Multi-loop sigma-delta quantization: spectral analysis

An exact spectral analysis of oversampled discrete-time L-loop ( L⩾2) sigma-delta modulator (ΣΔM) with DC (constant) input is presented. It is shown by a novel application of ergodic theory that the embedded quantizer error sequence is asymptotically uniformly distributed, white and uncorrelated with the input, a fact usually assumed a priori in the linearized models. It was found that for the following linear demodulators: (1) optimum FIR (finite-impulse response) filter, (2) sinc^k filters (k ⩾L+1), and (3) the ideal low-pass filter, the overall system (ΣΔM plus demodulator) can provide a normalized average quantization noise power inversely proportional to the (2L +1)-st power of the oversampling ratio, confirming the performance superiority of multiloop ΣΔM over single-loop ΣΔM