# A New RIS Architecture with a Single Power Amplifier: Energy Efficiency and
Error Performance Analysis

Recep A. Tasci, Fatih Kilinc, Ertugrul Basar, and George C. Alexandropoulos
Recep A. Tasci, Fatih Kilinc, Ertugrul Basar are with Communications Research
and Innovation Laboratory (CoreLab), Department of Electrical and Electronics
Engineering, Koç University, 34450 Istanbul, Turkey (e-mail: {ratasci20,
fkilinc20<EMAIL_ADDRESS>C. Alexandropoulos is with Department of
Informatics and Telecommunications, National and Kapodistrian University of
Athens, 15784 Athens, Greece (email: alexandg@ieee.org)Manuscript received
December 20, 2021; revised March 15, 2022.

###### Abstract

Reconfigurable intelligent surface (RIS)-assisted communication have recently
attracted the attention of the wireless communication community as a potential
candidate for the next $6$-th generation (6G) of wireless networks. Various
studies have been carried out on the RIS technology, which is capable of
enabling the control of the signal propagation environment by network
operators. However, when an RIS is used in its inherently passive structure,
it appears to be only a supportive technology for communications, while
suffering from a multiplicative path loss. Therefore, researchers have lately
begun to focus on RIS hardware designs with minimal active elements to further
boost the benefits of this technology. In this paper, we present a simple RIS
hardware architecture including a single and variable gain amplifier for
reflection amplification to confront the multiplicative path loss. The end-to-
end signal model for communication systems assisted with the proposed
amplifying RIS design is presented, together with an analysis focusing on the
capacity maximization and theoretical bit error probability performance, which
is corroborated by computer simulations. In addition, the major advantages of
the proposed amplifying RIS design compared to its passive counterpart are
discussed. It is shown that the proposed RIS-based wireless system
significantly eliminates the double fading problem appearing in conventional
passive RIS-assisted systems and improves the communication energy efficiency.

###### Index Terms:

Reconfigurable intelligent surface (RIS), active RIS, amplifying RIS, energy
efficiency, performance analysis.

##  I Introduction

The number of mobile users trying to keep up with emerging technologies is
increasing day by day. This situation has an impact on many things around the
world, especially on communication networks. Recent generations of wireless
communication have been developed to meet the high data demand arising from
these technological developments [1]. For these reasons, the fact that
existing communication systems will start to become insufficient in the
future, as it was before, will make the development of next generation of the
wireless communication, 6G and beyond, inevitable. 6G focuses further
enhancing data rate, security, consistency, and mobility, compared to its
predecessors. In order to meet the future demands, numerous studies are being
carried out towards 6G wireless networks. As a result, the use of advanced
technologies such as extreme multiple-input multiple-output (MIMO) systems
with power efficient front-ends [2], as well as Terahertz and millimeter-wave
communications might play an important role in supplying these demands.

In recent years, reconfigurable intelligent surface (RIS)-empowered
communication has become one of the most popular developments towards 6G
networks [3, 4, 5, 6, 7, 8, 9]. Supporting high wireless channel capacity,
expanding the signal coverage, reducing the bulkiness of the multiple antenna
systems as well as the energy consumption, low-cost implementation and
mitigating several negative effects of the wireless channel, such as multipath
fading and the Doppler effect can be given as some of the main reasons why
RISs has been investigated thoroughly[10, 11, 12].

Unlike massive antenna arrays that include multiple radio frequency (RF)
chains attached to the antennas, RISs include only a certain number of
reflecting elements that reflect the incoming signals with a specific phase.
This operation can be done by changing the electromagnetic properties of their
reflecting elements, such as the reflection coefficient, with the help of a
software-defined controller. In that way, the wireless propagation environment
becomes configurable. The potential of RISs stems from the fact that they
perform these operations while being almost passive because tiny reflecting
elements are used instead of power-hungry phase shifters. Thus, energy
efficient and low cost setups can be realized with the help of RISs.

Numerous studies have been conducted by combining RISs with many other
techniques. Assisting a relay system with an RIS [13], RIS-based index
modulation[14], RIS-enabled reflection modulation [15, 16], implementation of
non-orthogonal multiple access (NOMA) by using RISs [17, 18], and channel
modeling in the presence of an RIS [19, 20, 21] can be given as notable
examples. These studies have succeeded with the help of RISs but have not been
able to avoid the performance degradation caused by the double path loss
problem when RIS is not positioned close to terminals [22, 23]. Therefore,
passive RISs cannot go beyond being a supportive technology for communication
systems due to the double path loss problem. Some studies have been conducted
to overcome the double path loss effect by combining active elements with
RISs. In [24], a single receive RF chain that includes a low-noise amplifier
(LNA) was used to enable baseband reception, which in turn, was exploited for
channel estimation at the RIS side. Similarly, channel estimation using a
hybrid RIS architecture has been performed in [25] by deploying small numbers
of receive RF chains at the RIS side. Another RIS-based channel sensing
approach was studied in [26] by coupling some incident signals with a modified
RIS architecture, including receive RF chains. Furthermore, [2] analyzes
dynamic metasurface antennas (DMAs) from an analog and digital signal
processing perspective with less number of RF chains than the number of
metamaterials. In [27], a reflection-type amplifier has been proposed to be
used by each reflecting element of an RIS, and the proposed active RIS element
was designed, fabricated, and experimental results are provided. Another
active reflecting element design has been proposed in [28] such that the
magnitude of the reflection coefficient of the reflecting elements is greater
than unity. Connecting only a few elements of an RIS with RF chains and power
amplifiers (PAs) has been investigated in [29] as a hybrid relay-reflecting
intelligent surface.

Against this background, this paper proposes an amplifying RIS design that
incorporates a single PA and two RISs. According to our design, the signals
received by elements of the first RIS are combined after getting their phase
configured. Then, the combined signal is fed to the available PA, which feeds
it to the second RIS that transmits it with a controllable phase
configuration.

The proposed design works solely in the RF domain, similar to the waveguide-
based approach in [26], unlike full-duplex multi-antenna decode-and-forward
(DF) relays, which perform down-conversion and baseband processing [13].
Therefore, our design looks more similar to a full-duplex, multi-antenna, and
amplify-and-forward (AF) relay. However, they have some key differences.
Relays usually perform linear processing techniques, such as maximum ratio
combining (MRC), and realize power allocation optimization algorithms.
Furthermore, they include bulky phase-shifter networks for transmit
beamforming and are subject to loopback self-interference [30]. Our design is
simpler, does not require complex algorithms, and prevents loopback self-
interference via spatial separation in the form of back to back placement of
the different RISs. Note that simple self-interference cancellation techniques
in the RF domain can be applied [31].

TABLE I: Comparison of the proposed architecture with a passive RIS, a fully-
connected active RIS, and a multi-antenna AF relay

|  | Proposed
---
Design
| Passive
---
RIS
| Fully-
---
Connected
Active RIS
AF Relay
Complexity | Low | Low | High | High
| Power
---
Consumption
Low | Low | High | High
| Interference
---
Cancellation
| No
---
interference
| No
---
interference
| No
---
interference
| Signal
---
processing
required
Cost | Low | Low | High | High
Performance | High | Low | High | High

A detailed comparison of the proposed scheme with passive RISs, fully-
connected active RIS and AF relay is presented in Table I. Considering fully-
connected active designs, where each RIS element contains an active component,
there are some drawbacks, such as power consumption and cost. These
architectures may not be feasible for a large number of reflecting elements
because they contain numerous active components. The proposed design has the
key advantages of being energy efficient and low cost compared to other active
RIS designs since it includes a single PA. Furthermore, the fully-connected
active RIS architecture is more complex than the proposed system because the
gain of each active element needs to be configured. In contrast, the gain of a
single PA is adjusted in the proposed architecture. Although the fully-
connected active RIS design is more powerful because of the amplifiers being
connected to each reflecting element, the proposed scheme also benefits from
the passive beamforming of the receiving and transmitting parts of the RIS.
Compared to passive RISs, the proposed design enhances the system capacity in
a more energy efficient manner. Moreover, as our system diminishes the effect
of double path loss due to the amplification, it provides flexibility to place
the RIS at any point between the transmitter (Tx) and the receiver (Rx).
However, the proposed design is more complex because additional signal
processing procedures are required, such as adjusting the gain of the PA and
phase configurations of the RISs.

We note that a similar concept was considered in [32], assuming that the
received signal is amplified and fed back as leakage to the input; however, it
lacks a sufficient mathematical framework and does not present detailed
modeling of the system operation. On the other hand, in this paper, we propose
a novel amplifying RIS scheme with a solid mathematical framework and provide
a unified model for the system operation. Moreover, we analyze the theoretical
bit error probability (BEP), capacity, and energy efficiency (EE) of the
proposed design and compare it with a benchmark passive RIS structure.

The main contributions of this study are summarized as follows:

  * •

A new and energy efficient amplifying RIS design is proposed. The end-to-end
system model of the proposed amplifying RIS design is presented.

  * •

This work demonstrates a cheaper, less complex, and more energy efficient
architecture than fully connected active RIS designs and multi-antenna AF
relays.

  * •

The proposed design provides the flexibility to position the RIS at any
location, with an increase in performance compared to passive RISs.

  * •

The achievable rate and the EE of the system are examined under different
configurations, and an optimization problem is addressed to maximize the
system capacity. Moreover, the theoretical BEP is obtained under different
conditions, while being also verified through computer simulations.

  * •

The proposed active model is comprehensively compared with a passive RIS
setup.

The rest of the paper is organized as follows. In Section II, we present the
end-to-end system model of the proposed amplifying RIS design. Section III
presents the performance analysis of the system by investigating the BEP with
a theoretical approach. In Section IV, the EE analysis is performed by
considering the characteristics of the PAs. In Section V, we exhibit our
numerical results to evaluate the system performance. Finally, the paper is
concluded in Section VI.

##  II System Model

In this section, we propose an amplifying RIS-assisted system model. In
addition, a conventional passive RIS system model is given as the benchmark to
the proposed scheme.

###  II-A Amplifying RIS-assisted System

In this subsection, the considered single-input single-output (SISO)
amplifying RIS-assisted system model is introduced. In this scenario, there
are two passive RISs, denoted as RIS1 and RIS2, that are connected with a PA,
and both have $N$ number of reflecting elements while assuming there is no
direct link between the Tx and Rx as shown in Fig. 1. The vertical and
horizontal distances between the Tx and RISs, and the distance between the Tx
and Rx are denoted as $d_{v}$, $d_{h}$ and $d$, respectively. All of the
reflecting elements of RIS1 receive signals from the Tx and these signals are
combined by RIS1 to be directed to the PA as a single signal. Then, the signal
is amplified by the PA by taking into account that there is also some
additional noise to be amplified, which is due to all of the reflecting
elements of RIS1. Additionally, we consider the signal power at the output of
the PA is limited. The system passes the resulting amplified signal through
RIS2, such that the reflecting elements of RIS2 share and transmit the
amplified signal. One should consider that there might be leakage from the
signal transmitted by RIS2 to the signal received by RIS1. However, this issue
can be resolved by properly positioning the RISs, since an RIS is an one-sided
surface. In our design, RIS1 and RIS2 are placed back-to-back and separately
as shown in Fig. 1, so that the signals directed by RIS2 do not contaminate
the signal captured by RIS1.

In light of these, the received complex baseband signal can be expressed as

$y=\sqrt{\frac{G}{N}}\left(\boldsymbol{\phiup}^{\mathrm{T}}\mathbf{h}\sqrt{P_{t}}s+\sqrt{F}n_{\text{tot}}\right)\boldsymbol{\thetaup}^{\mathrm{T}}\mathbf{g}+n_{\text{rx}},$
(1)

where $s$, $y$, $P_{t}$, $G$, $F$, $n_{\text{tot}}$, and $n_{\text{rx}}$,
correspond to the transmitted and received signals, transmit power at the Tx,
the gain and noise figure of the PA, the total amount of noise at the input of
the PA, and the noise sample at the Rx, respectively.
$\boldsymbol{\phiup}\in\mathbb{C}^{N\times 1}$ and
$\boldsymbol{\thetaup}\in\mathbb{C}^{N\times 1}$ are the reflecting element
phase shift vectors of RIS1 and RIS2, respectively, where
$\boldsymbol{\phiup}=\left[\phi_{1},\phi_{2},\dots,\phi_{N}\right]^{\mathrm{T}}$,
$\boldsymbol{\thetaup}=\left[\theta_{1},\theta_{2},\dots,\theta_{N}\right]^{\mathrm{T}}$,
and $\phi_{i}$ and $\theta_{i}$ denote the phase shifts of the $i$th
reflecting element of RIS1 and RIS2 for $i=1,\dots,N$, respectively.

Figure 1: Generic system model for the considered amplifying RIS-assisted
scheme.

$\mathbf{h}\in\mathbb{C}^{N\times 1}$ is the channel between the Tx and RIS1,
where $\mathbf{h}=\left[h_{1},h_{2},\dots,h_{N}\right]^{\mathrm{T}}$, and
$h_{i}$ denotes the channel coefficient for Tx and $i$th reflecting element of
RIS1 for $i=1,\dots,N$. $\mathbf{g}\in\mathbb{C}^{N\times 1}$ is the channel
between RIS2 and the Rx, where
$\mathbf{g}=\left[g_{1},g_{2},\dots,g_{N}\right]^{\mathrm{T}}$, and $g_{i}$
represents the channel coefficient for $i$th reflecting element of RIS2 and
the Rx for $i=1,\dots,N$. Here,
$\boldsymbol{\phiup}^{\mathrm{T}}\mathbf{h}\sqrt{P_{t}}s$ stands for the
collected signal received by RIS1. This signal is scaled by $\sqrt{G}$ with
the help of the PA and divided by $\sqrt{N}$ due to the phase shift network of
RIS2 that is a kind of power-division circuit, so the total power is
distributed among all of the reflecting elements of RIS2. After that, the
amplified and distributed signal is steered to the Rx with the help of RIS2. A
similar scenario is also valid for $n_{\text{tot}}$, so that it is multiplied
with $\sqrt{F}$ and $\sqrt{G}$, then distributed among the reflecting elements
of RIS2. Here, the noise at RIS2 is ignored, where the noise at RIS1 is not
because it is subject to amplification.

The channels exhibit either Rayleigh or Rician fading depending on the line-
of-sight (LOS) probability, $p_{\text{LOS}}$, which is a function of the
distance. The channel coefficients are modeled as

$h_{i}=\sqrt{\frac{1}{\lambda^{h}}}\left(\sqrt{\frac{K_{1}}{K_{1}+1}}h^{L}_{i}+\sqrt{\frac{1}{K_{1}+1}}h^{NL}_{i}\right),$
(2)
$g_{i}=\sqrt{\frac{1}{\lambda^{g}}}\left(\sqrt{\frac{K_{2}}{K_{2}+1}}g^{L}_{i}+\sqrt{\frac{1}{K_{2}+1}}g^{NL}_{i}\right),$
(3)

where $K_{1}$, $K_{2}$, $\lambda^{h}$, $\lambda^{g}$, $h^{L}_{i}$,
$g^{L}_{i}$, $h^{NL}_{i}$ and $g^{NL}_{i}$ denote the Rician factors for Tx-
RIS1 and RIS2-Rx links, the path loss, LOS and non-line-of-sight (NLOS)
components for the channels $\mathbf{h}$ and $\mathbf{g}$, respectively,
$h^{NL}_{i},g^{NL}_{i}\sim\mathcal{CN}(0,1)$ for $i=1,\dots,N$, and
$\mathcal{CN}(0,1)$ stands for the complex Normal distribution with zero mean
and unit variance. If the channels do not include a LOS component, which
mostly refers to Rayleigh fading, we consider $K_{1}=K_{2}=0$. Furthermore,
the path loss components $\lambda^{h}$ and $\lambda^{g}$ depend on having NLOS
or LOS links, and calculated as follows by considering the Indoor Hotspot
(InH) environment in [33]:

$\lambda_{LOS}\text{[dB]}=32.4+17.3\log_{10}(d_{n})+20\log_{10}(f_{c}),$ (4)
$\lambda_{NLOS}\text{[dB]}=\max\left(\lambda_{LOS},32.4+31.9\log_{10}(d_{n})+20\log_{10}(f_{c})\right),$
(5)

where $d_{n}\in\\{d_{1},d_{2}\\}$ depends on the channel, and
$d_{1}=\sqrt{d_{v}^{2}+d_{h}^{2}}$, $d_{2}=\sqrt{d_{v}^{2}+(d-d_{h})^{2}}$,
and $f_{c}$ are denoted as the distances between the Tx-RIS1, RIS2-Rx, and
carrier frequency in GHz, respectively. The path loss components are the same
for each $h_{i}$ and $g_{i}$ because the RIS is located in the far-field of
the Tx and Rx. Furthermore, $p_{\text{LOS}}$ for the InH environment is given
as [33]

$p_{\text{LOS}}=\begin{cases}1,&d_{n}\leq 5,\\\
e^{-\left(\frac{d_{n}-5}{70.8}\right),}&5<d_{n}\leq 49,\\\
0.54e^{-\left(\frac{d_{n}-49}{211.7}\right),}&49<d_{n}.\end{cases}$ (6)

To determine the signal-to-noise ratio (SNR) of the amplifying RIS-assisted
sytem, (1) is expanded as

$y=\sqrt{\frac{GP_{t}}{N}}\left(\boldsymbol{\phiup}^{\mathrm{T}}\mathbf{h}\right)\left(\boldsymbol{\thetaup}^{\mathrm{T}}\mathbf{g}\right)s+\sqrt{\frac{GF}{N}}\left(\boldsymbol{\thetaup}^{\mathrm{T}}\mathbf{g}\right)n_{\text{tot}}+n_{\text{rx}}.$
(7)

From the received signal model given in (7), the SNR of the proposed system as
follows:

$\gamma_{\text{act}}=\dfrac{P_{t}\left|\sqrt{\dfrac{G}{N}}(\boldsymbol{\phiup}^{\mathrm{T}}\mathbf{h})(\boldsymbol{\thetaup}^{\mathrm{T}}\mathbf{g})\right|^{2}}{\left|\sqrt{\dfrac{GF}{N}}\boldsymbol{\thetaup}^{\mathrm{T}}\mathbf{g}\right|^{2}\sigma^{2}_{\text{tot}}+\sigma^{2}_{\text{rx}}},$
(8)

where $\sigma^{2}_{\text{tot}}$ and $\sigma^{2}_{\text{rx}}$ are the noise
powers at the input of the PA and Rx, respectively. $\gamma_{\text{act}}$ is
the instantaneous received SNR of the amplifying RIS-assisted system and the
achievable rate of the system is expressed as
$R_{\text{act}}=\log_{2}\left(1+\gamma_{\text{act}}\right)$.

Figure 2: Generic system model for a passive RIS-assisted scheme.

###  II-B Passive RIS-assisted System

This section is based on a passive RIS-assisted model shown in Fig. 2. In this
scenario, only one RIS is used and it contains $2N$ reflecting elements such
that it can be considered as a fair benchmark to the amplifying RIS-assisted
model. The signal model of the passive system is specified as

$y=\mathbf{g_{p}}^{\mathrm{T}}\mathbf{\Psi}\mathbf{h_{p}}\sqrt{P_{t}}s+n_{\text{rx}},\vspace{0.2cm}$
(9)

where $\mathbf{\Psi}\in\mathbb{C}^{2N\times 2N}$ is the reflecting element
phase shift matrix of the RIS and
$\mathbf{\Psi}=\mathrm{diag}\left(\begin{bmatrix}\psi_{1},\psi_{2},\dots,\psi_{2N}\end{bmatrix}\right)$
with $\psi_{i}$ representing the phase shift for the $i$th element of the RIS
for $i=1,\dots,2N$. $\mathbf{h_{p}}\in\mathbb{C}^{2N\times 1}$ is the channel
between the Tx and RIS, while $\mathbf{g_{p}}\in\mathbb{C}^{2N\times 1}$
stands for the channel between the RIS and Rx. They follow either Rician or
Rayleigh fading depending on the distances to the Tx and Rx similar to the
channels in the active model. Here, SNR of the passive model
$\gamma_{\text{pas}}$ is determined as

$\gamma_{\text{pas}}=\dfrac{P_{t}\left|\mathbf{g_{p}}^{\mathrm{T}}\mathbf{\Psi}\mathbf{h_{p}}\right|^{2}}{\sigma^{2}_{\text{rx}}},$
(10)

where the achievable rate for the passive model is
$R_{\text{pas}}=\log_{2}\left(1+\gamma_{\text{pas}}\right)$.

##  III Performance Analysis

In this section, we analyze the maximization of system capacity and the
distribution of the received SNR for the amplifying RIS-assisted system, and
present our theoretical BEP calculations accordingly. In order to maximize the
system capacity, the received SNR for the amplifying RIS-assisted system
should be maximized by finding optimum values of the phase shift vectors of
RIS1, RIS2 and the gain of the amplifier $G$. The corresponding optimization
problem is formulated as follows:

$\displaystyle\gamma_{\text{act}}=$
$\displaystyle\max_{G,\boldsymbol{\phiup},\boldsymbol{\thetaup}}\quad\dfrac{P_{t}\left|\sqrt{\dfrac{G}{N}}(\boldsymbol{\phiup}^{\mathrm{T}}\mathbf{h})(\boldsymbol{\thetaup}^{\mathrm{T}}\mathbf{g})\right|^{2}}{\left|\sqrt{\dfrac{GF}{N}}\boldsymbol{\thetaup}^{\mathrm{T}}\mathbf{g}\right|^{2}\sigma^{2}_{\text{tot}}+\sigma^{2}_{\text{rx}}}$
$\displaystyle\hskip 8.5359pt\text{s.t.}\hskip
12.80365pt\left|\phi_{i}\right|=\left|\theta_{i}\right|=1,\quad
i\in\\{1,2,\dots,N\\},$ $\displaystyle\hskip
35.28157ptGP_{t}\left|\boldsymbol{\phiup}^{\mathrm{T}}\mathbf{h}\right|^{2}\leq
P_{\text{max}},$ $\displaystyle\hskip 35.28157ptG\leq G_{\text{max}},$ (11)

where $G_{\text{max}}$ and $P_{\text{max}}$ stand for the maximum gain of the
amplifier and the maximum output power of the amplifier, respectively. Here,
the optimal value of $G$ depends on the phase shift matrix of RIS1 as well.
Therefore, the optimal solution to the phases of the reflecting elements
should be determined first. The signals should be constructively combined at
the PA and Rx to increase the received signal power. This can be done by
eliminating the phases of the channels $\mathbf{h}$ and $\mathbf{g}$. The
optimal solutions to reflecting element phases for RIS1 and RIS2 follow a
constructive combining strategy [34] by adjusting the phase shifts of the
reflecting elements as:

TABLE II: The shape ($k$) and scale ($\nu$) parameters of the fitted Gamma distribution |  | $N$ |  $P_{t}\text{(dBm)}$: | -10 | -5 | 0 | 5 | 10 | 15 | 20 | 25 | 30
---|---|---|---|---|---|---|---|---|---|---|---|---
$P_{\text{{max}}}\text{(dBm)}\quad\quad\quad\quad$ | $\vspace{-0.6cm}10$ | $\vspace{-0.3cm}64$ | $k$ | 44.8922 | 44.7180 | 44.7905 | 44.7109 | 44.8358 | 44.8963 | 48.0934 | 58.6428 | 58.7049
$\nu$ | 0.000405 | 0.001287 | 0.004063 | 0.012868 | 0.040595 | 0.128166 | 0.375257 | 0.329909 | 0.329483
$\vspace{-0.3cm}256$ | $k$ | 178.8281 | 178.8481 | 179.1008 | 178.2996 | 233.7027 | 234.8395 | 234.5761 | 233.9432 | 234.3112
$\nu$ | 0.006486 | 0.020508 | 0.064762 | 0.205720 | 0.330019 | 0.328426 | 0.328834 | 0.329725 | 0.329212
$\vspace{-0.6cm}20$ | $\vspace{-0.3cm}64$ | $k$ | 44.8284 | 44.7199 | 44.7593 | 44.84389 | 44.8633 | 44.7232 | 44.7599 | 44.7586 | 47.9274
$\nu$ | 0.000406 | 0.001286 | 0.004066 | 0.012835 | 0.040551 | 0.128680 | 0.406483 | 1.285651 | 3.765564
$\vspace{-0.3cm}256$ | $k$ | 178.2811 | 178.9068 | 178.4629 | 178.8242 | 178.8931 | 178.6688 | 234.2556 | 234.8709 | 233.5700
$\nu$ | 0.006505 | 0.020503 | 0.064994 | 0.205099 | 0.648377 | 2.052960 | 3.292076 | 3.283723 | 3.302386

$\displaystyle\phi_{i}=e^{-j\angle h_{i}},$ (12)
$\displaystyle\theta_{i}=e^{-j\angle g_{i}},$ (13)

where $\angle\cdot$ denotes the phase of a complex term. Next, we should
specify the value for an optimal $G$ by fixing the optimal value of
$\boldsymbol{\phiup}$ as in (12). It is mentioned earlier that there is a
maximum power level for the amplified signal. Therefore, an optimal gain value
should be determined so that the power of the input signal to the amplifier,
$P_{\text{in}}=P_{t}\left(\sum\nolimits_{i=1}^{N}\left|h_{i}\right|\right)^{2}$,
should not exceed the maximum output power when it is amplified. Thus, the
equation below should be satisfied to obtain the optimal value of $G$;

$\displaystyle\overline{G}_{\text{opt}}P_{t}\left(\sum\limits_{i=1}^{N}\left|h_{i}\right|\right)^{2}=P_{\text{max}},$
(14)

where $\overline{G}_{\text{opt}}$ stands for the optimal gain value without
the limitation of $G_{\text{max}}$. Considering that
$\overline{G}_{\text{opt}}$ can exceed $G_{\text{max}}$, we define the optimal
gain as:

$\displaystyle
G_{\text{opt}}=\min\left(\overline{G}_{\text{opt}},G_{\text{max}}\right).$
(15)

By substituting $G_{\text{opt}}$ in (15) to (8) for $G$ and arranging the
reflecting element phases of $\boldsymbol{\phiup}$ and $\boldsymbol{\thetaup}$
as in (12) and (13), respectively, the maximized received SNR for the
amplifying RIS-assisted system can be written as

$\displaystyle\gamma_{\text{act}}$
$\displaystyle=\dfrac{P_{t}\dfrac{P_{\text{max}}}{N}A^{2}B^{2}}{\dfrac{P_{\text{max}}F}{N}B^{2}\sigma_{\text{tot}}^{2}+P_{t}A^{2}\sigma_{\text{rx}}^{2}},$
(16)

where $A=\sum\nolimits_{i=1}^{N}\left|h_{i}\right|$ and
$B=\sum\nolimits_{i=1}^{N}\left|g_{i}\right|$. Here, one can easily observe
that the maximized received SNR increases proportional to $N$, $P_{t}$, and
$P_{\text{max}}$. However, there are also other constraints that can limit the
effect of the RIS size and the Tx power, such as $P_{\text{max}}$ and
$G_{\text{max}}$. The effects of these constraints are further investigated in
Section V. Moreover, the amplifying RIS model experiences multiplicative path
loss just like the passive RIS since (16) includes the multiplication of $A$
and $B$. Nevertheless, the gain factor significantly compensates this double
path loss effect.

Probability analysis of the maximized SNR in (16) gives us useful insights.
The terms $A$ and $B$ converge to Gaussian distributed random variables due to
the Central Limit Theorem (CLT) for sufficiently large $N$, where
$\left|h_{i}\right|$ and $\left|g_{i}\right|$ are Rayleigh or Rician
distributed random variables. Thus, $A^{2}$ and $B^{2}$ follow non-central
chi-square distribution with one degree of freedom. The denominator and
numerator of the SNR term in (16) are correlated and follow Gamma distribution
due to the combination of weighted non-central chi-square random variables.
The received SNR term in (16) becomes the ratio of two correlated Gamma random
variables and therefore, subject to the Gamma distribution [35]. To the best
of the authors’ knowledge, although there are a number of articles in the
literature examining the ratio of two Gamma random variables and ratio of
correlated chi-square random variables, no studies pointing to this kind of
distribution has been found[36, 37, 38]. Therefore, using Open Distribution
Fitter app (dfittool), we can find the Gamma parameters of the received SNR
distribution on MATLAB and prove that it follows the Gamma distribution.

Figure 3: The PDF of $\gamma_{\text{act}}$ and Gamma distribution fits for (a)
$N=10,\dots,20$ and (b) $N=300,\dots,400$.

The distribution of $\gamma_{\text{act}}$ is investigated for different $N$
values as seen in Fig. 3. Histograms represent the distributions of
$\gamma_{\text{act}}$ and are obtained by performing Monte Carlo simulations.
The overlapping lines stand for the fitted Gamma distributions. The Gamma
distribution fits exactly to the probability density function (PDF) of
$\gamma_{\text{act}}$. Thus, the PDF of $\gamma_{\text{act}}$ can be written
as[37]

$\displaystyle
f_{\Gamma}(\gamma_{\text{act}})=\frac{{\gamma_{\text{act}}^{k-1}e^{-\gamma_{\text{act}}/v}}}{v^{k}\Gamma(k)},$
(17)

where $\Gamma(\cdot)$, $k$, and $\nu$ represent the Gamma function, shape, and
scale parameters, respectively. The shape and scale parameters for the fitted
Gamma distribution are provided in Table II for various system
specifications111Interested readers can produce arbitrary results as well by
using the source code provided: https://github.com/recepakiftasci/Amplifying-
RIS. The parameters of the Gamma distribution vary for different
specifications such as $N$, $P_{\text{max}}$ and $P_{t}$. For instance, the
standard deviation of the distribution increases with $N$ as shown in Fig. 3.

Theoretical symbol error probability (SEP) is calculated by using the moment
generation function (MGF) of the Gamma distribution as follows [39]:

$\displaystyle M_{\gamma_{\text{act}}}(s)$ $\displaystyle=(1-\nu
s)^{-k},\hskip 5.69046pt\text{for}\hskip 5.69046pts<\cfrac{1}{\nu}.$ (18)

The average SEP for $M$-ary phase-shift keying ($M$-PSK) signaling by using
the MGF in (18) is given as follows:

$\displaystyle P_{s}$
$\displaystyle=\cfrac{1}{\pi}\bigintss_{0}^{(M-1)\pi/M}M_{\gamma_{\text{act}}}\left(\cfrac{-\text{sin}^{2}(\pi/M)}{\text{sin}^{2}x}\right)dx.$
(19)

We can numerically calculate the SEP in (19) by using the Gamma distribution
parameters and obtain BEP as $P_{e}\approx P_{s}/\log_{2}M$. Accordingly, the
BEP for binary phase-shift keying (BPSK) simplifies to

$\displaystyle P_{e}$
$\displaystyle=\cfrac{1}{\pi}\bigintss_{0}^{\pi/2}\left(1+\cfrac{\nu}{\text{sin}^{2}x}\right)^{-k}dx.$
(20)

##  IV Total Power Consumption Model and Energy Efficiency Analysis

In this section, we present power consumption models for the proposed
amplifying and passive RIS designs, and we analyze and compare the EE for both
systems. The main power consuming elements can be listed as transmitter,
receiver, amplifiers and RIS elements. There are two PAs in the system, one at
the Tx and one between the RISs. We assume ideal PAs whose power efficiency is
given as [40]

$\displaystyle\frac{P_{\text{out}}}{P_{\text{amp}}}=\eta_{\text{max}}\left(\frac{P_{\text{out}}}{P_{\text{max}}}\right)^{\varepsilon},$
(21)

where $P_{\text{amp}}$ and $P_{\text{out}}$ correspond to power consumed by
the amplifier and the output power of the amplifier, respectively. The maximum
output power of the amplifier is set to $P_{\text{max}}$ to ensure that it
operates in the linear region. Here, $\eta_{\text{max}}\in(0,1]$ is the
maximum efficiency of the amplifier and $\varepsilon$ is a parameter that
depends on the amplifier class. We assume $\varepsilon=0.5$ for more accurate
modeling as in [41]. The power consumed by the PA can be obtained by
reorganizing (21) as

$\displaystyle
P_{\text{amp}}=\frac{1}{\eta_{\text{max}}}\sqrt{P_{\text{out}}P_{\text{max}}}.$
(22)

The phase shift of each RIS element is arranged by programmable electronic
circuits that consume power as well. The power consumption of the RIS depends
on the phase resolution of RIS elements [42] and modeled as

$\displaystyle P_{\text{RIS}}=NP_{n}(b),$ (23)

where $P_{n}(b)$ is the power consumption of each RIS element which is a
function of bit-resolution. We consider $6$-bit phase resolution for each RIS
element, which consumes $7.8$ mW power according to [4]. The total power
consumption of the amplifying RIS-assisted system is expressed as follows:

$\displaystyle P_{\text{tot}}^{\text{act}}=\alpha
P_{t}+P_{\text{Tx}}+P_{\text{Rx}}+NP_{n}(b)+\beta\sqrt{P_{\text{out}}P_{\text{max}}},$
(24)

where $\alpha=\omega_{\text{max}}^{-1}$ and $\beta=\eta_{\text{max}}^{-1}$
with $\omega_{\text{max}}$ and $\eta_{\text{max}}$ represent the maximum
efficiency of the transmit PA and the PA between the RISs, respectively. We
assume that $P_{t}=P_{\text{max}}$ for the transmit PA. Here, $P_{\text{Tx}}$
and $P_{\text{Rx}}$ are the hardware dissipated static powers at Tx and Rx,
respectively and therefore they are constant and do not depend on the system
parameters. The values for the power consumption model parameters are given in
Table III as stated in [4]. Likewise, the power consumption of the passive
RIS-assisted system can be written as follows only by omitting the power
consumed by the PA between the RISs:

TABLE III: Power consumption model parameters Parameter | Value
---|---
$\alpha$ | 1.2
$\beta$ | 1.2
$P_{n}(b)$ | 7.8 mW
$P_{\text{Tx}}$ | 9 dBW
$P_{\text{Rx}}$ | 10 dBm

$\displaystyle P_{\text{tot}}^{\text{pas}}=\alpha
P_{t}+P_{\text{Tx}}+P_{\text{Rx}}+NP_{n}(b),$ (25)

The bit-per-joule energy efficiency $\left(\eta_{\text{EE}}\right)$ of a
system can be expressed as

$\displaystyle\eta_{\text{EE}}=\frac{R_{i}BW}{P_{\text{tot}}^{i}},$ (26)

where $BW$ is the communication bandwidth, $R_{i}$ is the achievable rate, and
$P_{\text{tot}}^{i}$ is the total consumed power by the system where
$i\in\\{\text{act},\text{pas}\\}$. The amplifying RIS-assisted system consumes
more power but provides higher capacity in return. On the other hand, passive
RIS-assisted system is less power consuming and provides less capacity due to
its fully passive nature. The EE analysis is an important criterion
determining which system performs better in terms of energy consumption. In
this study, we compute EE for different systems through computer simulations
without particularly focusing on the optimization of the EE.

##  V Numerical Results

In this section, we provide numerical results for both the amplifying and
passive RIS models to evaluate and compare the performances of them under
several configurations. Achievable rate, bit error rate (BER), power
consumption, and EE have been presented by computer simulations and discussed
in detail. We consider the systems as shown in Figs. 1 and 2 with parameters
$d_{v}=5$ m, $d_{h}=5$ m, $d=50$ m, $P_{t}=30$ dBm, $P_{\text{max}}=30$ dBm,
$f_{c}=28$ GHz, $BW=180$ kHz, $F=5$ dB, $K_{1}=K_{2}=5$, $N=128$,
$n_{\text{rx}}=n_{\text{tot}}=-100$ dBm, and $G_{\text{max}}=30$ dB, unless
specified otherwise[43, 44, 45]. Achievable rate and EE simulations are
performed with $10^{6}$ iterations.

Figure 4: BER results for (a) $P_{\text{max}}=10$ dBm and (b)
$P_{\text{max}}=20$ dBm.

###  V-A Performance Evaluation

In this subsection, numerical results for BER and achievable rate are
presented. For the BER analysis, two different setups with $P_{\text{max}}=10$
dBm and $P_{\text{max}}=20$ dBm are considered. Fig. 4(a) shows the BER
performance when $P_{\text{max}}=10$ dBm, while Fig. 4(b) exhibits the results
for $P_{\text{max}}=20$ dBm. Here, the corresponding theoretical values are
given, as well. From Fig. 4(a), one can easily observe that an error floor
occurs after $P_{t}=25$ dBm for $N=32$ case. On the other hand, an error floor
is not observed in Fig. 4(b). This is because of the limitation of
$P_{\text{out}}$ when the input signal becomes too strong such that the
amplifier cannot stay in the linear region if it continues to enhance the
signal with the current gain. In this case, when $P_{\text{out}}$ is fixed to
$P_{\text{max}}$, further increment in $P_{t}$ does not lead a better error
performance after a certain point, and thus we observe an error floor. On the
other hand, in Figs. 4(a) and (b) we do not observe any error floor when
$P_{\text{out}}$ is smaller than $P_{\text{max}}$. Furthermore, the better BER
performance is achieved with larger $N$ and higher $P_{t}$ in both cases as
long as $P_{\text{out}}$ does not reach $P_{\text{max}}$. Based on these
investigations, we can conclude that it may be beneficial to keep
$P_{\text{max}}$ high enough to prevent any error floor. In addition, it
should be emphasized that using a larger RIS can further strengthen the
received signal and an error floor appears for lower $P_{t}$ values due to the
limitation of $P_{\text{out}}$.

Figure 5: Achievable rates of the system for (a) $d_{h}=50$ m and (b)
$d_{h}=100$ m.

Achievable rates of both designs for varying $d_{h}$ with several $N$ values
are presented in Fig. 5. For this figure, the lines labeled as $N=128$, $256$,
and $512$ represent the achievable rates for the active model, where those
labeled with $2N=$ $256$, $512$, and $1024$ stand for the passive model. The
dashed lines show the performance behaviours when there is not any output
power limitation for the amplifier (i.e., results with a fixed $G$ value of
$30$ dB). In Fig. 5(a), the achievable rates for the active model overlap with
the corresponding dashed lines when $P_{\text{out}}\leq P_{\text{max}}$ and
$G_{\text{opt}}=G_{\text{max}}$. We observe that the intersection points with
the dashed lines occur at smaller values of $d_{h}$ as $N$ decreases. The
reason behind is that $G_{\text{opt}}$ begins to decrease after the
intersection points as the amplifying RIS gets closer to the Tx to keep
$P_{\text{out}}=P_{\text{max}}$. Moreover, it is a well known fact that
placing an RIS close to the Tx or Rx results in a better system performance.
This can be seen in both the Figs. 5(a) and (b), however, that is not the case
with the amplifying RIS design. Placing the amplifying RIS close to the Tx
provides lower achievable rates. We mentioned earlier that the reason behind
this is the limitation of $P_{\text{out}}$ and having a greater
$P_{\text{max}}$ would be a solution to this problem. Based on these
examinations, one can conclude that the amplifying RIS can overcome the common
problem that generally affects system performance unpleasingly in passive RIS
designs, which is the necessity of placing the RIS close to the Tx or Rx. As
seen in Fig. 5(a), the worst achievable rate is obtained when the RIS is
placed in the middle of the Rx and Tx for the passive RIS. This gets even
worse in Fig. 5(b) when the Tx and Rx are further apart. The amplifying RIS
design significantly reduces the multiplicative path loss effect by amplifying
the combined signal by RIS1 as it uses a PA between the two RISs. Using the
amplifying RIS instead of the passive one can compensate this performance
drop, even enhance the performance further.

Figure 6: Achievable rates of the amplifying RIS-assisted system for (a)
$P_{t}=20$ dBm and (b) $P_{t}=10$ dBm. The numbers on the markers indicate
$G_{\text{opt}}$ values.

Similar to passive RIS scenarios, the RIS size has a considerable impact on
achievable rate also for the active RISs. Fig. 6 illustrates the achievable
rates of the amplifying RIS design where Figs. 6(a) and (b) stand for
$P_{t}=20$ dBm and $P_{t}=10$ dBm, respectively. Here, the numbers for each
marker signify $G_{\text{opt}}$ for corresponding $N$. Here, $G_{\text{opt}}$
begins to decrease after a point for all of the cases except for
$P_{\text{max}}=20$ dBm in Fig. 6(b). Different from the previous case, the
reason behind the occurrence of the break points is the bigger $N$ values,
such that $P_{\text{in}}$ increases if more reflecting elements are used and
again limits $G_{\text{opt}}$. In Fig. 6(a), $P_{\text{out}}$ reaches
$P_{\text{max}}$ for large $N$ and amplifier cannot operate with
$G_{\text{max}}$ even if $P_{\text{max}}=20$ dBm. However, when $P_{t}$ is
reduced to 10 dBm as in Fig. 6(b), we see that $P_{\text{out}}$ cannot reach
$P_{\text{max}}=20$ dBm even for large $N$ and the amplifier can boost the
signal with $G_{\text{max}}$.

Figure 7: Achievable rates of the amplifying RIS-assisted system for different
$P_{t}$ and $P_{\text{max}}$.

Tx power and RIS size cause similar effects on the system performance because
both of them enhance $P_{\text{in}}$. As seen from Figs. 6(a) and (b),
increasing $P_{t}$ does not always have an extra constructive effect on the
system capacity because the achievable rates become identical for the same
$P_{\text{max}}$ after a certain point as clearly demonstrated in Fig. 7 as
well. In Fig. 7, the achievable rates are the same after $N=200$ and $N=60$
when $P_{\text{max}}=20$ dBm and $P_{\text{max}}=10$ dBm, respectively,
because $P_{\text{out}}=P_{\text{max}}$ at these $N$ values. Since
$P_{\text{max}}=10$ dBm can be reached by lower $P_{\text{in}}$ values, the
intersection point occurs at lower $N$ compared to the case
$P_{\text{max}}=20$ dBm. On the other hand, a considerable performance
difference is examined where $N$ is smaller than 100 and 50 for
$P_{\text{max}}=20$ dBm and $P_{\text{max}}=10$ dBm, respectively. These
observations indicate that the system performance is not enhanced with $P_{t}$
after the break points, nevertheless, it can be boosted slightly by using
larger $N$. This can be explained by the fact that there are two RISs at both
sides of the amplifier. Increasing $N$ for RIS1 can enhance $P_{\text{in}}$
just as $P_{t}$, however, $P_{\text{out}}$ will be the same if it is already
at $P_{\text{max}}$, so this increment in $N$ for RIS1 does not affect the
achievable rate. Despite this, if more reflecting elements are used for RIS2,
slightly higher achievable rates are achieved although the total power is the
same such that it is distributed among those reflecting elements. The reason
of this advance is the beamforming gain which increases with $N$.

Considering the results presented in this subsection, the main performance
limiting factors are the PA parameters, which are $P_{\text{max}}$ and
$G_{\text{max}}$. Therefore, $N$ and $P_{t}$ has a considerable effect on the
performance depending on the PA parameters.

Figure 8: (a) Energy efficiencies of the amplifying RIS-assisted system for
varying $N$ and (b) corresponding achievable rates and $P_{\text{tot}}$ values

###  V-B Energy Efficiency Evaluation

In this subsection, numerical results for EE and power consumption are
presented. Fig. 8(a) shows the EE values for varying $N$ and different
$P_{\text{max}}$ values where Fig. 8(b) stands for the corresponding
achievable rates and $P_{\text{tot}}$. In Fig. 8(a), the EE values start to
decline after some points because the achievable rates are almost at a
constant level for $P_{\text{max}}=20$ dBm and $P_{\text{max}}=10$ dBm while
$P_{\text{tot}}$ is still increasing as seen in the Fig. 8(b). When
$P_{\text{max}}=50$ dBm, a worse EE performance is obtained as the increase in
power consumption has a greater effect than the increase in the achievable
rate. Unless $P_{\text{max}}$ is too high, the total power consumption of the
active design is nearly the same as the passive RIS. Consequently, the active
design performs better in terms of EE.

EE is investigated for varying $P_{t}$ and different $P_{\text{max}}$ values
in Fig. 9(a). Achievable rates and $P_{\text{tot}}$ values are also given in
Fig. 9(b) for the same scenario. For $P_{\text{max}}=20$ dBm and
$P_{\text{max}}=10$ dBm, although the amplifying RIS does not consume high
amount of power, the EE begins to decrease after some points. The reason is
that increasing $P_{t}$ after these points does not affect the achievable
rate, but increases $P_{\text{tot}}$. For the case of $P_{\text{max}}=50$ dBm,
as we do not reach $P_{\text{max}}$, both the power consumption of the PA at
the base station and the PA between the RISs increases. Accordingly, we
observe that the EE starts to decrease after a certain point even if the
achievable rate keeps increasing.

Finally, Figs. 10(a) and (b) represent the EE, achievable rate, and
$P_{\text{tot}}$ for varying $P_{t}$ and different $P_{\text{max}}$ values.
Unlike Figs. 8 and 9, the break points occur when the amplifier cannot amplify
the signal more than $G_{\text{max}}=30$ dB. Thus, increasing the
$P_{\text{max}}$ only leads to higher power consumption as explained in (22)
while it does not have a positive effect on the achievable rate. This causes
the EE to show an interesting downward trend.

Figure 9: (a) Energy efficiencies for different $P_{\text{max}}$ values
varying with $P_{t}$ and (b) corresponding achievable rates and
$P_{\text{tot}}$ values

In the view of these observations, one can conclude that the EE is affected by
many parameters such as $P_{t}$, $P_{\text{max}}$, $N$, and $G_{\text{max}}$.
As shown in (22), $P_{\text{max}}$ and $P_{t}$ have a more clear effect on the
EE compared to the other parameters. Increasing these parameters to have a
higher achievable rate does not always lead to a more energy efficient system.
To reach a system with a maximum EE, all of these parameters should be jointly
optimized, which requires advanced optimization techniques and might be a
topic for the future works.

Figure 10: (a) Energy efficiencies for different $P_{t}$ values varying with
$P_{\text{max}}$ and (b) corresponding achievable rates and $P_{\text{tot}}$
values

##  VI Conclusion

In this paper, we have proposed an amplifying RIS design that utilizes a
single PA. The differences and advantages of the proposed design have been
clearly pointed out compared to the conventional passive RIS design. We have
presented its signal model for the amplifying RIS systems, and optimized the
amplifier gain and the phase response of RIS1 and RIS2 to maximize the
achievable rate. Ultimately, active RIS-assisted systems provide higher
communication capacity and better error rate performance than the passive RIS-
assisted systems. In addition, they enable the flexibility to place the
amplifying RIS anywhere in between Tx and Rx as it greatly reduces the effect
of double path loss, unlike passive RIS that should be placed closer to Tx or
Rx. On top of that, active RIS-assisted systems are more energy efficient,
although they consume more power.

This paper presented a new design for active RIS architectures by deploying a
single active component. Extending our design to MIMO systems introduces
complex optimization problems such as the joint optimization of transmit
precoding, phases of the reflecting elements, and gain of the PA needed.
Potentially, deep learning and alternating optimization algorithms can be used
to deal with these problems, however, these are beyond the scope of this first
study. Future works may also include joint uplink and downlink communication,
the optimization and extension of this system to MIMO systems, as well as
real-world experimental results.

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|  Recep A. Tasci (S’20) received his B.S degree in Electrical and Electronics
Engineering from Istanbul Medipol University in 2020. He is currently a
research and teaching assistant at Koç University. His research interests
include wireless communications, reconfigurable intelligent surfaces, channel
modeling, and signal processing.
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|  Fatih Kilinc (S’20) received his B.S degree from Istanbul Medipol
University in 2020. He is currently pursuing M.S. degree at Koç University. He
is a research and teaching assistant at Koç University. His research interest
include channel modeling, intelligent surfaces and signal processing for
wireless communications.
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|  Ertugrul Basar (S’09-M’13-SM’16) received his Ph.D. degree from Istanbul
Technical University in 2013. 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 beyond 5G
systems, index modulation, intelligent surfaces, waveform design, and signal
processing for communications. Dr. Basar currently serves as a Senior Editor
of IEEE Communications Letters and an Editor of IEEE Transactions on
Communications and Frontiers in Communications and Networks. He is a Young
Member of Turkish Academy of Sciences and a Senior Member of IEEE.
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|  George C. Alexandropoulos (S’07-M’10-SM’15) is an Assistant Professor with
the Department of Informatics and Telecommunications, National and
Kapodistrian University of Athens, Greece. His research interests span the
general areas of algorithmic design and performance analysis for wireless
networks with emphasis on multi-antenna transceiver hardware architectures,
active and passive reconfigurable metasurfaces, integrated communications and
sensing, millimeter wave and THz communications, as well as distributed
machine learning algorithms. He has received the best Ph.D. thesis award 2010,
the IEEE Communications Society Best Young Professional in Industry Award
2018, the EURASIP Best Paper Award of the Journal on Wireless Communications
and Networking 2021, the IEEE Marconi Prize Paper Award in Wireless
Communications 2021, and a Best Paper Award from the IEEE GLOBECOM 2021.
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