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

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 beam-forming problems. The examples we provide include the design of standard linear and nonlinear phase FIR filters, robust "chip" waveforms for wireless communications, and narrowband beam-formers 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.