Statistics of the binary quantizer error in single-loop sigma-delta modulation with white Gaussian input

Representations and statistical properties of the process e¯ defined by e¯_n+1=λ(e¯_n+ξ_n ), are given. Here λ(u):=u-b·sign(u)+m and {ξ_n}_n=0^+∞ is Gaussian white noise. The process e¯ represents the binary quantizer error in a model for single-loop sigma-delta modulation. The innovations variables are found and the existence and uniqueness 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 consider also e¯ as a random perturbation, for small values of the variance of ξ_n, Of the orbits of s_n+1=λ(s_n). Here s_n has the uniform invariant distribution on the interval [m-h, m+b]. Analytical approximations to the structure of the power spectrum of e¯ are obtained using a linear prediction in terms of the innovations variables and the perturbation approach