Abstract:
A receiver generates log-likelihood-ratio-based soft bit metrics of precoded quaternary continuous phase modulation signals using four state-constrained trellises and a streamlined maximum likelihood sequence estimation Viterbi algorithm requiring no survivor state storage elements for a preferred error correction-coded quaternary Gaussian minimum shift keying communication system employing reduced-complexity pulse-amplitude modulation matched-filtering and soft-decision decoding.

Description:
STATEMENT OF GOVERNMENT INTEREST  
       [0001]     The invention was made with Government support under contract No. F04701-00-C-0009 by the Department of the Air Force. The Government has certain rights in the invention. 
     
    
     FIELD OF THE INVENTION  
       [0002]     The invention relates to the field of continuous phase modulation communication systems. More particularly, the present invention relates to quaternary precoded continuous phase modulation and soft bit metric demodulation communication systems.  
       BACKGROUND OF THE INVENTION  
       [0003]     Continuous phase modulation (CPM) signals are phase-modulated signals having a spectral occupancy that can be tailored to fit limited transmission bandwidth through suitable pre-modulation filtering. Moreover, unlike non-constant envelope signals such as amplitude-modulated signals or filtered phase-modulated signals, the CPM signals are of constant envelope and allow saturated power amplifier operation for maximum power efficiency. These desirable signal properties, fueled by the rising premium being placed on bandwidth and power efficiency, have resulted in CPM signals such as binary Gaussian Minimum Shift Keying (GMSK) being deployed in operational terrestrial and satellite communication systems. The ever-rising premium of bandwidth efficiency further motivates extending binary CPM to higher alphabet signaling format such as quaternary GMSK.  
         [0004]     A quaternary CPM signal, despite its relatively poorer power performance than binary CPM, can be an attractive signaling format when used in conjunction with some forward error correction schemes. The use of soft-decision error correction decoding is particularly desirable in such a coded CPM communication system as it can significantly reduce the signal-to-noise ratio (SNR) needed for achieving an overall error rate performance. This SNR improvement, however, is predicated on the ability of the underlying demodulator to generate appropriate soft bit metrics for the soft-decision decoder. Conventional quaternary CPM signal demodulator produces hard decisions on symbols, hence on bits, by applying the maximum likelihood sequence estimation (MLSE) Viterbi algorithm and identify the most probable symbol sequences. Using such a hard-decision symbol demodulator in a coded quaternary CPM communication system brings only sub-optimal performance as only hard-decision error correction decoding is permissible.  
         [0005]     Mengali taught reduced-complexity quaternary CPM demodulator based upon the pulse-amplitude modulation (PAM) components of the complex envelope of a quaternary CPM signal as an extension of the PAM-based reduced-complexity receiver originally proposed by Kaleh for demodulating binary CPM signals. The design principle of such a reduced-complexity quaternary CPM demodulator is footed on a bank of matched-filters associated with the PAM components of the underlying CPM signal. The complexity reduction is achieved by using only a subset of the PAM matched-filters in the receiver. The total number of PAM components in a quaternary CPM signal is equal to 3.4 (L−1)  where L represents the memory of the CPM pre-modulation filter. The memory length L is generally in the order of the reciprocal of the bandwidth-time product BT, that is, L=1/BT. Typically, for moderate bandwidth-time product BT, only a small subset of these PAM components needs to be considered for demodulation purpose. For example, with BT=1/3, only the first three energy-dominate PAM components in the quaternary GMSK signal need to be considered out of a total of forty-eight PAM components, and hence, only three respective matched-filters are used for demodulation. Following Kaleh&#39;s work on demodulating binary CPM signals, Mengali taught hard-decision quaternary symbol demodulation using MLSE Viterbi algorithm. However, both Mangali and Kaleh have failed to construct an appropriate soft bit metric using the theoretical log-likelihood ratios and as such are less than optimal in performance. Additionally, Mangali and Kaleh taught hard-decision symbol demodulation requiring extensive memory elements for storing the survivor path states when implementing the MLSE Viterbi algorithm. These and other disadvantages are solved or reduced using the invention.  
       SUMMARY OF THE INVENTION  
       [0006]     An object of the invention is to provide a soft bit metric demodulator that generates log-likelihood ratios.  
         [0007]     Another object of the invention is to provide a soft bit metric demodulator that generates log-likelihood ratios for demodulating quaternary continuous phase modulated (CPM) signals.  
         [0008]     Yet another object of the invention is to provide a soft bit metric demodulator that generates log-likelihood ratios for demodulating quaternary CPM signals using a maximum likelihood sequence estimation (MLSE) Viterbi algorithm.  
         [0009]     Still another object of the invention is to provide a soft bit metric demodulator for demodulating quaternary CPM signals using a MLSE Viterbi algorithm where the soft metrics are log-likelihood ratios.  
         [0010]     The invention is directed to a soft bit metric demodulator that utilizes the maximum likelihood sequence estimation (MLSE) Viterbi algorithm to generate log-likelihood ratios. The demodulator can be used for demodulating precoded quaternary Gaussian minimum shift keying (GMSK) signals and, more generally, for demodulating precoded quaternary continuous phase modulation (CPM) signals. This quaternary soft bit metric (QSBM) demodulator is implemented as a streamlined MLSE Viterbi algorithm that requires no memory elements for storing the survivor path states. In the preferred GMSK form, the bandwidth-time product of the Gaussian pre-modulation shaping filter is BT=1/3, the modulation index is h=1/4, and the receiver uses three matched-filters. The QSBM demodulator can be used either in a stand-alone un-coded CPM system, or in a coded CPM system in conjunction with some forward-error-correction scheme such as the classical rate-1/2 convolution code with maximum likelihood Viterbi decoding. The QSBM demodulator permits soft-decision decoding in coded quaternary CPM communication systems by generating the log-likelihood ratios for the bits associated with the quaternary data symbols received over a noisy channel. In a typical rate-1/2 convolutional-coded quaternary GMSK system, the QSBM demodulator is able to provide an improvement of about 3.0 dB SNR over hard-decision error-correction decoding without the use of memory elements for storing the survivor path states. These and other advantages will become more apparent from the following detailed description of the preferred embodiment.  
     
    
     BRIEF DESCRIPTION OF THE DRAWINGS  
       [0011]      FIG. 1  is a block diagram of a precoded quaternary Gaussian minimum shift keying (GMSK) communication system having convolutional encoding, quaternary data preceding, and continuous phase modulation (CPM) in a GMSK transmitter, and having a quaternary soft bit metric (QSBM) demodulator in a GMSK receiver.  
         [0012]      FIG. 2  is a block diagram detailing the processing structure of the QSBM demodulator.  
         [0013]      FIG. 3A  depicts a constrained trellis wherein the most significant bit is restricted to a 1 (MSB- 1 ).  
         [0014]      FIG. 3B  depicts a constrained trellis wherein the most significant bit is restricted to a 0 (MSB- 0 ).  
         [0015]      FIG. 3C  depicts a constrained trellis wherein the least significant bit is restricted to a 1 (LSB- 1 ).  
         [0016]      FIG. 3D  depicts a constrained trellis wherein the least significant bit is restricted to a 0 (LSB- 0 ).  
         [0017]      FIG. 4  is a performance graph of a convolutional-coded quaternary GMSK system using the QSBM demodulator.  
     
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS  
       [0018]     An embodiment of the invention is described with reference to the figures using reference designations as shown in the figures. Referring to  FIG. 1 , a quaternary GMSK transmitter  10  comprises a binary data source  11 , a convolutional encoder  27 , a quaternary data precoder  12 , and a GMSK modulator  14 . The binary data source  11  continuously generates binary data bits b n  that are in turn convolutional-encoded into binary coded bits c n . The stream of binary coded bits is equivalently viewed, two coded bits at a time, as a stream of quaternary NRZ data symbols d n  chosen from an quaternary alphabet set {±1, ±3}. The mapping from two binary coded bits c 2n  and c 2n+1  to one quaternary NRZ symbol d n  may be, for example, d n =(1−2c 2n )·(1+2c 2n+1 ). The stream of quaternary NRZ source symbols d n  are then precoded by the quaternary data precoder  12  into quaternary precoded symbols α n ε{±1, ±3} that are in turn modulated by the GMSK modulator  14 . The GMSK modulator  14  includes a Gaussian filter  13 , a frequency modulator  15  with modulation index h, and a frequency converter  16  for upconverting the baseband signal z b (t) using a local oscillator  17 . The Gaussian filter  13  is defined by a bandwidth-time product (BT) that may be, for example, 1/3, where B is the one-sided 3 dB bandwidth in hertz of the Gaussian filter  13  and T is the quaternary channel symbol duration in seconds. For M-ary GMSK signals, both the main lobe bandwidth and the side-lobe amplitude decrease with a decreasing BT. The Gaussian filter  13  produces an output G(t) that is the cumulative sum of pulse responses resulting from the input quaternary precoded symbols α n , that is, G(t)=Σα n f(t−nT). The individual pulse response f(t), also known as the GMSK frequency pulse, depends on the BT product and is essentially zero except over a time interval of duration LT, where L is an integer representing the memory of the Gaussian filter  13 . The memory length L, generally on the order of (BT) −1 , is an integer greater than or equal to one. The frequency modulator  15  receives and frequency-modulates the Gaussian filter output G(t) by a predetermined modulation index h that may be, for example, 1/4. In general, lowering the modulation index h while keeping the BT product constant will further reduce the spectral occupancy of the M-ary GMSK signal. Preferably, the modulation index is set to h=1/M in order to facilitate M-ary data precoding. The frequency modulator  15  transforms the Gaussian filter output G(t) into a baseband GMSK signal z b (t) with a constant envelope of A=√(2E/T), that is, z b (t)=A·exp[jπh·Σα n g(t−nT)] where E denotes the data symbol energy. The function g(t), defined as the integral of the GMSK frequency pulse f(t), is the well-known GMSK phase pulse satisfying a boundary condition of g(LT)=1. The baseband GMSK signal z b (t) is then upconverted by the converter  16  using the carrier reference  17  and then transmitted over a communication channel  18  subject to additive white Gaussian noise (AWGN) and potential interference  19 . A quaternary GMSK receiver  20  equipped with a frequency downconverter  21  receives the transmitted GMSK signal, along with noise and interference. The downconverter  21  uses a locally generated carrier reference  22  to downconvert the received RF signal into a baseband signal z r (t). The received baseband signal z r (t) is then processed by a quaternary soft bit metric (QSBM) demodulator  24  to provide quaternary data estimates {circumflex over (d)} n  to a convolutional decoder  25  that matches the convolutional encoder  27  used in the transmitter  10 . The convolutional decoder  25  views each quaternary data estimate {circumflex over (d)} n  as a pair of binary coded bit estimates and produces a continuous stream of decoded bit estimates {circumflex over (b)} n  that is fed into a binary data sink  26 . The QSBM demodulator  24  includes a filter bank  28 , a sampler  30  operating at symbol rate, and a QSBM generator  32  generating the pair of binary coded bit estimates that constitute the quaternary data estimate {circumflex over (d)} n .  
         [0019]     Referring to  FIGS. 1, 2 ,  3 A,  3 B,  3 C, and  3 D, and more particularly to  FIG. 2 , the received baseband signal z r (t) is first filtered by the filter bank  28  consisting of at most F matched filters, where F=3·4 (L−1)  corresponds to the total number of PAM components in the quaternary GMSK signal. The filter bank  28  may be implemented as a matched filter bank or as an integrate-and-dump filter bank. As taught by Mengali, the transmitted quaternary GMSK baseband signal z b (t) for an N-symbol long quaternary data sequence {α n ; 0≦n≦N−1} has a quaternary PAM representation given by a z b (t) output equation. 
 
 z   b ( t )= A ·exp( jπhΣ   n =0 N−1 α n   ·g ( t−nT ))= A·Σ   k=0   F−1 Σ n=0   N−1   b   k,n   ·f   k ( t−nT ) 
 
         [0020]     In the z b (t) output equation, {b k,n }, known as the quaternary pseudo-symbols, are functions of the modulator input symbols {α n } ⊂ {±1, ±3}, and {f k (t)}, known as the quaternary PAM components, are explicitly defined in terms of the 2 (L−1)  binary PAM components originally taught by Kaleh. The filter bank 28 provides filtered signals r k (t) for 0≦k≦F−1, that are sampled by the sampler  30  at every symbol time boundaries t n =nT to produce discrete sample values r k,n . These discrete sample values are then processed by the QSBM generator  32  to provide the data estimates {circumflex over (d)} n  to the convolutional decoder  25 . In order to produce reliable data estimates {circumflex over (d)} n  the processing of the QSBM generator  32  must conform to the preceding performed by the quaternary precoder  12  on the source data symbols d n  in the transmitter  10 . The number of matched filters used in the filter bank  28  also dictates the exact processing structure of the QSBM generator  32 . For complexity reduction consideration, only the three most energy-dominant PAM components in the transmitted quaternary GMSK baseband signal z b (t) are considered when instituting the filter bank  28 . In this preferred form, the filter bank  28  consists of exactly three matched filters, that is, f 0 (−t), f 1 (−t) and f 2 (−t). The impulse response of these matched filters are time-reversibly related to the corresponding quaternary PAM components explicitly given by f 0 (t)=h 0 (t,h)h 0 (t,2h), f 1 (t)=h 0 (t+T,h)h 0 (t,2h), and f 2 (t)=h 0 (t,h)h 0 (t+T,2h) in terms of the dominant binary PAM component h 0 (t,a)=Π i=0   L c(t−iT,a) where c(t,a)=sin(πa−πag(|t|)/sin(πa) for |t|≦LT and c(t,a)=0 for |t|≧LT. The filtered signals r 0 (t), r 1 (t) and r 2 (t) are sampled by the sampler  30  and concurrently fed into the QSBM generator  32  as sampled inputs r 0,n , r 1,n  and r 2,n . In the preferred three-filter form, the sampled inputs r k,n , kε{0,1,2}, are fed into a most-significant-bit- 1  (MSB- 1 ) trellis  48  shown in  FIG. 3A , a most-significant-bit- 0  (MSB- 0 ) trellis  50  shown in  FIG. 3B , a least-significant-bit- 1  (LSB- 1 ) trellis  52  shown in  FIG. 3C , and a least-significant-bit- 0  (LSB- 0 ) trellis  54  shown in  FIG. 3D . As we shall soon explain in detail, the trellises  48 ,  50 ,  52 , and  54  are four-state constrained trellises, in that, either the MSB or the LSB of a quaternary state symbol un is restricted to take on a value of either 1 or 0. While the outputs of the MSB-constrained trellises  48  and  50  are subtracted at an adder  56  to form a ρ MSB  bit-metric stream  58 , the outputs of the LSB-constrained trellises  52  and  54  are subtracted at another adder  60  to form a ρ LSB  bit-metric stream  62 . The two bit-metric streams ρ MSB  and ρ LSB , together constituting a single stream of quaternary data estimates {circumflex over (d)} n , provide alternating soft bit metrics to the convolutional decoder  25 .  
         [0021]     The quaternary pseudo-symbols {b k,n } in the quaternary PAM representation of z b (t) are related to the modulator input data symbols {α n } ⊂ {±1, ±3} in such a manner that necessitates a differential decoding step when demodulating the CPM signal. The purpose of the quaternary data precoder  12  is to encode the source symbols {d n } ⊂ {±1, ±3} prior to the GMSK modulator  14  so that the resulting pseudo-symbols are directly related to the source symbols {d n }, thus avoiding the differential decoding step and improves demodulation performance in noise. Such a preceding scheme is achievable for all quaternary CPM signals generated using s frequency modulator  15  with modulation index h=1/4. This quaternary data precoding scheme is given by α n =[d n −d n−1 +1] mod8  (n≧0, d −1 =1) where [i] mod8  denotes a modified modulo-8 operation on integer i for which |[i] mod8 |&lt;4 is always maintained. With such a quaternary data precoding scheme applied in the preferred three-filter receiver form, the resulting three pseudo-symbols at any time t=nT involve only the two most recent source symbols {d n ,d n−1 } by quaternary pseudo-symbol equations. 
 
 b   0,n   =J   n   ·J   d is n  
 
 b   1,n   =J   n−1   ·J   d     n−1     +2α     n       (1)    
 
 b   2,n   =J   n−1   ·J   d     n−1     +α     n       (0)    
 
         [0022]     In the pseudo-symbol equations, J=exp(jπh)=(1+j)/√2 and {α n   (0) ,α n   (1) } ⊂ {±1} are the constituent binary symbols uniquely associated with the precoded quaternary symbol α n  through α n =α n   (0) +2α n   (1) . These precoded pseudo-symbols allow us to construct a four-state trellis, where the trellis state u n =a 1,n a 2,n ε(00,01,10,11} at time t=nT is defined as the two-bit pattern associated with the quaternary source symbol d n  at time t=nT, and apply the conventional MLSE Viterbi algorithm to identify the surviving state sequence and decide on the corresponding source symbol sequence. The branch metric u n , 0≦n≦N−1, needed for exercising this MLSE Viterbi algorithm in the preferred three-filter receiver is given by Re(r 0,n ·b* 0,n +r 1,n b* 1,n +r 2,n b* 2,n ). As mentioned earlier, because the MLSE Viterbi algorithm only produces sequence of estimated symbols, hence estimated bits, only hard-decision decoding is possible when this conventional MLSE demodulator is used in a coded quaternary CPM communication system.  
         [0023]     In contrast to the conventional MLSE Viterbi algorithm, the QSBM generator  32  is specifically devised to permit soft-decision decoding in a coded quaternary CPM communication system. With  u =(u 0 ,u 1 ,u 2 , . . . ,u N−1 ) denoting the equally probable quaternary source symbol sequence of length N, where u k =a 1,k a 2,k ε{00,01,10,11} and a 1,k , a 2,k  are, respectively, the most significant bit (MSB) and the least significant bit (LSB) of the k th  source symbol u k . With s(t, u ) denoting the transmitted GMSK waveform, and with r(t)=s(t, u )+n(t) denoting the corresponding RF waveform received over an AWGN channel with one-sided spectral density N 0 /2. The optimum soft bit metrics λ 1,k  and λ 2,k  for bits a 1,k  and a 2,k  of the k th  quaternary source symbol u k  of the transmitted N-symbol sequence  u  can be expressed as the log-likelihood ratios (LLR) as described in the following two LLR equations.  
               λ             ⁢     1   ,           ⁢   k         =     ln   ⁡     (               Pr   ⁢     (           ⁢         u             ⁢   k       ⁢           =           ⁢     10   ⁢           |           ⁢     r   ⁢     (   t   )           ,           ⁢     0   ⁢           ≤           ⁢   t   ⁢           ≤           ⁢   NT       )       ⁢           +                       ⁢     Pr   (           ⁢         u             ⁢   k       ⁢           =           ⁢     11   ⁢           |           ⁢     r   ⁢     (   t   )           ,           ⁢     0   ⁢           ≤           ⁢   t   ⁢           ≤           ⁢   NT       )                       Pr   ⁢     (           ⁢         u             ⁢   k       ⁢           =           ⁢     00   ⁢           |           ⁢     r   ⁢     (   t   )           ,           ⁢     0   ⁢           ≤           ⁢   t   ⁢           ≤           ⁢   NT       )       ⁢           +                       ⁢     Pr   (           ⁢         u             ⁢   k       ⁢           =           ⁢     01   ⁢           |           ⁢     r   ⁢     (   t   )           ,           ⁢     0   ⁢           ≤           ⁢   t   ⁢           ≤           ⁢   NT       )               )                   =     ln   ⁡     (                 ∑             ⁢     u   _         ⁢     exp   ⁡     (       2             ⁢     N             ⁢   0           ⁢       ∫   0             ⁢   NT       ⁢     r   ⁢     (   t   )     ⁢     s   ⁡     (     t   ,     u   _     ,       u             ⁢   k       =   10       )       ⁢           ⁢     ⅆ   t           )         +               exp   ⁡     (       2     N   0       ⁢       ∫   0   NT     ⁢       r   ⁡     (   t   )       ⁢     s   ⁡     (     t   ,     u   _     ,       u   k     =   11       )       ⁢           ⁢     ⅆ   t           )                         ∑             ⁢     u   _         ⁢     exp   ⁡     (       2             ⁢     N             ⁢   0           ⁢       ∫   0             ⁢   NT       ⁢     r   ⁢     (   t   )     ⁢     s   ⁡     (     t   ,     u   _     ,       u             ⁢   k       =   00       )       ⁢           ⁢     ⅆ   t           )         +               exp   ⁡     (       2     N   0       ⁢       ∫   0   NT     ⁢       r   ⁡     (   t   )       ⁢     s   ⁡     (     t   ,     u   _     ,       u   k     =   01       )       ⁢           ⁢     ⅆ   t           )               )                 
               λ     1   ,   k       =       ⁢     ln   ⁡     (               Pr   ⁡     (         u   k     =     10   |     r   ⁡     (   t   )           ,     0   ≤   t   ≤   NT       )       +               Pr   ⁡     (         u   k     =     11   |     r   ⁡     (   t   )           ,     0   ≤   t   ≤   NT       )                       Pr   ⁡     (         u   k     =     00   |     r   ⁡     (   t   )           ,     0   ≤   t   ≤   NT       )       +               Pr   ⁡     (         u   k     =     01   |     r   ⁡     (   t   )           ,     0   ≤   t   ≤   NT       )               )                   =       ⁢     ln   ⁡     (                 ∑     u   _       ⁢     exp   ⁡     (       2     N   0       ⁢       ∫   0   NT     ⁢       r   ⁡     (   t   )       ⁢     s   (     t   ,     u   _     ,       u   k     =   10       ⁢           )     ⁢     ⅆ   t           )         +               exp   ⁡     (       2     N   0       ⁢       ∫   0   NT     ⁢       r   ⁡     (   t   )       ⁢     s   ⁡     (     t   ,     u   _     ,       u   k     =   11       )       ⁢           ⁢     ⅆ   t           )                         ∑     u   _       ⁢     exp   ⁡     (       2     N   0       ⁢           ⁢       ∫   0   NT     ⁢               ⁢     r   ⁡     (   t   )         ⁢           ⁢     s   (     t   ,           ⁢     u   _     ,           ⁢       u   k     ⁢           =           ⁢   00       ⁢           )     ⁢           ⁢     ⅆ   t           )         +               exp   ⁡     (       2     N   0       ⁢       ∫   0   NT     ⁢       r   ⁡     (   t   )       ⁢     s   ⁡     (     t   ,     u   _     ,       u   k     =   01       )       ⁢           ⁢     ⅆ   t           )               )                 
 
         [0024]     In the LLR equations, the expression s(t, u ,u k =a 1,k a 2,k ) denotes the waveform corresponding to an N-symbol quaternary sequence  u  for which the MSB and LSB of the k th  quaternary symbol u k  is a 1,k  and a 2,k , respectively. As is apparent from the LLR equations, the optimum soft bit metrics λ 1,k  and λ 2,k  are impractical to implement because its computational complexity grows exponentially with sequence length N. Moreover, the LLR computations require the evaluation of nonlinear functions as well as the knowledge of N 0 , which may be difficult to ascertain in practice. To circumvent these implementation difficulties, simpler approximations to these LLR expressions can be used. Specifically, the numerator sum and denominator sum of both LLR expressions are approximated by their respective maximum terms. Upon replacing these sums with the respective maximum terms and omitting the constant factor (2/N 0 ), the following two approximate LLR equations are obtained. 
 
ρ 1,k =∫ 0   NT   r ( t ) s ( t, u     (i,is 1   )   , u   k =1 a   2,k   (i     1     ) ) dt −∫ 0   NT   r ( t ) s ( t, u     j     1     )   , u   k =0 a   2,k   (j     1     ) ) dt  
 
ρ 2,k =∫ 0   NT   r ( t ) s ( t, u     (i,is 2   )   , u   k   =a   1,k   (i     2     ) 1) dt −∫ 0   NT   r ( t ) s ( t, u     j     2     )   , u   k   =a   1,k   (j     2     ) 0) dt  
 
         [0025]     In the LLR equations,  u   (i)  for iε{i 1 , i 2 , j 1 , j 2 }, denote the source symbol sequences that achieve the respective maximums and (a 1,k   (i), a   2,k   (i) } denote the bit values associated with the k th  symbol u k  of sequence  u   (i) . As is apparent from the approximate LLR equations, the approximate soft bit metrics ρ 1,k  and ρ 2,k  are free of the implementation difficulties associated with the optimum soft bit metrics, and the respective bit decisions on a 1,k  and a 2,k , that are readily usable for hard-decision decoding, can be obtained simply by taking their algebraic signs. More importantly, as is also apparent from the approximate LLR equations, each of the four integrals defining the approximate soft bit metrics ρ 1,k  and ρ 2,k  can be efficiently computed by applying the conventional MLSE Viterbi algorithm over a constructed state-constrained trellis. This is due to the fact that, as the conventional MLSE Viterbi algorithm seeks maximum signal correlation over each of these four constrained trellises, the corresponding integral coincides exactly with the resulting survivor path metric. Referring to  FIG. 2 , the QSBM generator  32  applies the conventional MLSE Viterbi algorithm over two pairs of state-constrained trellises, that is, an MSB-constrained pair consisting of the MSB- 1  and MSB- 0  trellises respectively shown in  FIG. 3A  and  FIG. 3B , and an LSB-constrained pair consisting of the LSB- 1  and LSB- 0  trellises respectively shown in  FIG. 3C  and  FIG. 3D . While the MSB- 1  trellis  48  effectively computes the first integral of the ρ 1,k  equation, the MSB- 0  trellis  50  concurrently computes the second integral of the ρ 1,k  equation. These two MSB-constrained integrals are subtracted at the adder  56  to yield the approximate soft bit metric ρ 1,k , thus generating the ρ MSB  bit-metric stream  58 . In a similar manner, the QSBM generator  32  generates the ρ LSB  bit-metric stream  62  by applying the conventional MLSE Viterbi algorithm over the LSB- 1  trellis  52  and LSB- 0  trellis  54  and differencing the resulting surviving path metrics at the adder  60  to form the approximate soft bit metric ρ 2,k . As already mentioned, the two bit-metric streams ρ MSB  and ρ LSB , together constituting a single stream of quaternary data estimates {circumflex over (d)} n    64 , provide alternating soft bit metrics to the convolutional decoder  25 .  
         [0026]     Referring specifically to  FIGS. 3A, 3B ,  3 C and  3 D, both the MSB-constrained and LSB-constrained trellises have the same transitional structure as that of an unconstrained MLSE trellis over the observation window 0≦t&lt;nT. Over the observation window nT≦t&lt;(n+1)T, an MSB=1 constraint is imposed on the state symbol u n  in the MSB- 1  trellis, whereas an MSB=0 constraint is imposed in the MSB- 0  trellis. Similarly, an LSB=1 constraint is imposed in the LSB- 1  trellis, and an LSB=0 constraint is imposed in the LSB- 0  trellis. Each of these four constrained trellises reappears as an unconstrained MLSE trellis at time t=(n+1)T and continues to propagate in the same unconstrained manner for another (N d −1) symbol intervals, where N d ≧1 is a design parameter. The parameter N d  provides an avenue for performance optimization at the expense of introducing a decision delay of N d T seconds. In addition to being able to generate soft bit metrics that permit soft-decision decoding, another important processing advantage of the QSBM demodulator  24  is that, unlike conventional quaternary CPM demodulator, there is no need to store the survivor path states when applying the conventional MLSE Viterbi algorithm over each of the four constrained trellises because only the survivor path metrics need to be updated. Also, because the unconstrained MLSE trellis portion over the observation window 0≦t&lt;nT is common to all four constrained trellises, the associated add-and-compare processing needs only to be performed once. In general, the choice of the decision delay parameter N d  depends on the projected bit error rate (BER) at the decoder input, which in turn depends on the error correction code selected for the coded communication system and the overall objective BER. As an example, for a typical projected decoder input error rate of 10 −2  using a convolutional-coded quaternary GMSK signal with BT=1/3=1/L and h=1/4, a choice of N d &lt;3 results in significant performance degradation with respect to the conventional MLSE demodulator, whereas a choice of N d &gt;3 brings only negligible performance improvement with respect to the optimal choice of N d =3.  
         [0027]     Referring to all Figures, and particularly to  FIG. 4  of performance data, the QSBM demodulator  24  provides improved performance with respect to the conventional MLSE demodulator even when using only three matched filters in the filter bank  28  for a convolutional-coded GMSK communication system with a BT value of 1/3 or 1/2. The error correction code used by the encoder  27  and decoder  25  may be a rate-1/2 non-systematic non-recursive convolution code with constraint length 7 and code generators {133,171} in octal format. The transmitter  10  may include a 128×128 block interleaver between the convolutional encoder  27  and the quaternary precoder  12 , and the receiver  20  would include a corresponding deinterleaver between the QSBM generator  32  and the sixty-four-state maximum likelihood convolutional decoder  25 . The convolutional decoder  25  performs either soft- or hard-decision decoding depending upon whether the soft bit metrics ρ 1,k  and ρ 2,k  or their algebraic signs are outputted from the QSBM generator  32 . The decision delay parameter N d  is set at the optimal choice of N d =3, and the path length used by the convolutional decoder  25  is set at 32. The channel  18  is AWGN with no extraneous interference. The performance advantage of the proposed QSBM demodulator with respect to the conventional MLSE demodulator is measured by the difference in required SNR between using soft- or hard-decision decoding. With BT=1/3, for example, the proposed QSBM demodulator has a power advantage of about 3.1 dB over the conventional MLSE demodulator in achieving an overall objective BER of 10 −5 . Similar performance advantage is observed for BT=1/2. Here, the power advantage of using the proposed QSBM demodulator is about 2.7 dB.  
         [0028]     The invention is directed to a soft bit metric demodulator for quaternary CPM signals. The QSBM demodulator  24  is computationally as efficient as the conventional quaternary CPM demodulator, yet requires significantly fewer storage elements as it needs not store the survivor path states when generating the soft bit metrics using the QSBM generator  32 . The QSBM demodulator  24  can be applied to a convolutional-coded quaternary GMSK communication system with BT values of 1/3 and 1/2, resulting in a respective power advantage of 3.1 dB and 2.7 dB over the conventional quaternary CPM demodulator. The preferred QSBM demodulator  24  uses three matched filters in the filter bank  28 , but can be extended to a larger set of matched filters. The QSBM demodulator  24  is directly applicable to all precoded quaternary CPM signals. The quaternary data precoder  12  embedded in the quaternary GMSK modulator  10  is directly applicable to providing the necessary data precoding for general quaternary CPM signals. 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.