STATISTICS OF THE BINARY QUANTIZER ERROR IN SINGLE-LOOP SIGMA-DELTA MODULATION WITH WHITE GAUSSIAN INPUT

Representations and statistical properties of the process ($) over bar e defined by ($) over bar e(n+1) = lambda(($) over bar e(n) + xi(n)), are given. Here = lambda(u): =u - b . sign (u) + m and {xi(n)}(+infin ity)(n=0) is Gaussian white noise. The process ($) over bar e represen ts the binary quantizer error in a model for single-loop Sigma-Delta m odulation. The innovations variables are found and the existence and u niqueness of an invariant probability measure, ergodicity properties, as well as the existence of the exponential moment with respect to the invariant probability are proved using Markov process theory. We cons ider also ($) over bar e as a random perturbation, for small values of the variance of xi(n), of the orbits of s(n+1) = lambda(s(n)). Here s (n) has the uniform in;variant distribution on the interval [m - b, m + b]. Analytical approximations to the structure of the power spectrum of ($) over bar e are obtained using a linear prediction in terms of the innovations variables and the perturbation approach.