Sigma-delta modulation with leaky integration and constant input

Exact descriptions of the behavior of quantization noise for single-loop, multistage (cascade), and multiloop sigma-delta modulators for a variety of input signals have been found during recent years under the assumption of ideal integration. The techniques used to solve the ideal integrator case do not easily extend to the more realistic model of a leaky integrator sigma-delta. In this paper a dynamical system representation for the leaky integrator sigma-delta with a constant input is developed. Several properties of the resulting piecewise monotone and piecewise linear transformation T on the interval [0,1] are derived and applied to analyze the behavior of the corresponding sigma-delta modulator. It is shown that almost all constant inputs yield a periodic output sequence and an asymptotically periodic quantizer error sequence. These sequences are described. It is also found that unlike the ideal system, inputs of different value contained in a certain range can produce identical output sequences