An estimate of optimum performance is derived for a single-integration, adaptive delta modulator. Several simulations of adaptive delta modulators with single integrators have all produced signal-to-noise ratios near or below the estimate. The derivations presented here indicate that the performance of a single-integration delta modulator is dependent on the correlation between adjacent samples of the input signal and on the probability density function of its derivative. The relationship between the probability density of the derivative of the input signal and optimum performance, in turn, explains why signal-to-noise ratios taken on sine waves are greater than those recorded while processing speech signals.
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