Two-stage sigma-delta modulation

An exact analysis of the nonlinear difference equation describing a discrete time two-stage sigma-delta quantizer with DC and sinusoidal inputs is presented. It is shown that for DC inputs, the binary quantizer error of the second quantizer, which appears at the output of the modulator as a second-order difference, is asymptotically white, uniformly distributed, and uncorrelated with the input. For sinusoidal inputs, this result holds with one exception: the autocorrelation, which is a sum of weighted Bessel functions, does not correspond to that of a white noise process. At large oversampling ratios, the autocorrelation is approximately white and hence the white additive noise model is a good approximation for evaluating the first- and second-order moments. It is found that for both a sinc³ decoder filter and an ideal low-pass filter decoder, the average quantization noise power is inversely proportional to the fifth power of the oversampling ratio R for both types of inputs. This represents an improvement over the single-loop sigma-delta quantizer by a factor of R² . Simulation results that agree with the theoretical analysis are presented