On delta modulation

We show how the steady-state distribution and the mean squared error of a delta modulator with an ideal integrator can be computed exactly when the input signal to the modulator is a stationary Gaussian process with a rational power spectral density. Curves are presented for the mean squared error as a function of the quantizer step size and the sampling interval for several different input spectra. The mathematical development makes use of the Markov properties of the system and involves series ex-expansions in n-dimensional Hermite functions. The key integral equation is generalized to treat the case of a realizable filter in the feedback path, but an analytic method of solving this equation has not been found.