## 2 Modeling the Flat Fading Channel  
In the realms wireless communication systems, the fundamental performance measure is the average signal-to-noise ratio (SNR), which is the statistical average over the probability distribution of the fading. In simple mathematical terms, if \(\gamma\) denotes the instantaneous SNR, which is a random variable (RV) at the receiver output because of the effect of fading, then \(\overline{\gamma}\) is the average SNR, where \(f_{\gamma}\left(\gamma\right)\) denotes the probability density function (PDF) of \(\gamma\)[13].  
To begin with, let us first define the Moment Generating Function (MGF) associated with \(\gamma\) as  
\[M_{\gamma}\left(s\right)=\int_{0}^{\infty}f_{\gamma}\left(\gamma\right)\,e^{s \,\gamma}\,\mathrm{d}\gamma. \tag{1}\]  
Let us evaluate the average bit error rate (BER) of this system using a low complex but accurate method. Note that the BER depends on random fading, and it is a nonlinear function of the SNR.  
The nature of the nonlinearity is a function of the modulation/detection scheme employed [13]. For example, in the multichannel case, the average of the conditional BER over the fading statistics is not a simple average of the per channel performance measure as in average SNR.  
The average BER in the case of perfect phase matching between the transmitter and the receiver can be written as  
\[P_{b}\left(E\right)=\int_{0}^{\infty}f_{\gamma}\left(\gamma\right)\,P_{b}\left( E\setminus\gamma\right)\,\mathrm{d}\gamma, \tag{2}\]  
where \(P_{b}\left(E\setminus\gamma\right)\) is the conditional probability and may have different solutions depending on the channel fading type and the modulation method. When fading affects a narrow band system (flat), the received carrier amplitude is modulated by the fading gain \(\alpha\), where \(\alpha\) is an RV with a mean-square value \(\Omega=\overline{\alpha^{2}}\). Equivalently, the received instantaneous signal _power_ is modulated by \(\alpha^{2}\). Note that \(\alpha\) has a PDF \(f_{\alpha}\left(\alpha\right)\) that depends on the nature of the radio propagation environment.  
The signal is also affected by additive white Gaussian noise (AWGN), which is typically statistically independent of the fading amplitude. The noise is characterized by single-sided power spectral density of \(N_{0}\) (W/Hz). Therefore, we define the instantaneous SNR per symbol by \(\gamma=\frac{\alpha^{2}E_{s}}{N_{0}}\), and the average SNR per symbol by \(\overline{\gamma}=\Omega E_{s}/N_{0}\), where \(E_{s}\) is the energy per symbol [12].  
Depending on the nature of the radio propagation environment, there are different models describing the statistical behavior of the multipath fading envelope. The Rayleigh distribution is frequently used to model multipath fading with no direct line-of-sight (LOS) path. In this case, the channel fading amplitude is distributed according to Rayleigh distribution function and it is given by  
\[f_{\alpha}\left(\alpha\right)=\frac{\alpha}{\Omega}\,e^{\frac{\alpha^{2}}{2 \Omega}}. \tag{3}\]  
Likewise, the instantaneous SNR per symbol \(\gamma\) of the channel is distributed according to an exponential distribution given by [12].  
\[f_{\gamma}\left(\gamma\right)=\frac{1}{\overline{\gamma}}\,e^{\frac{\gamma}{ \overline{\gamma}}}. \tag{4}\]  
According to Equation (1), the MGF corresponding to this fading model is given by  
\[M_{\gamma}\left(s\right)=\left[1-\overline{\gamma}\,s\right]^{-1}. \tag{5}\]