Reconfigurable intelligent surfaces with solar energy harvesting

Abstract

In this paper, we compute the throughput of wireless communications using Reconfigurable Intelligent Surfaces (RIS) when the source harvests energy using a solar panel. Harvesting duration is also optimized to enhance the performance of wireless communications when RIS is used. We derive the statistics of the Signal to Noise Ratio (SNR). We show that the SNR is the product of a Gaussian and a chisquare random variables (r.v.). We consider solar energy harvesting for Rayleigh channels.

Appendix

The transmitted energy per symbol $E_s$ has the following Gaussian PDF

$$\begin{aligned} f_{E_s}(x)=\frac{1}{\sqrt{2\pi \sigma ^2}}exp(-\frac{[x-m(t)]^2}{2\sigma ^2}), \end{aligned}$$

(28)

The Mellin Tranform (MT) of $E_s$ is computed as

$$\begin{aligned}{} & {} M_{E_s}(s)=\int _0^{+\infty }f_{E_s}(x)x^{s-1}dx\nonumber \\{} & {} =\int _0^{+\infty }\frac{1}{\sqrt{2\pi \sigma ^2}}exp(-\frac{[x-m(t)]^2}{2\sigma ^2})x^{s-1}dx \end{aligned}$$

(29)

We deduce

$$\begin{aligned}{} & {} M_{E_s}(s)=\frac{1}{\sqrt{2\pi \sigma ^2}}exp(-\frac{m(t)^2}{2\sigma ^2})\int _0^{+\infty }exp(-\frac{x^2}{2\sigma ^2})exp(\frac{xm(t)}{\sigma ^2})x^{s-1}dx\nonumber \\{} & {} =\frac{1}{\sqrt{2\pi \sigma ^2}}exp(-\frac{m(t)^2}{2\sigma ^2})\sum _{k=0}^{+\infty }\frac{m(t)^k}{k!\sigma ^{2k}}\nonumber \\{} & {} \int _0^{+\infty }exp(-\frac{x^2}{2\sigma ^2})x^{k+s-1}dx \end{aligned}$$

(30)

Let $y=\frac{x^2}{2\sigma ^2}$, we deduce

$$\begin{aligned}{} & {} M_{E_s}(s)=\frac{1}{\sqrt{\pi }}exp(-\frac{m(t)^2}{2\sigma ^2})\sum _{k=0}^{+\infty }\frac{m(t)^k\sigma ^{s-k-2}}{k!} \sqrt{2}^{s+k-3}\int _0^{+\infty }exp(-y)y^{0.5k+0.5s-1}dx \nonumber \\{} & {} =\frac{1}{\sqrt{\pi }}exp(-\frac{m(t)^2}{2\sigma ^2})\sum _{k=0}^{+\infty }\sqrt{2}^{s+k-3}\frac{m(t)^k\sigma ^{s-k-2}}{k!}\Gamma (\frac{s+k}{2}) \end{aligned}$$

(31)

where $\Gamma (.)$ is the Gamma function.

The SNR $\gamma =\frac{\lambda _1\lambda _2E_sA^2}{N_0}=\frac{\lambda _1\lambda _2Y}{N_0}$ where $Y=A^2E_s$. $A^2$ depends on channels gains from the source to RIS and from RIS to destination D. $E_s$ depends on harvested power using a solar PV system. Therefore, $A^2$ and $E_s$ are independent and the MT of Y is the product of MT of $A^2$ and that of $E_s$:

$$\begin{aligned} M_{Y}(s)=M_{E_s}(s)M_{A^2}(s). \end{aligned}$$

(32)

We need to compute the MT of $A^2$ to deduce $M_Y(s)$ and by the inverse MT to obtain the PDF of Y and that of SNR $\gamma =\frac{\lambda _1\lambda _2Y}{N_0}$. The PDF of $A^2$ is written as [28]

$$\begin{aligned} f_{A^2}(y)=0.5e^{-0.5(y+\Delta ^2)}I_{-0.5}( \sqrt{\Delta ^2y})(\frac{y}{\Delta ^2})^{-0.25} \end{aligned}$$

(33)

where $\Delta ^2=\frac{m_A^2}{\sigma _A^2}$.

The Mellin transform of PDF of $A^2$ is equal to

$$\begin{aligned} M_{A^2}(s)=0.5e^{-\frac{\Delta ^2}{2}}\sqrt{\Delta }\int _0^{+\infty }I_{-0.5}( \sqrt{\Delta ^2y})y^{s-\frac{5}{4}}e^{-\frac{y}{2}}dy. \end{aligned}$$

(34)

We have

$$\begin{aligned} I_n(y)=\sum _{q=0}^{+\infty }\frac{y^{2q+n}}{2^{2q+n}q!\Gamma (q+n+1)} \end{aligned}$$

(35)

We deduce

$$\begin{aligned} M_{A^2}(s)=0.5e^{-\frac{\Delta ^2}{2}}\sqrt{\Delta }\sum _{q=0}^{+\infty }\frac{\Delta ^{2q-0.5}}{2^{q-0.5}q! \Gamma (q+0.5)}\int _0^{+\infty }y^{s+q-\frac{3}{2}}e^{-\frac{y}{2}}dy. \end{aligned}$$

(36)

We have

$$\begin{aligned} \int _0^{+\infty }e^{-0.5y}y^{s+q-1.5}dy=\Gamma (s+q-0.5)2^{s+q-0.5}. \end{aligned}$$

(37)

Therefore, we obtain

$$\begin{aligned} M_{A^2}(s)=e^{-\frac{\Delta ^2}{2}}\sum _{q=0}^{+\infty }\frac{\Delta ^{2q}}{q!\Gamma (q+0.5)}\Gamma (s+q-0.5)2^{s-q-1}. \end{aligned}$$

(38)

We deduce $M_Y(s)$ from (31)

$$\begin{aligned} M_Y(s)=\frac{1}{\sqrt{\pi }}exp(-\frac{m(t)^2}{2\sigma ^2})\sum _{k=0}^{+\infty }\sqrt{2}^{s+k-3}\frac{m(t)^k\sigma ^{s-k-2}}{k!}\Gamma (\frac{s+k}{2}) \end{aligned}$$

$$\begin{aligned} \times e^{-\frac{\Delta ^2}{2}}\sum _{q=0}^{+\infty }\frac{\Delta ^{2q}}{q!\Gamma (q+0.5)}\Gamma (s+q-0.5)2^{s-q-1} \end{aligned}$$

(39)

The PDF of Y is computed using the inverse MT

$$\begin{aligned} f_Y(y)=\frac{1}{2\pi j}\int _{e-j\infty }^{e+j\infty }y^{-s}M_Y(s)ds. \end{aligned}$$

(40)

Therefore, we have

$$\begin{aligned}{} & {} f_Y(y)=e^{-\frac{\Delta ^2}{2}}\frac{1}{\sqrt{\pi }}exp(-\frac{m(t)^2}{2\sigma ^2})\sum _{k=0}^{+\infty }\sqrt{2}^{k-3}\frac{m(t)^k\sigma ^{-k-2}}{k!} \sum _{q=0}^{+\infty }\frac{\Delta ^{2q}}{q!\Gamma (q+0.5)}2^{-q-1}\nonumber \\{} & {} \times \frac{1}{2\pi j}\int _{e-j\infty }^{e+j\infty }(\frac{y}{2 \sqrt{2}\sigma })^{-s}\Gamma (s+q-0.5)\Gamma (\frac{s+k}{2})ds. \end{aligned}$$

(41)

We deduce the expression of the PDF of Y

$$\begin{aligned}{} & {} f_Y(y)=e^{-\frac{\Delta ^2}{2}}\frac{1}{\sqrt{\pi }}exp(-\frac{m(t)^2}{2\sigma ^2})\sum _{k=0}^{+\infty }\sqrt{2}^{k-3}\frac{m(t)^k\sigma ^{-k-2}}{k!} \sum _{q=0}^{+\infty }\frac{\Delta ^{2q}}{q!\Gamma (q+0.5)}2^{-q-1}\nonumber \\{} & {} \times H_{1,1}^{1,1}(\frac{y}{2 \sqrt{2}\sigma }|_{q-0.5,1}^{1-0.5k,-0.5}). \end{aligned}$$

(42)

where $H_{m,n}^{p,q}()$ is the Fox H function [28].

We deduce the PDF of SNR $\gamma =\frac{\lambda _1\lambda _2Y}{N_0}$ [28]

$$\begin{aligned}{} & {} f_{\gamma }(y)=\frac{N_0}{\lambda _1\lambda _2}e^{-\frac{\Delta ^2}{2}}\frac{1}{\sqrt{\pi }}exp(-\frac{m(t)^2}{2\sigma ^2})\sum _{k=0}^{+\infty }\sqrt{2}^{k-3}\frac{m(t)^k\sigma ^{-k-2}}{k!} \sum _{q=0}^{+\infty }\frac{\Delta ^{2q}}{q!\Gamma (q+0.5)}2^{-q-1}\nonumber \\{} & {} \times H_{1,1}^{1,1}(\frac{yN_0}{\lambda _1\lambda _22 \sqrt{2}\sigma }|_{q-0.5,1}^{1-0.5k,-0.5}). \end{aligned}$$

(43)

We use the following result on Laplace Transform (LT) of the Fox H function [28]

$$\begin{aligned} LT[x^{\rho -1}H_{p,q}^{m,n}(ax^{\sigma }|^{a_p,A_p}_{b_q,B_q})] =u^{-\rho }H_{p+1,q}^{m,n+1}(au^{-\sigma }|^{1-\rho ,\sigma ,a_p,A_p}_{b_q,B_q}). \end{aligned}$$

(44)

We deduce the Moment Generating Function (MGF) of SNR as

$$\begin{aligned}{} & {} MGF_{\gamma }(u)=LT(f_{\gamma }(y))=\int _0^{+\infty }e^{-uy}f_{\gamma }(y)dy \nonumber \\ =\frac{N_0}{\lambda _1\lambda _2}e^{-\frac{\Delta ^2}{2}}\frac{1}{\sqrt{\pi }}exp(-\frac{m(t)^2}{2\sigma ^2})\sum _{k=0}^{+\infty }\sqrt{2}^{k-3}\frac{m(t)^k\sigma ^{-k-2}}{k!} \sum _{q=0}^{+\infty }\frac{\Delta ^{2q}}{q!\Gamma (q+0.5)}2^{-q-1}\nonumber \\{} & {} \times \frac{1}{u}H_{2,1}^{1,2}(\frac{N_0}{u\lambda _1\lambda _22 \sqrt{2}\sigma }|_{q-0.5,1}^{0,1,1-0.5k,-0.5}). \end{aligned}$$

(45)