**1. Introduction**

18 Digital Signal Processing

210 Applications of Digital Signal Processing

In this chapter, we consider four problems, FIR approximation and inverse FIR filtering of FIR/IIR filters by *H*<sup>∞</sup> and finite-frequency min-max, which are fundamental in signal processing. By using the KYP and generalized KYP lemmas, the problems are all solvable via semidefinite programming. We show MATLAB codes for the programming, and show

Anderson, B. D. O. (1967). A system theory criterion for positive real matrices, *Siam Journal on*

Iwasaki, T. & Hara, S. (2005). Generalized KYP lemma: unified frequency domain inequalities

Löfberg, J. (2004). Yalmip : A toolbox for modeling and optimization in MATLAB, *Proc. IEEE International Symposium on Computer Aided Control Systems Design* pp. 284–289.

Nagahara, M. & Yamamoto, Y. (2009). Optimal noise shaping in ∆Σ modulators via

Nagahara, M. & Yamamoto, Y. (2011). *H*<sup>∞</sup> optimal approximation for causal spline

Oppenheim, A. V. & Schafer, R. W. (2009). *Discrete-Time Signal Processing*, 3rd edn, Prentice

Rantzer, A. (1996). On the Kalman–Yakubovich–Popov lemma, *Systems & Control Letters*

Sturm, J. F. (2001). *Using SeDuMi 1.02, a MATLAB toolbox for optimization over symmetric cones*.

Tuqan, J. & Vaidyanathan, P. P. (1998). The role of the discrete-time

Vidyasagar, M. (1988). A state-space interpretation of simultaneous stabilization, *IEEE Trans.*

Yamamoto, Y., Anderson, B. D. O., Nagahara, M. & Koyanagi, Y. (2003). Optimizing FIR approximation for discrete-time IIR filters, *IEEE Signal Process. Lett.* 10(9).

Kalman-Yakubovitch-Popov lemma in designing statistically optimum FIR

Boyd, S. & Vandenberghe, L. (2004). *Convex Optimization*, Cambridge University Press.

with design applications, *IEEE Trans. Autom. Control* 50: 41–59.

generalized KYP lemma, *Proc. of IEEE ICASSP* III: 3381–3384.

**9. Conclusion**

**10. References**

Hall.

28(1): 7–10.

examples of designing FIR filters.

*Control and Optimization* 5: 171–182.

Francis, B. A. (1987). *A Course in H*∞ *Control Theory*, Springer.

URL: *http://users.isy.liu.se/johanl/yalmip/* Mathworks (2010). *Control System Toolbox Users Guide*.

URL: *http://www.mathworks.com/products/control/*

interpolation, *Signal Processing* 91(2): 176–184.

orthonormal filter banks, *Proc. of ISCAS* 5: 122–125.

Rugh, W. J. (1996). *Linear Systems Theory*, Prentice Hall.

URL: *http://sedumi.ie.lehigh.edu/*

*Autom. Control* 33(5): 506–508.

A medical Doppler ultrasound system has a spectrum display that indicates the blood flow direction, whether the blood flows forward or away from a probe. It also has Doppler audio outputs. In particular, the latter is a special process peculiar to the Doppler ultrasound system and separates the blood flow direction and outputs from the left and right speakers. Owing to this function, the existence of a blood flow is quickly detectable. When changing conventional analog signal-processing into digital signal-processing, we researched many processing systems of Doppler audio. First, target performances, such as a response time and direction separation, were set up, and six kinds of digital signal-processing systems were examined. Further, we investigated some new anti-aliasing processing systems unique to Doppler ultrasound system. We compared three kinds of anti-aliasing processing systems. Consequently, we clarified that a complex IIR (infinite impulse response) filter system has an excellent response and a low calculation load.
