢       c     2   ⁢           ⁢   i       ·     g   ⁡     (       0.5   ⁢           ⁢     nT   s       -     2   ⁢           ⁢     iT   s         )             ]       ⁢     sin   ⁡     (       π   ⁢           ⁢   n   ⁢           ⁢   Δ   ⁢           ⁢   f   ⁢           ⁢     T   s       +   ϕ     )         +             [       ∑     i   =   0       M   -   1       ⁢         (     -   1     )     i     ⁢       c       2   ⁢           ⁢   i     +   1       ·     g   ⁡     (       0.5   ⁢           ⁢     nT   s       -       [       2   ⁢           ⁢   i     +   1     ]     ⁢     T   s         )             ]     ⁢     cos   ⁡     (       π   ⁢           ⁢   n   ⁢           ⁢   Δ   ⁢           ⁢     fT   s       +   ϕ     )                     (     eqn   ⁢           ⁢   9     )               
where Δf is the carrier frequency error and φ is the carrier phase change.
 
     Phase rotator  218  enables the SQBL-MSK modulated signal to be serially demodulated, thus allowing the SYNC I and Q correlation operations to operate against the same SYNC symbol sequence versus having to separate the SYNC symbol sequence into even and odd symbols, as is typically performed by a parallel correlator. The two sample per symbol I and Q signals, out of phase rotator  218 , are sent to a SYNC detection module (for example  FIG. 6 ) which determines the timing control for selecting the proper sample and performs an adaptive SYNC threshold process based on a correlation segment of the SYNC sequence. 
     Phase rotator  218  may be implemented by a first phase rotator, as designated by  218   a  in  FIG. 4 , or a more simplified phase rotator, as designated by  218   b  in  FIG. 5 . The simplified phase rotator offers a lower complexity. Either phase rotator  218   a  or  218   b  may be used to implement the serial demodulation of the present invention. 
       FIG. 4  shows first phase rotator  218   a  for implementing the serial is demodulation. As shown, the I and Q symbol matched filter outputs, represented by x 2 (nT s /2) and y 2 (nT s /2), enter phase rotator  218   a  to be mixed with a cos(πn/4) signal by respective mixers  402  and  410  and a sin(πn/4) signal at mixers  406  and  408 . The outputs from mixers  402  and  406  are combined by summer  404  and the outputs from mixers  408  and  410  are combined by summer  412  (actually subtractor  412 ). 
       FIG. 5  shows simplified phase rotator  218   b  for implementing the serial demodulation. As shown, the I and Q symbol matched filter outputs, represented by x 2 (nT s /2) and y 2 (nT s /2), enter phase rotator  218   b  to be mixed with a cos[πINT(0.5 n)/2] signal by respective mixers  502  and  510  and a sin[πINT(0.5 n)/2] signal at mixers  506  and  508 , where INT represents an integer. The outputs from mixers  502  and  506  are combined by summer  504  and the outputs from mixers  508  and  510  are combined by summer  512  (actually subtractor  512 ). 
     The serial I and Q signals outputted from first phase rotator  218   a  ( FIG. 4 ) are given by the following equations: 
                       sx   ⁡     (     0.5   ⁢           ⁢     nT   s       )       =           x   2     ⁡     (       0.5   ⁢           ⁢     nT   s       +     Δ   ⁢           ⁢     T   s         )       ·     cos   ⁡     (         π   ⁢           ⁢   n     4     +   θ     )         +         y   2     ⁡     (       0.5   ⁢           ⁢     nT   s       +     Δ   ⁢           ⁢     T   s         )       ·     sin   ⁡     (         π   ⁢           ⁢   n     4     +   θ     )             ⁢     
     ⁢   and           (     eqn   ⁢           ⁢   10     )                   sy   ⁡     (     0.5   ⁢           ⁢     nT   s       )       =         -       x   2     ⁡     (       0.5   ⁢           ⁢     nT   s       +     Δ   ⁢           ⁢     T   s         )         ·     sin   ⁡     (         π   ⁢           ⁢   n     4     +   θ     )         +         y   2     ⁡     (       0.5   ⁢           ⁢     nT   s       +     Δ   ⁢           ⁢     T   s         )       ·     cos   ⁡     (         π   ⁢           ⁢   n     4     +   θ     )             ;           (     eqn   ⁢           ⁢   11     )               
where ΔT s  is the timing error (±T s /4 maximum) not removed by the SYNC timing correction and θ represents one of four possible timing conditions the phase rotator may have, conditions which depend on timing errors. The phase associated with the phase rotator is equal to either −90, 0, 90, or 180 degrees.
 
     The second, simplified phase rotator  218   b  ( FIG. 5 ) may be selected based upon simplification of the mathematical equations. Simplified phase rotator  218   b  introduces an additional phase change of −45 degrees every other sample. 
     Results derived from using the first phase rotator  218   a  ( FIG. 4 ) directly apply to simplified phase rotator  218   b  ( FIG. 5 ). Inserting the equations for x 2 (0.5 nT s ) and y 2 (0.5 nT s ) and applying simplifications to these equations provide the following equations: 
     
       
         
           
             
               
                 
                   
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                     ⁡ 
                     
                       ( 
                       
                         0.5 
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                                           ⁢ 
                                           
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                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             T 
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                                       ] 
                                     
                                     - 
                                     
                                       iT 
                                       s 
                                     
                                   
                                   ) 
                                 
                               
                               · 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       π 
                                       ⁡ 
                                       
                                         [ 
                                         
                                           
                                             0.5 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               nT 
                                               s 
                                             
                                           
                                           - 
                                           
                                             iT 
                                             s 
                                           
                                         
                                         ] 
                                       
                                     
                                     
                                       2 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         T 
                                         s 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ] 
                         
                         ⁢ 
                         
                           cos 
                           ⁡ 
                           
                             ( 
                             
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 
                                   π 
                                   ⁡ 
                                   
                                     [ 
                                     
                                       0.5 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       n 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       Δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         fT 
                                         s 
                                       
                                     
                                     ] 
                                   
                                 
                               
                               + 
                               
                                 θ 
                                 c 
                               
                             
                             ) 
                           
                         
                       
                       - 
                       
                           
                         
                           
                             [ 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   0 
                                 
                                 
                                   M 
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 
                                   c 
                                   i 
                                 
                                 · 
                                 
                                   g 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         [ 
                                         
                                           
                                             0.5 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               nT 
                                               s 
                                             
                                           
                                           + 
                                           
                                             Δ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               T 
                                               s 
                                             
                                           
                                         
                                         ] 
                                       
                                       - 
                                       
                                         iT 
                                         s 
                                       
                                     
                                     ) 
                                   
                                 
                                 · 
                                 
                                   sin 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         π 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             
                                               0.5 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 nT 
                                                 s 
                                               
                                             
                                             - 
                                             
                                               iT 
                                               s 
                                             
                                           
                                           ] 
                                         
                                       
                                       
                                         2 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           T 
                                           s 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     π 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         0.5 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         n 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         Δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           fT 
                                           s 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 + 
                                 
                                   θ 
                                   c 
                                 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             
 
                           
                           ⁢ 
                           and 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     eqn 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     12 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     sy 
                     ⁡ 
                     
                       ( 
                       
                         0.5 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           nT 
                           s 
                         
                       
                       ) 
                     
                   
                   = 
                   
                     - 
                     
                         
                       
                         
                           
                             [ 
                             
                               
                                 ∑ 
                                 
                                   i 
                                   = 
                                   0 
                                 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     M 
                                   
                                   - 
                                   1 
                                 
                               
                               ⁢ 
                               
                                 
                                   c 
                                   i 
                                 
                                 · 
                                 
                                   g 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         [ 
                                         
                                           
                                             0.5 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               nT 
                                               s 
                                             
                                           
                                           + 
                                           
                                             Δ 
                                             ⁢ 
                                             
                                                 
                                             
                                             ⁢ 
                                             
                                               T 
                                               s 
                                             
                                           
                                         
                                         ] 
                                       
                                       - 
                                       
                                         iT 
                                         s 
                                       
                                     
                                     ) 
                                   
                                 
                                 · 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       
                                         π 
                                         ⁡ 
                                         
                                           [ 
                                           
                                             
                                               0.5 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 nT 
                                                 s 
                                               
                                             
                                             - 
                                             
                                               iT 
                                               s 
                                             
                                           
                                           ] 
                                         
                                       
                                       
                                         2 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           T 
                                           s 
                                         
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     π 
                                     ⁡ 
                                     
                                       [ 
                                       
                                         0.5 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         n 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         Δ 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         
                                           fT 
                                           s 
                                         
                                       
                                       ] 
                                     
                                   
                                 
                                 + 
                                 
                                   θ 
                                   c 
                                 
                               
                               ) 
                             
                           
                         
                         - 
                         
                             
                           
                             
                               [ 
                               
                                 
                                   ∑ 
                                   
                                     i 
                                     = 
                                     0 
                                   
                                   
                                     M 
                                     - 
                                     1 
                                   
                                 
                                 ⁢ 
                                 
                                   
                                     c 
                                     i 
                                   
                                   · 
                                   
                                     g 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           [ 
                                           
                                             
                                               0.5 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 nT 
                                                 s 
                                               
                                             
                                             + 
                                             
                                               Δ 
                                               ⁢ 
                                               
                                                   
                                               
                                               ⁢ 
                                               
                                                 T 
                                                 s 
                                               
                                             
                                           
                                           ] 
                                         
                                         - 
                                         
                                           iT 
                                           s 
                                         
                                       
                                       ) 
                                     
                                   
                                   · 
                                   
                                     sin 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         
                                           π 
                                           ⁡ 
                                           
                                             [ 
                                             
                                               
                                                 0.5 
                                                 ⁢ 
                                                 
                                                     
                                                 
                                                 ⁢ 
                                                 
                                                   nT 
                                                   s 
                                                 
                                               
                                               - 
                                               
                                                 iT 
                                                 s 
                                               
                                             
                                             ] 
                                           
                                         
                                         
                                           2 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             T 
                                             s 
                                           
                                         
                                       
                                       ) 
                                     
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       π 
                                       ⁡ 
                                       
                                         [ 
                                         
                                           0.5 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           n 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           Δ 
                                           ⁢ 
                                           
                                               
                                           
                                           ⁢ 
                                           
                                             fT 
                                             s 
                                           
                                         
                                         ] 
                                       
                                     
                                   
                                   + 
                                   
                                     θ 
                                     c 
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
 
                             
                             ⁢ 
                             where 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   
                     eqn 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     13 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     θ 
                     c 
                   
                   = 
                   
                     θ 
                     + 
                     
                       ϕ 
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   
                     eqn 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     14 
                   
                   ) 
                 
               
             
           
         
       
     
     The phase error introduced by the down-conversion operation (φ) and the phase error introduced by the phase rotator (θ) may be combined into a total carrier phase error term (θ c ). From these equations, two key features of serial demodulation may be seen. First, the serial formatting factor (−1) i  seen in the modulation equation (eqn 1) can be removed. Secondly, it may be seen that the I and Q baseband signals consist of the filtered SYNC sequence multiplied by either a cosine or a sine weighting function. From  FIG. 3 , the QBL-MSK autocorrelation function  300  is nonzero for ±2.5 T s  about the ideal SYNC time of zero. Because the cosine weighting function forces the QBL-MSK autocorrelation function to zero at times −T s +ΔT s  and T s +ΔT s , only the QBL-MSK terms at −2T s +ΔT s , ΔT s , and 2T s +ΔT s  need to be considered for each cosine weighted QBL-MSK autocorrelation symbol response. 
     Similarly the sine weighting function forces the QBL-MSK autocorrelation function to zero at times −2T s +ΔT s , ΔT s , and 2T s +ΔT s , so only the QBL-MSK terms at −T s +ΔT s  and T s +ΔT s  need to be considered for each sine weighted QBL-MSK autocorrelation symbol response. 
     Using this information and the optimal symbol sample time, the equations for the serial I and Q signal can be rewritten as follows: 
                     sx   ⁡     (     nT   s     )       =         ∑     i   =   0         2   ⁢           ⁢   M     -   1       ⁢       c   i     ·     {         g   ⁡     (     Δ   ⁢           ⁢     T   s       )       ⁢       δ   ⁡     (     n   -   i     )       ·     cos   ⁡     (       2   ⁢   π   ⁢           ⁢   n   ⁢           ⁢   Δ   ⁢           ⁢     fT   s       +     θ   c       )           -       g   ⁡     (         -   2     ⁢           ⁢     T   s       +     Δ   ⁢           ⁢     T   s         )       ⁢       δ   ⁡     (     n   +   2   -   i     )       ·     cos   ⁡     (       2   ⁢           ⁢     π   ⁡     [     n   +   2     ]       ⁢   Δ   ⁢           ⁢     fT   s       +     θ   c       )           -       g   ⁡     (       2   ⁢           ⁢     T   s       +     Δ   ⁢           ⁢     T   s         )       ⁢       δ   ⁡     (     n   -   2   -   i     )       ·     cos   ⁡     (       2   ⁢           ⁢     π   ⁡     [     n   -   2     ]       ⁢   Δ   ⁢           ⁢     fT   s       +     θ   c       )             }         -             [       ∑     i   =   0         2   ⁢           ⁢   M     -   1       ⁢       c   i     ·     {         -     g   ⁡     (       -     T   s       +     Δ   ⁢           ⁢     T   s         )         ⁢       δ   ⁡     (     n   +   1   -   i     )       ·     sin   ⁡     (       2   ⁢     π   ⁡     [     n   +   1     ]       ⁢           ⁢   Δ   ⁢           ⁢     fT   s       +     θ   c       )           +       g   ⁡     (       T   s     +     Δ   ⁢           ⁢     T   s         )       ⁢       δ   ⁡     (     n   -   1   -   i     )       ·     sin   ⁡     (       2   ⁢           ⁢     π   ⁡     [     n   -   1     ]       ⁢   Δ   ⁢           ⁢     fT   s       +     θ   c       )             }         ]     ⁢     
     ⁢   and                 (     eqn   ⁢           ⁢   15     )                 sy   ⁡     (     nT   s     )       =       -     [       ∑     i   =   0         2   ⁢           ⁢   M     -   1       ⁢       c   i     ·     {         g   ⁡     (     Δ   ⁢           ⁢     T   s       )       ⁢       δ   ⁡     (     n   -   i     )       ·     sin   ⁡     (       2   ⁢   π   ⁢           ⁢   n   ⁢           ⁢   Δ   ⁢           ⁢     fT   s       +     θ   c       )           -       g   ⁡     (         -   2     ⁢           ⁢     T   s       +     Δ   ⁢           ⁢     T   s         )       ⁢       δ   ⁡     (     n   +   2   -   i     )       ·     sin   ⁡     (       2   ⁢           ⁢     π   ⁡     [     n   +   2     ]       ⁢   Δ   ⁢           ⁢     fT   s       +     θ   c       )           -       g   ⁡     (       2   ⁢           ⁢     T   s       +     Δ   ⁢           ⁢     T   s         )       ⁢       δ   ⁡     (     n   -   2   -   i     )       ·     sin   ⁡     (       2   ⁢           ⁢     π   ⁡     [     n   -   2     ]       ⁢   Δ   ⁢           ⁢     fT   s       +     θ   c       )             }         ]       -     [       ∑     i   =   0         2   ⁢           ⁢   M     -   1       ⁢       c   i     ·     {         -     g   ⁡     (       -     T   s       +     Δ   ⁢           ⁢     T   s         )         ⁢       δ   ⁡     (     n   +   1   -   i     )       ·     cos   ⁡     (       2   ⁢     π   ⁡     [     n   +   1     ]       ⁢           ⁢   Δ   ⁢           ⁢     fT   s       +     θ   c       )           +       g   ⁡     (       T   s     +     Δ   ⁢           ⁢     T   s         )       ⁢       δ   ⁡     (     n   -   1   -   i     )       ·     cos   ⁡     (       2   ⁢           ⁢     π   ⁡     [     n   -   1     ]       ⁢   Δ   ⁢           ⁢     fT   s       +     θ   c       )             }         ]               (     eqn   ⁢           ⁢   16     )               
where δ(n) is the unit impulse function, which is equal to 1 for n equal to zero and equal to 0 for all other values of n. The serial I and Q signals can be rewritten as follows:
 
     
       
         
           
             
               
                 
                   
                     
                       sx 
                       ⁡ 
                       
                         ( 
                         
                           nT 
                           s 
                         
                         ) 
                       
                     
                     = 
                     
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             0 
                           
                           
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               M 
                             
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   c 
                                   i 
                                 
                                 · 
                                 
                                   α 
                                   0 
                                 
                               
                               - 
                               
                                 
                                   c 
                                   
                                     i 
                                     + 
                                     2 
                                   
                                 
                                 · 
                                 
                                   α 
                                   2 
                                 
                               
                               - 
                               
                                 
                                   c 
                                   
                                     i 
                                     - 
                                     2 
                                   
                                 
                                 · 
                                 
                                   α 
                                   2 
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             δ 
                             ⁡ 
                             
                               ( 
                               
                                 n 
                                 - 
                                 i 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   n 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     fT 
                                     s 
                                   
                                 
                                 + 
                                 
                                   θ 
                                   c 
                                 
                               
                               ) 
                             
                           
                         
                       
                       - 
                       
                         
                           ∑ 
                           
                             i 
                             = 
                             0 
                           
                           
                             
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               M 
                             
                             - 
                             1 
                           
                         
                         ⁢ 
                         
                           
                             [ 
                             
                               
                                 
                                   - 
                                   
                                     c 
                                     
                                       i 
                                       + 
                                       1 
                                     
                                   
                                 
                                 · 
                                 
                                   α 
                                   
                                     - 
                                     1 
                                   
                                 
                               
                               + 
                               
                                 
                                   c 
                                   
                                     i 
                                     - 
                                     1 
                                   
                                 
                                 · 
                                 
                                   α 
                                   1 
                                 
                               
                             
                             ] 
                           
                           ⁢ 
                           
                             δ 
                             ⁡ 
                             
                               ( 
                               
                                 n 
                                 - 
                                 i 
                               
                               ) 
                             
                           
                           ⁢ 
                           
                             sin 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   π 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   n 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   Δ 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     fT 
                                     s 
                                   
                                 
                                 + 
                                 
                                   θ 
                                   c 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   and 
                 
               
               
                 
                   ( 
                   
                     eqn 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     17 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         sy 
                         ⁡ 
                         
                           ( 
                           
                             nT 
                             s 
                           
                           ) 
                         
                       
                       = 
                       
                         
                           - 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 0 
                               
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   M 
                                 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 [ 
                                 
                                   
                                     
                                       c 
                                       i 
                                     
                                     · 
                                     
                                       α 
                                       0 
                                     
                                   
                                   - 
                                   
                                     
                                       c 
                                       
                                         i 
                                         + 
                                         2 
                                       
                                     
                                     · 
                                     
                                       α 
                                       
                                         - 
                                         2 
                                       
                                     
                                   
                                   - 
                                   
                                     
                                       c 
                                       
                                         i 
                                         - 
                                         2 
                                       
                                     
                                     · 
                                     
                                       α 
                                       2 
                                     
                                   
                                 
                                 ] 
                               
                               ⁢ 
                               
                                 δ 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     i 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       2 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       π 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       n 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       Δ 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       
                                         fT 
                                         s 
                                       
                                     
                                     + 
                                     
                                       θ 
                                       c 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         - 
                         
                           
                             ∑ 
                             
                               i 
                               = 
                               0 
                             
                             
                               
                                 2 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 M 
                               
                               - 
                               1 
                             
                           
                           ⁢ 
                           
                             
                               [ 
                               
                                 
                                   
                                     - 
                                     
                                       c 
                                       
                                         i 
                                         + 
                                         1 
                                       
                                     
                                   
                                   · 
                                   
                                     α 
                                     
                                       - 
                                       1 
                                     
                                   
                                 
                                 + 
                                 
                                   
                                     c 
                                     
                                       i 
                                       - 
                                       1 
                                     
                                   
                                   · 
                                   
                                     α 
                                     1 
                                   
                                 
                               
                               ] 
                             
                             ⁢ 
                             
                               δ 
                               ⁡ 
                               
                                 ( 
                                 
                                   n 
                                   - 
                                   i 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     n 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       fT 
                                       s 
                                     
                                   
                                   + 
                                   
                                     θ 
                                     c 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     ; 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   where 
                 
               
               
                 
                   ( 
                   
                     eqn 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     18 
                   
                   ) 
                 
               
             
             
               
                 
                   
                     α 
                     n 
                   
                   = 
                   
                     
                       g 
                       ⁡ 
                       
                         ( 
                         
                           
                             nT 
                             s 
                           
                           + 
                           
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               T 
                               s 
                             
                           
                         
                         ) 
                       
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   
                     eqn 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     19 
                   
                   ) 
                 
               
             
           
         
       
     
     From these equations, it may be seen that the serial I and Q signals consist of a desired SYNC sequence multiplied by the sinusoidal carrier frequency error term and undesired cross-correlation property of the SYNC sequence. 
     For SYNC sequences with low cross-correlation property over the correlation segments, these cross-correlation terms are typically negligible. Removing these terms from the serial I and Q signals results in the following equations: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             
                               sx 
                               a 
                             
                             ⁡ 
                             
                               ( 
                               
                                 nT 
                                 s 
                               
                               ) 
                             
                           
                           = 
                             
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 0 
                               
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   M 
                                 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
                                 [ 
                                 
                                   
                                     c 
                                     i 
                                   
                                   · 
                                   
                                     α 
                                     0 
                                   
                                 
                                 ] 
                               
                               ⁢ 
                               
                                 δ 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     n 
                                     - 
                                     i 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               cos 
                               ⁢ 
                               
                                 ( 
                                 
                                   
                                     2 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     π 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     n 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     Δ 
                                     ⁢ 
                                     
                                         
                                     
                                     ⁢ 
                                     
                                       fT 
                                       s 
                                     
                                   
                                   + 
                                   
                                     θ 
                                     c 
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           = 
                             
                           ⁢ 
                           
                             
                               ∑ 
                               
                                 i 
                                 = 
                                 0 
                               
                               
                                 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   M 
                                 
                                 - 
                                 1 
                               
                             
                             ⁢ 
                             
                               
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     Performing the same simplification operations with the other symbol sample times results in the equation for the serial I and Q signals at the other sampling times: 
     
       
         
           
             
               
                 
                   
                     
                       
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       FIG. 6  shows SYNC detection module  600  which performs SYNC detection and includes an adaptive SYNC threshold adjustment, using the simplified (based on SYNC sequence assumptions given above) serial I and Q samples, shown in equations 20 to 23. The serial I and Q signals are sent to respective four 32 symbol digital matched filters (DMFs)  602  and  612 . Each DMF  602  and  612  includes four 32 symbol DMFs  604   a - d  (not shown for DMF  612 ), each of which implements a sliding correlation operation over the four 32 symbol sectors of the 128 symbol SYNC sequence. 
     The structure for each of 32 symbol DMFs  604   a - d  is the same as the 32 symbol I DMF for the SYNC sequence c 0  to c 31  (shown in  FIG. 7 ), except that the 32 symbol SYNC reference may be different. 
     Referring to  FIG. 7 , there is shown 32 symbol DMF  604   a  for SYNC sequence c 0  to c 31 . In an exemplary embodiment, time delay elements include flip-flops (F/F)  702 . Input serial I signal for DMF  604   a  is serially delayed using 64 F/Fs  702 . Because the input serial I and Q signals are each sampled at twice the symbol rate, the output of every other F/F  702  is provided to mixers  704  for multiplying with respective SYNC sequence coefficients c 0  to c 31 . The outputs from all 32 mixers  704  are summed by summer  706  to form the DMF output signal, I DMF  1 , for example. Similar operations are desirably performed with DMF  604   b - d  for the remaining input serial I signal components, and similar operations are desirably performed with the four 32 symbol DMF  612  for respective components of the input serial Q signal. 
     The four 32 symbol DMF sectors, shown in  FIG. 6 , minimize the correlation degradation introduced by the carrier frequency error (Δf). To minimize the correlation degradation to less than 3 dB, the phase drift across the 32 symbol DMF sector for a maximum frequency error is set to 45 degrees. The I and Q output signal for each of the 32 symbol DMF about the correlation peak, at n equal to zero, is given by the following equations: 
                         I   k     ⁡     (       0.5   ⁢           ⁢     nT   s       +       [     k   -   1     ]     ⁢     T   s         )       =     32   ·     [       sin   ⁡     (     32   ⁢           ⁢   π   ⁢           ⁢   Δ   ⁢           ⁢     fT   s       )         32   ·     sin   ⁡     (     π   ⁢           ⁢   Δ   ⁢           ⁢     fT   s       )           ]     ·     g   ⁡     (       nT   s     +     Δ   ⁢           ⁢     T   s         )       ·     cos   ⁡     (       2   ⁢           ⁢     π   ⁡     [     32   ⁢     (     k   -   1     )       ]       ⁢   Δ   ⁢           ⁢     fT   s       +     31   ⁢           ⁢   π   ⁢           ⁢   Δ   ⁢           ⁢     fT   s       +     π   ⁡     [     n   ⁢           ⁢   Δ   ⁢           ⁢     fT   s       ]       +       π   ⁢           ⁢   n     4     +     θ   c       )           ⁢     
     ⁢   and           (     eqn   ⁢           ⁢   24     )                     Q   k     ⁡     (       0.5   ⁢           ⁢     nT   s       +       [     k   -   1     ]     ⁢     T   s         )       =       -   32     ·     [       sin   ⁡     (     32   ⁢           ⁢   π   ⁢           ⁢   Δ   ⁢           ⁢     fT   s       )         32   ·     sin   ⁡     (     π   ⁢           ⁢   Δ   ⁢           ⁢     fT   s       )           ]     ·     g   ⁡     (       nT   s     +     Δ   ⁢           ⁢     T   s         )       ·     sin   ⁡     (       2   ⁢           ⁢     π   ⁡     [     32   ⁢     (     k   -   1     )       ]       ⁢   Δ   ⁢           ⁢     fT   s       +     31   ⁢           ⁢   π   ⁢           ⁢   Δ   ⁢           ⁢     fT   s       +     π   ⁡     [     n   ⁢           ⁢   Δ   ⁢           ⁢     fT   s       ]       +       π   ⁢           ⁢   n     4     +     θ   c       )           ;           (     eqn   ⁢           ⁢   25     )               
where k=1, 2, 3, and 4 represents the four 32 symbol DMF outputs from 32 symbol DMFs  604   a - d , n is the sample number about the peak QBL-MSK correlation function, at n equal to zero and the QBL-MSK correlation function represented by g( ) is non-zero when its argument is between the time interval of −3T s  and 3T s , as shown in  FIG. 8 .
 
     The [k−1] T s  term in equations 24 and 25 shows that the QBL-MSK correlation function is not phase aligned. The amplitude term multiplying the QBL-MSK correlation function {g( )} may be determined by the carrier frequency error, the number of symbol correlations in the DMF, and the symbol period. 
     The maximum carrier frequency for this approach may be set by the following equation: 
                       M   ⁢           ⁢   π   ⁢           ⁢   Δ   ⁢           ⁢     f   max     ⁢     T   s       =     π   4       ;           (     eqn   ⁢           ⁢   26     )               
where M is the number of correlation samples, which is equal to 32 for the exemplary embodiment. This relationship minimizes the reduction in the amplitude by less than 3 dB.
 
     It is desirable to reduce the number of symbols correlated in the DMF if the carrier frequency is larger than the maximum carrier frequency determined by this equation. For example, the DMF symbols may be reduced from 32 to 16, which increases the maximum carrier frequency error by a factor of 2. 
     By partitioning the 128 symbol SYNC sequence into small sectors, the maximum carrier frequency error is increased. The cross-correlation properties of the SYNC sequence increases, which increases the probability of a false SYNC detection. 
     Referring back to  FIG. 6 , the same four 32 symbol DMF structure  602  and  612  are implemented on the serial I and Q signal lines to produce four I DMF and four Q DMF output signals. As shown, the four 32 symbol DMF sectors are executed during the same time interval. The same 64 time delay elements, F/F  702  ( FIG. 7 ), are used to implement each of the four 32-symbol DMFs  604   a - d , because the DMF operates at twice the symbol rate. The same SYNC sequence is used for the four 32 symbol I DMF  602  and Q DMF  612 , because of the serially formatted waveform. 
     The four I and Q DMF output signals provide a correlation strength estimate for the adaptive SYNC threshold processing by taking a magnitude, or a squared magnitude, of the four I and four Q 32 symbol SYNC sectors using respective magnitude detectors  606 . The squared magnitude for each of the DMF I and Q output signal pairs can be obtained by squaring the DMF I and Q signal and then combining the squared values. The magnitude for each of the DMF I and Q output signal pairs is then obtained by taking the squared root of the squared magnitude value. 
     The squared magnitude approach results in a SYNC correlation response equal to the squared QBL-MSK correlation function, while the magnitude approach produces the QBL-MSK correlation function. The correlation response for the squared QBL-MSK autocorrelation function is sharper than the QBL-MSK autocorrelation function, as expected. 
       FIG. 8  shows the output signal for QBL-MSK autocorrelation function  802  and squared QBL-MSK autocorrelation function  804 , respectively. Either output may be used for correlation strength estimation, but typically the magnitude is used, because it can be implemented using the following equation: 
                         S   k     ⁡     (     0.5   ⁢           ⁢     nT   s       )       =       Max   ⁢     {       MAG   ⁡     [       IDMF   k     ⁡     (     0.5   ⁢           ⁢     nT   s       )       ]       ,     MAG   ⁡     [       QDMF   k     ⁡     (     0.5   ⁢           ⁢     nT   s       )       ]         }       +         1   2     ·   Min     ⁢     {       MAG   ⁡     [       IDMF   k     ⁡     (     0.5   ⁢           ⁢     nT   s       )       ]       ,     MAG   ⁡     [       QDMF   k     ⁡     (     0.5   ⁢           ⁢     nT   s       )       ]         }           ,       for   ⁢           ⁢   k     =   1     ,   2   ,   3   ,       &amp;           ⁢   4     ;             (     eqn   ⁢           ⁢   27     )               
where Max{ } is the maximum value of its two arguments, Min{ } is the minimum value of its two arguments, and Mag[ ] is the magnitude of its argument.
 
     Referring back to  FIG. 6 , magnitude detectors  606  are used to implement the correlation strength estimation for the four 32 symbol DMF outputs. The four correlation strength estimations (S 1 , S 2 , S 3 , and S 4 ) are sent to adaptive sync threshold adjustment module  616  to control the SYNC threshold level. Because the four correlation strength estimations are based on the DMF using four different sequences during the same time observation window, the correlation strength estimations consist of four QBL-MSK correlation functions each separated by 32 symbols, as the received SYNC sequence slides through the DMFs. 
     Accordingly, the adaptive control signal consists of the four QBL-MSK correlation functions from the respective outputs of magnitude detectors  606 , which are processed in adaptive sync threshold adjustment module  616  to provide the correlation strength estimation across each 32 symbol interval. Because each of the four QBL-MSK correlation functions provide the correlation strength estimations for only a few samples (as shown in  FIG. 8 ), signal processing may be used to maintain this estimation across the 32 symbol interval between QBL-MSK correlation functions, as shown in  FIGS. 12 and 14 . 
     For a low correlation strength estimate, the SYNC threshold level may be set to its lowest value, so SYNC detection may be achieved at the receiver. As the correlation strength estimate increases, the SYNC threshold may be increased to prevent false SYNC introduced by cross-correlation peaks, as the received SYNC sequence slides through the reference SYNC sequence. 
     The adjusted SYNC threshold level is desirably used as a reference to determine the occurrence of SYNC, by comparing it against the output of peak detector  614  in comparator  618 . Peak detector  614  compares the present correlation sample {COR(k)} to the past {COR{k−1)} and future {COR(k+1)} correlation samples to determine if the present correlation sample is a peak. If the detected peak is higher than the SYNC threshold level, SYNC is declared. 
     To determine the SYNC level, the DMF I and Q signals are desirably aligned so that the four QBL-MSK correlation functions add at the same point in time.  FIG. 9  shows DMF module  900  with four 32 symbol DMFs  902   a - d , in series, which provides this alignment. DMF module  900 , however, is not able to provide an adaptive control signal consisting of the four QBL-MSK correlation functions for four correlation strength estimations, as the received SYNC sequence slides through the DMF structures. It is thus more desirable to use the four parallel structures of DMFs  602  and  612  ( FIG. 6 ) for generation of the adaptive control signal, instead of the four serial structure, shown in  FIG. 9  by DMF  900 . 
     For SYNC detection, time alignment of the four QBL-MSK correlation functions is desired. This time alignment comes automatically with DMF  900  ( FIG. 9 ). For the parallel structure of DMF  602  and DMF  612 , however, three delay elements  608   a - c  are used for both the I and Q side to provide the proper time alignment, as shown in  FIG. 6 . 
     Still referring to  FIG. 6 , the 32 symbol DMF  604   a  with SYNC reference symbols c 0  to c 31  uses a 96 symbol delay element  608   a  (192 samples for 2 samples per symbol). The 32 symbol DMF  604   b  with SYNC reference symbols c 32  to c 63  uses a 64 symbol delay element  608   b  (128 samples for 2 samples per symbol). The 32 symbol DMF  604   c  with SYNC reference symbols c 64  to c 95  uses a 32 symbol delay element  608   c  (64 samples for 2 samples per symbol). 
     The properly time aligned four I and Q DMF outputs are then combined together by combiner network  610  to obtain the SYNC correlation signal. This combining operation may be a complicated process that corrects for the carrier frequency error, or it may be a simple process that adds the four magnitudes on the I and Q DMF outputs. 
       FIG. 10  shows an exemplary SYNC combiner  1000 , which uses the same magnitude detector  1002  for correlation strength estimation and SYNC detection as the system shown in  FIG. 6 . A serial I signal is provided to 32 symbol DMFs  604   a - d , as shown. After magnitude detectors  1002  performs correlation strength estimation, the outputs of magnitude detectors  1002  are provided to symbol delay elements  1006   a - c  to be delayed, as described above. Combiner  1004  combines the three outputs of symbol delay elements  1006   a - c  and the output of magnitude detector  1002  for the c 96  to c 127  correlation segment using summer  1008 . Peak detector  614  ( FIG. 6 ) receives the signal from summer  1008  and finds the peaks of the correlation samples by comparing the present correlation sample {COR(k)} to the past {COR{k−1)} and future {COR(k+1)} correlation samples. Correlation peaks provided by the peak detector are compared against the SYNC threshold, which is adjusted by the correlation signal strength estimations. If a peak level exceeds the SYNC threshold, SYNC is detected, as previously described. 
       FIG. 11  shows a detailed block diagram of an exemplary adaptive SYNC threshold adjustment module  616 , previously described with respect to  FIG. 6 . The adaptive SYNC threshold adjustment module is designated as  616   a  in  FIG. 11 . The maximum value of the four correlation strength estimations (S 1 , S 2 , S 3 , and S 4 ) is selected by selector  1102  on a sample by sample basis with the largest sample selected for further processing. Because the four correlation strength estimations (S 1 , S 2 , S 3 , and S 4 ) are magnitude values, which are positive values only, no sign information is involved and a direct magnitude level comparison can be done. 
     Because the QBL-MSK correlation function may provide correlation strength estimations for only a few samples, maximum signal selector  1104  and lowpass filter  1108  are provided to receive the correlation strength estimations. In operation, if the input signal, received from maximum selector  1102 , is longer than the output from lowpass filter  1108 , that input signal is selected by maximum signal selector  1104  and sent back to lowpass filter  1108  and forward to adaptive SYNC threshold mapping module  1106 . If, on the other hand, the input signal from maximum selector  1102  is smaller than the output from the lowpass filter, then the output from lowpass filter  1108  is selected by maximum signal selector  1104  and sent back to the lowpass filter and forward to adaptive SYNC threshold mapping module  1106 . Continued feedback to the lowpass filter results in the lowpass filter output signal decaying with time, based on a predetermined time constant. This exemplary operation provides a fast rise time, but a slow decay time from the peak signal. In the exemplary embodiment, the rise time is within one sample, while the decay time is set by the time constant in the lowpass filter. 
       FIG. 12  shows the adaptive SYNC threshold control signal, designated as  1202  having a specific decay time. As the decay time is increased, adaptive SYNC threshold control signal  1202  decays more slowly across the 32 symbol time interval and the threshold control signal may then be slow to respond to signal peaks of correlation strength estimation introduced by cross-correlation signals and interference signals. It will be appreciated that signal  1204  is the correlation strength estimation signal output from selector  1102  and signal  1202  is the Adaptive SYNC threshold control used to determine the SYNC threshold output from module  1106 . 
     The decay time may be reduced while providing less variation in the adaptive SYNC threshold control signal across the 32 symbol time interval by introducing a time shifted segment of the correlation strength estimator signal, as shown in  FIG. 13 . This figure shows another exemplary embodiment of adaptive SYNC threshold adjustment module  616  ( FIG. 6 ), designated as  616   b . As described above, correlation strength estimates S 1 , S 2 , S 3 , and S 4  are compared by maximum selector  1102 . Time delays by time shift modules  1302 ,  1304  and  1306  are applied to the output from maximum selector  1102 . A second maximum selector  1308  processes the outputs of maximum selector  1102  and the time delay outputs, respectively, from time shift modules  1302 ,  1304  and  1306 . The largest sample is selected as an output from selector  1308  for further processing. 
     Processing continues, as previously described, using maximum signal selector  1104  and lowpass filter  1108 . The output of maximum signal selector  1104  is sent to adaptive SYNC threshold mapping module  1106  for further processing, producing the adaptive SYNC threshold control signal, designated as  1402  in  FIG. 14 . The time shifted segments of the correlation strength estimator signal, designated as  1404  is used as an input signal to produce output signal  1402 . Because the correlation peaks are closer together, the decay time may be reduced, as shown. 
     Although in the exemplary embodiments a lowpass filter with a predetermined time constant is used to provide a slow decay time, it is also within the scope of the present invention to use a counter. The counter may hold the peak of the signal for a predetermined time period, unless a stronger peak occurs within the time period. In such case, the system may reset the counter. 
     Referring back to  FIG. 11  or  FIG. 13 , the adaptive SYNC threshold control signal is sent to adaptive SYNC threshold mapping module  1106 , which sets the SYNC threshold level based on the estimated correlation strength. For a low correlation strength estimate, the SYNC threshold level may be set to its lowest value, so SYNC detection may be achieved. As the correlation strength estimate increases, the SYNC threshold may be increased to prevent false SYNC introduced by cross-correlation peaks, as the received SYNC sequence slides through the reference SYNC sequence. Because the adaptive SYNC threshold control signal varies across the SYNC reception, the adaptive SYNC threshold values may be mapped using a look-up table (LUT) which is divided into different SYNC threshold values over an adaptive SYNC threshold control range. For example, the adaptive SYNC threshold mapping may include four ranges divided based on system design parameters. 
     In summary, the present invention provides a rapid synchronization (SYNC) process using an adaptive SYNC threshold, which may be implemented in hardware or software. The present invention substantially reduces false SYNC detections and false alarms by using a known synchronization (SYNC) pattern for a serially formatted QBL-MSK modulation signal. Serial formatting of the modulation waveform is selected to reduce SYNC correlation complexity. 
     The method of the present invention may be applied to other types of demodulator architectures, such as a demodulator architecture using four correlators, by making the appropriate changes to the correlation structure. Besides QBL-MSK, the adaptive SYNC approach may be applied to other quadrature modulation waveforms. To simplify the SYNC and adaptive SYNC threshold process, it is desirable that these waveforms be serially formatted. 
     The method of the present invention may also be applied to Binary Phase Shift Keying (BPSK) or filtered BPSK, where serial formatting is not used, to provide the reduced SYNC complexity. 
     For other similar and non-similar disclosures, please refer to the following five applications filed on the same day as this application. These five applications are TBD (and, respectively, correspond to the following five provisional applications 60/703,316; 60/703,180; 60/703,179; 60/703,373; and 60/703,320). These applications are all incorporated herein by reference in their entireties. 
     Although the invention is illustrated and described herein with reference to specific embodiments, the invention is not intended to be limited to the details shown. Rather, various modifications may be made in the details within the scope and range of equivalents of the claims and without departing from the invention.