On the robustness of single-loop sigma-delta modulation

Sigma-delta modulation, a widely used method of analog-to-digital (A/D) signal conversion, is known to be robust to hardware imperfections, i.e., bit streams generated by slightly imprecise hardware components can be decoded comparably well. We formulate a model for robustness and give a rigorous analysis for single-loop sigma-delta modulation applied to constant signals (DC inputs) for N time cycles, with an arbitrary (small enough) initial condition u_o, and a quantizer that may contain an offset error. The mean-square error (MSE) of any decoding scheme for this quantizer (with u_o and the offset error known) is bounded below by 1/96N^-3. We also determine the asymptotically best possible MSE as N→∞ for perfect decoding when u_o=0 and u_o=½. The robustness result is the upper bound that a triangular linear filter decoder (with both u_o and the offset error unknown) achieves an MSE of 40/3N^-3. These results establish the known result that the O(1/N³) decay of the MSE with N is optimal in the single-loop case, under weaker assumptions than previous analyses, and show that a suitable linear decoder is robust against offset error. These results are obtained using methods from number theory and Fourier analysis