Differential Pulse Code Modulation of Stationary Gaussian Inputs

This paper is concerned with an exact analysis of DPCM systems with stationary Gaussian inputs. We consider a class of digital communication systems in which DPCM is included as a member. An integral equation for the joint probability of the input and the state of the system is derived first. Solution of the equation is sought in the form of a power series in the elements of the covariance matrix of the input that involves generalized Hermite functions. Then the integral equation is reduced to a set of algebraic equations for the coefficients in the series that are solved recursively. The steady-state distribution of the input and the state is thus found. We are interested particularly in the mean-squared error of the system as a function of step size, the sampling interval and the number of quantization levels. Numerical results are shown for the GaussMarkov input; the MSE is calculated in terms of system parameters and performance of DPCM and PCM is compared with reference to the theoretical bound.