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Simultaneously transmitting and reflecting reconfigurable intelligent surfaces with energy harvesting from vibrations

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Abstract

Simultaneously Transmitting and Reflecting Reconfigurable Intelligent Surfaces (STAR-RIS) represent a cutting-edge technology in wireless communications, allowing surfaces to both transmit and reflect signals simultaneously. This dual functionality provides greater flexibility and efficiency in signal propagation and spectrum management. Integrating energy harvesting from vibrations into STAR-RIS creates an innovative and sustainable solution for powering these systems, enabling autonomous operation in environments where conventional power sources are unavailable. By converting ambient mechanical energy into electrical power, vibration-based energy harvesting supports the continuous operation of STAR-RIS without reliance on external energy sources. This advancement has significant implications for the deployment of future wireless networks, including smart cities, the Internet of Things, and remote sensing applications, offering a pathway to greener and more efficient communication infrastructure.

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Acknowledgements

This Research is funded by Researchers Supporting Project No. (RSPD2024R553), King Saud University, Riyadh, Saudi Arabia.

Funding

A fund from KSU has been received.

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This is the contribution of Prof. Hatem Boujemaa, Prof. Musaed Alhussein and Prof. Ghaya Rekaya.

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Correspondence to Hatem Boujemaa.

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Appendix A

Appendix A

The variable \(\mathcal {Y}=\alpha \times \mathcal {F}\) where \(\mathcal {F}\) is the frequency of mechanical vibrations has a Gaussian distribution written as

$$\begin{aligned} f_{\mathcal {Y}}(x)=\frac{1}{\sqrt{2\pi \alpha ^2\sigma _{\mathcal {F}}^2}}exp\left( -\frac{[x-\alpha m_{\mathcal {F}}]^2}{2\alpha ^2\sigma _{\mathcal {F}}^2}\right) , \end{aligned}$$
(25)

The Mellin Tranform (MT) of \(\mathcal {Y}\) is computed as

$$\begin{aligned} MGF_{\mathcal {Y}}(s)= & \int _0^{+\infty }f_{\mathcal {Y}}(x)x^{s-1}dx\nonumber \\= & \int _0^{+\infty }\frac{1}{\sqrt{2\pi \sigma _{\mathcal {F}}^2\alpha ^2}}exp\nonumber \\ & \quad \left( -\frac{[x-m_{\mathcal {F}}\alpha ]^2}{2\alpha ^2\sigma _{\mathcal {F}}^2}\right) x^{s-1}dx \end{aligned}$$
(26)

We deduce

$$\begin{aligned} MGF_{\mathcal {Y}}(s)= & \frac{1}{\sqrt{2\pi \alpha ^2\sigma _{\mathcal {F}}^2}}exp\left( -\frac{m_{\mathcal {F}}^2}{2\sigma _{\mathcal {F}}^2}\right) \nonumber \\ & \times \int _0^{+\infty }exp\left( -\frac{x^2}{2\alpha ^2\sigma _{\mathcal {F}}^2}\right) \nonumber \\ & \quad exp \left( \frac{xm_{\mathcal {F}}}{\alpha \sigma _{\mathcal {F}}^2}\right) x^{s-1}dx \nonumber \\= & \frac{1}{\sqrt{2\pi }}exp\left( -\frac{m_{\mathcal {F}}^2}{2\sigma _{\mathcal {F}}^2}\right) \sum _{k=0}^{+\infty }\frac{m_{\mathcal {F}}^k}{k!\alpha ^{k+1}\sigma _{\mathcal {F}}^{2k+1}}\nonumber \\ & \times \int _0^{+\infty }exp\left( -\frac{x^2}{2\alpha ^2\sigma _{\mathcal {F}}^2}\right) x^{k+s-1}dx \end{aligned}$$
(27)

Let \(y=\frac{x^2}{2\alpha ^2\sigma _{\mathcal {F}}^2}\), we deduce

$$\begin{aligned} MGF_{\mathcal {Y}}(s)= & \frac{1}{\sqrt{\pi }}exp \left( -\frac{m_{\mathcal {F}}^2}{2\sigma _{\mathcal {F}}^2}\right) \sum _{k=0}^{+\infty }\frac{m_{\mathcal {F}}^k\sigma _{\mathcal {F}}^{s-k-1}\alpha ^{s-1}}{k!} \nonumber \\ & \times \sqrt{2}^{s+k-3}\int _0^{+\infty }exp(-y)y^{0.5k+0.5s-1}dy\nonumber \\= & \frac{1}{\sqrt{\pi }}exp \left( -\frac{m_{\mathcal {F}}^2}{2\sigma _{\mathcal {F}}^2}\right) \sum _{k=0}^{+\infty }\frac{m_{\mathcal {F}}^k\sigma _{\mathcal {F}}^{s-k-1}\alpha ^{s-1}}{k!}\nonumber \\ & \sqrt{2}^{s+k-3}\varGamma \left( \frac{s+k}{2}\right) \end{aligned}$$
(28)

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

The variable \(\mathcal {X}=\mathcal {D}^2\) where \(\mathcal {D}\) is the mechanical deformation has a non-central zeta-square distribution with one degree of freedom and PDF given by

$$\begin{aligned} f_{\mathcal {X}}(x)&=\frac{2}{\sigma _{\mathcal {D}}}e^{-0.5\left( \frac{x}{\sigma _{\mathcal {D}}} +m_{\mathcal {D}}^2\right) }\left( \frac{x}{m_{\mathcal {D}}^2\sigma _{\mathcal {D}}}\right) ^{-0.25}I_{-0.5}\nonumber \\&\quad \left( m_{\mathcal {D}} \sqrt{\frac{x}{\sigma _{\mathcal {D}}}}\right) \end{aligned}$$
(29)

where \(I_m(x)\) is the modified Bessel function of the first kind and k-th order.

The Mellin transform of PDF of \(\mathcal {X}\) is equal to

$$\begin{aligned} MGF_{\mathcal {X}}(s)= & 2\sqrt{m_{\mathcal {D}}}\sigma _{\mathcal {D}}^{-0.75}e^{-\frac{m_{\mathcal {D}}^2}{2}}\int _0^{+\infty }I_{-0.5}\left( m_{\mathcal {D}}\sqrt{\frac{x}{\sigma _{\mathcal {D}}}}\right) \nonumber \\ & \times x^{s-1.25}e^{-\frac{x}{2\sigma _{\mathcal {D}}}}dx. \end{aligned}$$
(30)

We have

$$\begin{aligned} I_{-0.5}(y)=\sum _{q=0}^{+\infty }\frac{y^{2q-0.5}}{2^{2q+n}q!\varGamma (q+0.5)} \end{aligned}$$
(31)

We deduce

$$\begin{aligned} MGF_{\mathcal {X}}(s)=e^{-\frac{m_{\mathcal {D}}^2}{2}}\sum _{q=0}^{+\infty } \frac{\sigma _{\mathcal {D}}^{q+2s-2.5}2^{2s-0.5}}{q!\varGamma (q+0.5)}\varGamma (s+q-0.5) \end{aligned}$$
(32)

The transmitted energy per symbol \(E_s=\mathcal {X}\mathcal {Y}\). As \(\mathcal {X}\) and \(\mathcal {Y}\) are independent, we have

$$\begin{aligned} MGF_{E_s}(s)= & MGF_{\mathcal {X}}(s)MGF_{\mathcal {Y}}(s)\nonumber \\= & \frac{1}{\sqrt{\pi }}exp(-\frac{m_{\mathcal {F}}^2}{2\sigma _{\mathcal {F}}^2})e^{-\frac{m_{\mathcal {D}}^2}{2}}\sum _{q=0}^{+\infty }\frac{\sigma _{\mathcal {D}}^{q+2s-2.5}2^{2s-0.5}}{q!\varGamma (q+0.5)}\nonumber \\ & \varGamma (s+q-0.5)\nonumber \\ & \times \sum _{k=0}^{+\infty }\frac{m_{\mathcal {F}}^k\sigma _{\mathcal {F}}^{s-k-1}\alpha ^{s-1}}{k!}\sqrt{2}^{s+k-3}\varGamma \left( \frac{s+k}{2}\right) \nonumber \\ \end{aligned}$$
(33)

The PDF of \(E_s\) is computed using the inverse MT

$$\begin{aligned} f_{E_s}(y)=\frac{1}{2\pi j}\int _{e-j\infty }^{e+j\infty }y^{-s}MGF_{E_s}(s)ds. \end{aligned}$$
(34)

Therefore, we have

$$\begin{aligned} f_{E_s}(y)= & \frac{1}{\sqrt{\pi }}exp(-\frac{m_{\mathcal {F}}^2}{2\sigma _{\mathcal {F}}^2})e^{-\frac{m_{\mathcal {D}}^2}{2}}\sum _{q=0}^{+\infty }\frac{\sigma _{\mathcal {D}}^{q-2.5}}{q!\varGamma (q+0.5)}\nonumber \\ & \times \sum _{k=0}^{+\infty }\frac{m_{\mathcal {F}}^k\sigma _{\mathcal {F}}^{-k-1}}{\alpha k!}\sqrt{2}^{k-4}\int _{e-j\infty }^{e+j\infty }\left( \frac{y}{4\sigma _{\mathcal {D}}^2\alpha \sigma _{\mathcal {F}}}\right) ^{-s}\nonumber \\ & \times \varGamma \left( \frac{s+k}{2}\right) \varGamma (s+q-0.5)ds \end{aligned}$$
(35)

We deduce the expression of the PDF of \(E_s\)

$$\begin{aligned} f_{E_s}(y)= & \frac{1}{\sqrt{\pi }}exp(-\frac{m_{\mathcal {F}}^2}{2\sigma _{\mathcal {F}}^2})e^{-\frac{m_{\mathcal {D}}^2}{2}}\sum _{q=0}^{+\infty }\frac{\sigma _{\mathcal {D}}^{q-2.5}}{q!\varGamma (q+0.5)}\nonumber \\ & \times \sum _{k=0}^{+\infty }\frac{m_{\mathcal {F}}^k\sigma _{\mathcal {F}}^{-k-1}}{\alpha k!}\sqrt{2}^{k-4}H_{1,1}^{1,1}\nonumber \\ & \qquad \left( \frac{y}{4\sqrt{2}\sigma _{\mathcal {D}}^2\alpha \sigma _{\mathcal {F}}}|^{(1-0.5k,-0.5)}_{(q-0.5,1)}\right) \end{aligned}$$
(36)

where \(H_{m,n}^{p,q}()\) is the Fox H function [29].

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Boujemaa, H., Alhussein, M. & Rekaya, G. Simultaneously transmitting and reflecting reconfigurable intelligent surfaces with energy harvesting from vibrations. SIViP 19, 84 (2025). https://doi.org/10.1007/s11760-024-03643-x

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