Linear matrix inequality formulation of spectral mask constraints with applications to FIR filter design

Timothy N. Davidson, Zhi Quan Luo, Jos F. Sturm

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119 Scopus citations


The design of a finite impulse response (FIR) filter often involves a spectral "mask" that 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 that 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.

Original languageEnglish (US)
Pages (from-to)2702-2715
Number of pages14
JournalIEEE Transactions on Signal Processing
Issue number11
StatePublished - Nov 2002

Bibliographical note

Funding Information:
Manuscript received October 17, 2000; revised June 18, 2002. This work was supported in part by the Natural Sciences and Engineering Research Council of Canada under Grant OPG0090391. The second author was also supported by the Canada Research Chair program. The third author was also supported by Travel Grant R46-415 of the Netherlands Organization for Scientific Research (NWO). The associate editor coordinating the review of this paper and approving it for publication was Prof. Arnab K. Shaw.


  • Beamforming
  • FIR digital filter design
  • Optimization
  • Spectral masks


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