An efficient finite positivity test algorithm for statistical signal processing applications

A real finite sequence r[0,M] is positive if and only if its Fourier transform is positive over the entire frequency range. Testing positivity of r[0,M] using its Fourier transform or its autocorrelation matrix requires an infinite or very lengthy algorithm. As an alternative, an efficient finite algorithm to test positivity, non-negativity or negativity of the real finite sequence r[0,M] is presented. The need to test the positivity of a real sequence arises in many practical situations. For instance, though theoretically non-negative, practical considerations (or missing data) result in a non-positive spectral estimate, e.g. unbiased autocorrelation lag estimates (J.A. Proakis and D.G. Monalakis, 1996). The algorithm described in this paper is a finite algorithm based on Sturm´s theorem from the classical theory of equations. Performance analysis of the algorithm is presented together with simulation results for different positive and non-positive test cases