Deterministic analysis of oversampled A/D conversion and decoding improvement based on consistent estimates

Discusses the deterministic analysis of oversampled A/D conversion (ADC), the properties derivable from such an analysis, and the consequences on reconstruction using nonlinear decoding. Given a band-limited input X producing a quantized version C, the authors consider the set of all input signals that are band-limited and produce C. They call any element of this set a consistent estimate of X. Regardless of the type of encoder (simple, predictive, or noise-shaping), they show that this set is convex, and as a consequence, any nonconsistent estimate can be improved. They also show that the classical linear decoding estimates are not necessarily consistent. Numerical tests performed on simple ADC, single-loop, and multiloop ΣΔ modulation show that consistent estimates yield a mean square error (MSE) that decreases asymptotically with the oversampling ratio faster than the linear decoding MSE by approximately 3 dB/octave. This implies an asymptotic MSE of the order of 𝒪(R^-(2n+2)) instead of 𝒪(R^-(2n+1)) in linear decoding, where R is the oversampling ratio and n the order of the modulator. Methods of improvement of nonconsistent estimates based on the deterministic knowledge of the quantized signal are proposed for simple ADC, predictive ADC, single-loop, and multiloop ΣΔ modulation