Vector quantization analysis of ΣΔ modulation

When considering a class of finite-dimensional input signals such as bandlimited signals in the discrete Fourier sense within a finite time window [O,T] of observation, we show that a single-loop ΣΔ modulator behaves like the encoder of a vector quantizer. The intrinsic behavior of the modulator can be studied by analyzing the partition it generates in the input space. We show that this partition yields a particular structure that we call the “hyperplane wave structure”. As one consequence to this analysis, it can be proved for the considered class of bandlimited signals that the mean squared error (MSE) of reconstruction cannot asymptotically decrease with the oversampling ratio R faster than O(R^-4), regardless of the type of reconstruction used. We then generalize this analysis to n-loop ΣΔ modulators and show that the MSE cannot decrease faster than O(R^-(2n+2))