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 time cycles, with an arbitrary (small enough) initial condition 0, and a quantizer that may contain an offset error. The mean-square error (MSE) of any decoding scheme for this quantizer (with 0 and the offset error known) is bounded below by 1 96 . We also determine the asymptotically best possible MSE as for perfect decoding when 0 = 0 and 0 = 1 2 . The robustness result is the upper bound that a triangular linear filter decoder (with both 0 and the offset error unknown) achieves an MSE of 40 3 . These results establish the known result that the (1 ) decay of the MSE with 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.