Abstract:
A method of producing a correction signal includes receiving a predetermined data sequence ( 500 ). The data sequence is sampled at predetermined times, thereby producing a sampled data sequence ( 522, 532 ). The sampled data sequence is separated into first and second sampled data sequences. A ratio is calculated ( 550, 558 ) from the first and second sampled data sequences. A correction signal is produced ( 556, 564 ) in response to the ratio.

Description:
This application claims the benefit of U.S. Provisional Application No. 60/104,099, filed Oct. 13, 1998. 

   FIELD OF THE INVENTION 
   This invention relates to time division multiple access (TDMA) for a communication system and more particularly to a method for synchronizing carrier phase and symbol timing in a mobile receiver. 
   BACKGROUND OF THE INVENTION 
   Present time division multiple access (TDMA) systems are characterized by simultaneous transmission of different data signals over a common channel by assigning each signal a unique time period. These data signals are typically transmitted as binary phase shift keyed (BPSK) or quadrature phase shift keyed (QPSK) data symbols during such unique time periods. These unique periods are allocated to a selected receiver to determine the proper recipient of a data signal. Allocation of such unique periods establishes a communication channel between a transmitter and selected remote receivers for narrow band transmission. This communication channel may be utilized for cable networks, modem transmission via phone lines or for wireless applications. 
   A selected TDMA receiver must determine both carrier phase and symbol timing of its unique period from the received signal for data recovery. The carrier phase is necessary for generating a reference carrier with the same phase as the received signal. This reference carrier is used to coherently demodulate the received signal, thereby creating a baseband signal. Symbol timing synchronization of the receiver with the transmitter is necessary for the receiver to extract correct data symbols from the baseband signal. 
   Previous studies, such as J. G. Proakis,  Digital Communications  347–350 (1995), have utilized decision-directed phase locked loops (PLL) to estimate carrier phase. An exemplary decision-directed phase-locked loop (PLL) circuit of the prior art is shown at  FIG. 1 . The circuit receives baseband signal r(t) at lead  100 . Respective quadrature carriers at leads  104  and  108  developed from voltage-controlled oscillator (VCO) circuit  132  are multiplied by the received signal. The product signal is integrated over symbol time T by integrator  112  and sampled by circuit  114  according to the symbol time base circuit  116 . Decision circuit  118  produces output signal A(t) at lead  120 . A product signal from multiplier circuit  110  is delayed by circuit  124  to compensate for the decision circuit delay. The signals at leads  120  and  126  are multiplied by circuit  122  to produce error signal e(t) at lead  128 . This error signal is filtered by loop filter circuit  130  to eliminate double frequency components and applied to VCO circuit  132 . Problems with the PLL circuit of  FIG. 1  when used for phase estimation, however, include circuit complexity and likelihood of hang-up. Furthermore, the circuit of Proakis requires synchronization circuitry to correctly sample each symbol near the center of the respective symbol time. 
   Other studies determine maximum likelihood (ML) estimates for symbol timing by calculating a derivative of a matched filter output signal. Id. at 359–361. Referring to  FIG. 2 , there is, a circuit of the prior art that receives baseband signal r(t) on lead  100 . The baseband signal is filtered by matched filter  202 . Circuit  204  then calculates a derivative of the signal, which is then sampled by circuit  206  according to voltage-controlled clock (VCC) circuit  220 . Circuit  212  then multiplies the derivative at lead  208  by the known symbol sequence I n  at lead  210 . The product of this multiplication is summed by circuit  216  and applied to the VCC circuit  220 . A limitation of this circuit, however, is that calculation of a matched filter output derivative for symbol timing synchronization is not possible with modern digital receivers which work on sampled data input signals. Another study by L. E. Franks,  Carrier and Bit Synchronization in Data Communication - A Tutorial Review , IEEE Trans. on Communications, August 1980 1107, 1117, teaches a method for joint tracking of both carrier phase and symbol timing. Therein ( FIG. 9 ), Franks teaches a circuit that combines a PLL for carrier phase determination and a circuit to calculate a derivative of a low pass filter output. This method, therefore, is subject to the same limitations of the previously discussed methods. 
   SUMMARY OF THE INVENTION 
   These problems are resolved by a method of producing a correction signal by receiving a predetermined data sequence. The data sequence is sampled at predetermined times, thereby producing a sampled data sequence. The sampled data sequence is separated into first and second sampled data sequences. A ratio is calculated from the first and second sampled data sequences. A correction signal is produced in response to the ratio. 
   The present invention improves reception and reduces circuit complexity by providing maximum likelihood carrier phase and symbol timing correction signals. The method improves bit error rate compared to methods of the prior art and is comparable to the Cramer-Rao bound. 

   
     BRIEF DESCRIPTION OF THE DRAWINGS 
     A more complete understanding of the invention may be gained by reading the subsequent detailed description with reference to the drawings wherein: 
       FIG. 1  is a block diagram of a carrier recovery phase-locked loop circuit of the prior art; 
       FIG. 2  is a block diagram of a symbol time recovery circuit of the prior art; 
       FIG. 3A  is a sequence of training data that may be used for timing and carrier phase recovery; 
       FIG. 3B  is a diagram showing full and half sample sequences for I and Q signals corresponding to the training data of  FIG. 3A ; 
       FIG. 4  is a diagram showing full and half signal sample values at the output terminal of a raised cosine (RC) filter; 
       FIG. 5  is a block diagram of the carrier phase and symbol timing correction circuit of the present invention; 
       FIG. 6  is a diagram of the data in ROM lookup table  562  of  FIG. 5 ; 
       FIGS. 7A–7D  are simulations of the ML carrier phase estimate for various parameters compared to the Cramer-Rao bound; 
       FIGS. 8A–8D  are simulations of the ML symbol timing estimate for various parameters compared to the Cramer-Rao bound; and 
       FIGS. 9A–9B  are simulations of the RMS error of the present ML estimate compared to Gardner&#39;s method. 
   

   DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
   Referring now to  FIG. 5 , a received baseband signal y(t) given by equation [1] is applied to lead  500 . This baseband signal is preferably a quadrature phase shift keyed (QPSK) signal of discrete symbols received from a remote base station transmitter. Input signal samples I i  include in-phase (I real) and quadrature (Q imaginary) components of the sampled training data of  FIG. 3A . 
   This training data is a sequence of unique data words transmitted as a preamble or midamble by the remote base station to the receiver. Referring to  FIG. 3B , the samples are designated full samples w i  and y i  of the I and Q components and half samples x i  and z i  of the I and Q components, respectively, for the training data sequence { . . . (1+j),−(1+j),(1+j),−(1+j), . . . }. By convention, the full samples are assumed near the center of the symbol time T and the half samples are assumed near a boundary between symbols. Samples of the received signal are treated as an infinite series for purposes of the following discussion. 
               y   ⁡     (   t   )       =         ∑     i   =     -   ∞         i   =   ∞       ⁢       I   i     ⁢     g   ⁡     (     t   -   iT   -   τ     )       ⁢     ⅇ   jϕ         +     n   ⁡     (   t   )                 [   1   ]             
 
   The received baseband signal is filtered by a transmit pulse shaping filter  502  having a filter characteristic g(t) and having a shaping factor of α≧0.2. Simulations show small degradation of finite-length sequences compared to idealized infinite-length sequences with this shaping factor constraint. The filter is typically a square root raised cosine (RC) filter having a characteristic as in  FIG. 4 . The output of the RC filter is given in equation [2]. Both carrier phase φ and symbol timing τ must be determined from samples of the received signal to recover the I and Q components of the signal transmitted by the base station. The symbol timing error of  FIG. 4  showing a positive value for τ, indicates the time of the full sample prior to the center of the symbol time t/T. The range of τ is determined by (2i−1)/2≦(t−τ)/T≦(2i−1)/2, having an absolute value of τ≦T/2, where T is the symbol time period. The filter function of  FIG. 4  is given by equation [3] where −T/2≦t≦T/2. 
               r   ⁡     (   t   )       =           r   l     ⁡     (   t   )       +       jr   Q     ⁡     (   t   )         =         (       I   i   l     +     jI   i   Q       )     ⁢     f   ⁡     (     t   -   iT   -   τ     )       ⁢     ⅇ   jϕ       +     N   ⁡     (   t   )                   [   2   ]                 f   ⁢     (   t   )       =       ∑     i   =     -   ∞         i   =   ∞       ⁢       I   i   l     ⁢     h   ⁡     (     t   -   iT     )                   [   3   ]             
 
   The received signal is applied to multiplier circuits  506  and  514 . A free-running local oscillator circuit  516  produces a reference carrier signal on lead  512 . This reference carrier is multiplied by the received signal to produce a quadrature signal that is applied to low pass filter circuit  528 . A time synchronization circuit  524  produces a clock signal on lead  526  having twice the frequency of the symbol frequency transmitted signal from the base station. This clock signal on lead  526  is applied to analog-to-digital converter (ADC) circuit  530 . The ADC takes two samples of the quadrature signal corresponding to each symbol period T and produces a digital sample on lead  532  given by equation [4]. Likewise, the ADC circuit  520  takes two samples of the in-phase signal corresponding to each symbol period T and produces a digital sample on lead  522 . The samples on either of lead  522  and  532 , therefore, include sample sequences given by equations [5] and [6], corresponding to full-symbol and half-symbol samples and their respective noise terms.
 
 r ( l )= I   i ƒ( lT   S   −iT −τ) e   jφ   +N ( l )  [4]
 
 r   ƒ ( l )= I   1 ƒ(−τ) e   jφ   +N   ƒ ( l )  [5]
 
 r   h ( l )= I   1 ƒ( T   S −τ) e   jφ   +N   h ( l )  [6]
 
   Operation of sum circuits  534 ,  538 ,  566  and  568  and ratio circuits  550  and  558  will now be explained in detail. If the received signal is rewritten as a vector R including full and half samples as in equation [7], then the mean or expected value E[R] of these samples after filtering is given by equation [8]. Furthermore, the covariance H is given by equation [9]. The matrix I is an L×L unity matrix. The matrix B is an L×L correlation matrix with elements having an expected value given by β jk =h(2(j−k)+1)T S ), j, k=0, . . . ,L−1, where the function h(t)=g(t)*g(−t) is the RC filter response. The superscript *T in the following discussion denotes a conjugate transpose or Hermitian matrix. 
   The matrix inversion lemma of equation [10] is applied to equation [9] to produce inverted covariance matrix H −1  in equation [11], where Γ=(I−BB *T ) −1  and Ψ=(I−B *T B) −1 . 
             R   ⁢     ⌊         r   f     ⁡     (   0   )       ,       r   f     ⁡     (   1   )       ,   …   ⁢           ,       r   f     ⁡     (     L   -   1     )       ,       r   h     ⁡     (   0   )       ,       r   h     ⁡     (   1   )       ,   …   ⁢           ,       r   h     ⁡     (     L   -   1     )         ⌋             [   7   ]                 E   ⁡     [   R   ]       =       [         [       I   0     ,     I   1     ,   …   ⁢           ,     I     L   -   1         ]     ⁢     f   ⁡     (     -   τ     )         ,       [       I   0     ,     I   1     ,   …   ⁢           ,     I     L   -   1         ]     ⁢     f   ⁡     (       T   S     -   τ     )           ]     ⁢     ⅇ   jϕ               [   8   ]               H   =       E   ⁡     [         (     R   -     E   ⁡     [   R   ]         )             *     ⁢   T       ⁢     (     R   -     E   ⁡     [   R   ]         )       ]       =       N   0     ⁢              I         B           *     ⁢   T               B       I                          [   9   ]                   (     I   -       B           *     ⁢   T       ⁢   B       )       -   1       =     I   +           B           *     ⁢   T       ⁡     (     I   -     BB           *     ⁢   T         )         -   1       ⁢   B               [   10   ]                 H     -   1       =                I   +       B           *     ⁢   T       ⁢   Γ   ⁢           ⁢   B               -     B           *     ⁢   T         ⁢   Γ                 -   B     ⁢           ⁢   Ψ           I   +     B   ⁢           ⁢     ΨB           *     ⁢   T                              [   11   ]             
 
   The maximum likelihood (ML) estimate of φ is a value that satisfies equation [12]. Thus, the real part of the partial derivative in equation [13] must also be equal to zero. Since received vector R is independent of φ, its partial derivative is zero resulting in equation [14]. A substitution of equations [11] and [14] into equation [13] produces equation [15]. 
                 ∂     ∂   ϕ       ⁢     (       (     R   -     E   ⁡     [   R   ]         )     ⁢         H     -   1       ⁡     (     R   -     E   ⁡     [   R   ]         )               *     ⁢   T         )       =   0           [   12   ]                 R   ⁢     {       (       ∂     ∂   ϕ       ⁢     (     R   -     E   ⁡     [   R   ]         )       )     ⁢         H     -   1       ⁡     (     R   -     E   ⁡     [   R   ]         )                     *     ⁢   T           }       =   0           [   13   ]                   -     ∂     ∂   ϕ         ⁢     E   ⁡     [   R   ]         =     -     j   [         [       I   0     ,     I   1     ,   …   ⁢           ,     I     L   -   1         ]     ⁢     f   ⁡     (     -   τ     )         ,     
     ⁢         [       I   0     ,     I   1     ,   …   ⁢           ,     I     L   -   1         ]     ⁢     f   ⁡     (       T   S     -   τ     )       ⁢     ⅇ   jθ       =     -     jE   ⁡     [   R   ]                         [   14   ]                 R   ⁢     {       -     j   ⁡     (       ∂     ∂   ϕ       ⁢     E   ⁡     [   R   ]         )         ⁢                I   +       B           *     ⁢   T       ⁢   Γ   ⁢           ⁢   B               -     B           *     ⁢   T         ⁢   Γ                 -   B     ⁢           ⁢   Ψ           I   +     B   ⁢           ⁢   Ψ   ⁢           ⁢     B           *     ⁢   T                      ⁢       (     R   -     E   ⁡     [   R   ]         )             *     ⁢   T         }       =   0           [   15   ]             
 
   A simplification of equation [16] is applied to equation [15], thereby producing equation [17]. This simplification is appropriate, since sums of respective full and half samples of known training data alternate between +1 and −1. Thus, for large L, matrix products [I 0 , . . . , I L−1 ]B≈0 and [I 0 , . . . , I L−1 ]B T ≈0. 
                 E   ⁡     [   R   ]       ⁢     H     -   1         =             E   ⁡     [   R   ]         N   0       ⁢                I   +       B           *     ⁢   T       ⁢   Γ   ⁢           ⁢   B               -     B           *     ⁢   T         ⁢   Γ                 -   B     ⁢           ⁢   Ψ           I   ⁢           +     B   ⁢           ⁢   Ψ   ⁢           ⁢     B           *     ⁢   T                        ≈         E   ⁡     [   R   ]         N   0       ⁢              I       0           0       I                  =       E   ⁡     [   R   ]         N   0                 [   16   ]                 R   ⁢     {       -   j     ⁢           ⁢     E   ⁡     [   R   ]       ⁢     R           *     ⁢   T         }       =   0           [   17   ]             
 
   Sum circuits  534 , and  542  calculate respective I and Q sums for φ according to equation [18], where I 1 =I 1   1 +jI 1   Q , A,Bε{I,Q}, and Dε{f,h}. Thus, real and imaginary values of variables on the right side of equation [18] are indicated by I and Q subscripts, respectively. Substitution of equation [7] and [8] in summation form of equation [18] for respective matrices R and expected value E[R] yields equation [19]. Equation [19] is rewritten as equation [20] to further explain circuit operation. Ratio circuit  550  receives respective I and Q sums on leads  536  and  544 . The ratio circuit also receives current values for ƒ({circumflex over (τ)}) and ƒ(T S −τ) on lead  564  as will be explained in detail. The ratio circuit  550  then calculates the ratio on the right side of equation [20] and applies the calculated ratio to lead  552 . The ROM lookup table  554  receives the calculated ratio on lead  552  and responsively produces carrier phase estimate φ on lead  556 . 
               S   AB   D     =       ∑     l   =   0       L   -   1       ⁢         r   A   D     ⁡     (   l   )       ⁢     I   Bl   D                 [   18   ]                   sin   ⁢           ⁢   ϕ   ⁢     {         f   ⁡     (     -   τ     )       ⁢     (       S   ll   f     +     S   QQ   f       )       +       f   ⁡     (       T   S     -   τ     )       ⁢     (       S   ll   h     +     S   QQ   h       )         }       -     
     ⁢     cos   ⁢           ⁢   ϕ   ⁢     {         f   ⁡     (     -   τ     )       ⁢     (       S   OI   f     +     S   IQ   f       )       +       f   ⁡     (       T   S     -   τ     )       ⁢     (       S   QI   h     +     S   IQ   h       )         }         =   0           [   19   ]                 tan   ⁢           ⁢   ϕ     =           f   ⁡     (     -   τ     )       ⁢     (       S   QI   f     +     S   IQ   f       )       +       f   ⁡     (       T   S     -   τ     )       ⁢     (       S   QI   h     +     S   IQ   h       )               f   ⁡     (     -   τ     )       ⁢     (       S   II   f     +     S   QQ   f       )       +       f   ⁡     (       T   S     -   τ     )       ⁢     (       S   II   h     +     S   QQ   h       )                   [   20   ]             
 
   The desired ML estimate for τ is the value that satisfies equation [21] The real part of equation [21], therefore, must also be satisfied according to equation [22]. Substitution of equation [23] and the previously discussed simplification of equation [24] yields equation [25]. A further substitution of received matrix R full and half samples into equation [25] yields equation [26]. 
                 ∂     ∂   τ       ⁢     (       (     R   -     E   ⁡     [   R   ]         )     ⁢         H     -   1       ⁡     (     R   -     E   ⁡     [   R   ]         )         *   T         )       =   0           [   21   ]                 R   ⁢     {       (       ∂     ∂   τ       ⁢     (     R   -     E   ⁡     [   R   ]         )       )     ⁢         H     -   1       ⁡     (     R   -     E   ⁡     [   R   ]         )         *   T         }       =   0           [   22   ]                   ∂     ∂   τ       ⁢     (     R   -     E   ⁡     [   R   ]         )       =       -     j   ⁡     [         [       I   0     ,     I   1     ,   …   ⁢           ,     I     L   -   1         ]     ⁢       f   ′     ⁡     (     -   τ     )         ,       [       I   0     ,     I   1     ,   …   ⁢           ,     I     L   -   1         ]     ⁢       f   ′     ⁡     (       T   s     -   τ     )           ]         ⁢     ⅇ     j   ⁢           ⁢   θ                 [   23   ]                     (       ∂     ∂   τ       ⁢     E   ⁡     [   R   ]         )     ⁢     H     -   1         ≈       1     N   0       ⁢     (       ∂     ∂   τ       ⁢     E   ⁡     [   R   ]         )     ⁢              I       0           0       I                  =       1     N   0       ⁢     ∂     ∂   τ       ⁢     E   ⁡     [   R   ]                 [   24   ]                     f   ′     ⁡     (     -     τ   ^       )             ⁢           [                   -     (       S   II   f     -     S   QQ   f       )       ⁢   cos   ⁢           ⁢   ϕ     +       (       S   IQ   f     -     S   QI   f       )     ⁢   sin   ⁢           ⁢   ϕ     -       ∑     l   =   0       L   -   1       ⁢              I   l          2     ⁢     f   ⁡     (     -     τ   ^       )             ]     +     
     ⁢         f   ′     ⁡     (       T   S     -     τ   ^       )       ⁡     [         -     (       S   II   h     -     S   QQ   h       )       ⁢   cos   ⁢           ⁢   ϕ     +       (       S   IQ   h     -     S   QI   h       )     ⁢   sin   ⁢           ⁢   ϕ     -       ∑     l   =   0       L   -   1       ⁢              I   l          2     ⁢     f   ⁡     (       T   S     -     τ   ^       )             ]         =   0                 [   25   ]             
 
noise+[ƒ′(−{circumflex over (τ)})ƒ(−τ)+ƒ′( T   S −{circumflex over (τ)})ƒ( T   S −τ)]−[ƒ′(−{circumflex over (τ)})ƒ(−{circumflex over (τ)})+ƒ′( T   S −{circumflex over (τ)})ƒ( T   S −{circumflex over (τ)})]=0  [26]
 
   The terms ƒ′(−{circumflex over (τ)})ƒ(−{circumflex over (τ)})+ƒ′(T S −{circumflex over (τ)})ƒ(T S −{circumflex over (τ)}) of equation [26] are small and may be neglected. A further simplification of equation [26], given in equation [27], is possible for RC filters having a shaping factor α≧0.2 as previously described. This simplification yields equation [28]. Sum circuits  538  and  546  calculate respective symbol timing sums as previously described for the carrier phase estimate. The ratio circuit  558  receives these sums on leads  540  and  548  and calculates the ratio in the center term of equation [28]. The function q(−{circumflex over (τ)}), defined by equation [30], is substituted into equation [28] and yields quadratic equation [30]. This quadratic equation has one positive and one negative real root. The positive real root corresponds to the desired ML estimate for τ. This positive real root is calculated by ratio circuit  558  and applied to ROM lookup table  562  via lead  560 . The contents of ROM lookup table  562  correspond to values of the function q(−{circumflex over (τ)}) in  FIG. 6 . The ROM lookup table produces the corresponding τ on lead  564 . 
                 ∂     ∂   τ       ⁢     (         f   2     ⁡     (     -   τ     )       +       f   2     ⁡     (       T   S     -   τ     )         )       =             f   ′     ⁡     (     -   τ     )       ⁢     f   ⁡     (     -   τ     )         +         f   ′     ⁡     (       T   S     -   τ     )       ⁢     f   ⁡     (       T   S     -   τ     )           ≈   0             [   27   ]                     f   ⁡     (     -     τ   ^       )         f   ⁡     (       T   S     -     τ   ^       )         -       f   ⁡     (       T   S     -     τ   ^       )         f   ⁡     (     -     τ   ^       )           =             (       S   II   f     +     S   QQ   f       )     2     -       (       S   II   h     +     S   QQ   h       )     2     +               (       S   IQ   f     +     S   QI   f       )     2     -       (       S   IQ   h     +     S   QI   h       )     2                 (       S   II   f     +     S   QQ   f       )     ⁢     (       S   II   h     +     S   QQ   h       )       +       (       S   IQ   f     +     S   QI   f       )     ⁢     (       S   IQ   h     +     S   QI   h       )           ≡   Δ             [   28   ]                 q   ⁡     (     -     τ   ^       )       =       f   ⁡     (     -     τ   ^       )         f   ⁡     (       T   S     -     τ   ^       )                 [   29   ]                     q   2     ⁡     (     -     τ   ^       )       -     Δ   ⁢           ⁢     q   ⁡     (     -     τ   ^       )         -   1     =   0           [   30   ]             
 
   Interpolate circuits  566  and  568  receive respective I and Q signal samples on leads  522  and  532  together with the ML symbol estimate corresponding τ on lead  564 . The interpolate circuits correct the symbol timing of the signal samples according to the ML estimate of τ and produce corrected I and Q signal samples on leads  570  and  572 , respectively. Derotate circuit  574  receives the corrected signal samples together with the ML carrier phase estimate φ on lead  556 . The derotate circuit produces phase corrected I and Q signal samples on leads  576  and  578 , respectively, in response to the ML carrier phase estimate φ. 
   Turning now to  FIGS. 7A–7D , there are Monte-Carlo simulations of the ML carrier phase estimate of the present invention for various parameters compared to the Cramer-Rao bound. The Cramer-Rao bound is significant as a theoretical limit. The upper curves in each simulation show a 32-sample sequence compared to a 64-sample sequence. The 64-sample sequence improves the bit error rate by approximately 3 dB for each parameter set. Each simulation, however, shows performance of the present ML estimator is very close to the Cramer-Rao bound. Referring to  FIG. 8A–8D , corresponding Monte-Carlo simulations of the ML symbol timing estimate show approximately the same result. The worst-case difference of symbol timing estimate of  FIG. 8A  shows the present ML error is within 0.5 dB of the Cramer-Rao bound. Finally, referring to  FIGS. 9A–9B , performance of the present ML estimator is compared to Gardner&#39;s method, presented in Gardner, A BPSK/QPSK  timing - error detector for sampled receivers , IEEE Trans. on Communications, May 1986, at 423. The simulation of  FIG. 9A  for α=0.5, τ/T=0.1 and φ=π/8, shows a 4 dB improvement over Gardner&#39;s method. The simulation of  FIG. 9B  for α=0.5, τ/T=0.05 and φ=π/4, including a smaller symbol time error and a larger carrier phase error, shows a 2 dB improvement over Gardner&#39;s method. 
   The ML estimates of the present invention are highly advantageous with respect to methods of the prior art for several reasons. First, the bit error rate of the present ML estimate is substantially lower than previous methods. Second the present invention resolves all ambiguities of sampled data. Positions of the full and half data samples are inconsequential to the present method and long as the positive root of equation [30] is selected. Third, the present invention avoids the complexity of PLL circuits of the prior art and avoids hangup. Finally, the ML estimate signals are derived from a ratio of signal samples. Thus, they are insensitive to signal strength and do not require automatic gain control (AGC). 
   Although the invention has been described in detail with reference to its preferred embodiment, it is to be understood that this description is by way of example only and is not to be construed in a limiting sense. For example, the present invention may be easily applied to a BPSK system of alternating ones and zeros for the in-phase component and zero for the quadrature component. Moreover, many functions the present invention may be performed by a digital signal processor or other processor as will be understood by those of ordinary skill in the art having access to the present specification. 
   It is to be further understood that numerous changes in the details of the embodiments of the invention will be apparent to persons of ordinary skill in the art having reference to this description. It is contemplated that such changes and additional embodiments are within the spirit and true scope of the invention as claimed below.