# Optimization of Rate-Splitting Multiple Access with Integrated Sensing and
Backscatter Communication

Diluka Galappaththige, , Shayan Zargari, Chintha Tellambura, , and Geoffrey Ye
Li D. Galappaththige, S. Zargari, and C. Tellambura with the Department of
Electrical and Computer Engineering, University of Alberta, Edmonton, AB, T6G
1H9, Canada (e-mail: {diluka.lg, zargari, ct4}@ualberta.ca).
G. Y. Li is with the ITP Lab, the Department of Electrical and Electronic
Engineering, Imperial College London, SW7 2BX London, U.K.(e-mail:
geoffrey.li@imperial.ac.uk).

###### Abstract

An integrated sensing and backscatter communication (ISABC) system is
introduced herein. This system features a full-duplex (FD) base station (BS)
that seamlessly merges sensing with backscatter communication and supports
multiple users. Multiple access (MA) for the user is provided by employing
rate-splitting multiple access (RSMA). RSMA, unlike other classical orthogonal
and non-orthogonal MA schemes, splits messages into common and private
streams. With RSMA, the set of common rate forms can be optimized to reduce
interference. Optimized formulas are thus derived for communication rates for
users, tags, and the BS’s sensing rate, with the primary goal of enhancing the
transmission efficiency of the BS. The optimization task involves minimizing
the BS’s overall transmission power by jointly optimizing the BS’s beamforming
vectors, the tag reflection coefficients, and user common rates. The
alternating optimization method is employed to address this challenge.
Concrete solutions are provided for the received beamformers, and semi-
definite relaxation and slack-optimization techniques are adopted for transmit
beamformers and reflection coefficients, respectively. For example, the
proposed RSMA-assisted ISABC system achieves a 350 communication rate boost
over a non-orthogonal multiple access-assisted ISABC, with only a 24 increase
in transmit power, leveraging ten transmit/reception antennas at the BS.

###### Index Terms:

Backscatter communication (BackCom), integrated sensing and communication
(ISAC), passive tags, rate-splitting multiple access (RSMA).

##  I Introduction

Integrated sensing and communications (ISAC) networks are envisioned for 6G
wireless. In traditional radar-based sensing, a reflected signal from an
object/target is used to sense/detect the object [1]. Similarly, backscatter
communication (BackCom) uses radar-like load modulation to send data, i.e., a
backscatter device (or a tag) modulates its data onto an external radio
frequency (RF) signal and reflects it to a reader [2, 3, 4, 5]. However, all
the existing ISAC works do not consider backscatter scenarios, except for [6,
7]. In [6], by leveraging the similarities between BackCom and radar systems,
i.e., the use of reflected signals, a novel concept called Integrated Sensing
and Backscatter Communications (ISABC), has been proposed. This work
investigates the feasibility of integrating sensing into low-power Internet-
of-Things (IoT) devices. This new paradigm combines integrated sensing and
communication (ISAC) with BackCom features, resulting in concurrent sensing
and communication in ambient power-enabled IoT networks.

Table I: A comparison between ISAC and ISABC. Features | ISAC | ISABC
---|---|---
Target | ✓ | $\times$
Tag | $\times$ | ✓
Additional user data | $\times$ | ✓
BS Power allocation | ✓ | ✓
User decoding | Conventional | SIC
Sensing type | Active/Passive | Active

Combining the strengths of ISAC and BackCom, ISABC introduces backscatter tags
as replacements for the conventional sensing targets, enabling opportunistic
sensing in BackCom systems [6]. Although ISABC belongs to devices-based ISAC
with active sensing, this modification distinguishes ISABC from traditional
ISAC in several significant ways (Table I).

In ISAC, sensing targets can be passive objects, such as vehicles, which
neither transmit nor receive sensing signals, nor active devices, such as
mobile phones, which engage in transmitting and/or receiving. In contrast,
ISABC pivots around backscatter tags acting as sensing targets. These tags
relay environmental details to the base station (BS) and transmit additional
data to users or readers. For example, in a smart home, tags can be deployed
to monitor and map surroundings. Thus, the BS can use the tag-reflected
signals for sensing. This approach enhances communication and sensing
capabilities by harnessing both sensing and backscatter data. However, the
user/reader may need more advanced decoding mechanisms, such as successive
interference cancellation (SIC).

Applications of ISABC may include the following. For example, emerging IoT
networks demand highly accurate and robust sensing, high data throughput, low
latency, connectivity, reliability, and energy efficiency [8, 9].
Specifically, ambient power-enabled (i.e., battery-free) IoT is a key
contender for applications like smart homes, smart cities, industrial IoT, and
more, which require sensing, collecting, and exchanging data about the
environment without requiring explicit human input or control [2, 3, 4].
Moreover, BackCom-aided vehicular networks for navigation, traffic monitoring,
pedestrian monitoring, environmental surveillance, road safety, and autonomous
driving demand sensing capabilities [10, 11]. Thus, ISABC can facilitate
sensing requirements for such applications.

This research thus extends the notion of [6, 7] to a general framework with
enhanced spectrum efficiency by integrating it with rate-splitting multiple
access (RSMA) (see Section I-B).

###  I-A ISAC, RSMA, and BackCom

Since our study integrates these three technologies, this section briefly
describes them.

ISAC aims to reshape sixth-generation (6G) networks, significantly shifting
from traditional network architectures. It enables services such as
localization, activity monitoring, object detection, and environmental
mapping. Conversely, ISAC can enhance communication performance by using
environmental data to provide precise beamforming, rapid beam failure
recovery, and reduced channel estimation overhead [12].

Two main ISAC categories are [12]: (i) Device-free ISAC and (ii) Device-based
ISAC. The first refers to detecting unregistered external targets, such as
vehicles, animals, and humans, making the sensing approach independent of the
targets’ transmission and/or reception capability. In device-based ISAC,
network-registered devices, such as mobile phones and IoT sensors, provide
sensing functionality, resulting in a target transmission and/or reception-
based sensing approach. These categories can further be subdivided according
to the transmitter and receiver configurations (i.e., mono-, bi-, and multi-
static), as well as the type of sensing signal (i.e., active if the sensing
receiver uses its transmit signals or passive if it uses external sensing
signals) [12]. Active sensing employs full-duplex (FD) mode, which affects
sensing performance due to self-interference (SI) [13]. However, an FD ISAC
system must only perform IS cancellation for direct signal coupling between
transceiver antennas while keeping target reflections. Fortunately, machine
learning-based cancellation, digital cancellation, analog cancellation, and
antenna isolation can successfully suppress the SI [14].

RSMA has recently emerged as a viable multiple-access scheme with effective
interference control [15, 16, 17]. It relies on splitting user messages into
multiple parts and allowing for partial decoding of interference and partial
treatment of interference as noise [15, 16, 17]. In particular, RSMA divides
each user’s message into a common part and a private part, with the common
parts of all users encoded into a single common stream and the private part
encoded individually. Before decoding the corresponding private stream, each
user decodes and removes the common stream via SIC. RSMA offers superior
spectrum and energy efficiencies than other conventional multiple access
schemes, such as space-division multiple access (SDMA), which regards
interference as noise, and non-orthogonal multiple access (NOMA), which
decodes interference [15, 16, 17].

BackCom is a promising avenue for facilitating energy-efficient IoT networks
[2, 3, 4]. It enables wireless nodes (tags or IoT devices) to communicate by
reflecting external radio frequency (RF) signals. Thus, tags can operate
without active RF components. This approach may be a sustainable alternative
for battery-dependent IoT devices, which incur high maintenance costs,
environmental impact, and safety issues [2, 3, 4]. These tags, without active
RF components and reflecting external RF, achieve ultra-low cost and energy
use, ranging from nanowatts to microwatts [2].

###  I-B Motivation and Our Contribution

This study introduces a novel and generalized system model for ISABC: multiple
tags, multiple users, a backscatter reader, and an FD BS (Fig. 1). This
network combines primary users, tags, a reader, and sensing into a unified
framework, posing three critical challenges.

  1. 1.

The first challenge is the provision of multiple access for primary users.
Conventional options include time-division multiple access (TDMA), NOMA, and
SDMA. SDMA treats all undesired signals as interference, which saturates the
rates even with excess transmit powers if the number of transmit antennas is
insufficient or a resource block has multiple users [18]. In contrast, by
employing superposition coding and SIC, NOMA can serve multiple users over the
same resource block. However, the drawbacks are that NOMA has stringent
decoding and SIC requirements and that channel gains must have a significant
disparity [18]. RSMA outperforms these regarding spectrum and energy
efficiencies [18]. Due to these reasons, RSMA is proposed in this study and is
evaluated against NOMA and SDMA benchmarks.

  2. 2.

The second challenge is for the reader to detect the tag signals significantly
weaker than direct signals due to the double path loss effect [2, 3, 4].
Before decoding the tags’ data, the reader must reduce direct link
interference (DLI) [19]. While the reader and BS can be connected, DLI cannot
be eliminated due to the reader’s limited dynamic range. The reader thus
employs imperfect SIC to partially remove DLI before decoding the backscatter
data [19].

  3. 3.

The third challenge is the BS’s ability to sense the tag signals, while strong
SI affects its sensing [13]. To successfully sense the tag-reflected signal,
the BS must reduce SI power to a level comparable to the tags’ backscattered
signal power [13]. The SI suppression is not perfect. Thus, the BS will
operate with a residual SI.

Prior works [6] and [7] have several limitations. Specifically, [6] considers
only one backscatter tag and a user, a special case of our model (Fig. 1).
Thus, [6] does not consider multiple access for users. Reference [7] extends
[6] for multiple tags and a single user. However, it uses perfect SIC for
backscatter data decoding at the user and perfect SI cancellation at the BS
for acquiring sensing information. Conversely, this work generalizes the
system model and addresses the technical limitations in [6, 7] by
investigating multi-tag with EH, RSMA-enabled multiple access for the primary
users, and effects of imperfect SIC and SI cancellation. Thus, the
optimization framework differs radically from [7].

Inspired by the aforementioned challenges and potential applications of ISABC,
we present a novel RSMA-enabled ISABC system. Our contributions are summarized
as follows:

  1. 1.

Utilizing RSMA, the BS facilitates multiple access for the primary users to
enhance the primary communication performance. The tags reflect the BS’s
signal to communicate with the reader, while the BS uses the same reflected
signal to infer the tags’ environmental insights.

  2. 2.

The BS transmit power is minimized while preserving quality-of-service (QoS)
requirements for all nodes, including tags and users. The optimization
variables are the BS transmit/received beamforming, tag reflection
coefficients, and users’ common rates. Due to the appearance of the products
of the optimization variables, the proposed problem $\mathbf{P}_{1}$ (17) is
non-convex, and the widely available convex algorithms fail to solve it.

  3. 3.

To address this, an alternative optimization (AO) strategy [20] is employed,
beginning by optimizing the BS received beamforming for the tags’ signal using
minimum mean-squared error (MMSE) filtering and the generalized Rayleigh
quotient form of the signal-to-noise-to-interference ratio (SINR). The
semidefinite relaxation (SDR) approach is then used to determine the BS
transmit beamforming and the user common rates [21, 22]. Finally, the
feasibility problem of tag reflection coefficient optimization is transformed
into a slack-optimization problem to obtain an effective solution [23, 24].

  4. 4.

The benefits of the proposed RSMA-assisted ISABC are compared to NOMA-assisted
ISABC, RSMA-assisted BackCom, conventional BackCom, conventional ISABC, and
sensing-only schemes (with energy harvesting (EH)). The convergence and
complexity are also analyzed. With a configuration of ten BS antennas each for
transmission and reception, the proposed scheme offers a 350 communication
rate enhancement compared to the NOMA-assisted ISABC while only necessitating
a modest 24 transmit power rise.

Notation: Boldface lower case and upper case letters represent vectors and
matrices, respectively. For matrix $\mathbf{A}$, $\mathbf{A}^{\rm{H}}$ and
$\mathbf{A}^{\rm{T}}$ are Hermitian conjugate transpose and transpose of
matrix $\mathbf{A}$, respectively. $\mathbf{I}_{M}$ denotes the $M$-by-$M$
identity matrix. The Euclidean norm, absolute value, and expectation operators
are denoted by $\|\cdot\|$ and $|\cdot|$, and $\mathbb{E}\\{\cdot\\}$,
respectively. A circularly symmetric complex Gaussian (CSCG) random vector
with mean $\bm{\mu}$ and covariance matrix $\mathbf{C}$ is denoted by
$\sim\mathcal{C}\mathcal{N}(\bm{\mu},\,\mathbf{C})$. Besides,
$\mathbb{C}^{M\times N}$ and ${\mathbb{R}^{M\times 1}}$ represent $M\times N$
dimensional complex matrices and $M\times 1$ dimensional real vectors,
respectively. Further, $\mathcal{O}$ expresses the big-O notation. Finally,
$\mathcal{L}\triangleq\\{1,\ldots,L\\}$,
$\mathcal{K}\triangleq\\{1,\ldots,K\\}$,
$\mathcal{L}_{l}\triangleq\mathcal{L}\setminus\\{l\\}$,
$\mathcal{K}_{k}\triangleq\mathcal{K}\setminus\\{k\\}$,
$\mathcal{L}_{l}^{\prime}\triangleq\\{1,\ldots,l-1\\}$, and
$\mathcal{L}_{l}^{\prime\prime}\triangleq\\{l+1,\ldots,L\\}$.

##  II System Components

###  II-A System Model

An RSMA-assisted ISABC system (Fig. 1) is considered. It consists of an FD BS
as the primary transmitter having $M$ transmit and $N$ receiver uniform linear
array (ULA) antennas, $L$ single-antenna users ($U_{l}$ denotes the $l$-th
user), $K$ single-antenna tags ($T_{k}$ denotes the $k$-th tag), and a single-
antenna reader. The BS antennas are spaced at half-wavelengths [13]. Tags use
power-splitting (PS) protocol for EH and data backscattering (Section II-D).

The BS communicates and senses the surroundings using transmit beamforming,
whereas tags use the BS signal for EH and data backscattering via the PS
protocol. Cooperation is assumed between the BS and the backscatter reader via
a control link [25]. Thus, the reader subtracts the direct signal, i.e.,
$\mathbf{f}_{0}\mathbf{x}$, from its received signal before decoding the tag
data. This SIC operation minimizes the interference from the BS. The BS also
extracts environmental insights from unintentionally received backscatter
signals [6]. To reduce self-interference (SI), the BS uses two distinct
antenna sets for transmission and reception [13]. Time synchronization of the
nodes is assumed for simplicity [19].

###  II-B Channel Model

This study employs block flat-fading channel models. During each fading block,
$\mathbf{f}_{0}\in\mathbb{C}^{M\times 1}$,
$\mathbf{f}_{l}\in\mathbb{C}^{M\times 1}$, and
$\mathbf{g}_{f,k}\in\mathbb{C}^{M\times 1}$, respectively, represent the
channels between the BS and the reader, the BS and $U_{l}$, and the BS and
$T_{k}$. The channels between $T_{k}$ and $U_{l}$, and $T_{k}$ and the reader
are denoted by $v_{l,k}$ and $q_{k}$, respectively, while
$\mathbf{g}_{b,k}\in\mathbb{C}^{N\times 1}$ represents the channel between
$T_{k}$ and the BS receiver antennas. Within these channels, pure
communication channels, i.e., $\mathbf{f}_{0}$, $\mathbf{f}_{l}$, $v_{l,k}$
and $q_{k}$, are modeled as Rayleigh fading and given as
$\mathbf{a}=\zeta_{a}^{1/2}\tilde{\mathbf{a}}$, where
$\mathbf{a}\in\\{\mathbf{f}_{0},\mathbf{f}_{l},v_{l,k},q_{k}\\}$ and
$\zeta_{a}$ is the large-scale path-loss and shadowing, which stays constant
for several coherence intervals. Moreover,
$\tilde{\mathbf{a}}\sim\mathcal{CN}(\mathbf{0},\mathbf{I}_{A})$ accounts for
the small-scale Rayleigh fading, where $A\in\\{M,1\\}$.

Conversely, following the echo signal representation in multiple-input and
multiple-output (MIMO) radar systems, the channels between the BS and tags,
i.e., $\mathbf{g}{f,k}$ and $\mathbf{g}{b,k}$, are modeled as line-of-sight
(LoS) channels [13]. The transmit/receiver array steering vectors to the
direction $\theta_{k}$ are thus denoted as

$\displaystyle\mathbf{b}(\theta_{k})=\sqrt{\frac{\zeta_{b}}{B}}\left[1,e^{j\pi\sin(\theta_{k})},\ldots,e^{j\pi(B-1)\sin(\theta_{k})}\right]^{\rm{T}},$
(1)

where $\mathbf{b}\in\\{\mathbf{g}_{f,k},\mathbf{g}_{b,k}\\}$, $B\in\\{M,N\\}$,
$\theta_{k}$ is $T_{k}$’s direction for the BS and the reader direction, and
$\zeta_{b}$ denotes the path-loss. Finally,
$\mathbf{G}_{\rm{SI}}\in\mathbb{C}^{M\times N}$ is the SI channel matrix
between the transmitter and the receiver antennas of the BS and is modeled as
a Rician fading channel with a Rician factor of $K_{\rm{SI}}$ [14, 26].

Channel state information (CSI) is essential for BS beamforming, user/reader
data decoding, and other operations. To acquire CSI, the proposed system is
assumed to utilize the time-division duplexing mode. That means coherence time
is divided into two slots, one designated for channel estimation and one for
data transmission. In the first slot, pilot signals are transmitted, which can
yield high-quality CSI. Passive tags, however, have restricted pilot
transmission capabilities [27]. Instead, they can modulate pilot sequences
into the BS signal and reflect them to the reader, allowing the channel
estimation via least square (LS)/MMSE estimate and deep leaning [28, 29].
Although not the focus of this study, backscatter CSI estimation is complex,
and recent results [29] are promising. Thus, full CSI availability is a
reasonable assumption.

FD BS
---

Reader
---

$\mathbf{G}_{\rm{SI}}$
---

$U_{l}$
---

$T_{k}$
---

$T_{K}$
---

$T_{1}$
---

$\mathbf{f}_{l}$
---

$\mathbf{f}_{0}$
---

$\mathbf{g}_{f,k}$
---

$v_{l,k}$
---

$q_{k}$
---

$U_{L}$
---
$U_{1}$
---

$\mathbf{g}_{b,k}$
---

Figure 1: An RSMA-assisted ISABC system setup. Red lines denote interference
links.

###  II-C Transmission Model

The BS serves the primary users and simultaneously receives the tag-reflected
signals. The BS applies RSMA to send messages to the primary users. In
particular, the message $d_{l}$ of $U_{l}$ is split into a common part
$d_{c,l}$ and a private part $d_{p,l}$, i.e., $d_{l}=\\{d_{c,l},d_{p,l}\\}$
for $l\in\mathcal{L}$. The common parts of all users, i.e.,
$\\{d_{c,1},\ldots,d_{c,L}\\}$ are jointly encoded into the common stream
$x_{c}$ while the private parts, i.e., $\\{d_{p,1},\ldots,d_{p,L}\\}$ are
respectively encoded into private streams $\\{x_{1},\ldots,x_{L}\\}$, which
are assumed to be independent [15]. The common stream and the private streams
are then linearly precoded before transmission using the precoders
$\mathbf{w}_{c}\in\mathbb{C}^{M\times 1}$ and
$\mathbf{w}_{l}\in\mathbb{C}^{M\times 1}$ for $l\in\mathcal{L}$, respectively.

Conversely, the BS signal also includes sensing signal
$\mathbf{s}\in\mathbb{C}^{M\times 1}$ with the covariance matrix
$\mathbf{S}\triangleq\mathbb{E}\\{\mathbf{s}\mathbf{s}^{\rm{H}}\\}$. This
extends the degrees-of-freedom of the transmitted signal $\mathbf{x}$ to
achieve enhanced sensing performance [13]. The BS transmitted signal can be
expressed as

$\displaystyle\mathbf{x}=\mathbf{w}_{c}x_{c}+\sum\nolimits_{j\in\mathcal{L}}\mathbf{w}_{j}x_{j}+\mathbf{s}.$
(2)

In (2), it is assumed that $x_{c}$, $\\{x_{l}\\}_{l\in\mathcal{L}}$, and
$\mathbf{s}$ are independent of each other.

To backscatter data, tag $T_{k}$ uses a multi-level ($\tilde{M}$-ary)
modulation scheme [3, 4]. Tag data symbol $c_{k}$, with
$\mathbb{E}\\{|c_{k}|^{2}\\}=1$, is thus selected from a $\tilde{M}$-ary
constant-envelope modulation. Consequently, the users receive the BS signal
and the tags’ backscattered signals. Assuming the propagation delay difference
for all signals is negligible [30], the received signal at $U_{l}$ is given by

$\displaystyle
y_{l}^{\prime}=\mathbf{f}_{l}^{\rm{H}}\mathbf{x}+\sum\nolimits_{k\in\mathcal{K}}\sqrt{\alpha_{k}}\mathbf{h}_{l,k}^{\rm{H}}\mathbf{x}c_{k}+z_{u},$
(3)

where the first and second terms are the direct-link signals (BS-to-user) and
the backscatter-link signals (BS-to-tags-to-user), respectively,
$z_{u}\sim\mathcal{CN}(0,\sigma^{2})$ is the additive white Gaussian noise
(AWGN) at $U_{l}$, with $0$ mean and $\sigma^{2}$ variance, and
$\mathbf{h}_{l,k}$ is the effective backscatter channel at $U_{l}$ through
$T_{k}$, i.e., $\mathbf{h}_{l,k}=\mathbf{g}_{f,k}(\theta_{k})v_{l,k}$. Because
primary users and BS are integral to the primary networks, a pre-existing
connection permits the exchange of information, including sensing waveforms,
via a control link [25]. It is thus assumed that the users are aware of the
sensing waveform, however, partially eliminate it before decoding data. This
is due to the limited dynamic range of the users, even if the channels are
perfectly known. After the detecting signal is removed, the received signal is
given as

$\displaystyle y_{l}$ $\displaystyle=$
$\displaystyle\mathbf{f}_{l}^{\rm{H}}\mathbf{w}_{c}x_{c}+\sum\nolimits_{j\in\mathcal{L}}\mathbf{f}_{l}^{\rm{H}}\mathbf{w}_{j}x_{j}+\sqrt{\delta_{s}}\mathbf{f}_{l}^{\rm{H}}\mathbf{s}$
(4)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\sum\nolimits_{k\in\mathcal{K}}\sqrt{\alpha_{k}}\mathbf{h}_{l,k}^{\rm{H}}\bigg{(}\mathbf{w}_{c}x_{c}+\sum\nolimits_{j\in\mathcal{L}}\mathbf{w}_{j}x_{j}+\mathbf{s}\bigg{)}c_{k}+z_{u},\quad$

where $\delta_{s}\in[0,1]$ accounts for the SIC quality for the sensing
signal.

As the reader and the BS cooperate, the reader is aware of the BS transmitted
signal. The reader thus applies SIC before decoding tags data. The post-
processed received signal at the reader is given as

$\displaystyle y_{r}$ $\displaystyle=$
$\displaystyle\sqrt{\delta_{c}}\mathbf{f}_{0}^{\rm{H}}\mathbf{w}_{c}x_{c}+\sqrt{\delta_{p}}\sum\nolimits_{j\in\mathcal{L}}\mathbf{f}_{0}^{\rm{H}}\mathbf{w}_{j}x_{j}+\sqrt{\delta_{s}}\mathbf{f}_{0}^{\rm{H}}\mathbf{s}$
(5)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\sum\nolimits_{k\in\mathcal{K}}\sqrt{\alpha_{k}}\mathbf{h}_{k}^{\rm{H}}\bigg{(}\mathbf{w}_{c}x_{c}+\sum\nolimits_{j\in\mathcal{L}}\mathbf{w}_{j}x_{j}+\mathbf{s}\bigg{)}c_{k}+z_{r},\quad$

where $\mathbf{h}_{k}=\mathbf{g}_{f,k}(\theta_{k})q_{k}$ and
$z_{r}\sim\mathcal{CN}(0,\sigma^{2})$ is the AWGN at the reader. Moreover,
$\delta_{c}\in[0,1]$ and $\delta_{p}\in[0,1]$ are the SIC qualities for the
common stream and private streams, respectively.

Tag backscattered signals reach the users, reader, and BS. Thus, those signals
enable the BS to retrieve environmental information [6]. The BS received
signal, i.e., $\mathbf{y}_{b}\in\mathbb{C}^{N\times 1}$, is given as

$\displaystyle\mathbf{y}_{b}=\sum\nolimits_{k\in\mathcal{K}}\sqrt{\alpha_{k}}\mathbf{G}_{k}(\theta_{k})\mathbf{x}c_{k}+\mathbf{G}_{\rm{SI}}^{\rm{H}}\mathbf{x}+\mathbf{z}_{b},$
(6)

where
$\mathbf{G}_{k}(\theta_{k})\triangleq\mathbf{g}_{b,k}(\theta_{k})\mathbf{g}_{f,k}^{\rm{H}}(\theta_{k})$,
$\sqrt{\alpha_{k}}\mathbf{G}_{k}(\theta_{k})\mathbf{x}c_{k}$ is $T_{k}$’s
($k$-th target) reflection, and
$\mathbf{z}_{b}\sim\mathcal{CN}(\mathbf{0},\sigma^{2}\mathbf{I}_{N})$ is the
AWGN at the BS. In (6), the second term is the SI at the BS receiver due to
simultaneous transmission and reception. Due to the limited dynamic range of
the receiver, SI cannot be canceled perfectly even with the perfect CSI of the
SI channel [14]. After the SI cancellation, the BS applies the receiver
beamformer, $\mathbf{u}_{k}\in\mathbb{C}^{N\times 1}$ for $k\in\mathcal{K}$,
to the received signal (6) to acquire the desired reflected signal of $T_{k}$.
The post-processed signal for acquiring $T_{k}$ sensing information is thus
given as

$\displaystyle{y}_{b,k}$ $\displaystyle=$
$\displaystyle\sqrt{\alpha_{k}}\mathbf{u}_{k}^{\rm{H}}\mathbf{G}_{k}(\theta_{k})\mathbf{x}c_{k}+\sum\nolimits_{i\in\mathcal{K}_{k}}\sqrt{\alpha_{i}}\mathbf{u}_{k}^{\rm{H}}\mathbf{G}_{i}(\theta_{i})\mathbf{x}c_{i}$
(7)
$\displaystyle+\sqrt{\beta}\mathbf{u}_{k}^{\rm{H}}\mathbf{G}_{\rm{SI}}^{\rm{H}}\mathbf{x}+\mathbf{u}_{k}^{\rm{H}}\mathbf{z}_{b},$

where $\mathcal{K}_{k}\triangleq\mathcal{K}\setminus\\{k\\}$ and $0<\beta\ll
1$ is a constant denoting the SI cancellation ability of the FD BS, i.e., the
SI cancellation quality. FD radios can use SI cancellation techniques
involving hardware techniques, analog domain cancellation, and digital domain
cancellation of residual SI [14]. These can also be adapted for ISAC/ISABC
systems.

###  II-D Tags’ EH

As mentioned, passive tags do not generate RF signals and rely entirely on EH
to power their essential functions. To simultaneously perform EH and data
transfer operations, tags split the incident RF signal [31, 32] into two
streams. With this PS, $T_{k}$ reflects a fraction of the incident RF power,
i.e.,
$p_{k}^{\rm{in}}=|\mathbf{g}^{\rm{H}}_{f,k}\mathbf{w}|^{2}+\mathbf{g}^{\rm{H}}_{f,k}\mathbf{S}\mathbf{g}_{f,k}$,
and harvests the remainder based on the reflection coefficient, $\alpha_{k}$
[31]. In particular, $T_{k}$ reflects $\alpha_{k}p_{k}^{\rm{in}}$ or data
transmission, while absorbing the reminder, $(1-\alpha_{k})p_{k}^{\rm{in}}$,
for EH.

The harvested power at $T_{k}$, $p_{k}^{\rm{h}}$, can be modeled as a linear
or nonlinear function of $p_{k}^{\rm{in}}$. The harvested power of the widely
used linear model is given as
$p_{k}^{\rm{h}}=\eta(1-\alpha_{k})p_{k}^{\rm{in}}$, where $\eta\in(0,1]$ is
the power conversion efficiency. Despite its simplicity, the linear EH model
overlooks nonlinear characteristics of actual EH circuits, such as saturation
and sensitivity [33]. As a remedy, a parametric nonlinear EH model based on
the sigmoid function has been frequently employed [33]. It models the total
harvested power at $T_{k}$ as
$p_{k}^{\rm{h}}=\Phi((1-\alpha_{k})p_{k}^{\rm{in}})$, where $\Phi(\cdot)$ is a
function representing non-linear effects [33, eq. (4)].

Nevertheless, our problem formulation can handle linear and non-linear models
within one unified framework. The activation threshold, $p_{b}$, is an
essential parameter whether a linear or non-linear model is used. It is the
minimal amount of power necessary to activate the EH circuit, which is around
-20m for off-the-shelf passive tags [3]. This means that the harvested power
must exceed $p_{b}$ to activate the tag, i.e.,
$(1-\alpha_{k})p_{k}^{\rm{in}}\geq p_{b}^{\prime}$, where
$p_{b}^{\prime}\triangleq\Phi^{-1}(p_{b})$. The following utilizes the
nonlinear EH model to formulate the optimization problem and design the
resource allocation algorithm.

##  III Communication and Sensing Performance

This section derives the communication and sensing performance of the proposed
RSMA-assisted ISABC system.

###  III-A Primary Communication Performance

The users first decode the common message by treating private and backscatter
signals as interference. From (4), the rate of $U_{l}$ for decoding common
data is

$\displaystyle\mathcal{R}_{c,l}={\rm{log}}_{2}(1+\gamma_{c,l}),$ (8)

where $\gamma_{c,l}$ is the received SINR at $U_{l}$ for decoding $x_{c}$, and
given in (9).

$\displaystyle\gamma_{c,l}=\frac{|\mathbf{f}_{l}^{\rm{H}}\mathbf{w}_{c}|^{2}}{\sum_{j\in\mathcal{L}}|\mathbf{f}_{l}^{\rm{H}}\mathbf{w}_{j}|^{2}+\delta_{s}|\mathbf{f}_{l}^{\rm{H}}\mathbf{s}|^{2}+\sum_{k\in\mathcal{K}}\alpha_{k}\left(|\mathbf{h}_{l,k}^{\rm{H}}\mathbf{w}_{c}|^{2}+\sum_{j\in\mathcal{L}}|\mathbf{h}_{l,k}^{\rm{H}}\mathbf{w}_{j}|^{2}+|\mathbf{h}_{l,k}^{\rm{H}}\mathbf{s}|^{2}\right)+\sigma^{2}}$
(9)

To guarantee that all users successfully decode the common stream, the rate of
decoding the common stream $x_{c}$ should not exceed
$\mathcal{R}_{c}={\rm{min}}_{l\in\mathcal{L}}\mathcal{R}_{c,l}$ [17]. As
$\mathcal{R}_{c}$ is shared by $L$ users, it follows that
$\sum_{l\in\mathcal{L}}C_{l}=\mathcal{R}_{c}$, where $C_{l}$ is the portion of
the common rate at $U_{l}$ transmitting $d_{c,l}$.

Next, $U_{l}$ subtracts the decoded common message from the received signal
and decodes its desired private message, $x_{l}$. The private rate of $U_{l}$
is thus given by

$\displaystyle\mathcal{R}_{p,l}^{\prime}={\rm{log}}_{2}(1+\gamma_{p,l}),$ (10)

where $\gamma_{p,l}$ is the received SINR at $U_{l}$ for decoding $x_{l}$ and
given in (11). In (11) $\delta_{c}$ accounts for SIC qualities for removing
the common signal.

$\displaystyle\gamma_{p,l}=\frac{|\mathbf{f}_{l}^{\rm{H}}\mathbf{w}_{l}|^{2}}{\delta_{c}|\mathbf{f}_{l}^{\rm{H}}\mathbf{w}_{c}|^{2}+\sum_{j\in\mathcal{L}_{l}}|\mathbf{f}_{l}^{\rm{H}}\mathbf{w}_{j}|^{2}+\delta_{s}|\mathbf{f}_{l}^{\rm{H}}\mathbf{s}|^{2}+\sum_{k\in\mathcal{K}}\alpha_{k}\left(|\mathbf{h}_{l,k}^{\rm{H}}\mathbf{w}_{c}|^{2}+\sum_{j\in\mathcal{L}}|\mathbf{h}_{l,k}^{\rm{H}}\mathbf{w}_{j}|^{2}+|\mathbf{h}_{l,k}^{\rm{H}}\mathbf{s}|^{2}\right)+\sigma^{2}}$
(11)

The total achievable rate of $U_{l}$, including the portion of common rate
transmitting $d_{c,l}$ and private rate transmitting $d_{p,l}$, is thus
expressed as

$\displaystyle\mathcal{R}_{p,l}=C_{l}+\mathcal{R}_{p,l}^{\prime}.$ (12)

###  III-B BackCom Performance

The reader decodes the tags’ data. Using (5), the reader decodes $T_{k}$’s
data at the rate

$\displaystyle\mathcal{R}_{t,k}\approx{\rm{log}}_{2}(1+\gamma_{t,k}),$ (13)

where $\gamma_{t,k}$ is the $T_{k}$’s SINR at the reader and given in (14).

$\displaystyle\gamma_{t,k}=\frac{\alpha_{k}\left(|\mathbf{h}_{k}^{\rm{H}}\mathbf{w}_{c}|^{2}+\sum_{j\in\mathcal{L}}|\mathbf{h}_{k}^{\rm{H}}\mathbf{w}_{j}|^{2}+|\mathbf{h}_{k}^{\rm{H}}\mathbf{s}|^{2}\right)}{\delta_{c}|\mathbf{f}_{0}\mathbf{w}_{c}|^{2}+\delta_{p}\sum_{j\in\mathcal{L}}|\mathbf{f}_{0}^{\rm{H}}\mathbf{w}_{j}|^{2}+\delta_{s}|\mathbf{f}_{0}^{\rm{H}}\mathbf{s}|^{2}+\sum_{i\in\mathcal{K}_{k}}{\alpha_{i}}\left(|\mathbf{h}_{i}^{\rm{H}}\mathbf{w}_{c}|^{2}+\sum_{j\in\mathcal{L}}|\mathbf{h}_{i}^{\rm{H}}\mathbf{w}_{j}|^{2}+|\mathbf{h}_{i}^{\rm{H}}\mathbf{s}|^{2}\right)+\sigma^{2}}$
(14)

###  III-C Sensing Performance

Sensing performance is typically measured by the transmit beampattern gain or
mean squared error of the transmit beampattern [34, 35]. Although simple, they
do not account for the receiver’s beam pattern or multi-target interference.
This may cause ambiguity in multi-target detection due to interference between
the targets’ reflected signals [13, 36]. To overcome the limitations in such
sensing measures, the sensing SINR has been suggested to assess sensing
performance [13, 36]. The detection probability of a target (tag) is
proportional to its sensing SINR [13, 36]. In addition, the sensing SINR
enables target detection through both transmit and receiver beamforming,
reducing interference between targets [13, 36]. Due to these benefits, this
study employs sensing SINR to measure sensing performance.

Sensing is done by the BS, which utilizes the backscattered signal of tag
$T_{k}$. The BS applies the SI cancellation and the receiver beamformer,
$\mathbf{u}_{k}$, to received signal (6) to capture $T_{k}$’s reflected
signal. From (7), the sensing rate of $T_{k}$ is obtained as

$\displaystyle\mathcal{R}_{s,k}\approx{\rm{log}}_{2}(1+\Upsilon_{k}),$ (15)

where $\Upsilon_{k}$ is the sensing SINR of $T_{k}$ and given in (16).

$\displaystyle\Upsilon_{k}\\!\\!\\!\\!\\!$ $\displaystyle=$
$\displaystyle\\!\\!\\!\\!\\!\frac{\alpha_{k}\mathbb{E}\left\\{|\mathbf{u}_{k}^{\rm{H}}\mathbf{G}_{k}(\theta_{k})\mathbf{x}|^{2}\right\\}}{\\!\\!\\!\sum\limits_{i\in\mathcal{K}_{k}}\\!\\!\alpha_{i}\mathbb{E}\left\\{|\mathbf{u}_{k}^{\rm{H}}\mathbf{G}_{i}(\theta_{i})\mathbf{x}|^{2}\right\\}\\!+\\!\beta\mathbb{E}\left\\{|\mathbf{u}_{k}^{\rm{H}}\mathbf{G}_{\rm{SI}}^{\rm{H}}\mathbf{x}|^{2}\right\\}\\!+\\!\mathbb{E}\left\\{|\mathbf{u}_{k}^{\rm{H}}\mathbf{z}_{b}|^{2}\right\\}}\\!=\\!\frac{\alpha_{k}\mathbf{u}_{k}^{\rm{H}}\mathbf{G}_{k}(\theta_{k})\mathbf{R}_{x}\mathbf{G}_{k}^{\rm{H}}(\theta_{k})\mathbf{u}_{k}}{\mathbf{u}_{k}^{\rm{H}}\\!\left(\\!\sum\limits_{i\in\mathcal{K}_{k}}\\!\\!\alpha_{i}\mathbf{G}_{i}(\theta_{i})\mathbf{R}_{x}\mathbf{G}_{i}^{\rm{H}}\\!+\\!\beta\mathbf{G}_{\rm{SI}}\mathbf{R}_{x}\mathbf{G}_{\rm{SI}}^{\rm{H}}\\!+\\!\sigma^{2}\mathbf{I}_{N}\right)\\!\mathbf{u}_{k}}\qquad$
(16)

$\mathbf{R}_{x}\triangleq\mathbb{E}\\{\mathbf{x}\mathbf{x}^{\rm{H}}\\}=\mathbf{w}_{c}\mathbf{w}_{c}^{\rm{H}}+\sum_{j\in\mathcal{L}}\mathbf{w}_{j}\mathbf{w}_{j}^{\rm{H}}+\mathbf{S}$
in (16) is the BS transmitted signal covariance matrix [13].

##  IV Problem Formulation

Herein, the BS transmit power of the proposed RSMA-assisted ISABC network is
optimized. In particular, our objective is to minimize it by jointly
optimizing the BS receive beamforming,
$\\{\mathbf{u}_{k}\\}_{k\in\mathcal{K}}$, the BS transmit beamforming,
$\mathbf{w}_{c}$, $\\{\mathbf{w}_{l}\\}_{l\in\mathcal{L}}$, covariance matrix
$\mathbf{S}$, the tag reflection coefficients,
$\\{\alpha_{k}\\}_{k\in\mathcal{K}}$, and common rates of the users,
$\\{C_{l}\\}_{l\in\mathcal{L}}$. The set of all the optimization variables is
denoted as
$\mathcal{A}=\left\\{\\{\mathbf{u}_{k}\\}_{k\in\mathcal{K}},\mathbf{w}_{c},\\{\mathbf{w}_{l}\\}_{l\in\mathcal{L}},\mathbf{S}\succeq
0,\\{\alpha_{k}\\}_{k\in\mathcal{K}},\\{C_{l}\\}_{l\in\mathcal{L}}\right\\}$.

The communication rates for the users and the reader must exceed their
thresholds. The sensing and EH functionalities of the tags must be met. The
optimization problem is thus formulated as follows:

$\displaystyle\mathbf{P}_{1}:\underset{\mathcal{A}}{\rm{min}}$
$\displaystyle\|\mathbf{w}_{c}\|^{2}+\sum\nolimits_{l\in\mathcal{L}}\|\mathbf{w}_{l}\|^{2}+\text{Tr}{(\mathbf{S})},$
(17a) s.t.
$\displaystyle\mathcal{R}_{s,k}\geq\mathcal{R}_{s,k}^{\rm{th}},\leavevmode\nobreak\
\forall k,$ (17k)
$\displaystyle\mathcal{R}_{t,k}\geq\mathcal{R}_{t,k}^{\rm{th}},\leavevmode\nobreak\
\forall k,$
$\displaystyle\mathcal{R}_{c,l}\geq\mathcal{R}_{c,l}^{\rm{th}},\leavevmode\nobreak\
\forall l,$ $\displaystyle
C_{l}+\mathcal{R}_{p,l}^{\prime}\geq\mathcal{R}_{p,l}^{\rm{th}},\leavevmode\nobreak\
\forall l,$
$\displaystyle\sum\nolimits_{l\in\mathcal{L}}C_{l}\leq\mathcal{R}_{c},$
$\displaystyle 0\leq C_{l}\leq\mathcal{R}_{p,l}^{\rm{th}},\leavevmode\nobreak\
\forall l,$
$\displaystyle{p}_{k}^{\rm{in}}\geq\frac{\Phi^{-1}(p_{b})}{1-\alpha_{k}},\leavevmode\nobreak\
\forall k,$ $\displaystyle\mathbf{S}\succeq 0,$
$\displaystyle\|\mathbf{u}_{k}\|^{2}=1,\leavevmode\nobreak\ \forall k,$
$\displaystyle 0<\alpha_{k}<1,\leavevmode\nobreak\ \forall k,\qquad$

where constraints (17k) and (17k) ensure the required sensing and
communication rate requirement of each tag at the BS and the reader,
respectively, with $\mathcal{R}_{s,k}^{\rm{th}}$ and
$\mathcal{R}_{t,k}^{\rm{th}}$ representing the corresponding targeted sensing
and BackCom rates. On the other hand, (17k) and (17k) set the minimum rate
requirements for the user to decode common and private data, respectively, in
which $\mathcal{R}_{c,l}^{\rm{th}}$ and $\mathcal{R}_{p,l}^{\rm{th}}$ denote
the targeted common and private rates of respective user. Conditions (17k) and
(17k) are necessary to guarantee the successful decoding of the common stream.
Besides, (17k) guarantees the minimum tag power requirements for EH.
Constraint (17k) is the normalization constraint for the BS reception filter.
Finally, (17k) is the range of the tag reflection coefficient.

Since the user/tag communication/sensing rates are monotonically increasing
functions of their arguments, i.e., SINRs, they can be replaced with
respective SINRs before proceeding to the proposed solution. The optimization
problem, $\mathbf{P}_{1}$, is thus equivalently formulated as follows:

$\displaystyle\mathbf{P}_{2}:\underset{\mathcal{A}}{\rm{min}}$
$\displaystyle\|\mathbf{w}_{c}\|^{2}+\sum\nolimits_{l\in\mathcal{L}}\|\mathbf{w}_{l}\|^{2}+\text{Tr}{(\mathbf{S})},$
(18a) s.t.
$\displaystyle\Upsilon_{k}\geq\Upsilon_{k}^{\rm{th}},\leavevmode\nobreak\
\forall k,$ (18e)
$\displaystyle\gamma_{t,k}\geq\Gamma_{t,k}^{\rm{th}},\leavevmode\nobreak\
\forall k,$
$\displaystyle\gamma_{c,l}\geq\Gamma_{c,l}^{\rm{th}},\leavevmode\nobreak\
\forall l,$ $\displaystyle\eqref{P1_pvt_rate}-\eqref{P1_alpha},$

where $\Upsilon_{k}^{\rm{th}}\triangleq 2^{\mathcal{R}_{s,k}^{\rm{th}}}-1$,
$\Gamma_{t,k}^{\rm{th}}\triangleq 2^{\mathcal{R}_{t,k}^{\rm{th}}}-1$, and
$\Gamma_{c,l}^{\rm{th}}\triangleq 2^{\mathcal{R}_{c,l}^{\rm{th}}}-1$ are the
respective sensing/communication SINR thresholds.

##  V Proposed Solution

Since problem $\mathbf{P}_{2}$ is non-convex, widely available convex
algorithms cannot be directly applied. Thus, an AO strategy [20] is adopted to
handle it, where the problem is decomposed into three sub-problems [37]. It
turns out that the objective value of $\mathbf{P}_{2}$ is the same regardless
of the solutions of the first and third sub-problems. Thus, they are
feasibility problems. Feasibility problems aim to find any solution that
satisfies the constraints. However, these are converted into optimization
problems with explicit objectives for more effective solutions.

###  V-A Sub-Problem 1: Optimizing over $\mathbf{u}_{k}$

This develops the optimal reception filters of the BS while keeping other
variables constant. For given
$\\{\mathbf{w}_{c},\\{\mathbf{w}_{l}\\}_{l\in\mathcal{L}},\mathbf{S},\\{\alpha_{k}\\}_{k\in\mathcal{K}},\\{C_{l}\\}_{l\in\mathcal{L}}\\}$,
$\mathbf{P}_{2}$ becomes a feasibility problem. Therefore, any feasible value
of $\mathbf{u}_{k}$ that complies with (18e) and (17k) can be a potential
solution.

Although vector $\mathbf{u}_{k}$ may not have a direct impact on reducing the
BS transmit power, it is crucial for optimizing the sensing SINR at the BS for
each tag. Our methodology prioritizes the optimization of $\mathbf{u}_{k}$ to
meet the sensing performance criteria and indirectly contribute to transmit
power reduction. Enhanced SINR for tag signals reduces the need for higher
transmit power, thus aligning with our overarching aim of power minimization
[38, 39].

Using the unique structure of the sensing SINR for each tag (16), this sub-
problem is transformed into a generalized Rayleigh quotient problem, providing
closed-form optimal combiner vectors [38]. To this end, $\mathbf{P}_{2}$ is
reduced to the following optimization problem:

$\displaystyle\mathbf{P}_{\mathrm{u}}:\underset{\mathbf{u}_{k}}{\text{max}}$
(19a)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\frac{\alpha_{k}\mathbf{u}_{k}^{\rm{H}}\mathbf{G}_{k}\mathbf{R}_{x}\mathbf{G}^{\rm{H}}_{k}\mathbf{u}_{k}}{\mathbf{u}^{\rm{H}}_{k}\left(\sum\limits_{i\in\mathcal{K}_{k}}\alpha_{i}\mathbf{G}_{i}\mathbf{R}_{x}\mathbf{G}_{i}^{\rm{H}}+\beta\mathbf{G}_{\rm{SI}}\mathbf{R}_{x}\mathbf{G}_{\rm{SI}}^{\rm{H}}+\sigma^{2}_{k}\mathbf{I}_{N}\right)\mathbf{u}_{k}},\qquad$
s.t. $\displaystyle\|\mathbf{u}_{k}\|^{2}=1,\quad\forall k,$ (19b)

Next, the objective function, (19a), is rearranged to convert it into the
following optimization problem:

$\displaystyle\mathbf{P}_{\mathrm{u}1}:\underset{\mathbf{u}_{k}}{\text{max}}$
$\displaystyle\frac{\mathbf{u}_{k}^{\rm{H}}\tilde{\mathbf{G}}_{k}\tilde{\mathbf{G}}_{k}^{\rm{H}}\mathbf{u}_{k}}{\mathbf{u}_{k}^{\rm{H}}\mathbf{Q}_{k}\mathbf{u}_{k}},\qquad$
(20a) s.t. $\displaystyle\eqref{Pu_detector},$ (20b)

where
$\mathbf{Q}_{k}\triangleq\sum_{i\in\mathcal{K}_{k}}\\!{\alpha_{i}}\mathbf{G}_{i}\mathbf{R}_{x}\mathbf{G}_{i}^{\rm{H}}+\beta\mathbf{G}_{\rm{SI}}\mathbf{R}_{x}\mathbf{G}_{\rm{SI}}^{\rm{H}}+\sigma^{2}\mathbf{I}_{N}$,
and
$\tilde{\mathbf{G}}_{k}=\sqrt{\alpha_{k}}\mathbf{G}_{k}\left(\mathbf{w}_{c}+\sum_{j\in\mathcal{L}}\mathbf{w}_{j}+\mathbf{s}\right)$.
Problem $\mathbf{P}_{\mathrm{u}1}$ thus becomes a generalized Rayleigh ratio
quotient problem [38]. The optimal combiner vector for $T_{k}$ is
characterized by the Wiener or MMSE filter [32] and is given as

$\displaystyle\mathbf{u}_{k}^{*}=\frac{\mathbf{Q}_{k}^{-1}\tilde{\mathbf{G}}_{k}}{\|\mathbf{Q}_{k}^{-1}\tilde{\mathbf{G}}_{k}\|},\leavevmode\nobreak\
\forall k.$ (21)

###  V-B Sub-Problem 2: Optimizing over $\mathbf{w}_{c}$, $\mathbf{w}_{l}$,
$\mathbf{S}$, and $C_{l}$

This optimizes for the BS transmit beamforming, i.e., $\mathbf{w}_{c}$,
$\\{\mathbf{w}_{l}\\}_{l\in\mathcal{L}}$, and $\mathbf{S}$, and users’ common
rate, i.e., $\\{C_{l}\\}_{l\in\mathcal{L}}$ for given
$\\{\\{\mathbf{u}_{k}\\}_{k\in\mathcal{K}},\\{\alpha\\}_{k\in\mathcal{K}}\\}$.
However, owing to the interference within the SINRs, Constraints
$\mathbf{P}_{2}$ become non-convex for the BS transmit beamformers. The semi-
definite relaxation (SDR) method is employed to overcome this barrier. Thus,
$\mathbf{P}_{2}$ is reformulated as

$\displaystyle\mathbf{P}_{\mathrm{w}}:\underset{\mathbf{w}_{c},\mathbf{w}_{l},\mathbf{S},C_{l}}{\rm{min}}$
$\displaystyle\|\mathbf{w}_{c}\|^{2}+\sum\nolimits_{l\in\mathcal{L}}\|\mathbf{w}_{l}\|^{2}+\text{Tr}{(\mathbf{S})},$
(22a) s.t.
$\displaystyle\eqref{P2_sens_rate}-\eqref{P2_comm_rate},\leavevmode\nobreak\
\eqref{P1_pvt_rate}-\eqref{P1_S}.\qquad$ (22b)

$\mathbf{P}_{\mathrm{w}}$ can be effectively addressed by using the SDR method
[21, 22]. Define matrices
$\mathbf{W}_{c}\triangleq\mathbf{w}_{c}\mathbf{w}_{c}^{\rm{H}}$ and
$\mathbf{W}_{l}\triangleq\mathbf{w}_{l}\mathbf{w}_{l}^{\rm{H}}$ for
$l\in\mathcal{L}$. Here, $\mathbf{W}_{c}$ and $\mathbf{W}_{l}$ are semi-
definite matrices with rank one constraints, i.e.,
${\rm{Rank}}(\mathbf{W}_{c})=1$ and ${\rm{Rank}}(\mathbf{W}_{l})=1$. By
relaxing the highly non-convex rank one constraints, $\mathbf{P}_{\mathrm{w}}$
can be reformulated into a conventional semi-definite programming (SDP)
problem. However, as constraints (17k) and (17k) are still non-convex, they
are next converted into tractable convex forms. Define

$\displaystyle A_{l}$
$\displaystyle\triangleq\text{Tr}(\mathbf{f}_{l}\mathbf{f}_{l}^{\rm{H}}\mathbf{W}_{l}),$
(23a) $\displaystyle B_{l}$
$\displaystyle\triangleq\delta_{c}\text{Tr}(\mathbf{f}_{l}\mathbf{f}_{l}^{\rm{H}}\mathbf{W}_{c})+\sum\nolimits_{j\in\mathcal{L}_{l}}|\text{Tr}(\mathbf{f}_{l}\mathbf{f}_{l}^{\rm{H}}\mathbf{W}_{j})+\delta_{s}\text{Tr}(\mathbf{f}_{l}\mathbf{f}_{l}^{\rm{H}}\mathbf{S})$
$\displaystyle+\sum\nolimits_{k\in\mathcal{K}}\alpha_{k}\bigg{(}\text{Tr}(\mathbf{h}_{l,k}\mathbf{h}_{l,k}^{\rm{H}}\mathbf{W}_{c})+\sum\nolimits_{j\in\mathcal{L}}\text{Tr}(\mathbf{h}_{l,k}\mathbf{h}_{l,k}^{\rm{H}}\mathbf{W}_{j})$
$\displaystyle\hskip
85.35826pt+\text{Tr}(\mathbf{h}_{l,k}\mathbf{h}_{l,k}^{\rm{H}}\mathbf{S})\bigg{)}+\sigma^{2}.$
(23b)

The SINR in (17k) is now bounded such that $0\leq\mu_{l}\leq A_{l}/B_{l}$,
where $\mu_{l}$ is an auxiliary variable. Thereby introducing another
auxiliary variable, $\phi_{l}$, where $A_{l}\geq\mu_{l}\phi_{l}$ and
$B_{l}\leq\phi_{l}$. However, since the product $\mu_{l}\phi_{l}$ is non-
convex, Taylor-series linearization is used to convert it to a convex function
as

$\displaystyle\mu_{l}\phi_{l}=\mu_{l}^{(t)}\phi_{l}^{(t)}+\phi_{l}^{(t)}(\mu_{l}-\mu_{l}^{(t)})+\mu_{l}^{(t)}(\phi_{l}-\phi_{l}^{(t)})\triangleq
v_{l},\\!$ (24)

where $\\{\mu_{l},\phi_{l}\\}$ and $\\{\mu_{l}^{(t)},\phi_{l}^{(t)}\\}$ are
the current and the previous iteration values of $\\{\mu_{l},\phi_{l}\\}$,
respectively. The constraint (17k) is thus equivalent to

$\displaystyle C_{l}+\log_{2}(1+\mu_{l})\geq\mathcal{R}_{p,l}^{\rm{th}},$
$\forall l$, (25a) $\displaystyle A_{l}\geq v_{l},$ $\forall l$, (25b)
$\displaystyle B_{l}\leq\phi_{l},$ $\forall l$. (25c)

For constraint (17k), each rate value $\mathcal{R}{c}$ is replaced with
$\sum{l\in\mathcal{L}}C_{l}\leq\mathcal{R}{c,l}$ for $l\in\mathcal{L}$. Since
$\mathcal{R}{c,l}$ is not a convex function of the optimization variables, the
successive convex approximation (SCA) method is employed to linearize the
constraint as

$\displaystyle\sum\nolimits_{l\in\mathcal{L}}C_{l}\leq\digamma_{l}-\tilde{\digamma}_{l},$
(26)

where $\digamma_{l}$ and $\tilde{\digamma}_{l}$ are given in (27) and (28),
respectively,

$\displaystyle\digamma_{l}$ $\displaystyle\triangleq$
$\displaystyle\log_{2}\bigg{(}\text{Tr}(\mathbf{f}_{l}\mathbf{f}_{l}^{\rm{H}}\mathbf{W}_{c})+\sum_{j\in\mathcal{L}}\text{Tr}(\mathbf{f}_{l}\mathbf{f}_{l}^{\rm{H}}\mathbf{W}_{j})+\delta_{s}\text{Tr}(\mathbf{f}_{l}\mathbf{f}_{l}^{\rm{H}}\mathbf{S})$
(27)
$\displaystyle+\sum_{k\in\mathcal{K}}\alpha_{k}\bigg{(}\text{Tr}(\mathbf{h}_{l,k}\mathbf{h}_{l,k}^{\rm{H}}\mathbf{W}_{c})+\sum_{j\in\mathcal{L}}\text{Tr}(\mathbf{h}_{l,k}\mathbf{h}_{l,k}^{\rm{H}}\mathbf{W}_{j})+\text{Tr}(\mathbf{h}_{l,k}\mathbf{h}_{l,k}^{\rm{H}}\mathbf{S})\bigg{)}+\sigma^{2}\bigg{)}$
$\displaystyle\tilde{\digamma}_{l}$ $\displaystyle\triangleq$
$\displaystyle\log_{2}(\Lambda_{l})+\frac{\sum_{k\in\mathcal{K}}\alpha_{k}\text{Tr}(\mathbf{h}_{l,k}\mathbf{h}_{l,k}^{\rm{H}})}{\ln(2)\Lambda_{l}}\text{Tr}\left(\mathbf{W}_{c}-\mathbf{W}_{c}^{(t)}\right)+\sum_{j\in\mathcal{L}}\frac{\text{Tr}(\mathbf{f}_{l}\mathbf{f}_{l}^{\rm{H}})+\sum_{k\in\mathcal{K}}\alpha_{k}\text{Tr}(\mathbf{h}_{l,k}\mathbf{h}_{l,k}^{\rm{H}})}{\ln(2)\Lambda_{l}}\text{Tr}\left(\mathbf{W}_{j}-\mathbf{W}_{j}^{(t)}\right)$
(28)
$\displaystyle\frac{\delta_{s}\text{Tr}(\mathbf{f}_{l}\mathbf{f}_{l}^{\rm{H}})+\sum_{k\in\mathcal{K}}\alpha_{k}\text{Tr}(\mathbf{h}_{l,k}\mathbf{h}_{l,k}^{\rm{H}})}{\ln(2)\Lambda_{l}}\text{Tr}\left(\mathbf{S}-\mathbf{S}^{(t)}\right)$

and $\Lambda_{l}$ is defined as

$\displaystyle\Lambda_{l}$ $\displaystyle\triangleq$
$\displaystyle\sum_{j\in\mathcal{L}}\text{Tr}\left(\mathbf{f}_{l}\mathbf{f}_{l}^{\rm{H}}\mathbf{W}_{j}^{(t)}\right)+\delta_{s}\text{Tr}\left(\mathbf{f}_{l}\mathbf{f}_{l}^{\rm{H}}\mathbf{S}^{(t)}\right)$
(29)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\sum_{k\in\mathcal{K}}\alpha_{k}\bigg{(}\text{Tr}\left(\mathbf{h}_{l,k}\mathbf{h}_{l,k}^{\rm{H}}\mathbf{W}_{c}^{(t)}\right)+\sum_{j\in\mathcal{L}}\text{Tr}\left(\mathbf{h}_{l,k}\mathbf{h}_{l,k}^{\rm{H}}\mathbf{W}_{j}^{(t)}\right)$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!\\!+\text{Tr}\left(\mathbf{h}_{l,k}\mathbf{h}_{l,k}^{\rm{H}}\mathbf{S}^{(t)}\right)\bigg{)}+\sigma^{2}.$

In (29), $(\cdot)^{(t)}$ denotes the previous iteration values of respective
variables. Finally, $\mathbf{P}_{\mathrm{w}}$ can be reformulated to a relaxed
SDP problem as given in (30), which can be tackled using the CVX tool [40,
41].

$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\!\mathbf{P}_{\mathrm{w}1}:$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\underset{\mathbf{W}_{c},\mathbf{W}_{l},\mathbf{S},C_{l}}{\rm{min}}\text{Tr}{(\mathbf{W}_{c})}+\sum_{l\in\mathcal{L}}\text{Tr}{(\mathbf{W}_{l})}+\text{Tr}{(\mathbf{S})},$
(30a) s.t.
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\Upsilon_{k}^{\rm{th}}\mathbf{u}_{k}^{\rm{H}}\left(\sum_{i\in\mathcal{K}_{k}}\alpha_{i}\left(\text{Tr}(\mathbf{G}_{i}^{\rm{H}}\mathbf{G}_{i}\mathbf{W}_{c})+\sum_{j\in\mathcal{L}}\text{Tr}(\mathbf{G}_{i}^{\rm{H}}\mathbf{G}_{i}\mathbf{W}_{j})+\text{Tr}(\mathbf{G}_{i}^{\rm{H}}\mathbf{G}_{i}\mathbf{S})\right)+\sigma^{2}\mathbf{I}_{N}\right.$
(30g)
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\left.+\beta\left(\text{Tr}(\mathbf{G}_{\rm{SI}}^{\rm{H}}\mathbf{G}_{\rm{SI}}\mathbf{W}_{c})+\sum_{j\in\mathcal{L}}\text{Tr}(\mathbf{G}_{\rm{SI}}^{\rm{H}}\mathbf{G}_{\rm{SI}}\mathbf{W}_{j})+\text{Tr}(\mathbf{G}_{\rm{SI}}^{\rm{H}}\mathbf{G}_{\rm{SI}}\mathbf{S})\right)\right)\mathbf{u}_{k}$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!-\alpha_{k}\mathbf{u}_{k}^{\rm{H}}\left(\text{Tr}(\mathbf{G}_{k}^{\rm{H}}\mathbf{G}_{k}\mathbf{W}_{c})+\sum_{j\in\mathcal{L}}\text{Tr}(\mathbf{G}_{k}^{\rm{H}}\mathbf{G}_{k}\mathbf{W}_{j})+\text{Tr}(\mathbf{G}_{k}^{\rm{H}}\mathbf{G}_{k}\mathbf{S})\right)\mathbf{u}_{k}\leq
0,\leavevmode\nobreak\ \forall k,$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\Gamma_{t,k}^{\rm{th}}\left(\delta_{c}\text{Tr}(\mathbf{f}_{0}\mathbf{f}_{0}^{\rm{H}}\mathbf{W}_{c})+\delta_{p}\sum_{j\in\mathcal{L}}\text{Tr}(\mathbf{f}_{0}\mathbf{f}_{0}^{\rm{H}}\mathbf{W}_{j})+\delta_{s}\text{Tr}(\mathbf{f}_{0}\mathbf{f}_{0}^{\rm{H}}\mathbf{S})+\sigma^{2}+\sum_{i\in\mathcal{K}_{k}}\alpha_{i}\left(\text{Tr}(\mathbf{h}_{i}\mathbf{h}_{i}^{\rm{H}}\mathbf{W}_{c})+\sum_{j\in\mathcal{L}}\text{Tr}(\mathbf{h}_{i}\mathbf{h}_{i}^{\rm{H}}\mathbf{W}_{j})\right.\right.$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\left.\left.+\text{Tr}(\mathbf{h}_{i}\mathbf{h}_{i}^{\rm{H}}\mathbf{S})\right)\right)-\alpha_{k}\left(\text{Tr}(\mathbf{h}_{k}\mathbf{h}_{k}^{\rm{H}}\mathbf{W}_{c})+\sum_{j\in\mathcal{L}}\text{Tr}(\mathbf{h}_{k}\mathbf{h}_{k}^{\rm{H}}\mathbf{W}_{j})+\text{Tr}(\mathbf{h}_{k}\mathbf{h}_{k}^{\rm{H}}\mathbf{S})\right)\leq
0,\leavevmode\nobreak\ \forall k,$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\Gamma_{c,l}^{\rm{th}}\\!\\!\left(\sum_{j\in\mathcal{L}}\\!\text{Tr}(\mathbf{f}_{l}\mathbf{f}_{l}^{\rm{H}}\mathbf{W}_{j})\\!\\!+\\!\\!\delta_{s}\text{Tr}(\mathbf{f}_{l}\mathbf{f}_{l}^{\rm{H}}\mathbf{S})\\!+\\!\\!\sum_{i\in\mathcal{K}}\\!\\!\alpha_{i}\\!\\!\left(\\!\\!\text{Tr}(\mathbf{h}_{l,i}\mathbf{h}_{l,i}^{\rm{H}}\mathbf{W}_{c})\\!+\\!\\!\sum_{j\in\mathcal{L}}\\!\\!\text{Tr}(\mathbf{h}_{l,i}\mathbf{h}_{l,i}^{\rm{H}}\mathbf{W}_{j})\\!+\\!\text{Tr}(\mathbf{h}_{l,i}\mathbf{h}_{l,i}^{\rm{H}}\mathbf{S})\right)\\!\\!+\\!\sigma^{2}\\!\right)\\!\\!-\\!\\!\text{Tr}(\mathbf{f}_{l}\mathbf{f}_{l}^{\rm{H}}\mathbf{W}_{c})\\!\leq\\!0,\leavevmode\nobreak\
\forall l,\quad$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!P_{\rm{th}}-(1-\alpha_{k})\bigg{(}\text{Tr}(\mathbf{g}_{f,k}\mathbf{g}^{\rm{H}}_{f,k}\mathbf{W}_{c})+\sum_{j\in\mathcal{L}}\text{Tr}(\mathbf{g}_{f,k}\mathbf{g}^{\rm{H}}_{f,k}\mathbf{W}_{j})+\text{Tr}(\mathbf{g}^{\rm{H}}_{f,k}\mathbf{g}_{f,k}\mathbf{S})\bigg{)}\leq
0,\leavevmode\nobreak\ \forall k,$
$\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\\{\mathbf{W}_{c},\\{\mathbf{W}_{l}\\}_{l\in\mathcal{L}},\mathbf{S}\\}\succeq
0,$ $\displaystyle\\!\\!\\!\\!\\!\\!\\!\\!\eqref{P1_cl},\leavevmode\nobreak\
\eqref{Pw_pvt_rate_c1}-\eqref{Pw_pvt_rate_c3},\leavevmode\nobreak\
\eqref{eqn_ComRate_decod_c1}.$

If the solutions to the relaxed problem $\mathbf{P}_{\mathrm{w}1}$ are of rank
one, the optimal transmit beamformers are obtained by eigenvalue decomposition
[22]. Let the eigenvalue decomposition of $\mathbf{W}$ (i.e.,
$\mathbf{W}\in\\{\mathbf{W}_{c}^{*},\\{\mathbf{W}_{l}^{*}\\}_{l\in\mathcal{L}}\\}$)
to be $\mathbf{W}=\mathbf{U}\bm{\Sigma}\mathbf{U}^{\rm{H}}$, where
$\mathbf{U}$ is a unitary matrix and
$\bm{\Sigma}=\text{diag}(\lambda_{1},\dots,\lambda_{M})$ is a diagonal matrix.
If $\mathbf{W}^{*}$ is rank one, the optimal transmit beamformer,
$\mathbf{w}^{*}$, is the eigenvector for the maximum eigenvalue. If not, the
Gaussian randomization accounts for the relaxed rank-one constraint [22].
Specifically, a solution for $\mathbf{P}_{\mathrm{w}}$ is generated as
$\bar{\mathbf{W}}=\mathbf{U}\bm{\Sigma}^{1/2}\mathbf{r}$, with
$\mathbf{r}\in\mathcal{CN}(0,\mathbf{I}_{M})$. Such solutions are tested for
$10^{5}$ times, and the best one is selected. These numerous random
realizations of $\mathbf{r}$ with the SDR technique ensure an
$\frac{\pi}{4}$-approximation to the optimal value of
$\mathbf{P}_{\mathrm{w}}$ [21, 22].

###  V-C Sub-Problem 3: Optimizing over $\alpha_{k}$

This final sub-problem requires optimizing $\\{\alpha\\}_{k\in\mathcal{K}}$
for given
$\\{\mathbf{w}_{c},\\{\mathbf{w}_{l}\\}_{l\in\mathcal{L}},\mathbf{S},\\{\mathbf{u}_{k}\\}_{k\in\mathcal{K}},\\{C_{l}\\}_{l\in\mathcal{L}}\\}$.
Note that objective (18a) is not a function of the reflection coefficients of
the tags. Thus, Sub-Problem 3 reduces to a constraint-satisfaction problem of
$\mathbf{P}_{2}$. That is, any $\\{\alpha_{k}\\}_{k\in\mathcal{K}}$ that
satisfies the constraints (18e)–(18e) is considered to be an optimal solution.
However, to effectively handle Sub-Problem 3, a slack-optimization-based
problem is proposed [23]. While satisfying the other constraints, two new
slack variables are introduced to optimize further the tags’ SINR and EH
margins, i.e., $t_{1}$ and $t_{2}$, respectively [23]. The proposed problem is
thus given in (31),

$\displaystyle\mathbf{P}_{\alpha}:$
$\displaystyle\\!\\!\\!\\!\underset{\alpha_{k},t_{1},t_{2}}{\rm{min}}\lambda_{1}t_{1}+\lambda_{2}t_{2},$
(31a) s.t.
$\displaystyle\\!\\!\\!\\!\alpha_{k}\mathbf{u}_{k}^{\rm{H}}\mathbf{G}_{k}\mathbf{R}_{x}\mathbf{G}^{\rm{H}}_{k}\mathbf{u}_{k}\geq\Upsilon_{k}^{\text{th}}{\mathbf{u}^{\rm{H}}_{k}\left(\sum_{i\in\mathcal{K}_{k}}\alpha_{i}\mathbf{G}_{i}\mathbf{R}_{x}\mathbf{G}_{i}^{\rm{H}}+\beta\mathbf{G}_{\rm{SI}}\mathbf{R}_{x}\mathbf{G}_{\rm{SI}}^{\rm{H}}+\sigma^{2}\mathbf{I}_{N}\right)\mathbf{u}_{k}},\leavevmode\nobreak\
\forall k,$ (31g)
$\displaystyle\\!\\!\\!\\!\alpha_{k}\left(|\mathbf{h}_{k}^{\rm{H}}\mathbf{w}_{c}|^{2}+\sum_{j\in\mathcal{L}}|\mathbf{h}_{k}^{\rm{H}}\mathbf{w}_{i}|^{2}+\mathbf{h}_{k}^{\rm{H}}\mathbf{S}\mathbf{h}_{k}\right)\geq\Gamma_{k}^{\rm{th}}\left(\delta_{c}|\mathbf{f}_{0}^{\rm{H}}\mathbf{w}_{c}|^{2}+\delta_{p}\sum_{j\in\mathcal{L}}|\mathbf{f}_{0}^{\rm{H}}\mathbf{w}_{j}|^{2}+\delta_{c}\mathbf{f}_{0}^{\rm{H}}\mathbf{S}\mathbf{f}_{0}\right.$
$\displaystyle\\!\\!\\!\\!\left.+\sum_{i\in\mathcal{K}_{k}}{\alpha_{i}}\left(|\mathbf{h}_{i}^{\rm{H}}\mathbf{w}_{c}|^{2}+\sum_{j\in\mathcal{L}}|\mathbf{h}_{i}^{\rm{H}}\mathbf{w}_{j}|^{2}+\mathbf{h}_{i}^{\rm{H}}\mathbf{S}\mathbf{h}_{i}\right)+\sigma^{2}\right)+t_{1},\leavevmode\nobreak\
\forall k,$
$\displaystyle\\!\\!\\!\\!|\mathbf{f}_{l}^{\rm{H}}\mathbf{w}_{c}|^{2}\geq\tilde{\Gamma}_{c,l}^{\rm{th}}\left(\sum_{j\in\mathcal{L}}|\mathbf{f}_{l}^{\rm{H}}\mathbf{w}_{j}|^{2}+\delta_{s}|\mathbf{f}_{l}^{\rm{H}}\mathbf{s}|^{2}+\sum_{k\in\mathcal{K}}\alpha_{k}\left(|\mathbf{h}_{l,k}^{\rm{H}}\mathbf{w}_{c}|^{2}+\sum_{j\in\mathcal{L}}|\mathbf{h}_{l,k}^{\rm{H}}\mathbf{w}_{j}|^{2}+|\mathbf{h}_{l,k}^{\rm{H}}\mathbf{s}|^{2}\right)+\sigma^{2}\right),\leavevmode\nobreak\
\forall l,\quad$
$\displaystyle\\!\\!\\!\\!|\mathbf{f}_{l}^{\rm{H}}\mathbf{w}_{l}|^{2}\\!\geq\\!\tilde{\Gamma}_{p,l}^{\rm{th}}\\!\\!\left(\\!\delta_{c}|\mathbf{f}_{l}^{\rm{H}}\mathbf{w}_{c}|^{2}\\!+\\!\\!\sum_{j\in\mathcal{L}_{l}}|\mathbf{f}_{l}^{\rm{H}}\mathbf{w}_{j}|^{2}\\!+\\!\delta_{s}|\mathbf{f}_{l}^{\rm{H}}\mathbf{s}|^{2}\\!+\\!\\!\sum_{k\in\mathcal{K}}\\!\\!\alpha_{k}\\!\\!\left(\\!|\mathbf{h}_{l,k}^{\rm{H}}\mathbf{w}_{c}|^{2}\\!+\\!\sum_{j\in\mathcal{L}}|\mathbf{h}_{l,k}^{\rm{H}}\mathbf{w}_{j}|^{2}\\!+\\!|\mathbf{h}_{l,k}^{\rm{H}}\mathbf{s}|^{2}\\!\\!\right)+\\!\\!\sigma^{2}\\!\right),\leavevmode\nobreak\
\forall l,$
$\displaystyle\\!\\!\\!\\!(1-\alpha_{k})p_{k}^{\rm{in}}\geq{\Phi^{-1}(p_{b})}+t_{2},\leavevmode\nobreak\
\forall k,$ $\displaystyle\\!\\!\\!0<\alpha_{k}<1,\leavevmode\nobreak\ \forall
k,$

where
$\tilde{\Gamma}_{p,l}^{\rm{th}}\triangleq\max\left(\Gamma_{c,l}^{\rm{th}},2^{\sum_{l\in\mathcal{L}}C_{l}}-1\right)$
and $\tilde{\Gamma}_{p,l}^{\rm{th}}\triangleq
2^{\mathcal{R}_{p,l}^{\rm{th}}-C_{l}}-1$. In (31a), $\lambda_{1}$ and
$\lambda_{2}$ are positive constants. $\mathbf{P}_{\alpha}$ is convex, and
thus can be efficiently solved by solvers such as CVX [41]. Introducing slack
variables in $\mathbf{P}_{\alpha}$ converts strict constraints into adjustable
ones with a definable margin. This facilitates the convergence process by
setting a more tangible minimization goal and aligns well with the convergence
strategies of iterative solvers like CVX due to the explicit objective guiding
the solution path [23, 22].

Algorithm 1 presents the overall steps to solve problem $\mathbf{P}_{1}$ (17).
It begins by setting
$\\{\\{\alpha_{k}\\}_{k\in\mathcal{K}},\mathbf{w}_{c},\mathbf{w}_{l},\mathbf{S},C_{l}\\}$
to random feasible values that satisfy the constraints in $\mathbf{P}_{1}$,
and then refines the received/transmit beamforming, reflection coefficients,
and common rate iteratively until the normalized reduction in the total
transmit power is less than $\epsilon=10^{-3}$.

Variable initialization assumes a crucial role in determining algorithm
convergence. Closer initialization to optimal values typically accelerates
convergence. However, complex optimization problems with numerous constraints
pose challenges to finding a near-optimal starting guess, potentially
escalating overall computing complexity [40]. Researchers often resort to
randomly generating feasible initial values [22, 42], a strategy also embraced
in this study. Nonetheless, simulation results show that our overall and
individual algorithms achieve rapid convergence regardless of the random
initial values.

Algorithm 1 Overall Algorithm

1: Input: Set the iteration counter $t=0$, the convergence tolerance
$\epsilon>0$, initial feasible solution
$\\{\\{\alpha_{k}\\}_{k\in\mathcal{K}},\mathbf{w}_{c},\mathbf{w}_{l},\mathbf{S},C_{l}\\}$.
Initialize the objective function value $F^{(0)}=0$.

2: while $\frac{F^{(t+1)}-F^{(t)}}{F^{(t+1)}}\geq\epsilon$ do

3: Solve (21) to derive optimal received beamformer, $\mathbf{u}_{k}^{(t+1)}$.

4: Solve $\mathbf{P}_{\mathrm{w}1}$ (30) to obtain suboptimal transmit
beamformers,
$\\{\mathbf{w}^{(t+1)}_{c},\mathbf{w}^{(t+1)}_{l},\mathbf{S}^{(t+1)},C^{(t+1)}_{l}\\}$
by applying Gaussian randomization to recover the rank-one solution.

5: Solve $\mathbf{P}_{\alpha}$ (31) to obtain the suboptimal power reflection
coefficients, $\alpha_{k}^{(t+1)}$.

6: Calculate the objective function value $F^{(t+1)}$.

7: Set $t\leftarrow t+1$;

8: end while

9: Output: Optimal solutions $\mathcal{A}^{*}$.

######  Remark 1.

This study utilizes the AO method, and each associated subproblem yields a
local solution. The convergence principle of the AO method is well-documented
[20]. Simply put, if the individual sub-problems reach convergence, then the
entire optimization process will converge as well [20]. This study employs the
SDR and slack-optimization techniques to determine
$\\{\mathbf{w}_{c},\mathbf{w}_{l},\mathbf{S},C_{l}\\}$ and
$\\{\alpha_{k}\\}_{k\in\mathcal{K}}$, respectively. Meanwhile,
$\\{\mathbf{u}_{k}\\}_{k\in\mathcal{K}}$ is directly solved via the Rayleigh
ratio quotient method. Both the SDR and slack-optimization are established
methods with guaranteed convergence [22], reinforcing the convergence of our
AO approach. Our simulation outcomes further validate this assertion (Fig. 2).

######  Theorem 1.

Algorithm 1 iterations yield a non-increasing sequence of objective values
with guaranteed convergence.

###### Proof.

Please see Appendix A. ∎

###  V-D Complexity of Proposed Algorithm

The computational complexity of Algorithm 1 is analyzed.

####  V-D1 Optimization over $\mathbf{u}_{k}$

Here, the Rayleigh quotient method provides the optimal closed-form received
beamformers. It requires $\mathcal{O}(N^{3})$ complexity for computing the
inverse of the matrix $\mathbf{Q}$ while the MMSE filters for the $K$ tags
(21) need complexity of $\mathcal{O}(KN^{2})$. The total complexity for this
problem is thus $\mathcal{O}(KN^{2}+N^{3})$.

####  V-D2 Optimization over
$\\{\mathbf{w}_{c},\mathbf{w}_{l},\mathbf{S},C_{l}\\}$

This SDR sub-problem is tackled using SDP, specifically the interior-point
method. From [43, Th. 3.12], the order of computational complexity for a SDP
problem with $m$ SDP constraints with an $n\times n$ positive semi-definite
(PSD) matrix is given by
$\mathcal{O}\left(\sqrt{n}\log\left({1}/{\epsilon}\right)(mn^{3}+m^{2}n^{2}+m^{3})\right)$,
where $\epsilon>0$ is the solution accuracy. In our proposed problem,
$\mathbf{P}_{\mathrm{w}}$ (22), $n=M$ and $m=3(K+L)+2$, and the approximate
computational complexity is
$\mathcal{O}\left((K+L)M^{3}\sqrt{M}\log\left({1}/{\epsilon}\right)\right)$.

####  V-D3 Optimization over $\alpha_{k}$

The difference of convex (d.c.) method and the interior point method are
employed [23]. The number of iterations for convergence is given by
$\frac{\left({\log(C)}/{t^{0}\delta}\right)}{\log\epsilon}$, where $C$ is the
overall number of constraints, $t^{0}$ signifies the initial approximation for
the interior point method’s accuracy, and $0<\delta\ll 1$ is the stopping
criterion [40].

####  V-D4 Algorithm 1

Finally, the proposed Algorithm 1’s computational complexity can be expressed
asymptotically as
$\mathcal{O}\\!\left(\\!I_{o}\\!\left(\\!(K\\!+\\!N)N^{2}\\!+\\!(K\\!+\\!L)M^{3}\sqrt{M}\log\\!\left(\\!\frac{1}{\epsilon}\\!\right)\\!+\\!\frac{\left({\log(C)}/{t^{0}\delta}\right)}{\log\epsilon}\\!\right)\\!\right)$
where $I_{o}$ is the overall number of iterations for the Algorithm 1 to
converge. Despite its higher-order polynomial time complexity, the suggested
approach exhibits noteworthy real-world performance for datasets up to a
certain size, particularly when $N$ and $M$ are kept below a predefined
threshold. For large datasets, adopting optimization measures such as parallel
processing can dramatically enhance performance [44].

##  VI Simulation Results

Next, simulation results are presented to assess the performance of the
proposed ISABC network. The 3GPP urban micro (UMi) model is chosen to depict
the path-loss values $\zeta_{a},\zeta_{b}$, operating at a frequency of
$f_{c}=\qty{3}{}$ [45, Table B.1.2.1]. Additionally, the AWGN variance,
symbolized as $\sigma^{2}$, is modeled as
$\sigma^{2}=10\log_{10}(N_{0}BN_{f})$ m. In this equation, $N_{0}$ is given by
$\qty{-174}{m/}$, while $B$ represents the bandwidth, and $N_{f}$ denotes the
noise figure. Table II lists the main simulation parameters unless mentioned
otherwise. All simulations are evaluated for ${10}^{3}$ iterations.

Considering smart home/city scenarios, the BS and the mobile reader are
positioned at coordinates $\\{0,0\\}$ and $\\{12,0\\}$, respectively.
Meanwhile, tags are sporadically placed within a circle whose center is
$\\{6,-4\\}$ and which has a radius of 3 [46]. Furthermore, the users are
randomly scattered within a circle with a radius of 5 and a center of
$\\{55,0\\}$.

Table II: Simulation parameters. Parameter | Value | Parameter | Value
---|---|---|---
$f_{c}$ |  3 | $\mathcal{R}_{t,k}^{\rm{th}}$ |  1bps/
$B$ |  10 | $\\{\mathcal{R}_{c,l}^{\rm{th}},\mathcal{R}_{p,k}^{\rm{th}}\\}$ |  2bps/
$N_{f}$ |  10 | $K_{\rm{SI}}$ |  3
$M=N$ | $8$ | $\beta$ |  -90
$L$ | $3$ | $\\{\delta_{c},\delta_{p},\delta_{s}\\}$ |  -10
$K$ | $2$ | $p_{b}$ |  -20m
$\mathcal{R}_{s,k}^{\rm{th}}$ |  1bps/ | $\epsilon$ | $10^{-3}$

For comparative evaluation purposes, the following benchmark schemes are
considered:

####  VI-1 NOMA-assisted ISABC

This benchmark uses NOMA to serve the primary users. For a fair comparison,
BackCom and sensing at the BS are also considered, i.e.,
$\mathbf{x}=\sum_{j\in\mathcal{L}}\mathbf{w}_{j}x_{j}+\mathbf{s}$. Without
loss of generality, the channel gains are sorted in descending order, i.e.,
$\|{\mathbf{f}}_{1}\|^{2}\geq\ldots\geq\|{\mathbf{f}}_{L}\|^{2}$. Thus,
$U_{l}$ first decodes the data for users with high channel gains and applies
SIC before decoding its data signal. The SINR of $U_{l}$ with NOMA is given in
(32),

$\displaystyle\gamma_{l}=\frac{|\mathbf{f}_{l}^{\rm{H}}\mathbf{w}_{l}|^{2}}{\delta_{p}\sum_{j\in\mathcal{L}_{l}^{\prime}}|\mathbf{f}_{l}^{\rm{H}}\mathbf{w}_{j}|^{2}+\sum_{j\in\mathcal{L}_{l}^{\prime\prime}}|\mathbf{f}_{l}^{\rm{H}}\mathbf{w}_{j}|^{2}+\delta_{s}|\mathbf{f}_{l}^{\rm{H}}\mathbf{s}|^{2}+\sum_{k\in\mathcal{K}}\alpha_{k}\left(\sum_{j\in\mathcal{L}}|\mathbf{h}_{l,k}^{\rm{H}}\mathbf{w}_{j}|^{2}+|\mathbf{h}_{l,k}^{\rm{H}}\mathbf{s}|^{2}\right)+\sigma^{2}}$
(32)

where $\mathcal{L}_{l}^{\prime}\triangleq\\{1,\ldots,l-1\\}$ and
$\mathcal{L}_{l}^{\prime\prime}\triangleq\\{l+1,\ldots,L\\}$. Moreover, the
tags’ SINRs and sensing SINRs can be easily derived by removing the common
beamforming in (14) and (16), respectively. This benchmark facilitates the
investigation of the impacts of RSMA compared to conventional NOMA.

####  VI-2 RSMA-assisted BackCom

This benchmark considers multiple RSMA-enabled primary users and multiple
tags. However, the BS has no sensing capabilities, i.e.,
$\mathbf{x}=\mathbf{w}_{c}x_{c}+\sum_{j\in\mathcal{L}}\mathbf{w}_{j}x_{j}$.
This benchmark helps to evaluate the cost of adding sensing.

####  VI-3 Conventional BackCom

This benchmark considers a conventional AmBC system with $K$-tags and
$L$-users. Specifically, the BS does not perform rate-splitting and sensing
and uses SDMA to serve the users. Thus, the BS transmit signal becomes
$\mathbf{x}=\sum_{j\in\mathcal{L}}\mathbf{w}_{j}x_{j}$. These tags operate
solely on EH and send data to the reader by reflecting $\mathbf{x}$. This
benchmark helps to evaluate the cost of incorporating sensing and rate-
splitting.

####  VI-4 Conventional ISABC

This benchmark uses SDMA to serve the primary users. The BS senses
environmental information from $K$ tags. Thus, the BS transmit signal becomes
$\mathbf{x}=\sum_{j\in\mathcal{L}}\mathbf{w}_{j}x_{j}+\mathbf{s}$. The
benchmark helps to gauge the difference between SDMA and RSMA.

####  VI-5 Sensing-only scheme

This benchmark scheme evaluates a system with $K$-targets/tags without primary
and backscatter communications. The targets/tags use EH, allowing for a fair
comparison. For example, targets may be wireless sensors/tags that do not send
data but harvest energy to remain active.

These benchmark schemes are special cases of Algorithm 1, which can easily
handle them with minor modifications.

###  VI-A Convergence Rate of Algorithm 1

$1$$2$$3$$4$$5$$6$$7$$8$$9$$10$$26$$30$$34$$38$$42$$46$Number of iterationsBS
transmit power
[dBm]$M=N=4,K=2,L=2$$M=N=4,K=3,L=2$$M=N=8,K=2,L=3$$M=N=8,K=3,L=3$

Figure 2: Convergence of the proposed Algorithm 1.

Algorithm 1 outputs all variable set $\mathcal{A}$ in several iterations.
Thus, the saturation of the BS transmit power is a convergence test, and Fig.
2 thus plots it as a function of the number of iterations. The stopping
criterion is that the normalized objective function increases less than
$\epsilon=10^{-3}$. The BS transmit power decreases rapidly with each
iteration and saturates after approximately three iterations, regardless of
$M$, $L$, or $K$. This implies quick convergence and validates the proposed
algorithm’s effectiveness. Extensive simulations reveal that Algorithm 1
requires only three iterations to attain adequate performance with any system
setup, hardly improving performance beyond three iterations.

###  VI-B Beampattern Gains

$-80$$-60$$-40$$-20$$0$$20$$40$$60$$80$$-40$$-20$$0$Angle (degrees)Combined
beampattern $p_{3}(\theta)$Tag 1Tag 2BSReaderUser 1User 2User
3$-80$$-60$$-40$$-20$$0$$20$$40$$60$$80$$-20$$-10$$0$Transmit beampattern
$p_{1}(\theta)$$-80$$-60$$-40$$-20$$0$$20$$40$$60$$80$$-40$$-20$$0$Beampattern
gain (dB)Receive beampattern $p_{2}(\theta)$

Figure 3: Beampattern of radar functionality of Algorithm 1.

$-80$$-60$$-40$$-20$$0$$20$$40$$60$$80$$-20$$0$Angle (degrees)Combined
beampattern $p_{3}(\theta)$Tag 1Tag
2BSReader$-80$$-60$$-40$$-20$$0$$20$$40$$60$$80$$-15$$-10$$-5$Transmit
beampattern
$p_{1}(\theta)$$-80$$-60$$-40$$-20$$0$$20$$40$$60$$80$$-30$$-20$$-10$$0$Beampattern
gain (dB)Receive beampattern $p_{2}(\theta)$

Figure 4: Beampattern of radar functionality of sensing-only scheme.

The direct transmission and reception of beams in precise directions help to
extract more information. To this end, Algorithm 1 functions as a linchpin,
controlling the formation and direction of these beams. Beamforming refers to
merging signals from an array of antennas to produce a directed “beam” or
“lobe.” This beam can be electronically steered while the antennas stay
stationary. This steering feature improves signal quality, increases
backscatter tag detection, and reduces possible interference significantly.

The BS transmit signal, $\mathbf{x},$ represents the outward-projected energy
to illuminate the targets. The received beamformer is designed for clear
reception, catching echoes or reflections off the backscatter tags. The three
crucial beampatterns are

$\displaystyle p_{1}(\theta)$ $\displaystyle=$
$\displaystyle\left|\mathbf{b}^{\rm{H}}(\theta_{k})\mathbf{x}^{*}\right|^{2},$
(33a) $\displaystyle p_{2}(\theta)$ $\displaystyle=$
$\displaystyle\left|(\mathbf{u}_{k}^{*})^{\rm{H}}\mathbf{b}(\theta_{k})\right|^{2},$
(33b) $\displaystyle p_{3}(\theta)$ $\displaystyle=$
$\displaystyle\left|(\mathbf{u}_{k}^{*})^{\rm{H}}\mathbf{b}(\theta_{k})\mathbf{b}^{\rm{H}}(\theta_{k})\mathbf{x}^{*}\right|^{2}.$
(33c)

Here, (33a) indicates how transmitted energy disperses as a function of angle
$\theta$, while (33b) captures the sensitivity across multiple angles while
receiving reflected energy. Moreover, (33c) provides a composite
representation incorporating the impacts of transmission and subsequent
reflection processing.

Fig. 3 and Fig. 4 illustrate the aforementioned three beampatterns generated
using Algorithm 1 and the sensing-only approach, respectively. The first sub-
figure in Fig. 3 shows that Algorithm 1 aims the transmit beams at tags and
users. In contrast, the second sub-figure depicts the directions of
interference from other tags for a specific tag with relatively deep nulls,
separating the individual sensing information at the BS. The overall
beampattern (third sub-figure) combines the transmit and receive beampatterns.
Compared to the sensing-only scheme (Fig. 4), Algorithm 1 additionally
provides users with communication capabilities while alleviating interference
between targets/tags for sensing.

Fig. 3 and Fig. 4 help sense operations as they depict the radar/beamgain
patterns. These figures offer a practical perspective, helping to better
understand the intricacies and efficacy of beamforming algorithms.

###  VI-C Effects of Number of BS Antennas

$5$$10$$15$$20$$25$$30$$15$$20$$25$$30$$35$$40$Number of transmit and receive
antenna at BS, $M=N$BS transmit power [dBm]RSMA ISABCRSMA BackComNOMA
ISABCConventional ISABCConventional BackComSensing-only

Figure 5: BS transmit power versus the number of BS transmit/receive antennas,
$M=N$.

$5$$10$$15$$20$$25$$30$$0$$2$$4$$6$$8$$10$$12$$14$Number of transmit and
receive antenna at BS, $M=N$User rate [bps/Hz]RSMA ISABC -
$\bar{\mathcal{R}}_{c}$RSMA ISABC - $\bar{\mathcal{R}}_{p}$RSMA BackCom -
$\bar{\mathcal{R}}_{c}$RSMA BackCom - $\bar{\mathcal{R}}_{p}$NOMA ISABC -
$\bar{\mathcal{R}}_{p}$Conventional ISABC -
$\bar{\mathcal{R}}_{p}$Conventional BackCom - $\bar{\mathcal{R}}_{p}$

Figure 6: Average user rate versus the number of BS transmit/receive antennas,
$M=N$.

Fig. 5 and Fig. 6 show the influence of the number of BS antennas, $M=N$, on
the BS transmit power and primary user rates, respectively. Fig. 5 shows that
increasing $M$ has an inverse effect on the required power. Specifically,
increasing $M$ decreases the transmit power for all the schemes. This trend
suggests the spatial diversity benefits for ISABC, namely (1) increased
communication rates and (2) significant power savings. Additionally, NOMA and
sensing-only schemes consume the lowest amount of transmit power. This is
because NOMA has no common rate requirements (i.e., constraint (17k)), and the
sensing-only approach lacks communication. As a result, NOMA achieves a
significantly lower user rate than RSMA (Fig 6). In contrast, conventional
BackCom and ISABC consume the most power resulting from multi-user multi-tag
interference. Conversely, the proposed setup and RSMA-assisted BackCom have
moderate transmit power requirements. Importantly, the proposed IMBO algorithm
enables sensing, a critical feature for IoT networks, with only a modest
transmit power increase.

Fig. 6 compares the average primary user rates of all the schemes, i.e.,
$\bar{\mathcal{R}}_{c}\triangleq 1/L\sum_{l\in\mathcal{L}}\mathcal{R}_{c,l}$
and $\bar{\mathcal{R}}_{p}\triangleq
1/L\sum_{l\in\mathcal{L}}\mathcal{R}_{p,l}$. The figure shows the proposed
scheme achieves the highest communication rate among all. The reason is that
RSMA enables the optimization of the common rates, which increases them. For
example, just for a $\qty{24}{}$ transmit power increase ($M=N=$10$$), our
scheme achieves a $\qty{350}{}$ rate gain over NOMA-assisted ISABC.

###  VI-D Transmit Power Versus Number of Tags

$1$$2$$3$$4$$5$$20$$25$$30$$35$$40$$45$$50$Number of tags, $K$BS transmit
power [dBm]RSMA ISABCRSMA BackComNOMA ISABCConventional ISABCConventional
BackComSensing-only

Figure 7: Transmit power versus the number of tags, $K$.

Fig. 7 exhibits the intricate relationship between the number of tags and the
BS transmit power requirements. From Fig. 7, these two variables show an
evident linear correlation across all schemes. Although the NOMA and sensing-
only methods reduce the transmit power, they also have inferior communication
performance (Fig. 6).

Conversely, both conventional BackCom and conventional ISABC demand higher BS
transmit power to satisfy communication (and/or sensing) rates and EH while
mitigating multi-user multi-tag interference. In contrast, both the proposed
and RSMA-assisted BackCom utilize a moderate BS transmit power. Nonetheless,
compared to RSMA-assisted BackCom, the proposed ISABC approach requires a low
power increase and promotes both communication and sensing by transforming
passive tags into proactive participants.

###  VI-E Impacts of SIC and SI cancellation

$-30$$-24$$-18$$-12$$-6$$0$$26$$26.5$$27$$27.5$$28$$28.5$$29$$29.5$$30$$30.5$$31$SIC
quality, $\delta$ [dB]BS transmit power [dBm]$\beta=-80$ dBm$\beta=-90$
dBm$\beta=-100$ dBmPerfect SI cancellation

Figure 8: Transmit power versus SIC imperfection $\delta$ for various residual
SI values.

Impairments, such as SIC errors and SI cancellation quality, can degrade the
performance and reliability. SIC errors harm signal reception and detection.
SI cancellation errors occur when FD nodes cannot fully remove SI, hindering
incoming signal reception. Both errors have a significant impact.

Fig. 8 examines the effects of imperfect SIC and SI cancellation on Algorithm
1. The BS transmit power is plotted as a function of SIC quality, i.e.,
$\delta=\delta_{c}=\delta_{p}=\delta_{s}$, for various SI cancellation
qualities, i.e., $\beta=\\{-80,-90,-100\\}\,\qty{}{}$ and prefect SI
cancellation, at the BS.

According to Fig. 8, severe SIC imperfection, i.e., $\delta\rightarrow
1\,(\qty{0}{})$ requires the BS to transmit additional power to alleviate the
adverse effects of SIC imperfection while maintaining tag EH requirements and
communication/sensing rates. Conversely, when $\delta\rightarrow 0$ or perfect
SIC, less power is required to sustain these demands. Imperfect SI
cancellation has bad consequences. Variable $\beta\in[0,1]$ reflects the
degree of imperfection at the BS. Further experiments offer insight into
transmission power with varying residual SI values. Like SIC errors, the BS
transmit power is sacrificed as $\beta$ increases.

##  VII Conclusion

This paper presents a novel framework integrating RSMA and ISABC into a
cohesive system. Within this framework, the FD BS assumes dual functionality:
first, it acts as a sensor to detect information from backscatter tags, and
second, it employs RSMA for efficient communication with multiple users.
Conversely, the backscatter reader is responsible for detecting multi-tag
backscatter data.

An advanced AO-based algorithm is introduced to precisely regulate
communication channels, both primary and backscatter, while concurrently
reducing the transmit power of the BS. This innovation facilitates passive
tags’ opportunistic sensing and communication within forthcoming IoT networks.
To evaluate its performance, the proposed system is benchmarked against NOMA
and SDMA, as well as sensing-only and communication-only approaches.

Through rigorous comparison, the RSMA-based system demonstrates significant
advantages over NOMA and SDMA counterparts. Furthermore, the study delves into
the impact analysis of SIC errors and the quality of SI cancellation. Looking
ahead, numerous avenues for future research emerge, particularly machine
learning techniques tailored for channel estimation and interference
mitigation within ISABC systems.

##  Appendix A Proof of Theorem 1

The main problem is divided into three sub-problems, which optimize tag
reflections coefficients
$\left(\bm{\alpha}:=\\{\alpha_{k}\\}_{k\in\mathcal{K}}\right)$, received
beamforming $\left(\mathbf{U}:=\\{\mathbf{u}_{k}\\}_{k\in\mathcal{K}}\right)$,
and transmit beamforming
$\left(\mathbf{P}=\\{\mathbf{w}_{c},\\{\mathbf{w}_{l}\\}_{l\in\mathcal{L}},\mathbf{S},\\{C_{l}\\}_{l\in\mathcal{L}}\\}\right)$,
via solving problems (21), (30), and (31), while keeping the other two blocks
of variables fixed. Let us define $F(\mathbf{U},\bm{\alpha},\mathbf{P})$ as a
function of $\mathbf{U}$, $\bm{\alpha}$, and $\mathbf{P}$ for the objective
value of (17a). First, in step $3$ of Algorithm 1 with fixed variables
$\bm{\alpha}^{(i)}$ and $\mathbf{P}^{(i)}$, $\mathbf{U}^{(i+1)}$ is the
optimal solution that minimizes the value of the objective function.
Accordingly, the following holds:

$F(\bm{\alpha}^{(i)},\mathbf{U}^{(i+1)},\mathbf{P}^{(i)})\leq
F(\bm{\alpha}^{(i)},\mathbf{U}^{(i)},\mathbf{P}^{(i)}).$ (34)

Next, in step $4$ of Algorithm 1, $\mathbf{P}^{(i+1)}$ is the optimal transmit
beamformers with given variables $\bm{\alpha}^{(i)}$ and $\mathbf{U}^{(i+1)}$
to minimize $F$ via solving (22a). Thus, it guarantees that

$F(\bm{\alpha}^{(i)},\mathbf{U}^{(i+1)},\mathbf{P}^{(i+1)})\leq
F(\bm{\alpha}^{(i)},\mathbf{U}^{(i+1)},\mathbf{P}^{(i)}).$ (35)

Finally, in step $5$ of Algorithm 1 with the given $\mathbf{P}^{(i+1)}$ and
$\mathbf{U}^{(i+1)}$, problem (30) is solved to obtain an optimal solution for
$\bm{\alpha}^{(i)}$, which yields:

$F(\bm{\alpha}^{(i+1)},\mathbf{U}^{(i+1)},\mathbf{P}^{(i+1)})\leq
F(\bm{\alpha}^{(i)},\mathbf{U}^{(i+1)},\mathbf{P}^{(i+1)}).$ (36)

According to (34)–(36), it follows that

$\\!\\!\\!\\!F(\bm{\alpha}^{(i+1)},\mathbf{U}^{(i+1)},\mathbf{P}^{(i+1)})\leq
F(\bm{\alpha}^{(i+1)},\mathbf{U}^{(i+1)},\mathbf{P}^{(i+1)}).$ (37)

The objective values of Algorithm 1 monotonically decrease with each
iteration, always remaining non-negative. This consistency, combined with the
design choice where each iteration starts from the previous one’s end, ensures
the algorithm’s convergence. Thus, the proof is completed.

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