A Mixture Gamma Distribution to Model the SNR of Wireless Channels

Composite fading (i.e., multipath fading and shadowing together) has increasingly been analyzed by means of the K channel and related models. Nevertheless, these models do have computational and analytical difficulties. Motivated by this context, we propose a mixture gamma (MG) distribution for the signal-to-noise ratio (SNR) of wireless channels. Not only is it a more accurate model for composite fading, but is also a versatile approximation for any fading SNR. As this distribution consists of N (≥ 1) component gamma distributions, we show how its parameters can be determined by using probability density function (PDF) or moment generating function (MGF) matching. We demonstrate the accuracy of the MG model by computing the mean square error (MSE) or the Kullback-Leibler (KL) divergence or by comparing the moments. With this model, performance metrics such as the average channel capacity, the outage probability, the symbol error rate (SER), and the detection capability of an energy detector are readily derived.

I. Introduction

Fundamental wireless propagation effects include macroscopic (large scale or slow) fading and microscopic (small scale or fast) fading. Macroscopic fading results from the shadowing effect by buildings, foliage and other objects. Microscopic fading results from multipath, which occurs in indoor environments, and also both macrocellular and microcellular outdoor environments [2]. Shadowing can significantly impact satellite channels, point-to-point long distance microwave links and macrocellular outdoor environments [3]. Both microscopic and macroscopic fading together are modeled by composite shadowing/fading distributions, of which Rayleigh-lognormal (RL) and Nakagami-lognormal (NL) are the two most common models [2]. But the probability density function (PDF) of these two composite models are not in closed form, making performance analysis of some applications difficult or intractable. Hence, several other composite models have been developed including the Suzuki distribution, the and generalized- distributions, the distribution, and the Gamma distribution [4]–[8]. Note that these models are approximations of the RL and NL models.