RIS-aided Cooperative FD-SWIPT-NOMA Performance Over Nakagami-m Channels

Abstract

In this work, we investigate Reconfigurable Intelligent Surface (RIS)-aided Full-Duplex (FD)-Simultaneous Wireless Information Power Transfer (SWIPT)-Cooperative non-Orthogonal Multiple Access (C-NOMA) consisting of two paired devices. The device with better channel conditions ($D_1$) is designated to act as a FD relay to assist the device with poor channel conditions ($D_2$). We assume that $D_1$ does not use its own battery energy to cooperate but harvests energy by utilizing SWIPT. A practical non-linear Energy Harvesting (EH) model is considered. We first approximate the harvested power as a Gamma Random Variable (RV) via the Moment Matching (MM) technique. This allows us to derive analytical expressions for Outage Probability (OP) and ergodic rate (ER) that are simple to compute yet accurate for a wide range of system parameters, such as EH coefficients and residual Self-Interference (SI) levels, being extensively validated by numerical simulations. The OP and ER expressions reveal how important it is to mitigate the SI in the FD relay mode since, for reasonable values of residual SI coefficient, its detrimental effect on the system performance, is extremely noticeable. Also, numerical results reveal that increasing the number of RIS elements can benefit the cooperative system much more than the non-cooperative one.

Appendices

Appendix A. Proof of Lemma 1

Our objective is to approximate the power utilized in the relaying step, $P_H$, defined by in Eq. (7), as a Gamma RV using the method of moments. For this reason, let us define $Y=\frac{1}{1+e^{-a(\rho P_{in}-b)}}$, thus the first moment of Y can be lower bounded as

$$\begin{aligned} \mathbb {E}[Y] \ge \frac{1}{1+\mathbb {E}\left[ e^{-a(\rho P_{in}-b)}\right] } = \frac{1}{1 + \frac{e^{ab}}{\left( 1 + \theta _1 a \rho P_t \beta _{ss} \beta _{s1}\right) ^{k_1}}} \end{aligned}$$

(40)

where, utilizing Eq. (8), we obtain

$$\begin{aligned} \mathbb {E}\left[ e^{-a(\rho P_{in}-b)}\right]= & {} e^{ab} \int _{0}^{\infty } e^{-a\rho P_t \beta _{ss} \beta _{s1} x} f_{X_1^2}(x) dx \nonumber \\\overset{(i)}{=} & {} \frac{e^{ab}}{\Gamma (k_1)\theta _1^{k_1}} \int _{0}^{\infty } x^{k1-1} e^{-x\left( \frac{1}{\theta _1}+a \rho P_t \beta _{ss}\beta _{s1} \right) } \nonumber \\= & {} \frac{e^{ab}}{ \left( 1 + \theta _1 a\rho P_t \beta _{ss} \beta _{s1} \right) ^{k_1} } \end{aligned}$$

(41)

where [44, 3.351.3] is utilized in (i). By proceeding with the same steps above, similarly, the second moment of Y can be given by

$$\begin{aligned} \mathbb {E}[Y^2] \ge \frac{1}{1 + \frac{2e^{ab}}{\left( 1+\theta _1 a\rho P_t \beta _{ss} \beta _{s1}\right) ^{k_1}} + \frac{e^{2ab}}{\left( 1+2\theta _1 a\rho P_t \beta _{ss} \beta _{s1}\right) ^{k_1}} }. \end{aligned}$$

(42)

The shape $k_{P_H}$ and scale $\theta _{P_H}$ parameters of $P_H$, which will be approximated as Gamma RV, are given by [40]

$$\begin{aligned} k_{P_H}= & {} \frac{\left( \mathbb {E}[Y]-\frac{1}{1+e^{ab}}\right) ^2}{\mathbb {E}[Y^2]-\mathbb {E}[Y]^2}, \end{aligned}$$

(43)

$$\begin{aligned} \theta _{P_H}= & {} \frac{\mathbb {E}[Y^2]-\mathbb {E}[Y]^2}{\mathbb {E}[Y]-\frac{1}{1+e^{ab}}}. \end{aligned}$$

(44)

By defining, $\chi \triangleq \theta _1 a\rho P_t \beta _{ss} \beta _{s1}$ and plugging (40) and (42) in (43) and (44); (25) (26) are obtained and this completes the proof.

Appendix B. Proof of Theorem 2

After performing some mathematical manipulations, Eq. (31) can be written as follows

$$\begin{aligned} P_\textrm{out}^{D_1} = \textrm{Pr}\left( X_1^2 < \frac{\xi (|h_{SI}|^2P_H+\sigma ^2)}{P_t(1-\rho )\beta _{ss}\beta _{s1}} \right) , \end{aligned}$$

(45)

where $\xi = \max \left( \frac{\gamma _{th}}{\alpha _1},\frac{\gamma _{th2}}{ (\alpha _2 - \alpha _1\gamma _{th2})} \right)$. We observe that when $\rho =0$ or $\omega =0$, it leads to $P_H=0$ or $|h_{SI}|^2=0$, respectively. In both of these particular cases, we can compute the OP of $D_1$ as the following

$$\begin{aligned} P_{\textrm{out}}^{D_1} = \frac{1}{\Gamma (k_1)}\gamma \left( k_1, \frac{\xi \sigma ^2}{\theta _1P_t(1-\rho )\beta _{ss}\beta _{s1}} \right) , \end{aligned}$$

(46)

where we utilized the CDF of $X_1^2$ as defined in (24).

When we have $\rho \ne 0$ and $\omega \ne 0$, (45) should be further analyzed. Firstly, we should notice that according to Lemma 1, $P_H$ is a Gamma RV with shape and scale parameters given by $k_{PH}$ and $\theta _{PH}$ respectively. Thus (45) can be written as

$$\begin{aligned} P_\textrm{out}^{D_1}= & {} 1 - \textrm{Pr}\left( \underbrace{\frac{|h_{ SI}|^2\xi P_H}{P_t(1-\rho )}}_{V} -\underbrace{X_1^2\beta _{ss}\beta _{s1}}_{Z} <- \frac{\xi \sigma ^2}{P_t(1-\rho )}\right) , \end{aligned}$$

(47)

let us define $V\triangleq \frac{|h_{ SI}|^2 \xi P_H}{P_t(1-\rho )}$ and $Z\triangleq X_1^2\beta _{ss}\beta _{s1}$. Since $P_H$ range from low values lower than nano-Watts to $P_{th}$, which assumes values till mili-Watts, its values are lower than $|h_{SI}|^2$, which is parameterized in $\omega$ and undergo an Exponential distribution, due to this, here, we propose to approximate V as an exponential RV, i.e., $V \overset{\textrm{approx}}{\sim } \textrm{Exponential}\left( \frac{\omega \xi \widetilde{P}_H}{P_t(1-\rho )}\right)$. Furthermore, since Z is the product of $X_1$ with a constant, utilizing the scale property of Gamma distribution, $Z \sim \textrm{Gamma}(k_1,\theta _1\beta _{ss}\beta _{s1})$, hence, (47) can be written as

$$\begin{aligned} P_\textrm{out}^{D_1}= & {} 1 - \int _{\frac{\xi \sigma ^2}{P_t(1-\rho )}}^{\infty } F_{V}\left( z-\frac{\xi \sigma ^2}{P_t(1-\rho )}\right) f_Z(z) dz\nonumber \\\overset{(i)}{=} & {} 1 -\int _{\frac{\xi \sigma ^2}{P_t(1-\rho )}}^{\infty } f_Z(z) dz +e^{\frac{\sigma ^2}{\omega \widetilde{P}_H}} \int _{\frac{\xi \sigma ^2}{P_t(1-\rho )}}^{\infty }e^{- \frac{z P_t(1-\rho )}{\omega \xi \widetilde{P}_H}} f_Z(z) \nonumber \\= & {} \frac{\gamma \left( k_1,\frac{\xi \sigma ^2}{\theta _1\beta _{ss}\beta _{s1}P_t(1-\rho )}\right) }{\Gamma (k_1)}\nonumber \\{} & {} + \frac{e^{\frac{\sigma ^2}{\omega \widetilde{P}_H}}}{\Gamma (k_1) } \frac{\Gamma \left( k_1, \frac{\xi \sigma ^2}{P_t(1-\rho )\theta _1\beta _{ss} \beta _{s1}} + \frac{\sigma ^2}{\omega \widetilde{P}_H} \right) }{ \left( \frac{P_t(1-\rho )\theta _1\beta _{ss}\beta _{s1}}{\omega \widetilde{P}_H \xi } + 1 \right) ^{k_1}}, \end{aligned}$$

(48)

where in (i) we utilized the CDF of exponential RV V, given by $F_V(z) = 1 - e^{-\frac{z Pt(1-\rho )}{\omega \xi \widetilde{P}_H}}$, for first integral, and we apply [44, 3.381.9] for solving the second integral.

Appendix C. Proof of Theorem 3

To obtain the OP of $D_2$, it is reasonable to consider two different cases: I) when the $D_1$ does not operate as relay $(\rho = 0)$; II) when $D_1$ operates as a relay $(\rho \ne 0)$.

1.1 I) $\rho = 0$ ($D_1$ Does Not Act as a Relay)

By recalling the established definition in subsection 3.1, we have that $|h_2|^2 = \beta _{ss}\beta _{s2}X_2^2$, after some manipulations, we can written Eq. (33) as

$$\begin{aligned} P_\textrm{out}^{D_2}= & {} \textrm{Pr}\left( X_2^2 < \frac{\sigma ^2 \gamma _{th2} }{\beta _{ss} \beta _{s2} P_t(\alpha _2-\alpha _1\gamma _{th2})} \right) , \end{aligned}$$

(49)

according to subsection 3.1, $X_2$ follows a Rayleigh RV with scale parameter $\sqrt{\frac{N}{2}}$, thus, its squared magnitude follows an exponential distribution with rate parameter N. By using Eq. (24), we obtain the following expression for the OP of $D_2$ when $D_1$ does not operate as a relay

$$\begin{aligned} P_\textrm{out}^{D_2} = 1 - e^{\frac{-\sigma ^2 \gamma _{th2} }{N P_t\beta _{ss}\beta _{s2}(\alpha _2-\alpha _1\gamma _{th2})}}. \end{aligned}$$

(50)

1.2 II) $\underline{\rho \ne 0}$ ($D_1$ Operates as a Relay)

By recalling the established definition in Sect. 3.1, we have that $|h_1|^2 = \beta _{ss}\beta _{s1}X_1^2$, similarly to the subsection I) of this Appendix, after some manipulations, we can write Eq. (33) as

$$\begin{aligned} P_\textrm{out}^{D_2}= & {} \textrm{Pr} \left( X_1^2< \frac{\gamma _{th2}(|h_{SI}|^2 P_H+\sigma ^2)}{\beta _{ss} \beta _{s1}P_t(1-\rho )(\alpha _2 - \alpha _1\gamma _{th2})}, \right. \nonumber \\{} & {} \left. X_2^2< \frac{\sigma ^2 \gamma _{th2} }{\beta _{ss} \beta _{s2}P_t(\alpha _2-\alpha _1\gamma _{th2})} \right) \nonumber \\{} & {} + \textrm{Pr}\left( X_1^2 \ge \frac{\gamma _{th2}(|h_{SI}|^2 P_H+\sigma ^2)}{\beta _{ss} \beta _{s1}P_t(1-\rho )(\alpha _2 - \alpha _1\gamma _{th2})}, \right. \nonumber \\{} & {} \left. \frac{P_t X_2^2\beta _{ss} \beta _{s2}}{\sigma ^2} < \frac{\gamma _{th2} - \frac{P_H}{\sigma ^2} |h_{1,2}|^2}{\alpha _2-\alpha _1(\gamma _{th2}-\frac{P_H}{\sigma ^2} |h_{1,2}|^2)} \right) . \end{aligned}$$

(51)

We can straightforwardly observe that the first term given by $\textrm{Pr} \left( X_1^2 < \frac{\gamma _{th2}(|h_{SI}|^2 P_H+\sigma ^2)}{\beta _{ss} \beta _{s1}P_t(1-\rho )(\alpha _2 - \alpha _1\gamma _{th2})} \right)$ has already been derived in the Appendix B. Similarly, the second term given by $\textrm{Pr} \left( X_2^2 < \frac{\sigma ^2 \gamma _{th2} }{\beta _{ss} \beta _{s2}P_t(\alpha _2-\alpha _1\gamma _{th2})} \right)$, has been previously derived earlier in subsection I) of this Appendix.

Therefore, here, we should focus on computing the term $\textrm{Pr} \left( \frac{P_t X_2^2 \beta _{ss} \beta _{s2}}{\sigma ^2 } < \frac{\gamma _{th2} - \frac{P_H}{\sigma ^2} |h_{1,2}|^2}{\alpha _2-\alpha _1(\gamma _{th2}-\frac{P_H}{\sigma ^2} |h_{1,2}|^2)} \right)$. For easiness, let us define the following variables, $Q \triangleq \frac{P_t X_2^2 \beta _{ss} \beta _{s2}}{\sigma ^2 }$ and $W \triangleq \frac{P_H}{\sigma ^2} |h_{1,2}|^2$. Since Q is $X_2$ scaled by constants, its distribution is given as $Q \sim \textrm{exp}\left( \frac{N Pt \beta _{ss}\beta _{s2}}{ \sigma ^2}\right)$. Moreover, It can be observed once again that the condition $P_{th} < \beta _{12}$, leads to $P_H$ values generally lower than $|h_{12}|^2$. In other words, the harvested power is highly likely to be lower than path-loss in D2D communication, as $D_1$ is typically located in close proximity to $D_2$. Therefore, as $|h_{12}|^2$ follows an exponential distribution, we again proposed to approximate W as exponentialRV, $W \sim \textrm{exp}\left( \frac{\widetilde{P}_H\beta _{12}}{\sigma ^2}\right)$, hence we must compute the following integral

$$\begin{aligned}{} & {} {\Pr } \left( Q < \frac{\gamma _{th2} - W}{\alpha _2-\alpha _1(\gamma _{th2}- W)} \right) \nonumber \\{} & {} \quad = \int _{0}^{\gamma _{th2}} F_{Q}\left( \frac{\gamma _{th2} - w}{\alpha _2-\alpha _1(\gamma _{th2}- w)}\right) f_{W}(w) dw \nonumber \\{} & {} \quad = F_{W}(\gamma _{th2}) \nonumber \\{} & {} \qquad - \frac{\sigma ^2}{\beta _{12}\widetilde{P}_H} \underbrace{\int _{0}^{\gamma _{th2}} e^{\left( \frac{-\sigma ^2}{P_tN\beta _{ss}\beta _{s2}}\right) \left( \frac{\gamma _{th2} - w}{\alpha _2-\alpha _1(\gamma _{th2}-w)} \right) -\frac{\sigma ^2}{\beta _{12}\widetilde{P}_H} w} dw }_{I}, \end{aligned}$$

(52)

Since $D_2$ assume low rate values, and $\alpha _2 > \alpha _1$ due to implementation of NOMA, we have $\alpha _2 \gg \alpha _1 \gamma _{th2}$, thus, the integral in I can be approximated as

$$\begin{aligned} I\approx & {} e^{\frac{-\gamma _{th2}\sigma ^2}{P_t\beta _{ss}\beta _{s2}N\alpha _2}} \int _{0}^{\gamma _{th2}} e^{ y \left( \frac{\sigma ^2 }{P_{t}N\beta _{ss}\beta _{s2}\alpha _2 } -\frac{\sigma ^2}{\beta _{12}\widetilde{P}_H}\right) } dy, \end{aligned}$$

(53)

whose solution is found trivially. Substituting the solution of Eq. (53) in Eq. (52) we obtain

$$\begin{aligned}{} & {} \textrm{Pr} \left( \frac{P_t X_2^2\beta _{ss} \beta _{s2}}{\sigma ^2} < \frac{\gamma _{th2} - \frac{P_H}{\sigma ^2} |h_{1,2}|^2}{\alpha _2-\alpha _1(\gamma _{th2}-\frac{P_H}{\sigma ^2} |h_{1,2}|^2)} \right) \nonumber \\{} & {} \quad \approx 1 - e^{-\frac{\gamma _{th2}\sigma ^2}{\beta _{12}\widetilde{P}_H}} - \frac{\sigma ^2}{\beta _{12}\widetilde{P}_H} \left( \frac{e^{ -\frac{\gamma _{th2}\sigma ^2}{\beta _{12}\widetilde{P}_H} } - e^{\frac{-\gamma _{th2}\sigma ^2}{P_t\beta _{ss}\beta _{s2}N\alpha _2}} }{ \left( \frac{\sigma ^2 }{P_{t}N\beta _{ss}\beta _{s2}\alpha _2 } -\frac{\sigma ^2}{\beta _{12}\widetilde{P}_H}\right) } \right) \end{aligned}$$

(54)

Hence, by utilizing Eq. (54) and the remaining analytical results derived for Eq. (51) in the previous Appendix, Eq. (34) is obtained. This completes the proof.

Appendix D. Proof of Theorem 4

We have that the ER of $D_1$ can be computed as

$$\begin{aligned} \bar{R}_1= & {} \mathbb {E} \left[ \log _2\left( 1 + \frac{|h_1|^2 (1-\rho ) P_t \alpha _1 }{ |h_{SI}|^2 P_H + \sigma ^2} \right) \right] \nonumber \\\overset{(i)}{\le } & {} \log _2 \left( \mathbb {E} \left[ 1 + \frac{|h_1|^2 (1-\rho ) P_t \alpha _1 }{ |h_{SI}|^2 P_H + \sigma ^2} \right] \right) , \end{aligned}$$

(55)

where Jensen’s inequality has been utilized in (i). Here will take the conditional expectation of Eq. (55) with respect to $|h_1|^2$, therefore, Eq. (55) can be written as

$$\begin{aligned} \bar{R}_1 \le \log _2 \left( 1 + \frac{N^2\beta _{ss}\beta _{s1}\mu _{ss}^2\mu _{s1}^2 (1-\rho ) P_t \alpha _1 }{ |h_{SI}|^2 P_H + \sigma ^2} \right) . \end{aligned}$$

(56)

By rewriting Eq. (56) we have

$$\begin{aligned} \bar{R}_1\le & {} \log _2\left( \frac{N^2 \beta _{ss} \beta _{s1} \mu _{ss}^2\mu _{s1}^2 (1-\rho )P_t\alpha 1 + \sigma ^2}{ \widetilde{P}_H} + |h_{\textrm{SI}}|^2 \right) \nonumber \\{} & {} - \log _2\left( \frac{\sigma ^2}{\widetilde{P}_H} + |h_{\textrm{SI}}|^2 \right) , \end{aligned}$$

(57)

defining $A \triangleq \frac{N^2 \beta _{ss} \beta _{s1} \mu _{ss}^2\mu _{s1}^2 (1-\rho )P_t\alpha 1 + \sigma ^2}{ \widetilde{P}_H}$ and $B \triangleq \frac{\sigma ^2}{\widetilde{P}_H}$, we should computing the expectation with respect the variable $|h_{SI}|^2$, therefore we have that

$$\begin{aligned} \mathbb {E}_{|h_{SI}|^2}[R_1]\le & {} \int _{0}^{\infty } \log _2\left( A + x \right) f_{|h_{SI}|^2}(x) dx \nonumber \\{} & {} - \int _{0}^{\infty } \log _2(B+x) f_{|h_{SI}|^2}(x) dx, \nonumber \\= & {} \frac{\ln \left( \frac{A}{B}\right) + e^{\frac{A}{\omega }} \Gamma \left( 0,\frac{A}{\omega }\right) - e^{\frac{B}{\omega }} \Gamma \left( 0,\frac{B}{\omega }\right) }{\log (2)}, \end{aligned}$$

(58)

It is essential to note that

$$\begin{aligned} \underset{x\rightarrow \infty }{\lim }\ e^{x} \Gamma \left( 0,x\right) = \underset{x\rightarrow \infty }{\lim } \frac{\Gamma (0,x)}{\frac{1}{e^{x}}} \overset{L'H}{=}\ \frac{-e^{-x} x^{-1}}{ -\frac{1}{e^{x}}} = \frac{1}{x} = 0, \end{aligned}$$

(59)

where the L’Hospital’s rule has been utilized, with [45, 06.06.20.0003.01]. Since the term $\frac{A}{\omega }$ assumes high values, we neglect it, therefore, the ER of $D_1$ can be given as

$$\begin{aligned} \bar{R}_1 \le \log _2\left( \frac{A}{B}\right) - \frac{e^{\frac{B}{\omega }} \Gamma \left( 0,\frac{B}{\omega }\right) }{\log (2)}, \end{aligned}$$

(60)

substituting the values of A and B in Eq. (60), Eq. (37) is obtained and this completes the proof.

Appendix E. Proof of Theorem 5

The ER of $D_2$ is given as

$$\begin{aligned} \bar{R}_2 = \mathbb {E} \left[ \log _2\left( 1 + \textrm{SINR}_{D_2,S}^{x_2} + \textrm{SNR}_{D_2,D_1}^{x_2}\right) \right] , \end{aligned}$$

(61)

Let us focus firstly when $\rho =0$, then by utilizing Eq. (11), we can rewritten Eq. (61) as

$$\begin{aligned} \bar{R}_2= & {} \mathbb {E}\left[ \log _2\left( 1 + \frac{|h_2|^2 P_t \alpha _2 }{|h_2|^2 P_t \alpha _1 + \sigma ^2} \right) \right] , \end{aligned}$$

(62)

By utilizing Jensen‘s inequality, we can obtain an upper bound for the ER of $D_2$, given as

$$\begin{aligned} \bar{R}_2\le & {} \log _2\left( 1 + \mathbb {E} \left[ \frac{|h_2|^2 P_t \alpha _2 }{|h_2|^2 P_t \alpha _1 + \sigma ^2} \right] \right) \nonumber \\\le & {} \log _2 \left( 1 + \frac{N\beta _{ss}\beta _{s2}P_t\alpha _2}{N\beta _{ss}\beta _{s2}P_t\alpha _1 + \sigma ^2} \right) . \end{aligned}$$

(63)

When $\rho \ne 0$, we should consider the term $\textrm{SNR}_{D_2,D_1}^{x_2}$ in Eq. (61). By taking the conditional expectation with respect to the RV $P_H$, we obtain

$$\begin{aligned} \bar{R}_2\le & {} \mathbb {E}_{P_H}\left[ \log _2\left( 1 + \frac{ N\beta _{ss}\beta _{s2} P_t \alpha _2 }{N\beta _{ss}\beta _{s2} P_t \alpha _1 + \sigma ^2} + \frac{P_H |h_{12}|^2}{\sigma ^2} \right) \right] \nonumber \\\le & {} \log _2\left( 1 + \frac{ N\beta _{ss}\beta _{s2} P_t \alpha _2 }{N\beta _{ss}\beta _{s2} P_t \alpha _1 + \sigma ^2} + \frac{ \mathbb {E}\left[ P_H\right] |h_{12}|^2}{\sigma ^2} \right) \nonumber \\= & {} \log _2\left( 1 + \frac{ N\beta _{ss}\beta _{s2} P_t \alpha _2 }{N\beta _{ss}\beta _{s2} P_t \alpha _1 + \sigma ^2} + \frac{ \widetilde{P}_H |h_{12}|^2}{\sigma ^2} \right) , \end{aligned}$$

(64)

Now, we turn our focus on calculating the expectation of Eq. (64), with respect to the variable $|h_{12}|^2$, rewriting Eq. (64) in a conventional way, we wave

$$\begin{aligned} \bar{R}_2\le & {} \mathbb {E}\left[ \log _2\left( \frac{\sigma ^2}{\widetilde{P}_H}\left( 1 + \frac{N\beta _{ss}\beta _{s2} P_t \alpha _2 }{N\beta _{ss}\beta _{s2} P_t \alpha _1 + \sigma ^2}\right) + |h_{12}|^2\right) \right. \nonumber \\{} & {} \left. +\log _2\left( \frac{\widetilde{P}_H}{\sigma ^2}\right) \right] , \end{aligned}$$

(65)

let us define $A \triangleq \frac{\sigma ^2}{\widetilde{P}_H}\left( 1 + \frac{N\beta _{ss}\beta _{s2} P_t \alpha _2 }{N\beta _{ss}\beta _{s2} P_t \alpha _1 + \sigma ^2}\right)$. According to Section 2, $h_{12}$ is a complex Gaussian RV of zero mean and variance $\beta _{12}$. Therefore, the square magnitude of $h_{12}$, follows an exponential distribution whose PDF is given as $f_{|h_{12}|^2} = \frac{e^{-\frac{x}{\beta _{12}}}}{\beta _{12}}$. Based on this, we should compute the following integral

$$\begin{aligned}{} & {} \int _{0}^{\infty } \log _2(A + x) f_{|h_{12}|^2}(x) dx = \frac{1}{\beta _{12}}\int _{0}^{\infty } \log _2(A + x) e^{-\frac{x}{\beta _{12}}} dx \nonumber \\{} & {} \quad \overset{(i)}{=}\ \log _2\left( A \right) + \frac{e^{ \frac{A}{\beta _{12}} }}{\log (2)} \Gamma \left( 0, \frac{A}{\beta _{12}} \right) , \end{aligned}$$

where in (i), [44, 4.337.1] and [44, 8.359.1] have been utilized. Finally, we conclude the proof by substituting the value of A in Eq. (66) and the result into Eq. (65) and performing a few basic mathematical manipulations.