Quantization noise in sigma-delta modulations driven by deterministic inputs

The authors consider the asymptotic statistics of quantization error sequences that arise in oversampling sigma-delta modulators driven by time-varying deterministic inputs. For a wide class of such inputs, the ergodic theory of maps of the torus provides not only a mechanism for proving that such statistics exist but also appealing closed-form representations for them. Among the cases discussed are standard single- and two-stage modulators driven by sampled periodic inputs and the constant-input behavior of two-stage modulators in which one of the discrete-time integrators has nonzero leakage