Optimal Design of Delta-Sigma Modulators

Control theory sometimes gives a powerful tool for solving problems in signal processing. LMI (Linear Matrix Inequality) is one of them. Today, I'd like to introduce my paper (preprint PDF) on LMI application to delta-sigma modulation:

M. Nagahara and Y. Yamamoto, Frequency Domain Min-Max Optimization of Noise-Shaping Delta-Sigma Modulators, IEEE Trans. on Signal Processing, Vol. 60, No. 6, pp. 2828-2839, 2012.

What is delta-sigma modulation? The Wikipedia entry reads:

Delta-sigma (ΔΣ; or sigma-delta, ΣΔ) modulation is a method for encoding analog signals into digital signals or higher-resolution digital signals into lower-resolution digital signals. The conversion is done using error feedback, where the difference between the two signals is measured and used to improve the conversion. The low-resolution signal typically changes more quickly than the high-resolution signal and it can be filtered to recover the high-resolution signal with little or no loss of fidelity. 

To filter out the quantization noise from low-resolution quickly-changing signals, it is essential to push away the quantization noise out of the frequency band of the original signal. In other words, it should be sufficiently attenuated the noise transfer function (NTF) from the quantization noise to the modulator output (before the lowpass filter).

Although a delta-sigma modulator ia a highly nonlinear system, designing is a linear problem like filter design; shape the frequency response of NTF.

Conventionally, NTF is shaped such that the squared norm of the magnitude (i.e., H² norm) of NTF on the frequency band of interest. As I noted in my past entry that an H²-based system shows very nice performance for almost all frequencies but is fragile for a frequency, say f₀ [Hz] (see the picture below).

To avoid this, we introduced the H^∞ norm to uniformly attenuate the magnitude on a frequency band. A difficulty is that this problem is not a standard H^∞ problem since it does not optimize over the whole frequency band [0,π) but on a subset [0,Ω), Ω<π. You cannot use the standard hinfsyn
command in MATLAB. How to solve it?

The solver is found in control theory, generalized KYP (GKYP) lemma. This solves our problem via LMI (linear matrix inequality).

The picture below shows two plots: the Bode magnitude plot of NTF designed by GKYP (red line) and by H²-based zero optimization (blue line). The magnitude by GKYP is uniformly attenuated over the low frequency range, while that by H² shows peaks in this band. The difference between the two maximal magnitudes at the frequency ω = π/32 is approximately 11.2 (dB), and the difference at low frequencies is about 12.4 (dB).

The paper is available here.
MATLAB codes are available here.
GKYP for signal processing is also discussed in this article.

A related blog entry is here.