An Optimization Framework for the Design of Noise Shaping Loop Filters with Improved Stability Properties
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An Optimization Framework for the Design of Noise Shaping Loop Filters with Improved Stability Properties Brett C. Hannigan1 Guy A. Dumont4
· Christian L. Petersen2
· A. Martin Mallinson3 ·
Received: 29 April 2019 / Revised: 15 May 2020 / Accepted: 18 May 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract A framework using semidefinite programming is proposed to enable the design of sigma delta modulator loop filters at the transfer function level. Both continuous-time and discrete-time, low-pass and band-pass designs are supported. For performance, we use the recently popularized Generalized Kalman–Yakuboviˇc–Popov (GKYP) lemma to place constraints on the H∞ norm of the noise transfer function (NTF) in the frequency band of interest. We expand the approach to incorporate common stability criteria in the form of H2 and 1 norm NTF constraints. Furthering the discussion of stability, we introduce techniques from control systems to improve the robustness of the feedback system over a range of quantizer gains. The performance-stability trade-off is examined using this framework and motivated by simulation results. Keywords Sigma delta modulation · Semidefinite programming · Noise transfer function · Generalized KYP lemma · Analog/digital conversion
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Brett C. Hannigan [email protected] Christian L. Petersen [email protected] A. Martin Mallinson [email protected] Guy A. Dumont [email protected]
1
School of Biomedical Engineering, The University of British Columbia, 251-2222 Health Sciences Mall, Vancouver, BC V6T 1Z3, Canada
2
ESS Technology Inc., 601-1726 Dolphin Ave., Kelowna, BC V1Y 9R9, Canada
3
SiliconIntervention Inc., 403-460 Doyle Ave., Kelowna, BC V1Y 0C2, Canada
4
Department of Electrical and Computer Engineering, The University of British Columbia, 3023-2332 Main Mall, Vancouver, BC V6T 1Z4, Canada
Circuits, Systems, and Signal Processing
1 Introduction Sigma delta modulators (Ms) are nonlinear feedback systems containing a noise shaping filter and coarse quantizer element. Applied as an A/D converter, the systems operate on an oversampled input and produce a discrete-time, sampled value output. The feedback loop contains a noise shaping loop filter that pushes the error introduced by quantization out of the signal band, where it can be removed by a digital decimation filter outside the loop. The sigma delta architecture is widely used to digitize signals with moderate frequency content because of high resolution and reliance on less expensive digital circuitry rather than precision analog components. However, the presence of a nonlinearity in the feedback system makes analysis difficult and higher-order systems are prone to instability. The design of the loop filter transfer function may be done in many ways. Often, a linearized model is used, where the nonlinear quantizer is replaced by a fixed gain and an additive “quantization noise” signal. A loop filter of just one or two pure integrators is provably stable for dc inpu
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