Compressed Sensing for Communication Systems

Compressed sensing becomes popular in the field of communications.

This week, there was a small and nice conference, APWCS (Asia Pacific Wireless Communications Symposium), in Kyoto University, JAPAN. Researchers and engineers of wireless communications discussed on compressed sensing for communication systems at a special session called Compressed Sensing for Communication Systems. I felt compressed sensing is a powerful tool for a number of problems in communications.

The following is the summary of the session:

S4: Special Session "Compressed Sensing for Communication Systems"
Thursday, August 23, 11:10 - 12:50, Room: Conference Room III
Chair: Kazunori Hayashi (Kyoto University, Japan)

Session Summary: Compressed sensing has drawing much attention in various fields of applications. This is not only because it enables us to obtain an exact solution from an underdetermined linear system under a certain condition on the measurement matrix taking advantage of the sparsity of the solution, but also because the solution can be obtained via computationally efficient algorithms. In this special session, we focus on the communication applications of compressed sensing, such as OFDM systems, network tomography and networked control systems, as well as a brief introduction to compressed sensing. Moreover, we also have a couple  of talks on the problems with  underdetermined systems, where we discuss a possibility to apply compressed sensing to the problems.

S4.1  A Brief Introduction to Compressed Sensing
Kazunori Hayashi (Kyoto University, Japan); Masaaki Nagahara (Kyoto University, Japan)

S4.2  Application of Compressed Sensing to Inter-Symbol Interference Cancellation in OFDM Systems
Masato Saito (University of the Ryukyus, Japan)

S4.3  Path Construction for Compressed Sensing-Based Network Tomography
Kazushi  Takemoto (Osaka University, Japan); Takahiro Matsuda (Osaka University, Japan); Tetsuya Takine (Osaka University, Japan)

S4.4  Sparsity-Promoting Methods in Remote Control Systems
Masaaki Nagahara (Kyoto University, Japan)

S4.5  A Successive Detector with Virtual Channel Detection for Cooperative Multiuser MIMO Systems
Akihito Taya (Kyoto University, Japan); Satoshi Denno (Okayama University, Japan); Koji Yamamoto (Kyoto University, Japan); Masahiro Morikura (Kyoto University, Japan); Daisuke Umehara (Kyoto Institute of Technology, Japan); Hidekazu Murata (Kyoto University, Japan); Susumu Yoshida (Graduate School of Informatics, Kyoto University, Japan)

S4.6  Nested Array and Its Applications
Masashi Tsuji (Tokyo University of Agriculture and Technology, Japan); Takashi Morimoto (Tokyo University of Agriculture and Technology, Japan); Kenta Umebayashi (Tokyo University of Agriculture and Technology, Japan); Yasuo Suzuki (Tokyo University of Agriculture and Technology, Japan)


Related entries:
Compressed sensing meets control systems
Recent papers on compressed sensing for control systems

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.