Oversampled Sigma-Delta Modulation

Oversampled sigma-delta modulation has been proposed as a practical implementation for high rate analog-to-digital conversion because of its simplicity and its robustness against circuit imperfections. To date, mathematical developments of the basic properties of such systems have been based either on simplified continuous-time approximate models or on linearized discrete-time models where the quantizer is replaced by an additive white uniform noise source. In this paper, we rigorously derive several basic properties of a simple discrete-time single integrator loop sigma-delta modulator with an accumulate-and-dump demodulator. The derivation does not require any assumptions on the correlation or distribution of the quantizer error, and hence involves no linearization of the nonlinear system, but it does show that when the input is constant, the state sequence of the integrator in the encoder loop can be modeled exactly as a linear system in an appropriate space. Two basic properties are developed: 1) the behavior of the sigma-delta quantizer when driven by a constant input and its relation to uniform quantization, and 2) the rate-distortion tradeoffs between the oversampling ratio and the average mean-squared quantization error.