# RIS-Assisted Energy Harvesting Gains for Bistatic Backscatter Networks:
Performance Analysis and RIS Phase Optimization

Diluka Galappaththige*, , Fatemeh Rezaei*, , Chintha Tellambura, Sanjeewa
Herath *D. Galappaththige and F. Rezaei contributed equally to this work.D.
Galappaththige, F. Rezaei, and C. Tellambura with the Department of Electrical
and Computer Engineering, University of Alberta, Edmonton, AB, T6G 1H9, Canada
(e-mail: {diluka.lg, rezaeidi, ct4}@ualberta.ca).
S. Herath is with the Huawei Canada, 303 Terry Fox Drive, Suite 400, Ottawa,
Ontario K2K 3J1 (e-mail: sanjeewa.herath@huawei.com).

###### Abstract

Inexpensive tags powered by energy harvesting (EH) can realize green (energy-
efficient) Internet of Things (IoT) networks. However, tags are vulnerable to
energy insecurities, resulting in poor communication ranges, activation
distances, and data rates. To overcome these challenges, we explore the use of
a reconfigurable intelligent surface (RIS) for EH-based IoT networks. The RIS
is deployed to enhance RF power at the tag, improving EH capabilities. We
consider linear and non-linear EH models and analyze single-tag and multi-tag
scenarios. For single-tag networks, the tag’s maximum received power and the
reader’s signal-to-noise ratio with the optimized RIS phase-shifts are
derived. Key metrics, such as received power, harvested power, achievable
rate, outage probability, bit error rate, and diversity order, are also
evaluated. The impact of RIS phase shift quantization errors is also studied.
For the multi-tag case, an algorithm to compute the optimal RIS phase-shifts
is developed. Numerical results and simulations demonstrate significant
improvements compared to the benchmarks of no-RIS case and random RIS-phase
design. For instance, our optimal design with a $200$-element RIS increases
the activation distance by 270 and 55 compared to those benchmarks. In
summary, RIS deployment improves the energy autonomy of tags while maintaining
the basic tag design intact.

###### Index Terms:

Bistatic backscatter communication (BiBC), Reconfigurable intelligent surface
(RIS), Performance analysis.

##  I Introduction

###  I-A The Problems with Energy Harvesting Backscatter Tags

Parcel tracking using passive electronic tags is one of the many potential
applications of the Internet of Things (IoT). The global parcel volume
surpassed $131$ billion in $2020$, showing a 27 year-over-year increase. In
the United States alone, $59$ million parcels were generated daily in $2021$,
projected to reach $25$-$40$ billion with a 5-10 annual growth rate from
$2022$-$2027$. Similar growth trends are observed globally. Barcode-based
tracking is currently employed, but electronic tag-based tracking offers
advantages such as enhanced labor productivity, throughput, warehousing
efficiency, and real-time data accuracy for quality control. Tags without
batteries are particularly suitable because of their cost-effectiveness and
compact size. Their applications include medical and healthcare, agriculture,
livestock, logistics, retail chains, and passive IoT networks [1, 2, 3]. These
tags have low-cost and low-power circuits with limited processing
capabilities. They rely on backscatter modulation, a process described in
detail in [4, 5], where they reflect radio-frequency (RF) signals to
communicate with the reader.

Passive tags encounter two main problems related to their reliance on RF
energy harvesting (EH) for power: activation failure and energy outage (EO).
Activation failure occurs when the tag fails to reach the activation threshold
($P_{b}$), typically around -20m [6], required to initiate the EH circuitry
[7]. Imperfections in the matching network between the tag’s antenna and the
EH circuit can cause this failure. The matching network aims to align the
complex impedance of the EH circuit with the antenna’s impedance, optimizing
power transfer and minimizing signal reflections. However, the EH circuit’s
impedance depends on the incident input power due to nonlinear devices,
leading to reduced circuit efficiency with changes in input power. The second
problem is EO. Ambient energy sources are unpredictable with RF power density
values as low as $1$$\sim$100/^2 and varying with distance [3]. As a result,
there is a risk of an EO where the tag does not reach the activation
threshold. These problems cause ultra-low power (n-), short communication
ranges ($\leq\qty{6}{}$), short activation distances, and low data rates
($\leq\qty{1}{bps/}$). It is clear that all these problems are initiated
whenever the incident RF energy is low. Addressing that issue is the main
focus of this paper.

Backscatter networks can be categorized into three types: monostatic,
bistatic, and ambient. In monostatic systems, the reader and emitter are co-
located, resulting in doubled path loss [8]. Ambient systems rely on
reflecting existing RF signals, which are highly unpredictable. Bistatic
systems, on the other hand, offer better support for applications such as
warehouses (Fig.1). These systems deploy dedicated RF emitters, either single
or multiple, to provide energy to the tags and enable backscatter modulation.
By optimizing the locations of multiple emitters, a larger area can be covered
(Fig.1), maximizing coverage and performance. Dedicated emitters have
advantages over ambient signals, including predictability, reduced
interference, control over the system, and knowledge of ambient signal
parameters [9]. However, the high cost, complexity, and transmit powers
associated with dedicated emitters can be problematic.

These considerations motivate the following questions: 1) what is the best way
to increase the chance of the incident RF power on the tag exceeding $P_{b}?$
2) How can that goal be reached without increasing dedicated RF emitters’ cost
and energy expenditure?

Tag
---
Emitter
---
RIS
---
Figure 1: A warehouse use case of BiBC network. $\mathcal{E}$
---

$\mathbf{h}(d_{h})$
---

$f_{1}(d_{f_{1}})$
---

$(d_{x})$
---

$(d_{y})$
---

$\mathbf{\Theta}$
---

$\mathbf{g}_{1}(d_{g_{1}}\\!)$
---

$\mathcal{R}$
---

$u_{1}(d_{u_{1}})$
---

$f_{K}(d_{f_{K}})$
---

$\mathbf{g}_{K}(d_{g_{K}}\\!)$
---

$u_{K}(d_{u_{K}})$
---

RIS
---
Blockage
---
$\mathcal{T}_{1}$
---
$\mathcal{T}_{K}$
---
Figure 2: An RIS-assisted BiBC. Red lines denote interference signals and
$d_{f_{k}},d_{\mathbf{g}_{k}},d_{u_{k}}$ and $d_{\mathbf{h}}$ are respectively
$\mathcal{E}$-$\mathcal{T}_{k}$, RIS-$\mathcal{T}_{k}$,
$\mathcal{T}_{k}$-$\mathcal{R}$, and $\mathcal{E}$-RIS distances.

###  I-B Existing Solutions

Various solutions can enhance the performance of passive tags, including the
use of multi-antenna configurations [10], energy beamforming techniques, RF
energy harvesters with improved activation thresholds [11], and channel coding
methods [5, 12]. Additionally, active tunnel diodes in the tags have been
explored as a solution [13]. However, these techniques may not be feasible for
passive tags due to their limited processing capability, power constraints,
and cost limitations. Increasing the transmit power at the emitter or
deploying multiple emitters is also not an energy-efficient (green) solution.

To overcome these issues, study [14] is the first to propose the use of a
reconfigurable intelligent surface (RIS) in BiBC networks. A RIS is a
synthetic surface consisting of numerous passive reflectors, also known as
meta-material elements [15, 16]. These reflectors have the ability to
independently modify the characteristics of incoming electromagnetic waves,
such as phase, amplitude, frequency, or polarization [15, 16]. Through real-
time adaptation, the reflectors can be programmed to dynamically adjust to the
wireless channel conditions. Each reflector is equipped with one or more
switches, with a switching frequency of up to 5, which ensures minimal
switching time compared to the channel coherence time [17]. By dynamically
adjusting the reflectors, the RIS controller coordinates their actions to
create constructive and destructive interference patterns in the reflected
waves as required [15, 16].

A RIS consumes a few watts during the reconfiguration states and much less
during idle states, e.g., 6 per element for $4$-bits resolution phase shifting
[18]. A RIS can provide significant gain even without an amplifier, ranging
from 3040 relative to the isotropic radiation depending on the size of the
surface and frequency [15, 16]. Typically, the size of a single reflector is
much smaller than the signal wavelength ($\lambda$), ranging between
$\lambda/10$ and $\lambda/5$ [16]. The RIS integrates with existing networks
without modifying the basic network design [19].

###  I-C Problem Statement and Contributions

We address the problem of maximizing the number of passive tags supported in a
given area, such as a large warehouse (Fig.1) while minimizing the number of
carrier emitters required. As mentioned before, passive tags have limited
activation/communication ranges and data rates, relying on a minimum RF power
(activation threshold) for self-activation. Existing solutions like multiple-
antenna tags, coding, batteries, multiple emitters, repeaters, and relays have
certain limitations. To overcome these challenges, we adopt the approach
proposed in [14] by deploying a RIS in the BiBC network (Fig.2). The RIS
increases the incident RF power on the tags, allowing them to harvest more
energy and improving their reliability and communication range. Optimal
placement and configuration of the phase shift elements in the RIS can be
determined to achieve maximum gains. Moreover, continuous phase shifts (i.e.,
infinite levels in $[-\pi,\pi]$) are not viable because more switches per
reflector will cost more for the RIS. For $16$ phase shifts, $\log_{2}(16)=4$
switches are needed [20]. Hence, to keep the size, cost, and power consumption
of the RIS down, phase quantization is essential.

We propose using a RIS in the BiBC network (Fig. 2) to enhance its energy
security and performance. Initially, we consider the case of a single tag and
assume a linear energy EH model for the tag. The RIS acts as a reflector,
increasing the power delivered to the passive tags by reflecting the RF
emitter signal. By optimizing the phase shifts of the RIS, we aim to improve
harvested power, communication range, and data rate. Our results provide
insights into the improvements in harvested energy, data rate, and reliability
achieved through the use of the RIS. Furthermore, to support passive IoT
applications involving multiple tags, we develop RIS optimization techniques
that aid tag activation while enhancing communication performance.

Our study differs from [14] in several aspects. Firstly, we focus on
increasing the energy security of passive tags, resulting in improved
activation distance, communication range, and data rates. In contrast, [14]
primarily aims to enhance the communication performance of passive/semi-
passive tags. We achieve this by employing a RIS in the source-to-tag links,
which enhances the incident energy at the tags. In contrast, [14] uses a RIS
to increase the signal-to-noise ratio (SNR) at the reader, serving a different
purpose. Secondly, our study utilizes fully passive tags with reflective
modulation schemes such as binary phase-shift keying (BPSK), implemented by
switching the tag’s impedance between two levels. These passive modulation
schemes can be easily extended to higher-order quadrature phase-shift keying
(QPSK) and $M$-quadrature amplitude modulation (QAM) [4, 5]. On the other
hand, [14] employs semi-passive and passive tags with frequency-shift keying
(FSK) modulation to enable multiple access. However, implementing FSK requires
more from the tags as they need to generate distinct carrier frequencies.
Therefore, achieving higher-order FSK may not be feasible with simple tags.
Thirdly, in [14], the beamformer at the multi-antenna emitter and the RIS
phase shifts are optimized to minimize the transmit power at the emitter. In
our study, we aim to maximize the sum rate of multiple tags by optimizing RIS
phase shifts with a single-antenna emitter that maintains constant transmit
power.

As well, there are analytical and algorithmic differences between our study
and [14]. While [14] uses a minorization maximization (MM) algorithm with a
semi-passive tag, we propose a closed-form solution for RIS phase shifts that
simultaneously maximize the received power of the passive tag and the SNR at
the reader. This analytical framework allows us to quantify the benefits of
utilizing an RIS for a single passive tag. We evaluate system efficiency,
reliability, and error performance through metrics such as harvested power,
achievable rate, outage probability, bit error rate (BER), achievable
diversity order, and RIS phase shift quantization errors. Additionally, in the
multi-tag scenario, [14] extends the single-tag MM algorithm with minor
modifications. In contrast, we employ a fractional programming-based
optimization approach that enables multiple access for passive tags. Our
approach focuses on maximizing the sum rate of the tags while ensuring tag
activation and reducing tag-to-tag interference.

Specifically, the main contributions of this paper can be summarized as
follows:

  1. 1.

In the single-tag network, we optimize the RIS phase shifts to maximize the
received signal power at the tag. This has a closed-form solution, allowing us
to determine the optimal RIS phase shifts directly. Furthermore, we analyze
the impact of the RIS on the EH process by deriving the average harvested
power at the tag. This quantifies the role of the RIS in enhancing the EH
performance.

  2. 2.

To analyze the performance of the system, we require the probability density
function (PDF) and cumulative distribution function (CDF) of the optimal SNR
at the reader. However, obtaining analytical expressions for these functions
appears intractable, even in the single-tag case. To overcome this challenge,
we test Gaussian and Gamma approximations for the PDF and CDF by assuming a
Nakagami-$m$ fading. Among the two approximations, we select the Gamma
approximation as it provides a more accurate analysis of the harvested power,
achievable rate, outage probability, BER, and achievable diversity order.

  3. 3.

We also investigate the effect of RIS phase shift errors to quantify the
effects of hardware limitations and imperfect channel estimation in practical
scenarios.

  4. 4.

We further explore the optimization of the RIS for a multi-tag scenario. To
this end, we introduce an iterative algorithm that aims to maximize the
achievable sum rate while ensuring that each tag meets its minimum power
requirement. This goal is achieved by optimizing the RIS phase shifts
iteratively.

  5. 5.

Our simulations and numerical results validate the accuracy of the derived
analytical results, demonstrating a close agreement between them. These
simulations provide valuable insights for system design and assessment, aiding
in the practical implementation and evaluation of the proposed approach.

We show that RIS optimization provides a green solution for energizing and
improving the activation distance and communication range of passive tags,
enabling massive connectivity without modifications or additional processing
at the tag. This makes passive tags suitable for forming the backbone of
passive IoT networks. The analytical results presented in this paper offer
valuable insights for practical applications. Before proceeding to the
technical contributions, we provide an overview of related works in the field
of RIS-assisted BackCom literature.

###  I-D Previous Contributions on RIS-Assisted BackCom

Existing works often overlook the tag activation requirement and primarily
concentrate on utilizing RIS to enhance the tag-reflected signal for improved
communication. However, this approach assumes either the use of active tags
powered by batteries, where energy harvesting is unnecessary, or tags that can
be activated with extremely low power ($P_{b}=-\infty$), which is not
representative of actual tags. These works optimize the RIS without
considering the EH constraint, aiming to improve outage, sum rates, error
probabilities, or other metrics. References [21, 22, 23, 24, 25, 26, 27, 28]
fall into this category and differ significantly from our contribution. A
summary of relevant and recent contributions is provided in Table I.

TABLE I: Summary of related works.

Setup | Reference | Tag type | EH constraint | Contribution
---|---|---|---|---
MoBC | [21] | – | ✗ | BER analysis
[22] | – | ✗ | BER, outage, and rate analysis
AmBC | [23] | Passive | ✗ | BER and coverage analysis
[24] | – | ✗ | BER analysis
[29] | Passive/Semi-passive | ✓ | Throughput and coverage analysis
[25] | Semi-passive | ✗ | Outage and spectral efficiency analysis
[26] | Passive | ✗ | Outage and BER analysis
[30] | Passive | ✓ | Rate maximization
[28]∗ | – | ✗ | Transmit power minimization
BiBC | [27] | – | ✗ | Rate maximization
[14] | Semi-passive/Passive | ✓ | Transmit power minimization
This Paper | Passive | ✓ |  | BER, outage, rate, and harvested power analysis
---
Rate maximization

  * ∗

Symbiotic radio - the reader supports both the primary and the backscatter
transmission

In contrast to the works [21, 22, 23, 24, 25, 26, 27, 28], a few studies [29,
14, 30] have explored the use of RIS to enhance the EH potential of
backscatter networks. In [14], RIS was applied to a BiBC network, serving as
the inspiration for our study. The main purpose of the RIS in [14] was to
improve the signal-to-noise ratio (SNR) at the reader. In contrast, we utilize
the RIS to enhance the RF signal received by the tags. Earlier, we discussed
the differences between [14] and our study.

Another study [29] utilizes a RIS to enable direct tag-to-tag communication
when the direct source-to-tag links are blocked. If the tag receives signal
power exceeding a certain threshold $P_{b}$, it first harvests energy and then
reflects its data. Otherwise, it operates using a battery. Hence, [29]
considers the use of semi-passive tags, which is fundamentally different from
ours. In our previous work [30], we also explored the use of a RIS to energize
a single tag in an AmBC network, where the RIS is optimized to satisfy the
tag’s EH constraint while minimizing interference from the ambient source.
Once again, the system model in that study is distinct from the one considered
here.

Structure: This paper is structured as follows. Section II introduces the
system and channel model, including the tag and signal models. The analysis of
the single-tag system is presented in Section III. Section IV explores the
multi-tag scenario. The analytical results are validated through simulation
examples in Section V. Finally, Section VI provides the conclusion of the
paper and outlines potential future research directions.

Notation: For random variable $X$, $f_{X}(\cdot)$ and $F_{X}(\cdot)$ denote
PDF and CDF. $\mathbb{E}\\{\cdot\\}$ and $\mathbb{V}\rm{ar}\\{\cdot\\}$ denote
the expectation and variance. Lowercase bold and uppercase bold denote vectors
and matrices. $\mathbf{A}^{\mathrm{T}}$, $\mathbf{A}^{\mathrm{H}}$ denote
transpose and Hermitian transpose of matrix $\mathbf{A}$. Moreover, the
positive part of real $x$ is denoted by $[x]^{+}=\max(0,x)$. The gamma
function $\Gamma(a)$ is given in [31, Eq. (8.310.1)], $D_{v}(x)$ is parabolic
cylinder function [31, eq. (9.240)], and $\gamma(n,x)$ is the lower incomplete
gamma function [31, Eq. (8.350)]. The complementary error function is
${\rm{erfc}}(x)$ [31, eq. (8.25.4)] and
$Q(x)=\frac{1}{2}{\rm{erfc}}(x/\sqrt{2})$ is the Gaussian $Q$-function.
Finally, $\mathcal{CN}(\bm{\mu},\mathbf{R})$ is a complex Gaussian vector with
mean $\bm{\mu}$ and co-variance matrix $\mathbf{R}$.

##  II System Model and Preliminaries

###  II-A System and Channel Models

We consider a RIS-assisted BiBC setup consisting of a single-antenna emitter
($\mathcal{E}$), $K$-single-antenna passive tags, a single-antenna reader
($\mathcal{R}$), and a RIS with $N$ passive reflective elements (Fig. 2). We
use $\mathcal{T}_{k}$ to denote the $k$th tag. Since tags are batteryless and
entirely rely on EH, we deploy a RIS to deliver as much RF power as possible.
To do this, the RIS controller sets the states of individual reflectors to
adjust the phase shifts intelligently to maximize the received power at each
tag. For this, the controller requires channel state information (CSI) for all
the channels in Fig. 2. We assume that the controller has a backhaul
connection between the RIS and $\mathcal{E}$, which can provide all such
necessary information [32]. For brevity, we index the set of RIS passive
elements as $\mathcal{N}=\\{1,\ldots,N\\}$ and the set of tags as
$\mathcal{K}=\\{1,\ldots,K\\}$.

We consider a block, flat-fading channel model where the channel response
remains constant over the duration of a block and changes independently from
block to block [33]. During each fading block, the direct channel coefficients
in the $\mathcal{E}$-$\mathcal{T}_{k}$ link and the
$\mathcal{T}_{k}$-$\mathcal{R}$ link are denoted as $f_{k}$ and $u_{k}$,
respectively. Moreover, the channel coefficient vectors in the
$\mathcal{E}$-RIS link and the RIS-$\mathcal{T}_{k}$ link are denoted as
$\mathbf{g}_{k}=[g_{k,1},\ldots,g_{k,N}]^{\rm{T}}\in\mathbb{C}^{N\times 1}$
and $\mathbf{h}=[h_{1},\ldots,h_{N}]^{\rm{T}}\in\mathbb{C}^{N\times 1}$,
respectively. Here, $g_{k,n}$ and $h_{n}$ for $n\in\mathcal{N}$ and
$k\in\mathcal{K}$ are the channels between the $n$th element of the RIS and
$\mathcal{T}_{k}$, and the $n$th element of the RIS and $\mathcal{E}$,
respectively. All channel envelopes are assumed to be independent Nakagami-$m$
distributed, where $m$ is the shape parameter [34]. A unified representation
of all four channels $\mathcal{A}=\\{f_{k},u_{k},h_{n},g_{k,n}\\}$ is thus
given as

$\displaystyle a=\alpha_{a}\exp({j\theta_{a}}),$ (1)

where $\alpha_{a}$ is the envelope of the $a$ and $\theta_{a}\in[-\pi,\pi]$ is
the phase of $a$. The PDF of $\alpha_{a}$ is given as

$\displaystyle
f_{\alpha_{a}}(x)=\frac{2m_{a}^{m_{a}}x^{2m_{a}-1}}{\Gamma{(m_{a})}\Omega_{a}^{m_{a}}}\exp{\left(\frac{-m_{a}x^{2}}{\Omega_{a}}\right)},$
(2)

where $m_{a}$ is the shape parameter and $\Omega_{a}=m_{a}\zeta_{a}$ is the
scaling parameter, in which $\zeta_{a}$ accounts for the large-scale
fading/path-loss. It should be noted that, since the RIS reflective elements
are co-located, the large-scale fading parameters are the same for all of
them, i.e., $\zeta_{g_{k,n}}=\zeta_{g_{k}}$ and $\zeta_{h_{n}}=\zeta_{h}$ for
$n\in\mathcal{N}$.

######  Remark 1.

The Nakagami-$m$ model is versatile to represent a variety of propagation
environments. For instance, $m=1$ represents Rayleigh fading, and
$m\rightarrow\infty$ represents the no fading scenario. Hence, our performance
analysis covers the special case of Rayleigh fading channels [21, 35].

We make the following key assumptions:

  1. A1:

Perfect channel state information (CSI) is available for the
$\mathcal{E}$-$\mathcal{T}_{k}$, $\mathcal{E}$-RIS, RIS-$\mathcal{T}_{k}$,
$\mathcal{T}_{k}$-$\mathcal{R}$, and $\mathcal{E}$-$\mathcal{R}$ channels in
the system. However, channel estimation poses challenges due to the passive
nature of the RIS and tags, as well as the large number of RIS elements.
Sophisticated methods and novel pilot designs are required for accurate
estimation. While a comprehensive treatment of channel estimation is beyond
the scope of this work, a possible estimation strategy for the system in Fig.
2 can be envisioned. Initially, the RIS is set to the non-reflecting state,
allowing estimation of the direct $\mathcal{E}$-$\mathcal{R}$ link, the
$\mathcal{E}$-$\mathcal{T}_{k}$-$\mathcal{R}$ link, and the
$\mathcal{E}$-$\mathcal{T}_{k}$ link using existing methods [36, 37].
Subsequently, the cascaded $\mathcal{E}$-RIS-$\mathcal{R}$ channel can be
resolved into its $\mathcal{E}$-RIS and RIS-$\mathcal{R}$ components using
appropriate techniques [38]. Finally, the RIS-$\mathcal{T}_{k}$ channels can
be obtained using information from other available paths. Thus, this
assumption is justified because emerging techniques can address the challenges
associated with channel estimation.

  2. A2:

We assume that the RIS-$\mathcal{R}$ link (blue line in Fig. 2) is blocked or
negligible because the RIS focuses the reflected beam toward the tags [39].
The $\mathcal{E}$-$\mathcal{R}$ signal (red line in Fig. 2) is interference
felt at $\mathcal{R}$. When $\mathcal{R}$ has CSI and knows the signal of
$\mathcal{E}$, it can cancel this interference signal [14, 40, 41]. If full
CSI is unavailable or cancellation is impossible, our analytical results serve
as fundamental limits or upper bounds of achievable performance.

  3. A3:

We consider the linear model for the EH process at $\mathcal{T}_{k}$. Even
though practical EH circuits have non-linear operating characteristics and an
activation threshold, the linear model can be accurate in certain operating
regions.

###  II-B Passive Tag Operation

As mentioned before, we consider the use of passive tags only as they have the
potential to enable massive connectivity at a low cost. Thus,
$\mathcal{T}_{k}$ does not have batteries and entirely relies on the harvested
energy from the RF signal transmitted by $\mathcal{E}$. Specifically, the
harvested energy powers $\mathcal{T}_{k}$’s circuit operation and enables it
to backscatter data to $\mathcal{R}$ simultaneously [42]. Thus, the amount of
harvested energy is the fundamental parameter that ensures the success or
failure of $\mathcal{T}_{k}$ to communicate. We next set up the basics
necessary to estimate the EH ability of $\mathcal{T}_{k}$.

Let $P_{T,k}$ be the received power at $\mathcal{T}_{k}$’s antenna. The
reflection coefficient of $\mathcal{T}_{k}$ is then given as
$\sqrt{\beta}q_{m},$ where $q_{m}$ is the normalized backscatter symbol
selected from a multi-level (${M}$-ary) modulation (i.e., $|q_{{m}}|^{2}\leq
1$) and $0<\beta<1$ is the fraction of power reflected at $\mathcal{T}_{k}$.
Therefore, when $\mathcal{T}_{k}$ reflects the RF signal, $\beta P_{T,k}$ of
the RF power is reflected while the rest $P_{l,k}=(1-\beta)P_{T,k}$ is
absorbed for EH purposes. The EH circuits convert the RF power $P_{l,k}$ into
direct current (DC) power by using a rectifier. With linear and nonlinear
models for the energy harvester, the harvested DC power ($P_{h,k}$) can be
defined as [7]

$\displaystyle P_{h,k}=\begin{cases}\phi P_{l,k},&\text{Linear},\\\
\Phi(P_{l,k}),&\text{Nonlinear},\end{cases}$ (3)

where $\phi$ is the power conversion efficiency, typically measured around
31.861.4% at 2.45 [43], independent of the RF power $P_{l,k}$. Moreover,
$\Phi(\cdot)$ represents the nonlinear EH function [7]. Although the linear
model offers simplicity, non-linear EH models are also widely used (see [7,
44] and references therein). However, our solution can be easily extended to
such models as well. We omit the details for brevity.

###  II-C Signal Model

Node $\mathcal{E}$ transmits an unmodulated carrier signal $\sqrt{P}s$ to
energize tags, where $s$ is the carrier signal satisfying
$\mathbb{E}\\{|s|^{2}\\}=1$ and $P$ is the carrier transmission power. In
general, $s$ should be designed to maximize the EH potential of tags. Thus,
waveforms with high peak-to-average power ratio (PAPR) are used as these
increase the RF-to-DC conversion efficiency of the tag’s EH circuit [45].
Hence, $s$ can be white noise, with a flat power spectral density and high
PAPR [46]. Other designs include orthogonal frequency-division multiplexing
(OFDM), chaotic, and multisine waveforms [45, 46].

The signal $\sqrt{P}s$ is received at $\mathcal{T}_{k}$ through the direct
channel, $f_{k}$, and the reflective channels of the RIS, $\mathbf{h}$ and
$\mathbf{g}_{k}$. The received signal at $\mathcal{T}_{k}$ is thus given
as111Note that, since $\mathcal{T}_{k}$ is a passive device without active RF
components, the noise contributed by $\mathcal{T}_{k}$ can be neglected [47].

$\displaystyle
y_{k}=\sqrt{P}f_{k}s+\sqrt{P}\mathbf{g}_{k}^{\rm{T}}\mathbf{\Theta}\mathbf{h}s.$
(4)

The first term is due to the direct path and the second is from the RIS’s
reflective elements. In (4), $\mathbf{\Theta}\in\mathbb{C}^{N\times N}$ is a
diagonal matrix that captures the reflection properties (the magnitude of
attenuation and the phase shift) of the RIS elements, i.e.,
$\mathbf{\Theta}={\rm{diag}}\left(\eta_{1}\exp{(j\theta_{1})},\dots,\eta_{N}\exp{(j\theta_{N})}\right)$,
where $\eta_{n}\exp{(j\theta_{n})}$ is the reflection coefficient of the
$n$-th RIS element with the magnitude of attenuation $\eta_{n}$ and the phase
shift $\theta_{n}\in[-\pi,\pi]$. Discrete phase shifts, resulting in a phase
quantization error, will be treated in Section III-H. Moreover, here we only
consider a passive RIS without active amplification, i.e., $\eta_{k}\leq
1,\forall k.$ However, an active RIS that can amplify and reflect incident RF
signals, i.e., $\eta_{n}>1$, is a potential future extension of this work.

Using (4), the received power at $\mathcal{T}_{k}$ is given as

$\displaystyle
P_{T,k}=P|f_{k}+\mathbf{g}_{k}^{\rm{T}}\mathbf{\Theta}\mathbf{h}|^{2}.$ (5)

$\mathcal{T}_{k}$ harvests energy from the received power, $P_{T,k}$ (5), and
modulates the received signal with its normalized ${M}$-ary backscatter
signal, $q_{k},\mathbb{E}\\{|q_{k}|^{2}\\}=1$, to be transmitted to
$\mathcal{R}$. The received signal at $\mathcal{R}$ is thus given as

$\displaystyle y_{R}=\sqrt{\beta
P}\sum\nolimits_{k\in\mathcal{K}}u_{k}(f_{k}+\mathbf{g}_{k}^{\rm{T}}\mathbf{\Theta}\mathbf{h})sq_{k}+z,$
(6)

where $z\sim\mathcal{CN}(0,\sigma^{2}_{z})$ is additive white Gaussian noise
(AWGN) with mean $0$ and variance $\sigma^{2}_{z}$.

##  III Single-Tag System

Here, we consider $K=1$, i.e., the single-tag setup. The RIS phase shifts
should be optimized to maximize both the received signal power at
$\mathcal{T}$ and the achievable rate at $\mathcal{R}$. To this end, the
optimization problem is formulated as follows:

$\displaystyle\mathcal{P}_{S,1}:\underset{\mathbf{\Theta}}{\text{maximize}}$
$\displaystyle{{\rm{log}}_{2}(1+\gamma)},$ (7a) subject to
$\displaystyle(1-\beta)P_{T}\geq P_{b}^{\prime},$ (7c)
$\displaystyle|\bar{\theta}_{n}|\leq 1,$

where $\bar{\theta}_{n}=\eta_{n}\exp{(j\theta_{n})}$ and
$P_{b}^{\prime}\triangleq P_{b}/\phi$ is the respective threshold value. Note
that this constraint is equivalent to the nonlinear EH case with
$P_{b}^{\prime}=\Phi^{-1}(P_{b})$. Hence, we adopt the linear EH model (3) due
to its tractability. Here, $P_{T}$ is the received signal power at
$\mathcal{T}$ and given as

$\displaystyle
P_{T}={P}\left|\left(\alpha_{f}e^{j\theta_{f}}+\sum\limits_{n\in\mathcal{N}}{\eta_{n}\alpha_{g_{n}}\alpha_{h_{n}}e^{j(\theta_{g_{n}}+\theta_{h_{n}}+\theta_{n})}}\right)\right|^{2}.$
(8)

When the tag reflects this power level, the corresponding received SNR at
$\mathcal{R}$ is given as

$\displaystyle\gamma$ $\displaystyle=$
$\displaystyle\bar{\gamma}|u(f+\mathbf{g}^{\rm{T}}\mathbf{\Theta}\mathbf{h})|^{2}$
(9) $\displaystyle=$
$\displaystyle\bar{\gamma}\left|\alpha_{u}e^{j\theta_{u}}\\!\left(\\!\alpha_{f}e^{j\theta_{f}}\\!+\\!\sum\limits_{n\in\mathcal{N}}{\\!\eta_{n}\alpha_{g_{n}}\alpha_{h_{n}}e^{j(\theta_{g_{n}}+\theta_{h_{n}}+\theta_{n})}}\\!\right)\right|^{2}\\!,\quad$

where $\bar{\gamma}=P\beta/\sigma^{2}_{z}$.

###  III-A Proposed Solution

To maximize the received power at $\mathcal{T}$ (which will also maximize the
received SNR at $\mathcal{R}$ if the interference is negligible), the received
signal at $\mathcal{T}$ through the RIS (the signal terms inside the summation
of (8)), should be constructively added to the signal received from the direct
path $f$, by compensating the phase distortion effect of the multipath
channel, i.e., in (8), $\theta_{f}=\theta_{g_{n}}+\theta_{h_{n}}+\theta_{n}$.
Thus, the phase shift at the $n$th RIS element should be adjusted as [32]

$\displaystyle\theta^{\star}_{n}=\max_{-\pi\leq\theta_{n}\leq\pi}P_{T}=\theta_{f}-(\theta_{g_{n}}+\theta_{h_{n}}),~{}n\in\mathcal{N}.\quad$
(10)

Thereby, the optimal received power at $\mathcal{T}$ is derived as

$\displaystyle
P_{T}^{\star}={P}\left|\left(\alpha_{f}+\sum\nolimits_{n\in\mathcal{N}}{\eta_{n}\alpha_{g_{n}}\alpha_{h_{n}}}\right)\right|^{2}.$
(11)

Similarly, by using (10), the optimal received SNR at $\mathcal{R}$ is derived
as

$\displaystyle\gamma^{\star}=\bar{\gamma}\left|\alpha_{u}\left(\alpha_{f}+\sum\nolimits_{n\in\mathcal{N}}{\eta_{n}\alpha_{g_{n}}\alpha_{h_{n}}}\right)\right|^{2}.$
(12)

Next, we derive the approximate distributions of the optimal received power
(11) and SNR (12) for the single-tag system. We develop Gaussian and Gamma
approximations and compare them briefly, ultimately selecting the Gamma
approximation for further analysis. Using it, we analyze various aspects of
the single-tag system, including harvested power, achievable rate, outage
probability, bit error rate (BER), and diversity order. Additionally, we
investigate the impact of phase quantization error on the achievable rate.

###  III-B Average Harvested Power

We first analyze the impact of the EH process before proceeding to the
performance analysis section. As mentioned before, $\mathcal{T}$ activation
depends on the amount of harvested power, i.e., the sensitivity threshold,
$P_{b}$, which is about -20m for commercial passive RFID tags [6]. If
$\mathcal{T}$ cannot exceed this threshold, it will not be operational to
reflect the RF signal to communicate its data. Thus, we can envision an EO,
which will cripple the system. To understand this scenario, we will
analytically determine the average harvested power below. Note that this
derivation is limited to the linear EH model only because of its simplicity.
However, the achievable rate, outage, and BER depend on the amount of
reflected power at $\mathcal{T}$ and hence the reflection coefficient,
$\alpha$.

The instantaneous harvested power at $\mathcal{T}$ is given as
$P_{h}=\phi(1-\beta)P_{T}^{\star}$, where $\phi$ is in (3) and $P_{T}^{\star}$
is the received power at $\mathcal{T}$ (11). The average harvested power is
given as

$\displaystyle\bar{P}_{h}=\phi(1-\beta)\mathbb{E}\\{P_{T}^{\star}\\}.$ (13)

To derive $\mathbb{E}\\{P_{T}^{\star}\\}$, first, we define
$X\triangleq\sum_{n\in\mathcal{N}}{\eta_{n}\alpha_{g_{n}}\alpha_{h_{n}}}$ and
then, $\mathbb{E}\\{P_{T}^{\star}\\}$ is given as

$\displaystyle\mathbb{E}\\{P_{T}^{\star}\\}=P\left(\sigma_{\alpha_{f}}^{2}+\mu_{\alpha_{f}}^{2}+\sigma_{X}^{2}+\mu_{X}^{2}+2\mu_{\alpha_{f}}\mu_{X}\right),$
(14)

where $\mu_{X}$ and $\sigma^{2}_{X}$ are respectively given as

$\displaystyle\mu_{X}$ $\displaystyle=$
$\displaystyle\sum\nolimits_{n\in\mathcal{N}}{\eta_{n}\mu_{\alpha_{g_{n}}}\mu_{\alpha_{h_{n}}}},\quad$
(15a) $\displaystyle\sigma^{2}_{X}$ $\displaystyle=$
$\displaystyle\sum\nolimits_{n\in\mathcal{N}}{\left(\mu^{(2)}_{x_{n}}-\mu_{x_{n}}^{2}\right)},$
(15b)

in which
$\mu^{(2)}_{x_{n}}={\eta_{n}^{2}\mu^{(2)}_{\alpha_{g_{n}}}\mu^{(2)}_{\alpha_{h_{n}}}}$,
$\mu_{x_{n}}={\eta_{n}\mu_{\alpha_{g_{n}}}\mu_{\alpha_{h_{n}}}}$, and

$\displaystyle\mu_{\alpha_{a}}^{(m)}$ $\displaystyle=$
$\displaystyle\frac{\Gamma(m_{a}+\frac{m}{2})}{\Gamma(m_{a})}\left(\frac{\Omega_{a}}{m_{a}}\right)^{\frac{m}{2}},\quad$
(16a) $\displaystyle\sigma^{2}_{\alpha_{a}}$ $\displaystyle=$
$\displaystyle\Omega_{a}\left(1-\frac{1}{m_{a}}\left(\frac{\Gamma(m_{a}+\frac{1}{2})}{\Gamma(m_{a})}\right)^{2}\right),$
(16b)

for $a\in\mathcal{A}$. Besides, $\mu_{V}=\mathbb{E}\\{V\\}$,
$\sigma^{2}_{V}=\mathbb{V}\text{ar}\\{V\\}$ and
$\mu^{(m)}_{V}=\mathbb{E}\\{V^{m}\\}$.

###  III-C Statistics of the Optimal Received Power and SNR

To derive the distribution of the optimal SNR (12), we consider
$\gamma^{\star}=\bar{\gamma}\Lambda^{2}$, where $\Lambda\triangleq\alpha_{u}Y$
and $Y\triangleq\alpha_{f}+X$. The $\alpha_{u},\alpha_{f},\alpha_{h_{n}}$, and
$\alpha_{g_{n}},\forall n$ are independent Nakagami-$m$ variables, and the
exact derivations of the PDFs of $Y$, $\Lambda$, $P_{T}^{\star}$, and
$\gamma^{\star}$ are analytically intractable. To overcome this challenge, we
will develop two approximations, i.e., Gaussian and Gamma, of $Y$ and
$\Lambda$ to derive the approximate distribution of $P_{T}^{\star}$ and
$\gamma^{\star}$. We will verify these via simulations.

######  Theorem 1.

Gaussian Approximation: Using the moment matching technique, $Y$ and $\Lambda$
can be approximated as Gaussian variables with
$Y\sim\mathcal{N}(\mu_{Y},\sigma^{2}_{Y})$ and
$\Lambda\sim\mathcal{N}(\mu_{\Lambda},\sigma^{2}_{\Lambda})$, within the
interval $[0,\infty)$, where $\mu_{Y}=\mu_{\alpha_{f}}+\mu_{X}$,
$\sigma^{2}_{Y}=\sigma^{2}_{\alpha_{f}}+\sigma^{2}_{X}$, and
$\mu_{\Lambda}=\mu_{\alpha_{u}}\mu_{Y}$,
$\sigma^{2}_{\Lambda}=\mu^{(2)}_{\alpha_{u}}\mu^{(2)}_{Y}-\mu_{Y}^{2}\mu_{\alpha_{u}}^{2}$.

###### Proof.

See Appendix -A. ∎

######  Theorem 2.

Gamma Approximation: $Y$ and $\Lambda$ can be approximated as Gamma variables
with $Y\sim\Gamma(k_{Y},\lambda_{Y})$ and
${\Lambda}\sim\Gamma(k_{\Lambda},\lambda_{\Lambda})$, where
$k_{C}=\mu^{2}_{C}/\sigma^{2}_{C}$ and $\lambda_{C}=\sigma^{2}_{C}/\mu_{C}$
for $C\in\\{Y,\Lambda\\}$ are the first and second moments [48, 49]. Hence,
the CDF and PDF of $Y$ and $\Lambda$ are given as

$\displaystyle F_{C}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(k_{C})}\gamma(k_{C},\frac{r}{\lambda_{C}}),\quad$
(17a) $\displaystyle f_{C}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma(k_{C})\lambda_{C}^{k_{C}}}r^{k_{C}-1}\exp\left(-\frac{r}{\lambda_{C}}\right),$
(17b)

for $r\geq 0$.

Since $P_{T}^{\star}=PY^{2}$ and $\gamma^{\star}=\bar{\gamma}\Lambda^{2}$, we
have

$\displaystyle F_{D}(r)$ $\displaystyle=$ $\displaystyle
F_{d}(\sqrt{r/\bar{d}}),\quad$ (18a) $\displaystyle f_{D}(r)$ $\displaystyle=$
$\displaystyle\frac{1}{(2\sqrt{\bar{d}r})}f_{d}(\sqrt{r/\bar{d}}).$ (18b)

Here, for $D=P_{T}^{\star}$, $d=Y$ and $\bar{d}=P$, and for
$D=\gamma^{\star}$, $d=\Lambda$ and $\bar{d}=\bar{\gamma}$.

######  Remark 2.

For Rayleigh fading channels, i.e., $m_{a}=1,a\in\mathcal{A}$,
$\mu_{\alpha_{a}}=\sqrt{\pi\Omega_{a}}/2$,
$\mu_{\alpha_{a}}^{(2)}=\Omega_{a}$, and
$\sigma^{2}_{\alpha_{a}}=\Omega_{a}(4-\pi)/4$ (16a). Besides, for
$\eta_{n}=\eta$, $\mu_{X}=\pi/4N\eta\sqrt{\Omega_{g}\Omega_{h}}$,
$\sigma^{2}_{X}=(1-(\pi/4)^{2})N\eta^{2}\Omega_{g}\Omega_{h},$ and
$\mu_{\Lambda}$ and $\sigma^{2}_{\Lambda}$ can be found accordingly as

$\displaystyle\mu_{\Lambda}$ $\displaystyle=$
$\displaystyle\frac{\pi}{4}\sqrt{\Omega_{u}\Omega_{f}}+\frac{\pi^{3/2}}{8}N\eta\sqrt{\Omega_{g}\Omega_{h}\Omega_{u}},\quad$
(19a) $\displaystyle\sigma^{2}_{\Lambda}$ $\displaystyle=$
$\displaystyle\Omega_{u}\sigma^{2}_{Y}+\frac{(4-\pi)}{4}\Omega_{u}\mu_{Y}^{2},$
(19b)

where $\mu_{Y}=\sqrt{\pi\Omega_{f}}/2+\mu_{X}$ and
$\sigma^{2}_{Y}=\Omega_{f}(4-\pi)/4+\sigma^{2}_{X}$.

$0$$5$$10$$15$$20$$25$$30$$35$$40$$45$$50$$0$$2\cdot 10^{-2}$$4\cdot
10^{-2}$$6\cdot 10^{-2}$$8\cdot
10^{-2}$$0.1$$0.12$$0.14$$0.16$$0.18$$0.2$$0.22$$0.24$$r$$f_{\gamma^{\star}}(r)$SimulationGaussianGamma$N=100$$N=200$$N=400$

Figure 3: The PDF of $\gamma^{\star}$ for $N=\\{100,200,400\\}$,
$P=\qty{10}{m}$, $d_{f}=\qty{10}{}$, $d_{u}=\qty{5}{}$, $d_{h}=\qty{5}{}$,
$d_{g}=\qty{6}{}$, $m_{f}=3$, $m_{u}=5$, $m_{h}=3$, $m_{g}=4$, $\beta=0.6$,
and $\eta_{n}=0.8,\,\forall n$.

$0$$5$$10$$15$$20$$25$$30$$35$$40$$45$$50$$0$$0.1$$0.2$$0.3$$0.4$$0.5$$0.6$$0.7$$0.8$$0.9$$1$$r$$F_{\gamma^{\star}}(r)$SimulationGaussianGamma$N=100$$N=400$$N=200$

Figure 4: The CDF of EO at $\mathcal{T}$ as a function of $d_{f}$ for
$P=\qty{20}{m}$, $d_{u}=\qty{5}{}$, $d_{h}=\qty{1}{}$,
$d_{g}=\sqrt{d_{h}^{2}+d_{f}^{2}}$, $m_{f}=m_{u}=m_{h}=m_{g}=3$, $\beta=0.6$,
$\phi=0.8$, and $\eta_{n}=0.8,\,\forall n$.

To verify these approximations, we plot the PDF of $\gamma^{\star}$ in Fig. 3
for different RIS sizes, $N$. The Gamma approximation demonstrates better
agreement with the exact simulation compared to the Gaussian approximation. As
a result, we exclusively utilize it due to its superior accuracy.

######  Remark 3.

Fig. 4 plots the CDF of EO at $\mathcal{T}$ for the cases with and without a
RIS. The CDF is evaluated as

$\displaystyle F_{\rm{EO}}(P_{b})={\rm{Pr}}\left((1-\beta)P_{T}^{\star}\leq
P_{b}^{\prime}\right).$ (20)

When there is no RIS to power up the tag, the range is just $\sim\qty{6}{}$,
and beyond 6, the tag goes to an EO. However, a RIS can lower the EO and
achieve higher ranges than without a RIS case. For example, a RIS with $100$
and $400$ reflecting elements extends the tag activation range to 13 and 34
metres, respectively. Therefore, a RIS can prevent the risk of EO at the tag.

###  III-D Average Achievable Rate

The achievable rate describes the bit rate a network can reliably transmit
under a long-term, stable fading regime. This measure provides valuable
insights into the use of specific coding schemes. In the context of BiBC, it
may help to identify feasible applications. The average achievable rate is the
mean Shannon capacity given by

$\displaystyle\mathcal{R}=\mathbb{E}\\{\log_{2}\left(1+\gamma^{\star}\right)\\}.$
(21)

The exact derivation of the average achievable rate given in (21) seems
intractable. However, by invoking Jensen’s inequality, the tight upper and
lower bounds are given as [50]

$\displaystyle\mathcal{R}_{\rm{lb}}\leq\mathcal{R}\leq\mathcal{R}_{\rm{ub}},$
(22)

where $\mathcal{R}_{\rm{lb}}$ and $\mathcal{R}_{\rm{ub}}$ are respectively
defined as

$\displaystyle\mathcal{R}_{\rm{lb}}$ $\displaystyle=$
$\displaystyle\log_{2}\left(1+\left[\mathbb{E}\left\\{{1}/{\gamma^{\star}}\right\\}\right]^{-1}\right),\quad$
(23a) $\displaystyle\mathcal{R}_{\rm{ub}}$ $\displaystyle=$
$\displaystyle\log_{2}\left(1+\mathbb{E}\left\\{{\gamma^{\star}}\right\\}\right).$
(23b)

where
$\mathbb{E}\left\\{{\gamma^{\star}}\right\\}=\bar{\gamma}\mu^{(2)}_{\Lambda}$,
and

$\displaystyle\mathbb{E}\left\\{{1}/{\gamma^{\star}}\right\\}=\frac{1}{\mathbb{E}\left\\{{\gamma^{\star}}\right\\}}+\frac{\sigma^{2}_{{\gamma}^{\star}}}{\left[\mathbb{E}\left\\{{\gamma^{\star}}\right\\}\right]^{3}},$
(24)

in which
$\sigma^{2}_{{\gamma}^{\star}}=\bar{\gamma}^{2}\sigma^{2}_{{\Lambda}^{2}}$. To
calculate $\sigma^{2}_{{\gamma}^{\star}}$, we need to calculate
$\sigma^{2}_{{\Lambda}^{2}}=\mu^{(4)}_{\Lambda}-[\mu^{(2)}_{\Lambda}]^{2}$ of
$\Lambda$. Specifically, when $\Lambda$ is approximated with Gamma
distribution, ${\Lambda}\sim\Gamma(k_{\Lambda},\lambda_{\Lambda})$, [49]

$\displaystyle\mu^{(m)}_{\Lambda}=k_{\Lambda}^{m}\frac{\Gamma(m+k_{\Lambda})}{\Gamma(k_{\Lambda})}.$
(25)

######  Remark 4.

To gain further insights, we let the number of RIS elements grow without a
bound. We observe that the transmit power can be scaled inversely proportional
to the square of the number of IRS elements in this operating regime, i.e.,
$\lim_{N\rightarrow\infty}P=P_{A}/N^{2}$. This is the well-known squared power
gain provided by the RIS [51]. Hence, the lower and upper rate bounds can be
shown to approach an asymptotic limit using this transmit power scaling law as
(Appendix -B)

$\displaystyle\lim_{N\rightarrow\infty}\mathcal{R}_{\rm{lb}}=\mathcal{R}_{\infty},\quad\text{and}\quad\lim_{N\rightarrow\infty}\mathcal{R}_{\rm{ub}}=\mathcal{R}_{\infty},$
(26)

where $\mathcal{R}_{\infty}$ is given in (27)

$\displaystyle\mathcal{R}_{\infty}=\log_{2}\left(1+\bar{\gamma}_{A}(\sigma_{\alpha_{u}}^{2}+\mu_{\alpha_{u}}^{2})\left(\eta\frac{\Gamma(m_{g}+1/2)\Gamma(m_{h}+1/2)}{\Gamma(m_{g})\Gamma(m_{h})}\sqrt{\frac{\Omega_{g}\Omega_{h}}{m_{g}m_{h}}}\right)^{2}\right)$
(27)

with $\bar{\gamma}_{A}=\beta P_{A}/\sigma_{z}^{2}$ and $\eta=\eta_{n},\forall
n$.

###  III-E Outage Probability

The EO at $\mathcal{T}$ and rate/SNR outage at $\mathcal{R}$ both affect the
ability to deliver information reliably. An EO occurs when the received signal
power falls below $\mathcal{T}$’s activation threshold ($P_{b}$). In contrast,
the SNR outage probability is the probability that the instantaneous SNR falls
below a threshold ($\gamma_{\rm{th}}$). Mathematically, the outage is
expressed as

$\displaystyle
P_{\rm{out}}=P_{\rm{out}}^{\rm{EO}}+\left(1-P_{\rm{out}}^{\rm{EO}}\right)P_{\rm{out}}^{\rm{SNR}},$
(28)

where $P_{\rm{out}}^{\rm{EO}}={\rm{Pr}}\\{(1-\beta)P_{T}^{\star}\leq P_{b}\\}$
and $P_{\rm{out}}^{\rm{SNR}}=\rm{Pr}\\{\gamma\leq\gamma_{\rm{th}}\\}$. A tight
approximation to the outage (28) is obtained as

$\displaystyle P_{\rm{out}}\approx
F_{P_{T}^{\star}}(\gamma_{\rm{th}})+\left(1-F_{P_{T}^{\star}}(\gamma_{\rm{th}})\right)F_{\gamma^{\star}}(\gamma_{\rm{th}}),$
(29)

where $F_{\gamma^{\star}}(r)$ and $F_{P_{T}^{\star}}(r)$ are given in (18a).

###  III-F Average Bit/Symbol Error Rate

The average BER is the ratio between the number of bits received in error and
the number of bits transmitted. This measure is widely used to assess the
reliability and quality of wireless links. In this section, we derive the BER
of the system for BPSK modulation, a simple yet robust modulation scheme. The
conditional error probability of BPSK is given as
${P_{\rm{BER}}}\left(\gamma\right)=\lambda{Q}({\sqrt{\nu\gamma}})$, with
$\lambda=1$ and $\nu=2$ [52]. The average BER of the system with BPSK, using
Gamma distribution (17a), is thus given as (Appendix -C)

$\displaystyle\bar{P}_{\rm{BER}}^{\rm{Gamma}}$ $\displaystyle=$ $\displaystyle
A_{1}(2\bar{A})^{-\frac{k_{\Lambda}}{2}}\Gamma(k_{\Lambda})\exp{\left(\frac{\hat{B}_{1}^{2}}{8\bar{A}}\right)}D_{-k_{\Lambda}}\left(\frac{\hat{B}_{1}^{2}}{\sqrt{2\bar{A}}}\right),$

where $\bar{A}=A\nu\bar{\gamma},\bar{B}=B\sqrt{\nu\bar{\gamma}}$,
$A_{1}=\exp{(-C)}/(\Gamma(k_{\Lambda})\lambda_{\Lambda}^{k})$,
$\hat{B}_{1}=\bar{B}+1/\lambda_{\Lambda}$, in which $A=0.3842,B=0.7640,$ and
$C=0.6964$. Note that the BER analysis above can be extended for higher-order
modulation schemes such as QPSK and $M$-QAM [53, 54]. For brevity, such
analyses are omitted here.

###  III-G Achievable Diversity Order

The diversity order reveals how the BER or SNR outage probability decays with
the SNR. It is defined as the negative slope of those measures in the high SNR
regime as follows:

$\displaystyle
G_{d}=-\lim_{\bar{\gamma}\rightarrow\infty}\frac{\log(P_{\rm{out}}^{\rm{SNR}})}{\log(\bar{\gamma})}=-\lim_{\bar{\gamma}\rightarrow\infty}\frac{\log(\bar{P}_{\rm{BER}})}{\log(\bar{\gamma})}.$
(30)

We next derive the outage probability and BER in the high SNR domain. We use
the first-order polynomial expansion of the PDF approximations.

######  Lemma 1.

The asymptotic Outage Probability and BER are respectively given as

$\displaystyle
P_{\rm{out}}^{\infty}=\frac{1}{\Gamma(k_{\Lambda}+1)\lambda_{\Lambda}^{k_{\Lambda}}}\left(\frac{\gamma_{\rm{th}}}{\bar{\gamma}}\right)^{{k_{\Lambda}}/{2}}+\mathcal{O}\left(\bar{\gamma}^{-({k_{\Lambda}}/{2}+1)}\right),$
(31)

and

$\displaystyle\bar{P}_{\rm{BER}}^{\infty}$ $\displaystyle=$
$\displaystyle({\lambda_{E}}\bar{\gamma})^{-k_{\Lambda}/2}+\mathcal{O}\left(\bar{\gamma}^{-(k_{\Lambda}/2+1)}\right),$
(32)

where
$\lambda_{E}=\nu\left[\left(2^{k_{\Lambda}/2}/3+(3/2)^{k_{\Lambda}/2}\right)C_{1}\right]^{2/k_{\Lambda}}$
and
$C_{1}={\lambda}\Gamma(k_{\Lambda}/2)/(8\Gamma(k_{\Lambda})\lambda_{\Lambda}^{k_{\Lambda}})$.

###### Proof.

See Appendix -D. ∎

Using (31), the diversity order $G_{d}$ and the coding gain, $O_{c}$, are
respectively given as $G_{d}=k_{\Lambda}/2$, and
$O_{c}={1}/({\Gamma(k_{\Lambda}+1)\lambda_{\Lambda}^{k_{\Lambda}}})$ [55].
Besides, from (32) the array gain becomes $G_{a}=\lambda_{E}$.

######  Remark 5.

According to Section III-C,
${k_{\Lambda}}={\mu^{2}_{\Lambda}}/{\sigma^{2}_{\Lambda}}$ and
$\lambda_{\Lambda}=\sigma^{2}_{\Lambda}/\mu_{\Lambda}$. Thus, the diversity
order $G_{d}$ can be given in (33),

$\displaystyle
G_{d}=\frac{1}{2}\frac{\Upsilon_{u}^{2}\zeta_{u}\left(\Upsilon_{f}\zeta_{f}^{1/2}+N\hat{\Upsilon}\zeta_{g}^{1/2}\zeta_{h}^{1/2}\right)^{2}}{\zeta_{u}\left(\bar{\Upsilon}_{u}+\Upsilon_{u}^{2}\right)\left(\bar{\Upsilon}_{f}\zeta_{f}+N{\Upsilon}^{\prime}\zeta_{g}\zeta_{h}\right)+\bar{\Upsilon}_{u}\zeta_{u}\left(\Upsilon_{f}\zeta_{f}^{1/2}+N\hat{\Upsilon}\zeta_{g}^{1/2}\zeta_{h}^{1/2}\right)^{2}},$
(33)

where we assume that $\eta_{n}=\eta,n\in\mathcal{N}$. In (33), we define
$\mu_{\alpha_{a}}\triangleq\Upsilon_{a}\zeta_{a}^{1/2}$, and
$\sigma^{2}_{\alpha_{a}}\triangleq\bar{\Upsilon}_{a}\zeta_{a}$, where
$\Upsilon_{a}={\Gamma(m_{a}+{1}/{2})}/{\Gamma(m_{a})}$ and
$\bar{\Upsilon}_{a}=m_{a}-\Upsilon_{a}^{2}$ (16a). Besides, for
$\eta_{n}=\eta,n\in\mathcal{N}$,
$\mu_{X}=N\hat{\Upsilon}\zeta_{g}^{1/2}\zeta_{h}^{1/2}$,
$\sigma^{2}_{X}=N{\Upsilon}^{\prime}\zeta_{g}\zeta_{h}$, where
$\hat{\Upsilon}\triangleq\eta\Upsilon_{g}\bar{\Upsilon}_{h}$ and
${\Upsilon}^{\prime}\triangleq\eta^{2}\bar{\Upsilon}_{h}\bar{\Upsilon}_{g}$.

To gain further insight, we let the number of RIS elements increase to a large
limit, i.e., $N\rightarrow\infty$. After several mathematical manipulations,
we have222Note that $G_{d}$ is derived using the Gamma approximation and
letting $N$ grow large. Hence, $G_{d}$ may not be exact. However, since the
exact derivation is mathematically intractable, this approximation provides
valuable insights.

$\displaystyle\lim_{N\rightarrow\infty}G_{d}=\frac{1}{2}\frac{\Upsilon_{u}^{2}}{\bar{\Upsilon}_{u}}=\frac{1}{2}\left(m_{u}\left(\frac{\Gamma(m_{u})}{\Gamma(m_{u}+\frac{1}{2})}\right)^{2}-1\right)^{-1}.\quad$
(34)

From (34), unexpectedly, we observe that the diversity order of the proposed
RIS-assisted system (Fig. 2) is constant in the high regime of $N$ and solely
depends on the $\mathcal{T}$-$\mathcal{R}$ channel parameters. This behavior
is because $\mathcal{T}$ acts as a keyhole/pinhole [4]. Hence, regardless of
the number of RIS elements or antennas of $\mathcal{E}$, the rank of combined
channels $\mathcal{E}$-$\mathcal{T}$ and $\mathcal{E}$-RIS-$\mathcal{T}$
becomes one. Therefore, this combined channel does not contribute to the
diversity order because of keyhole channel properties [56].

The coding gain and array gain are similarly derived as

$\displaystyle\lim_{N\rightarrow\infty}O_{c}$ $\displaystyle=$
$\displaystyle\frac{1}{\Gamma\left(\frac{\Upsilon_{u}^{2}}{\bar{\Upsilon}_{u}}+1\right)}\left({N\hat{\Upsilon}\sqrt{\zeta_{u}\zeta_{g}\zeta_{h}}\bar{\Upsilon}_{u}{\Upsilon}_{u}^{-1}}\right)^{-\frac{\Upsilon_{u}^{2}}{\bar{\Upsilon}_{u}}},$
(35) $\displaystyle\lim_{N\rightarrow\infty}G_{a}$ $\displaystyle=$
$\displaystyle\lambda_{\bar{E}}{\left({N\hat{\Upsilon}\sqrt{\zeta_{u}\zeta_{g}\zeta_{h}}\bar{\Upsilon}_{u}{\Upsilon}_{u}^{-1}}\right)}^{-2},$
(36)

where
$\lambda_{\bar{E}}=\nu\left[\left(2^{{\Upsilon_{u}^{2}}/{(2\bar{\Upsilon}_{u})}}/3+(3/2)^{{\Upsilon_{u}^{2}}/{(2\bar{\Upsilon}_{u})}}\right)C_{\bar{1}}\right]^{\frac{2\Upsilon_{u}^{2}}{\bar{\Upsilon}_{u}}}$,
and
$C_{\bar{1}}=\lambda\Gamma({\Upsilon_{u}^{2}}/{(2\bar{\Upsilon}_{u})})/(8\Gamma({\Upsilon_{u}^{2}}/{\bar{\Upsilon}_{u}}))$.

From (35) and (36), while the diversity order is independent of the number of
RIS elements, the coding gain and the array gain are both functions of it -
see Fig. 8 and Fig. 9.

###  III-H Effect of RIS Phase Quantization Errors

Due to hardware limitations and imperfect CSI, adopting continuous RIS phase
shifts over $[-\pi,\pi]$ might be infeasible. Thus, phase shifts would be
quantized to a set of discrete values [20], i.e., each RIS element uses only a
finite number of phase shifts. Accordingly, the discrete phase of $n$-th RIS
element is given as $\hat{\theta}^{\star}=\kappa\pi/2^{D}$, where $D$
represents the number of quantization bits, $\kappa=\min_{\varrho\in\\{0,\pm
1,\ldots,\pm 2^{D-1}\\}}|{\theta}^{\star}-\varrho\pi/2^{D}|$, and
${\theta}^{\star}$ is the optimal phase shift, given in (10). Thus, the
quantization error is
$\epsilon_{n}={\theta}^{\star}_{n}-\hat{\theta}^{\star}_{n},n\in\mathcal{N},$
which can be shown to be uniformly distributed for large quantization levels,
i.e., $\epsilon_{n}\sim\mathcal{U}[-\tau,\tau)$ and $\tau=\pi/2^{D}$ [57, 58].
Therefore, the optimal received SNR $\gamma^{\star}$ given in (12) with the
discrete phase shifts may be expressed as

$\displaystyle\gamma^{\star}$ $\displaystyle=$
$\displaystyle\bar{\gamma}\left|\alpha_{u}\left(\alpha_{f}+\sum\nolimits_{n\in\mathcal{N}}{\eta_{n}\alpha_{g_{n}}\alpha_{h_{n}}e^{j\epsilon_{n}}}\right)\right|^{2}$
(37) $\displaystyle=$
$\displaystyle\bar{\gamma}\alpha_{u}^{2}((\alpha_{f}+X_{R})^{2}+X_{I}^{2}),$

where
$X_{R}\triangleq\sum_{n\in\mathcal{N}}{\eta_{n}\alpha_{g_{n}}\alpha_{h_{n}}\cos(\epsilon_{n})}$
and
$X_{I}\triangleq\sum_{n\in\mathcal{N}}{\eta_{n}\alpha_{g_{n}}\alpha_{h_{n}}\sin(\epsilon_{n})}$.

We investigate the effect of phase quantization errors at the RIS on the
average achievable rate. According to Section III-D, we can derive an upper
bound for the average achievable rate, given in (23a), with discrete phase
shifts as follows:

$\displaystyle\bar{\mathcal{R}}_{\rm{ub}}\\!=\log_{2}\\!\left(1\\!+\\!\bar{\gamma}\mu_{\alpha_{u}}^{(2)}\left(\mu_{\alpha_{f}}^{(2)}\\!+\\!\mu_{X_{R}}^{(2)}\\!+\\!2\mu_{\alpha_{f}}\mu_{X_{R}}\\!+\\!\mu_{X_{I}}^{(2)}\right)\right),$
(38)

where $\mu_{X_{R}}^{(2)}=\sigma^{2}_{X_{R}}+\mu_{X_{R}}^{2}$,
$\mu_{X_{I}}^{(2)}=\sigma^{2}_{X_{I}}+\mu_{X_{I}}^{2}$, and

$\displaystyle\mu_{X_{R}}$ $\displaystyle=$
$\displaystyle\sum\nolimits_{n\in\mathcal{N}}{\eta_{n}\mu_{\alpha_{g_{n}}}\mu_{\alpha_{h_{n}}}}\frac{\sin(\tau)}{\tau},\quad$
(39a) $\displaystyle\sigma^{2}_{X_{R}}$ $\displaystyle=$
$\displaystyle\sum\nolimits_{n\in\mathcal{N}}{\eta_{n}^{2}\mu_{\alpha_{g_{n}}}^{(2)}\mu_{\alpha_{h_{n}}}^{(2)}\left(\frac{1}{2}+\frac{\sin\left({2\tau}\right)}{{4\tau}}\right)}-\mu_{X_{R}}^{2},\qquad$
(39b) $\displaystyle\sigma^{2}_{X_{I}}$ $\displaystyle=$
$\displaystyle\sum\nolimits_{n\in\mathcal{N}}{\eta_{n}^{2}\mu_{\alpha_{g_{n}}}^{(2)}\mu_{\alpha_{h_{n}}}^{(2)}\left(\frac{1}{2}-\frac{\sin({2\tau})}{{4\tau}}\right)},$
(39c)

where $\mu_{X_{I}}=0$, and $\mu_{\alpha_{a}}^{(m)},a\in\mathcal{A}$ is given
in (16a).

######  Remark 6.

If a RIS is placed in the $\mathcal{T}$-$\mathcal{R}$ link of a BiBC setup, we
can investigate the network’s performance following a similar approach for the
setup with RIS in the $\mathcal{E}$-$\mathcal{T}$ link. However, when both the
$\mathcal{E}$-$\mathcal{T}$ and $\mathcal{T}$-$\mathcal{R}$ links use RIS, the
Gaussian and Gamma approximations may not match well with the exact PDF and
CDF of the received SNR. We can still derive the upper and lower bounds of the
average achievable rate using a similar method given in Section III-D.

##  IV Multi-Tag System

This section develops the analysis of multiple ($K>1$) single-antenna tags in
Fig. 2. Here, by optimizing the RIS phase shifts, we simultaneously maximize
the received signal power at $\mathcal{T}_{k}$ and the achievable sum rate at
$\mathcal{R}$. To this end, the problem is formulated as follows:

$\displaystyle\mathcal{P}_{M,1}:\underset{\mathbf{\Theta}}{\text{maximize}}$
$\displaystyle\sum\limits_{k\in\mathcal{K}}{{\rm{log}}_{2}(1+\gamma_{k})},$
(40a) subject to $\displaystyle(1-\beta)P_{T,k}\geq P_{b}^{\prime},$ (40c)
$\displaystyle|\bar{\theta}_{n}|\leq 1,$

where $P_{T,k}$ is given in (5), and $\gamma_{k}$ is the signal-to-
interference-plus-noise ratio (SINR) of $\mathcal{T}$k at $\mathcal{R}$. Using
(6), $\gamma_{k}$ is obtained as

$\displaystyle\gamma_{k}=\frac{\beta
P|u_{k}(f_{k}+\mathbf{g}_{k}^{\rm{T}}\mathbf{\Theta}\mathbf{h})|^{2}}{\sum\nolimits_{i\in\mathcal{K}/k}{\beta
P|u_{i}(f_{i}+\mathbf{g}_{i}^{\rm{T}}\mathbf{\Theta}\mathbf{h})|^{2}}+\sigma^{2}_{z}}.$
(41)

In (40), the constraint (40c) guarantees the minimum power for $\mathcal{T}$k.

Because of the non-convex objective function and the constraint,
$\mathcal{P}_{M,1}$ is not amenable to popular convex algorithms. Therefore,
we develop a solution based on fractional programming.

###  IV-A Proposed Solution

First, we define
$\mathbf{a}_{k}\triangleq{\rm{diag}}(u_{k}\mathbf{g}_{k}^{\rm{T}})\mathbf{h}$
and $b_{k}\triangleq u_{k}f_{k}$. Thereby, the SINR (41) is rearranged as

$\displaystyle\gamma_{k}=\frac{\beta
P|b_{k}+\bm{\theta}^{\rm{H}}\mathbf{a}_{k}|^{2}}{\sum\nolimits_{i\in\mathcal{K}/k}{\beta
P|b_{i}+\bm{\theta}^{\rm{H}}\mathbf{a}_{i}|^{2}}+\sigma^{2}_{z}},$ (42)

where $\bm{\theta}=[\bar{\theta}_{1},\ldots,\bar{\theta}_{N}]^{\rm{T}}$. Then,
$\mathcal{P}_{M,1}$ can be treated as a fractional programming problem [59,
30]. We next apply a quadratic transform to the objective function of
$\mathcal{P}_{M,1}$ as

$\displaystyle\\!\\!\\!f(\bm{\theta},\lambda)\\!$ $\displaystyle=$
$\displaystyle\\!\sum\limits_{k\in\mathcal{K}}\\!{\rm{log}_{2}}\\!\left(\\!1\\!+\\!2\lambda_{k}\sqrt{\beta
P}{\rm{Re}}\left\\{b_{k}\\!+\\!\bm{\theta}^{\rm{H}}\mathbf{a}_{k}\right\\}\right.$
(43) $\displaystyle\left.\\!-\\!\lambda_{k}^{2}\\!\left(\\!\beta
P\\!\sum\limits_{i\in\mathcal{K}/k}\\!|b_{i}+\bm{\theta}^{\rm{H}}\mathbf{a}_{i}|^{2}\\!+\\!\sigma_{z}^{2}\right)\right)\\!,$

where $\bm{\lambda}=[\lambda_{1},\ldots,\lambda_{K}]^{\rm{T}}$ is auxiliary
variables introduced by the quadratic transformation. Thereby, we
alternatively optimize $\bm{\theta}$ and $\bm{\lambda}$. For a given
$\bm{\theta}$, the optimal $\lambda_{k}$ is found in closed-form as [59, 30]

$\displaystyle\lambda_{k}^{\star}=\frac{\sqrt{\beta
P}{\rm{Re}}\left\\{b_{k}+\bm{\theta}^{\rm{H}}\mathbf{a}_{k}\right\\}}{\ln(2)\left(\beta
P\sum\nolimits_{i\in\mathcal{K}/k}|b_{i}+\bm{\theta}^{\rm{H}}\mathbf{a}_{i}|^{2}+\sigma_{z}^{2}\right)}.$
(44)

######  Remark 7.

Without losing generality, we constrain the phase-shift vector, $\bm{\theta}$,
with the channel responses to obtain a non-negative real desired signal term,
i.e.,
$\left|b_{k}+\bm{\theta}^{\rm{H}}\mathbf{a}_{k}\right|\approx{\rm{Re}}\\{b_{k}+\bm{\theta}^{\rm{H}}\mathbf{a}_{k}\\}$.
Our simulation results also support the validity of this approach. This is due
to the fact that our method iteratively maximizes the achievable rate/SINR by
co-phasing the desired signal component while reducing interference.

Next, we must optimize $\bm{\theta}$ for a given $\bm{\lambda}$. First, by
expanding $|b_{k}+\bm{\theta}\mathbf{a}_{k}|^{2}$ and then applying several
mathematical manipulations, the objective function in (43) can be rearranged
as

$\displaystyle
f(\bm{\theta})=\sum\nolimits_{k\in\mathcal{K}}{\rm{log}}_{2}\left(1-\bm{\theta}^{\rm{H}}\mathbf{U}_{k}\bm{\theta}+2{\rm{Re}}\left\\{\bm{\theta}^{\rm{H}}\mathbf{v}_{k}\right\\}+c_{k}\right),\quad$
(45)

where $\mathbf{U}_{k}$, $\mathbf{v}_{k}$, and $c_{k}$ are defined as

$\displaystyle\mathbf{U}_{k}$ $\displaystyle\triangleq$ $\displaystyle
2(\lambda_{k}^{\star})^{2}\beta
P\sum\nolimits_{i\in\mathcal{K}/k}\mathbf{a}_{i}\mathbf{a}_{i}^{\rm{H}},\quad$
(46a) $\displaystyle\mathbf{v}_{k}$ $\displaystyle\triangleq$
$\displaystyle\lambda_{k}^{\star}\sqrt{\beta
P}\mathbf{a}_{k}-(\lambda_{k}^{\star})^{2}\beta
P\sum\nolimits_{i\in\mathcal{K}/k}b_{i}^{*}\mathbf{a}_{i},$ $\displaystyle
c_{k}$ $\displaystyle\triangleq$
$\displaystyle\\!2\lambda_{k}^{\star}\sqrt{\beta
P}{\rm{Re}}\\{b_{k}\\}\\!-\\!(\lambda_{k}^{\star})^{2}\\!\left(\\!\beta
P\\!\sum\nolimits_{i\in\mathcal{K}/k}\\!|b_{i}|^{2}\\!+\\!\sigma_{z}^{2}\\!\right)\\!.\qquad$
(46b)

Next, the corresponding optimization problem is given as

$\displaystyle\mathcal{P}_{M,2}:\underset{\bm{\theta}}{\text{maximize}}$
$\displaystyle f(\bm{\theta}),$ (47a) subject to
$\displaystyle(1-\beta)P_{{\rm{T}},k}^{\rm{Lin}}\geq P_{b}^{\prime},$ (47c)
$\displaystyle|\bar{\theta}_{n}|\leq 1,$

where $P_{{\rm{T}},k}^{\rm{Lin}}$ is the linearized received signal power at
$\mathcal{T}_{k}$, given as

$\displaystyle P_{\rm{T},k}^{\rm{Lin}}$ $\displaystyle=$
$\displaystyle\bm{\theta}_{j-1}^{\rm{H}}\bar{\mathbf{U}}_{k}\bm{\theta}_{j-1}+2{\rm{Re}}\left\\{\bm{\theta}_{j-1}^{\rm{H}}\bar{\mathbf{v}}_{k}\right\\}+\bar{c}_{k}$
(48)
$\displaystyle\\!\\!+\left(\left(\bar{\mathbf{U}}_{k}+\bar{\mathbf{U}}_{k}^{\rm{H}}\right)\bm{\theta}_{j-1}+2{\rm{Re}}\left\\{\bar{\mathbf{v}}_{k}\right\\}\right)^{\rm{H}}\left(\bm{\theta}_{j}-\bm{\theta}_{j-1}\right),\quad\,\,$

where
$\bar{\mathbf{a}}_{k}\triangleq{\rm{diag}}(\mathbf{g}_{k}^{\rm{T}})\mathbf{h}$,
$\bar{b}_{k}\triangleq f_{k}$, $\bar{\mathbf{U}}_{k}\triangleq
P\bar{\mathbf{a}}_{k}\bar{\mathbf{a}}_{k}^{\rm{H}}$,
$\bar{\mathbf{v}}_{k}\triangleq P\bar{b}_{k}^{*}\bar{\mathbf{a}}_{k}$, and
$\bar{c}_{k}\triangleq P|\bar{b}_{k}|^{2}$. Besides, $\bm{\theta}_{j}$ and
$\bm{\theta}_{j-1}$ are the current and the previous iteration values of
$\bm{\theta}$.

Because $\mathbf{a}_{k}\mathbf{a}_{k}^{\rm{H}}$ is a positive-definite matrix,
$\mathbf{U}_{k}$ is also a positive-definite matrix. Hence, the objective
function, $f(\bm{\theta})$, is a quadratic concave function of $\bm{\theta}$.
Thus, $\mathcal{P}_{2}$ can be solved as a quadratically constrained quadratic
program (QCQP) [60], see Algorithm 1.

Algorithm 1 : Algorithm for phase shift optimization.

Initialize $\bm{\theta}$ to a feasible value.

Step 1: Update $\bm{\lambda}$ by (44).

Step 2: Update $\bm{\theta}$ by solving $\mathcal{P}_{2}$ in (47).

Until the value of the objective function converges.
Output: The optimal phase shift matrix $\mathbf{\Theta}^{o}$.

######  Remark 8.

The proposed optimization approach for solving $\mathbf{\Theta}$ is presented
in Algorithm 1 after the original problem, $\mathcal{P}_{M,1}$, is transformed
into a convex problem. $\mathcal{P}_{M,1}$ is solved iteratively using an
alternate optimization technique. First, we calculate the SINR in (42) after
we initiate $\mathbf{\Theta}$ to a feasible value, and then we update a better
solution for $\mathbf{\Theta}$ in each iteration. This process is repeated
until the normalized objective function increases by less than
$\epsilon=10^{-4}$.

######  Remark 9.

Because the tags are passive devices with limited power, $\mathcal{P}_{M,1}$
considers fixed reflection coefficients, $\beta_{k}$’s, to keep the tags’
power, cost, and form factor to a minimum [61]. $\beta_{k}$’s, however, can be
optimized at the expense of a simple tag architecture. We can easily utilize
alternative optimization because $\bm{\Theta}$ and $\beta_{k}$’s are
independent variables. Thereby, $\bm{\Theta}$ and $\beta_{k}$’s are
alternately optimized till convergence. For the sake of brevity, we omit
details.

###  IV-B Computational Complexity

The proposed algorithm is an alternating optimization solution with iterative
multiple stages. The outer loop optimizes $\mathbf{\Theta}$. This sub-problem
requires iterative updates. The computational complexity of Algorithm 1 is
centered in step 2. As CVX (MATLAB) uses the SDPT3 solver for this, the
computational complexity of Algorithm 1 is $\mathcal{O}(N^{3})$ [62]. Thus,
the total complexity is $\mathcal{O}(I_{\theta}N^{3})$, where $I_{\theta}$ is
the number of iterations of Algorithm 1.

##  V Simulation Results

TABLE II: Simulation settings. Parameter | Value | Parameter | Value
---|---|---|---
$f_{c}$ |  3 | $\eta_{n},\forall n$ | $0.8$
$B$ |  10 | $d_{f},d_{u}$ |  $10,5$ m
$N_{f}$ |  10 | $d_{g},d_{h}$ |  $3,8$ m
$m_{a},a\in\mathcal{A}$ | $3$ | $\beta,\phi$ | $0.6,0.8$

We adopt the 3GPP UMi model to model the large-scale fading $\zeta_{a}$ for
$a\in\mathcal{A}$ with $f_{c}=\qty{3}{}$ operating frequency [63, Table
B.1.2.1]. The AWGN variance, $\sigma_{z}^{2}$, is modeled as
$\sigma_{z}^{2}=10\log_{10}(N_{0}BN_{f})$ dBm, where $N_{0}=\qty{-174}{m/}$,
$B$ is the bandwidth, and $N_{f}$ is the noise figure. Unless otherwise
specified, Table II gives the simulation parameters. To ensure statistical
consistency, we generate the simulation curves by averaging over $10^{5}$
iterations except for outage and BER, where we use $10^{8}$ iterations. As a
benchmark, we also consider random RIS phase-shifts, i.e.,
$\theta_{n}\in\mathcal{U}[-\pi,\pi]$.

###  V-A Single-Tag Scenario

####  V-A1 RIS Location

$0$$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$-10$$-8$$-6$$-4$$-2$$0$$2$$4$$6$Distance
($d_{x}$) [m]Harvested Power [dBm]Simulation -OptimalAnalysis -
OptimalSimulation - Random$N=100$$N=200$$N=400$

Figure 5: The harvested power for $P=\qty{20}{m}$, $d_{f}=\qty{10}{}$,
$d_{u}=\qty{5}{}$, $d_{h}=\sqrt{d_{y}^{2}+d_{x}^{2}}$,
$d_{g}=\sqrt{d_{y}^{2}+(d_{f}-d_{x})^{2}}$, and $d_{y}=\qty{1}{}$.

$0$$5$$10$$15$$20$$25$$30$$35$$40$$-50$$-40$$-30$$-20$$-10$$0$$10$$20$$d_{f}$
[m]Received Power [dBm]Simulation - OptimalAnalysis - OptimalSimulation -
RandomSimulation - Without RIS$N=100$$N=200$$N=400$Tag Activation Threshold

Figure 6: The average received signal power at the tag as a function of
$\mathcal{E}$-$\mathcal{T}$ distance ($d_{f}$) for $P=\qty{20}{m}$,
$d_{h}=\qty{1}{}$, and $d_{g}=\sqrt{d_{h}^{2}+(d_{f})^{2}}$.

Fig. 5 explores the optimal RIS location using both optimal and random phase-
shift designs. The harvested power at $\mathcal{T}$, which is computed using
(14), is plotted as a function of the RIS placement in the power-up link.
Monte-Carlo simulation is performed to validate the accuracy of the harvested
power analysis. The results demonstrate that the optimal placement of the RIS
should be near either $\mathcal{E}$ or $\mathcal{T}$, while placing the RIS in
the middle of the $\mathcal{E}$-$\mathcal{T}$ link leads to the lowest
harvested power and, consequently, the poorest rate performance. Optimal RIS
phase-shifts result in a substantial improvement in harvested power compared
to the random phase-shift design.

####  V-A2 Passive Tag Activation Range

In Fig. 6, we present the received signal power at $\mathcal{T}$ as a function
of the $\mathcal{E}$-$\mathcal{T}$ distance ($d_{f}$) for different RIS sizes.
We include a comparison case without RIS. To provide valuable insights, we
depict the activation threshold of $\mathcal{T}$ (typically around -20m for
commercial passive RFID tags [6]). Without a RIS, the range of $\mathcal{T}$
is limited to less than 6 as the received power at the tag is insufficient for
activation beyond this distance. However, the presence of an RIS between
$\mathcal{E}$ and $\mathcal{T}$ significantly increases this range. For RIS
sizes $100$ and $400$ and optimized phase shifts, $\mathcal{T}$’s activation
distance increases 13 and 34, respectively. However, with random phase shifts,
it reduces to 9 and 20 respectively. These gains highlight the effectiveness
of our proposed design.

####  V-A3 Performance Evaluation

Fig. 7 depicts the average rate gain offered by the RIS against the non-RIS
scenario as a function of the number of RIS elements $N$ for different
$\mathcal{E}$’s transmit power $P$. The achievable rate gain is calculated as
$\mathcal{R}_{\rm{ub}}^{\rm{RIS}}-\mathcal{R}_{\rm{ub}}^{\rm{non-RIS}}$, where
$\mathcal{R}_{\rm{ub}}^{\rm{RIS}}$ is the achievable rate upper bound with a
RIS in the forward link and $\mathcal{R}_{\rm{ub}}^{\rm{non-RIS}}$ is the
achievable rate upper bound of non-RIS setup. As observed, deploying a RIS in
the forward link ($\mathcal{E}$-$\mathcal{T}$ link) can considerably increase
the achieved rate of $\mathcal{T}$. Moreover, with increasing $N$,
$\mathcal{T}$ can achieve a higher rate gain due to the increased received
power at $\mathcal{T}$, which eventually increases the received SNR. This
suggests that a large RIS is more beneficial. Specifically, at
$P=\qty{10}{m}$, a RIS with $200$ and $300$ elements provides respectively a
gain of 1.5bps/ and 2.1bps/.

Fig. 8 depicts the outage probability as a function of transmit power $P$ for
various numbers of RIS elements. It includes the analytical outage curves
obtained using the closed-form expression (29), as well as the Monte-Carlo
simulated curves and the Gaussian approximation for comparison. The outage
probability of $\mathcal{E}$-$\mathcal{T}$-$\mathcal{R}$ transmission in a
non-RIS setup is provided as a reference. The figure demonstrates that the
Gaussian and Gamma approximations closely align with the exact outage curves,
particularly for low-to-moderate transmit power levels of the emitter. As
expected, the proposed RIS-aided system significantly improves the outage
performance compared to the non-RIS counterpart. Moreover, with optimal RIS
phase-shifts, the outage performance is further enhanced, providing a minimum
power gain of 5m and 14m when compared to the random phase shift design and
the non-RIS case.

$0$$50$$100$$150$$200$$250$$300$$350$$400$$0$$0.5$$1$$1.5$$2$$2.5$$3$$3.5$$4$$N$Achievable
Rate Gain [bps/Hz]Simulation$P=0$ dBm$P=10$ dBm$P=20$ dBm$P=30$ dBm

Figure 7: The rate gain compared to the non-RIS case for $d_{f}=\qty{10}{}$,
$d_{u}=\qty{5}{}$, $d_{h}=\qty{8}{}$, and $d_{g}=\qty{3}{}$.

$0$$5$$10$$15$$20$$25$$30$$35$$40$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$Transmit
Power [dBm]Outage ProbabilitySimulation - OptimalGaussianGammaSimulation -
RandomSimulation - Without RIS$N=100$$N=200$$N=400$

Figure 8: The outage probability for $\gamma_{th}=\qty{0}{}$,
$d_{f}=\qty{8}{}$, $d_{u}=\qty{4}{}$, $d_{h}=\qty{1}{}$, and
$d_{g}=\qty{8}{}$.

In Fig. 9, we present the average BER of BPSK as a function of the transmit
power $P$ for different numbers of RIS elements. The analytical BER curves,
obtained using the closed-form expression (III-F), are plotted along with the
analytical BER curves for the Gaussian approximation. Additionally, the exact
BER curves are generated through Monte Carlo simulations. We include the BER
performance of $\mathcal{E}$-$\mathcal{T}$-$\mathcal{R}$ transmission in a
non-RIS setup for comparison. The asymptotic behavior of the average BER in
the high power (SNR) regime is also examined using (32). From Fig. 9, it can
be observed that the analytical BER curves, under both approximations, closely
match the exact BER curve for low-to-moderate transmit power of $\mathcal{E}$.
For high transmit power of $\mathcal{E}$, the BER under Gaussian and Gamma
approximations serve as upper and lower bounds, respectively, for the exact
average BER of the RIS-assisted system. The asymptotic BER probability curves
provide insights into the achievable diversity orders.

Fig. 8 and Fig.9 demonstrate that the proposed system outperforms the non-RIS
setup in terms of the outage and BER performance given the same transmit
power, or the former achieves the same outage and BER performance as the
latter with much less transmit power. For instance, achieving a $10^{-2}$ BER
requires 19m in the non-RIS setup, whereas the same can be achieved with a RIS
($N=100$) at only 5m. This amounts to a power saving of 14 dB. Furthermore,
increasing the size of the RIS enhances the outage and BER performance. From a
green perspective, using a RIS to energize the tags improves reliability,
reduces the required transmit power for a specific BER, and extends the
communication range.

$-10$$-5$$0$$5$$10$$15$$20$$25$$30$$10^{-7}$$10^{-6}$$10^{-5}$$10^{-4}$$10^{-3}$$10^{-2}$$10^{-1}$$10^{0}$Transmit
Power [dBm]Average
BERSimulationGaussianGammaAsymptotic$N=400$$N=200$$N=100$Without RIS

Figure 9: BER of BPSK for $d_{f}=\qty{10}{}$, $d_{u}=\qty{5}{}$,
$d_{h}=\qty{8}{}$, and $d_{g}=\qty{3}{}$.

$-10$$-5$$0$$5$$10$$15$$20$$25$$30$$55$$60$$65$$70$$75$$80$$85$$90$$95$$100$Transmit
Power [dBm]Percentage Rate Ratio
[%]Simulation$N=100$$N=200$$N=400$4-bits2-bits1-bit

Figure 10: The effect of phase quantization on the rate for
$D=\\{1,2,4\\}\qty{}{}$, $d_{f}=\qty{10}{}$, $d_{u}=\qty{5}{}$,
$d_{h}=\qty{8}{}$, and $d_{g}=\qty{3}{}$.

####  V-A4 The effect phase discretization

Fig. 10 examines the impact of RIS phase quantization error on the average
achievable rate for different numbers of quantization bits ($D$) and RIS
elements ($N$). The plot shows the percentage rate ratio against the average
transmit power, which is defined as
$\hat{\mathcal{R}}=\bar{\mathcal{R}}_{\rm{ub}}/{\mathcal{R}}_{\rm{ub}}\times\%100$,
where $\bar{\mathcal{R}}_{\rm{ub}}$ is the average achievable rate upper bound
with phase shift quantization errors given in (38) and
${\mathcal{R}}_{\rm{ub}}$ denotes the upper bound of the achievable rate with
continuous phase shifts in (22b). The results are validated through Monte-
Carlo simulations. As the number of quantization bits ($D$) increases, the
effect of quantization error becomes negligible. For $D=4$, nearly 100% of the
rate with continuous phase shifts can be achieved, rendering the quantization
error negligible. Additionally, as expected, the impact of phase quantization
error intensifies with increasing RIS size and for fixed quantization level.

###  V-B Multi-Tag Scenario

$0$$5$$10$$15$$20$$25$$30$$35$$40$$10^{-1}$$10^{0}$Transmit Power [dBm]Outage
ProbabilityOptimalRandom$N=100$$N=200$$N=400$Without RIS

Figure 11: The outage probability for $K=2$, $\gamma_{th}=\qty{0}{}$,
$d_{f_{k}}=\\{4,5\\}\qty{}{}$, $d_{u_{k}}=\\{5,5\\}\qty{}{}$,
$d_{h}=\qty{2}{}$, and $d_{g_{k}}=\\{4.5,5.4\\}\qty{}{}$.

$0$$2$$4$$6$$8$$10$$12$$14$$16$$18$$20$$22$$24$$26$$28$$30$$10^{-1}$$10^{0}$Transmit
Power [dBm]Average BEROptimalRandom$N=100$$N=200$$N=400$Without RIS

Figure 12: BER of BPSK for $K=2$, $d_{f_{k}}=\\{4,5\\}\qty{}{}$,
$d_{u_{k}}=\\{5,5\\}\qty{}{}$, $d_{h}=\qty{2}{}$, and
$d_{g_{k}}=\\{4.5,5.4\\}\qty{}{}$.

Fig. 11 and Fig. 12 show the outage and BER performance of our proposed RIS-
aided system with two tags ($K=2$), which significantly improves their outage
and rate performances compared to the non-RIS setup. Our optimal RIS phase-
shift design for a RIS with $100$ elements clearly outperforms the other
options, e.g., resulting in a reduction in the outage of 37.2 and 55.1
compared to the random phase-shift case and the no-RIS case at a transmit
power of $P=\qty{30}{m}$, respectively. Furthermore, when the RIS elements
increase to $200$ and $400$ with optimal phase-shift design, an additional
outage reduction of 108.6 and 368.0 is achieved at the same transmit power. On
the other hand, increasing the number of elements with the random phase-shift
design does not yield significant improvement. Therefore, the proposed optimal
phase-shift design, RIS placement in the emitter-to-tag path, and increasing
the number of its elements prove to be an effective and green solution for
passive IoT networks.

Figs. 11 and 12 also indicate that the reliability of the system in the high
SNR regime is limited by interference between tags. Increasing the size of the
RIS alone does not provide a solution. However, one effective approach is to
employ a multi-antenna emitter and reader. This enables optimal energy
beamforming and receive filtering in the spatial domain, allowing interference
suppression and performance enhancement. This is a fertile future research
direction.

##  VI Conclusion

Passive tags face drawbacks such as activation failures, limited activation
distances, and low data rates due to energy scarcity. To address these issues,
we explored the use of a RIS to enhance the RF signal power received by the
tags. We employed a linear EH model and developed analytical techniques to
quantify the benefits of RIS deployment. Specifically, we statistically
analyzed the maximum received power at the tag and optimized the SNR at the
reader. We derived performance metrics such as achievable rate, outage
probability, BER, and diversity order for the system. Additionally, we
examined the impact of RIS phase shift errors and proposed a RIS optimization
algorithm for multi-tag networks. Overall, the use of a RIS can significantly
enhance the received power at the tag, thereby increasing activation
distances, achievable rates, and communication ranges. Importantly, these
improvements can be achieved while maintaining the fundamental design of the
tags, which preserves their cost, size, form factor, and batteryless
advantages. Some of the specific insights into the use of the RIS are as
follows.

  1. 1.

The low activation distance (that is, less than 6) can be improved (Fig. 6).
For example, $100$ and $400$ elements improve it to 13 and 34 for a single
tag. RIS also improves rate, outage, and BER. Compared to the non-RIS setup,
for $P=20$ dBm, $200$ RIS elements and a BPSK tag can achieve a rate gain of
2.2bps/ (Fig. 7), an outage probability gain of $\sim 10^{2}$ (Fig. 8), and a
BER gain of $\sim 10^{4.6}$ (Fig. 9).

  2. 2.

The diversity order is independent of the number of RIS elements, $N$.
However, the coding and array gains depend on $N$ (Fig. 9). The optimal
placement of the RIS is found to be as close to the emitter or the tags as
possible (Fig. 5).

  3. 3.

When imperfect channel estimates and hardware limitations are present,
continuous phase shifts over the entire interval of $[0,2\pi]$ are not
feasible. Discrete phase shifts with various quantization levels can then be
employed. Fortunately, even $D=4$ has negligible performance degradation
(Figure 10).

These insights address the rate requirement, reliability, and coverage, for
establishing passive IoT. Our study spurs further research into the following
topics:

  1. 1.

Imperfect CSI and Interference Cancellation: The assumption of perfect CSI
yields the upper bounds of the performance metrics. However, especially for
multiple tags, accurate CSI acquisition is challenging. introduces residual
interference in the direct $\mathcal{E}$-$\mathcal{R}$ link and the reflected
RIS-$\mathcal{R}$ link. With our analytical tools, interference cancellation
techniques could be developed (Section II-A). The imperfect CSI case should be
researched.

  2. 2.

Active RIS: Using an active RIS or a hybrid RIS which combines both active and
passive elements, can further improve the tag activation and system
performance. However, the total energy consumption must be considered
especially for green networks. This is a potential future extension of this
work.

  3. 3.

Multiple-Antenna Nodes: We considered only signal-antenna tags and the emitter
(Fig. 2). However, the use of multiple antennas offers spatial diversity gains
that can be exploited to enhance performance. Thus, energy beamforming and
reception filters can be designed at the emitter and reader to not only
deliver more power to the passive tags but also suppress the mutual
interference between tags and enhance performance.

###  -A Gaussian Approximation

First, we find the mean and the variance of $\Lambda$. Since $\Lambda$ is a
product of two independent variables, we find that
$\mu_{\Lambda}=\mu_{\alpha_{u}}\mu_{Y}$, and
$\sigma^{2}_{\Lambda}=\mu^{(2)}_{\alpha_{u}}\mu^{(2)}_{Y}-\mu_{Y}^{2}\mu_{\alpha_{u}}^{2}$,
where $\mu_{Y}=\mu_{\alpha_{f}}+\mu_{X}$,
$\mu^{(2)}_{\alpha_{u}}=\sigma^{2}_{\alpha_{u}}+\mu_{\alpha_{u}}^{2}$
(16a)333For Nakagami-${m_{a}}$ $\alpha_{a}$ given in (2),
$\alpha_{a}^{2}\sim\Gamma(k,\lambda)$, where $k=m_{a}$ and
$\lambda=\Omega_{a}/m_{a}$, with $\mu_{\alpha_{a}^{2}}=k\lambda$ and
$\sigma^{2}_{\alpha_{a}^{2}}=k\lambda^{2}$.,
$\mu^{(2)}_{Y}=\sigma^{2}_{Y}+\mu_{Y}^{2}$, and
$\sigma^{2}_{Y}=\sigma^{2}_{\alpha_{f}}+\sigma^{2}_{X}$.

Direct Gaussian approximations do not work here. Thus, our approach involves
two steps: (i) we use the moment matching technique to approximate $Y$ and
$\Lambda$ as Gaussian variables with
$Y\sim\mathcal{N}(\mu_{Y},\sigma^{2}_{Y})$ and
$\Lambda\sim\mathcal{N}(\mu_{\Lambda},\sigma^{2}_{\Lambda})$. (ii) however,
since $Y$ and $\Lambda$ are positive random variables, i.e., $Y,\Lambda\geq
0$, we truncate the approximated Gaussian distribution such that it only lies
within the interval $[0,\infty)$, i.e., positive real axis [49, 48]. Hence the
approximated CDF and PDF of $Y$ and $\Lambda$ are respectively given as

$\displaystyle F_{C}(r)$ $\displaystyle=$ $\displaystyle 1-\Psi
Q\left(\frac{r-\mu_{C}}{\sigma_{C}}\right),\quad$ (49a) $\displaystyle
f_{C}(r)$ $\displaystyle=$
$\displaystyle\frac{\Psi}{\sqrt{2\pi\sigma^{2}_{C}}}\exp{\left(-\frac{(r-\mu_{C})^{2}}{2\sigma^{2}_{C}}\right)},$
(49b)

for $r\geq 0$ and $C\in\\{Y,\Lambda\\}$. Here, $\Psi=1/Q(-\mu_{C}/\sigma_{C})$
achieves normalization: $\int_{0}^{\infty}f_{C}(x)dx=1$.

###  -B Asymptotic Achievable Rate

When the RIS size increases indefinitely, i.e., $N\rightarrow\infty$,
$\mu_{X}$ and $\sigma_{X}^{2}$ can be approximated as

$\displaystyle\\!\\!\\!\mu_{X}\\!$ $\displaystyle\approx$
$\displaystyle\\!N\\!\left(\\!\eta\frac{\Gamma(m_{g}+1/2)\Gamma(m_{h}+1/2)}{\Gamma(m_{g})\Gamma(m_{h})}\sqrt{\frac{\Omega_{g}\Omega_{h}}{m_{g}m_{h}}}\right)\\!=\\!N\bar{\mu}_{X},\quad$
(50) $\displaystyle\\!\\!\\!\sigma_{X}^{2}\\!$ $\displaystyle\approx$
$\displaystyle\\!N\\!\left(\\!\eta^{2}\frac{\Gamma(m_{g}+1)\Gamma(m_{h}+1)}{\Gamma(m_{g})\Gamma(m_{h})}{\frac{\Omega_{g}\Omega_{h}}{m_{g}m_{h}}}-\bar{\mu}_{X}^{2}\right)\\!=\\!N\bar{\sigma}_{X}^{2},\quad$
(51)

where $\eta=\eta_{n},\forall n$. Next, the SNR in $\mathcal{R}_{\rm{ub}}$ can
be asymptotically evaluated as

$\displaystyle\\!\\!\\!\lim_{N\rightarrow\infty}\\!\gamma_{\rm{ub}}$
$\displaystyle=$
$\displaystyle\lim_{N\rightarrow\infty}\\!N^{2}\bar{\gamma}(\sigma_{\alpha_{u}}^{2}\\!+\\!\mu_{\alpha_{u}}^{2})\\!$
(52)
$\displaystyle\times\left(\frac{\sigma_{\alpha_{f}}^{2}}{N^{2}}+\frac{\bar{\sigma}_{X}^{2}}{N}+\frac{\mu_{\alpha_{f}}^{2}}{N^{2}}+\frac{2\mu_{\alpha_{f}}\bar{\mu}_{X}}{N}+\bar{\mu}_{X}^{2}\right)$
$\displaystyle=$
$\displaystyle\bar{\gamma}_{A}(\sigma_{\alpha_{u}}^{2}\\!+\\!\mu_{\alpha_{u}}^{2})\bar{\mu}_{X}^{2},$

where $\bar{\gamma}_{A}=\beta P_{A}/\sigma_{z}^{2}$ and
$\lim_{N\rightarrow\infty}P=P_{A}/N^{2}$. Similarly, we can show that the SNR
of the rate lower bound converges for the same asymptotic SNR, i.e.,
$\lim_{N\rightarrow\infty}\gamma_{\rm{lb}}=\bar{\gamma}_{A}(\sigma_{\alpha_{u}}^{2}\\!+\\!\mu_{\alpha_{u}}^{2})\bar{\mu}_{X}^{2}$.
Then, the asymptotic achievable rate can be derived as given in (27).

###  -C Average BER

The average BER of the system with BPSK is given as

$\displaystyle\bar{P}_{\rm{BER}}$ $\displaystyle=$
$\displaystyle\mathbb{E}\\{\lambda{Q}({\sqrt{\nu\gamma}})\\}\overset{(a)}{\mathop{=}}\\!\int_{0}^{\infty}\\!\\!\\!\lambda{Q}(x{\sqrt{\nu\bar{\gamma}}}){{f}_{\Lambda}}\left(x\right)dx$
(53) $\displaystyle\overset{(b)}{\mathop{=}}$
$\displaystyle\\!\int_{0}^{\infty}\\!\\!\\!\lambda\exp{\left(-A\nu\bar{\gamma}x^{2}-B\sqrt{\nu\bar{\gamma}}x-C\right)}{{f}_{\Lambda}}\left(x\right)dx,~{}~{}\quad$

where $(a)$ is due to submitting $\gamma=\bar{\gamma}\Lambda^{2}$ and
$f_{\Lambda}(r)$ is given in (17a). Besides, $Q$ function is substituted with
its tight approximation: $Q(x)=\exp{(-Ax^{2}-Bx-C)}$, where
$A=0.3842,B=0.7640$ and $C=0.6964$ [64].

By using the PDF of $\Lambda$ given in (17a), we have

$\displaystyle\\!\bar{P}_{\rm{BER}}^{\rm{Gamma}}\\!$ $\displaystyle=$
$\displaystyle\\!A_{1}\\!\int_{0}^{\infty}\\!\\!\\!\exp{\left(-\bar{A}x^{2}-\bar{B}x\right)}x^{k_{\Lambda}-1}\exp{\left(-\frac{x}{\lambda_{\Lambda}}\right)}dx$
(54) $\displaystyle=$
$\displaystyle\\!A_{1}\\!\int_{0}^{\infty}\\!\\!\\!\exp{\left(-\bar{A}x^{2}-\hat{B}_{1}x\right)}x^{k_{\Lambda}-1}dx$
$\displaystyle\overset{(d)}{\mathop{=}}$ $\displaystyle
A_{1}(2\bar{A})^{-\frac{k_{\Lambda}}{2}}\Gamma(k_{\Lambda})\exp{\left(\frac{\hat{B}_{1}^{2}}{8\bar{A}}\right)}D_{-k_{\Lambda}}\left(\frac{\hat{B}_{1}^{2}}{\sqrt{2\bar{A}}}\right),\quad$

where $\bar{A}=A\nu\bar{\gamma},\bar{B}=B\sqrt{\nu\bar{\gamma}}$,
$A_{1}=\exp{(-C)}/(\Gamma(k_{\Lambda})\lambda_{\Lambda}^{k})$, and
$\hat{B}_{1}=\bar{B}+1/\lambda_{\Lambda}$. Beside, $(d)$ is obtained by using
the integral [31, eq. (3.462.1)].

###  -D Asymptotic Outage probability and BER

In order to derive the outage probability and BER in the high SNR domain, we
use the first-order polynomial expansion of the PDF approximations.

Asymptotic SNR Outage Probability: this can be approximated as [55]

$\displaystyle\lim_{\bar{\gamma}\rightarrow\infty}P_{\rm{out}}^{\rm{SNR}}=P_{\rm{out}}^{\infty}\approx
O_{c}\left(\frac{\gamma_{\rm{th}}}{\bar{\gamma}}\right)^{G_{d}}+\mathcal{O}\left(\bar{\gamma}^{-(G_{d}+1)}\right),$
(55)

where $G_{d}$ is the diversity order and $O_{c}$ is a measure of the coding
gain [55]. By approximating the PDF of $\Lambda$ for $r\\!\rightarrow\\!0^{+}$
as
$f_{\Lambda}^{0^{+}}\\!(r)\\!=\\!1/{\Gamma(k_{\Lambda})\lambda_{\Lambda}^{k_{\Lambda}}}r^{k_{\Lambda}-1}\\!+\\!\mathcal{O}(r^{k_{\Lambda}})$,
$P_{out}^{\infty}$ is derived as

$\displaystyle
P_{\rm{out}}^{\infty}=\frac{1}{\Gamma(k_{\Lambda}+1)\lambda_{\Lambda}^{k_{\Lambda}}}\left(\frac{\gamma_{\rm{th}}}{\bar{\gamma}}\right)^{{k_{\Lambda}}/{2}}+\mathcal{O}\left(\bar{\gamma}^{-({k_{\Lambda}}/{2}+1)}\right),$
(56)

where the diversity order, $G_{d}=k_{\Lambda}/2$, and the coding gain,
$O_{c}={1}/({\Gamma(k_{\Lambda}+1)\lambda_{\Lambda}^{k_{\Lambda}}})$.

Asymptotic Average BER: The BER can be asymptotically approximated as [55]

$\displaystyle\lim_{\bar{\gamma}\rightarrow\infty}P_{\rm{BER}}=\bar{P}_{\rm{BER}}^{\infty}\approx\left(G_{a}\bar{\gamma}\right)^{-G_{d}}+\mathcal{O}\left(\bar{\gamma}^{-(G_{d}+1)}\right),$
(57)

where $G_{a}$ is the array gain. Thereby, the asymptotic BER is derived as

$\displaystyle\bar{P}_{\rm{BER}}^{\infty}$ $\displaystyle=$
$\displaystyle\int_{0}^{\infty}\lambda{Q}(x{\sqrt{\nu\bar{\gamma}}}){{f}_{\Lambda}^{0+}}\left(x\right)dx$
(58) $\displaystyle\overset{(e)}{\mathop{=}}$
$\displaystyle({\lambda_{E}}\bar{\gamma})^{-k_{\Lambda}/2}+\mathcal{O}\left(\bar{\gamma}^{-(k_{\Lambda}/2+1)}\right),$

where
$\lambda_{E}=\nu\left[\left(2^{k_{\Lambda}/2}/3+(3/2)^{k_{\Lambda}/2}\right)C_{1}\right]^{2/k_{\Lambda}}$
and
$C_{1}={\lambda}\Gamma(k_{\Lambda}/2)/(8\Gamma(k_{\Lambda})\lambda_{\Lambda}^{k_{\Lambda}})$.
Moreover, $(e)$ is obtained by substituting $Q$ function with its tight
approximation: $Q(x)=\exp{(-x^{2}/2)}/12+\exp{(-2x^{2}/3)}/4$ [64]. Then, the
array gain, $G_{a}=\lambda_{E}$.

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