Formal Spectral Theory for Ideal Sigma-Delta Quantization with Stationary Time-Varying Inputs

Because a ΣΔ modulator is a nonlinear feedback system, its rigorous signal analysis escapes from the standard theories used in signal processing and communications. We introduce mathematical tools that are new to these two areas and that lead to systematic and high-level methods for the spectral error analysis of ΣΔ modulators. They include the use of dynamical system techniques, of the roots of ergodic theory and the spectral properties of unitary operators in a Hubert space. In this paper, we show their application to the case of ideal ΣΔ modulation, thus providing a formal and concise theory for the extensive derivations performed by Gray et al. and He et al. in this special case some 15 years ago. We finally point to the potential extension of these new methods to a substantially larger class of ΣΔ modulators.