Perturbative corrections to stochastic resonant quantizers

Abstract

This communication considers perturbative effects on 2-level and 3-level stochastic resonant (SR) quantizers. Such quantizers are briefly reviewed in the small input signal-to-noise ratio (SNR) limit. First order perturbative corrections to the optimal SNR gain and normalized threshold due to small, non-zero input SNRs and a drift in the noise probability density function (PDF) are derived. The noise PDF is assumed to belong to the family of generalized Gaussians indexed by the parameter p ∈ [ 1 , ∞ ) . For p > 1 , it is established that these corrections are: (i) bounded, indicating that SR quantizers are stable to such perturbative effects and (ii) can be evaluated numerically using standard mathematical functions and improper integrals. For p → 1 + , the corrections are found to be singular, indicating that regular perturbation theory becomes inapplicable for such PDFs. In the limit of heavy-tailed noise PDFs two important results are as follows: (i) the corrections to the SNR gains of 2-level and 3-level quantizers due to a variation in the PDF are equal; (ii) the correction to the normalized threshold of the 2-level quantizer due to a variation in the PDF vanish, but that of the 3-level quantizer do not, implying that 2-level quantizers are stabler than 3-level quantizers to variations in the PDF.