# Novel KLD-based Resource Allocation for Integrated Sensing and Communication

Yousef Kloob, , Mohammad Al-Jarrah, , Emad Alsusa, and Christos Masouros Y.
Kloob, M. Al-Jarrah, E. Alsusa are with the Department of Electrical and
Electronic Engineering, University of Manchester, Manchester M13 9PL, U.K.
(e-mail: {yousef.kloob, mohammad.al-jarrah, e.alsusa}@manchester.ac.uk), and
C. Masouros is with Department of Electronic and Electrical Engineering,
University College London, London WC1E 6BT, U.K (e-mail:
c.masouros@ucl.ac.uk).

###### Abstract

In this paper, we introduce a novel resource allocation approach for
integrated sensing-communication (ISAC) using the Kullback–Leibler divergence
(KLD) metric. Specifically, we consider a base-station with limited power and
antenna resources serving a number of communication users and detecting
multiple targets simultaneously. First, we analyze the KLD for two possible
antenna deployments, which are the separated and shared deployments, then use
the results to optimize the resources of the base-station through minimising
the average KLD for the network while satisfying a minimum predefined KLD
requirement for each user equipment (UE) and target. To this end, the
optimisation is formulated and presented as a mixed integer non-linear
programming (MINLP) problem and then solved using two approaches. In the first
approach, we employ a genetic algorithm, which offers remarkable performance
but demands substantial computational resources; and in the second approach,
we propose a rounding-based interior-point method (RIPM) that provides a more
computationally-efficient alternative solution at a negligible performance
loss. The results demonstrate that the KLD metric can be an effective means
for optimising ISAC networks, and that both optimisation solutions presented
offer superior performance compared to uniform power and antenna allocation.

###### Index Terms:

Integrated sensing and communication, multiple-input-multiple-output (MIMO),
radar, zero-forcing, Kullback–Leibler divergence, power allocation, antenna
allocation, mixed integer non-linear programming (MINLP).

##  I Introduction

The field of communication technology is undergoing rapid evolution, with 6G
being the next generation of wireless communication networks that promises to
bring about transformative changes [1, 2, 3]. As we approach this new era of
wireless communication, the demand for advanced and efficient communication
techniques has become increasingly apparent [4]. 6G networks are expected to
support various services and applications, such as holographic communications,
smart grids, digital twins, and integrating radar technology into
communication systems which is the interest of this paper.

Integrated sensing and communication (ISAC), when properly implemented, is
exceptionally beneficial, as both systems could use the same hardware and
network resources. For instance, the large antenna arrays which form massive
multiple-input-multiple-output (mMIMO) can be shared among both subsystems,
which reduces the cost immensely [5, 6, 7, 8, 9, 10, 11]. Generally speaking,
ISAC systems can be implemented in two ways, namely, the separated deployment
in which the base station (BS) antennas are distributed among each sub-system
and the shared deployment in which antennas are exploited for both sub-
systems. Additionally, incorporating both systems and utilising the same
spectrum bands leads to improved system capacity, higher data rates and better
sensing capabilities due to increased spectral efficiency [12]. Furthermore,
radar systems are generally used for detection, localisation, tracking, and
navigation, which have the potential to provide precise data about the
surrounding environment, and thus numerous applications such as autonomous
vehicles and drones can benefit from such information to take appropriate
actions in order to accomplish their tasks [13, 14, 15].

One of the challenging factors which should be carefully considered for
effective ISAC functionality is the allocation of available resources at the
BS. Remarkable efforts have been devoted in the literature to achieve this
objective. For instance, the optimisation of power allocation while maximising
the signal-to-interference-noise-ratio (SINR) for the radar system or the
communication system capacity, either as cooperative or coexistence design,
has been discussed in literature [16, 17, 18, 19]. However, the measures
considered in these articles to evaluate the performance of the communication
and sensing subsystems are not homogeneous, i.e., different measures for each
subsystem. Consequently, it is difficult to assess the performance gained by
the power optimisation on the functionality of the overall system.

In [16, 17], power allocation and transmission strategies are proposed with
the aim of optimising the performance of coexisting radar and communication
systems. The work in [16] focuses on orthogonal frequency division
multiplexing (OFDM) communication systems and sparse sensing radars in
cluttered environments and develops a joint transmit design framework
accordingly, where the SINR of the radar is maximised with constraints on the
throughput of the communication system. The optimisation problem relies on
prior knowledge of the communication system signal-to-noise ratio (SNR),
interference-to-noise-ratio (INR), and radar system clutter-to-noise-ratio
(CNR), and the target SNR up to an unknown constant common to all subcarriers.
In [17], the authors present an optimisation framework that maximises the
performance metric of one system while meeting the service constraint of the
other system where both radar-centric and communication-centric designs are
studied. OFDM dual functional radar and communication (DFRC) based systems
that serve the dual purpose of radar and communication are considered in [18]
and [19]. The authors in [18] have presented an optimal power allocation
method for monostatic OFDM systems by maximising the mutual information (MI)
between the radar target and communication users while minimising the radar
detection error probability. Moreover, [19] proposes an algorithm that
balances the competing requirements of radar detection and communication data
rate, aiming at optimising the power and maximising performance of the DFRC
system.

Apparently, none of these papers explored the joint allocation antenna and
power resources in both of the antenna deployment configurations for ISAC
systems and the comparison between the two models (e.g. shared and separated
deployments), and how these models affect the transmission and reception of
both the communications and radar. However, incorporating the optimal antenna
allocation for each subsystem into the problem adds a new dimension to the
problem where power and antennas for each system are controllable to achieve
the highest performance. In addition, ISAC systems aim at integrating the
communication and sensing tasks in a more holistic manner than DFRC or a
coexistent operation.

Figure 1: Antenna deployments for ISAC systems. (a) Separated deployment; (b)
Shared deployment.

Interestingly, in our previous works [20, 21], we have shown that a unified
performance evaluation measure using Kullback–Leibler divergence (KLD) can be
effectively used to characterise ISAC systems holistically. The relation
between KLD from one side and the symbol error rate (SER) and detection
performance of the communication system and radar system, respectively, from
the other side, has been thoroughly investigated. Moreover, the trade-off
between both subsystems using the introduced KLD has been well studied.
Obviously, KLD can be utilised as a design reference for the overall ISAC
system as a single entity and put both subsystems on one scale, unlike
conventional designs, which rely on two different measures, i.e., one measure
for each subsystem. Furthermore, unlike SER, detection probability and false
alarm probability, KLD measure is independent of the detection process and
detection threshold applied at the receiver side, which makes it an ideal
candidate for system level design and resource allocation [22, 23].

Inspired by the benefits of KLD, this paper investigates utilizing such metric
for optimal resource allocation in ISAC networks. We do not only focus on a
typical ISAC setup, but also explore the models used in [6, 24], for shared
and separated deployment modes. As shown in Fig. 1.a, in shared deployment,
communication and radar systems use the entire antenna array, and therefore
optimization is only for power allocation. In contrast, the separated
deployment in Fig. 1.b, in which each subsystem is allocated a certain subset
of antennas, requires the optimization of both power and antenna allocation
for communication and radar systems. We present an optimisation framework for
power and antenna allocation for each node, i.e., communication user or
sensing target, which results in the maximum achievable network-level KLD
while satisfying a minimum KLD requirement that needs to be satisfied for each
device. The formulated problem is found to be a constrained non-convex mixed
integer non-linear programming (MINLP) for the separated deployment, and
nonlinear programming (NLP) for the shared deployment. Therefore, we propose a
low-complexity algorithm for solving the separated deployment optimisation
problem based on the interior-point method (IPM), which is referred to
rounding-based interior-point method (RIPM) algorithm. The achievable
performance of the proposed algorithm is compared to a genetic algorithm (GA)
as a benchmark, as well as, both algorithms are compared with the uniform
resource allocation (URA) scenario as an additional benchmark. The results
show the effectiveness of the proposed algorithm when compared to GA and URA,
where the obtained results demonstrate that both GA and RIPM significantly
outperform the URA scenario. Moreover, as compared to GA, the proposed RIPM
provides comparable detection capability with more computational efficiency.
The primary contributions of this paper are summarized as follows:

  * •

A generalized system model with multiple communication users and multiple
targets is introduced, which seamlessly integrates the impact of clutter and
imperfect interference cancellation. In addition, both the separated and
shared ISAC antenna deployments are investigated thoroughly.

  * •

Providing the analysis of the KLD to characterize the achievable detection
capability for radar and communication systems within the shared deployment of
ISAC systems.

  * •

Proposing a novel approach for power and antenna allocation using the derived
KLD. The formulated optimization problems are found to be constrained non-
convex MINLP for the separated deployment and as a constrainted non-convex NLP
for the shared deployment. A low complex optimization algorithm, that includes
RIPM algorithm based on the IPM to solve the MINLP, is developed. While the
shared deployment NLP is solved using IPM.

  * •

A detailed comparative analysis between RIPM, GA, and URA, are conducted to
emphasize the computational efficiency and convergence rates.

  * •

Extensive simulated validations are provided that underscore the performance
benefits of RIPM and GA over the URA methodology. Furthermore, the efficacy of
both shared and separated deployment methods is evaluated, drawing attention
to the superior performance outcomes of shared deployment when assessed using
the KLD metric.

In summation, this research underscores the relevance of KLD as an effective
metric for system optimization in ISAC systems. The results shows that RIPM
algorithm matches the performance of the established GA, with significant
reduction in the computational complexity, where also, both the GA and RIPM
methods surpass the URA in effectiveness. Importantly, when comparing the
shared and separated deployment methods, the shared deployment yielded
superior results based on the KLD measure. This shows the potential benefits
of progressing towards a more unified ISAC system.

The remainder of this paper is organised as follows. In section II, the system
model is presented. Section III introduces a summary of the derivation of the
radar KLD, the communication system’s KLD, and the average KLD for the overall
ISAC system. In section IV, the optimisation problem for power and antenna
allocation is formulated. Section V shows the numerical results. Section VI
the complexity analysis for the optimisation techniques are explored and
Section VII concludes the work.
$Notation$: The following notations are used in this paper. Bold uppercase
letters (e.g. $\mathbf{S}$) specify matrices, while bold lowercase letters
(e.g. $\mathbf{s}$) specify vectors. Superscripts ${(\cdot)}^{*}$,
${(\cdot)}^{T}$ and ${(\cdot)}^{H}$ specify the conjugate, transpose and
Hermitian transpose, respectively. Subscripts ${(\cdot)}_{\mathrm{c}}$ and
${(\cdot)}_{\mathrm{r}}$ relate the corresponding parameter to the
communication or radar system, respectively. The absolute value and the trace
operator are denoted by $|\cdot|$ and $\mathrm{tr}\\{\cdot\\}$, respectively.

##  II System Model

This paper is concerned in a multi-user multi-target (MUMT) MIMO scenario. The
ISAC system model comprises an $N$ antenna MIMO-BS, these antennas are
utilized for detecting ${T}$ number of radar targets, and serving $K$ number
of single-antenna communication user equipments (UEs) in downlink. The ISAC
system is analyzed for two system deployments, the separated and shared
deployment as shown in Fig.1.

The total transmitted power available at BS is $P_{\mathrm{T}}$ which is
utilised for both sensing and data communication functionalities. The portions
of the power that are allocated to the radar and communication subsystems are
respectively denoted as $P_{\mathrm{r}}$ and $P_{\mathrm{c}}$, where
$P_{\mathrm{T}}=P_{\mathrm{c}}+P_{\mathrm{r}}$. Zero-forcing (ZF) beamforming
is employed at BS to precode the information of communication UEs and
eliminate the cross-interference caused by signals intended for other UEs
[25]. Moreover, the radar waveform is designed such that the covariance matrix
is
$\mathbf{R}_{x}\triangleq\frac{1}{L}\sum_{l=1}^{L}\mathbf{x}_{l}\mathbf{x}_{l}^{H}\in\mathbb{C}^{N_{\mathrm{r}}\times
N_{\mathrm{r}}}$, where $L$ is the total number of snapshots, and
$\mathbf{x}_{l}\in\mathbb{C}^{N_{\mathrm{r}}\times 1}$ is the radar waveform
vector for the $l$-th snapshot.

###  II-A Separated Deployment

In this case, as shown in Fig.1.a for the separated deployment antenna model,
the MIMO-BS antennas are divided into two subsets. The first subset contains a
number of $N_{\mathrm{r}}$ antennas applied for detecting ${T}$ number of
radar targets, whereas the second one utilises the remaining
$N_{\mathrm{c}}=N-N_{\mathrm{r}}$ to serve $K$ number of single-antenna
communication user equipments (UEs) in the downlink.

####  II-A1 Communication System

At each $l$ instance, a data symbol $s_{k,l}$ intended for the $k$-th UE is
drawn from a certain normalised constellation, i.e,
$\mathbb{E}[\left|s_{k,l}\right|^{2}]=1$. Given the channel matrix from MIMO-
BS to UEs, $\mathbf{H}_{l}$, these symbols, i.e., $s_{k,l}\forall k$, are
precoded using a ZF precoder with a precoding matrix
$\mathbf{W}_{\mathrm{c},l}\in\mathbb{C}^{N_{\mathrm{c}}\times K}$ that is
normalised using instantaneous matrix normalisation scheme for which
$\mathbf{W}_{\mathrm{c},l}=\frac{\mathbf{\tilde{W}}_{\mathrm{c},l}}{\sqrt{\mathbf{s}_{l}^{H}\>\mathbf{\tilde{W}}_{\mathrm{c},l}\>\mathbf{\tilde{W}}_{\mathrm{c},l}^{H}\>\mathbf{s}_{l}}}$
with
$\mathbf{\tilde{W}}_{\mathrm{c},l}=\mathbf{H}_{l}^{H}\>\left(\mathbf{H}_{l}\>\mathbf{H}_{l}^{H}\right)^{-1}$
is the non-normalised ZF precoding matrix. This sort of normalisation ensures
that the communication system transmit power satisfies the power constraint.
The received signal $\mathbf{y}_{l}\in\mathbb{C}^{K\times 1}$, at the $l$-th
instance, can be shown as follows,

$\mathbf{y}_{l}=\mathbf{D}_{\mathrm{c}}\>\mathbf{H}_{l}^{T}\>\mathbf{W}_{\mathrm{c},l}\>\mathbf{P}_{\mathrm{c}}\>\mathbf{s}_{l}\mathbf{+}\>\mathbf{D}_{\mathrm{c}}\>\sqrt{\frac{P_{\mathrm{r}}}{N_{\mathrm{r}}}}\mathbf{F}_{l}^{T}\mathbf{x}_{l}\mathbf{+}\mathbf{n}_{l},$
(1)

where
$\mathbf{P}_{\mathrm{c}}=\mathrm{diag}(P_{\mathrm{c},1},P_{\mathrm{c},2},\cdots,P_{\mathrm{c},K})$
is a power control matrix for UEs,
$\mathbf{H}_{l}\in\mathbb{C}^{N_{\mathrm{c}}\times
K}\sim\mathcal{C}\mathcal{N}\left(0,2\sigma_{H}^{2}\right)$ is the BS-UEs
communication channel matrix which is modeled as flat Rayleigh fading,
$\mathbf{F}_{l}^{T}\in\mathbb{C}^{N_{\mathrm{r}}\times
T}\sim\mathcal{C}\mathcal{N}\left(0,2\sigma_{F}^{2}\right)$ is the radar-
communication interference channel matrix modeled as flat Rayleigh fading from
the radar antennas to each of the UEs, and
$\mathbf{n}_{l}\in\mathbb{C}^{K\times
1}\sim\mathcal{C}\mathcal{N}\left(0,2\sigma_{n}^{2}\right)$ is the additive
white Gaussian noise (AWGN). The matrix
$\mathbf{D}_{\mathrm{c}}=\mathrm{diag}(d_{\mathrm{c},1}^{-\eta/2},d_{\mathrm{c},2}^{-\eta/2},\cdots,d_{\mathrm{c},K}^{-\eta/2})$
counts for the free space pathloss, where $\eta$ is the pathloss exponent, and
$d_{\mathrm{c},k}$ is the $k$-th UE distance to BS. Throughout the paper, the
channels are assumed to be independent and identically distributed (iid).

####  II-A2 Radar System

For this system, we are targeting the scenario in which each target is located
in a separate radar bin [26, 27], allowing for easy identification for the
number of targets in the environment. We further assume that BS has knowledge
about the number of possible targets in the area through historical records.
MIMO radar makes it possible to generate multiple beams simultaneously by
combining multiple orthogonal signals. Accordingly, the transmitted signal
vector at the output of the antennas can be represented as,

$\mathbf{x}_{l}=\mathbf{W}_{\mathrm{r},l}\>\mathbf{P}_{\mathrm{r}}\>\Phi,$ (2)

where $\Phi=\left[\phi_{1},\phi_{2},\cdots,\phi_{T}\right]^{T}$ is a set of
$T$ orthonormal baseband waveforms [28],
$\mathbf{P}_{\mathrm{r}}=\mathrm{diag}\left(\sqrt{P_{\mathrm{r},1}/{N_{\mathrm{r}}}},\sqrt{P_{\mathrm{r},2}/{N_{\mathrm{r}}}},\cdots,\sqrt{P_{\mathrm{r},T}/{N_{\mathrm{r}}}}\right)$
is the power allocation matrix which is used to control the amount of power to
be emitted towards each target, and
$\mathbf{W}_{\mathrm{r},l}\in\mathbb{C}^{N_{\mathrm{r}}\times T}$ is the
precoding matrix for the radar at the $l$-th signalling period. The precoding
matrix for the radar system can be tailored to enhance the radar performance
or fulfil a specific radar covariance matrix requirement. For instance, a
radar covariance matrix with desired characteristics can be achieved through
the design of appropriate precoding matrix
${\mathbf{R}}_{\mathrm{w}}\triangleq\frac{1}{L}\mathbf{W}_{\mathrm{r}}\times\mathbf{W}_{\mathrm{r}}^{H}=\mathbf{I}_{N_{\mathrm{r}}\times
N_{\mathrm{r}}}$ which is typically used for omnidirectional radar.

The radar return from all targets is processed through a bank of matched
filters with the signal waveform $\phi_{t}\>\forall
t=\left\\{1,2,\cdots,T\right\\}$ which is tuned to a certain radar angular-
range-Doppler bin. Thereafter, since $\phi_{t}\perp\phi_{i}\>\forall t\neq i$,
the radar return from different targets can be separated, and thus the
detection of each target can be performed independently. The binary hypothesis
problem associated with each target can be defined as
$\mathcal{H}_{q}\>\forall q\in\\{0,1\\}$, where $q=0$ and $q=1$ denote the
absence and presence of a target, respectively. Mathematically, the received
radar signal from target $t$ under hypothesis $\mathcal{H}_{q}$ can be written
as

$\mathbf{y}_{\mathrm{r},t,l|\mathcal{H}_{q}}=\sqrt{\frac{P_{\mathrm{r},t}}{N_{\mathrm{r}}}}d_{\mathrm{r},t}^{-\eta/2}\>\alpha_{t}\>\textbf{A}\left(\theta_{\mathrm{t}}\right)\mathbf{w}_{\mathrm{r},t,l}\>q+\mathbf{\tilde{\omega}}_{\mathrm{r}},$
(3)

where $d_{\mathrm{r},t}^{-\eta}$ is the two-way channel pathloss from BS to
the target and backwards with $d_{\mathrm{r},t}$ is the two-way distance,
$\alpha_{t}$ represents the target cross-section, and
$\textbf{A}\left(\theta_{t}\right)=\textbf{a}_{\mathrm{T}}\left(\theta_{t}\right)\times\textbf{a}_{\mathrm{R}}\left(\theta_{t}\right)$
is the equivalent array manifold with
$\textbf{a}_{\mathrm{T}}\left(\theta_{t}\right)$ and
$\textbf{a}_{\mathrm{R}}\left(\theta_{t}\right)$ represent the transmit and
receive steering vector for the $t$-th target, respectively. In this paper, it
is assumed that
$\textbf{a}\left(\theta_{t}\right)\triangleq\textbf{a}_{\mathrm{T}}\left(\theta_{t}\right)=\textbf{a}_{\mathrm{R}}\left(\theta_{t}\right)$.
After employing interference cancellation (IC) at BS, the noise plus
interference residue due to the communication signal is denoted as
$\mathbf{\tilde{\omega}}_{\mathrm{r}}\triangleq\mathbf{\omega}_{\mathrm{r}}+\mathbf{n}_{\mathrm{r},l}\sim\mathcal{CN}\left(0,2\sigma_{\tilde{\omega}}^{2}\mathbf{I}_{N_{\mathrm{r}}}\right)$,
with total variance
$\sigma_{\tilde{\omega}}^{2}=\sigma_{\omega}^{2}+\sigma_{n}^{2}$, where
$\mathbf{n}_{\mathrm{r},l}\in\mathbb{C}^{N_{\mathrm{r}}\times
1}\sim\mathcal{C}\mathcal{N}\left(0,2\sigma_{n}^{2}\>\mathbf{I}_{N_{\mathrm{r}}}\right)$
is AWGN with $\mathbf{I}_{N_{\mathrm{r}}}$ is an identity matrix. Here,
$\mathbf{\omega}_{\mathrm{r}}\in\mathbb{C}^{N_{\mathrm{r}}\times
1}\triangleq\mathbf{G}_{\mathrm{err}}\>\mathbf{P}_{\mathrm{c}}\>\mathbf{W}_{\mathrm{c},l}\>\mathbf{s}_{l}$
represents the residual error of the imperfect IC that is employed to mitigate
interference caused by the communication signal, with
$\mathbf{G}_{\mathrm{err}}\in\mathbb{C}^{N_{\mathrm{r}}\times N_{\mathrm{c}}}$
is the channel estimation error matrix at BS. The residual error of imperfect
IC, $\mathbf{\omega}_{\mathrm{r}}$, is approximated to a complex Gaussian
distribution by assuming that the channel estimation errors are normally
distributed with each entry in $\mathbf{G}_{\mathrm{err}}$ being
$\mathcal{CN}\left(0,2\sigma_{\mathrm{err}}^{2}\right)$, where
$2\sigma_{\mathrm{err}}^{2}$ is the variance of the channel estimator. Each
element of $\mathbf{\omega}_{\mathrm{r}}$ is a sum of independent
$KN_{\mathrm{c}}$ random variables. Therefore, central limit theorem (CLT) can
be applied to approximate the density of the elements of
$\mathbf{\omega}_{\mathrm{r}}$ for large $KN_{\mathrm{c}}$. This approximation
leads to the conclusion that the errors caused by imperfect IC are complex
Gaussian distributed. Specifically, $\mathbf{\omega}_{\mathrm{r}}$ follows a
complex Gaussian distribution with mean zero and covariance matrix
$2\sigma_{\omega}^{2}\mathbf{I}_{N_{\mathrm{r}}}$, where
$\sigma_{\omega}^{2}=\sigma_{\mathrm{err}}^{2}\sigma_{\mathrm{w}}^{2}N_{\mathrm{c}}\>P_{\mathrm{c}}$.
Here, $\sigma_{\mathrm{w}}^{2}$ is the variance of the elements of
$\mathbf{w}_{\mathrm{c},k,l}$, which are the elements of the precoding matrix
$\mathbf{W}_{\mathrm{c},l}\triangleq[\mathbf{w}_{\mathrm{c},1,l},...,\mathbf{w}_{\mathrm{c},k,l}]^{T}$.

###  II-B Shared Deployment

In this scenario, as shown in Fig.1.b, the MIMO-BS antennas are not divided
into two subsets for the shared deployment model. Instead, the whole the
antenna array with number of elements $N$ is utilized by both systems for
detecting ${T}$ number of radar targets, and serving $K$ number of single-
antenna communication user equipments (UEs) in downlink. It is important to
notice that the transmitted signal in shared deployment is a combined
superimposed signal of both radar and communication signals, which can be
represented as

$\mathbf{\dot{W}}_{\mathrm{c},l}\>\mathbf{P}_{\mathrm{c}}\>\mathbf{s}_{l}+\mathbf{\dot{x}}_{l},$
(4)

where
$\mathbf{\dot{x}}_{l}=\mathbf{\dot{W}}_{\mathrm{r},l}\>\mathbf{P}_{\mathrm{r}}\>\Phi$
has similar properties to the communication signal of the separated deployment
in (2) but with different dimensions as the shared deployment utilizes $N$
antennas rather than $N_{\mathrm{r}}$. Therefore, the precoding matrix for the
communication and radar systems are
$\mathbf{\dot{W}}_{\mathrm{c},l}\in\mathbb{C}^{N\times K}$, respectively, and
$\mathbf{\dot{W}}_{\mathrm{r},l}\in\mathbb{C}^{N\times T}$, and the power
control diagonal matrix for the radar system is
$\mathbf{\dot{P}}_{\mathrm{r}}=\mathrm{diag}\left(\sqrt{P_{\mathrm{r},1}/{N}},\sqrt{P_{\mathrm{r},2}/{N}},\cdots,\sqrt{P_{\mathrm{r},T}/{N}}\right)$.

####  II-B1 Communication system

The received signal $\mathbf{y}_{l}\in\mathbb{C}^{K\times 1}$, at the $l$-th
instance, can be shown as follows,

$\mathbf{y}_{l}=\mathbf{D}_{\mathrm{c}}\>\mathbf{\dot{H}}_{l}^{T}\>\mathbf{\dot{W}}_{\mathrm{c},l}\>\mathbf{P}_{\mathrm{c}}\>\mathbf{s}_{l}+\mathbf{D}_{\mathrm{c}}\>\sqrt{\frac{P_{\mathrm{r}}}{N}}\>\left(\mathbf{\dot{H}}_{l}^{T}+\mathbf{\dot{F}}^{T}\right)\>\mathbf{\dot{x}}_{l}\mathbf{+}\mathbf{n}_{l},$
(5)

the first term in the received signal represents the useful part of the
transmitted signal after the going through the fading channel and the
pathloss, while the second term represents the radar interference from clutter
and the transmit signal. $\mathbf{\dot{H}}_{l}\in\mathbb{C}^{N\times K}$ is
the BS-UEs communication channel matrix which is modeled as flat Rayleigh
fading,, and $\mathbf{\dot{F}}_{l}^{T}\in\mathbb{C}^{N\times T}$is the radar-
communication interference channel matrix modeled as flat Rayleigh fading from
the radar antennas to each of the UEs, and
$\mathbf{n}_{l}\in\mathbb{C}^{K\times
1}\sim\mathcal{C}\mathcal{N}\left(0,2\sigma_{n}^{2}\right)$ is the additive
white Gaussian noise (AWGN). The clutter term, noise and the transmit signal
radar interference can be encapsulated as
$\mathbf{\zeta}\in\mathbb{C}^{K\times
1}\triangleq\mathbf{D}_{\mathrm{c}}\>\sqrt{\frac{P_{\mathrm{r}}}{N}}\>\left(\mathbf{\dot{H}}_{l}^{T}+\mathbf{\dot{F}}^{T}\right)\>\mathbf{x}_{l}\mathbf{+}\mathbf{n}_{l}$
each element of $\zeta$ has a variance of
$\sigma_{\zeta}^{2}=d_{\mathrm{c},k}^{-\eta}P_{\mathrm{r}}\sigma_{\dot{H}}^{2}+d_{\mathrm{c},k}^{-\eta}P_{\mathrm{r}}\sigma_{\dot{F}}^{2}+\sigma_{n}^{2}$.

####  II-B2 Radar system

The binary hypothesis problem associated with each target can be defined as
$\mathcal{H}_{q}\>\forall q\in\\{0,1\\}$, where $q=0$ and $q=1$ denote the
absence and presence of a target, respectively. Mathematically, the received
radar signal from target $t$ under hypothesis $\mathcal{H}_{q}$ can be written
as

$\mathbf{y}_{\mathrm{r},t,l|\mathcal{H}_{q}}=\alpha_{t}\>\textbf{A}\left(\theta_{\mathrm{t}}\right)\left(\sqrt{\frac{P_{\mathrm{r},t}}{N}}d_{\mathrm{r},t}^{-\eta/2}\mathbf{\dot{w}}_{\mathrm{r},t,l}\>q+\mathbf{\dot{W}}_{\mathrm{c},l}\>\mathbf{P}_{\mathrm{c}}\>\mathbf{s}_{l}\right)\\\
+\mathbf{G}_{\mathrm{err_{1}}}\>\left(\sqrt{\frac{P_{\mathrm{r}}}{N}}\>\mathbf{\dot{x}}_{l}\>q+\mathbf{\dot{W}}_{\mathrm{c},l}\>\mathbf{P}_{\mathrm{c}}\>\mathbf{s}_{l}\right)+\mathbf{n}_{\mathrm{r},l}.$
(6)

The first part of the signal in parentheses is the transmitted signal after
reception, while the second term in parentheses is the clutter signal
reflected back to the BS. Similar to the derivation of (3), (6) can be reduced
to

$\mathbf{y}_{\mathrm{r},t,l|\mathcal{H}_{q}}=\sqrt{\frac{P_{\mathrm{r},t}}{N}}d_{\mathrm{r},t}^{-\eta/2}\>\alpha_{t}\>\textbf{A}\left(\theta_{\mathrm{t}}\right)\mathbf{\dot{w}}_{\mathrm{r},t,l}\>q+\mathbf{\tilde{\zeta}},$
(7)

where
$\tilde{\zeta}=\alpha_{t}\>\textbf{A}\left(\theta_{\mathrm{t}}\right)\>\mathbf{W}_{\mathrm{c},l}\>\mathbf{P}_{\mathrm{c}}\>\mathbf{s}_{l}+\mathrm{G_{err_{1}}}\>\left(\sqrt{\frac{P_{\mathrm{r}}}{N}}\mathbf{x}_{l}\>q+\mathbf{W}_{\mathrm{c},l}\>\mathbf{P}_{\mathrm{c}}\>\mathbf{s}_{l}\right)+\mathbf{n}_{\mathrm{r},l}$
is the interference from the clutter, plus the interference from the combined
transmitted signal, and the noise. There is two operations of imperfect IC
that are important to note, one that involves the clutter part of the signal
which follows the same explanation done before in the separated deployment
model and is represented with the channel estimation error matrix at the BS
$\mathbf{G}_{\mathrm{err_{1}}}\in\mathbb{C}^{N\times
N}\sim\mathcal{CN}\left(0,2\sigma_{\mathrm{err_{1}}}^{2}\right)$. While the
other IC involves the interference from the transmitted signal itself which is
a combined signal of both systems, this IC depends on the estimation of the
target cross section $\alpha_{t}$ and the equivalent array manifold
$\textbf{A}\left(\theta_{\mathrm{t}}\right)$, where the estimation is
considered imperfect where we considered a combined Gaussian error
$\mathbf{G}_{\mathrm{err_{2}}}\in\mathbb{C}^{N\times
N}\sim\mathcal{CN}\left(0,2\sigma_{\mathrm{err_{2}}}^{2}\right)$ with variance
for both of the parameters represented as $\sigma_{\mathrm{err_{2}}}^{2}$.
Following these approximations leads to the conclusion that all the imperfect
IC errors along with the complex gaussian noise can be combined into a complex
gaussian with zero mean and $2\sigma_{\tilde{\zeta}}^{2}\mathbf{I}_{N}$
covariance matrix, where
$\sigma_{\mathbf{\tilde{\zeta}}}^{2}=\sigma_{\mathrm{n}}^{2}+\sigma_{\mathrm{err_{1}}}^{2}\>\left(\sigma_{\mathrm{\dot{w}}}^{2}\>N\>\mathrm{P}_{\mathrm{c}}+\mathrm{tr}\\{R_{x}\\}\>\mathrm{P}_{\mathrm{r}}\>/N\right)+\sigma_{\mathrm{err_{2}}}^{2}\>\sigma_{\mathrm{\dot{w}}}^{2}\>N\>\mathrm{P}_{\mathrm{c}}$.
Here, $\sigma_{\mathrm{\dot{w}}}^{2}$ is the variance of the elements of
$\mathbf{\dot{w}}_{\mathrm{c},k,l}$, which are the elements of the precoding
matrix
$\mathbf{\dot{W}}_{\mathrm{c},l}\triangleq[\mathbf{\dot{w}}_{\mathrm{c},1,l},...,\mathbf{\dot{w}}_{\mathrm{c},k,l}]^{T}$.

##  III Kullback–Leibler divergence

This section presents the derivation of KLD for both the separated and shared
deployment which is to be used in the formulation of the optimisation problem
for both systems.

###  III-A Separated Deployment

####  III-A1 KLD fo Communication System

In the case of ZF precoding, the precoding matrix is given in the section
introducing the communication system. The KLD shown below accommodates any
type of constellations and has been modified to account for the long-term
pathloss effect as follows,

$\mathrm{KLD}_{\mathrm{c},k}=\frac{\lambda\>\alpha_{\mathrm{ZF}}^{2}\>P_{\mathrm{c},k}\>d_{\mathrm{c},k}^{-\eta}}{2\>M\left(M-1\right)\left(P_{\mathrm{r}}\>\sigma_{F}^{2}\>d_{\mathrm{c},k}^{-\eta}+\sigma_{n}^{2}\right)\ln
2},$ (8)

where $M$ is the modulation order,
$\alpha_{\mathrm{ZF}}=\sqrt{N_{\mathrm{c}}-K+1}$ is the normalisation factor
for ZF precoding, and $\lambda$ is a constant that depends on the
constellation, for example, in the case of $M$-ary Phase Shift Keying (M-PSK)
it would be $\lambda_{\mathrm{MPSK}}=M^{2}$.

####  III-A2 KLD for Radar System

The sufficient statistics of the generalised likelihood ratio test is denoted
as $\xi(\theta_{k})$. As $L$ increases, $\xi(\theta_{k})$ distribution
approaches a Chi-squared distribution [29, Eq. 54]. Hence,
$\xi(\theta_{\mathrm{t}})$ can be expressed as,

$\mathrm{\xi}\left(\theta_{t}\right)\sim\left\\{\begin{array}[]{l}\mathcal{H}_{1}:\mathcal{X}_{2}^{2}\left(\lambda_{t}\right)\\\
\mathcal{H}_{0}:\mathcal{X}_{2}^{2}\left(0\right)\end{array}\right.,$ (9)

where $\mathcal{X}_{2}^{2}\left(\lambda_{t}\right)$ denotes a noncentral Chi-
squared random variable with $2$ degrees of freedom and a noncentrality
parameter of

$\lambda_{t}=\frac{\alpha_{t}\>d_{\mathrm{r},t}^{-\eta}\>P_{\mathrm{r},t}\>|\textbf{a}^{H}(\theta_{t})\>\textbf{R}_{t}\>\textbf{a}(\theta_{t})|^{2}}{N_{\mathrm{r}}\>(\sigma_{n}^{2}+\sigma_{\mathrm{err}}^{2}\sigma_{\mathrm{w}}^{2}N_{\mathrm{c}}\>P_{\mathrm{c}}\>d_{\mathrm{r},t}^{-\eta})},$
(10)

where
$\mathbf{R}_{t}=\frac{1}{L}\sum_{l=1}^{L}\mathbf{w}_{\mathrm{r},t,l}\mathbf{w}_{\mathrm{r},t,l}^{H}$.which
can be reduced to
$\lambda_{t}=\dfrac{\alpha_{t}\>N_{\mathrm{r}}\>d_{\mathrm{r},t}^{-\eta}\>P_{\mathrm{r},t}}{(\sigma_{n}^{2}+\sigma_{\mathrm{err}}^{2}\sigma_{\mathrm{w}}^{2}N_{\mathrm{c}}\>P_{\mathrm{c}}\>d_{\mathrm{r},t}^{-\eta})}$
for the case of orthogonal waveforms.

The KLD for the radar from $\mathrm{\xi}_{\mathcal{H}_{1}}$ to
$\mathrm{\xi}_{\mathcal{H}_{0}}$ is found to be [20],

$\mathrm{KLD}_{\mathrm{\xi}_{\mathcal{H}_{0}}\\!\parallel\mathrm{\xi}_{\mathcal{H}_{1}}\\!}\\!=\\!\\!\frac{1}{2}\\!\left(\\!1.4427\lambda_{t}\\!-\\!\frac{1}{\ln
2}\\!\\!\int\limits_{0}^{\infty}\\!\\!\\!\mathrm{e}^{-0.5\xi}\ln\left(I_{0}\left(\\!\\!\sqrt{\lambda_{t}\xi}\\!\right)\\!\right)\\!\right)\\!\\!d\xi,$
(11)

where $I_{0}(.)$ is a modified bessel function of the first kind. That is, the
KLD for the radar from $\mathrm{\xi}_{\mathcal{H}_{0}}$ to
$\mathrm{\xi}_{\mathcal{H}_{1}}$ is derived as,

$\mathrm{KLD}_{\mathrm{\xi}_{\mathcal{H}_{1}}\parallel\mathrm{\xi}_{\mathcal{H}_{0}}}=\frac{-0.5\lambda_{t}}{\ln
2}+\frac{\mathrm{e}^{-0.5\lambda_{t}}}{2\ln 2}\mathcal{I},$ (12)

where
$\mathcal{I}\\!=\\!\\!\int\limits_{0}^{\infty}\mathrm{e}^{-0.5\xi}I_{0}\left(\sqrt{\lambda_{t}\xi}\right)\ln\left(I_{0}\left(\sqrt{\lambda_{t}\xi}\right)\right)d\xi$
can be solved with numerical methods such as trapezoidal integration.
Consequently, for the $t$th target, the average KLD can be written as
$\mathrm{KLD}_{\mathrm{r},t}\\!\\!=\\!0.5\left(\>\mathrm{KLD}_{\mathrm{\xi}_{\mathcal{H}_{0}}\parallel\mathrm{\xi}_{\mathcal{H}_{1}}}+\mathrm{KLD}_{\mathrm{\xi}_{\mathcal{H}_{1}}\parallel\mathrm{\xi}_{\mathcal{H}_{0}}}\right)$.

###  III-B Shared Deployment

####  III-B1 KLD for the Communication system

The KLD here differs from separated deployment as the radar interference is
essentially doubled as shown in (5), as there is two sources of radar
interference, the first is the transmit signal radar interference, and the
second is from the radar clutter. For multivariate Gaussian distributed random
variables having mean vectors of $\mathbf{\mu}_{m}$ and $\mathbf{\mu}_{n}$ and
covariance matrices of $\Sigma_{m}$ and $\Sigma_{n}$, the KLD can be derived
as

$\mathrm{KLD}_{n\rightarrow m}=\frac{1}{2\ln
2}\left(\mathrm{tr}\left(\Sigma_{n}^{-1}\Sigma_{m}\right)-2+\left(\mathbf{\mu}_{k,n}-\mathbf{\mu}_{k,m}\right)^{T}\right.\\\
\;\left.\times\;\Sigma_{n}^{-1}\left(\mathbf{\mu}_{k,n}-\mathbf{\mu}_{k,m}\right)+\ln\frac{\left|\Sigma_{n}\right|}{\left|\Sigma_{m}\right|}\right).\vspace{-0.04in}$
(13)

By noting that $\Sigma_{n}=\Sigma_{m}=\sigma_{\zeta}^{2}\mathbf{I}_{2}$, and
given that
$\mathbf{\mu}_{k,m}=\left[\sqrt{P_{\mathrm{c},k}}\>d_{\mathrm{c},k}^{-\eta/2}\alpha_{\mathrm{ZF}}\cos\phi_{k,m},\sqrt{P_{\mathrm{c},k}}\>d_{\mathrm{c},k}^{-\eta/2}\alpha_{\mathrm{ZF}}\sin\phi_{k,m}\right]$,
and
$\mathbf{\mu}_{k,n}=\left[\sqrt{P_{\mathrm{c},k}}\>d_{\mathrm{c},k}^{-\eta/2}\alpha_{\mathrm{ZF}}\cos\phi_{k,n},\sqrt{P_{\mathrm{c},k}}\>d_{\mathrm{c},k}^{-\eta/2}\alpha_{\mathrm{ZF}}\sin\phi_{k,n}\right]$
Then by substituting $\Sigma_{n}$, $\Sigma_{m}$, $\mathbf{\mu}_{k,m}$, and
$\mathbf{\mu}_{k,n}$ in (13), then the KLD for shared deployment communication
system can be found as[20, Corrolary 1],

$\mathrm{KLD}_{\mathrm{c},k}=\frac{\lambda\>\alpha_{\mathrm{ZF}}^{2}\>P_{\mathrm{c},k}\>d_{\mathrm{c},k}^{-\eta}}{2\>M\left(M-1\right)\left(P_{\mathrm{r}}\>\sigma_{H}^{2}\>d_{\mathrm{c},k}^{-\eta}+P_{\mathrm{r}}\>\sigma_{F}^{2}\>d_{\mathrm{c},k}^{-\eta}+\sigma_{n}^{2}\right)\ln
2},$ (14)

####  III-B2 KLD for the Radar System

After collecting $L$ snapshots, the received signal matrix can be formulated
as

$\displaystyle\\!\\!\\!\mathbf{Y}_{\mathrm{r},t|\mathcal{H}_{q}}\\!\\!$
$\displaystyle\\!\\!=\\!\\!$
$\displaystyle\\!\\!\sqrt{\frac{P_{\mathrm{r},t}}{N}}\alpha_{t}\textbf{A}\left(\theta_{t}\right)\\!\mathbf{W}_{\mathrm{r},t}\>q\\!+\\!\mathbf{\tilde{Z}}_{\mathrm{r}},$
(15)

where $\mathbf{{W}}_{t,\mathrm{r}}\in\mathbb{C}^{N\times
L}=\left[\mathbf{w}_{t,\mathrm{r},1},\mathbf{w}_{t,\mathrm{r},2},\cdots\right.$$\left.,\mathbf{w}_{t,\mathrm{r},l}\right]$
and $\mathbf{\tilde{Z}}_{\mathrm{r}}\in\mathbb{C}^{N\times
L}=\left[\mathbf{{\tilde{\zeta}}}_{t,\mathrm{r},1},\mathbf{{\tilde{\zeta}}}_{t,\mathrm{r},2},\cdots,\right.$$\left.\mathbf{{\tilde{\zeta}}}_{t,\mathrm{r},l}\right]$.
By noting that
$\mathbf{y}_{\mathrm{r},t|H_{1}}\left[l\right]\sim\mathcal{CN}\left(\sqrt{\frac{P_{\mathrm{r}}}{N}}\alpha_{t}\mathbf{A}\left(\theta_{t}\right)\mathbf{w}_{\mathrm{r},t,l},2\sigma_{{\tilde{\zeta}}}^{2}\mathbf{I}_{N}\right)$,
then using this the KLD can be derived from (13) using (15) as follows,

$\mathrm{KLD_{r,t}}=\frac{|\alpha_{t}|^{2}\>P_{\mathrm{r},t}\>d_{\mathrm{r},t}^{-\eta}\>|a(\theta_{t})\>\mathrm{R_{x}}\>a(\theta_{t})^{H}|^{2}}{2\>N\>\sigma_{\tilde{\zeta}}^{2}\>\ln
2}$ (16)

By considering the derived KLD for all radar targets and all served
communication UEs, the weighted average KLD for the overall ISAC network can
be formulated as

$\mathrm{KLD}_{\mathrm{avg}}=\>\left(\mathrm{c_{com}}\sum_{k=1}^{K}{\mathrm{KLD}_{\mathrm{c},k}}+\mathrm{c_{rad}}\sum_{t=1}^{T}{\mathrm{KLD}_{\mathrm{r},t}}\right),$
(17)

where $\mathrm{c_{com}}$ and $\mathrm{c_{rad}}$ are the weights for each
system, with $K\>\mathrm{c_{com}}+T\>\mathrm{c_{rad}}=1$. The weights gives
priority to the desired system which allows for more flexibility and trade-
off.

##  IV Power and Antenna Allocation

Effective power and antenna allocation is a critical design factor for ISAC
systems, as it directly impacts the system’s sensing and communication
capabilities. As well as, allocating more resources to one system does not
only reduce the resources of the other subsystem, but also could result in a
considerable amount of interference which may lead to system breakdown.
Nevertheless, by optimising these parameters through maximizing the system’s
$\mathrm{KLD}_{\mathrm{avg}}$, a significant improvement on the detection of
radar targets and the quality of communication among UEs can be achieved.
This, in turn, results in better overall system performance and effective
resource utilisation. In this section, we formulate two optimisation problems
for separated deployment and shared deployment. It is noteworthy mentioning
that the shared deployment utilizes all of the antennas for both systems,
therefore, the required computational resources are reduced by eliminating the
need for allocating the antennas, while in the separated deployment the
optimization algorithm needs to take into account the allocation of both
antennas and power.

###  IV-A Separated Deployment

The optimisation problem is formulated to maximise the average KLD by
optimally allocating a limited budget of power and antennas among UEs and
radar targets while satisfying certain KLD requirements for every served
device (e.g. communication UEs and sensed targets). With the aid of (17), the
optimisation problem can be formulated as follows,

$\displaystyle\mathcal{P}_{1}:\max_{P_{\mathrm{c},k},P_{\mathrm{r},t},N_{\mathrm{c}},N_{\mathrm{r}}}\quad\\!\\!$
$\displaystyle\mathrm{KLD}_{\mathrm{avg}}(P_{\mathrm{c},k},P_{\mathrm{r},t},N_{\mathrm{c}},N_{\mathrm{r}})$
(18a) s.t. $\displaystyle
A_{t,\mathrm{L}}\leq{\mathrm{{KLD}}_{\mathrm{r},t}}\leq
A_{t,\mathrm{U}},\,\,\,\forall t\in T,$ (18b) $\displaystyle
B_{k,\mathrm{L}}\leq{\mathrm{{KLD}}_{\mathrm{c},k}}\leq
B_{k,\mathrm{U}},\forall k\in K,$ (18c)
$\displaystyle\sum_{k=1}^{K}{P_{\mathrm{c},k}}+\sum_{t=1}^{T}{P_{\mathrm{r},t}}=P_{\mathrm{T}},$
(18d) $\displaystyle{N_{\mathrm{c}}}+{N_{\mathrm{r}}}=N.$ (18e)

where $A_{t,\mathrm{L}}$ is the minimum KLD for the $t$-th radar target, and
$B_{k,\mathrm{L}}$ is the minimum KLD for the $k$-th UE, an upper limit for
the KLD can also be enforced with the use of $A_{t,\mathrm{U}}$, and
$B_{k,\mathrm{U}}$, for both targets and UEs, respectively. The formulated
$\mathcal{P}_{1}$ is a compact optimisation problem for our system model,
while the non-compact problem can be found in (19), denoted as
$\mathcal{P}_{2}$. This optimisation problem maximises
$\mathrm{KLD}_{\mathrm{avg}}$ by optimally allocating the power and the number
of antennas among UEs and targets in both systems while maintaining a certain
minimum KLD for each. Because the antennas are allocated through the precoding
process, they can be allocated per subsystem, whereas a certain amount of
power can be allocated to each target and UE.

$\displaystyle\mathcal{P}_{2}:\max_{P_{\mathrm{c},k},P_{\mathrm{r},t},N_{\mathrm{c}},N_{\mathrm{r}}}$
$\displaystyle\\!\\!\\!\\!\left(\mathrm{c_{com}}\>\sum_{k=1}^{K}{\frac{M\>(N_{\mathrm{c}}-K+1)\>P_{\mathrm{c},k}\>d_{\mathrm{c},k}^{-\eta}}{2\left(M-1\right)\left(P_{\mathrm{r}}\>\sigma_{F}^{2}\>d_{\mathrm{c},k}^{-\eta}+\sigma_{n}^{2}\right)\ln
2}}\\!\right.+\\!\\!\>\mathrm{c_{rad}}\sum_{t=1}^{T}{\\!\\!\frac{1}{4}\\!\left(\\!1.4427\lambda_{t}\\!-\\!\frac{1}{\ln
2}\\!\\!\int\limits_{0}^{\infty}\\!\\!\\!\mathrm{e}^{-0.5\xi}\ln\left(I_{0}\left(\\!\\!\sqrt{\lambda_{t}\xi}\\!\right)\\!\right)\\!\right)\\!\\!d\xi}$
(19) $\displaystyle\\!+\left.\frac{1}{2}\left(\frac{-0.5\lambda_{t}}{\ln
2}+\frac{\mathrm{e}^{-0.5\lambda_{t}}}{2\ln
2}\int\limits_{0}^{\infty}\mathrm{e}^{-0.5\xi}I_{0}\left(\sqrt{\lambda_{t}\xi}\right)\ln\left(I_{0}\left(\sqrt{\lambda_{t}\xi}\right)\right)d\xi\right)\right)$
s.t. (18b), (18c), (18d), and (18e) $\displaystyle\mathcal{P}_{4}:\text{
}\text{ }\text{ }\text{ }\text{ }\text{ }\text{
}\max_{P_{\mathrm{c},k},P_{\mathrm{r},t}}$
$\displaystyle\\!\\!\\!\\!\>\Bigg{(}\mathrm{c_{com}}\sum_{k=1}^{K}{\frac{\lambda\>\alpha_{\mathrm{ZF}}^{2}\>P_{\mathrm{c},k}\>d_{\mathrm{c},k}^{-\eta}}{2\>M\left(M-1\right)\left(P_{\mathrm{r}}\>\sigma_{H}^{2}\>d_{\mathrm{c},k}^{-\eta}+P_{\mathrm{r}}\>\sigma_{F}^{2}\>d_{\mathrm{c},k}^{-\eta}+\sigma_{n}^{2}\right)\ln
2}}$
$\displaystyle+\mathrm{c_{rad}}\sum_{t=1}^{T}{\frac{|\alpha_{t}|^{2}\>P_{\mathrm{r},t}\>d_{\mathrm{r},t}^{-\eta}\>|a(\theta_{t})\>\mathrm{R_{x}}\>a(\theta_{t})^{H}|^{2}}{2\>ln(2)\>(\sigma_{\mathrm{n}}^{2}+\sigma_{\mathrm{err_{1}}}^{2}\>d_{\mathrm{r},t}^{-\eta}\>\left(\sigma_{\mathrm{w}}^{2}\>N\>\mathrm{P}_{\mathrm{c}}+\>\mathrm{P}_{\mathrm{r}}\right)+\sigma_{\mathrm{err_{2}}}^{2}\>d_{\mathrm{r},t}^{-\eta}\>\sigma_{\mathrm{w}}^{2}\>N\>\mathrm{P}_{\mathrm{c}})}}\Bigg{)}$
s.t. (18b), (18c), and (18d)

The discrete nature of antenna allocation and the continuous nature of power
allocation make optimisation challenging. This problem can be formulated as a
constrained MINLP problem, which is known to be computationally expensive to
solve [30]. One possible approach for solving this problem is using the GA
approach, a metaheuristic optimisation technique inspired by natural selection
and genetics. GA generates a population of candidate solutions and iteratively
improves them by applying operations such as mutation, crossover, and
selection. This process continues until a stopping criterion is met, such as
reaching a maximum number of iterations or achieving a satisfactory solution.
GA can provide a global optimum, but it requires significant computational
power resources and may not be suitable for real-time applications [31].

Therefore, in this paper, we propose RIPM which is a low-complex heuristic and
very effective algorithm. The proposed RIPM manages to provide comparative
results with much less computational complexity when compared to GA. RIPM
algorithm aims at solving the antenna and power allocation problem by
exploiting an IPM with integer constraints as a base structure. Accordingly,
RIPM treats the discrete nature of antenna allocation as a continuous variable
and uses a continuous optimisation approach to find the solution, where in
each iteration, the number of antennas $N_{\mathrm{r}}$ is rounded and the
number of communication antennas is found as
$N_{\mathrm{c}}=N-N_{\mathrm{r}}$. A pseudo-code for the introduced RIPM is
provided below in Algorithm 1 which explains the computations executed to
realise this algorithm. It is worth noting that RIPM is an iterative algorithm
that moves towards the solution by following a central path in the feasible
region while satisfying the required constraints [32].

Algorithm 1 Rounding-based Interior-Point Method for MINLP

0: $F(x)$: objective function defined in (18a),$g(x)$: linear and nonlinear
constraints defined in (18b),(18c),(18d), and
(18e),$x=\\{P_{\mathrm{c},k},P_{\mathrm{r}ad,t},N_{\mathrm{c}},N_{\mathrm{r}}\\}$:
continuous and integer optimisation variables

1: Set $\epsilon_{\mathrm{opt}}=10^{-6}$,
$\epsilon_{\mathrm{constr}}=10^{-6}$, and $\mathrm{max\\_iter}=1000$

2: Define a modified objective function $F_{m}(x)$:

3: Function $F_{m}(x)$:

4: Round integer variables in $x$ to their nearest integer  values, obtaining
$x_{r}$

5: Compute $F(x_{r})$ using the original objective function  $F(x)$

6: Return $F(x_{r})$

7: End Function

8: Relax integer variables in the MINLP problem, creating a relaxed problem

9: Initialize IPM parameters and starting point $x^{0}$ for the relaxed
problem

10: Replace the original objective function $F(x)$ with the modified objective
function $F_{m}(x)$ in the relaxed problem

11: Set iteration counter $k\leftarrow 0$

12: repeat

13: Apply the IPM to the relaxed problem with the modified objective function,
updating the solution $x^{k}$

14: Compute $||x^{k}-x^{k-1}||$, the optimality tolerance $\mathrm{opt\\_tol}$
and constraint tolerance $\mathrm{constr\\_tol}$ for the current solution
$x^{k}$

15: $k\leftarrow k+1$

16: until $(\mathrm{opt\\_tol}<\epsilon_{\mathrm{opt}}$ and
$\mathrm{constr\\_tol}<\epsilon_{\mathrm{constr}})$ or
$k\geq\mathrm{max\\_iter}$

17: Obtain the final solution $x^{*}$ of the relaxed problem with the modified
objective function

18: Round the integer variables in $x^{*}$

19: return $x^{*}$

###  IV-B Shared Deployment

In shared deployment, only the power is optimally allocated as both systems
utilizes the whole antenna array, the optimisation problem can be formulated
similar to $\mathcal{P}_{1}$, as follows,

$\displaystyle\mathcal{P}_{3}:\max_{P_{\mathrm{c},k},P_{\mathrm{r},t}}\quad\\!\\!$
$\displaystyle\mathrm{KLD}_{\mathrm{avg}}(P_{\mathrm{c},k},P_{\mathrm{r},t})$
(20a) s.t. $\displaystyle
A_{t,\mathrm{L}}\leq{\mathrm{{KLD}}_{\mathrm{r},t}}\leq
A_{t,\mathrm{U}},\,\,\,\forall t\in T,$ (20b) $\displaystyle
B_{k,\mathrm{L}}\leq{\mathrm{{KLD}}_{\mathrm{c},k}}\leq
B_{k,\mathrm{U}},\forall k\in K,$ (20c)
$\displaystyle\sum_{k=1}^{K}{P_{\mathrm{c},k}}+\sum_{t=1}^{T}{P_{\mathrm{r},t}}=P_{\mathrm{T}}.$
(20d)

The non-compact problem can be found in (19), denoted as $\mathcal{P}_{4}$.
This problem is solved using the IPM algorithm, which has a very low
computational complexity. It is important to note that the shared deployment
optimisation problem has less complexity from the separated deployment
optimisation problem and that is due to the fact that only power allocation is
required for the shared deployment, which makes this optimisation problem a
NLP, rather than a MINLP. The results will show that not only the shared
deployment has better performance but lower resource allocation complexity as
well.

##  V Complexity analysis

Optimization problems often necessitate the need for algorithms that provide a
good trade-off between solution accuracy and computational efficiency.
Complexity analysis gives a theoretical perspective on this, while empirical
analysis provides practical insights. Here, we delve into the complexity of
the RIPM and contrast it with that of GA.

####  V-1 RIPM Complexity Analysis

The RIPM introduces a hybrid approach, effectively capitalizing on the
deterministic search strategy of IPM while incorporating a rounding technique
adept for MINLP problems.

For $I$ iterations, RIPM’s estimated complexity is:

$O(E+I\times(i+n^{3}+(k+1)\times E+n+m)),$ (21)

where $E$ is the cost of evaluating the objective function and its
derivatives, this is assumed to be of constant time complexity, $n$ is the
total number of variables, $m$ is the number of constraints, $i$ is number of
integer variables, $I$ is the number of IPM iterations and it is assumed with
empirical verification that for the given problem domain, the RIPM and IPM
will converge within a feasible number of iterations, and $k$ is the maximum
number of objective function evaluations per iteration.

The complexity of RIPM predominantly escalates with the cube of the number of
variables ($n^{3}$), contingent on dense matrix operations. Nevertheless,
given the problem structure, matrix operations remain tractable, and the
rounding technique, which could have been a potential source of
discontinuities, operates seamlessly.

####  V-2 GA Complexity Analysis

GA are recognized for their heuristic nature and widespread search potential.
The computational demands of GA, for $G$ generations with a population of size
$P$, is estimated as [31],

$O(G\times P\times L\times F(n)),$ (22)

where $L$ is the length of the chromosome (often proportional to $n$), $F(n)$
is the time taken for a single fitness evaluation which is a function of the
number of variables.

The complexity of GA is linearly related to the population size, and the
generation count, also the $F(n)$ differs from problem to problem and its
influence is more dominant in our problem. It’s noteworthy that the
probabilistic and explorative nature of GA, while powerful, can sometimes lead
to broader, less-directed searches, which might not always be efficient.

When contrasting the complexities of RIPM and GA, it becomes apparent that the
RIPM’s structured and deterministic approach holds a tangible advantage. The
RIPM complexity, while escalating polynomially with variables, remains
moderated by the IPM’s determinism and the efficient integration of the
rounding technique. Meanwhile, the GA linear dependence on the population size
and generations can render it demanding, especially for larger populations or
extended generations. Our findings further accentuate this observation,
showcasing RIPM’s superior efficiency for the problem at hand.

####  V-3 IPM

The IPM for nonconvex NLP benefits from a systematic search strategy that
offers determinism, even in the midst of nonconvexities. The IPM’s complexity,
for a given $I$ iterations, can be depicted as [32],

$O(I\times(E+n^{3}+m\times n^{2})),$ (23)

where the variables have same definition for RIPM. Our empirical observations
accentuate IPM’s ability to approach solutions swiftly, minimizing the implied
computational toll of the $n^{3}$ factor. GA has same complexity analysis for
MINLP as in NLP but the single fitness evaluation will go higher in MINLP due
to more variables and integers, GA heuristic fabric require exhaustive
function evaluations, especially when precision is paramount. This inherent
characteristic was underscored in our experiments, where GA fell out compared
to IPM in execution rapidity and efficiency.

##  VI Numerical Results

This section presents the performance results of the above-introduced
optimisation problem for both ISAC systems. The optimisation is carried out
for single-UE-single-target (SUST) and MUMT scenarios. Mixed-integer GA and
RIPM are compared for the first scenario only, e.g., SUST, as running the
mixed-integer GA requires extremely high computational complexity for the case
of MUMT, which makes it unsuitable for real-time applications compared to the
RIPM. In the simulations, GA population size is set to $50$ times the number
of decision variables, with a crossover fraction of $0.8$ and a mutation rate
of $0.2$. The stopping criteria for GA include reaching the maximum number of
generations, which is set to $100$ times the number of decision variables, or
a stall in the best fitness value for a predetermined number of generations.
While for the RIPM simulations, the stopping criteria for RIPM is based on the
maximum number of iterations which is set to $1000$. Furthermore, the
optimality tolerance and the constraint tolerance are set to $10^{-6}$. For
MUMT scenario, a number of $3$ UEs and $3$ targets have been considered for
two cases; the first case without upper limits, whereas upper limits on
individual KLDs have been enforced in the second one. In all cases, the
antenna separation is set to half wavelength, the total transmit power is
fixed at $P_{\mathrm{T}}=1$, the covariance matrix for the radar in the case
of separated deployment is $\mathbf{R}_{x}=\mathbf{I}_{N_{\mathrm{r}}}$, and
in the case of shared deployment $\mathbf{R}_{x}=\mathbf{I}_{N}$, the target
cross section is normalised to $\alpha_{t}=1\>\forall t$, and QPSK modulation
is used throughout the results. Moreover, the variance of the channel
estimation error of the IC process is
$\sigma_{\mathrm{err}}=\sigma_{\mathrm{err_{1}}}=0.01$, the pathloss exponent
is $\eta=3$, which is considered to model the effect of large-scale fading,
and the total number of antennas at BS is fixed at $N=20$. In the figures
provided, the $\mathrm{P}_{\mathrm{R}}/{N}_{o}$ is chosen, so the optimisation
problem is in the feasible region. Unless stated otherwise,
$\mathrm{c_{com}}=\mathrm{c_{rad}}=\frac{1}{K+T}$.

###  VI-A Separated Deployment

Figure 2: RIPM and GA optimisation methods comparison for SUST scenario, the
minimum KLD requirement for the UE and target are $B_{1,\mathrm{L}}=20$ and
$A_{1,\mathrm{L}}=5$, respectively. The distances are
$d_{\mathrm{UE}_{1}}=150$ m, and $d_{t_{1}}=220$ m.  Figure 3: RIPM
optimisation is used for MUMT scenario, the minimum KLD for each UE is
$B_{1,\mathrm{L}}=10$, $B_{2,\mathrm{L}}=5$, and $B_{3,\mathrm{L}}=15$, and
for each target is $A_{1,\mathrm{L}}=10$, $A_{2,\mathrm{L}}=15$, and
$A_{3,\mathrm{L}}=5$, where $d_{\mathrm{UE}_{1}}=150$ m,
$d_{\mathrm{UE}_{2}}=210$ m, $d_{\mathrm{UE}_{3}}=100$ m, $d_{t_{1}}=250$ m,
$d_{t_{2}}=180$ m, and $d_{t_{3}}=300$ m.

In Fig. 2, $\mathrm{KLD}_{\mathrm{avg}}$ and the individual KLD for both the
UE and the target are plotted using both RIPM and GA methods for optimisation.
As can be seen from the figure, $\mathrm{KLD}_{\mathrm{avg}}$ for both
optimisation methods is much higher than the $\mathrm{KLD}_{\mathrm{avg}}$ of
URA. Therefore, both optimisation methods show significant improvement over
URA method by maximising the $\mathrm{KLD}_{\mathrm{avg}}$ and satisfying the
minimum KLD requirements. In addition, RIPM with integer constraint rounding
and GA performances are identical. It is important to note that because the UE
is closer to BS and has a higher minimum KLD requirement than the radar
target, it is allocated the rest of the resources after the radar target’s
minimum KLD requirement is satisfied.

As shown in Fig. 3.a, the system is generalised to multiple UEs and targets
using the RIPM optimisation method. The performance of GA has not been
considered in this figure because it is computationally untraceable for this
scenario. It is evident from the figure that $\mathrm{UE}_{1}$ and
$\mathrm{UE}_{3}$ for the URA case are in an outage for the whole range of
$\mathrm{P}_{\mathrm{R}}/N_{o}$ as they do not satisfy the minimum KLD
requirement, as well as, targets $\mathrm{T}_{1}$ and $\mathrm{T}_{3}$ are in
an outage at $\mathrm{P}_{\mathrm{R}}/N_{o}=28$ dB. The optimal
$\mathrm{KLD}_{\mathrm{avg}}$ improvement over the URA case is still
significant for MUMT scenario. It is important to note that changing the KLD
requirements will change the gained improvement, as the minimum KLD
requirement for UEs and targets affects $\mathrm{KLD}_{\mathrm{avg}}$. It will
decrease whenever a higher minimum requirement is enforced on distant targets
or UEs. In addition, Fig. 3.a shows that the minimum KLD requirement is
satisfied for all UEs and targets, and the rest of the resources are given to
the target or UE that will provide the highest $\mathrm{KLD}_{\mathrm{avg}}$.
In the considered scenario, $\mathrm{T}_{2}$ is the target that provides the
highest $\mathrm{KLD}_{\mathrm{avg}}$, which is the closest target to BS, even
though $\mathrm{UE}_{3}$ and $\mathrm{UE}_{1}$ are closer to BS than
$\mathrm{T}_{2}$, this is because the communication system is very sensitive
to radar-interference as there is no IC applied at UEs.

In Fig. 3.b and Fig. 3.c, the same MUMT setup as Fig. 3.a is used with the
addition of upper limits applied on the KLD of each of UEs and targets. The
upper limits are introduced to ensure fairness between UEs and targets. Once
$\mathrm{T}_{2}$, the device with the highest KLD, as shown in Fig. 3.a,
reaches its KLD upper limit, which is set to $50$ bits for both
$B_{k,\mathrm{U}}$ and $A_{t,\mathrm{U}}$ in this simulation setup, the extra
resources will be allocated to the other UEs and targets in the order of their
contribution in the maximisation of $\mathrm{KLD}_{\mathrm{avg}}$. The KLD for
$\mathrm{UE_{1}}$ and $\mathrm{UE_{3}}$ in the case of URA are in outages as
they are below the minimum KLD requirement.

###  VI-B Shared Deployment

The shared deployment MUMT ISAC system power is optimally allocated by
utilising the KLD in (19). Fig.4 shows that the performance of shared
deployment is superior to the performance of separated deployment when the
$\sigma_{\mathrm{err_{2}}}^{2}$ is in reasonable range, at the point
$\mathrm{P}_{\mathrm{R}}/N_{o}=30$ dB, the separated deployment
$\mathrm{KLD_{avg.}}$ is approximately 30 bits, while in the case of
$\sigma_{\mathrm{err_{2}}}^{2}=0.01$, the $\mathrm{KLD_{avg.}}$ is
approximately 45 bits. This shows that shared deployment system and moving to
more integration of both systems is the way forward. The estimation error of
radar parameter which is dictated by the error variance
$\sigma_{\mathrm{err_{2}}}^{2}$, affects the amount of information received
from the target which is reflected on radar KLD and consequently on the
$\mathrm{KLD_{avg.}}$, more error leads to less information, and as shown in
the figure this gives decrease in the KLD measure. It is also still evident
from the figure that $\mathrm{UE}_{1}$ and $\mathrm{UE}_{3}$ for the URA case
are in an outage for the whole range of $\mathrm{P}_{\mathrm{R}}/N_{o}$ as
they do not satisfy the minimum KLD requirement, as the whole communication
system in URA saturate at around 5.8 bits.

Figure 4: RIPM optimisation is used for shared deployment MUMT scenario, the
configuration is the same as Fig.3, it is done for multiple cases of
$\sigma_{\mathrm{err_{2}}}^{2}$ Figure 5: RIPM optimisation is used for shared
deployment MUMT scenario, the configuration is the same as Fig.3, with
$\sigma_{\mathrm{err_{2}}}^{2}=0.01$

In Fig. 5, the same setup as Fig. 4.b is used with the addition of upper
limits applied on the KLD of each of UEs and targets. The upper limits are
introduced to ensure fairness between UEs and targets just as shown in 3.b and
Fig. 3.c for the separated deployment. This is done for shared deployment and
the same thing is noticed where once $\mathrm{T}_{2}$, the device with the
highest KLD, reaches its KLD upper limit, which is set to $80$ bits for both
$B_{k,\mathrm{U}}$ and $A_{t,\mathrm{U}}$ in this simulation setup, the extra
resources will be allocated to the other UEs and targets in the order of their
contribution in the maximisation of $\mathrm{KLD}_{\mathrm{avg}}$.

Figure 6: ROC curves for the MUMT scenario with upper limit of 80 bits using
RIPM and URA, at $\mathrm{P}_{\mathrm{R}}/N_{o}=28$ dB. (a) Separated
deployment; (b) Shared deployment.

For the sake of completeness, the receiver operating characteristic (ROC)
curves are shown in Fig. 6 for all the targets using the same MUMT parameters
in Fig.3.a but with an upper limit of 80 bits, this is done for both separated
and shared deployment antenna configuration. It is clear from the figure that
the performance of RIPM is superior to URA in terms of radar detection
capability, for all targets in the separated deployment and for targets
$\mathrm{T_{1}}$ and $\mathrm{T_{2}}$ for the shared deployment, this is
because the URA basically only focuses on one system which is the one least
susceptible to interference while the other system is totally cut-off, this is
why a power optimisation is essential to ensure that both systems work
properly, the upper limit introduced makes sure more fairness to not single
out one UE or target.

As seen from the figure, $\mathrm{T_{2}}$ has the best probability of
detection trade-off with the probability of false alarm because it is the
closest target and allocated more resources. Furthermore, the bit error rates
(BERs) of UEs achieved by using RIPM for both deployments at
$\mathrm{P}_{\mathrm{R}}/N_{o}=40$ dB for both deployment configurations are
$0.0175,0.0620,$ and $0.0009$, respectively, whereas a BER of $0.0610$ for all
UEs is obtained using URA, it is important to note that the BER for each UE is
determined by the set KLD minimum requirement for that UE, which can be
changed to accommodate different BER requirements. By comparing the results in
this figure with those in Fig.3.a, it can be realised that KLD measure can
effectively characterise ISAC system. This sort of characterisation can be
observed by the fact that a higher KLD results in better detection capability
for the radar subsystem and lower BER for the communication subsystem.
Apparently, the obtained results in this figure reinforce the fact that the
usage of KLD as a unifying and homogeneous measure for ISAC system that
provides a clear system level trade-off between both subsystems, and thus it
can be effectively utilized as a standard measure for ISAC system design
purposes.

Figure 7: Probability of error for the radar in the MUMT scenario with upper
limit of 80 bits using RIPM and URA. (a) Separated deployment; (b) Shared
deployment.

In Fig. 7, the probability of error can be observed for the radar in both
antenna deployment configurations, the optimisation works to increase the
total $\mathrm{KLD_{avg.}}$, satisfying the constraints, minimum KLD
requirements, and the upper KLD fairness constraint. $T_{2}$ is the nearest
between the targets and UEs, therefore, it will contribute the highest to the
$\mathrm{KLD_{avg.}}$, but to ensure it is not singled out, the upper limit is
introduced. In Fig. 7.b, the upper limit is reached for $T_{2}$ at 26 dB, then
the extra power resources will go to the next target $T_{1}$, and when that
target reaches the upper limit the extra resources will go to the next target
$T_{3}$ at around 31 dB, similar thing happens in Fig.7.a. but the upper limit
for each target is reached at a higher $\mathrm{P}_{\mathrm{R}}/N_{o}$ which
shows that the shared deployment configuration has superiority over the
separated deployment. The URA can be a bit better than the RIPM for $T_{1}$
and $T_{3}$ before the point that they are given more resources, because the
URA does not have any upper or lower limit for either the radar or
communication systems, and the optimisation focuses the minimum requirement
for both systems and their equipments.

Figure 8: Convergence evaluation for both RIPM and GA optimisation methods for
both deployment configurations. (a) Separated deployment for SUST; (b) Shared
deployment for SUST.

For the sake of computational complexity argument, the convergence,
optimality, and complexity of both optimisation methods, i.e., GA and RIPM,
are investigated, as well. The time and computational resources consumed by GA
are generally much more than RIPM due to the nature of its search process
(e.g. selection, crossover, mutation). In Fig. 8, the convergence criteria is
shown through plotting $\mathrm{KLD}_{\mathrm{avg}}$ versus the numbers of
iterations that are required for the optimisation techniques to find the
optimal solution. For Fig.8.a, the convergence is explored for separated
deployment at $\mathrm{P}_{\mathrm{R}}/{N}_{o}=20$ dB, using the same
simulation environment used for SUST configuration in Fig. 2, where the RIPM
method requires $10$ iterations to find the solution, while GA needs $164$
iterations to find the same solution. This clearly depicts that there is a
massive difference in terms of the number of iterations. Moreover, the number
of times the objective function is evaluated during the optimisation process
using GA is $6275$ calculations, whereas it is $51$ calculations for the case
of RIPM. Obviously, RIPM is a significantly better candidate to achieve the
allocation objective. Generally speaking, the quality of the final solution
determines the optimality condition. It can be noticed from the figure that
the final solution for GA is $41.007$ bits, whereas it is $40.736$ for RIPM,
and thus the percentage difference between them is as little as $0.6631$%.
This shows that GA gives a very tiny improvement over the RIPM regarding the
final solution. However, this tiny amount is not as proportional to the amount
of the functions calculations and the iterations difference between the two
methods.

In Fig. 8.b, the shared deployment convergence for both the GA and RIPM is
shown, this is done at $\mathrm{P}_{\mathrm{R}}/{N}_{o}=20$ dB, using the same
simulation environment used for the SUST configuration in Fig. 2. This shows
that GA falls in terms of efficiency in the shared deployment as well, this is
observed in the figure as the GA starts with infeasible points and only starts
searching in the right direction after 100 iteration, reaching its feasible
solution at around 140 iterations. The number of objective function
calculations for GA reaches around 2520 calculations in total, while the RIPM
reaches the solution at 9 iterations with only 27 objective function
calculations.

Figure 9: CPU time simulation for both RIPM and GA optimisation methods for
both deployment configurations.

In Fig. 9, the CPU time versus the input size which is $K+T$, this figure
shows the average CPU time it takes to perform optimisation for both RIPM/IPM
and GA in separated and shared deployment configurations by varying the input
size i.e the number of continuous optimisation variables. This shows in both
deployment configuration how the RIPM and IPM performs more efficiently in
terms of time when compared with the GA technique, which also backs our
convergence rate results.

Figure 10:  $\mathrm{KLD_{c}}$ versus $\mathrm{KLD_{r}}$ trade-off for shared
deployment at $\mathrm{P}_{\mathrm{R}}/{N}_{o}=44$ dB, same MUMT system
configuration from Fig.4.b, with relaxed constraints.

Lastly, it is important to show the trade-off performance between the two
systems in response to the weights of each system $\mathrm{c_{com}}$ and
$\mathrm{c_{rad}}$ in (17). This kind of trade-off is achieved by varying
$\mathrm{c_{com}}$ from 0 to $\frac{2}{T+K}$, calculating $\mathrm{c_{rad}}$
using $\mathrm{c_{rad}}=\frac{1-K\>\mathrm{c_{com}}}{T}$, and then plotting
the achievable $\mathrm{KLD_{c}}$ against $\mathrm{KLD_{r}}$, where
$\mathrm{KLD_{c}}=\mathrm{c_{com}}\sum_{k=1}^{K}{\mathrm{KLD}_{\mathrm{c},k}}$
and
$\mathrm{KLD_{r}}=\mathrm{c_{rad}}\sum_{t=1}^{T}{\mathrm{KLD}_{\mathrm{r},t}}$.
As can be observed from Fig. 10, which depicts this trade-off for the shared
deployment scenario, the IPM optimised method provides a better trade-off
between the radar and communication system, and much more controllability when
compared to the URA or any fixed power approach. When observing the boundary
which is the maximum achievable performance trade-off using the fixed power
approach and the IPM optimised trade-off, the optimisation clearly provides a
much better trade-off in comparison. For example, the maximum achievable
$\mathrm{KLD_{c}}$ in Fig. 10 is about $50$ bits achieved when
$\mathrm{KLD_{r}}$ approaches 0, whereas a $\mathrm{KLD_{c}}$ of $90$ bits is
obtained using IPM at $\mathrm{P}_{\mathrm{R}}/{N}_{o}=44$ dB. Some sort of
discontinuity can be seen in Fig.10, in the $x$-axis range of (3,21). This
behaviour is attributed to the fact that optimisation prioritises a system
over the other after satisfying the minimum KLD requirement to get the maximum
KLD gain, this will make the system at a certain weight switching the priority
from one system to the other to provide optimal KLD gain. This zone is
considered infeasible and does not provide maximum performance gain. In
Fig.10, it is shown that the IPM optimised method provides better trade-off
between the radar and communication system.

##  VII Conclusion

In this paper, we explored new means for the allocation of antenna and power
resources in downlink ISAC system to optimize its functionality. Unlike
existing literature, we applied the KLD metric as a unifying measure to merge
both subsystems into a singular integrated system, thereby facilitating the
allocation of antennas and power resources in a novel manner. It was shown
that this unifying KLD measure can provide an efficient, clearer and more
leaner ISAC design.

Both separated and shared ISAC modes of operation have been considered. The
findings were decisive in showing that the shared deployment outperformed the
separated one which suggests that deeper integration offers better rewards and
hence a more promising direction for future ISAC systems.

Furthermore, the proposed RIPM algorithm was benchmarked against the GA and
URA approaches. Notably, we delved into the MUMT scenario and a SUST one. The
obtained results demonstrated that the proposed RIPM provides comparable
performance to the optimal GA with significant reduction in computational
complexity. The results highlighted that both GA and RIPM have an edge over
URA, especially in the realms of KLD, BER, and probability of detection.

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