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

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1 Author(s)
Dumitrescu, B. ; Tampere Int. Center for Signal Process., Tampere Univ. of Technol.

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

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Signal Processing, IEEE Transactions on  (Volume:54 ,  Issue: 11 )