Optimum Timing Phase for an Infinite Equalizer

A digital equalizer for data transmission linearly combines a sequence of samples b(r + kt), k = 0, ± 1, ··· of the received data wave to mitigate the effects of intersymbol interference and noise. A natural question is, How will the performance of such a system depend on the timing phase τ, 0 ≦ τ ≦ T? We examine this problem in considerable detail for an infinite equalizer using a mean-square measure of performance. Excess bandwidth results in a significant difference in performance between the best and worst timing phase. Under practical noise conditions, it is estimated that the excess bandwidth must be down to 1 or 2 percent before the timing effect becomes insignificant. With a 10-percent roll-off, a 3-dB penalty can be incurred by choosing a bad timing epoch. Our main result is that under conditions likely to be encountered on channels similar to voiceband telephone channels, the optimum sampling instants will be accurately approximated by the consecutive maxima and minima of the sine wave that result when an alternating sequence of positive and negative pulses (dotting sequence) is transmitted. There are no additional local minima of the minimum mean-square error as the timing phase is varied.