Timothy N. Davidson, Zhi-Quan (Tom) Luo and Jos F. Sturm.
Linear matrix inequality formulation of spectral mask constraints
with applications to FIR filter design.
IEEE Transactions on Signal Processing,
50(11):2702-2715, November 2002.
The design of a finite impulse response (FIR) filter often involves a spectral `mask' which the magnitude spectrum must satisfy. The mask specifies upper and lower bounds at each frequency, and hence yields an infinite number of constraints. In current practice, spectral masks are often approximated by discretization, but in this paper we will derive a result which allows us to precisely enforce piecewise constant and piecewise trigonometric polynomial masks in a finite and convex manner via linear matrix inequalities. While this result is theoretically satisfying in that it allows us to avoid the heuristic approximations involved in discretization techniques, it is also of practical interest because it generates competitive design algorithms (based on interior point methods) for a diverse class of FIR filtering and narrowband beamforming problems. The examples we provide include the design of standard linear and nonlinear phase FIR filters, robust `chip' waveforms for wireless communications, and narrowband beamformers for linear antenna arrays. Our main result also provides a contribution to system theory, as it is an extension of the well-known Positive-Real and Bounded-Real Lemmas.
The paper, as published, is available as a pdf file
Since the paper contains many equations that are quite long, it can be a little difficult to read the standard IEEE two-column format provided above. Therefore, I have also made the paper available in a single column format, as a pdf file.
A summary of this paper appears in Proceedings of the 2001 International Conference on Acoustics, Speech, and Signal Processing, Salt Lake City, UT, USA, May 2001.
The formulation described in this paper has been used to design prototype filters for a certain subband adaptive filtering system. A description of that application, and a Matlab m-file which implements that instance of the formulation are available here.
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