Linear matrix inequality formulation of spectral mask constraints

The design of a finite impulse response filter often involves a spectral 'mask' which the magnitude spectrum must satisfy. This constraint can be awkward because it yields an infinite number of inequality constraints (two for each frequency point). In current practice, spectral masks are often approximated by discretization, but in this paper we will show that piecewise constant masks can be precisely enforced in a finite and convex manner via linear matrix inequalities. This facilitates the formulation of a diverse class of filter and beamformer design problems as semidefinite programmes. These optimization problems can be efficiently solved using recently developed interior point methods. Our results can be considered as extensions to the well-known Positive-Real and Bounded-Real Lemmas from the systems and control literature.