# On the Second Order Statistics of the Multihop Rayleigh Fading Channel

## TL;DR

This paper develops new analytical expressions for second order statistics, specifically level crossing rate and average fade duration, in multihop Rayleigh fading channels, aiding in the design of wireless systems.

## Contribution

It introduces a novel analytical framework for evaluating second order statistics of multihop Rayleigh channels, including closed-form approximations and validation.

## Key findings

- Derived new expressions for LCR and AFD of N*Rayleigh channels.
- Validated accuracy of formulas through simulations.
- Provided efficient approximations using multivariate Laplace theorem.

## Abstract

Second order statistics provides a dynamic representation of a fading channel and plays an important role in the evaluation and design of the wireless communication systems. In this paper, we present a novel analytical framework for the evaluation of important second order statistical parameters, as the level crossing rate (LCR) and the average fade duration (AFD) of the amplify-and-forward multihop Rayleigh fading channel. More specifically, motivated by the fact that this channel is a cascaded one and can be modeled as the product of N fading amplitudes, we derive novel analytical expressions for the average LCR and the AFD of the product of N Rayleigh fading envelopes (or of the recently so-called N*Rayleigh channel). Furthermore, we derive simple and efficient closed-form approximations to the aforementioned parameters, using the multivariate Laplace approximation theorem. It is shown that our general results reduce to the corresponding ones of the specific dual-hop case, previously published. Numerical and computer simulation examples verify the accuracy of the presented mathematical analysis and show the tightness of the proposed approximations.