Output Filter Aware Optimization of the Noise Shaping Properties of ΔΣ Modulators Via Semi-Definite Programming

Abstract

Abstract-The Noise Transfer Function (NTF) of ΔΣ modulators is typically designed after the features of the input signal. We suggest that in many applications, and notably those involving D/D and D/A conversion or actuation, the NTF should instead be shaped after the properties of the output/reconstruction filter. To this aim, we propose a framework for optimal design based on the Kalman-Yakubovich-Popov (KYP) lemma and semi-definite programming. Some examples illustrate how in practical cases the proposed strategy can outperform more standard approaches.

Unified LMI-based design of ΔΣmodulators

“…, our minimization based on the H 2 system norm is equivalent to the minimization of [12]. Then, the difference between the proposed method and [12] lies in the usage of different optimization procedures for solving the H 2 norm objective function.…”

“…Then, the difference between the proposed method and [12] lies in the usage of different optimization procedures for solving the H 2 norm objective function. As pointed out in [12], if one knows the system H S [z] connected to the modulator, one should set…”

“…However, the method in [11] cannot incorporate the system connected to the quantizer into its design, while we consider a nonideal output filter to minimize the quantization noise. In [12,13], the optimization problem based on H 2 norm is formulated as a convex quadratic optimization problem where the weighting function (output filter) impulse response in truncated to finite number of samples.…”

“…If one imposes a constraint on the filter, then there are nine combinations for the design, three types of objectives, and three types of constraints, which can be handled by LMIs. One of these nine combinations, H 2 norm subjected to the Lee criterion, is similar to the design criteria of the method in [12,13]. However, for our proposed H 2 norm design, we provide an alternate approach based on expressing the H 2 norm by using LMIs.…”

“…However, for our proposed H 2 norm design, we provide an alternate approach based on expressing the H 2 norm by using LMIs. The similarity lies in the fact that our proposed design and the method in [12,13] use the idea of incorporating the non-ideal output filter for the minimization of the quantization noise.…”