Positive Trigonometric Polynomials and Signal Processing Applications by Bogdan Dumitrescu

Author:Bogdan Dumitrescu
Language: eng
Format: epub, pdf
Publisher: Springer International Publishing, Cham

(4.33)

If the inequality

(4.34)

is satisfied, then there exist matrices , , such that the relations (4.16) and

(4.35)

hold, where is the vector of the coefficients of the filter defined by (3.​27).

Reciprocally, relations (4.16) and (4.35) imply .

Proof

The inequality (4.34) holds if and only if , which is the same with , , where is defined in (4.31). If the latter inequality holds, then, by applying Lemma 4.25 with , there exists a Gram set associated with such that and , , where are the Gram matrices from (4.32), associated with . This means that (4.16) holds for , . Moreover, the inequality

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