Statistical Properties of a Sum of Sinusoids and Gaussian Noise and its Generalization to Higher Dimensions

Abstract

This paper investigates the statistical properties of the sum, S, of an n-dimensional Gaussian random vector, N, plus the sum of M vectors, X <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</inf> , …, X <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">M</inf> , having random amplitudes and independent arbitrary orientations in n-dimensional space. We derive expressions for the probability density function (p.d.f.) and distribution function (d.f.) of S and of its length, |S|, as series expansions involving only the moments of |X <inf xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">i</inf> |, i = 1, …, M. In addition, we find the p.d.f. and d.f. of the projection of S onto 1-dimensional space. Our results are generalizations of the n = 2-dimensional problem of finding the statistical properties of a sum of constant-amplitude sinusoids having independent uniformly distributed phase angles plus Gaussian noise. The latter problem has been treated by Rice <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">1</sup> and Esposito and Wilson, <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> but our results can also deal with sinusoids having random amplitudes. When n = 3, our findings treat, in the presence of a Gaussian vector, the classical problem of “random flights” dating back to Rayleigh. Some calculations for the 2- and 3-dimensional problem are presented, and an application to coherent phase-shift-keying communications systems is discussed.