sdm-opt

Code for designing sigma delta modulator loop filters with optimal properties. Developed during my masters thesis research at the Digital Health Innovation Lab.

Requirements

• MATLAB (tested in versions 2018a-2022a)
• YALMIP
• Delta Sigma Toolbox
• Optimization solver such as Mosek or Sedumi (optional, the built-in LMILAB seems to work better for this problem)

Getting Started

This repository includes the main function sdsyn.m, its dependencies, as well as a large amount of other code developed during the thesis. Much of this is not well documented and was used for testing ideas. However, there are some robust control synthesis functions that may be useful in the \Control subfolder.

The process of designing a loop filter has the following steps:

1. Define the optimization constraints for performance and stability. This is described at length in [1].
2. Obtain an initial starting condition loop filter. The optimization is quite sensitive to initial conditions so it is recommended to have a good initial guess.
3. Run the sdsyn.m function.

Reproducing Examples from the Thesis and Paper

H-Inifinity Optimal Design (Lee Criterion)

First, define the optimization objectives using the defineSynOpt.m function. This function takes in five parameters per objective:

1. Target value of the norm (-1 to minimize, Inf to not optimize).
2. Norm (1 for $\ell_1$, 2 for $\mathcal{H}2$, or Inf for $\mathcal{H}\infty$).
3. Frequency interval, as a 2-element vector for the digital frequency range to be optimized ([0 pi] to span the entire frequency range, otherwise GKYP lemma used to optimize over a finite frequency interval).
4. Input channel, an index to select which generalized plant disturbance input should be optimized.
5. Output channel, an index to select which generalized plant performance output channel should be optimized

The channels for 4. and 5. are summarized in this table, adapted from Table 2.1 in [1]. See also the function sdsyn_buildplant in sdsyn.m, line 638.

I\O	1. z	2. e	3. u	4. y
1. w	Quantizer gain robustness	Not used	Not used	Not used
2. r	Not used	NTF performance	Constraints on quantizer input	STF constraints for CT design

The optimization objectives for the H-Infinity case are defined as such:

synOpt_51 = defineSynOpt(-1, Inf, [0 pi/32], 2, 2, 1.5, Inf, [0 pi], 2, 2);

Next, an initial guess of the loop filter is required. For this, it is convenient to use the popular synthesizeNTF command of the Delta Sigma Toolbox.

S0 = synthesizeNTF(5, 32, 1, 0.5, 0);

synthesizeNTF produces a NTF, which must be converted into loop filter form following Equation 1.3 from [1]. The poles must also be slightly scaled so that the loop filter is strictly stable, otherwise sdsyn.m will produce an error.

H0 = zpk(minreal((1 - S0)/S0));
H0.p{1} = H0.p{1}*0.995;

The main sdsyn function may now be run:

[H, S, CL, normz, diagn, iterProgress] = sdsyn(H0, synOpt_51, 'display', 'on');

Once the optimization is complete, the NTF may be evaluated using the predictSNR function from the Delta Sigma Toolbox.

[snr, amp] = predictSNR(S, 32);
figure; 
plot(amp, snr, '.-k')
title('SQNR vs. Amplitude for H-Infinity Design')
xlabel('Amplitude (dB)')
ylabel('SQNR (dB)')

The result is a modulator stable to about -5 dBFS and with SQNR of 77 dB. Results are often dependent on inital condition and choice of optimization hyperparameters, such as those on line 195 of sdsyn.m.

This example and others may be seen in the file sdsyn_examples.m.

More Information

[1] B. C. Hannigan, "On the Design of Stable, High Performance Sigma Delta Modulators", MASc. thesis, University of British Columbia, 2018.

[2] B. C. Hannigan, C. L. Petersen, A. M. Mallinson, and G. A. Dumont, "An Optimization Framework for the Design of Noise Shaping Loop Filters with Improved Stability Properties", Circuits, Systems, and Signal Processing, vol. 39, no. 12, pp. 6276–6298, 2020.