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/sub o/, and a quantizer that may contain an offset error. The mean-square error (MSE) of any decoding scheme for this quantizer (with u/sub o/ and the offset error known) is bounded below by 1/96N/sup -3/. We also determine the asymptotically best possible MSE as N/spl rarr//spl infin/ for perfect decoding when u/sub o/=0 and u/sub o/= 1/2 . The robustness result is the upper bound that a triangular linear filter decoder (with both u/sub o/ and the offset error unknown) achieves an MSE of 40/3N/sup -3/. These results establish the known result that the O(1/N/sup 3/) 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.