A Nakagami-N-gamma Model for Shadowed Fading Channels

Abstract

Wireless channels are subject to short term fading and shadowing. Such shadowed fading channels are described using a Nakagami-lognormal process, with the Nakagami-m (short term fading) and lognormal distributions (shadowing). This approach does not result in a closed form solution for the density function of the signal-to-noise ratio (SNR) making wireless systems analysis difficult. It was suggested that a gamma or an inverse Gaussian distribution can be used in place of the lognormal distribution providing an analytical framework. The match of these two distributions to the lognormal was less than ideal. Invoking shadowing as multiplicative process, the distribution of the product of N gamma variables is proposed in place of the lognormal pdf resulting in the Nakagami-N-gamma model. It is shown that this model leads to simple solutions to the density and distribution functions as well as error rates for coherent phase shift keying modems. The outage probabilities and error rates based on the Nakagami-lognormal (NL) and Nakagami-N-gamma (NNG) models were compared. Results showed excellent match at levels of shadowing generally observed in wireless systems. While values of N as low as 3 was sufficient for low values of m and weak to moderate shadowing, values of N in the range of 7---9 provided better match for higher levels of shadowing and higher values of m. By varying N, it is also possible to get the NNG pdf to move closer to the NL pdf making the new model an ideal one for the shadowed fading channels with its flexibility and availability of analytical expressions.