Positive trigonometric polynomials and signal processing by Bogdan Dumitrescu

By Bogdan Dumitrescu

Positive and sum-of-squares polynomials have got a different curiosity within the most up-to-date decade, because of their connections with semidefinite programming. hence, effective optimization tools could be hired to resolve assorted difficulties regarding polynomials. This publication gathers the most fresh effects on optimistic trigonometric polynomials inside of a unitary framework; the theoretical effects are acquired in part from the overall conception of actual polynomials, in part from self-sustained advancements. The optimization purposes hide a box diverse from that of actual polynomials, normally in sign processing difficulties: layout of 1-D and 2-D FIR or IIR filters, layout of orthogonal filterbanks and wavelets, balance of multidimensional discrete-time systems.

Positive Trigonometric Polynomials and sign Processing purposes  has elements: idea and functions. the idea of sum-of-squares trigonometric polynomials is gifted unitarily in accordance with the concept that of Gram matrix (extended to Gram pair or Gram set). The presentation begins via giving the most effects for univariate polynomials, that are later prolonged and generalized for multivariate polynomials. The purposes half is prepared as a set of similar difficulties that use systematically the theoretical effects. the entire difficulties are dropped at a semidefinite programming shape, able to be solved with algorithms freely on hand, like these from the library SeDuMi.

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2y0 Since the above quadratic form is nonnegative for any h ∈ Rn+1 , it follows that the matrix Toep(2y0 , y1 , . . , yn ) is positive semidefinite. Knowing the form of the dual cone, we can build easier the duals of optimization problems with nonnegative trigonometric polynomials. 22), where, for simplicity we take Γ = I. 33) is g(y) = inf f (r) − y T r , r where the Lagrangean multiplier y belongs to the dual cone. The minimum is obtained trivially for y = 2(r − rˆ ) and so 1 g(y) = − y T y − y T rˆ .

2  .. ..  i=k k=0 yn yn−1 . . 2y0 Since the above quadratic form is nonnegative for any h ∈ Rn+1 , it follows that the matrix Toep(2y0 , y1 , . . , yn ) is positive semidefinite. Knowing the form of the dual cone, we can build easier the duals of optimization problems with nonnegative trigonometric polynomials. 22), where, for simplicity we take Γ = I. 33) is g(y) = inf f (r) − y T r , r where the Lagrangean multiplier y belongs to the dual cone. The minimum is obtained trivially for y = 2(r − rˆ ) and so 1 g(y) = − y T y − y T rˆ .

Q 0, y ≤ α This is a standard SQLP problem in equality form. 13 (Complexity issues) As discussed in Appendix A, the complexity of an SDP problem in equality form is O(n2 m2 ), where n × n is the size of the variable positive semidefinite matrix and m is the number of equality constraints. 23), respectively) do not change significantly the complexity. Since the size of the Gram matrix Q is (n+1)×(n+1) and the number of equality constraints is n+1, we can appreciate that the complexity of the three problems—Most positive Gram matrix, Min poly value and Nearest autocorrelation—formulated in SDP form in this section is O(n4 ).

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