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 u0, and a quantizer that may contain an offset error. The mean-square error (MSE) of any decoding scheme for this quantizer (with u0 and the offset error known) is bounded below by 1/96 N-3. We also determine the asymptotically best possible MSE as N → ∞ for perfect decoding when u0 = 0 and u0 = 1/2. The robustness result is the upper bound that a triangular linear filter decoder (with both u0 and the offset error unknown) achieves an MSE of 40/3 N-3. These results estab lish 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.
- Dynamical systems
- Oversampled quantization
- Sigma-delta modulation
ASJC Scopus subject areas
- Information Systems
- Computer Science Applications
- Library and Information Sciences