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On the robustness of single-loop sigma-delta modulation

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3 Author(s)
Güntürk, C.S. ; Program in Appl. & Comput. Math., Princeton Univ., NJ, USA ; Lagarias, Jeffrey C. ; Vaishampayan, V.A.

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 uo, and a quantizer that may contain an offset error. The mean-square error (MSE) of any decoding scheme for this quantizer (with uo 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 uo=0 and uo=½. The robustness result is the upper bound that a triangular linear filter decoder (with both uo 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

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Information Theory, IEEE Transactions on  (Volume:47 ,  Issue: 5 )