# On the Performance of Underlay Device-to-Device Communications
## 4 Performance Analysis
### The Amount of Fading (AoF) of D2D Link  
The amount of fading is a fundamental metric to quantify the fading severity of a specific channel and is measured by the ratio of the variance to the square of the mean of the received signal. In particular, the AoF of the \(k\)-th D2D receiver of the \(s\in\{equ,ran,pl\}\) scheme is given as [59]\[\text{AoF}_{k}^{s}= \frac{\text{Var}\{\Xi_{k}^{s}\}}{\left(\mathbb{E}\{\Xi_{k}^{s}\} \right)^{2}}\] \[\text{Var}\{\Xi_{k}^{s}\}= \mathbb{E}\Big{\{}(\Xi_{k}^{s})^{2}\Big{\}}-\left(\mathbb{E}\{\Xi_ {k}^{s}\}\right)^{2}\] \[\mathbb{E}\Big{\{}(\Xi_{k}^{s})^{2}\Big{\}}= \Bigg{(}A_{k,c}^{s}\Bigg{(}\prod_{i=1,i\
eq k}^{N}B_{k,i}^{s} \Bigg{)}\Bigg{)}\Bigg{(}C_{k,c}^{s}H_{3}\Bigg{(}\frac{\sigma_{k}^{2}}{P_{D_{k} }^{s}\Omega_{d_{\text{r}},d_{k}}},A_{k,c}^{s}\Bigg{)}\] \[+\sum_{i=1,j\
eq k}^{N}D_{k,i}^{s}H_{3}\Bigg{(}\frac{\sigma_{k}^ {2}}{P_{D_{k}}^{s}\Omega_{d_{\text{r}},d_{k}}},B_{k,i}^{s}\Bigg{)}\Bigg{)} \tag{17}\] \[\mathbb{E}\{\Xi_{k}\}= \Bigg{(}A_{k,c}^{s}\Bigg{(}\prod_{i=1,i\
eq k}^{N}B_{k,i}^{s} \Bigg{)}\Bigg{)}\Bigg{(}C_{k,c}^{s}H_{2}\Bigg{(}\frac{\sigma_{k}^{2}}{P_{D_{k} }^{s}\Omega_{d_{\text{r}},d_{k}}},A_{k,c}^{s}\Bigg{)}\] \[+\sum_{i=1,j\
eq k}^{N}D_{k,i}^{s}H_{2}\Bigg{(}\frac{\sigma_{k}^ {2}}{P_{D_{k}}^{s}\Omega_{d_{\text{r}},d_{k}}},B_{k,i}^{s}\Bigg{)}\Bigg{)},\]  
where \(\text{Var}\{X\}\) is the variance of RV \(X\), \(H_{3}(a,b)\) is defined as  
\[H_{3}(a,b)= \int\limits_{0}^{\infty}\frac{2x}{x+b}\exp(-ax)dx=2\Bigg{[}\int \limits_{0}^{\infty}\left(1-\frac{b}{x+b}\right)\exp(-ax)dx\Bigg{]}\] \[= 2\bigg{[}\frac{1}{a}-H_{2}(a,b)\bigg{]}. \tag{18}\]  
Proof. : To prove (17), let us first compute the expectation of the e2e SINR as follows  
\[\mathbb{E}\{\Xi_{k}^{s}\}= \int\limits_{0}^{\infty}\text{Pr}\{X>x\}dx=\int\limits_{0}^{ \infty}\overline{F}_{X}(x)dx\] \[\overset{(a)}{=} \Bigg{(}A_{k,c}^{s}\Bigg{(}\prod_{i=1,i\
eq k}^{N}B_{k,i}^{s} \Bigg{)}\Bigg{)}\int\limits_{0}^{\infty}\exp\left(-\frac{x\sigma_{k}^{2}}{P_{ D_{k}}^{s}\Omega_{d_{\text{r}},d_{k}}}\right)\] \[\text{Var}\{\Xi_{k}^{s}\}= \mathbb{E}\Big{\{}(\Xi_{k}^{s})^{2}\Big{\}}-\left(\mathbb{E}\{\Xi_ {k}^{s}\}\right)^{2}\]0 \[\text{Var}\{\Xi_{k}^{s}\}= \mathbb{E}\Big{\{}(\Xi_{k}^{s})^{2}\Big{\}}-\left(\mathbb{E}\{\Xi_ {k}^{s}\}\right)^{2}\]1 \[\text{Var}\{\Xi_{k}^{s}\}= \mathbb{E}\Big{\{}(\Xi_{k}^{s})^{2}\Big{\}}-\left(\mathbb{E}\{\Xi_ {k}^{s}\}\right)^{2}\]2  
where \(H_{2}(a,b)\) is provided (15). Next, we are going to compute the \(\mathbb{E}\{(\Xi_{k}^{s})^{2}\}\) as follows\[\text{Var}\{\Xi_{k}^{s}\}= \mathbb{E}\Big{\{}(\Xi_{k}^{s})^{2}\Big{\}}-\left(\mathbb{E}\{\Xi_ {k}^{s}\}\right)^{2}\]3 \[\text{Var}\{\Xi_{k}^{s}\}= \mathbb{E}\Big{\{}(\Xi_{k}^{s})^{2}\Big{\}}-\left(\mathbb{E}\{\Xi_ {k}^{s}\}\right)^{2}\]4 \[\text{Var}\{\Xi_{k}^{s}\}= \mathbb{E}\Big{\{}(\Xi_{k}^{s})^{2}\Big{\}}-\left(\mathbb{E}\{\Xi_ {k}^{s}\}\right)^{2}\]5 \[\text{Var}\{\Xi_{k}^{s}\}= \mathbb{E}\Big{\{}(\Xi_{k}^{s})^{2}\Big{\}}-\left(\mathbb{E}\{\Xi_ {k}^{s}\}\right)^{2}\]6 \[\text{Var}\{\Xi_{k}^{s}\}= \mathbb{E}\Big{\{}(\Xi_{k}^{s})^{2}\Big{\}}-\left(\mathbb{E}\{\Xi_ {k}^{s}\}\right)^{2}\]7  
Combining (19) and (20), we obtain (17). Q.E.D.