Statistical Properties of a Sum of Sinusoids and Gaussian Noise and its Generalization to Higher Dimensions
01 April 1974
Copyright © 1974, American Telephone and Telegraph Company. Printed in U.S.A. Statistical Properties of a Sum of Sinusoids and Gaussian Noise and its Generalization to Higher Dimensions By JOEL G O L D M A N ( M a n u s c r i p t received September 6, 1 9 7 3 ) This paper investigates the statistical properties of the sum, S, of an n-dimensional Gaussian random vector, N, plus the sum of M vectors, Xi, ···, Xm, 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 |Xi|, 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 Rice1 and Esposito and Wilson,2 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 phaseshift-keying communications systems is discussed.