Lower bounds on the MSE of the single and double loop sigma delta modulators

It is shown that the single- and double-loop sigma-delta (ΣΔ) encoders with constant input can be analyzed as nonuniform quantizers. Specifically, expressions are derived for the transition points, i.e. the input values at which the output sequence changes from one codeword to another. This is used to find upper bounds on the number of codewords that the encoders are capable of producing as source coders, given the initial state or states. The upper bound is O(N²) for the single-loop encoder, and O(N³) for the double-loop encoder, where N is the oversampling ratio. It is shown that the mean squared error (MSE) for both single- and double-loop decoders is lower-bounded by O(N^-3). The optimal decoders based on table look-up decoding are shown to have O(N^-3 ) MSE performance. For the double-loop encoder, an optimal decoder with O(N^-5) MSE performance can be achieved by a slight reduction in the dynamic range of the input