 
Summary: Page 1 of 21
On the Utility of Laguerre Series for the Envelope PDF in Multipath Fading Channels
Ali Abdi
Abstract  It is well known that multipath fading significantly affects the performance of communication
systems. In order to incorporate the impact of this random phenomenon on system analysis and design, in many
cases we need to calculate the probability density function (PDF) of the received signal envelope in multipath fading
channels. In this paper, we consider a general multipath fading channel with arbitrary number of paths, where the
amplitudes of multipath components are arbitrary correlated positive random variables, independent of phases,
whereas the phases are independent and identically distributed random variables with uniform distributions. Since
the integral form of the envelope PDF for such a general channel model is too complicated to be used for analytic
calculations, we propose two infinite expansions for the PDF, a Laguerre series and a power series. Based on the
tight uniform upper bounds on the truncation error of these two infinite series, we show that the Laguerre series is
superior to the power series due to the fact that for a fixed number of terms, it yields a smaller truncation error.
This Laguerre series with a finite number of terms, which expresses the envelope PDF just in terms of simple
polynomialexponential kernels, is particularly useful for mathematical performance prediction of communication
systems in those indoor and outdoor multipath propagation environments, where the number of strong multipath
components is small.
Index TermsFading channels, Wireless channels, Multipath propagation, Rayleigh fading, Radar clutter, Light
scattering, Envelope distribution, Random vectors, Sum of sinusoids, Infinite expansions, Laguerre polynomials,
Truncation error, Performance analysis, Bit error rate.
