Optimal Design of ∆Σ Modulators via Generalized KYP Lemma

∆Σ modulators [1] are widely used in AD (Analog-toDigital) and DA (Digital-to-Analog) converters, in which high performance can be obtained with coarse quantizers. A fundamental issue in designing ∆Σ modulators is noise shaping in the frequency domain [1]. A usual solution to this is to insert accumulator(s) in the feedback loop to attenuate the gain of the noise transfer function (NTF) in low frequencies. This methodology looks like PID (Proportional-Integral-Derivative) control [2], in which the performance of the designed system depends on the amount of experiences of the designer. That is, the conventional design is of an ad hoc nature. Let us consider a general ∆Σ modulator shown in Fig. 1. In this modulator, Q is a quantizer and H = [H1, H2] is a linear filter with 2 inputs and 1 output. The filter H1 shapes the signal transfer function (STF) from the input u to the output y to have a unity gain in the frequency band of interest. On the other hand, the filterH2 eliminates the in-band quantization noise by shaping the NTF. To shape optimally the NTF in the frequency band of interest, say [0,Ω], the NTF zero optimization [1] can be used. This method is to minimize the normalized noise power, given by the integral of the squared magnitude of the NTF over [0,Ω]. On the other hand, we minimize the maximum of the gain of the NTF in [0,Ω]. This is related to a minimax optimization (or an H one), and more effective than the NTF zero optimization in terms of uniform attenuation of the frequency response over the band. We have proposed an H optimization in [3], in which we have to choose a suitable weighting function to obtain a good performance. On the other hand, we propose in this article more useful method with no weighting function, by generalized Kalman-YakubovicPopov (KYP) lemma [4]. Then the optimization can be reduced to one with a linear matrix inequality (LMI). The