# A Novel RIS-Assisted Modulation Scheme

Liang Yang, Fanxu Meng, Mazen O. Hasna, and Ertugrul Basar L. Yang and F. Meng
are with the College of Computer Science and Electronic Engineering, Hunan
University, Changsha 410082, China<EMAIL_ADDRESS>mengfx@hnu.edu.cn).M. O. Hasna is with the Department of Electrical
Engineering, Qatar University, Doha 2713, Qatar. (e-mail: hasna@qu.edu.qa).E.
Basar is with the CoreLab, Department of Electrical and Electronics
Engineering, Koç University, Istanbul 34450, Turkey (e-mail:
ebasar@ku.edu.tr).

###### Abstract

In this work, in order to achieve higher spectrum efficiency, we propose a
reconfigurable intelligent surface (RIS)-assisted multi-user communication
uplink system. Different from previous work in which the RIS only optimizes
the phase of the incident users’s signal, we propose the use of the RIS to
create a virtual constellation diagram to transmit the data of an additional
user signal. We focus on the two-user case and develop a tight approximation
for the cumulative distribution function (CDF) of the received signal-to-noise
ratio of both users. Then, based on the proposed statistical distribution, we
derive the analytical expressions of the average bit error rate of the
considered two users. The paper shows the trade off between the performance of
the two users against each other as a function of the proposed phase shift at
the RIS.

###### Index Terms:

RIS, Average BER, Spectrum Efficiency.

##  I Introduction

Reconfigurable intelligent surfaces (RISs) are man-made surfaces composed of
electromagnetic (EM) materials, which are highly controllable by leveraging
electronic devices. In essence, an RIS can deliberately control the
reflection/scattering characteristics of the incident wave to enhance the
signal quality at the receiver, and hence converts the propagation environment
into a smart one [1].

Owing to their promising gains, recently RISs have been extensively
investigated in the literature. In particular, the authors in [2] proposed a
practical phase shift model for RISs. In [3], the authors studied the
beamforming optimization of RIS-assisted wireless communication under the
constraints of discrete phase shifts, while in [4], the authors studied the
coverage and signal-to-noise ratio (SNR) gain of RIS-assisted communication
systems. In [5], the authors proposed highly accurate closed-form
approximations to channel distributions of two different RIS-based wireless
system setups. Recently, RISs have been used in many scenarios and have shown
superior performance over systems not employing RISs. For instance, in [6], an
intelligent reflecting surface (IRS)-assisted multiple-input single-output
communication system is considered, and in [7], the physical layer security of
RIS-assisted communication with an eavesdropping user is studied. In [8], the
authors proposed RIS-assisted dual-hop unmanned aerial vehicle (UAV)
communication systems while in [9], the authors used the RIS for downlink
multi-user communication from a multi-antenna base station, and developed an
energy-efficient designs for transmit power allocation and phase shifts of the
surface reflection elements.

In all of the above studies, the advantages of RISs are mainly used to enhance
the quality of the signal, and the reflection patterns were not used to carry
additional information. i.e., the role of an RIS has been mainly based on the
mitigation of the phase shifts of the involved channels, without any
additional purpose of controlling those phase shifts. In this paper, we
propose a novel modulation scheme utilizing the phase shifts of the RIS in a
spectrally efficient way to superimpose the data of an additional user 2 (U2)
on that of the ordinary user 1 (U1).

More specifically, we consider a multi-user uplink scenario and mainly
consider the feasibility of uploading the data of two users simultaneously
through the RIS, where U1 sends the data to the base station through a direct
link and is given a chance to utilize an available RIS-assisted link to
enhance its signal, but with the condition of having the data of U2 embedded
with its data through the RIS. Basically, the RIS optimizes the phase of the
incident U1’s signal to mitigate the phase shifts of the cascaded link as well
as its direct link, and additionally to embed the data of U2 through creating
a modified virtual constellation diagram. Hence, we assume that U2’s data is
known when the RIS optimizes the phase of the incident U1’s signal, and then,
a virtual constellation diagram is created by the RIS to embed U2’s data.
Consequently, the signal reflected by the RIS contains the data of both users.
In summary, the main contributions of this work include the following: (i) we
propose a novel and spectrallly efficient RIS-assisted modulation scheme, (ii)
for the proposed system, we develop two tight approximate statistical
distributions for the received SNR of the two considered users, (iii) based on
the proposed statistical distributions, closed-form expressions for the
average bit error rates (BER) are derived and analysed.

The remaining of this letter is organized as follows. Section II presents the
system and channel models. The performance analysis is presented in section
III, and the numerical and simulation results are detailed in section IV.
Finally, conclusions are drawn in section V.

##  II System and Channel Models

We consider the uplink system shown in Fig. 1, where U1 is communicating with
the base station (BS) directly and with the help of an RIS to boost its
connectivity. In the same time, U2 is in the vicinity of the RIS and is
communicating with the BS through superimposing its signal on that of U1 using
the RIS. It is assumed that the RIS can obtain perfect channel state
information (CSI) through a control link that enables it to optimize the phase
shifts of the reflected signals. As will be discussed later, the phase shifts
will be utilized to superimpose U2 data on that of U1 in a way to efficiently
utilize the same spectrum. This is in return of allowing U1 to take advantage
of the RIS to improve its connectivity to the BS. At the receiving end, the BS
first decodes the signal of U1 and extracts it from the composite received
signal. In the second step, the remaining signal is processed to get the data
of U2.

Figure 1: RIS-assisted multi-user uplink.

###  II-A Analysis of User 1 SNR

As mentioned above, U1 is communicating with the BS through the RIS-assisted
dual-hop link and a direct link. A binary phase shift keying (BPSK) symbol $x$
with average power $E_{s}$ is sent to the RIS with $N$ reflecting elements
through a set of channels $h_{i}=\alpha_{i}e^{j\theta_{i}}$ where
$\alpha_{i}$’s are independent and identically distributed (i.i.d.) Rayleigh
random variables (RVs) with mean $\sqrt{\pi}/2$, variance $(4-\pi)/4$, and
uniformly distributed phase $\theta_{i}$. Meanwhile, the direct link to the BS
is denoted as $h_{D}=\varepsilon e^{j\eta}$, where $\varepsilon$ is a Rayleigh
random variable (RV) with mean $\sqrt{\pi}/2$, variance $(4-\pi)/4$ and phase
$\eta$. The RIS elements are assumed to have a line of sight with the BS
through channels $g_{i}=\beta_{i}e^{j\psi_{i}}$ where $\beta_{i}$ are i.i.d.
Rician RVs with Rician factor $K$ and phases $\psi_{i}$. With the knowledge of
the different channels CSI (U1-RIS, RIS-BS, U1-BS), the RIS optimizes the
incident signals in a way to create a virtual constellation diagram by
embedding the signal of U2. The overall received signal at the BS including
that of the direct link can be expressed as

$\displaystyle y_{1}$
$\displaystyle=\left(\sqrt{\frac{E_{s}}{L_{1}}}h_{D}+\sqrt{\frac{E_{s}}{L_{2}}}\sum_{i=1}^{N}h_{i}g_{i}{\rm{e}}^{j\phi_{i}}\right)x+n$
$\displaystyle=\left(\sqrt{\frac{E_{s}}{L_{1}}}\varepsilon+\sqrt{\frac{E_{s}}{L_{2}}}\sum_{i=1}^{N}\alpha_{i}\beta_{i}{\rm{e}}^{jw_{m}}\right)x+n,$
(1)

where $L_{1}$ and $L_{2}$ are the path losses of the direct link and the RIS-
assisted dual-hop link, respectively,
$\phi_{i}=(w_{m}-\theta_{i}-\psi_{i}+\eta)$ is the adjustable phase introduced
by the $i$th reflecting element of the RIS to mitigate the channels’ phase
shifts, $w_{m}$ is the message-dependent phase introduced by the RIS to carry
the information of U2 where $w_{m}$ represents a binary symbol of 1 and
$-w_{m}$ represents 0, and $n\sim\mathcal{C}\mathcal{N}(0,N_{0})$ is the
additive white Gaussian noise (AWGN) signal.

Then, the received SNR can be written as

$\displaystyle\gamma_{1}=$
$\displaystyle\frac{\left(\sqrt{\frac{E_{s}}{L_{1}}}\varepsilon+\sqrt{\frac{E_{s}}{L_{2}}}\sum_{i=1}^{N}\alpha_{i}\beta_{i}\cos(w_{m})\right)^{2}}{N_{0}}$
$\displaystyle=$
$\displaystyle\left(\sqrt{\bar{\gamma}_{1}}\varepsilon+\sqrt{\bar{\gamma}_{2}}R_{1}\right)^{2}=Z_{1}^{2},$
(2)

where $R_{1}=\sum_{i=1}^{N}\alpha_{i}\beta_{i}\cos(w_{m})$ ,
$\bar{\gamma}_{1}={\bar{\gamma}}/{L_{1}}$,
$\bar{\gamma}_{2}={\bar{\gamma}}/{L_{2}}$ and $\bar{\gamma}=E_{s}/N_{0}$
denotes the average SNR.

For a sufficiently large number of reflecting elements $N$, and relying on the
central limit theorem (CLT), $R_{1}$ can be assumed to follow a Gaussian RV.
Thus, the probability density function (PDF) of $R_{1}$ can be simply
expressed as

$f_{R_{1}}(r)=\frac{1}{\sqrt{2\pi\sigma_{1}^{2}}}\exp\left(-\frac{\left(r-\mu_{1}\right)^{2}}{2\sigma_{1}^{2}}\right),$
(3)

where
$\mu_{1}=\frac{N\Gamma\left(\frac{3}{2}\right)\sqrt{\pi}}{2\sqrt{1+K}e^{K}}{}_{1}F_{1}(\frac{3}{2},1,K)\cos(w_{m})$,
$\sigma_{1}^{2}=N\left[\frac{{}_{1}{\rm{F}}_{1}(2,1,K)}{(1+K)e^{K}}-\left(\frac{\mu_{1}}{N\cos(w_{m})}\right)^{2}\right]\cos^{2}(w_{m})$,
${}_{1}F_{1}(\cdot)$ is the Degenerate hypergeometric function, and
$\Gamma(\cdot)$ is the Gamma function [10]. Then, we obtain the cumulative
distribution function (CDF) of $Z_{1}$ as

$\displaystyle F_{Z_{1}}(z)$
$\displaystyle=\Pr(\sqrt{\bar{\gamma}_{1}}\varepsilon+\sqrt{\bar{\gamma}_{2}}R_{1}<Z)$
$\displaystyle=\int_{0}^{\frac{z}{\sqrt{\bar{\gamma}_{2}}}}\frac{1}{\sqrt{2\pi\sigma_{1}^{2}}}\exp\left(-\frac{\left(r-\mu_{1}\right)^{2}}{2\sigma_{1}^{2}}\right)dr$
$\displaystyle-\int_{0}^{\frac{z}{\sqrt{\bar{\gamma}_{2}}}}\frac{1}{\sqrt{2\pi\sigma_{1}^{2}}}{\rm{e}}^{\left(-\frac{\left(r-\mu_{1}\right)^{2}}{2\sigma_{1}^{2}}-\frac{\left(z-\sqrt{\bar{\gamma}_{2}}r\right)^{2}}{\bar{\gamma}_{1}}\right)}dr.$
(4)

From [10, Eq.(2.33.1)], the CDF of $\gamma_{1}$ is given by

$\displaystyle
F_{\gamma_{1}}(\gamma)=\frac{1}{2}\left[\rm{erf}\left(\frac{C_{2}\sqrt{\gamma}-\mu_{1}}{\sqrt{2\sigma_{1}^{2}}}\right)+\rm{erf}\left(\frac{\mu_{1}}{\sqrt{2\sigma_{1}^{2}}}\right)\right]$
$\displaystyle-\frac{1}{2\sqrt{2}C_{3}}\rm{e}^{\left(-\frac{\left(C_{1}C_{2}\sqrt{\gamma}-C_{1}\mu_{1}\right)^{2}}{2C_{3}^{2}C_{2}^{2}}\right)}\left[\rm{erf}\left(\frac{C_{2}\sqrt{\gamma}-\mu_{1}}{2C_{3}\sigma_{1}}\right)\right]$
$\displaystyle-\frac{1}{2\sqrt{2}C_{3}}\rm{e}^{\left(-\frac{\left(C_{1}C_{2}\sqrt{\gamma}-C_{1}\mu_{1}\right)^{2}}{2C_{3}^{2}C_{2}^{2}}\right)}\left[\rm{erf}\left(\frac{C_{4}\sqrt{\gamma}+\mu_{1}}{2C_{3}\sigma_{1}}\right)\right],$
(5)

where $C_{1}=\frac{1}{\sqrt{\bar{\gamma}_{1}}}$,
$C_{2}=\frac{1}{\sqrt{\bar{\gamma}_{2}}}$,$C_{3}=\sqrt{\frac{\sigma^{2}L_{1}}{L_{2}}+\frac{1}{2}}$,
$C_{4}=\frac{2\sigma_{1}^{2}C_{1}^{2}}{C_{2}}$ and ${\rm{erf}}(\cdot)$ is the
error function [10].

$\displaystyle
A_{2}{=}\sqrt{\frac{\bar{\gamma}_{1}\gamma}{2\sigma_{2}^{2}\bar{\gamma}_{2}{+}\bar{\gamma}_{1}\gamma}}\exp\left(\frac{1}{\bar{\gamma}_{1}}{-}\frac{\mu_{2}^{2}}{2\sigma_{2}^{2}}\right)\left[\sum_{l=0}^{L}\sum_{k=0}^{l-1}\frac{(\gamma\mu_{2}^{2})^{l}\left(\frac{1}{2}{-}l\right)_{l{-}k{-}1}({-}1)^{l{+}k{-}1}\left(\frac{1}{\bar{\gamma}_{1}}{+}\frac{\gamma}{2\bar{\gamma}_{2}\sigma_{2}^{2}}\right)^{k{+}\frac{1}{2}}}{l!2^{l}\Gamma(l{+}\frac{1}{2})\left(\gamma\sigma_{2}^{2}{+}\frac{2\bar{\gamma}_{2}\sigma_{2}^{4}}{\bar{\gamma}_{1}}\right)^{l}\exp\left(\frac{1}{\bar{\gamma}_{1}}{+}\frac{\gamma}{2\bar{\gamma}_{2}\sigma_{2}^{2}}\right)}{+}\sum_{l=0}^{L}\frac{(\gamma\mu_{2}^{2})^{l}{\rm{erfc}}\left(\frac{1}{\bar{\gamma}_{1}}{+}\frac{\gamma}{2\bar{\gamma}_{2}\sigma_{2}^{2}}\right)}{l!2^{l}\left(\gamma\sigma_{2}^{2}{+}\frac{2\bar{\gamma}_{2}\sigma_{2}^{4}}{\bar{\gamma}_{1}}\right)^{l}}\right]$
(12)

$\displaystyle I_{1}=$
$\displaystyle\frac{\sqrt{\pi}}{2}\left[1+{\rm{erf}}(\frac{\mu_{1}}{\sqrt{2\sigma_{1}^{2}}})-2{\rm{erf}}(\frac{\mu_{1}}{C_{2}})\right]+\sum_{i=1}^{4}\frac{S_{i}}{2}\sqrt{\frac{2\pi\sigma_{1}^{2}}{2\sigma_{1}^{2}+T_{i}C_{2}^{2}}}\exp(\frac{T_{i}^{2}C_{2}^{2}\mu_{1}^{2}}{4\sigma_{1}^{4}+2T_{i}C_{2}^{2}\sigma_{1}^{2}}-\frac{T_{i}\mu_{1}^{2}}{2\sigma_{1}^{2}})$
$\displaystyle\times\left[2{\rm{erf}}\left(\sqrt{\frac{\mu_{1}^{2}}{C_{2}^{2}}+\frac{T_{i}^{2}\mu_{1}^{2}}{2\sigma_{1}^{2}}}-\frac{T_{i}C_{2}\mu_{1}}{\sqrt{4\sigma_{1}^{4}+2T_{i}C_{2}^{2}\sigma_{1}^{2}}}\right)-{\rm{erfc}}\left(\frac{T_{i}C_{2}\mu_{1}}{\sqrt{4\sigma_{1}^{4}+2T_{i}C_{2}^{2}\sigma_{1}^{2}}}\right)\right]$
(16)

$\displaystyle I_{2}=$
$\displaystyle\frac{1}{2}\sqrt{\frac{\pi}{2C_{3}^{2}+C_{1}^{2}}}\exp\left(-\frac{\mu_{1}^{2}C_{1}^{2}}{2C_{2}^{2}C_{3}^{2}+C_{1}^{2}C_{2}^{2}}\right)\left(1+\rm{erf}\left(\frac{C_{1}^{2}\mu_{1}}{\sqrt{4C_{2}^{2}C_{3}^{4}+2C_{1}^{2}C_{2}^{2}C_{3}^{2}}}\right)\right)$
$\displaystyle-\sum_{i=1}^{4}\frac{S_{i}}{\sqrt{2}}\sqrt{\frac{\pi\sigma_{1}^{2}}{4C_{3}^{2}\sigma_{1}^{2}+2C_{1}^{2}\sigma_{1}^{2}+T_{i}C_{4}^{2}}}\exp\left(C_{5}-\frac{\mu_{1}^{2}C_{1}^{2}}{2C_{2}^{2}C_{3}^{2}}-\frac{\mu_{1}^{2}T_{i}}{4C_{3}^{2}\sigma_{1}^{2}}\right)\left(1+{\rm{erf}}\left(\sqrt{C_{5}}\right)\right)$
(17)

$\displaystyle I_{3}=$
$\displaystyle\frac{1}{2}\sqrt{\frac{\pi}{2C_{3}^{2}+C_{1}^{2}}}\exp\left(-\frac{\mu_{1}^{2}C_{1}^{2}}{2C_{2}^{2}C_{3}^{2}+C_{1}^{2}C_{2}^{2}}\right)\left[{\rm{erfc}}\left(\frac{\mu_{1}C_{1}^{2}}{\sqrt{4C_{2}^{2}C_{3}^{4}+2C_{1}^{2}C_{2}^{2}C_{3}^{2}}}\right)-2{\rm{erf}}\left(\sqrt{C_{7}}-\frac{\mu_{1}C_{1}^{2}}{\sqrt{4C_{2}^{2}C_{3}^{4}+2C_{1}^{2}C_{2}^{2}C_{3}^{2}}}\right)\right]$
$\displaystyle+\sum_{i=1}^{4}\frac{S_{i}}{\sqrt{2}}\sqrt{\frac{\pi\sigma_{1}^{2}}{4C_{3}^{2}\sigma_{1}^{2}+2C_{1}^{2}\sigma_{1}^{2}+T_{i}C_{2}^{2}}}\exp\left(C_{6}-\frac{C_{1}^{2}\mu_{1}^{2}}{2C_{2}^{2}C_{3}^{2}}-\frac{T_{i}\mu_{1}^{2}}{4C_{3}^{2}\sigma_{1}^{2}}\right)\left[2{\rm{erf}}\left(\sqrt{C_{7}}-\sqrt{C_{6}}\right)-{\rm{erfc}}\left(\sqrt{C_{6}}\right)\right]$
(18)

###  II-B Analysis of User 2 SNR

Assuming that the signal of U1 can be successfully decoded (represented by
$\hat{x}$), the received signal can now be expressed as

$\displaystyle y_{2}$ $\displaystyle=y_{1}{\rm{e}}^{j\hat{x}\pi}$
$\displaystyle=\sqrt{\frac{E_{s}}{L_{2}}}\sum_{i=1}^{N}\alpha_{i}\beta_{i}{\rm{e}}^{jw_{m}}x{\rm{e}}^{j\hat{x}\pi}+\sqrt{\frac{E_{s}}{L_{1}}}\varepsilon
x{\rm{e}}^{j\hat{x}\pi}+n{\rm{e}}^{j\hat{x}\pi}$
$\displaystyle=\sqrt{\frac{E_{s}}{L_{2}}}\sum_{i=1}^{N}\alpha_{i}\beta_{i}{\rm{e}}^{jw_{m}}+\sqrt{\frac{E_{s}}{L_{1}}}\varepsilon+n{\rm{e}}^{j\hat{x}\pi}.$
(6)

Thus, the signal of U2 can be regarded as a biased BPSK signal with an initial
phase of $\frac{\pi}{2}$, and an offset angle of $\frac{\pi}{2}-w_{m}$. Then,
the SNR of U2 can be written as

$\displaystyle\gamma_{2}=\frac{\frac{\bar{\gamma}}{L_{2}}\left(\sum_{i=1}^{N}\alpha_{i}\beta_{i}\cos(\frac{\pi}{2}-w_{m})\right)^{2}}{\frac{\bar{\gamma}}{L_{1}}\left|\varepsilon\right|^{2}+1}\
=\frac{\bar{\gamma}_{2}R_{2}^{2}}{\bar{\gamma}_{1}\left|\varepsilon\right|^{2}+1}.$
(7)

where $R_{2}=\sum_{i=1}^{N}\alpha_{i}\beta_{i}\cos(\frac{\pi}{2}-w_{m})$.
Similar to $R_{1}$, and relying again on the CLT, $R_{2}$ is assumed to follow
the Gaussian distribution with mean $\mu_{2}$ and variance $\sigma_{2}^{2}$.
The PDF of $R_{2}$ can be written as

$f_{R_{2}}(r)=\frac{1}{\sqrt{2\pi\sigma_{2}^{2}}}\exp\left(-\frac{\left(r-\mu_{2}\right)^{2}}{2\sigma_{2}^{2}}\right),$
(8)

where
$\mu_{2}=\frac{N\Gamma\left(\frac{3}{2}\right)\sqrt{\pi}}{2\sqrt{1+K}e^{K}}{}_{1}F_{1}(\frac{3}{2},1,K)\cos(\frac{\pi}{2}-w_{m})$,
and
$\sigma_{2}^{2}=N\left[\frac{{}_{1}{\rm{F}}_{1}(2,1,K)}{(1+K)e^{K}}-\left(\frac{\mu_{1}}{N\cos(\frac{\pi}{2}-w_{m})}\right)^{2}\right]\cos^{2}(\frac{\pi}{2}-w_{m})$.

Let $Y=\bar{\gamma}_{1}\left|\varepsilon\right|^{2}+1$, then the PDF of $Y$
can be readily written as
$f_{Y}(y)=\frac{1}{\bar{\gamma}_{1}}e^{-\frac{y-1}{\bar{\gamma}_{1}}}$. Thus
the CDF of $\gamma_{2}$ can be calculated as

$\displaystyle F_{\gamma_{2}}(\gamma)=$
$\displaystyle\Pr\left(\frac{\bar{\gamma}_{2}R_{2}^{2}}{Y}<\gamma\right)$
$\displaystyle=$ $\displaystyle
1-Q_{\frac{1}{2}}\left(\frac{\mu_{2}}{\sqrt{\sigma_{2}^{2}}},\sqrt{\frac{\gamma}{\sigma_{2}^{2}\bar{\gamma}_{2}}}\right)$
$\displaystyle+\sqrt{\frac{\bar{\gamma}_{1}\gamma}{2\sigma_{2}^{2}\bar{\gamma}_{2}+\bar{\gamma}_{1}\gamma}}\exp\left(\frac{1}{\bar{\gamma}_{1}}-\frac{\mu_{2}^{2}\bar{\gamma}_{2}}{\bar{\gamma}_{1}\gamma+2\sigma_{2}^{2}\bar{\gamma}_{2}}\right)$
$\displaystyle\times
Q_{\frac{1}{2}}\left(\sqrt{\frac{\mu_{2}^{2}\gamma}{\sigma_{2}^{2}\gamma+\frac{2\bar{\gamma}_{2}\sigma_{2}^{4}}{\bar{\gamma}_{1}}}},\sqrt{\frac{2}{\bar{\gamma}_{1}}+\frac{\gamma}{\bar{\gamma}_{2}\sigma_{2}^{2}}}\right),$
(9)

where $Q_{\frac{1}{2}}(\cdot,\cdot)$ is the Marcum Q-function [10].

Using (9) to calculate the average BER is difficult, however, from [11, Eq.
(16)], we have

$Q_{\frac{1}{2}}\left(a,b\right)\approx\sum_{l=0}^{L}\frac{a^{2l}\Gamma(l+\frac{1}{2},\frac{b^{2}}{2})}{l!\Gamma(l+\frac{1}{2})2^{l}{\rm{e}}^{\frac{a^{2}}{2}}},$
(10)

where $\Gamma(\cdot,\cdot)$ is the incomplete gamma function [10]. Then, with
the help of [12, Eq. (06.06.03.0005.01)], the CDF of $\gamma_{2}$ can be
written as

$\displaystyle F_{\gamma_{2}}(\gamma)\approx A_{1}+A_{2},$ (11)

where
$A_{1}=1-\sum_{l=0}^{L}\frac{(\frac{\mu_{2}}{\sigma_{2}})^{2l}\Gamma(l+\frac{1}{2},\frac{\gamma}{2\bar{\gamma}_{2}\sigma_{2}^{2}})}{l!\Gamma(l+\frac{1}{2})2^{l}{\rm{exp}}({\frac{\mu^{2}}{2\sigma_{2}^{2}}})}$,
$A_{2}$ is shown at the bottom of this page, and $\rm{erfc}(\cdot)$ is the
complementary error function [10].

$\displaystyle I_{4}=$
$\displaystyle\Gamma\left(\frac{1}{2}\right)-\sum_{l=0}^{L}\left(\frac{\left(\frac{\mu_{2}^{2}}{\sigma_{2}^{2}}\right)^{l}\sqrt{2\bar{\gamma}_{2}\sigma_{2}^{2}}}{2^{l-1}\Gamma(l+\frac{1}{2})e^{\frac{\mu_{2}^{2}}{2\sigma_{2}^{2}}}(1+2\bar{\gamma}_{2}\sigma_{2}^{2})^{1+l}}{}_{2}{\rm{F}}_{1}\left(1,l+1;\frac{3}{2};\frac{2\bar{\gamma}_{2}\sigma_{2}^{2}}{1+2\bar{\gamma}_{2}\sigma_{2}^{2}}\right)\right)$
(21)

$\displaystyle I_{5}=$
$\displaystyle\exp\left(\frac{1}{\bar{\gamma}_{1}}-\frac{\mu_{2}^{2}}{2\sigma_{2}^{2}}\right)\sum_{l=0}^{L}\sum_{i=1}^{4}\frac{S_{i}(\frac{\mu_{2}}{\sigma_{2}})^{2l}}{2^{l}(1+\frac{T_{i}}{2\bar{\gamma}_{2}\sigma_{2}^{2}})^{3/4}(\frac{2\bar{\gamma}_{2}\sigma_{2}^{2}}{\bar{\gamma}_{1}})^{1/4}}\exp\left(\frac{\bar{\gamma}_{2}\sigma_{2}^{2}}{\bar{\gamma}_{1}}-\frac{T_{i}}{2\bar{\gamma}_{1}}\right){\rm{W}}_{-l-\frac{1}{4},-\frac{1}{4}}\left(\frac{2\bar{\gamma}_{2}\sigma_{2}^{2}}{\bar{\gamma}_{1}}+\frac{T_{i}}{\bar{\gamma}_{1}}\right)$
(22)

$\displaystyle I_{6}=$
$\displaystyle\exp\left(\frac{1}{\bar{\gamma}_{1}}-\frac{\mu_{2}^{2}}{2\sigma_{2}^{2}}\right)\sum_{l=0}^{L}\sum_{k=1}^{l-1}\frac{\mu_{2}^{2l}(\bar{\gamma}_{2}\sigma_{2}^{2})^{k/2}\left(\frac{1}{2}-l\right)_{l-k-1}(-1)^{k+l-1}}{\Gamma(l+\frac{1}{2})2^{l}\sigma_{2}^{2l}(\bar{\gamma}_{2}\bar{\gamma}_{1}\sigma_{2}^{2})^{k+\frac{1}{2}}\sqrt{\bar{\gamma}_{1}/2}}\exp\left(\frac{\bar{\gamma}_{2}\sigma_{2}^{2}}{\bar{\gamma}_{1}}-\frac{1}{2\bar{\gamma}_{1}}\right)\left(\frac{2}{\bar{\gamma}_{1}}+\frac{1}{\bar{\gamma}_{2}\bar{\gamma}_{1}\sigma_{2}^{2}}\right)^{-k/2-1}$
$\displaystyle\times{\rm{W}}_{\frac{k-2l}{2},\frac{-k-1}{2}}\left(\frac{2\bar{\gamma}_{2}\sigma_{2}^{2}}{\bar{\gamma}_{1}}+\frac{1}{\bar{\gamma}_{1}}\right)$
(23)

##  III Performance Analysis

In this section, we analyze the performance of the proposed scheme by deriving
closed-form expressions for the average BER. For different binary modulation
schemes, a unified average BER expression is given by [13]

$P_{e}=\frac{q^{p}}{2\Gamma(p)}\int_{0}^{\infty}\exp(-q\gamma)\gamma^{p-1}F_{\gamma}(\gamma)d\gamma,$
(13)

where $F_{\gamma}(\gamma)$ is the CDF of $\gamma$, and the parameters $p$ and
$q$ are modulation schemes dependent. In this work, we consider BPSK
modulation, and hence we use $p=\frac{1}{2}$ and $q$ = 1.

####  III-1 Average BER of User 1

From (5) and (13), $P_{e1}$ can be formulated as

$\displaystyle P_{e1}=(I_{1}-I_{2}-I_{3})/(2\Gamma(0.5)),$ (14)

where $I_{1}$, $I_{2}$ and $I_{3}$ are derived next. Using the expression in
(5) to evaluate $P_{e1}$ is difficult. Hence, we opt to utilize an alternative
expression for the erf function [14] as

$\begin{split}\rm{erf}(x)\approx\left\\{\begin{matrix}1-\sum_{i=1}^{4}S_{i}{\rm{e}}^{-T_{i}x^{2}}x\geq
0\\\
-1+\sum_{i=1}^{4}S_{i}{\rm{e}}^{-T_{i}x^{2}}x<0\end{matrix}\right.\end{split}$
(15)

where $S=[\frac{1}{8},\frac{1}{4},\frac{1}{4},\frac{1}{4}]$ and
$T=[1,2,\frac{20}{3},\frac{20}{17}]$.

Then, with the help of [10, Eq. (2.33.1)], closed form expressions for
$I_{1}$, $I_{2}$, and $I_{3}$ are shown at the bottom of the previous page,
where
$C_{5}=\frac{\left(2C_{1}^{2}\mu_{1}\sigma_{1}^{2}-T_{i}C_{2}C_{4}\mu_{1}\right)^{2}}{16C_{2}^{2}C_{3}^{4}\sigma_{1}^{4}+8C_{1}^{2}C_{2}^{2}C_{3}^{2}\sigma_{1}^{4}+4T_{i}C_{2}^{2}C_{3}^{2}C_{4}^{2}\sigma_{1}^{2}}$,
$C_{6}=\frac{\left(2C_{1}^{2}\mu_{1}\sigma_{1}^{2}+T_{i}C_{2}^{2}\mu_{1}\right)^{2}}{16C_{2}^{2}C_{3}^{4}\sigma_{1}^{4}+8C_{1}^{2}C_{2}^{2}C_{3}^{2}\sigma_{1}^{4}+4T_{i}C_{2}^{4}C_{3}^{2}\sigma_{1}^{2}}$,
and
$C_{7}=\frac{\mu_{1}^{2}}{C_{2}^{2}}+\frac{\mu_{1}^{2}C_{1}^{2}}{2C_{2}^{2}C_{3}^{2}}$.

####  III-2 Average BER of User 2

As the decoding of U2 signal follows that of U1, it is usually difficult to
ensure that the decoded signal is completely correct. Therefore, after
processing the received signal, we might get some inverted information bits of
U2. Then, the practical average BER of U2 can be expressed as

$\displaystyle P_{e2}=P_{e2}^{ideal}(1-P_{e1})+P_{e1}(1-P_{e2}^{ideal}),$ (19)

where $P_{e2}^{ideal}$ denotes U2’s average BER with ideal conditions (i.e.
assuming the decoded U1’s data is completely correct). From (11) and (13), and
using [10, Eq. (6.455)], the ideal U2’s average BER can be calculated as

$\displaystyle P_{e2}^{ideal}=(I_{4}+I_{5}+I_{6})/(2\Gamma(0.5)),$ (20)

where $I_{4}$, $I_{5}$ and $I_{6}$ can be shown to be given as in (21), (22)
and (23), where ${}_{2}F_{1}(\cdot)$ is the Gauss hypergeometric function, and
$W_{a,b}(\cdot)$ is the Whittaker hypergeometric function [10].

Figure 2: The constellation diagram of U1 and U2. Figure 3: Average BER of U1
versus $w_{m}$. Figure 4: Average BER of U2 versus $w_{m}$. Figure 5: Average
BER of U1 versus $\bar{\gamma}$ with and without U2.

##  IV Numerical and Simulation Results

In this section, we present some numerical results to verify our analysis. The
parameters used in the figures are $k$ = 3, $p=\frac{1}{2}$, $q=1$, $L_{1}=20$
dB, and $L_{3}=30$ dB.

In Fig. 2 we plot the constellation diagram of U1 and U2 where
$\bar{\gamma}=20$ dB, $N=50$ and $w_{m}=\pi/4$. It can be deduced from Fig. 2
that superimposing U2’s data on that of U1 causes the constellation diagram of
U1’s to shift based on the value of $w_{m}$ used. This causes the BER of U1 to
increase, simply because the separation of the two constellation points is
lower at this time. From the constellation of U2, it can be deduced that the
processed signal is similar to a BPSK signal where the initial phase is
$-\pi/2$.

In Fig. 3, we plot the average BER of U1 as a function of $w_{m}$. When
$w_{m}=0$, this is equivalent to the case where there is no U2 and only the
data of U1 is transmitted in the uplink, and its constellation is a pure BPSK
one. As we increase $w_{m}$, the system needs to utilize the spectrum
resources of U1 to transmit the signal of U2. Consequently, U1’s average BER
increases. When $w_{m}$ reaches $\pi/2$, it leads to the lowest data accuracy
of U1. In addition, and as expected, it can be seen from Fig. 3 that
increasing $N$ can bring performance improvements to U1. This means that we
can reduce the negative effect of $w_{m}$ by increasing $N$.

In Fig. 4, we plot the average BER of U2 versus $w_{m}$. It is clear from the
figure that by increasing $w_{m}$, the average BER of U2 decreases first and
then increases. When $w_{m}$ is small, $P_{e1}$ is small, and hence the system
performance is mainly determined by $P_{e2}^{ideal}$. Similar to the
observations in Fig. 3, increasing $w_{m}$ causes $P_{e1}$ to increase, and
when $w_{m}$ is large, one cannot get a clean U2 signal due to the large
average BER of U1. Hence, the system performance is dominated in this case by
$P_{e1}$. This means that the choice of $w_{m}$ is critical to ensure
reasonable performance of both users.

In Fig. 5, we plot the average BER of U1 versus $\bar{\gamma}$ with and
without U2. As expected, increasing the average SNR leads to decreasing the
average BER. In addition, the effect of having U2 on the average BER of U1 is
clear from the figure. The figure shows also the performance of U1 when using
the direct link only. It is clear that the incentive given to U1 through the
RIS usage to allow the superposition of U2 data is worth it to U1 from
performance point of view. Finally, we observe a close match between the
derived expressions and the simulation results which confirms the accuracy of
the analytical expressions.

##  V Conclusion

In this letter, we proposed an RIS-assisted multi-user uplink communication
system employing a novel modulation scheme. More specifically, we derived the
analytical expression of average BER and tight approximation on the CDF of the
received SNR of the case of two users sharing the same spectrum with the help
of the RIS. Numerical results show that we can obtain U2’s data with higher
accuracy while ensuring the accuracy of U1’s data by setting an appropriate
phase shift and large enough number of surface elements.

## References

  * [1] E. Basar, M. Di Renzo, J. de Rosny, M. Debbah, M.-S. Alouini, and R. Zhang, “Wireless communications through reconfigurable intelligent surfaces,” _IEEE Access_ , vol. 7, pp. 116753–116773, Sep. 2019.
  * [2] S. Abeywickrama, R. Zhang, Q. Wu, and C. Yuen, “Intelligent reflecting surface: practical phase shift model and beamforming optimization,” _IEEE Trans. Commun._ , vol. 68, no. 9, pp. 5849-5863, Sep. 2020.
  * [3] Q. Wu, and R. Zhang, “Beamforming optimization for wireless network aided by intelligent reflecting surface with discrete phase shifts,” _IEEE Trans. Commun._ , vol. 68, no. 3, pp. 1838-1851, Mar. 2020.
  * [4] L. Yang, Y. Yang, M. O. Hasna, and M. Alouini, “Coverage, probability of SNR gain, and DOR analysis of RIS-aided communication systems,” _IEEE Wireless Commun. Lett._ , vol. 9, no. 8, pp. 1268-1272, Aug. 2020.
  * [5] L. Yang, F. Meng, Q. Wu, D. B. da Costa and M. Alouini, “Accurate closed-form approximations to channel distributions of RIS-aided wireless systems,” _IEEE Wireless Commun. Lett._ , Early Access, DOI: 10.1109/LWC.2020.3010512.
  * [6] X. Hu, J. Wang, and C. Zhong, “Statistical CSI based design for intelligent reflecting surface assisted MISO systems,” _Sci. China Inf. Sci._ , Early Access, DOI: 10.1007/s11432-020-3033-3.
  * [7] L. Yang, J. Yang, W. Xie, M. Hasna, T. Tsiftsis and M. Di Renzo, “Secrecy performance analysis of RIS-aided wireless communication systems,” _IEEE Trans. Veh. Technol._ , Early Access, DOI: 10.1109/TVT.2020.3007521.
  * [8] L. Yang, F. Meng, J. Zhang, M. O. Hasna and M. Di Renzo, “On the performance of RIS-assisted dual-hop UAV communication systems,” _IEEE Trans. Veh. Technol._ , Early Access, DOI: 10.1109/TVT.2020.3004598.
  * [9] C. Huang, A. Zappone, G. C. Alexandropoulos, M. Debbah and C. Yuen, “Reconfigurable intelligent surfaces for energy efficiency in wireless communication,” _IEEE Trans. Wireless Commun._ , vol. 18, no. 8, pp. 4157-4170, Aug. 2019.
  * [10] I. S. Gradshteyn, and I. M. Ryzhik, _Table of integrals, series, and products,_ 7th ed. San Diego, CA, USA: Academic, 2007.
  * [11] P. C. Sofotasios, T. A. Tsiftsis, Y. A. Brychkov, S. Freear, M. Valkama and G. K. Karagiannidis, ”Analytic expressions and bounds for special functions and applications in communication theory,” _IEEE Trans. Inf. Theory._ , vol. 60, no. 12, pp. 7798-7823, Dec. 2014.
  * [12] (2001). Wolfram, Champaign, IL, USA. _The Wolfram functions site,_ [Online]. Available: http://functions.wolfram.com.
  * [13] I. S. Ansari, S. Al-Ahmadi, F. Yilmaz, M. Alouini, and H. Yanikomeroglu, “A new formula for the BER of binary modulations with dual-branch selection over generalized-K composite fading channels,” _IEEE Trans. Commun._ , vol. 59, no. 10, pp. 2654-2658, Oct. 2011
  * [14] D. Sadhwani, and R. Narayan Yadav, “Tighter bounds on the Gaussion Q function and its application in Nakagami-m fading channel,” _IEEE Wireless Commun. Lett._ , vol. 6, no. 5, pp. 574-577, Oct. 2017