Semidefinite Programming in Analysis and Optimization of Performance of Sigma-Delta Modulators for Low Frequencies

This paper concerns analysis and synthesis of Sigma-Delta modulators with a one-bit quantizer. The Sigma-Delta modulator model in this paper is a feedback system consisting of a linear dynamical system in the forward loop (loop filter) and a one-bit quantizer (sign function) in the feedback, with the zeros of the loop filter being the main design parameters. An upper bound on the average power of the quantization error within a specific frequency range is formulated. This upper bound is based on a worst-case performance criterion and is directly proportional to the maximum amplitude of the output before quantization and inversely proportional to the minimum magnitude of the transfer function of the loop filter within the frequency range of interest. Minimization of this upper bound is desirable for optimization of performance. Quadratic Lyapunov functions and the finite frequency KYP lemma are used to formulate the design problem as an optimization problem. It is shown that after various convex relaxations this problem can be set up as a semidefinite program. Several second order Sigma- Delta modulators are designed and their performances are compared with a benchmark Sigma-Delta modulator. Our designs performed better than the benchmark in various simulations.