5.1 Introduction

Here we consider the following question: given a time series obtained by regular sampling of some waveform, how do we form the time series of a delayed version of the waveform? Clearly there is no real problem for a delay that is a multiple of the sampling period-instead of the current sample from the undelayed waveform, we just take the correctly delayed sample. The required series could be obtained from a shift register clocked at the sampling rate. Thus, we are left with the problem of generating series corresponding to delays of less than a sampling period. We consider only sampled analytic signals (complex time series), and we show that considerable benefits, in terms of reduced computation, are given if the waveform is sampled at a rate above the minimum required to retain all its information (see Chapter 4)-the case of oversampling.

First, in Section 5.2, we investigate the weights on the taps of a transversal filter required to give the series for the delayed waveform, which are derived without reference to the waveform. This filter is thus suitable for the general case, where any waveform (subject to it being within a given bandwidth) may be taken and where its power spectrum is not necessarily known. We start with the case of the minimum sampling rate and then explore the gains possible with an oversampled waveform. In Section 5.3, we find the weights that give the optimum series in the sense of the least mean square error (or...