Delta Modulation of Time-Discrete Processes with i.i.d. Increments Having a Rational Characteristic Function

We investigate the mean-squared error (MSE) performance of a perfectly integrating delta modulator driven by an input sequence of independent and identically distributed (i.i.d.) increments having a rational characteristic function
\phi Y^{(u)}
. Although the input process is nonstationary, the mean-squared error (MSE) can be finite under some conditions. The limiting characteristic function of the error sequence is found by the method of Wiener-Hopf, and then a formula for the MSE is given in terms of the coefficients of the defining polynomials for
\phi Y^{(u)}
, roots of transcendental equations involving
\phi Y^{(u)}
, and certain quantization parameters. Curves are presented of normalized MSE versus the distribution parameter or quantization parameter, where the increments are one- or two-sided gamma distributed. The results obtained here give some insight to the asymptotic performance of delta modulation of stationary Markov processes having an amplitude distribution belonging to a wide class.