Design of Noise Transfer Functions for Delta-Sigma Modulators Using the Least-pth Norm

The design of an efficient noise transfer function (NTF) is necessary for the synthesis of a delta-sigma modulator (DSM). In one of the popular methods of designing an NTF, its zeros are optimally placed in order to minimize the noise in the band of interest. The poles, however, are those of a maximally flat all-pole filter. Such a method does not yield the optimal design since the numerator and denominator are considered separately. In this brief, we describe a design method wherein the energy in the band of interest is minimized using a least-pth norm while satisfying all the realizability constraints of a DSM. Our formulation provides the flexibility in assigning both the on- and off-the-unit-circle zeros of the NTF. By way of examples we show that the signal-to-quantization-noise ratio of the NTF designed by our method is substantially higher than that designed by one of the well-known methods. In terms of the signal-to-noise ratio, the NTF designed with our method is better for a wide amplitude range of the DSM. We have also shown the flexibility of our design by incorporating a weighting function that is popular in the audio industry.