Author: Ali Abdi

Supervisor: Said Nader-Esfahani Advisor: Homayoun Hashemi

Granting Institution: University of Tehran, Dept. of Elec. and Comp. Eng.

Acceptance Date: February 1996

Contact Information:

Dept of Elec and Comp Eng
University of Minnesota
4-174 EE/CSci Bldg
200 Union St SE
Minneapolis, MN 55455
Fax: (612) 625 4583
Tel: (612) 625 7542 (Lab)
(612) 624 4887 (Office)

Index Terms: Random vectors, sum of sinusoids, envelope distribution, fading, clutter, multipath propagation, random interference, infinite expansions, Laguerre polynomials, truncation error.


There are various applications in physics and engineering sciences (specially communications) where one encounters the sum of several random sinusoidal signals; and it is desired to obtain the probability density function (PDF) of the resulting signal envelope. Multipath fading in communication channels, clutter and target cross section in radars, interference in communication systems, wave propagation in random media, light scattering, etc. are phenomena that can be described using the sum of several random sinusoidal signals. According to the correspondence between a random sinusoidal signal and a random vector, the sum of random vectors can be considered as an abstract mathematical model for the above phenomena. It is desired to obtain the PDF of the length of the resulting vector, under various conditions and assumptions.

Assuming uniform distributions for the angles of vectors, many researchers have obtained the PDF of the length of the resulting vector. However, by considering other simplifying assumptions, they have only studied special cases. In this thesis and for the first time, this PDF is obtained for the most general case; i.e. the case in which the lengths of vectors are dependent random variables. This PDF is in the form of a definite integral, which may be inappropriate for analytic manipulations and numerical computations. Thus, appropriate infinite expansions are presented for this PDF; and their numerical properties such as truncation error, convergence rate, etc. are investigated thoroughly.

Nonuniform distribution of the angles of vectors is an important case which appears in a large number of applications. However only a limited number of researchers have studied it. In this thesis, it is shown that under this condition, the PDF of the length of the resulting vector can also be interpreted as a definite integral. In addition, a general model is proposed for multipath propagation environments having nonuniform phases; and based on the above result, it is shown that the signal envelope PDF in such environments becomes the well-known Rice PDF. Thus in this thesis and for the first time, a simple and useful PDF is introduced for the signal envelope in multipath propagation environments having nonuniform phases.

Finally, a large number of suggestions are proposed that refer to the other aspects of the sum of random vectors, and are useful for future investigations.