Abstract:
Data aided carrier phase and symbol timing synchronizers are implemented at baseband as digital modulators isolating input signal inphase and quadrature component signals fed into inphase and quadrature. Laurent transforms that function as data detector to provide odd and even data bit multiplexed output data signal while cross coupling the inphase and quadrature transformed outputs for removing data modulation in error signals to correct phase errors and timing errors in the received signal so as to provide reliable data demodulation of noisy received signals having dynamic carrier phase and symbol timing errors as found in continuous phase modulation communications systems such as Gaussian minimum shift keying communications systems.

Description:
STATEMENT OF GOVERNMENT INTEREST 
     The invention was made with Government support under contract No. F04701-93-C-0094 by the Department of the Air Force. The Government has certain rights in the invention. 
    
    
     REFERENCE TO RELATED APPLICATION 
     The present application is related to applicant&#39;s copending application entitled Data Aided Carrier Phase Timing Tracking System for Precoded Continuous Phase Modulated Signals, Ser. No. 09/694,650, filed Oct. 24, 2000, by the same inventors. 
     FIELD OF THE INVENTION 
     The invention relates to the field of continuous phase modulation communications systems. More particularly, the present invention relates to symbol time tracking for continues phase modulations communications systems, such as Gaussian minimum shift keying communications systems having small bandwidth time products. 
     BACKGROUND OF THE INVENTION 
     In synchronous digital data communication systems, the carrier phase and symbol timing of the received signal must be acquired and tracked by the receiver in order to respectively demodulate the received signal and to recover the transmitted data from the received signal. Typically, receivers require carrier phase tracking for signal demodulation and symbol time tracking for data detection for generating received data streams. 
     Continuous phase modulation (CPM) provides a class of digital phase modulation signals that have a constant envelope. The spectral occupancy of a CPM signal can be controlled or tailored to the available bandwidth of a transmission channel. The constant envelope CPM signals allow saturated power amplifier operation for maximum power efficiency. The use of CPM signals in communications systems can potentially achieve significant improvement in both power and spectral efficiency over other conventional modulation techniques, at the cost of a moderate increase in receiver complexity. Bit error rate reduction has been achieved using trellis CPM demodulation with ideal synchronization. There is a continuing need to develop hardware implementation of the symbol time and carrier phase synchronizers that provides required tracking functions for the coherent CPM receiver. Often, symbol time tracking and carrier phase tracking limit the performance of CPM systems. 
     A particular type of CPM system is a Gaussian minimum shift keying (GMSK) system where a data sequence is precoded and the precoded data symbols are used for continuous phase modulation. The GMSK received signals are filtered using Laurent filters and samplers for providing data samples subjected to trellis demodulation for generating an estimate of the data sequence. Carrier phase tracking loops are used for demodulating the received signal by tracking the carrier phase, and symbol time tracking loops are used for synchronized sampling of Laurent matched filter signals for generating the data samples that used to generate estimates of the transmitted bit stream using trellis demodulation. These carrier phase and symbol time tracking loops are often referred to as synchronizer. These synchronizers often lose track during noisy communications. 
     A binary continuous phase modulation signal can be described by complex envelop equations. 
               z   ⁡     (   t   )       =     Re   ⁡     (         z   b     ⁡     (   t   )       ⁢     ⅇ         j   ⁢     2   ⁢   πf       c     ⁢   t         )                       z   b     (   t   )     =         2   ⁢       E   b     /   T         ⁢     ⅇ       j   ⁢   ϕ     ⁡     (     t   ,   α     )                         ϕ   ⁡     (     t   ,   α     )       =     πh   ⁢       ∫     -   ∞     t     ⁢       ∑     n   =   0       N   -   1       ⁢           ⁢       α   n     ⁢     f   (     t   -   nT     )     ⁢           ⁢     ⅆ   t                         =     πh   ⁢       ∑     n   =   0       N   -   1       ⁢           ⁢       α   n     ⁢     g   ⁡     (     t   -   nT     )                       
 
     The term Z b (t) is called the complex envelope of the CPM signal, f c  is the carrier frequency, E b  is the bit energy, T is the bit duration, and N is the transmitted data length in bits, α=(α 0 α 1  . . . α N−1 ,)α i ∈{±1}, represents one of 2 N  equally probable data sequences. The parameter h is the modulation index, f(t) is the pulse response of the smoothing filter in the CPM modulator, and g(t) is the CPM phase response defined in terms of the f(t) pulse response. 
         g   ⁡     (   t   )       =       ∫     -   ∞     t     ⁢       f   (   s   )     ⁢           ⁢     ⅆ   s             
 
     The pulse response f(t) is limited to the time interval [0,LT] for some integer L and having the properties that f(t)=f(LT−t) and g(LT)=1. The pulse amplitude modulation (PAM) representation of signal CPM envelope is well known. Laurent has shown that the complex envelope z b (t) can be expressed as a double summation. 
           z   b     (   t   )     =         2   ⁢       E   b     /   T         ⁢       ∑     k   =   0       2     L   -   1         ⁢           ⁢       ∑     n   =   0       N   -   1       ⁢           ⁢       α     k   ,   n       ⁢       h   k     ⁡     (     t   -   kT     )                   
 
     In this PAM representation of the baseband CPM signal envelope, also referred to as the Laurent decomposition, the a k,n  values are known as pseudo data symbols and are related to the modulated data symbols generally by a pseudo data symbol equation. 
         α     k   ,   n       =     exp   ⁡     (       j   ⁢   hπ     ⁡     [         ∑     m   =   0     n     ⁢     α   m       -       ∑     i   =   0                 L   -   1         ⁢       α     n   -   i       ⁢     β     k   ,   i             ]       )           
 
     In the pseudo data symbol equation, for all k, 0≦k≦2 L−1 , β k,0 =0 and β n  is 0 or 1 digit in the binary expansion of 
       k   =       ∑     i   =   1                 L   -   1         ⁢       2     i   -   1       ⁢     2     i   -   1       ⁢       β     k   ,   i       .             
 
These pseudo data symbols take on values in the set {±1,±j} when the modulation index h equals ½. In general, the first two pseudo data symbols, a 0h  and a 1,n  can be written in an expanded form. 
                 a     0   ,   n       =       exp   ⁡     (       j   ⁢   hπ     ⁢       ∑     m   =   0     n     ⁢     α   m         )       =       a     0   ,     n   -   1         ⁢     J     α   n             ,                   a     0   ,     -   1         =   1     ,     J   =         e   ixh     ⁢     a     1   ,   n         =       a     0   ,     n   -   L         ⁢     J     α   n       ⁢     J     α     n   -   2         ⁢     J     α     n   -   3         ⁢       …   ⁢   J       α     n   -   L   +   1                         
 
     The set of pulse functions {b k (t)}, termed Laurent pulse functions, have a real value and are finite in duration, and are formed by an h k (t) equation. 
           h   k     ⁡     (   t   )       =       ∑     i   =   0                 L   -   1         ⁢     c   (     t   +   iT   +       (       β     k   ,   i       -   1     )     ⁢   LT       )           
     where     
         c   (   t   )     =     (             sin   ⁡     (     πh   -     πhg   ⁡     (        t        )         )       /     sin   ⁡     (   πh   )                    t        ≤   LT               0   ,         elsewhere         )         
 
     Among these h k (t) pulses, most of the signal energy is carried by the principal Laurent pulse h 0 (t), which has a duration of L+1 bit times. Another property of the principal Laurent pulse h 0 (t) is that it is symmetrical about t=(L+1) T/2. The principal Laurent function h 0 (t) output provides a gross estimate of the transmitted symbol sequence. These properties of the principal Laurent pulse function h 0 (t) have not yet been exploited in developing the error signals for the symbol time and carrier phase tracking loops. These and other disadvantages are solved or reduced using the invention. 
     SUMMARY OF THE INVENTION 
     An object of the invention is to provide data aided symbol timing tracking in continuous phase modulation communication systems. 
     Another object of the invention is to provide data aided symbol timing tracking in a Gaussian minimum shift keying communications systems. 
     Yet another object of the invention is to provide data aided carrier phase tracking in continuous phase modulation communication systems. 
     Still another object of the invention is to provide data aided carrier phase tracking in a Gaussian minimum shift keying communications systems. 
     Still another object of the invention is to provide data aided carrier phase synchronizers and symbol time synchronizers in Gaussian minimum shift keying communications systems using principal Laurent responses for generating carrier phase and symbol time errors. 
     The present invention is directed to data aided synchronization in digital carrier phase and symbol timing synchronizers applicable to precoded continuous phase modulation (CPM) signal formats, such as in Gaussian minimum shift keying (GMSK) communications systems having, for example, a modulation index of ½ with a bandwidth time product (BT) of ⅕. The imbedded synchronizers enable simple implementations for data demodulation for CPM signals, such as GMSK signals with small BT values. Data aided tracking is applied in one form to symbol time tracking, and in another form, to carrier phase tracking. An advantage of the proposed data aided symbol timing synchronizer is the combination of both symbol timing tracking and data demodulation functions into an integrated process obviating the need for a separate data demodulator in the receiver. For example, for GMSK signals with BT values of ⅓ and larger, the data demodulation performance in the symbol timing synchronizer can provide optimum performance. An advantage of the data aided carrier phase synchronizer is the combination of both carrier phase tracking and data demodulation functions into one integrated process obviating a need for separate data demodulator in the receiver. For example, for GMSK signals with BT values of ⅓ and larger, the data demodulated performance provided by the carrier phase synchronizer can also be optimum. 
     In the first form, the symbol time tracking synchronizer includes a data aided symbol timing error discriminator that extracts the timing error of the received CPM signal from the principal Laurent amplitude modulation component by an early and late gating operation followed by a multiplication of the data decision to remove the data modulation in the error signal. This symbol timing error signal is then tracked by a second order digital loop operating at the symbol rate. In the second form, the carrier phase tracking synchronizer includes a data aided phase error discriminator that extracts the phase error of the received CPM signal from the principal Laurent amplitude modulation component by a cross correlation operation with the data decision produced by a serial data demodulator. This error signal is then tracked by a second order digital loop also operating at the symbol rate. 
     These digital synchronizers are use to track the symbol timing or carrier phase of a continuous phase modulation signal received in the presence of noise with the receiver operating in a data demodulation mode. These synchronizers have a nondegraded bit error rate (BER) performance with reduced design complexity. The GMSK signal with a BT=⅕ can be used as a typical partial response CPM signal. The hardware implementation of such a GMSK receiver with both synchronizers can be modeled for providing simulated BER performance. With data precoding of the original data bit stream prior to transmission of the CPM signal, the synchronizers can function as serial demodulators that achieve absolute phase data detection. The data preceding and data aided synchronization approach for detecting symbol timing and carrier phase error is central to providing accurate symbol time and carrier phase tracking in the synchronizers with reduced design complexity. These and other advantages will become more apparent from the following detailed description of the preferred embodiment. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1A  is a block diagram of a symbol time synchronized data demodulator. 
         FIG. 1B  is a block diagram of a symbol time synchronizer. 
         FIG. 2A  is a block diagram of a carrier phase synchronized data demodulator. 
         FIG. 2B  is a block diagram of a carrier phase synchronizer. 
         FIG. 3  is a graph depicting Laurent pulse functions. 
         FIG. 4  is a graph depicting an early-late gate function. 
         FIG. 5  is a plot of a symbol time error discriminator curve. 
         FIG. 6  is a plot of a carrier phase discriminator curve. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     An embodiment of the invention is described with reference to the figures using reference designations as shown in the figures. Referring to  FIG. 1A , a symbol time synchronized data demodulator includes a symbol time synchronizer  10  for data demodulating an r(t) received signal  11  sampled by input sampler  12  using a generated t n  timing signal  13 . The r(t) received signal  11  is a combination of the transmitted signal z b (t) and noise n(t) and is converted into an r n  sampled input signal  14 . The synchronizer  10  receives the sampled input signal  14  and provides a d n  estimate  15  of the received data sequence of the r n  sampled input  14  as well as generating a L mN  timing signal  17  and t n  timing signal  13 . The r n  sampled input  14  can be communicated to conventional Laurent matched filters such as a principal Laurent matched filter  18  and a secondary Laurent matched filter  19  having respective principal and secondary matched filter outputs respectively sampled by samplers  20  and  21  for providing respective filter samples into a Viterbi algorithm demodulator  22  that provides a d m  estimate  23 . The matched filters  18  and  19 , samplers  20  and  21 , and demodulator  22  are used to generate the d m  estimate  23  of the original data sequence using the symbol timing of the L mN    17  timing signal generated by the symbol time synchronizer  10 . The filters  18 ,  19  samplers  20  and  21 , and demodulator  22  providing the am data estimate  23  represents conventional data demodulation. 
     Referring to  FIGS. 1A and 1B , and more particularly to the symbol time synchronizer of  FIG. 1B , a real component and an imaginary component of the r n  sampled input signal  14  are respectively isolated by an inphase component isolator  24  and a quadrature component isolator  26  respectively providing inphase and quadrature sample signals to an odd timing error detector  32  and an even timing error detector  34 , that in turn, provide respective odd data and even data signals to a data demultiplexer  36  that provides the {circumflex over (d)} n  estimated data sequence  15 . The odd timing error detector  32  and even timing error detector  34  receive the inphase and quadrature sampled signals that are respectively communicated to early-late gates  44 a and  44 b and Laurent transformers h D (t)  46 a and  46 b isolating principal Laurent components. The Laurent transformer outputs of the transformers  46 a and  46 b are sampled by samplers  47 a and  47 b providing transformed sampled outputs. The early-late gate outputs of the early-late gates  44 a and  44 b are sampled by gate samplers  48 a and  48 b providing gate sampled outputs, respectively. The transformer sampled outputs of the transformer samplers  47 a and  47 b are respectively communicated to hard limiters  50 a and  50 b. The gate sampled outputs of the gate samplers  48 a and  48 b are respectively communicated to mixers  52 a and  52 b. The hard limiters  52 a and  52 b respectively provide the odd data and even data to the data multiplexer  36  that provides the {circumflex over (d)} n  estimated data  15 . The mixers  52 a and  52 b respectively mix odd and even data with the gate sampled outputs of gate samplers  48 a and  48 b to respectively provide e 2k+1  odd and e 2k  even timing signals that drive a loop filter  53 , that in turn, controls a voltage controlled oscillator  54  used for generating the t n  timing signal. The e 2k+1  odd and e 2k  even timing signals are alternately processed and combined by the loop filter  53  for controlling the voltage controlled oscillator  54 . The t n  timing signal  13  is further communicated to a modulo N counter  55  that provides the I mN  timing signals as well as generating the e(2k+1)N odd and e(2k)N even sampling signals that respectively control the samplers  47 a and  47 b, and,  48 a and  48 b. As may now be apparent, the synchronizer  10  operates in a timing loop extending through samplers  47 ab, limiters  50 ab, mixers  52 ab, loop filter  53 , VCO  54  and counter  55  for synchronized generation of the odd and even data and the t n  and t mN  timing signals,  13  and  17 , respectively, while generating the d n  data estimates  15 . 
     Referring to  FIGS. 1A ,  1 B,  2 A and  2 B, and more particularly to  FIGS. 2A and 2B , the carrier phase synchronizer demodulator of FIG.  2 A and specifically the carrier phase synchronizer  60  of  FIG. 2B , the carrier phase synchronizer  60  generates a e −jδ  phase adjustment signal  59  for adjusting the phase of the r(t) input signal  11 . The carrier phase synchronizer  60  also receives an r n e −jδ input sample signal  61  from a carrier phase sampler  62 . The r(t) received input signal  11  and e −jθ  phase adjustment signal are mixed by a mixer  63  that provide an input mixed signal that is sampled by a carrier phase sampler  62  at the rate of the t n  timing signal for providing the r n e −jθ  sampled input signal  61  to the carrier phase synchronizer  60 . The r n e −jθ  input sampled signal  61  can be fed into a conventional principal Laurent matched filter  64  and a secondary Laurent filter  66  providing matched filters outputs respectively to and sampled by matched filtered samplers  68  and  70  sampled at the rate of the t nN  symbol timing signals for providing matched filter inputs into a Viterbi algorithm demodulator  72  that generates a d m  estimate  73  of the original data sequence. The carrier phase synchronizer  60  can also be used to generate the d n  data estimate is. 
     The carrier phase synchronizer  60  receives the t n  timing signal that may originate from the symbol time synchronizer  10  in the preferred form, or from a convention symbol timing tracking loop, now shown. The r n e −jθ  sample input signal  61  is communicated to an inphase component isolator  74  and a quadrature component isolator  76 . The inphase component output of isolator  74  and the quadrature component output of isolator  76  are respectively sampled by an inphase sampler  80  and a quadrature sampler  82  at the rate of the t n  symbol timing signal  13  that also drives a modulo N counter  84  providing 2kN even and (2k+1)N odd timing sampling signals. The inphase sampler  80  provides a sampled inphase signal to an inphase transformer  86  as the quadrature sampler  82  provides a sampled quadrature signal to a quadrature transformer  88 , providing respectively inphase and quadrature transformed signals to hard limiters  90 a and  90 b, and by cross coupling, to mixers  92 b and  92 a. The hard limiters  90 a and  90 b respectively provide inphase and quadrature hard limited signals to hard limiter samplers  94 a and  94 b that respectively sample at rates of the 2kN even and (2k+1)N odd timing sampling signals from the modulo N counter  84 . The hard limiter samplers  94 a and  94 b respectively provide odd and even data signals that are fed into a data multiplexers  94  for generating the {circumflex over (d)} n  data estimate  15 . The odd data and even data are respectively mixed with the quadrature and inphase transformed signals from the transformer  88  and  86 , respectively, by the mixer  92 a and  92 b, for generating e 2k+1  odd and −e 2k  even timing error signals. The −e 2k  timing error signal is inverted by inverter  96  for generating an e 2k  even timing signal. The e 2k  even and e 2k+1  odd timing error signals are alternately processed and combined by the loop filter  97  to form the e −jθ  phase adjustment signal  59 . The e 2k  even and e 2k+1  odd timing error signals drive a loop filter  97  that in turn controls a VCO  98  that generates the e −jθ  phase adjustment signal  59 . As may now be apparent, the carrier phase synchronizer  60  is part of a loop between the e −j0  phase adjustment signal  59  and the r n e −jθ  input sampled signal  61  with the loop extending through the isolators  74  and  76 , samplers  80  and  82 , transformers  86  and  88 , hard limiters  90 a and  90 b, samplers  94 a and  94 b, mixers  92 a and  92 b, loop filter  97  and VCO  98  for providing the e −jθ  phase adjustment signal  59 , while concurrently generating the {circumflex over (d)} n  data estimate  15 . 
     Referring to all the Figures, the Laurent phase function is shown in  FIG. 3  for the principal h 0  pulse function, the h 1 (t) secondary pulse function and the h 2 (t) tertiary pulse function. The inphase component isolators  24  and  74  isolate the real component of the r n  input signal as the quadrature component isolators  16  and  76  isolate the imaginary component of the r n  input signal. The inphase Laurent transformers  46 a and  86  isolate the energy of the principal Laurent pulse component of the real component of the r n  input signal as the quadrature. Laurent transformers  46 b and  88  isolate the energy of the principal Laurent pulse component of the imaginary component of the r n  input signal. The early-late gate function is shown in  FIG. 4  for providing a digital transition in synchronism with Laurent components as isolated by the isolators  24  and  26 . In the symbol timing synchronizer  10 , the early-gates  44 a and  44 b operate on the respective isolated real and imaginary component energy for indicating the magnitude of the symbol timing error. The early-late gates  44 a and  44 b ideally have a positive value and a negative value on early and late respective sides of the center of the principal Laurent pulse function. These +/− values are combined with respective sides of the principal Laurent pulse function to provide two equal but opposite products that ideally sum to a zero magnitude error. As the principal Laurent pulse function early or late shifts relative to the current timing of the +/− gate function, the magnitude error increases positively or negatively. The area under the principal Laurent pulse function is multiplied by the gate function to produce a cross correlation of the gate function and principal Laurent pulse function for generating the magnitude error value that is used to adjust the timing signal to be in synchronism with the current symbol time of the received signal.  FIG. 5  shows symbol timing errors for the symbol timing synchronizer  10 . 
     The carrier phase synchronizer  60  uses the Laurent transformers  86  and  88  for isolating the energy of the principal Laurent pulse component for generating the magnitude of the carrier phase error. The carrier phase synchronizer  60  also uses cross coupled principal Laurent pulse energy for indicating the sign of the carrier phase error.  FIG. 6  shows the carrier phase errors of the carrier phase synchronizer  60 . 
     The symbol time synchronization data demodulator includes the symbol time synchronizer  10  for generating the t n  timing signal  13  as well as the {circumflex over (d)} n  data estimates  15 . The carrier phase synchronizer  60  receives the t n  symbol timing signal  13  for sampling the real and imaginary isolated components as well as for generating the odd and even data of the {circumflex over (d)} n  data estimate  15 . Hence, both of the synchronizers  10  and  60  operate as serial data demodulators for generating the {circumflex over (d)} n  data estimates  15 . Both of the symbol timing and carrier phase serial demodulators of synchronizers  10  and  60  operate respective modulo N counters  55  and  84  at the rate of N counts per symbol period of T seconds clocked at the rate of the t n  symbol timing signal  13 . The complex envelope Z b (t) of the CPM input signal  11  is sampled at a uniform rate of N samples per symbol period. These r n  samples are simultaneously applied to the Laurent transformers  46 a,  46 b,  86 , and  88  that function as data detection filters. 
     In the symbol timing synchronizer  10 , the early-late gates  44 a and  44 b function as impulse response filters. At each symbol decision instant of t=KN sample counts, for odd values of K, i.e., K=2k+1, the timing error between the receiver t n  timing signal  13  and the timing of the received signal is formed by respectively multiplying the output of the early-late gate  44 a the algebraic sign of the respective data decision filter, that is, the transformer  46 a and hard limiters  50 a. For even values of K, i.e., K=2k, the even timing error detector  34  operates similar to the odd time error detector  32 . The algebraic sign of the data detection filter outputs, that is, the output of the hard limiters  50 a and  50 b, is a data decision on the received data symbol for precoded binary CPM received signals. The timing error formed by the detectors  32  and  34  is then filtered by the loop filter  53 , integrated by the VCO  54 , and quantized into sample counts by the modulo N counter  55  to produce an adjustment to the sampling timing at symbol epoch i.e., at time instants of a multiple of N counts. The symbol timing signal  13  as well as the sampling signals are delayed or advanced by the timing adjustment according to whether the adjustment is positive or negative. No more than N most recent signal samples need to be stored by the synchronizer to allow for the advancing of the sampling timing at the symbol time in the tracking mode. 
     During data demodulation, the transmitted data symbol can be obtained by differentially decoding two successively received pseudo data symbols a 0,n . For a CPM modulation index of h=0.5, the data stream is precoded into a data stream d k  fed into the data modulator having an input symbol stream α k  with α k =(−1) k d k−1 d k . The pseudo data symbol a 0,n  becomes a 0,n =J(n)d n  with J(n)=1 for n being odd and J(n)=j for n being even. Thus, with data precoding, either a conventional trellis demodulator or a serial demodulator of the synchronizers  10  and  60  can be used to demodulate the received CPM signal without differential decoding. A CPM modem using precoding can achieve a performance improvement from 0.5 dB to nearly 2.0 dB over a modem without precoding. 
     Because the Laurent pulse function h 0 (t) is the dominant pulse function in a CPM signal, the symbol timing error of the received signal relative to the receiver clock can be detected by using the early-late gating on the received baseband signal in conjunction with serial data demodulation of the synchronizers  10  and  60 . The timing error is produced by respectively multiplying the data decisions generated by the serial demodulation of the transformers  46 a and  46 b and the hard limiters  50 a and  50 b with the output of the early-late gate  44 a and  44 b. Respective multiplication by mixers  52 a and  52 b of the early-late gate output with hard limited data decisions is needed to eliminate the data modulation so that a consistent timing error can be formed. With ideal elimination of the data modulation, the detected timing error is given by a detection equation. 
           D   t     ⁡     (   τ   )       =       ∫   0       (     L   +   1     )     ⁢   T       ⁢       G   ⁡     (   s   )       ⁢       h   0     (     s   -   τ     )     ⁢           ⁢     ⅆ   s             
 
     The early-late gate function G(t) provides an ideal timing error detection curve D s (τ) for a given CPM signal, such as a BT=⅕ GMSK signal. 
     Carrier phase error detection is formulated based on a unit amplitude CPM signal received in the absence of channel noise with a carrier phase offset θ. The phase offset complex signal envelope is defined by an r(t,θ) equation. 
               r   (     t   ,   θ     )     =         z   b     (   t   )     ⁢     ⅇ     j   ⁢   θ                     =       {       ∑     k   =   0       Q   -   1       ⁢       ∑     n   =   0       N   -   1       ⁢       a     k   ,   n       ⁢       h   k     ⁡     (     t   -   nT     )             }     ⁢     ⅇ     j   ⁢   θ                   
 
     When the r(t,θ) signal is applied to the transformed and hard limited serial demodulator, the demodulator output at time t=mT is defined by an r m  equation. 
               r   m     =       ∫     -   ∞     ∞     ⁢       r   (     t   ,   θ     )     ⁢       h   0     ⁡     (     t   -   mT     )       ⁢           ⁢     ⅆ   t                     =       {       ∑     k   =   0       Q   -   1       ⁢       ∑     n   =   0       N   -   1       ⁢       a     k   ,   n       ⁢       R     0   ,   k       ⁡     (     m   -   n     )             }     ⁢     ⅇ     j   ⁢   θ                     =         J   ⁡     (   m   )       ⁢     d   m     ⁢     ⅇ     j   ⁢   θ       ⁢       R     0   ,   0       ⁡     (   0   )         +       {       ∑     k   =   0       Q   -   1       ⁢       ∑       n   =   0         (       n   ≠   m     ,           k   =   0     )           N   -   1       ⁢       a     k   ,   n       ⁢       R     0   ,   k       ⁡     (     m   -   n     )             }     ⁢     ⅇ     j   ⁢   θ       ⁢           ⁢   where                 
           R     0   ,   k       ⁡     (   p   )       =       ∫     -   ∞     ∞     ⁢         h   0     ⁡     (   t   )       ⁢       h   k     ⁡     (     t   +   pT     )       ⁢           ⁢     ⅆ   t             
 
     With the data d k  being equally probable, the averaged value of d m a kn  is zero for all integers m, when k≠0, and also for all integers m≠n when k=0. Thus, with the carrier phase error θ being small and when the serial demodulators can correctly demodulate the m-th transmitted bit d m , then, by multiplying the serial demodulated bit by the complex conjugate of J(m)d m  and taking the imaginary part of the product obtains a random variant whose mean value is D φ (θ)=R 0,0 (0)sin(θ)=R 0,0 (0)θ. The randomness is due to the intersymbol interference, which is data pattern dependent. 
     Because both timing and carrier phase error detection use serial demodulation to provide the required data decision for error generation, the transformed and hard limited serial demodulator, such as in the synchronizers  10  and  60 , can be used for both the tracking error generation and data detection. The error signals produced at every receiver symbol time are applied to the respective loop filter  53  and  97  and voltage control oscillator  54  and  98  to adjust the sampling timing instants or the carrier phase to the received signal. Data reliability of a trellis demodulator is usually better than that of a serial demodulator such as the synchronizers  10  and  60 , particularly when the signal memory span L is large. However, if L is small or if an equalizer is used in cascade with the principal Laurent pulse filter, the simple serial receiver can perform particularly as well as the more complex trellis demodulator for the purpose of tracking error generation. Thus, an equivalent variation of the synchronizers  10  and  60  is to feedback the data decisions from the trellis demodulator to the error detectors, provided that the processing delay of the trellis demodulator is properly compensated for and that tracking performance is not unduly compromised by the delay. 
     The mean error output or discriminator characteristics of the symbol timing error and carrier phase error detectors is shown for the BT=⅕ GMSK signal, in FIG.  5  and  FIG. 6 , respectively. These characteristics are obtained by computing in random data the averaged detector output for a given error offset with the other offset error set at zero. For small errors, the linear slope of the timing error discriminator curve is about −1.5 and that of the phase error discriminator curve is about 1.0. The deviation of these characteristics from their ideal S curves, at large offset errors, is attributed to the feedback of erroneous data decisions caused by the intersymbol interference in the GMSK signal. 
     Both the symbol time synchronizer  10  and carrier phase synchronizer  60  have a linear continuous time model that can be implemented digitally for use in performance simulations of the GMSK receiver. The linear model is appropriate because the tracking error is typically small when the receiver is in a tracking mode. The loop filter, used in each synchronizer  10  and  60 , is of a proportional and integral type with a transfer function in the form of F(s)=α+β/s and the VCO transfer function in the form of K v /s where K v  is the VCO gain. The closed loop transfer function of the synchronizers  10  and  60  is defined by an H(s) equation. 
         H   (   s   )     =           ϕ   0     ⁡     (   s   )           ϕ   1     ⁡     (   s   )         =         2   ⁢     ςω   n     ⁢   s     +     ω   n   2           s   2     +     2   ⁢     ςω   n     ⁢   s     +     ω   n   2               
 
     In the H(s) equation, the term ζ is the damping factor and ω n  is the natural frequency of the synchronizers  10  and  60 . These parameters are related to the loop filter and gain parameters by α=2ζω n /K D K v  and β=ω n   2 /K D K v , where K D  is the slope of the error discriminator curves shown in  FIGS. 5 and 6 . The one-sided equivalent noise bandwidth of the synchronizers  10  and  60  is B L =(ω n /8ζ 2 )(1+4ζ 2 ). Each of the second order synchronizers  10  and  60  can be digitally implemented with the integrator 1/s approximated by the digital accumulator 1/( 1−z   −1 ) where z −1  represents a unit bit time delay. In a digital implementation, the natural frequency and loop bandwidth parameters should be regarded as parameters normalized by the bit rate. Using the loop parameters K D =1, K v =1 and ζ=1/√2 for the carrier phase synchronizer  60  and K D =√1.5, K v =1 and ζ=1/√2 for the symbol time synchronizer  10 , the step error response of the carrier phase synchronizer  60  to a 20 degree phase step and that of the symbol time synchronizer  10  to a half bit time step are stimulated and compared to the theoretical step error response. The ramp error responses for both synchronizers  10  and  60  are also simulated and compared to the theoretical ramp error responses. The dispersion of the simulated error responses from the theoretical is due to the intersymbol interference in the received signal. 
     The symbol time synchronizer  10  and carrier phase synchronizer  60  are characterized as providing error signals generated from quadrature Laurent pulse response components of a receiving signal modulated by symbols generated from a precoded data sequence. In the preferred form, the principal Laurent components indicates the original digital bit sequence of the precoded bit stream. The precoding functions to precondition the transmitted symbol sequence so that the principal Laurent function indicates the original data bit stream that is alternately disposed on the I and Q channels of the transmitted CPM signal. 
     The precoded PCM signal allows the use of the principal Laurent pulse response for extracting the sign of the symbol timing error or carrier phase error that is also the data of the original data uncoded sequence. In the symbol time synchronizer  10 , the early-late gates  44 a and  44 b will extract the magnitude of the symbol timing error. The early-late gates  44 a and  44 b are sampled at the current symbol t n  timing signal  13 . As the timing of the received signal  11 , varies from the current timing of the timing signal  13 , the early-late gates  44 a and  44 b provide an indication of the magnitude of the current timing error. The CPM signal will carry the data information in one symbol time in the inphase component signal and in the next symbol instance in the quadrature component signal, as the data bit information content alternates between the inphase and quadrature components. The timing synchronization  10  in combination with data preceding enable efficient synchronization timing and data extraction at the expense of requiring the use of both 1 &amp; Q component signals that might otherwise be used to communicate two independent data streams. The loop filter  53  functions to smooth the timing error signal generated by the detectors  32  and  34 . The smoothed timing error from the loop filter  53  then drives the VCO that in turn provides the smoothly varying t n  timing clock signal. The precoded data provides the sign of the timing error, and hence, the symbol timing synchronizer  10  is data aided, and hence also provides an estimater  15  of the original data sequence. 
     In the carrier phase synchronizer receives the t n  timing signal and the received signal r n  and operates on the phase error 0 generated from the r(t,0) equation that describes the phase error. The carrier phase synchronizer  60  also uses the isolated I &amp; Q principal Laurent components and determines the sign of the phase error. But, rather than determining a magnitude of the phase error using early-late gates, the carrier phase synchronizer drifts the phase error depending on the sine of the phase error having a sign that is also the original uncoded data sequence. The § term represents the carrier phase error that is generated using cross-coupling of the Laurent components generating the e 2k  and e 2k+1  error signals with the sign of θ indicating the direction of the phase error drift. 
     The symbol timing synchronizer  10  and the carrier phase synchronizer  60  offer an efficient mechanism for generating timing and phase error signal while also providing an indication of the uncoded data sequence however requiring data precoding having symbol modulated on both I and Q channels. Those skilled in the art can make enhancements, improvements, and modifications to the invention, and these enhancements, improvements, and modifications may nonetheless fall within the spirit and scope of the following claims.