Consider L independent and identically distributed exponential random variables (r.vs) X1,X2,...,XL, and positive scalars b1,b2,...,bL. In this letter, we present the probability density function (pdf), cumulative distribution function and the Laplace transform of the pdf of the composite r.v Z = (X1 + X2 + ... + XL)2 /(b1*X1 + b2*X2 + ... + bL*XL). We show that the r.v Z appears in various communication systems such as 1) maximal ratio combining of signals received over multiple channels with mismatched noise variances, 2) M-ary phase-shift keying with spatial diversity and imperfect channel estimation, and 3) coded multi-carrier code-division multiple access reception affected by an unknown narrow-band interference, and the statistics of the r.v Z derived here enable us to carry out the performance analysis of such systems in closed-form.
IEEE Transactions on CommunicationsThis work may not be copied or reproduced in whole or in part for any commercial purpose. Permission to copy in whole or in part without payment of fee is granted for nonprofit educational and research purposes provided that all such whole or partial copies include the following: a notice that such copying is by permission of Mitsubishi Electric Research Laboratories, Inc.; an acknowledgment of the authors and individual contributions to the work; and all applicable portions of the copyright notice. Copying, reproduction, or republishing for any other purpose shall require a license with payment of fee to Mitsubishi Electric Research Laboratories, Inc. All rights reserved. Abstract-Consider L independent and identically distributed exponential random variables (r.vs) X1, X2, . . . , XL and positive scalars b1, b2, . . . , bL. In this letter, we present the probability density function (pdf), cumulative distribution function and the Laplace transform of the pdf of the composite r.We show that the r.v Z appears in various communication systems such as i) maximal ratio combining of signals received over multiple channels with mismatched noise variances, ii) M -ary phase-shift keying with spatial diversity and imperfect channel estimation, and iii) coded multi-carrier code-division multiple access reception affected by an unknown narrow-band interference, and the statistics of the r.v Z derived here enable us to carry out the performance analysis of such systems in closed-form.Index Terms-Exponential random variables, distribution of ratio of two random variables, bivariate Laplace transform, mismatched statistics, partial-band interference.