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