# Reconfigurable Intelligent Surface-Based Index Modulation:
A New Beyond MIMO Paradigm for 6G

Ertugrul Basar Manuscript received August 5, 2019; revised December 13, 2019;
accepted January 28, 2020. This work was supported in part by the Scientific
and Technological Research Council of Turkey (TUBITAK) under Grant 117E869,
the Turkish Academy of Sciences (TUBA) GEBIP Programme, and the Science
Academy BAGEP Programme. The associate editor coordinating the review of this
article and approving it for publication was Prof. J. Choi.The author is with
the Communications Research and Innovation Laboratory (CoreLab), Department of
Electrical and Electronics Engineering, Koç University, Sariyer 34450,
Istanbul, Turkey (e-mail: ebasar@ku.edu.tr).Codes available at
https://corelab.ku.edu.tr/tools Color versions of one or more of the figures
in this article are available online at http://ieeexplore.ieee.org.Digital
Object Identifier XXX

###### Abstract

Transmission through reconfigurable intelligent surfaces (RISs), which control
the reflection/scattering characteristics of incident waves in a deliberate
manner to enhance the signal quality at the receiver, appears as a promising
candidate for future wireless communication systems. In this paper, we bring
the concept of RIS-assisted communications to the realm of index modulation
(IM) by proposing RIS-space shift keying (RIS-SSK) and RIS-spatial modulation
(RIS-SM) schemes. These two schemes are realized through not only intelligent
reflection of the incoming signals to improve the reception but also
utilization of the IM principle for the indices of multiple receive antennas
in a clever way to improve the spectral efficiency. Maximum energy-based
suboptimal (greedy) and exhaustive search-based optimal (maximum likelihood)
detectors of the proposed RIS-SSK/SM schemes are formulated and a unified
framework is presented for the derivation of their theoretical average bit
error probability. Extensive computer simulation results are provided to
assess the potential of RIS-assisted IM schemes as well as to verify our
theoretical derivations. Our findings also reveal that RIS-based IM, which
enables high data rates with remarkably low error rates, can become a
potential candidate for future wireless communication systems in the context
of beyond multiple-input multiple-output (MIMO) solutions.

###### Index Terms:

6G, index modulation (IM), reconfigurable intelligent surface (RIS), smart
reflect-array, software-defined surface, space shift keying (SSK), spatial
modulation (SM).

##  I Introduction

As of the first quarter of 2020, the first commercial fifth generation (5G)
wireless networks have been already deployed, in part or as a whole, in
certain countries while the first 5G compatible mobile devices are being
introduced to the market. Although the initial 5G standard, which was
completed in 2018, has brought more flexibility to the physical layer by
exploiting millimeter-waves and multiple orthogonal frequency division
multiplexing (OFDM) numerologies, researchers have already started to explore
the potential of alternative technologies for later releases of 5G. These
technologies include index modulation (IM), non-orthogonal multiple access,
alternative/advanced waveforms, low-cost massive multiple-input multiple-
output (MIMO) variants, terahertz communications, and new antenna
technologies. At the first glance, the future 6G technologies may seem as the
extension of their 5G counterparts [1], however, new user requirements,
completely new applications/use-cases, and new networking trends of 2030 and
beyond may bring more challenging communication engineering problems, which
necessitate radically new communication paradigms in the physical layer [2].

Within the context of unconventional wireless communication paradigms, there
has been a growing interest in controlling the reflection, scattering, and
refraction characteristics of electromagnetic waves, that is, controlling the
propagation, in order to increase the quality of service and/or achievable
rate. IM-based emerging schemes such as media-based modulation [3, 4, 5],
spatial scattering modulation [6], and beam index modulation (IM) [7], use the
variations in the signatures of received signals by exploiting reconfigurable
antennas or scatterers to transmit additional information bits in rich
scattering environments [8]. On the other hand, reconfigurable intelligent
surfaces/walls/reflect-arrays/metasurfaces are smart devices that
intentionally control the propagation environment to boost the signal quality
at the receiver [9, 10, 11].

As a matter of fact, the reconfigurable intelligent surface (RIS)-based
transmission concept, in which the large number of small, low-cost, and
passive elements on an RIS only reflect the incident signal with an adjustable
phase shift without requiring a dedicated energy source for radio frequency
(RF) processing, decoding, encoding, or retransmission, is completely
different from existing MIMO, beamforming, amplify-and-forward relaying, and
backscatter communication paradigms. Inspired by the definition of software-
defined radio, which is given as “radio in which some or all of the physical
layer functions are software defined” and considering the interaction of the
intelligent surface with incoming waves in a software-defined fashion, we may
also use the term of software-defined surface (SDS) for these intelligent
surfaces. In other words, due to the fact that reflection characteristics of
these intelligent surfaces/walls/arrays in the physical layer can be
controlled by a software, they can be termed as SDS.

Transmission through intelligent walls is proposed in one of the early works
by utilizing active frequency-selective surfaces to control the signal
coverage [12]. The promising concept of communications over smart reflect-
arrays with passive reflector elements is proposed in [13] as an alternative
to beamforming techniques that require large number of antennas to focus the
transmitted or received signals. It has been also demonstrated that reflect-
arrays can be used in an effective way to change the phase of incoming signals
during smart reflection without buffering or processing them and the received
signal quality can be enhanced through the adjustment of the phase shift of
each reflector element on the reflect-array. The authors of [14, 15, 16]
considered an RIS-assisted downlink transmission scenario to support multiple
users and focused on the maximization of sum-rate and energy efficiency. The
selection of optimum RIS phases is also investigated and low complexity
algorithms are considered for the formulated non-convex optimization problems.
Recently, the interesting problem of joint active and passive beamforming is
investigated in [17] and [18], and the user’s average received power is
investigated. Even more recently, the researchers focused on RIS architectures
based on sparse channel sensors [19], in which some of the existing RIS units
are active, and RIS-assisted physical-layer security schemes [20]. Finally, we
provided a mathematical framework in [21] for the calculation of average
symbol error probability (SEP) of RIS-assisted systems. Furthermore, we
proposed the promising concept of using the RIS itself as an access point (AP)
by utilizing an unmodulated carrier for intelligent reflection. Interested
readers are referred to [11, 22, 10] and references therein for a recent
overview of RIS empowered communications.

Figure 1: Three conceptual RIS-based IM system realizations: a) IM for source
(S) transmit antennas, b) IM for RIS reflector regions, c) IM for destination
(D) receive antennas.

The emerging IM concept also falls to the category of potential beyond 5G
technologies and has been widely recognized by both academia and industry
during the past few years [23, 24, 8, 25]. Contrary to the traditional
modulation formats, the indices of the available transmit entities, such as
transmit antennas for space modulation techniques [25] and subcarriers for
OFDM with IM [26], are used to convey information for an IM scheme. The
undeniable potential of both IM- and RIS-based communication schemes has been
the main motivation of this study. With this purpose, we investigated three
conceptual RIS-based IM system realizations in Figs. 1(a)-(c), in which we
consider IM for source (S) transmit antennas, RIS regions, and destination (D)
receive antennas, respectively. Since the first concept requires the knowledge
of activated transmit antenna indices of S at RIS for optimum reflection,
i.e., requires an extra signaling link between S and the RIS, and the second
concept reduces the effective gain of the RIS by activating a part of the
available reflectors, i.e., reduces the effective received signal power, we
decided to focus on the third approach in this preliminary work.

In this paper, we propose the visionary concept of RIS-based IM as a potential
beyond MIMO solution by amalgamating the techniques of transmission over RISs
and IM for receive antenna indices to achieve high reliability along with high
spectral efficiency. Different from emerging massive MIMO systems with fully
digital or hybrid beamforming, our design neither requires multiple RF chains
nor analog phase shifters at transmitter/receiver sides and exploits the
inherent randomness of the propagation environment. First, considering the RIS
as an AP [21, 11], we propose RIS-space shift keying (RIS-SSK) and RIS-spatial
modulation (RIS-SM) schemes by exploiting the RIS not only to boost the signal
quality in hostile fading channels but also to realize IM by the selection of
a particular receive antenna index according to the information bits. Second,
we formulate the greedy and maximum likelihood (ML) detectors of both schemes
and investigate their complexity. Third, we present a unified framework for
the calculation of the theoretical error performance of the proposed schemes
and provide useful insights. Finally, extensive computer simulations are given
to assess the potential of the RIS-SSK and RIS-SM schemes.

The rest of the paper is organized as follows. In Section II, we introduce the
system model of RIS-based SSK/SM schemes and formulate their detectors.
Sections III and IV respectively focuses on the error performance of greedy
and ML detectors. Computer simulation results and comparisons are given in
Section V. Finally, conclusions are given in Section VI111Notation: Bold,
lowercase and capital letters are used for column vectors and matrices,
respectively. $(\cdot)^{*}$ $(\cdot)^{\mathrm{T}}$, and $(\cdot)^{\mathrm{H}}$
denote complex conjugation, transposition, and Hermitian transposition,
respectively. The real and imaginary parts of a complex variable $X$ are
denoted by $X_{\Re}$ (or $\Re\left\\{X\right\\}$) and $X_{\Im}$ (or
$\Im\left\\{X\right\\}$), respectively. $\mathrm{det}(\cdot)$ and
$(\cdot)^{-1}$ stand for the determinant and the inverse of a matrix, and
$\mathrm{diag}(\cdot)$ returns a diagonal matrix from a vector. The $n\times
n$ identity matrix is denoted by $\mathbf{I}_{n}$.
$X\sim\mathcal{N}(\mu,\sigma^{2})$ stands for the real Gaussian distribution
of $X$ with mean $\mathrm{E}[X]=\mu$ and variance
$\mathrm{Var}[X]=\sigma^{2}$, while $\mathcal{CN}(0,\sigma^{2})$ represents
circularly symmetric complex Gaussian distribution with variance $\sigma^{2}$.
$P(\cdot)$ stands for the probability of an event. $f_{X}(x)$, $F_{X}(x)$,
$\Psi_{X}(w)$, and $M_{X}(s)$ respectively stand for the probability density
function (PDF), cumulative distribution function (CDF), characteristic
function (CF), and moment generating function (MGF) of a random variable (RV)
$X$. $Q(\cdot)$ is the Gaussian $Q$-function and $j=\sqrt{-1}$. .

Figure 2: RIS-based IM schemes: a) RIS-SSK, b) RIS-SM.

##  II System Model

In this section, we present the system models of the proposed RIS-based SSK
and SM schemes and investigate the problem of optimal RIS phase selection to
achieve the best error performance. We build the proposed schemes on the
concept of RIS-AP introduced in [21, 11], where the RIS, which is a part of
the transmitter, reflects the signals generated by a near RF source in a
deliberate manner to convey information bits. We assume that the RIS is
consisting of $N$ passive and low-cost reflector elements (reconfigurable
meta-surfaces), while D, which lies in the far-field of the RIS and does not
receive transmission from the RF source, is equipped with $n_{R}$ receive
antennas. The wireless fading channel between the $l$th receive antenna of D
and $i$th reflector element is characterized by
$g_{l,i}=\beta_{l,i}e^{-j\psi_{l,i}}$ for $l=1,2,\ldots,n_{R}$ and
$i=1,2,\ldots,N$, and follows $\mathcal{CN}(0,1)$ distribution under the
assumption of flat Rayleigh fading channels [16]. We also assume that all
wireless channels are uncorrelated and perfect channel state information (CSI)
is available at D if it is required by the utilized detector. It is worth
noting that more practical channel modeling and phase shift design [27] along
with effective channel estimation techniques [28] are open research topics
with the context of RISs and the considered model may shed light on
performance limits of future schemes. Particularly, some level of spatial
correlation is likely to be expected among reflector elements and this may
affect the ultimate performance limits. The phase induced by the $i$th
reflector is shown by $\phi_{i}$ for $i=1,2,\ldots,N$. For intelligent
reflection, the RIS has the knowledge of channel phases $\psi_{l,i}$ for all
$l$ and $i$, which is a common assumption in the literature to model the
optimum system behavior [14, 15, 16, 17, 18].

###  II-A RIS-Assisted Space Shift Keying

In this scenario, the unmodulated carrier signal generated by the RF source,
is reflected to D with the aim of maximizing the instantaneous received SNR at
a specific receive antenna, which is selected according to the incoming
$\log_{2}n_{R}$ information bits, as shown in Fig. 2(a). In other words, the
RIS phase terms are adjusted in such a way that the SNR at the target receive
antenna is maximized, while the task of D is to detect the index of its
receive antenna with the maximized instantaneous received SNR.

For this system, the transmitted signal is reflected by an RIS that is made of
$N$ reflecting elements. Ignoring the fading effects between the RF source and
the RIS, the received baseband signal at the $l$th receive antenna of D is
given as

$r_{l}=\sqrt{E_{s}}\left[\sum_{i=1}^{N}g_{l,i}e^{j\phi_{i}}\right]+n_{l},\quad
l\in\left\\{1,\ldots,n_{R}\right\\}$ (1)

where $E_{s}$ is the transmitted signal energy of the unmodulated carrier and
$n_{l}$ is the additive white Gaussian noise (AWGN) sample at the $l$th
receiver, which follows $\mathcal{CN}(0,N_{0})$ distribution. Here, the
reflector phases $\left\\{\phi_{i}\right\\}_{i=1}^{N}$ are adjusted according
to the information bits to maximize the received SNR at a specific receive
antenna. For this purpose, the incoming $\log_{2}n_{R}$ bits specify the index
$m$ of a receive antenna and the RIS adjusts its phases according to this
selected receive antenna as $\phi_{i}=\psi_{m,i}$ for $i=1,2,\ldots,N$. More
specifically, the received instantaneous SNR at the $l$th receive antenna is
expressed as

$\gamma_{l}=\frac{\big{|}\sum_{i=1}^{N}\beta_{l,i}e^{j(\phi_{i}-\psi_{l,i})}\big{|}^{2}E_{s}}{N_{0}}.$
(2)

Considering

$\left|\sum_{i=1}^{N}z_{i}e^{j\xi_{i}}\right|^{2}=\sum_{i=1}^{N}z_{i}^{2}+2\sum_{i=1}^{N}\sum_{k=i+1}^{N}z_{i}z_{k}\cos(\xi_{i}-\xi_{k})$
(3)

which can be maximized by ensuring $\xi_{i}=\xi$ for all $i$, to maximize the
instantaneous SNR at the $m$th receive antenna, we select
$\phi_{i}=\psi_{m,i}$. This results in the following maximum SNR value at the
selected receive antenna:

$\gamma_{m}=\frac{\big{|}\sum_{i=1}^{N}\beta_{m,i}\big{|}^{2}E_{s}}{N_{0}}.$
(4)

Later, we will also show that this selection of RIS phases is optimal in terms
of error performance. We introduce two different detectors for the RIS-SSK
scheme, which are commonly used for existing IM-based systems.

i) Greedy Detector: This simple yet effective detector eliminates the need for
channel estimation at D, that is, performs non-coherent detection, and simply
detects the selected receive antenna as the one with the highest instantaneous
energy:

$\hat{m}=\arg\max_{m}\left|r_{m}\right|^{2}.$ (5)

As seen from (5), this detector does not require CSI, which can be prohibitive
at D for increasing $N$ and $n_{R}$.

In order to shed light on the derivation of the optimum RIS phases, let us
consider the selection of the $m$th receive antenna of D at the RIS and its
erroneous detection as $\hat{m}$. For this case, the corresponding pairwise
error probability (PEP) can be easily expressed from (5) as

$\displaystyle P(m\rightarrow\hat{m})$
$\displaystyle=P(\left|r_{m}\right|^{2}<\left|r_{\hat{m}}\right|^{2})$
$\displaystyle=\\!P\\!\left(\left|\\!\sqrt{E_{s}}\sum_{i=1}^{N}\\!g_{m,i}e^{j\phi_{i}}\\!+\\!n_{m}\right|^{2}\\!\\!\\!<\left|\\!\sqrt{E_{s}}\sum_{i=1}^{N}\\!g_{\hat{m},i}e^{j\phi_{i}}\\!+\\!n_{\hat{m}}\right|^{2}\right)$
(6)

where $\left\\{\phi_{i}\right\\}_{i=1}^{N}$ is determined according to the
$m$th receive antenna (with a predefined method). As seen from (II-A), a
logical selection of $\phi_{i}$’s should minimize this PEP for the selected
receive antenna $m$. The above phase optimization problem can be reformulated
as follows by ignoring the noise terms:

$\min_{\left\\{\phi_{i}\right\\}_{i=1}^{N}}\\!\\!P\left(\left|\sum_{i=1}^{N}\\!\beta_{m,i}e^{j(\phi_{i}-\psi_{m,i})}\right|^{2}\\!\\!\\!<\left|\sum_{i=1}^{N}\\!\beta_{\hat{m},i}e^{j(\phi_{i}-\psi_{\hat{m},i})}\right|^{2}\right)\\!.$
(7)

As seen from (7), even for a specific pair of $m$ and $\hat{m}$, this
optimization is not a straightforward task. Consequently, we aim to maximize
the first term by letting $\phi_{i}=\psi_{m,i}$ for all $i$ to minimize this
probability. We will also show later that this selection maximizes the mean of
the first term in (7), while providing a zero-mean for the second one.

ii) ML Detector: The ML detector of the RIS-SSK scheme considers the received
signals at all receive antennas of D and performs the detection as follows:

$\hat{m}=\arg\min_{m}\sum_{l=1}^{n_{R}}\left|r_{l}-\sqrt{E_{s}}\left[\sum_{i=1}^{N}g_{l,i}e^{j\psi_{m,i}}\right]\right|^{2}.$
(8)

In terms of computational complexity, comparing (5) and (8), we observe that
the ML detector requires not only CSI but also $\sim\\!\\!Nn_{R}^{2}$ real
multiplications (RMs), while the greedy detector requires only $\\!n_{R}$
squared complex modulus operations ($\sim n_{R}$ RMs). As we will show in
later sections, the price paid for this increased complexity can be
compensated with the improved error performance.

###  II-B RIS-Assisted Spatial Modulation

For the RIS-SM scheme, ordinary $M$-ary modulation formats are also considered
at the RF source to further improve the spectral efficiency. As shown in Fig.
2(b), the incoming $\log_{2}n_{R}+\log_{2}M$ information bits are partitioned
into two groups. While the first group of $\log_{2}n_{R}$ bits adjusts the RIS
phases according to the selected receive antenna with index $m$ as done for
the RIS-SSK scheme, i.e., $\phi_{i}=\psi_{m,i}$ for $i=1,2,\ldots,N$, the
second group of $\log_{2}M$ bits is passed to the RF source for the generation
of an amplitude/phase modulated signal through an RF chain. Consequently, the
received signal at the $l$th receive antenna of D is expressed as

$r_{l}=\left[\sum_{i=1}^{N}g_{l,i}e^{j\phi_{i}}\right]x+n_{l},\quad
l\in\left\\{1,\ldots,n_{R}\right\\}$ (9)

where $x$ is the data symbol selected from $M$-QAM/PSK constellations,
$\mathrm{E}[\left|x\right|^{2}]=E_{s}$, and $n_{l}\sim\mathcal{CN}(0,N_{0})$
is the noise term. In a similar way, we propose the greedy and ML detectors of
RIS-SM in the sequel.

i) Greedy Detector: This detector simplifies the receiver design by detecting
the selected receive antenna index and the transmitted symbol in a sequential
fashion. For this purpose, the selected receive antenna is determined similar
to the RIS-SSK scheme: $\hat{m}=\arg\max_{m}\left|r_{m}\right|^{2}$. After the
detection of the selected receive antenna index as $\hat{m}$, this detector
demodulates the transmitted data symbol as

$\hat{x}=\arg\min_{x}\left|r_{\hat{m}}-\left(\sum_{i=1}^{N}\beta_{\hat{m},i}\right)x\right|^{2}.$
(10)

Here, compared to RIS-SSK, an additional (but minor) complexity comes during
the search for the constellation point $x$, however, due to disjoint detection
of $m$ and $x$, the overall complexity still linearly increases with $n_{R}$
and $M$. Furthermore, this detector requires only channel amplitudes for
detection. It is worth noting that for constant-envelope constellations such
as $M$-PSK, the transmitted symbol can be detected even without channel
amplitudes since $\sum_{i=1}^{N}\beta_{\hat{m},i}$ is a real variable:

$\hat{x}=\arg\min_{x}\left|r_{\hat{m}}-x\right|^{2}.$ (11)

ii) ML Detector: This detector performs a joint search for the selected
receive antenna index $m$ and the transmitted data symbol $x$ by considering
all received signals as

$(\hat{m},\hat{x})=\arg\min_{(m,x)}\sum_{l=1}^{n_{R}}\left|r_{l}-\left[\sum_{i=1}^{N}g_{l,i}e^{j\psi_{m,i}}\right]x\right|^{2}.$
(12)

As seen from (12), the ML detector of the RIS-SM scheme requires the full CSI
along with $\sim\\!(N+M)n_{R}^{2}$ RMs to make a joint decision on $(m,x)$,
while its greedy detector requires only $\sim(n_{R}+M)$ RMs.

Finally, it is worth noting that for large $N$ and $n_{R}$, the greedy
detector would be a more practical choice since it does not require dedicated
channel estimation for D.

##  III Greedy Detection: Performance Analysis

In this section, we investigate the theoretical bit error probability (BEP) of
the proposed RIS-SSK and RIS-SM schemes in the presence of greedy detection.
We also provide useful insights regarding the asymptotic behavior of the
proposed schemes with this type of detection.

###  III-A Performance of RIS-SSK

Based on the detection rule given in (5), the corresponding PEP is given in
(II-A) for the erroneous detection of the selected receive antenna index $m$
as $\hat{m}$. Considering $\phi_{i}=\psi_{m,i}$ for $i=1,2,\ldots,N$, (II-A)
simplifies to

$\displaystyle
P(m\rightarrow\hat{m})=P\\!\left(\left|\sqrt{E_{s}}B+n_{m}\right|^{2}\\!\\!<\\!\left|\\!\sqrt{E_{s}}\hat{B}+n_{\hat{m}}\right|^{2}\right)$
(13)

where $B=\sum_{i=1}^{N}\\!\beta_{m,i}$ and
$\hat{B}=\sum_{i=1}^{N}\beta_{\hat{m},i}e^{j(\psi_{m,i}-\psi_{\hat{m},i})}$.
Here, we resort to the central limit theorem (CLT) under the assumption of
$N\gg 1$ for the calculation of this PEP. Under the CLT, $B$ and $\hat{B}$
follow Gaussian distribution regardless of the distributions of their
components. Specifically, since $\beta_{m,i}$ is a Rayleigh distributed RV
with $\mathrm{E}[\beta_{m,i}]=\sqrt{\pi}/2$ and
$\mathrm{Var}[\beta_{m,i}]=(4-\pi)/4$, we have
$B\sim\mathcal{N}(N\sqrt{\pi}/2,N(4-\pi)/4)$. Since $\psi_{m,i}$ and
$\psi_{\hat{m},i}$ are independent and uniformly distributed in $(0,2\pi)$,
the distribution of $\bar{\psi_{i}}=\psi_{m,i}-\psi_{\hat{m},i}$ is obtained
as

$f_{\bar{\psi_{i}}}(x)=\begin{cases}\frac{1}{2\pi}\left(1+\frac{x}{2\pi}\right),\quad-2\pi<x<0\\\
\frac{1}{2\pi}\left(1-\frac{x}{2\pi}\right),\quad\quad\,\,0<x<2\pi.\end{cases}$
(14)

Then defining $\hat{B}_{i}=\beta_{\hat{m},i}e^{j\bar{\psi_{i}}}$, we have
$\mathrm{E}[\hat{B}_{i}]=0$ and
$\mathrm{Var}[(\hat{B}_{i})_{\Re}]=\mathrm{Var}[(\hat{B}_{i})_{\Im}]=0.5$, due
to the symmetry of cosine and sine functions. As a result, we obtain
$\hat{B}\sim\mathcal{CN}(0,N)$ with independent and identically distributed
(iid) real and imaginary parts. In light of this information, we obtain

$\displaystyle(\sqrt{E_{s}}B+n_{m})_{\Re}$
$\displaystyle\sim\mathcal{N}\big{(}\tfrac{N\sqrt{\pi
E_{s}}}{2},\tfrac{N(4-\pi)E_{s}}{4}+\tfrac{N_{0}}{2}\big{)}$
$\displaystyle(\sqrt{E_{s}}B+n_{m})_{\Im}$
$\displaystyle\sim\mathcal{N}\left(0,\tfrac{N_{0}}{2}\right)$
$\displaystyle(\sqrt{E_{s}}\hat{B}+n_{\hat{m}})_{\Re}$
$\displaystyle\sim\mathcal{N}\left(0,\tfrac{NE_{s}+N_{0}}{2}\right)$
$\displaystyle(\sqrt{E_{s}}\hat{B}+n_{\hat{m}})_{\Im}$
$\displaystyle\sim\mathcal{N}\left(0,\tfrac{NE_{s}+N_{0}}{2}\right).$ (15)

Considering (III-A), we may re-express (13) as

$P(m\rightarrow\hat{m})=P(Y<0)=P(Y_{1}+Y_{2}-Y_{3}<0)$ (16)

where $Y_{1}=(\sqrt{E_{s}}B+n_{m})_{\Re}^{2}$ is non-central chi-square
$(\chi^{2})$ RV with one degree of freedom,
$Y_{2}=(\sqrt{E_{s}}B+n_{m})_{\Im}^{2}$ is a central $\chi^{2}$ RV with one
degree of freedom, and
$Y_{3}=\big{|}\\!\sqrt{E_{s}}\hat{B}+n_{\hat{m}}\big{|}^{2}$ is a central
$\chi^{2}$ RV with two degrees of freedom. Due to the complexity of the
distribution of $Y_{1}+Y_{2}$ [29, Eq. (5.45)], we use the Gil-Pelaez’s
inversion formula [30, Eq. (4.4.1)]

$F_{Y}(y)=\frac{1}{2}-\int_{0}^{\infty}\frac{\Im\left\\{e^{-jwy}\Psi_{Y}(w)\right\\}}{w\pi}dw$
(17)

where $F_{Y}(y)=P(Y\leq y)$ is the CDF and $\Psi_{Y}(w)=\mathrm{E}[e^{jwY}]$
is the CF of $Y$. Since the CF of the sum of independent RVs (thanks to zero-
mean RVs in (III-A), we satisfy this condition) is the multiplication of their
individual CFs, we obtain
$\Psi_{Y}(w)=\Psi_{Y_{1}}(w)\Psi_{Y_{2}}(w)\Psi_{(-Y_{3})}(w)$. Consider the
generic CF of a non-central $\chi^{2}$ RV $X$ with $n$ degrees of freedom,
which is given as

$\Psi_{X}(w)=\left(\frac{1}{1-2jw\sigma^{2}}\right)^{n/2}\exp\left(\frac{jw\mu^{2}}{1-2jw\sigma^{2}}\right)$
(18)

where $X=\sum_{k=1}^{n}X_{k}^{2}$ and
$X_{k}\sim\mathcal{N}(\mu_{k},\sigma^{2})$ and
$\mu^{2}=\sum_{k=1}^{n}\mu_{k}^{2}$. It is worth noting that for a central
$\chi^{2}$ RV, we have $\mu=0$ in (18). Substituting the values of (III-A) in
(18) for $Y_{1},Y_{2}$, and $Y_{3}$, considering
$\Psi_{(-Y_{3})}(w)=\Psi_{(Y_{3})}(-w)$, and evaluating the integral in (17)
numerically222For numerical integration, the infinity in the upper limit of
the integral in (17) is replaced by $10^{3}$ to avoid numerical calculation
errors. for $y=0$, we obtain the corresponding exact PEP, i.e.,
$P(m\rightarrow\hat{m})=F_{Y}(0)$.

To gain further insights, considering the fact that
$\mathrm{E}[Y_{1}]\gg\mathrm{E}[Y_{2}]$ for large $N$, the PEP in (16) can be
upper bounded by

$P(m\rightarrow\hat{m})<P(Y_{1}-Y_{3}<0).$ (19)

Defining $\tilde{Y}=Y_{1}-Y_{3}$ and considering the PDF of the difference of
a non-central and a central $\chi^{2}$ RV [29, Eq. (4.35)], we obtain

$P(m\rightarrow\hat{m})<\\!\int_{-\infty}^{0}\\!\\!\\!\\!f_{\tilde{Y}}(\tilde{y})d\tilde{y}=\exp\left(\frac{-m_{1}^{2}}{2(\sigma_{1}^{2}+\sigma_{2}^{2})}\right)\sqrt{\frac{\sigma_{2}^{2}}{\sigma_{1}^{2}+\sigma_{2}^{2}}}$
(20)

where $m_{1}=\frac{N\sqrt{\pi E_{s}}}{2}$,
$\sigma_{1}^{2}=\frac{N(4-\pi)E_{s}}{4}+\frac{N_{0}}{2}$ and
$\sigma_{2}^{2}=\frac{NE_{s}+N_{0}}{2}$. Simple manipulations give

$P(m\rightarrow\hat{m})<\left(\frac{1+\frac{NE_{s}}{N_{0}}}{2+\frac{NE_{s}(6-\pi)}{2N_{0}}}\right)^{\\!\frac{1}{2}}\exp\left(\frac{\frac{-N^{2}\pi
E_{s}}{N_{0}}}{8+\frac{2NE_{s}(6-\pi)}{N_{0}}}\right).$ (21)

According to (21), for the SNR range of interest $\frac{NE_{s}}{N_{0}}\ll 10$
(although the SNR ($E_{s}/N_{0}$) is relatively small, considerably low BEP
values are possible for RIS-based systems in this region with increasing $N$
[21]), we obtain

$P(m\rightarrow\hat{m})\propto\exp\left(\frac{-N^{2}\pi E_{s}}{8N_{0}}\right)$
(22)

which suggests a superior index detection probability for the RIS-SSK scheme
with increasing $N$.

It is important to note that for binary signaling, i.e., $n_{R}=2$, the above
PEP yields the exact BEP. However, for the general case of $n_{R}>2$, we use
the following union bound:

$\displaystyle P_{b}$
$\displaystyle\\!\leq\\!\frac{1}{\log_{2}\\!n_{R}}\\!\sum_{\hat{m}}\\!P(m\rightarrow\hat{m})e(m,\hat{m})\\!=\\!\frac{n_{R}}{2}P(m\rightarrow\hat{m})$
(23)

where $e(m,\hat{m})$ is the Hamming distance between the binary
representations of $m$ and $\hat{m}$. Here, we considered the fact that the
resulting PEP is independent of $m$ and $\hat{m}$, and identical for all pairs
(uniform error probability) and
$\sum_{\hat{m}}e(m,\hat{m})=(n_{R}/2)\log_{2}n_{R}$ for all $m$ due to bit
symmetry. For simplicity, we adopt natural mapping for receive antenna
indices.

Remark 1: We observe from the exact and upper-bounded PEP expressions of (16)
and (22) that the resulting PEP is independent of $n_{R}$ for greedy detection
of RIS-SSK. As seen from (23), doubling $n_{R}$ doubles $P_{b}$ in high SNR.

###  III-B Performance of RIS-SM

To derive the theoretical BEP of the RIS-SM scheme with greedy detection, we
consider the following approximation:

$P_{b}\approx\frac{P_{c}(m)P_{s}}{\log_{2}(Mn_{R})}+0.5P_{e}(m)$ (24)

where $P_{c}(m)$ is the average correct detection probability of the selected
receive antenna with index $m$ for RIS-SM, which is the same for all $m$,
$P_{s}$ is the average symbol error probability (SEP) conditioned on correct
index detection, and $P_{e}(m)=1-P_{c}(m)$ is the erroneous index detection
probability. Here, we followed a conservative approach by assuming that
approximately $50\%$ of the transmitted bits are erroneously detected if the
receive antenna index is erroneously estimated, which is a valid assumption
for IM-based systems due to error propagation. Resorting to the union bound on
error probability with uniform PEP values, we obtain

$P_{e}(m)\leq\sum_{\hat{m}=1,\hat{m}\neq
m}^{n_{R}}\\!\\!\bar{P}(m\rightarrow\hat{m})=(n_{R}-1)\bar{P}(m\rightarrow\hat{m})$
(25)

where $\bar{P}(m\rightarrow\hat{m})$ is the PEP associated with index
detection averaged over all data symbols for RIS-SM. From (25), $P_{c}(m)$ is
obtained as $P_{c}(m)\geq 1-(n_{R}-1)\bar{P}(m\rightarrow\hat{m})$.

Assuming that $x$ is transmitted, the PEP for erroneous detection of the
selected receive antenna index $m$ as $\hat{m}$ is given as follows
considering (5), which is also valid for RIS-SM:

$\displaystyle
P(m\rightarrow\hat{m}\\!\left.\right|\\!x)=P\\!\left(\big{|}Bx+n_{m}\big{|}^{2}<\big{|}\hat{B}x+n_{\hat{m}}\big{|}^{2}\right).$
(26)

Here, $B$ and $\hat{B}$ are as defined in (13). In what follows, we calculate
(26) for different constellations.

i) BPSK: For this case, we have $x\in\left\\{\pm\sqrt{E_{s}}\right\\}$ and we
obtain the same PEP derived from (17) or (21) for RIS-SSK, which is
independent of $x$.

ii) $M$-QAM: For this case, we have $x=x_{\Re}+jx_{\Im}$ with
$\mathrm{E}[\left|x\right|^{2}]=E_{s}$, and we may express the corresponding
PEP as

$P(m\rightarrow\hat{m}\\!\left.\right|\\!x)=P(B_{1}^{2}+B_{2}^{2}-B_{3}^{2}-B_{4}^{2}<0)=P(D<0)$
(27)

where $B_{1}=(Bx+n_{m})_{\Re}$, $B_{2}=(Bx+n_{m})_{\Im}$,
$B_{3}=(\hat{B}x+n_{\hat{m}})_{\Re}$, and
$B_{4}=(\hat{B}x+n_{\hat{m}})_{\Im}$, and all follow Gaussian distribution.
Unfortunately, $B_{1}$ and $B_{2}$ are correlated through $B$ due to non-zero
values of $x_{\Re}$ and $x_{\Im}$, and we have to express $D$ in the quadratic
form of Gaussian RVs to derive its statistics. Considering
$D=\mathbf{x}^{\mathrm{T}}\mathbf{A}\mathbf{x}$ for
$\mathbf{x}=\begin{bmatrix}B_{1}&B_{2}&B_{3}&B_{4}\end{bmatrix}^{\mathrm{T}}$
and $\mathbf{A}=\mathrm{diag}(\begin{bmatrix}1&1&-1&-1\end{bmatrix})$, the
mean vector and covariance matrix of $\mathbf{x}$ are respectively calculated
as

$\mathbf{m}=\begin{bmatrix}\frac{N\sqrt{\pi}x_{\Re}}{2}&\frac{N\sqrt{\pi}x_{\Im}}{2}&0&0\end{bmatrix}^{\mathrm{T}}$
(28)
$\mathbf{C}=\\!\begin{bmatrix}\frac{N(4-\pi)x_{\Re}^{2}}{4}\\!+\\!\frac{N_{0}}{2}&\sigma_{1,2}&0&0\\\
\sigma_{1,2}&\frac{N(4-\pi)x_{\Im}^{2}}{4}\\!+\\!\frac{N_{0}}{2}&0&0\\\
0&0&\frac{NE_{s}+N_{0}}{2}&0\\\ 0&0&0&\frac{NE_{s}+N_{0}}{2}\end{bmatrix}$
(29)

where $\sigma_{1,2}$ is the covariance of $B_{1}$ and $B_{2}$, given as

$\sigma_{1,2}=\mathrm{E}[B_{1}B_{2}]-\mathrm{E}[B_{1}]\mathrm{E}[B_{2}]=\tfrac{N(4-\pi)x_{\Re}x_{\Im}}{4}.$
(30)

Then the MGF of $D$ can be calculated from [30, Eq. (3.2a.1)] as

$\displaystyle M_{D}(s)$
$\displaystyle=(\det(\mathbf{I}-2s\mathbf{AC}))^{-\frac{1}{2}}$
$\displaystyle\times\exp\left(-\frac{1}{2}\mathbf{m}^{\mathrm{T}}\left[\mathbf{I}-(\mathbf{I}-2s\mathbf{AC})^{-1}\right]\mathbf{C}^{-1}\mathbf{m}\right).$
(31)

The CF of $D$ $(\Psi_{D}(w))$ can be obtained by replacing $s$ by $jw$ in
(III-B). Finally, Gil-Pelaez’s inversion formula (17) can be used to calculate
the PEP.

Due to the symmetry of the real and imaginary parts of $x$ for QPSK ($4$-QAM),
i.e., $x_{\Re}^{2}=x_{\Im}^{2}=E_{s}/2$, it can be proved that the above PEP
is independent of $x$. However, for $M$-QAM with $M>4$,
$P(m\rightarrow\hat{m}\\!\left.\right|\\!x)$ becomes dependent on $x$ and the
average PEP for index detection can be obtained via

$\bar{P}(m\rightarrow\hat{m})=\frac{1}{M}\sum_{x}P(m\rightarrow\hat{m}\\!\left.\right|\\!x).$
(32)

Substitution of (32) in (25) yields an upper bound on the average erroneous
index detection probability $P_{e}(m)$.

For the calculation of the average SEP ($P_{s}$) under the condition of
correct index detection, we may consider the following equivalent signal model
from (9):

$r_{m}=\left[\sum_{i=1}^{N}\beta_{m,i}\right]x+n_{m}=Bx+n_{m},$ (33)

where the MGF of the instantaneous received SNR $\gamma=E_{s}B^{2}/N_{0}$ is
derived in [21, Eq. (19)], as

$M_{\gamma}(s)=\left(\dfrac{1}{1-\frac{sN(4-\pi)E_{s}}{2N_{0}}}\right)^{\\!\frac{1}{2}}\\!\\!\exp\left(\dfrac{\frac{sN^{2}\pi
E_{s}}{4N_{0}}}{1-\frac{sN(4-\pi)E_{s}}{2N_{0}}}\right).$ (34)

Using this MGF, $P_{s}$ can be easily calculated for BPSK and square $M$-QAM
constellations [31], respectively, as

$P_{s}=\frac{1}{\pi}\int_{0}^{\pi/2}M_{\gamma}\left(\frac{-1}{\sin^{2}\\!\eta}\right)d\eta,$
(35) $\displaystyle P_{s}=\,\,$
$\displaystyle\,\frac{4}{\pi}\left(1-\frac{1}{\sqrt{M}}\right)\int_{0}^{\pi/2}M_{\gamma}\left(\frac{-3}{2(M-1)\sin^{2}\\!\eta}\right)d\eta$
$\displaystyle-\frac{4}{\pi}\left(1-\frac{1}{\sqrt{M}}\right)^{2}\int_{0}^{\pi/4}M_{\gamma}\left(\frac{-3}{2(M-1)\sin^{2}\\!\eta}\right)d\eta.$
(36)

Finally, substituting (25) and (35) (or (III-B)) in (24) provides the desired
$P_{b}$.

Remark 2: Our observations indicate that for high SNR, which is relative for
RIS-based schemes, $P_{c}(m)P_{s}\ll P_{e}(m)$, and $P_{b}$ is dominated by
$P_{e}(m)$, i.e., $P_{b}\propto P_{e}(m)$, which deteriorates with increasing
$n_{R}$ as well.

##  IV Maximum Likelihood Detection: Performance Analysis

In this section, we extend our theoretical analyses to the ML detection of
RIS-SSK and RIS-SM schemes. To provide a more concise and intuitive
presentation, we first deal with the RIS-SM scheme and then extend our
theoretical derivations to RIS-SSK by simply assuming
$x=\hat{x}=\sqrt{E_{s}}$.

###  IV-A Performance of RIS-SM

To derive the theoretical BEP of the RIS-SM scheme for ML detection, we
consider the underlying PEP for joint detection of the selected receive
antenna index $m$ and the transmitted data symbol $x$. From (12), conditioned
on channel coefficients, this PEP can be expressed as follows:

$\displaystyle
P(m,x\rightarrow\hat{m},\hat{x})=P\left(\sum_{l=1}^{n_{R}}\big{|}r_{l}-G_{l}x\big{|}^{2}>\sum_{l=1}^{n_{R}}\big{|}r_{l}-\hat{G}_{l}\hat{x}\big{|}^{2}\right)$
(37)

where $G_{l}=\sum_{i=1}^{N}g_{l,i}e^{j\psi_{m,i}}$ and
$\hat{G}_{l}=\sum_{i=1}^{N}g_{l,i}e^{j\psi_{\hat{m},i}}$. After simple
manipulations, we obtain

$\displaystyle P(m,x\rightarrow\hat{m},\hat{x})$
$\displaystyle=P\\!\left(\sum_{l=1}^{n_{R}}\big{|}G_{l}\big{|}^{2}\big{|}x\big{|}^{2}\\!\\!-\\!\big{|}\hat{G}_{l}\big{|}^{2}\big{|}\hat{x}\big{|}^{2}\\!-\\!2\Re\\!\left\\{r_{l}^{*}\big{(}G_{l}x-\hat{G}_{l}\hat{x}\big{)}\right\\}\\!>\\!0\right)$
$\displaystyle=P\\!\left(\sum_{l=1}^{n_{R}}-\big{|}G_{l}x-\hat{G}_{l}\hat{x}\big{|}^{2}-2\Re\\!\left\\{n_{l}^{*}\big{(}G_{l}x-\hat{G}_{l}\hat{x}\big{)}\right\\}\\!>\\!0\right)$
$\displaystyle=P(G>0)$ (38)

where we considered the fact that $r_{l}=G_{l}x+n_{l}$ for all $l$. Here,
$G\sim\mathcal{N}(\mu_{G},\sigma_{G}^{2})$ with
$\mu_{G}=-\sum_{l=1}^{n_{R}}\big{|}G_{l}x-\hat{G}_{l}\hat{x}\big{|}^{2}$ and
$\sigma_{G}^{2}=\sum_{l=1}^{n_{R}}2N_{0}\big{|}G_{l}x-\hat{G}_{l}\hat{x}\big{|}^{2}$.
Consequently, from $P(G>0)=Q(-\mu_{G}/\sigma_{G})$, we arrive at

$P(m,x\rightarrow\hat{m},\hat{x})=Q\left(\sqrt{\frac{\sum_{l=1}^{n_{R}}\big{|}G_{l}x-\hat{G}_{l}\hat{x}\big{|}^{2}}{2N_{0}}}\right)$
(39)

which is analogous to classical SM-based systems [32]. Defining
$\Gamma\triangleq\sum_{l=1}^{n_{R}}\big{|}G_{l}x-\hat{G}_{l}\hat{x}\big{|}^{2}$
and considering the alternative form of the $Q$-function, the unconditional
(averaged over channel coefficients) PEP can be calculated as follows:

$\displaystyle\bar{P}(m,x\rightarrow\hat{m},\hat{x})=\int_{0}^{\infty}Q\left(\sqrt{\frac{\Gamma}{2N_{0}}}\right)f_{\Gamma}(\Gamma)d\Gamma$
$\displaystyle\hskip
28.45274pt=\int_{0}^{\infty}\frac{1}{\pi}\int_{0}^{\pi/2}\exp\left(\frac{-\Gamma}{4\sin^{2}\eta
N_{0}}\right)f_{\Gamma}(\Gamma)d\eta d\Gamma$ $\displaystyle\hskip
28.45274pt=\frac{1}{\pi}\int_{0}^{\pi/2}M_{\Gamma}\left(\frac{-1}{4\sin^{2}\eta
N_{0}}\right)d\eta.$ (40)

Here, we need the MGF of $\Gamma$ $(M_{\Gamma}(s))$ to perform this
integration. This MGF can be derived by considering the general quadratic form
of correlated Gaussian RVs and depends on erroneous or correct detection of
the receive antenna index $m$.

i) First Case $(m\neq\hat{m})$: Let us rewrite $\Gamma$ as
$\Gamma=\Gamma_{1}+\Gamma_{2}+\Gamma_{3}$, where

$\displaystyle\Gamma_{1}$
$\displaystyle=\left|\left[\sum_{i=1}^{N}\beta_{m,i}\right]\\!x-\left[\sum_{i=1}^{N}g_{m,i}e^{j\psi_{\hat{m},i}}\right]\\!\hat{x}\right|^{2}\\!\\!=\big{|}G_{m}x-\hat{G}_{m}\hat{x}\big{|}^{2}$
$\displaystyle\Gamma_{2}$
$\displaystyle=\left|\left[\sum_{i=1}^{N}g_{\hat{m},i}e^{j\psi_{m,i}}\right]\\!x-\left[\sum_{i=1}^{N}\beta_{\hat{m},i}\right]\\!\hat{x}\right|^{2}\\!\\!=\big{|}G_{\hat{m}}x-\hat{G}_{\hat{m}}\hat{x}\big{|}^{2}$
$\displaystyle\Gamma_{3}$ $\displaystyle=\sum_{l=1(l\neq
m,l\neq\hat{m})}^{n_{R}}\big{|}G_{l}x-\hat{G}_{l}\hat{x}\big{|}^{2}.$ (41)

Here, $\Gamma_{1}$, $\Gamma_{2}$, and $\Gamma_{3}$ respectively stand for
$l=m$, $l=\hat{m}$, and $l\neq m,l\neq\hat{m}$ in $\Gamma$. Different
distributions of $G_{l}$ and $\hat{G}_{l}$ with respect to $l$ as well as the
correlation among them necessitate the quadratic form of Gaussian RVs to
derive $M_{\Gamma}(s)$.

Considering $g_{l,i}=\beta_{l,i}e^{-j\psi_{l,i}}$, let us rewrite $\Gamma_{1}$
and $\Gamma_{2}$ as

$\displaystyle\Gamma_{1}$
$\displaystyle=\left|\gamma_{1}\right|^{2}=(\gamma_{1})_{\Re}^{2}+(\gamma_{1})_{\Im}^{2}=\left|\sum_{i=1}^{N}\beta_{m,i}\left(x-e^{-j\bar{\psi_{i}}}\hat{x}\right)\right|^{2}$
$\displaystyle\Gamma_{2}$
$\displaystyle=\left|\gamma_{2}\right|^{2}=(\gamma_{2})_{\Re}^{2}+(\gamma_{2})_{\Im}^{2}=\left|\sum_{i=1}^{N}\beta_{\hat{m},i}\left(xe^{j\bar{\psi_{i}}}-\hat{x}\right)\right|^{2}$
(42)

where $\bar{\psi_{i}}=\psi_{m,i}-\psi_{\hat{m},i}$ has a triangle-shaped PDF
defined in (14). It is obvious from (IV-A) and the CLT that $\gamma_{1}$ and
$\gamma_{2}$ follow complex Gaussian distribution for increasing $N$, however,
we need to consider the correlation among their components. After tedious but
straightforward calculations, the mean vector and the covariance matrix of
$\mathbf{g}=\begin{bmatrix}(\gamma_{1})_{\Re}&(\gamma_{1})_{\Im}&(\gamma_{2})_{\Re}&(\gamma_{2})_{\Im}\end{bmatrix}^{\mathrm{T}}$
are obtained respectively as follows:

$\mathbf{m}=\begin{bmatrix}\frac{N\sqrt{\pi}x_{\Re}}{2}&\frac{N\sqrt{\pi}x_{\Im}}{2}&-\frac{N\sqrt{\pi}\hat{x}_{\Re}}{2}&-\frac{N\sqrt{\pi}\hat{x}_{\Im}}{2}\end{bmatrix}^{\mathrm{T}}$
(43)
$\mathbf{C}=\\!\begin{bmatrix}\sigma_{1}^{2}&\sigma_{1,2}&\sigma_{1,3}&\sigma_{1,4}\\\
\sigma_{1,2}&\sigma_{2}^{2}&\sigma_{2,3}&\sigma_{2,4}\\\
\sigma_{1,3}&\sigma_{2,3}&\sigma_{3}^{2}&\sigma_{3,4}\\\
\sigma_{1,4}&\sigma_{2,4}&\sigma_{3,4}&\sigma_{4}^{2}\end{bmatrix}$ (44)

where

$\displaystyle\sigma_{1}^{2}$
$\displaystyle=\tfrac{N(4-\pi)x_{\Re}^{2}}{4}\\!+\\!\tfrac{N\left|\hat{x}\right|^{2}}{2},\,\,\,\sigma_{2}^{2}=\tfrac{N(4-\pi)x_{\Im}^{2}}{4}\\!+\\!\tfrac{N\left|\hat{x}\right|^{2}}{2}$
$\displaystyle\sigma_{3}^{2}$
$\displaystyle=\tfrac{N(4-\pi)\hat{x}_{\Re}^{2}}{4}\\!+\\!\tfrac{N\left|x\right|^{2}}{2},\,\,\,\sigma_{4}^{2}=\tfrac{N(4-\pi)\hat{x}_{\Im}^{2}}{4}\\!+\\!\tfrac{N\left|x\right|^{2}}{2}$
$\displaystyle\sigma_{1,2}$
$\displaystyle=\tfrac{N(4-\pi)x_{\Re}x_{\Im}}{4},\,\,\,\sigma_{3,4}=\tfrac{N(4-\pi)\hat{x}_{\Re}\hat{x}_{\Im}}{4}$
$\displaystyle\sigma_{1,3}$
$\displaystyle=\tfrac{N\pi(-x_{\Re}\hat{x}_{\Re}+x_{\Im}\hat{x}_{\Im})}{8},\,\,\,\sigma_{1,4}=-\tfrac{N\pi(x_{\Re}\hat{x}_{\Im}+\hat{x}_{\Re}x_{\Im})}{8}$
$\displaystyle\sigma_{2,3}$
$\displaystyle=\sigma_{1,4},\,\,\,\sigma_{2,4}=-\sigma_{1,3}.$

Substituting (43) and (44) in the MGF of the quadratic form of Gaussian RVs
given in (III-B) for $\mathbf{A}=\mathbf{I}_{4}$, yields the MGF of
$\Gamma_{1}+\Gamma_{2}=\mathbf{g}^{\mathrm{T}}\mathbf{A}\mathbf{g}$. On the
other hand, $\Gamma_{3}$ can be rewritten as

$\Gamma_{3}=\sum_{l=1(l\neq
m,l\neq\hat{m})}^{n_{R}}\left|\sum_{i=1}^{N}g_{l,i}\left(xe^{j\psi_{m,i}}-\hat{x}e^{j\psi_{\hat{m},i}}\right)\right|^{2}.$
(45)

Fortunately, due to zero means of $g_{l,i}$ and
$x_{i}=xe^{j\psi_{m,i}}-\hat{x}e^{j\psi_{\hat{m},i}}$, and their independence
for all $l\,(l\neq m,l\neq\hat{m})$ and $i$, we have
$\mathrm{Var}[(g_{l,i}x_{i})_{\Re}]=\mathrm{Var}[(g_{l,i}x_{i})_{\Im}]=\frac{\left|x\right|^{2}+\left|\hat{x}\right|^{2}}{2}$.
Consequently, for large $N$, $\Gamma_{3}$ can be expressed as the sum of
$n_{R}-2$ independent central $\chi^{2}$ RVs with two degrees of freedom, and
has the following simple MGF:

$M_{\Gamma_{3}}(s)=\left(\frac{1}{1-sN(\left|x\right|^{2}+\left|\hat{x}\right|^{2})}\right)^{n_{R}-2}.$
(46)

Finally, substituting the MGF of $\Gamma$, obtained from the product of MGFs
of $\Gamma_{1}+\Gamma_{2}$ and $\Gamma_{3}$, in (IV-A) and evaluating this
simple integral numerically yields the desired unconditional PEP. It is worth
noting that the unconditional PEP is independent of $m$ and $\hat{m}$.

ii) Second Case $(m=\hat{m})$: For the calculation of PEP in case of correctly
detected receive antenna indices, considering $G_{l}=\hat{G}_{l}$, we can
rewrite $\Gamma$ as

$\Gamma=\sum_{l=1}^{n_{R}}\big{|}G_{l}(x-\hat{x})\big{|}^{2}=\big{|}x-\hat{x}\big{|}^{2}\left(G_{m}^{2}+\sum_{l=1(l\neq
m)}^{n_{R}}\left|G_{l}\right|^{2}\right).$ (47)

Keeping in mind that $G_{m}\sim\mathcal{N}(N\sqrt{\pi}/2,N(4-\pi)/4)$ and
$G_{l}\sim\mathcal{CN}(0,N)$ for $l\neq m$ (see Section III.A), we obtain

$\displaystyle M_{\Gamma}(s)$
$\displaystyle=\left(\frac{1}{1-\frac{sN(4-\pi)\left|x-\hat{x}\right|^{2}}{2}}\right)^{\\!\frac{1}{2}}\exp\left(\frac{\frac{sN^{2}\left|x-\hat{x}\right|^{2}\pi}{4}}{1-\frac{sN(4-\pi)\left|x-\hat{x}\right|^{2}}{2}}\right)$
$\displaystyle\hskip
28.45274pt\times\left(\frac{1}{1-sN\left|x-\hat{x}\right|^{2}}\right)^{n_{R}-1}.$
(48)

Substituting this MGF in (IV-A) and performing numerical integration provides
the desired PEP.

Finally, the PEP values obtained from (IV-A) for both cases are used to derive
the following union bound on BEP:

$P_{b}\\!\leq\\!\frac{1}{Mn_{R}}\sum_{m}\sum_{\hat{m}}\sum_{x}\sum_{\hat{x}}\frac{\bar{P}(m,x\rightarrow\hat{m},\hat{x})e(m,x\rightarrow\hat{m},\hat{x})}{\log_{2}(Mn_{R})},$
(49)

where $e(m,x\rightarrow\hat{m},\hat{x})$ stands for the number of bits in
error for the corresponding pairwise error event.

Remark 3: The above analysis is general and can be considered for all
constellations. Derivation of simplified expressions for BPSK and QPSK (or
$M$-PSK in general) is left to interested readers.

Figure 3: Theoretical BEP performance of RIS-SSK and RIS-SM systems with
increasing $n_{R}$.

###  IV-B Performance of RIS-SSK

Considering $x=\hat{x}=\sqrt{E_{s}}$ (i.e., an unmodulated carrier in the
baseband) in (39), we obtain the conditional PEP of the RIS-SSK scheme as

$P(m\rightarrow\hat{m})=Q\left(\sqrt{\frac{\sum_{l=1}^{n_{R}}E_{s}\big{|}G_{l}-\hat{G}_{l}\big{|}^{2}}{2N_{0}}}\right).$
(50)

In light of this information, the analyses in Section IV.A (for the case of
$m\neq\hat{m}$) is also valid for ML detection of RIS-SSK and the
unconditional PEP $\bar{P}(m\rightarrow\hat{m})$ can be easily derived from
(IV-A) with suitable modifications in $M_{\Gamma}(s)$. Then the BEP upper
bound of RIS-SSK can be calculated similar to (23) as

$P_{b}\leq\frac{n_{R}}{2}\bar{P}(m\rightarrow\hat{m}).$ (51)

Remark 4: We observe that unlike the greedy detector, increasing $n_{R}$ for
ML detection improves the PEP performance of RIS-SM and RIS-SSK schemes
through the MGF terms of (46) and (IV-A), which include $n_{R}$ in their
exponents. Since increasing $n_{R}$ also improves the data rate along with
increasing number of bit errors in (49) and (51), we face an interesting
trade-off among complexity, performance, and data rate. In Fig. 3, we show the
theoretical BEP performance of RIS-SSK and RIS-SM schemes calculated from (49)
and (51) for $N=128$ reflectors and increasing $n_{R}$, with respect to
$E_{s}/N_{0}$. As seen from Fig. 3, increasing $n_{R}$ eventually improves the
BER performance (since $n_{R}$ appears in (51)), while providing a higher data
rate, which is quite unusual for legacy communication systems not exploiting
IM. In other words, increasing $n_{R}$ both improves the spectral efficiency
and overall BER performance for RIS-SSK/SM systems in the presence of ML
detection.

Figure 4: Theoretical and computer simulation results for RIS-SSK with greedy
detection.

##  V Simulation Results

In this section, we provide computer simulation results for the proposed RIS-
based SSK and SM schemes and make comparisons with our theoretical results and
reference schemes. We consider $E_{s}/N_{0}$ as the SNR, similar to the
classical diversity combining and space modulation schemes. For clarity, as
usual practice, the large-scale path-loss is not considered since it is
implicitly taken into account in the received SNR.

In Figs. 4 and 5, we provide BER performance curves of the RIS-SSK and RIS-SM
systems for greedy detection and make comparisons with the theoretical results
obtained from (23) and (24), respectively. As seen from the given results, our
theoretical findings are quite accurate for both schemes and the performance
of RIS-SSK and RIS-SM schemes degrade with increasing bits per channel use
(bpcu), or equivalently $n_{R}$, values (see Remarks 1 and 2). It is worth
noting that BER performance of both schemes significantly improves by
increasing the $N$ value from $64$ to $128$, which is consistent with (22).

Figure 5: Theoretical and computer simulation results for RIS-SM with greedy
detection. Figure 6: Theoretical and computer simulation results for RIS-SSK
and RIS-SM with ML detection. Figure 7: BER performance comparisons for greedy
and ML detectors of RIS-SSK and RIS-SM schemes.

In Fig. 6, we focus on the performance of RIS-SSK and RIS-SM schemes with ML
detection and make comparisons with the theoretical results obtained from (49)
and (51). As seen from Fig. 6, while increasing $n_{R}$ does not cause a
remarkable BER degradation for the RIS-SSK scheme, the effect of increasing
$M$ is more evident for the RIS-SM scheme. It is worth noting that the
performance of the ML detector improves with $N$ as well due to (46) and
(IV-A).

We compare the BER performances of greedy and ML detectors for both RIS-SSK
and RIS-SM systems at various bpcu values in Fig. 7. We observe that the ML
detector of RIS-SSK provides approximately $2$ dB improvement in the required
SNR for the considered two setups ($N=64,n_{R}=2$ and $N=128,n_{R}=8$).
Although we observe a similar improvement for RIS-SM in case of $N=64$, the
difference in BER performances of greedy and ML detectors is relatively
smaller for the case of $N=128$. The gap between the greedy and ML detectors
of RIS-SSK can be attributed to its nature, which considers only indices to
convey information.

Finally, in Fig. 8, we present BER performance comparison results for RIS-SSK,
RIS-SM, RIS-AP [21], and conventional fully-digital (zero-forcing) precoding-
based receive SSK (RSSK) [33] at $3,4,$ and $6$ bpcu values with ML detection.
We have several important observations from Fig. 8. First, an interesting
trade-off exists between the receiver cost and the BER performance for RIS-SSK
and RIS-SM schemes: while the former provides a better BER performance, the
latter exhibits a slight degradation by using a less number of receive
antennas at the same bpcu. Second, the RIS-AP scheme proposed in [21], which
utilizes a single receive antenna, cannot compete with RIS-SSK/SM schemes
since it creates a virtual $M$-PSK constellation by altering RIS phases and
suffers at high bpcu values. Third, compared to RIS-based new schemes, a more
than $15$ dB difference in required SNR is observed for the conventional RSSK-
MIMO scheme. Although utilizing a massive MIMO system, RSSK forces the MIMO
channels into zero to realize a pure SSK-like reception, while RIS-based
SM/SSK schemes constructively exploit the wireless channels to boost the
signal quality at the intended receive antenna. In other words, RIS-based
schemes exploit the inherent randomness of fading channels in a more effective
way by the constructive alignment of reflected signals. However, although both
solutions have similar level of algorithmic complexity, RIS-based schemes may
have higher signaling overhead due to the training of RISs, while traditional
schemes have a much higher transmitter complexity.

Figure 8: BER performance comparison of RIS-SSK, RIS-SM, RIS-AP [21], and
conventional RSSK [33] schemes for $3$, $4$, and $6$ bpcu. Figure 9:
Performance of RIS-SSK and RIS-SM schemes in the presence of channel phase
estimation errors.

In Fig. 9, we investigate the performance of proposed RIS-SSK and RIS-SM
schemes with erroneous channel phase estimates at the RIS. Here, the phase
estimation error at the RIS is modelled according to von Mises (circular
normal) distribution, whose concentration parameter $\kappa$ captures the
accuracy of the estimation [34]. As seen from Fig. 9, while the degradation in
BER is not significant for $\kappa=10$ (higher accuracy estimation), a
noticeable difference is observed for $\kappa=5$ for both proposed schemes.

##  VI Conclusions

The general concept of RIS-assisted IM has been proposed in this paper as a
new beyond massive MIMO paradigm for next-generation (potentially 6G or
beyond) wireless networks. It has been shown by comprehensive theoretical
derivations as well as computer simulations that the proposed RIS-SSK and RIS-
SM schemes have the potential to provide considerably high spectral efficiency
at extremely low SNR values through a smart and RIS-assisted indexing
mechanism for available receive antennas. We conclude that the effective use
of RIS-assisted communication schemes may be a game-changing paradigm for
next-generation communication networks by eliminating the need for
sophisticated massive MIMO schemes that require expensive and power-hungry
components. The extremely low SNR regions of operation may also be a remedy to
the increasing need for advanced channel coding schemes to achieve ultra-
reliable communications. As also discussed in Introduction, potential
applications of IM for transmit antennas and/or RIS regions along with other
advanced/generalized schemes, the design of low-complexity receiver
architectures, and analyses in the presence of potential system imperfections
and more sophisticated channel/correlation models, remain as interesting and
open research problems. We also hope that the theoretical insights provided in
this work may be useful for future implementation and hardware impairment
related challenges in the context of emerging RISs.

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|  Ertugrul Basar (S’09-M’13-SM’16) received the B.S. degree (Hons.) from
Istanbul University, Turkey, in 2007, and the M.S. and Ph.D. degrees from
Istanbul Technical University, Turkey, in 2009 and 2013, respectively. He is
currently an Associate Professor with the Department of Electrical and
Electronics Engineering, Koç University, Istanbul, Turkey and the director of
Communications Research and Innovation Laboratory (CoreLab). His primary
research interests include MIMO systems, index modulation, intelligent
surfaces, waveform design, visible light communications, and signal processing
for communications. Dr. Basar currently serves as an Editor of the IEEE
Transactions on Communications and Physical Communication (Elsevier), and as a
Senior Editor of the IEEE Communications Letters.
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