Trigonometric Polynomials Positive on Frequency Domains and Applications to 2-D FIR Filter Design

We propose a characterization of multivariate trigonometric polynomials that are positive on a given frequency domain. The positive polynomials are parameterized as a linear function of sum-of-squares polynomials and so semidefinite programming (SDP) is applicable. The frequency domain is expressed via the positivity of some trigonometric polynomials. We also give a bounded real lemma (BRL) in which a bounding condition on the magnitude of the frequency response of a multidimensional finite-impulse-response (FIR) filter is expressed as a linear matrix inequality (LMI). This BRL avoids the problem of a lack of spectral factorization in the multidimensional case. All the proposed theoretical contributions can be implemented only as sufficient conditions, due to degree limitations on the sum-of-square polynomials. However, the two-dimensional (2-D) FIR filter designs we study numerically suggest that these limitations have negligible impact on the optimality