New properties of sigma-delta modulators with DC inputs

New properties of single- and double-loop sigma-delta modulators with constant inputs are derived by exploiting the inherent structure os the output sequences or codewords that the modulators are capable of producing. Upper bounds are derived on the number of N-bit codewords for the single- and double-loop modulators. Analytical lower bounds on the mean squared error (MSE) obtainable by any decoder, linear or nonlinear, in approximating the constant input are also derived. Optimal nonlinear decoders for constant inputs based on a table lookup approach which operates directly on the nonuniform quantization intervals are considered. Using simulations is is found that the optimal nonlinear decoders perform better than linear decoders, by about 3 and 20 dB for the single- and double-loop modulators, respectively. A cascade structure specifically for constant inputs is introduced, and its corresponding decoding algorithm is derived. It is shown that for a fixed latency, the MSE performance of the cascade structure is 12 dB superior, and its throughput is twice that of the conventional two-stage MASH modulator