Timing recovery in PAM systems

It is shown how various timing recovery schemes are reasonable approximations of the maximum likelihood strategy for estimating an unknown timing parameter in additive white gaussian noise. These schemes derive an appropriate error signal from the received data which is then used in a closed-loop system to change the timing phase of a voltage-controlled oscillator. The technique of stochastic approximation is utilized to cast the synchronization problem as a regression problem and to develop an estimation algorithm which rapidly converges to the desired sampling time. This estimate does not depend upon knowledge of the system impulse response, is independent of the noise distribution, is computed in real time, and can be synthesized as a feedback structure. As is characteristic of stochastic approximation algorithms, the current estimate is the sum of the previous estimate and a time-varying weighted approximation of the estimation error. The error is approximated by sampling the derivative of the received signal, and the mean-square error of the resulting estimate is minimized by optimizing the choice of the gain sequence. If the receiver is provided with an ideal reference (or if the data error rate is small) it is shown that both the bias and the jitter (mean-square error) of the estimator approach zero as the number of iterations becomes large. The rate of convergence of the algorithm is derived and examples are provided which indicate that reliable synchronization information can be quickly acquired.