Abstract:
The present invention provides a system and method of timing estimation for use in a digital, receiver within a communication system. This method exploits the complementary information available from the use of two different nonlinearities to estimate the timing offset following asynchronous sampling of the received signal at a rate of two samples per symbol. The present method operates in a feedforward configuration thereby avoiding issues associated with feedback configurations such as hangup. The present method uses a magnitude square nonlinearity as well as a delay, complex conjugate and multiply nonlinearity to calculate the timing offset. The choice of these nonlinearities is influenced by the need for the estimator to operate in the presence of a phase offset on the received signal. Timing estimators which can tolerate the presence of a slowly varying phase offset over the observation interval on the received signal are especially important in all-digital feedforward receiver design. Separating the two sample per symbol outputs from the nonlinearities into odd and even samples provides four signals which, when suitably manipulated, yield an expression for the timing estimate. This timing offset is then fed to a timing correction unit which calculates the data samples corresponding to the sampling clock phase and removes the redundant samples. The resultant sampled signal is then forwarded to additional synchronization and functional units in the receiver. The system may be utilized in a variety of digital receivers employing CDMA, TDMA, FDMA and/or any combination of the principles of the above or other technologies.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The invention relates to communication systems, and more particularly to an apparatus for achieving synchronization in a receiver. 
     2. Related Art 
     In synchronous digital transmission, information is conveyed by uniformly spaced pulses and the function of any receiver is to isolate these pulses as accurately as possible. However, due to the noisy nature of the transmission channel, the received signal undergoes changes during transmission and therefore, a complete estimation of certain reference parameters is necessary prior to data detection. The unknown parameters may cover such factors as the optimum sampling or the phase offset introduced in the channel or induced by the instabilities of the transmitter and receiver oscillators. 
     Estimation theory proposes various techniques for the estimation of these parameters; one such technique is Maximum Likelihood. However, ad-hoc techniques which are either totally unrelated to the optimum estimation derived from Maximum Likelihood or are at most loosely related, can also give excellent performance. 
     Generally, if the transmitter does not generate a pilot synchronization signal, the receiver must derive symbol timing from the received signal. The term symbol is used in this context to refer to transmitted signals that are phase modulated with discrete phase relationships. Both the transmitter and the receiver employ separate clocks which drift relative to each other, and any symbol synchronization technique must be able to track such drift. 
     Choosing the proper sampling instants is critical for reliable data detection. Failure to sample at the correct instants leads to Inter-Symbol Interference (ISI), which can be especially severe in sharply bandlimited signals. The term ISI refers to two or more symbols that are superimposed upon each other which makes phase detection of each symbol extremely difficult. Incorrect sampling implies the receiver is inadvertently sampling where the influence of the previous data symbol is still present (J. G. Proakis, “ Digital Communications, ” Third Edition, McGraw-Hill Publishers, pp. 536-537, 1995). 
     In a conventional digital receiver, the signal following demodulation is first passed through an anti-aliasing filter, which is used to limit the bandwidth of the received signal, and is subsequently sampled asynchronously. Asynchronous sampling implies there is no control over the instant at which the sampling of the continuous time signal occurs. 
     FIGS. 1- a  and  1 - b  illustrate the concept of oversampling a continuous signal at four and two samples per symbol, respectively. The optimum sampling instants correspond to the maximum eye opening and are located approximately at the peaks of the signal pulses. FIG. 1- a  shows that oversampling at four samples per symbol provides more information about the received signal than oversampling at two samples per symbol (FIG. 1- b ). 
     The term “eye opening” refers to the amplitude variations of the signal at the output of the pulse-shaping filter. An “eye” pattern is formed by superimposing the output of the pulse-shaping filter for each symbol upon the other until the central portion takes on the shape of an “eye”. This is illustrated in FIGS. 2- a  and  2 - b  for a BPSK (Binary Phase Shift Keying) modulation scheme for high and low signal to noise conditions. Note that at high signal to noise conditions the “eye” is open, whereas at low signal to noise conditions, the “eye” is closed. 
     Among synchronization techniques, a distinction is made between feedforward and feedback systems. A feedback system uses the signal available at the system output to update future parameter estimates. Feedforward systems process the received signal to generate the desired estimate without explicit use of the system output. 
     In an error tracking feedback loop, the timing estimator constantly adjusts the phase of a local clock oscillator to minimize the phase error between the estimated and the optimum.sampling instant as illustrated in FIG.  3 . In a feedforward-timing loop, as illustrated in FIG. 4, the incoming signal is sampled asynchronously and applied to a timing estimator. This timing estimate is subsequently fed to an interpolator to estimate the received signal at the instant of the estimated timing offset phase. Interpolation estimates the signal value at the optimum sampling instant using the timing phase from the timing phase estimation unit (H. Meyr, M. Moeneclaey and S. A. Fechtel, “ Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing, ” John Wiley Publishers, Chapter 9, pp. 505-532, 1998). Redundant samples are removed using a decimator. Both feedforward and feedback techniques are currently in use; however, it should be noted that there are advantages and disadvantages associated with both approaches. 
     Problems with feedback techniques include the length of the acquisition time, the high probability of hangup and cycle slips associated with phase locked loop (PLL) based structures, especially in the presence of channel fading. Fading occurs when signal components arriving via different propagation paths add destructively. Hangup occurs when the initial phase error of the estimator is close to an unstable equilibrium point, which can result in an extremely long acquisition time (i.e., a long time for the loop to adjust to the correct phase); in fact, the loop may never recover. Hangup is very serious as it can even occur in perfect channel conditions (H. Meyr, M. Moeneclaey and S. A. Fechtel, “ Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing, ” John Wiley Publishers, pp. 94-97, 1998). Cycle slips are very harmful to the reliability of the receiver&#39;s decisions, because a cycle slip corresponds to the repetition or omission of a channel symbol (H. Meyr, M. Moeneclaey and S. A. Fechtel, “ Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing, ” John Wiley Publishers, pp. 385-399, 1998). 
     These problems can be circumvented through the use of feedforward estimation. The advantages of feedforward estimation are that acquisition time is solely dependent on loop bandwidth and is not influenced by channel conditions. In addition, hangup does not occur and implementation costs are lower, as feedforward designs are more suited to VLSI (Very Large Scale Integration) implementation. Flexibility in the design of the synchronization unit has increased with the advent of increasingly powerful silicon chips. 
     However, feedforward techniques in the literature generally require a higher oversampling ratio than is prevalent in sampled feedback estimators (F. M. Gardner, “ A BPSK/QPSK Timing Error Detector for Sampled Receivers, ” IEEE Transactions on Communications, COM-44, pp. 399-406, March 1996), which generally require an oversampling rate of one or two samples per symbol for reliable operation. In feedforward designs, the oversampling rate is generally four or more samples per symbol (H. Meyr, M. Moeneclaey and S. A. Fechtel, “ Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing, ” John Wiley Publishers, pp. 289-295, 1998). This in turn is in contrast to analog feedback methods, which require a continuous waveform (J. G. Proakis, “ Digital Communications, ” Third Edition, McGraw-Hill Publishers, pp. 358-365, 1995). Digital synchronization methods recover timing by operating only on samples taken at a suitable rate. Digital implementation of an estimator has enormous appeal in communications technology and influences the design of all modem receivers. 
     There are two distinct stages involved in timing estimation: first, the estimation of the timing phase offset and second, the use of this estimate in the interpolationldecimation process. The estimated sampling instant within a symbol is the timing phase. The configuration of the feedforward timing estimator loop is very different from that of a feedback loop. FIG. 4 illustrates that, in a feedforward arrangement, sampling is asynchronous and the subsequent processing in the interpolator and decimator units must choose the optimum data samples for the received signal using an estimate of the timing offset based on the received signal samples. An algorithm is applied to estimate the timing phase in the timing phase estimation unit. What is desired is an improved timing phase estimation unit, wherein, for example, only two samples per symbol of the recovered signal are used to estimate the timing offset. Two forms of processing can be applied within the timing phase estimating unit depending on how the data present on the received signal is exploited to assist in the timing estimation. The first is data-aided (DA) estimation wherein known data within the data stream is exploited to improve the estimation performance. Alternatively, non-data-aided (NDA) estimation is possible wherein the arbitrary data is considered a nuisance parameter, which is removed by averaging the received signal over the statistics of the arbitrary data. 
     The received noisy signal samples contains no periodic components because the information symbols have zero mean. However, if the samples are passed through an appropriate nonlinearity, such as a square law operation, a cyclostationary process results which is observed as spectral lines in the frequency domain at multiples of the symbol rate. The phase of the spectral line at the symbol rate is related to the normalized timing offset. Exploitation of this and similar nonlinearities has been presented in the literature (M. Morelli, A. N. D&#39;Andrea and U Mengali, “ Feedforward ML - Based Timing Estimation with PSK Signals, ” IEEE Communications Letters, 1(3): pp. 80-82, 1997; E. Panayirci and E. Y. Bar-Ness “ A New Approach for Evaluating the Performance of a Symbol Timing Recovery System Employing a General Type of Nonlinearity, ” IEEE Transactions on Communications; 44(1): pp. 29-33, 1996). 
     Timing estimators utilizing a nonlinearity are known as spectral line generating synchronizers. The expectation of the resulting signal at the output of the nonlinearity is a periodic signal with a period equal to the symbol rate T. This periodic signal can be composed as the sum of sinusoidal components (spectral lines) occurring at 1/T and multiples thereof. The signal at the nonlinearity output enters either a PLL or a narrowband bandpass filter. In the case of a narrowband bandpass filter tuned to the channel symbol rate 1/T, the sinusoidal component that is present at the output of the nonlinearity is isolated and serves as a clock signal for the sampling device. 
     An alternative to using a PLL or a narrowband bandpass filter involves evaluating the Fourier component of the spectral line occurring at the symbol rate. The accuracy of the estimate depends on the length of the observation interval over which the Fourier component is formed. The jitter on the estimate reduces as the observation interval increases. Exploitation of these and similar nonlinearities has been presented in the literature. However, these typically use a minimum of four samples per symbol. Therefore, there is a need for an estimator that has a feedforward configuration and can operate at a lower sampling rate. 
     If the timing estimator is among the first modules in a baseband receiver system, then the performance of the estimator should be independent of the presence of a phase offset on the sampled received signal which is deterministic or, at most, slowly varying over the observation interval. 
     Consequently, what is needed is to provide a timing estimation method that can give comparable performance in the presence of a slowly varying or static phase offset as alternative algorithms which require the removal of the phase offset beforehand. Timing estimation would be provided based on asynchronously sampling the pulse shaping filter output at a rate of two samples per symbol. The timing estimation would use a NDA technique and a feedforward configuration that avoids the problems associated with feedback design approaches such as hangup. The proposed estimator would employ two nonlinearities to generate spectral lines in the frequency domain with a period equal to the channel symbol rate. The phase of the spectral line generated at the symbol rate would be related to τ/T, where τ is the timing offset. Two samples per symbol would provide sufficient information to isolate the timing offset as the argument of an expression formed using the outputs of the two nonlinearities. An embodiment of the estimator could then be used in conjunction with any of the established methods of timing correction available in the literature. 
     SUMMARY OF THE INVENTION 
     One aspect of the present invention includes a system and method of timing estimation for use in a digital receiver within a communication system. A timing estimation block or timing estimator is provided within a digital receiver where the inputted signal stream may be processed at two samples per symbol and the estimator operates in a feedforward manner. The invention implements an ad-hoc timing estimation technique whose performance is unaffected by the presence of a phase offset on the received signal samples. Furthermore, NDA estimation is utilized in the timing estimator. 
     This timing estimation method exploits the complementary information available from the use of two different nonlinearities to estimate the timing offset using two samples per symbol. The method uses a squaring nonlinearity as well as a delay, complex conjugation and multiply nonlinearity to generate the information to calculate the timing offset. Separating the two sample per symbol outputs from the two nonlinearities into odd and even samples yields four signal branches which, when suitably manipulated, give an expression for the timing estimate. This timing offset is then fed to a timing correction unit, which calculates the data samples corresponding to the sampling clock phase and removes the redundant samples. The resultant sampled signal is then forwarded to additional synchronization and functional units. 
     This method is provided for a variety of digital receivers employing Code Division Multiple Access (CDMA), in which a transmitted signal is spread over a band of frequencies much wider than the minimum bandwidth required to transmit the signal, Time Division Multiple Access (TDMA) where the users share the radio spectrum in the time domain, Frequency Division Multiple Access (FDMA) where a user is allocated at least one unique frequency for communication without interference with users in the same frequency spectrum, and/or any combination of the principles of the above or other technologies. 
     In accordance with another aspect of the invention, a digital receiver system comprises an anti-aliasing filter, a sampling unit, a filtering block, a timing estimation block, a timing correction block and additional functional blocks. The filtering block comprises a pulse-shaping filter. The timing estimation subsystem comprises a squaring nonlinearity, a delay, complex conjugation and multiply unit, two demultiplexers, two summation blocks, two subtractors and a phase calculator. The filtering block receives signals from an Intermediate Frequency (IF) block which have been demodulated to baseband. In one embodiment, these signals are sampled at the Analog to Digital Converter (ADC) with a fixed clock (Sampling Clock=46.7 MHz). It is necessary to note that the signals received by the filtering block within the digital receiver might not be sampled, and that the sampling may take place only after the filtering block. The outputted signals from the filtering block are then fed to the timing estimation subsystem for further processing. The resulting timing estimate is then fed to the timing correction unit to correctly estimate the received data stream. The output from the timing correction unit is subsequently fed to additional functional blocks for further processing. 
     In another aspect of the present invention there is a method of feedforward timing estimation for use in a digital receiver within a communication system, the method comprising splitting an input data stream into a first data stream and a second data stream; applying a first nonlinearity to the first data stream and a second nonlinearity to the second data stream; averaging the instantaneous values of the first nonlinearly derived data stream; averaging the instantaneous values of the second nonlinearly derived data stream; and determining a timing offset as a function of the complex components of the averaged instantaneous values. 
     In yet another aspect of the present invention there is a feedforward timing estimation system for use in a digital receiver within a communication system, the timing estimation system comprising a pulse-shaping filter receiving a data stream sampled at a minimum rate of two samples per symbol; a timing estimator, receiving the filtered data stream, and being capable of generating a timing offset; a processing delay, receiving the filtered data stream, and delaying the filtered data stream; and a timing correction subsystem, receiving the delayed data stream and the timing offset, and being capable of generating corrected data samples according to the timing offset. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     The above and other aspects, features and advantages of the invention will be better understood by referring to the following detailed description of the preferred embodiment, which should be read in conjunction with the accompanying drawings, in which: 
     FIG. 1- a  is a graph illustrating the difference between an analog signal and the same signal sampled at four samples per symbol. 
     FIG. 1- b  is a graph illustrating the same signal as in FIG. 1- a  using two samples per symbol. 
     FIG. 2- a  is a graph illustrating the formation of an eye pattern at the output of a pulse-shaping filter for high signal to noise conditions. 
     FIG. 2- b  is a graph illustrating the formation of an eye pattern at the output of a pulse-shaping filter for low signal to noise conditions. 
     FIG. 3 is a block diagram of a feedback estimator configuration. 
     FIG. 4 is a block diagram of a feedforward estimator configuration. 
     FIG. 5 is a functional block diagram of a digital receiver using the proposed timing estimation method as implemented in one embodiment of the invention. 
     FIG. 6 is a block diagram of the proposed timing estimation subsystem as shown in FIG.  5 . 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     The following detailed description of the preferred embodiments presents a description of certain specific embodiments of the present invention. However, the present invention can be embodied in a multitude of different ways as defined and covered by the claims. The following description is not intended to limit the enumerated claims, but to serve as a particular example thereof. In this description, reference is made to the drawings wherein like parts are designated with like numerals throughout. 
     FIGS. 1- a  and  1 - b  illustrate the difference between a continuous time and discrete time representation of a signal for the situation of taking four and two samples per symbol, respectively. The information available about the received signal while sampling at four samples per symbol (FIG. 1- a ) is much improved over that while sampling at two samples per symbol (FIG. 1- b ). The challenge in using a digital receiver is to provide the same or similar performance with limited information about the received signal as for an analog receiver. 
     FIGS. 2- a  and  2 - b  illustrate the concept of an eye opening at high (30 dB) and low (5 dB) signal to noise ratios, respectively. Superimposing portions of a signal, equivalent to the duration of one or more symbols, onto itself forms an eye diagram. At high signal to noise conditions there are few fluctuations on the signal so the central portion of the superimposed signal remains clear and tends to form the outline of an eye. This is illustrated in FIG. 2- a  at a 30 dB signal-to-noise ratio. FIG. 2- b  illustrates how the eye closes at lower signal to noise conditions, e.g., 5 dB, due to fluctuations of the amplitude of the signal. 
     Referring now to FIG. 3, a feedback timing estimator configuration  300  will be described. The feedback timing estimator  300  comprises a pulse-shaping filter block  310 , a sampler block  315 , a timing error detector block  320 , additional functional blocks  325 , a loop filter  330  and a Voltage Controlled Clock (VCC)  335 . The pulse-shaping filter  310  may be either an analog or digital component. The pulse shaping filter output is subsequently fed to the sampling device  315 . The phase of the sampling device  315  is controlled by the VCC  335 . The sampling rate of the VCC  335  with respect to the data rate depends on the algorithm used in the timing error detector  320 . The VCC  335  causes the received signal input to be synchronously sampled. The sampled received signal at the output of the sampling device  315  is fed to the timing error detector  320 , which produces an instantaneous error signal based on the sampler output samples. The output of the timing error detector block  320 , is fed to the loop filter  330  which averages the instantaneous error signals from the timing error detector  320  to produce a smoothed error response. The longer the averaging interval in the loop filter, the more accurate the timing estimate. This averaged error signal is subsequently fed to the VCC  335  to alter the phase of the sampling clock that samples the continuous received signal at the input of the sampling device  315 . The data samples at the output of the sampling device are subsequently fed to additional functional blocks  325  for further processing. 
     Referring now to FIG. 4, a feedforward timing estimator  400  configuration will be described. The feedforward timing estimator  400  comprises an ADC  410 , a pulse shaping filter  415 , a processing delay  420 , an interpolation/decimation unit  425 , additional functional blocks  430  and a timing estimator unit  435 . In the feedforward configuration, the first element is the ADC  410  which asynchronous samples the received signal at a rate of two or more samples per symbol. Typically, the sampling rate is four or more samples per symbol. The sampled output signal from the ADC  410  is fed to a digital pulse-shaping filter  415 . The data stream from the output of the pulse-shaping filter  415  is fed to the timing estimation unit  435  as well as to a processing delay unit  420 , which compensates for the delay in the calculation of the timing offset in the timing estimator unit  435 . The timing offset is then fed to the interpolation/decimation unit  425 . The interpolation/decimation unit  425  first interpolates the sampled received signal at the instant corresponding to the timing offset. The decimation unit then removes the redundant samples in the decimation unit. The decimated output signal is then fed to additional functional blocks  430  in the receiver. 
     Referring to FIG. 5, a digital receiver  500  using a timing estimation method will be described. In one embodiment, the digital receiver  500  comprises a filtering subsystem  510 , a processing delay compensation module  515 , a timing estimation subsystem or estimator  600 , an interpolation/decimation subsystem  520  and additional subsystems  525 . The filtering subsystem  510  comprises a pulse-shaping filter which receives the data stream  505  from an Intermediate Frequency (IF) subsystem (not shown) of the receiver. This data stream  505  may have already been asynchronously sampled within the IF subsystem. In addition, the IF subsystem may also contain an analog anti-aliasing filter to limit the bandwidth of the received signal. 
     The sampling rate of the data stream is two samples per symbol, as shown in FIG. 1- b.  The sampled data stream  505  is then fed into the pulse-shaping filter within the filtering subsystem  510 . The pulse-shaping filter is preferably matched to the pulse-shaping filter used at the transmitter. The pulse-shaping filter provides a Nyquist pulse shape which gives optimal performance in the presence of Additive White Gaussian Noise (AWGN). In other embodiments, the filtering subsystem  510  may also comprise additional components. 
     The output of the filtering subsystem  510  is then fed into the timing estimation subsystem  600  to estimate the timing offset present on the received sampled signal. The output of the filtering subsystem  510  is also fed to a processing delay  515  which compensates for the delay between the output of the filtering subsystem and the calculation of the timing estimate, which takes place in the estimator  600 . The output of the processing delay  515  is fed to an interpolation and decimation subsystem or module  520 , which interpolates the samples using the timing estimate from the timing estimator  600 . The timing estimate correction unit  520  estimates the data samples corresponding to the estimated timing offset. The estimated data samples are then fed to additional synchronization and functional blocks  525 . 
     FIG. 6 illustrates a functional block diagram of the timing estimation subsystem  600  used in the digital receiver  500 . The timing estimator  600  comprises a magnitude squarer  605  operator; delay  615   a,  complex conjugation  615   b  and multiply  610  operators; two demultiplexers  620   a  and  620   b;  four summation units  625   a,    625   b,    625   c  and  625   d;  two subtraction units  630  and  635 ; a constant gain factor  640 ; and an angle calculator  645 . The angle calculator  645  may be implemented as a look-up table. The inputted data stream from the filtering block  510  is fed into the timing estimation subsystem  600 . 
     The sampled received signal at the output of the pulse-shaping matched filter  510  may be represented as follows:                  r        (     mT   s     )       =         ∑   k                       a   k          g        (       mT   s     -   kT   -   τ     )            exp        (   jφ   )           +     b        (     mT   s     )           ,           Equation                 1                                
     where r(mT s ) represents the received signal sampled at time instants mT s  following the pulse-shaping filter, oversampling rate T s  is related to the data symbol rate T as T s =T/2, a k  is the arbitrary transmitted data, g(t) represents the pulse-shaping matched filter, τ is the timing offset, b(mT s ) represents the filtered sampled AWGN of noise spectral density N o , and exp(jφ) represents an exponential phase offset. The baseband received signal is first passed through an anti-aliasing filter in the IF subsystem and then asynchronously sampled. It is known to an engineer in the technology that when the sampling frequency is greater than twice the maximum frequency of a bandlimited signal, the resulting samples contain the same information as the continuous time signal. In one embodiment, the received signal is sampled at two samples per symbol such that T s =T/2. 
     The filtered received data stream  510  is fed into two nonlinearities. The first is a magnitude squarer operator  605  and the second nonlinearity is a delay  615   a,  complex conjugation  615   b  and multiply  610  operations subsystem. In one embodiment, a magnitude square nonlinearity and a delay, complex conjugation and multiply nonlinearity are used. However, other nonlinearities or combinations of nonlinearities may also be used. The advantage of using a magnitude square or delay, complex conjugation and multiply nonlinearity is that any phase offset, which may have been present on the received signal, is removed at its output. Therefore, this estimator  600  can be used in the presence of a phase offset on the received signal. 
     The average value of the received signal defined in Equation 1 at the output of a magnitude squaring nonlinearity is a cyclostationary process A (mT s ), which is exploited to estimate the timing offset. A cyclostationary process implies A( 0 )=A(mT s ), A(T/2)=A((2m+1)T s /2) for all values of m. A cyclostationary process generates spectral lines in the frequency domain. However, due to the pulse-shaping filter, only those terms at 1/T and 1/T and the energy component at DC (zero Hertz) are relevant to timing estimation. Evaluating A(mT s ) at two samples per symbol at mT s =0 and mT s =T/2 gives,                      A        (   0   )       =       1   T          [       c   0     +     2        c   1          cos        (       2                 π                 τ     T     )           ]                       A        (     T   2     )       =       1   T          [       c   0     -     2        c   1          cos        (       2      πτ     T     )           ]         ,                 Equation                 2                                
     where c 0  is the energy component at DC (i.e., zero Hertz) and c 1  is the energy of the pulse shaping filter g(t) when overlapped with a symbol rate shifted replica of itself as shown in Equation 3 as follows:                      c   0     =       ∫     -   ∞     ∞              G   2          (   f   )                          f                       c   1     =         ∫     -   ∞     ∞            G        (     f   +     1     2      T         )            G        (     f   -     1     2      T         )                          f         =       β                 T     8                     Equation                 3                                
     Expansion of the c 1  term gives βT/8 for the case of a raised cosine filter, where β is the roll-off factor which takes on values in the range [0,1]. 
     From Equation 2, an expression for the Cosine function of the timing offset is generated. However, an expression for the Sine function of the timing offset is necessary to isolate an expression for the timing offset. A suitable nonlinearity is a unit delay, complex conjugation and multiply operation as illustrated at blocks  615   a,    615   b  and  610 . The output of this nonlinearity is also a cyclostationary process B(mT s ). The sampled average values from the delay  615   a,  complex conjugation  615   b  and multiply unit  610  at mTs=0 and mTs=T/2, at the output of the summation blocks  625   c  and  625   d,  are:                      B        (   0   )       =       1   T          [       c   0     -     2        c   2          sin        (       2                 π                 τ     T     )           ]                       B        (     T   2     )       =       1   T          [       c   0     +     2        c   2          sin        (       2      πτ     T     )           ]         ,                 Equation                 4                                
     Where c 2  is                  c   2     =       ∫     -   ∞     ∞            G        (     f   +     1     2      T         )            G        (     f   -     1     2      T         )            cos        (     π                 f                 T     )                          f           ,           Equation                 5                                
     And G(f) is the Fourier transform of the pulse shape g(t) in Equation 1. Expansion of c 2  in Equation 5 gives a value very close to c 1 , where the approximation c 1 =αc 2  is appropriate if α is chosen to be a value very close to one (1). Note that A( 0 ), A(T/2), B( 0 ) and B(T/2) are four equations with the timing offset being the unknown parameter. 
     The next stage in the timing estimator  600  is to average the instantaneous values at the outputs of both the square-law nonlinearity  605  and the delay  615   a,  complex conjugation  615   b  and multiply nonlinearity  610 , to form A( 0 ), A(T/2), B( 0 ) and B(T/2). However, this first requires isolating the even and odd samples from both nonlinearities to form the expressions for A( 0 ), A(T/2), B( 0 ) and B(T/2). The even samples at the output of the two nonlinearities are those corresponding to mT s =nT where n is an integer and the odd samples are those at mT s =nT+T/2. The demultiplexers in blocks  620   a  and  620   b  isolate the even and odd samples from both nonlinearities and are controlled using the sampling clock. 
     The samples are then averaged over the observation interval to estimate A( 0 ), A (T/2), B( 0 ) and B(T/2) as illustrated at blocks  625   a,    625   b,    625   c  and  625   d.  Equation 2 and Equation 4 can now be solved as four simultaneous equations with two unknowns, namely cos(2πτ/T) and sin(2πτ/T). 
     The cos(2πτ/T) term is formed as the difference of A( 0 ) and A(T/2) as can be noted by examining Equation 2. The sin(2πτ/T) term is formed as the difference of B(T/2) and B( 0 ) as can be seen by examining Equation 4. The cos(2πτ/T) term is formed at subtraction unit  630  and the sin(2πτ/T) term is formed at subtraction unit  635 . Calculating the angle formed by the complex components derived at units  630  and  635  gives an expression for the timing offset. The timing estimate is formed as follows:                  τ   ^     T     =       1     2      π            arg   [       α   (       A   (   0   )     -     A   (     T   2     )       )     +     j        (       B        (     T   2     )       -     B        (   0   )         )         ]               Equation                 6                                
     Where α is a gain factor of value close to unity, which is ideally c 1 /c 2 . The estimator  600  thus provides immunity to the effect of phase offsets on the input signal. This gives a designer much more flexibility in the design of an all digital receiver. 
     The advantage of the estimator is especially apparent for digital communication systems having high data rates. In such systems, oversampling the received signal at more than two samples per symbol may not always be possible or desirable ( Tayebi et al., “Wireless Multimedia Carrier System, ” U.S. patent application Ser. No. 08/954,217, filed Oct. 20, 1997). The estimator of the present invention uses a similar approach (i.e., feedforward design with asynchronous sampling) as used for an alternative algorithm in the literature. The known algorithm requires a minimum oversampling rate of four samples per symbol (H. Meyr, M. Moeneclaey and S. A. Fechtel, “ Digital Communication Receivers: Synchronization, Channel Estimation and Signal Processing, ” John Wiley Publishers, pp. 289-295, 1998). However, with the addition of a delay, complex conjugation and multiply nonlinearities, the present estimator avoids the requirement of having a minimum oversampling rate of four samples per symbol. Yet even with reduced sampling, the estimator is immune to the presence of a slowly varying phase offset over the observation interval. 
     While the above detailed description has shown; described, and pointed out the fundamental novel features of the invention as applied to various embodiments, it will be understood that various omissions and substitutions and changes in the form and details of the system illustrated may be made by those skilled in the art, without departing from the intent of the invention.