Abstract:
A symbol epoch tracking circuit and method for a Continuous Phase Modulation (CPM) receiver. A phase tracking circuit performs carrier phase tracking with little extra computation for a Viterbi decoder and has an excellent tracking performance. The symbol epoch tracking circuit is implemented in a CPM receiver, and could also be implemented as a software defined process for use in any CPM demodulator employing a Viterbi algorithm for accurate data detection.

Description:
BACKGROUND OF THE INVENTION 
   Field of the Invention 
   The present invention pertains generally to continuous wave modulation, and more particularly relates to a reduced complexity tracking algorithm for the symbol timing synchronization of a continuous phase modulation (CPM) input data signal using a Viterbi algorithm for demodulation. 
   Continuous phase modulation (CPM) is a modulation format which generates a constant envelope signal and which possesses advantageous spectral properties. CPM has found utility in systems where it is desirable to use a constant envelope modulation, such as where high power nonlinear amplifiers are used. The constant envelope constraint introduces memory at the transmitter, effectively making a CPM signal more difficult to detect. Typically a Viterbi algorithm is used to perform sequence detection on the received symbol sequence. In order to compute the proper branch metric signals required in the Viterbi algorithm, the receiver must have an accurate estimate of the phase of the transmitter&#39;s symbol clock signal with respect to the received signal. This is known as symbol timing synchronization. The present invention describes a reduced complexity-tracking algorithm for the symbol timing synchronization of a CPM modulated signal where the Viterbi algorithm is used for demodulation. 
   Trellis decoders or demodulators are frequently used to demodulate signals modulated by continuous phase modulation (CPM). In the context of the present invention, coding/decoding and modulation/demodulation are analogous, and may be viewed as corresponding. CPM modulation has an advantage of providing a signal having substantially constant power, which is a marked advantage when transmitting the modulated signals over a nonlinear channel, as the constant power tends to reduce the generation in such a channel of unwanted distortion products which obscure the signals. A further advantage of CPM modulation is that the bandwidth of the transmitted signal is easily maintained, and the frequency spectrum exhibits low sidelobes, which is advantageous for situations in which a plurality of signals traverse a channel, as the signal spectrum for one of the signals traversing the channel has little frequency overlap with the signal next adjacent in frequency. In other words, the channels may be closer together in frequency. 
   CPM modulation is performed, in general, by converting the information or signal to m-ary quantized form, if not already in the desired form. For the simple case in which m=2, the signal is converted into binary form. The m-ary signal is applied to a shift register array having a particular length. As the signal bits are applied to the input end of the shift register array, the previously-applied signals are clocked and propagate through the shift register array, altering the states of the registers in succession. A combinatorial or functional logic arrangement is coupled to the output of each register of the array, and applies some function to the combination of register logic states. The applied function results in one or more output signals, which depend upon the combinatorial function, and also depend upon the current state of each register in the array. The current state of each register in the array in turn depends upon the history of the input signal. 
   The demodulation of a signal modulated in the above-described fashion can be accomplished by a trellis demodulator. The “trellis” represents, by “nodes”, the possible states of the registers of the modulator, and by lines joining the nodes, the possible paths by which transitions between states can be made. The trellis demodulator is often implemented as a Viterbi algorithm which performs sequence detection on the received symbol sequence. Demodulation using a Viterbi algorithm requires an accurate estimate of the phase of the transmitter&#39;s symbol clock signal with respect to the received signal. The process of obtaining an accurate estimate of the phase of the transmitter&#39;s symbol clock signal with respect to the received signal is known as symbol timing synchronization. 
   SUMMARY OF THE INVENTION 
   It is therefore an object of the present invention to provide a method of and apparatus for accurately estimating the timing of a transmitter&#39;s symbol clock signal with respect to a received signal. 
   It is another object of the present invention to realize a new symbol epoch tracking circuit for CPM receivers which overcomes the shortcomings of prior art conventional CPM receivers. 
   To that end, a symbol epoch tracking circuit performs symbol epoch tracking with minimal extra computation for a Viterbi decoder using a Viterbi algorithm and with excellent tracking performance. 
   The symbol epoch tracking circuit is implemented in a CPM receiver, and could also be implemented as a software defined process for use in any CPM demodulator employing a Viterbi algorithm for accurate data detection. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
       FIG. 1  is a simplified block diagram of a maximum likelihood symbol timing estimator which functions in accordance with the teachings of the present invention. 
       FIG. 2  is a simplified block diagram of a maximum likelihood symbol timing estimator of reduced complexity wherein a summation term is implemented as a lookup table. 
       FIG. 3  is a simplified block diagram of a maximum likelihood phase tracking synchronizer demodulator which includes a Viterbi decoder. 
   

   DETAILED DESCRIPTION OF THE INVENTION 
   A maximum likelihood estimator for the transmitted symbol timing epoch can be derived using maximum likelihood theory. 
   An appropriate likelihood function L(t, θ, τ, {overscore (d)}) for estimating the symbol timing epoch in a CPM demodulator is defined by equation (1): 
               L   ⁡     (     t   ,   θ   ,   τ   ,   d     )       =     C   ⁢           ⁢   exp   ⁢     {       -     1     N   o         ⁢       ∫     T   o       ⁢         (       y   ⁡     (   t   )       -     s   ⁡     (     t   ,   θ   ,   τ   ,   d     )         )     2     ⁢     ⅆ   t           }               (   1   )             
 
where N 0  is the noise power, T 0  is the observation interval, y(t) is the received signal, C is a constant responsive to the amplitude of the received signal, and s(t, θ, τ, {overscore (d)}) is the transmitted signal. The parameters of the transmitted signals are θ, the carrier phase, τ, the symbol timing reference, and {overscore (d)}, the transmitted data sequence. {overscore (d)} is a vector and is referred to herein as d or {overscore (d)}. Taking logarithms and expanding the squared term in equation (1) gives the log-likelihood function as set forth in equation (2): 
               l   ⁡     (     t   ,   θ   ,   τ   ,     d   _       )       =       ln   ⁡     (   C   )       -       1     N   o       ⁢       ∫     T   o       ⁢       (     s   ⁡     (     t   ,   θ   ,   τ   ,   d     )       )     2         +       (     y   ⁡     (   t   )       )     2     -     2   ⁢     y   ⁡     (   t   )       ⁢     s   ⁡     (     t   ,   θ   ,   τ   ,   d     )       ⁢     ⅆ   t                 (   2   )             
 
   The constant first term of equation (2) and the second term within the integral of equation (2) are independent of the parameter τ and may be dropped. For a constant envelope scheme such as CPM, the first term within the integral of equation (2) is also independent of the carrier phase reference θ. The equivalent log-likelihood function to be maximized is therefore given by equation (3): 
               l   ⁡     (     t   ,   θ   ,   τ   ,   d     )       =       1     N   o       ⁢       ∫     T   o       ⁢     2   ⁢     y   ⁡     (   t   )       ⁢     s   ⁡     (     t   ,   θ   ,   τ   ,   d     )       ⁢     ⅆ   t                   (   3   )             
 
   A necessary condition for a maximum of the equivalent log-likelihood function of equation (3) is that the derivative be zero at the maximum. Differentiating equation (3) with respect to the symbol timing reference, τ, and setting the result equal to zero, gives likelihood equation (4) for the estimation of the symbol timing epoch. 
             0   =       2     N   o       ⁢       ∫     T   o       ⁢       y   ⁡     (   t   )       ⁢       ∂     s   ⁡     (     t   ,   θ   ,   τ   ,     d   _       )           ∂   τ       ⁢     ⅆ   t                   (   4   )             
 
   The transmitted signal in a CPM arrangement can be expressed as s(t, θ, τ, d) in equation (5): 
               s   ⁡     (     t   ,   θ   ,   τ   ,   d     )       =     Re   ⁡     [         (       2   ⁢   E     T     )       1   /   2       ⁢     exp   ⁡     (     j   ⁡     (         ω   o     ⁡     (     t   +   τ     )       +   θ   +     π   ⁡     (       ∑     i   =     -   ∞         n   -   L       ⁢           ⁢       d   i     ⁢   h       )       +     2   ⁢   π   ⁢       ∑     i   =     n   -   L   +   1       n     ⁢           ⁢       d   i     ⁢     hq   ⁡     (     t   -   iT   +   τ     )               )       )         ]               (   5   )             
 
where {overscore (d)} is the data vector, E is the transmit energy, T is the symbol period, ω 0  is the carrier frequency, and {overscore (d)}=(d_, d n−2 , d n−1 , d n ) is the transmit information or data sequence. In equation (5), the parameter q(t−iT+τ) is the phase pulse, L is the duration of the phase pulse, and h is the modulation index. Substituting the definition of the transmitted signal of equation (5) into the partial derivative of equation (4) one obtains equation (6): 
                 ∂     s   ⁡     (     t   ,   θ   ,   τ   ,     d   _       )           ∂   τ       =     Re   ⁡     [         j   ⁡     (       2   ⁢   E     T     )         1   /   2       ⁢   2   ⁢   π   ⁢       ∑                 i   =     n   -   L   +   1       ⁢               n   ⁢               ⁢     d   i     ⁢   h   ⁢       ∂     q   ⁡     (     t   -   iT   +   τ     )           ∂   τ       ⁢     exp   ⁡     (     j   ⁡     (         ω   o     ⁡     (     t   +   τ     )       +   θ   +     π   ⁡     (       ∑     i   =     -   ∞         n   -   L       ⁢           ⁢       d   i     ⁢   h       )       +     2   ⁢   π   ⁢       ∑     i   =     n   -   L   +   1       n     ⁢           ⁢       d   i     ⁢     hq   ⁡     (     t   -   iT   +   τ     )               )       )         ]               (   6   )             
 
   Ignoring the constants, the likelihood equation associated with equation (6) is equation (7): 
             0   =       ∫     T   o       ⁢       Re   ⁡     [     j   ⁢           ⁢     y   ⁡     (   t   )       ⁢       ∑               i   =     n   -   L   +   1         n   ⁢               ⁢     d   i     ⁢       ∂     q   ⁡     (     t   -   iT   +   τ     )           ∂   τ       ⁢     exp   ⁡     (     j   ⁡     (         ω   o     ⁡     (     t   +   τ     )       +   θ   +     π   ⁡     (       ∑     i   =     -   ∞         n   -   L       ⁢           ⁢       d   i     ⁢   h       )       +     2   ⁢   π   ⁢       ∑     i   =     n   -   L   +   1       n     ⁢           ⁢       d   i     ⁢     hq   ⁡     (     t   -   iT   +   τ     )               )       )         ]       ⁢     ⅆ   t                 (   7   )             
 
   The derivative of the phase pulse q(t) with respect to the symbol timing epoch is equal to the frequency pulse g(t) so that: 
             0   =       ∫     T   o       ⁢       Re   ⁡     [     j   ⁢           ⁢     y   ⁡     (   t   )       ⁢       ∑               i   =     n   -   L   +   1         n   ⁢               ⁢     d   i     ⁢     g   ⁡     (     t   -   iT   +   τ     )       ⁢     exp   ⁡     (     j   ⁡     (         ω   o     ⁡     (     t   +   τ     )       +   θ   +     π   ⁡     (       ∑     i   =     -   ∞         n   -   L       ⁢           ⁢       d   i     ⁢   h       )       +     2   ⁢   π   ⁢       ∑     i   =     n   -   L   +   1       n     ⁢           ⁢       d   i     ⁢     hq   ⁡     (     t   -   iT   +   τ     )               )       )         ]       ⁢     ⅆ   t                 (   8   )             
 
   To derive a structure from the above equation, we can make a few assumptions. The data sequence vector d, and the carrier transmit phase, θ, are not known to the receiver. However, if the receiver is in a tracking mode so that carrier tracking errors are small, and the signal to noise ratio is high enough so that the detected data sequence is usually correct, then the receiver&#39;s estimates can be substituted for these parameters. The right hand side of equation (8) can then be used as an error signal to correct the current estimate of the symbol timing epoch reference. Furthermore, the term: 
             Re   ⁡     [     jexp   ⁡     (     j   ⁡     (         ω   o     ⁡     (     t   +     τ   ^       )       +   θ   +     π   ⁡     (       ∑     i   =     -   ∞         n   -   L       ⁢           ⁢         d   ^     i     ⁢   h       )       +     2   ⁢   π   ⁢       ∑     i   =     n   -   L   +   1       n     ⁢           ⁢         d   ^     i     ⁢     hq   ⁡     (     t   -   iT   +     τ   ^       )               )       )       ]             (   9   )             
 
is just the receiver&#39;s estimate of the transmitted signal phase shifted by 90°.
 
     FIG. 1  is a simplified block diagram of a Maximum Likelihood (ML) symbol timing estimator  10  which functions in accordance with equations (1)-(9). An input signal y(t) is provided to a CPM detector  11  and a multiplier  12 . The multiplier  12  multiplies the input signal y(t) with js(t), a regenerated transmit signal s(t) produced by a transmit signal regenerator  16  which is shifted by 90° through a phase shifter  19  and g(t), the transmit frequency signal. The output of the multipier  12  is passed through a low pass filter (LPF)  13  to provide an input control signal for a VCO  14 . The VCO  14  provides an output symbol timing reference signal τ to the CPM detector  11 , to the transmit signal regenerator  16 , and to the transmit signal frequency estimator  17 . 
   Based upon the tracking mode receiver as explained above, a phase estimator  18  provides a phase signal θ to the CPM detector  11  and also to the transmit signal regenerator  16 . The CPM detector  11  outputs a vector signal d to both the transmit signal regenerator  16  and the transmit frequency estimator  17 . 
   Much of the complexity in the Maximum Likelihood (ML) symbol timing tracking circuit of  FIG. 1  is due to complexities of the transmit signal regenerator  16  and the multiplier  12 . A reduced complexity symbol timing estimation algorithm can be implemented which operates with a CPM signal that is demodulated using the Viterbi algorithm. The maximum likelihood symbol timing estimator performs a correlation similar to the correlation performed to compute the branch metric signals of the Viterbi algorithm. The branch metric signals in the trellis of the CPM signal are computed using 
               λ   ⁡     (       a   ^     ,   m     )       =       ∫     t   =   mT       mT   +   1       ⁢       y   ⁡     (   t   )       ⁢     Re   ⁡     [     exp   ⁡     (     j   ⁡     (         ω   o     ⁡     (     t   +   τ     )       +   θ   +     π   ⁡     (       ∑     i   =     -   ∞         m   -   L       ⁢           ⁢       d   i     ⁢   h       )       +     2   ⁢   π   ⁢       ∑     i   =     m   -   L   +   1       m     ⁢           ⁢       d   i     ⁢     hq   ⁡     (     t   -   iT   +   τ     )               )       )       ]       ⁢     ⅆ   t                 (   10   )             
 
   A distinct branch metric signal is computed for each branch of the trellis. We now define 
               Θ     m   -   L       =     π   ⁡     (       ∑     i   =     -   ∞         m   -   L       ⁢           ⁢       d   i     ⁢   h       )               (   11   )             
 
as a phase state of the branch, and
 
 a   m =( d   m−L+1   , d   m−L+2   , . . . d   m )  (12)
 
as the correlative state of the branch. There are usually more than two distinct phase states, so that the computation of the branch metric signals for the same correlative state can be performed by using a complex correlator for the correlative state, and then by applying a phase rotation of this complex value to obtain the branch metric signal for each phase state. Two correlators compute the values. 
                   λ   I     ⁡     (     d   ,   m     )       =       ∫     t   =   mT       mT   +   1       ⁢       y   ⁡     (   t   )       ⁢     Re   ⁡     [     exp   ⁡     (     j   ⁡     (         ω   o     ⁡     (     t   +   τ     )       +   θ   +     2   ⁢   π   ⁢       ∑     i   =     m   -   L   +   1       m     ⁢           ⁢       d   i     ⁢     hq   ⁡     (     t   -   iT   +   τ     )               )       )       ]       ⁢           ⁢     ⅆ   t           ⁢     
     ⁢   and           (   13   )                   λ   Q     ⁡     (     d   ,   m     )       =       ∫     t   =   nT       nT   +   1       ⁢       y   ⁡     (   t   )       ⁢     Re   ⁡     [     jexp   ⁡     (     j   ⁡     (         ω   o     ⁡     (     t   +   τ     )       +   θ   +     2   ⁢   π   ⁢       ∑     i   =     m   -   L   +   1       m     ⁢           ⁢       d   i     ⁢     hq   ⁡     (     t   -   iT   +   τ     )               )       )       ]       ⁢           ⁢     ⅆ   t                 (   14   )             
 
The computed values are multiplied by a complex number representing each of the possible values of equation 11. The real component after this multiplication is the desired branch metric signal of equation 10. The complex component is normally discarded. The complex component is defined as Q (a,m) and is 
               Q   ⁡     (     a   ,   m     )       =       ∫     t   =   mT       mT   +   1       ⁢       y   ⁡     (   t   )       ⁢     Re   ⁡     [     jexp   ⁡     (     j   ⁡     (         ω   o     ⁡     (     t   +   τ     )       +   θ   +     π   ⁡     (       ∑     i   =     -   ∞         m   -   L       ⁢           ⁢       d   i     ⁢   h       )       +     2   ⁢   π   ⁢       ∑     i   =     m   -   L   +   1       m     ⁢           ⁢       d   i     ⁢     hq   ⁡     (     t   -   iT   +   τ     )               )       )       ]       ⁢           ⅆ               (   15   )             
 
   To facilitate the description of this reduced complexity timing epoch estimator, the error signal for estimation of the timing epoch is rewritten as 
               τ   err     =       ∫     T   o       ⁢         ∑                 i   =     m   -   L   +   1       ⁢               m   ⁢               ⁢         ⅆ   ^     i     ⁢     g   ⁡     (     t   -   iT   +     τ   ^       )         ⁢     Re   ⁡     [     j   ⁢           ⁢     y   ⁡     (   t   )       ⁢     exp   ⁡     (     j   ⁡     (         ω   o     ⁡     (     t   +     τ   ^       )       +     θ   ^     +     π   ⁡     (       ∑     i   =     -   ∞         m   -   L       ⁢           ⁢         ⅆ   ^     i     ⁢   h       )       +     2   ⁢   π   ⁢       ∑     i   =     m   -   L   +   1       m     ⁢           ⁢         d   ^     i     ⁢     hq   ⁡     (     t   -   iT   +     τ   ^       )               )       )         ]       ⁢     ⅆ   t                 (   16   )             
 
   Equation 16 can be simplified by assuming that the first summation is constant over a T symbol period so that the error signal can be separated into two terms. Then the error signal is approximated as 
               τ   err     ≈         [       ∑     i   =     m   -   L   +   1       m     ⁢           ⁢         d   ^     i     ⁢     g   ⁡     (     t   -   iT   +     τ   ^       )           ]     _     ⁢       ∫     t   =   mT       mT   +   1       ⁢       Re   ⁡     [       jy   ⁡     (   t   )       ⁢     exp   ⁡     (     j   ⁡     (         ω   o     ⁡     (     t   +     τ   ^       )       +     θ   ^     +     π   ⁡     (       ∑     i   =     -   ∞         m   -   L       ⁢           ⁢         d   ^     i     ⁢   h       )       +     2   ⁢   π   ⁢       ∑     i   =     m   -   L   +   1       m     ⁢           ⁢         d   ^     i     ⁢     hq   ⁡     (     t   -   iT   +     τ   ^       )               )       )         ]       ⁢           ⁢     ⅆ   t                   (   17   )             
 
   The integral term is the quantity Q (a,m) introduced above, and is computed in the branch metric signal calculations in the Viterbi algorithm. 
     FIG. 2  illustrates a reduced complexity symbol timing estimator  10 ′ in which in equation (17), the summation term, 
                 ∑     i   =     m   -   L   +   1       m     ⁢           ⁢       d   i     ⁢     g   ⁡     (     t   -   iT   +   τ     )           _           (   18   )               
is precalculated and stored in a lookup table  20 . This summation term depends only on the last L output symbols of the Viterbi decoder. The values are precomputed and are stored in the lookup table  20  having a size M L .
 
   A multiplier  21  then multiplies the output of the lookup table  20  with the output Q p,s  of the CPM detector  11 . The output of the multiplier  21  is filtered by the low pass filter  13  and controls the phase of the VCO  14  output to produce the symbol timing reference signal τ. 
     FIG. 3  illustrates a CPM receiver  30  which uses a Viterbi algorithm to perform sequence detection on a received symbol sequence in a CPM signal. The CPM receiver computes the branch metric signals required in the Viterbi algorithm, which requires an accurate estimate of the phase of the transmitter&#39;s symbol clock signal with respect to the received signal, which is termed symbol timing epoch synchronization. The CPM receiver  30  uses a Viterbi trellis (termed trellis because it looks like an interweaved trellis) decoder or demodulator  36  to demodulate the CPM signals. The decoder  36  represents, by nodes, the possible states of the shift registers of the modulator, and by lines joining the nodes, the possible paths by which transitions between states can be made. The decoder  36  computes a distinct branch metric signal for each branch of the trellis, which is representative of the likelihood that that branch is in a modulator path. The decoder  36  uses the computed branch metric signals to select one path through the trellis having the highest probability of representing the CPM modulated data signal. 
   Referring to  FIG. 3 , a received input signal y(t) is applied to a first input port of a correlative branch calculator  32 . An estimated transmitter phase reference θ from phase estimator  18  and an estimated symbol timing signal τ from VCO  14  are applied to second and third input ports of the calculator  32 , which calculates and produces output signals λ i  (A,m) and λ q  (A,m) according to equation (10). The calculated signals λ i  (A,m) and λ q  (A,m) are applied to a phase rotator  34  which rotates the phase thereof to produce outputs which are applied to the Viterbi decoder  36 . The Viterbi decoder  36  performs the usual determination of the most likely trellis state, and produces an output {overscore (d)} of the estimated data sequence. The Viterbi decoder  36  also keeps track of the phase transitions occurring in the trellis which are associated with paths leading to each state. A set of these phase transitions are associated with each current state. The Viterbi decoder selects the most likely or most probable state, and also outputs an associated set of phase transitions. 
   The present invention departs from a conventional CPM receiver by phase shifting the computed branch metric signals by π/2, to thereby generate phase shifted branch metric signals associated with each path. At each symbol interval, a symbol timing estimator selects the associated phase shifted branch metric signal for that one path having the highest probability. The symbol timing estimator multiplies the selected phase shifted branch metric signal by a term representative of a summation of a plurality of weighted frequency pulses, and uses the resultant product to produce a symbol timing reference signal τ, which the CPM detector uses to adjust the timing epoch. 
   The embodiment of  FIGS. 1 and 2  are related and merely use different signals which are generated and already computed in the CPM detector. 
   The embodiment of  FIG. 1  takes the computed data vector signal {overscore (d)} of the CPM detector, and uses the symbol timing reference output signal τ and the estimated phase θ of the transmitter symbol clock signal to produce a highest probability brand metric signal s(t), which is then 90° phase shifted to produce js(t), one input to multiplier  12 . The transmitted signal frequency estimator uses the same two input signals of {overscore (d)} and τ to produce g(t). The multiplier  12  then multiplies the input signal y(t) by each of the two signals js(t) and g(t), and the product controls the frequency of VCO  14  which produces the symbol timing reference signal τ, which is an input to the CPM detector  11 . 
   The embodiment of  FIG. 2  uses two signals which are computed by the CPM detector, the data vector signal {overscore (d)}, and Qps which is merely the selected highest probability branch metric signal js(t) multiplied by the CPM input signal y(t). the data signal {overscore (d)} is input to a lookup data table  2 , which then provides g(t), the transmit frequency signal g(t), which is multiplied by multiplier  21  with Qps=js(t)x y(t), to produce an output signal which controls the VCO to produce the symbol timing estimator signal τ. 
   Both embodiments are related by performing a similar multiplication of the input signal y(t), the transmit frequency signal g(t), and the phase-shifted highest probability signal js(t). 
   The assumption that the summation of the frequency pulses is constant over a symbol interval will in general cause a degradation of the estimator with respect to the maximum likelihood estimator. Simulations have shown that the degradation is modest for many types of CPM modulations. 
   While several embodiments and variations of the present invention for a simplified symbol timing tracking circuit for a CPM modulated signal are described in detail herein, it should be apparent that the disclosure and teachings of the present invention will suggest many alternative designs to those skilled in the art.