**10. References**


188 Applications of Digital Signal Processing

The work reported in this chapter started with our work on compressive sensing for direction of arrival (DOA) detection with a phased array (Shaw and Valley, 2010). In that work, we realized that most work in compressive sensing concerned recovering signals on a sparse grid. In the DOA domain, that meant that targets had to be on a set of grid angles, which of course never happens in real problems. We found a recovery solution for a single target in that work by scanning the sparsifying DFT over an offset index that was a measure of the sine of the target angle but the solution was time consuming because the penalized ell-1 norm recovery algorithm had to be run multiple times until the best offset and best sparse solution was found and the procedure was not obviously extendable to multiple targets. This work led us to the concepts of orthogonal matching pursuit and removing one target (or sinusoid) at a time. But we still needed a reliable method to find arbitrary frequencies or angles not on a grid. The next realization was that nonlinear least squares could be substituted for the linear least squares used in most versions of OMP. This has proved to be an extremely reliable method and we have now performed 10's of thousands of calculations with this method. Since OMP is not restricted to finding sinusoids, it is natural to ask if OMP with NLS embedded in it works for other functions as well. We have not tried to prove this generally, but we have performed successful calculations using OMP-NLS with signals composed of multi-dimensional sinusoids such as would be obtained with 2D phased arrays (see also Li et al., 2001), signals composed of multiple sinusoids multiplied by

Z

This work was supported under The Aerospace Corporation's Independent Research and

Baraniuk, R. G.; (2007). Compressive sensing, *IEEE Signal Processing Mag.*, Vol.24, No.4,

Candes, E. J.; & Tao, T., (2006). Near optimal signal recovery from random projections: Universal encoding strategies? *IEEE Trans. Inform. Theo*ry, Vol.52, pp. 5406-5425. Candes, E. J.; & Wakin, M. B., (2008). An introduction to compressive sampling, *IEEE Signal* 

Candes, E. J.; Eldar, Y. C., Needell, D., & Randall, P., (2011). Compressed sensing with

Chan, K. W.; & So, H. C., (2004). Accurate frequency estimation for real harmonic sinusoids,

Christensen, M. G.; & Jensen, S. H., (2006). On perceptual distortion minimization and

coherent and redundant dictionaries, submitted to *Applied and Computational* 

nonlinear least-squares frequency estimation, *IEEE Trans. Audio, Speech, Language* 

*kt+bkt2 )* and multiple Gaussian pulses within

**8. Conclusion** 

chirps (i.e. sums of terms of the form *akexp(i*

the same time window.

**9. Acknowledgment** 

Development Program.

pp.118-120, 124.

*Harmonic Analysis*.

*Processing Magazine*, Vol.21, pp. 21-30.

*Processing*, Vol.14, No.1, pp. 99-109.

*IEEE Signal Processing Lett.*, Vol.11, No.7, pp. 609-612.

**10. References** 


**Part 3** 

**DSP Filters** 


**Part 3** 

**DSP Filters** 

190 Applications of Digital Signal Processing

Tropp, J. A.; & Gilbert, A. C., (2007). Signal recovery from random measurements via

Vetterli, M.; Marziliano, P., & Blu, T., (2002). Sampling signals with finite rate of innovation,

*IEEE Trans. Signal Processing*, Vol.50, No.6, pp. 1417-1428.

4655-4666.

orthogonal matching pursuit, *IEEE Trans. Information Theory*, Vol.53, No.12, pp.

**0**

**10**

*Japan*

**Min-Max Design of FIR Digital Filters by**

*Robustness* is a fundamental issue in signal processing; unmodeled dynamics and unexpected noise in systems and signals are inevitable in designing systems and signals. Against such uncertainties, *min-max optimization*, or *worst case optimization* is a powerful tool. In this light, we propose an efficient design method of FIR (finite impulse response) digital filters for approximating and inverting given digital filters. The design is formulated by *min-max optimization* in the frequency domain. More precisely, we design an FIR filter which minimizes

This design has a direct relation with *H*<sup>∞</sup> *optimization* (Francis, 1987). Since the space *H*<sup>∞</sup> is not a Hilbert space, the familiar projection method in conventional signal processing cannot be applied. However, many studies have been made on the *H*<sup>∞</sup> optimization, and nowadays the optimal solution to the *H*<sup>∞</sup> problem is deeply analysed and can be easily obtained by numerical computation. Moreover, as an extension of *H*<sup>∞</sup> optimization, a min-max optimization on a *finite* frequency interval has been proposed recently (Iwasaki & Hara, 2005). In both optimization, the *Kalman-Yakubovich-Popov (KYP) lemma* (Anderson, 1967; Rantzer, 1996; Tuqan & Vaidyanathan, 1998) and the *generalized KYP lemma* (Iwasaki & Hara, 2005) give an easy and fast way of numerical computation; *semidefinite programming* (Boyd & Vandenberghe, 2004). Semidefinite programming can be efficiently solved by

In this chapter, we consider two fundamental problems of signal processing: FIR approximation of IIR (infinite impulse response) filters and inverse FIR filtering of FIR/IIR filters. Each problems are formulated in two types of optimization: *H*<sup>∞</sup> optimization and finite-frequency min-max one. These problems are reduced to semidefinite programming in a similar way. For this, we introduce state-space representation. Semidefinite programming is obtained by the generalized KYP lemma. We will give MATLAB codes for the proposed

In this chapter, we frequently use notations in control systems. For readers who are not familiar to these, we here recall basic notations and facts of control systems used throughout

the chapter. We also show MATLAB codes for better understanding.

the maximum gain of the frequency response of an error system.

**1. Introduction**

numerical optimization softwares.

design, and will show design examples.

**2. Preliminaries**

**Semidefinite Programming**

Masaaki Nagahara *Kyoto University*
