Adaptive Delta Modulation with a One-Bit Memory

Abstract

We propose a delta modulator which, at every sampling instant r, adapts its step-size (for a staircase approximation to the input signal) on the basis of a comparison between the two latest channel symbols, Cr and Cr-1. Specifically, the ratio of the modified step-size mr to the previous step size mr-1 is either +P or — Q depending on whether Cr and Cr-1 are equal or not. (We recall that, in delta modulation, Cr represents the polarity of the difference, at the sampling instant r, between the input signal Xr and the latest staircase approximation to it, Yr-1.) A simulation of the delta modulator with a band-limited speech input has revealed that PQ = 1 and P ⋍ 1.5 represent optimal adaptation characteristics, on the basis of signal-to-error ratios, over an important range of sampling frequencies; and that at 60 kHz, delta modulation with these adaptation parameters compares favorably with 7-bit logarithmic PCM, which reproduces speech with good telephone quality. We present several graphical results from this simulation, and include an evaluation of the effect of independent channel errors on the adaptive delta modulator. We proceed to suggest a heuristic theory of the delta modulator which explains the optimality of the condition PQ = 1, and develops an upper bound of 2 for the optimum value of P. We conclude with a summary of results from a video simulation which revealed that aforementioned optima for P and Q apply to a video signal