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In this paper a novel theory and algorithm for spectral factorization is presented. It is based on a criterion for minimal extraction of a so-called "elementary factor." Although not all positive para-hermitian matrices can be minimally factored into elementary factors, still the method can be adapted to fit the general case by increasing the degree in a well-controlled way and removing the nonminimal units of degree at the end. The method is, in this sense, strictly minimal. Moreover, the algorithm produces the spectral factor in ali cases where such a factorization does exist. Also, an independent proof of the famous spectral factorization result of Youla is obtained, so that the completeness of the method is ascertained. The procedure results in a workable and optimally minimal algorithm.