# FD-JCAS Techniques for mmWave HetNets:
Ginibre Point Process Modeling and Analysis

Christodoulos Skouroumounis, , Constantinos Psomas, , and Ioannis Krikidis
Christodoulos Skouroumounis, Constantinos Psomas, and Ioannis Krikidis are
with the IRIDA Research Centre for Communication Technologies, Department of
Electrical and Computer Engineering, University of Cyprus, Cyprus,
e-mail:{cskour03, psomas<EMAIL_ADDRESS>
###### Abstract

In this paper, we study the co-design of full-duplex (FD) radio with joint
communication and radar sensing (JCAS) techniques in millimeter-wave (mmWave)
heterogeneous networks (HetNets). Spectral co-existence of radar and
communication systems causes mutual interference between the two systems,
compromising both the data exchange and sensing capabilities. Focusing on the
detection performance, we propose a cooperative detection technique, which
exploits the sensing information from multiple base stations (BSs), aiming at
enhancing the probability of successfully detecting an object. Three combining
rules are considered, namely the OR, the Majority and the AND rule. In real-
world network scenarios, the locations of the BSs are spatially correlated,
exhibiting a repulsive behavior. Therefore, we model the spatial distribution
of the BSs as a $\beta$-Ginibre point process ($\beta$-GPP), which can
characterize the repulsion among the BSs. By using stochastic geometry tools,
analytical expressions for the detection performance of $\beta$-GPP-based FD-
JCAS systems are expressed for each of the considered combining rule.
Furthermore, by considering temporal interference correlation, we evaluate the
probability of successfully detecting an object over two different time slots.
Our results demonstrate that our proposed technique can significantly improve
the detection performance when compared to the conventional non-cooperative
technique.

###### Index Terms:

Full-duplex, millimeter-wave, cooperative detection, Ginibre point process,
stochastic geometry.

##  1 Introduction

Emerging applications such as smart cars, unmanned aerial vehicles (UAV) and
enhanced localization, lead to an ever-increasing demand for systems with both
communication and radar sensing capabilities [1, 2]. In order to address this
demand, joint communication and radar sensing (JCAS) techniques have been
developed, integrating the two operations of communication and sensing over a
shared spectrum. As an emerging research topic, the spectrum sharing of JCAS
techniques enables the efficient use of the available spectrum, and provides a
new way for designing and modeling novel network architectures and protocols
that can benefit from the synergy of communication and radar systems.

As a straightforward approach for achieving the spectral co-existence of
communication and radar sensing systems, the authors in [3, 4] consider an
opportunistic spectrum sharing scheme between a pulsed radar and a cellular
network, where the communication system transmits if and only if its
transmission will not compromise the operation of the radar system. While such
approach achieves low implementation complexity, the simultaneous operation of
communication and radar sensing systems is unattainable. The main challenge
for the joint operation of communication and radar sensing systems over a
shared spectrum is the negative effects of the mutual interference between the
two systems, which significantly alleviates the network performance [5].
Therefore, the employment of interference mitigation techniques is of
paramount importance in the context of JCAS systems. The impact of mutual
interference on the coverage performance of communication systems is well
investigated in the literature, however, its impact on the sensing
capabilities of radar systems has not fully elucidated. From this standpoint,
the authors in [6] study the spectral co-existence of a power-controlled
cellular network with rotating radar devices, demonstrating that the power
control is of critical importance for the mitigation of the mutual
interference. Moreover, the spectrum sharing between a multi-input-multi-
output (MIMO) radar and a wireless communication system is analyzed in [7];
the results show that by exploiting the degrees of freedom offered by the MIMO
technology, the mutual interference can be mitigated from the main/side lobe
without compromising the radar system’s performance. Similarly, beamforming
techniques are proposed in [8] for a MIMO JCAS system, where each remote radio
unit can operate as a radar and as a communication transmitter by
simultaneously sensing nearby objects and communicating with downlink (DL)
users, respectively. However, the concept of cooperation between multiple base
stations (BSs) in the context of FD-JCAS systems, aiming to mitigate the
overall interference and enhance the network’s performance is missing from the
current literature.

Efficient spectrum utilization is another key issue for JCAS systems that
needs to be addressed. Spectrum efficiency in wireless communication systems
can be significantly improved by operating in a full-duplex (FD) mode, which
is being considered for the next-generation wireless systems. Specifically,
the FD technology could potentially double the spectral efficiency with
respect to the half-duplex counterpart, as it employs simultaneous
transmission and reception using non-orthogonal channels [9]. However, the
non-orthogonal operation creates a self-interference (SI) link between the
input and the output antennas. Owing to the overwhelming negative effect of
the SI at a transceiver, the FD technology has been previously regarded as an
unrealistic approach in wireless communications. Fortunately, with the recent
advancements in transceiver design and signal processing techniques, the SI
signal can be successfully suppressed below the noise floor, and therefore the
FD technology becomes feasible [10, 11]. However, the performance of the FD
radio in large-scale multi-cell networks is additionally compromised by the
increase of the intra- and out-of-cell co-channel interference. Several
research efforts have been carried out to study the effect of multi-user
interference on the FD performance for large-scale wireless networks, and
several techniques have been proposed to mitigate the additional interference
caused by the FD operation [10, 12, 13].

The requirements of next generation cellular networks in massive connectivity
and high throughput, motivate the operation of millimeter-wave (mmWave)
frequency bands and their heterogeneous network (HetNet) deployment. mmWave
communications are considered as a suitable environment for integrating FD-
JCAS systems due to their unique features, such as the large available
bandwidth and the antenna directivity, which can boost the quality of the
direct link [14]. In particular, the large available bandwidth of mmWave
communications can lead to multi-Gbps rates, which is essential to satisfy the
high-capacity requirements of emerging applications in FD-JCAS systems, such
as fully-connected vehicles, UAV and robotics [15]. Furthermore, recent
studies have shown that the higher path-losses of the mmWave signals and their
sensitivity to blockages, can improve the network performance by mitigating
the overall interference [13, 16]. Therefore, the modeling and the analysis of
FD-JCAS systems in the context of mmWave cellular networks, is of critical
importance in order to support the massive data-rate demands of emerging
applications and combat the severe multi-user interference. Even though FD-
JCAS systems are well-investigated for sub-$6$ GHz applications, few works
deal with their operation in higher frequency bands. Recent studies suggest
exploiting the radar function to support vehicle-to-everything communications
based on the IEEE $802.11$ad wireless local area network protocol, which
operates in the $60$ GHz band [14, 15]. Similarly, the authors in [17]
investigate the implementation of radar devices that operate in the mmWave
frequency bands in the context of vehicular systems, and study the impact of
radar interference on the probability of successful range estimation. In
addition, the authors in [18] characterize the blockage detection probability
achieved by the mmWave radar devices, and provide guidelines for their
efficient deployment. However, the above studies only focus on the co-design
of FD-JCAS in the mmWave cellular networks, neglecting the negative effects of
the SI link on the overall network performance.

Motivated by the mathematical tractability of the Poisson point process
(PPP)-based abstraction model, modeling and analysis of JCAS systems with the
aid of stochastic geometry has gained a lot of interest [19, 20, 21, 22].
However, the PPP is only suitable for the networks where nodes are deployed in
a fully unplanned fashion. Moreover, in [23], the authors shown that by
modeling the spatial distribution of the network’s nodes as a PPP, the
achieved performance of the actual deployment of cellular networks is
underestimated. In many practical networks, the locations of the nodes are
determined to alleviate the interference or extend the coverage region, and
therefore there exists a form of repulsion among the network’s nodes. In this
context, the Ginibre point processes (GPPs) [25], which is a special case of
the determinantal point processes (DPPs) [24], have gained much attention as
models for wireless networks where there exists a form of repulsion among the
nodes in the networks. The $\alpha$-GPP ($-1\leq\alpha\leq 0$) [26] is a
superposition of $-1/\alpha$ independent GPPs with an intensity scaled by
$\sqrt{-\alpha}$. By modeling the BSs’ locations of a primary network as an
$\alpha$-GPP, the authors in [26] investigate the performance of the ambient
backscatter communications in the context of wireless-powered HetNets, where
the secondary transmitters utilize the traffic resources offered by the
primary network. On the other hand, the $\beta$-GPP ($0<\beta\leq 1$) [27] is
a thinned and rescaled GPP which is obtained by deleting each point of the GPP
independently with probability $1-\beta$. In [27], the authors analyzed the
secure communication performance by using artificial noise from the source and
cooperative jamming for single antenna nodes, where the network’s nodes are
spatially distributed as $\beta$-GPP. In addition to the spatial correlation
between the network’s nodes, the presence of common randomness in the
locations of the interfering nodes induces temporal correlation in the
observed interference [28]. Although there has been substantial work
quantifying interference correlation in wireless network [28, 29], the
investigation of temporal correlation in FD-JCAS systems has been disregarded.

To the best of our knowledge, the spectrum co-existence of FD-JCAS systems in
mmWave HetNets, where the network’s nodes experience a repulsion between each
other, is overlooked from the literature. In addition, the effect of the SI
and the temporal interference correlation on the achieved performance of the
FD-JCAS deployments has not been investigated. Hence, the aim of this work is
to fill this gap by modeling and analyzing such networks and by providing new
analytical results for the network performance in a stochastic geometry
framework. Specifically, the main contributions of this paper are summarized
as follows:

  * •

We develop a mathematical framework based on stochastic geometry, which
comprises the modeling of spectrum sharing FD-JCAS systems in the context of
heterogeneous mmWave cellular networks. Specifically, we consider a JCAS
system, where all BSs exhibit radar sensing capabilities, of which a fraction
also exhibit DL communication capabilities by exploiting the concept of FD
radio. As real network deployments form a more regular point pattern than the
PPP, the developed framework takes into account the correlation among the
locations of the BSs by modeling their distribution as a $\beta$-GPP.

  * •

A novel cooperative multi-point radar detection (CoMRD) technique is proposed,
aiming at providing an enhanced detection probability of objects sensing by
the BSs’ radar system. Three hard-decision combining rules, namely the OR, the
Majority and the AND rules, are analyzed. Using stochastic geometry tools,
analytical expressions for the detection performance are derived for each of
the considered hard-decision combining rules. Moreover, we derive simplified
analytical expressions for the scenario where the BSs are independently
deployed within the network area, i.e. $\beta\rightarrow 0$. Finally, the
temporal correlation of the interference is investigated and analytical
expressions of the joint detection probability over different time slots are
derived.

  * •

The behavior of the detection performance achieved by our proposed technique
for the considered combining rules is analyzed for different system
parameters. We demonstrate that our cooperative technique can significantly
improve the detection performance in the context of the considered network
deployments, when compared to the conventional non-cooperative radar detection
technique. In addition, our numerical results illustrate that a repulsive
deployment of BSs is beneficial for the detection performance, since by
increasing the distance of an interfering link results in a reduction of the
caused interference. Furthermore, we show that the temporal interference
correlation significantly impacts the network’s detection performance.

The rest of the paper is organized as follows: Section 2 describes the
considered system model together with the channel, blockage, antenna, power
allocation, detection and communication models, as well as the main
definitions and properties of the $\beta$-GPP. The proposed CoMRD technique is
explicitly described in Section 3. Section 4 provides the main results for the
detection performance for each hard-decision combining rule in the context of
mmWave HetNets, where the network’s nodes are distributed based on a
$\beta$-GPP. In Section 5, we consider the effect of the interference
correlation on the detection performance, providing analytical expressions for
the joint detection probability over different time slots. Numerical results
are presented in Section 6, followed by our conclusions in Section 7.

TABLE I: Summary of Notations

Notation | Description | Notation | Description
---|---|---|---
$K$ | Total number of network tiers | $\mu,\sigma^{2}_{\rm SI}$ | Nagakami-$\mu$ fading parameters
$\Phi_{k},\lambda_{k}(r)$ |  $\beta$-GPP of BSs in $k$-th tier of density $\lambda_{k}(r)$ | $p_{\rm L}(r)$ | LoS probability function
$\widetilde{\Phi}_{k},\widetilde{\lambda}_{k}$ |  $\beta$-GPP of interfering BSs in $k$-th tier of density $\widetilde{\lambda}_{k}$ | $p_{\rm L},{R}$ | parameters of LoS probability function
$\beta_{k}$ | repulsion parameter of BSs in the $k$-th tier | $\phi,G$ | Main-lobe beamwidth and gain
$\chi_{k}$ | fraction of FD-JCAS BSs in $k$-th tier | $P_{k}$ | Maximum transmit power of BSs
$\mathcal{C}$ | set of $K$ cooperative BSs | ${P}_{k}(d)$ | Allocated BSs’ transmit power for the DL communication
$d$ | distance between a BS and its associated user | $\rho,\varepsilon$ | Power control parameters
$h(\tau)$ | power of the channel fading at time slot $\tau$ | $A$ | Cross-section area of the point-of-interest
$L(X,Y)$ | large-scale path-loss between nodes $X$ and $Y$ | $\sigma_{n}^{2}$ | variance of the thermal noise
$a,\epsilon$ | large-scale path-loss parameters | $\delta_{k}$ | Binary decision of the cooperative BS from the $k$-th tier
$g_{\rm SI}$ | channel power gain of the residual SI | $\mathds{1}_{A}$ | Indicator function that the event $A$ holds

Notation: $r_{x}$ denotes the Euclidean norm of $x\in\mathbb{R}^{d}$ i.e.,
$r_{x}=||x||$, $B(x,r)$ represents a circle of radius $r$ centered at
$x\in\mathbb{R}^{2}$, $\binom{n}{k}=\frac{n!}{k!(n-k)!}$ is the binomial
coefficient, $\mathcal{G}(a,b)$ is a gamma random variable (r.v.) with shape
parameter $a$ and scale parameter $b$; $\Gamma[a]$ and $\Gamma[a,x]$ denote
the complete and incomplete gamma function, respectively, $\mathds{1}_{X}$ is
the indicator function where $\mathds{1}_{X}=1$ if $X$ is true, otherwise
$\mathds{1}_{X}=0$, and ${}_{2}F_{1}[\cdot,\cdot,\cdot,\cdot]$ denotes the
hypergeometric function.

##  2 System model

In this section, we provide details of the considered system model. The
network is studied from a large-scale point-of-view leveraging tools from
stochastic geometry. In order to model and analyze the spectral coexisting
communication and radar systems in the context of mmWave HetNets, an FD-JCAS
system is investigated, where the mmWave BSs have two operations i.e., DL
transmission and object detection. The main mathematical notation related to
the system model is given in Table I.

###  2.1 Network topology

Figure 1: Network topology of a two-tier FD-JCAS system in mmWave cellular
networks.

We consider a HetNet composed by $K$ network tiers, consisting of randomly
located BSs that operate in the mmWave frequency band. In this context, we
model the spatial distribution of the BSs that belong to the $k$-th tier i.e.,
$k\in\\{1,\dots,K\\}$, according to a $\beta$-GPP denoted as
$\Phi_{k}\\!=\\!\\{X_{j,k}\\!\in\mathbb{R}^{2},j\in\mathbb{N}\\}$ with spatial
density $\lambda_{k}$ and repulsion parameter $\beta_{k}$ i.e.,
$\Phi_{k}\sim{\rm GPP}(\lambda_{k},\beta_{k})$. The locations of the points-
of-interest (PoIs) at which it is necessary to detect whether or not an object
exists, follow a PPP $\Phi_{\rm PoI}$ with density $\lambda_{PoI}$, where at
each PoI there is an object with a probability $\zeta$, where $\zeta\in[0,1]$.

The knowledge of such information is essential in many communication
techniques, such as beamforming, handover and localization techniques,
especially for systems that operate in high frequency bands i.e., mmWave and
Tera-Hz communication [14, 15, 17], where the detection of an object is
crucial for their operations. Although the acquisition of this knowledge can
be accomplished by a plethora of systems, the demand for a new infrastructure
prohibitively increases the implementation cost, making such techniques
unsuitable for the next-generation systems [17]. Instead of having two
separate systems, we develop a JCAS technique that integrates the two
functions into one by sharing hardware and signal processing modules. With
such techniques, the implementation cost is significantly reduced, and the
network performance can be further enhanced from the mutual sharing of
information [14, 15, 17].

We assume that all BSs exhibit radar sensing capabilities and are responsible
for determining whether or not there is an object in a PoI. In this work, a
round-robin scheduling mechanism is employed where the detection of all PoIs
within the coverage area of a BS is randomly scheduled at different time
slots. In addition to the sensing capabilities, a fraction of BSs is also
exhibits communication capabilities, serving their associated users in the DL
direction by implementing the FD-JCAS scheme. Specifically, a BS that belongs
to th $k$-th tier, exhibits both communication and radar sensing capabilities
based on a predefined probability $\chi_{k}$, otherwise, it solely exhibits
sensing capabilities. We assume that the two-dimensional space
$\mathbb{R}^{2}$ is divided into several large sub-regions
$V\subset\mathbb{R}^{2}$, where all BSs of each sub-region $V$ are connected
to the dedicated central unit, also known as fusion center (FC), of that
particular sub-region through an ideal report channel [20]. The focus of this
work is the performance investigation of BSs that belong in a single sub-
region $V\in\mathbb{R}^{2}$, while the cooperation between different sub-
regions is an interesting future work. Fig. 1 shows a realization of a two-
tier mmWave cellular network i.e, $K=2$, consisting of macro-cell and micro-
cell BSs, where solid and dashed lines represent the communication signals and
the radar sensing of a BS, respectively. Based on Slivnyak’s theorem and
without loss of generality, we focus on detecting the existence or absence of
an object at the typical PoI, which is assumed to be located at the origin.

All wireless signals are assumed to experience both large-scale path-loss
effects and small-scale fading. Since the small-scale fading in mmWave links
is less severe than the conventional systems due to deployment of directional
antennas, all links are assumed to be subject to independent Nakagami fading.
Hence, the power of the channel fading is a Gamma r.v. with shape parameter
$\nu$ and scale parameter $1/\nu$. For the large-scale path-loss between nodes
$X$ and $Y$, we assume a bounded path-loss model which is based on the
distance $d=\|X-Y\|$, in the form $L(d)=1/\left(\epsilon+d^{a}\right)$, where
$a>2$ denotes the path-loss exponent and $\epsilon>0$. The spectrum co-
existence of radar and communication systems using non-orthogonal channels
causes the existence of SI between the output and the input antennas of the
communication and radar systems, respectively. Regarding the channel between
radar and communication antennas, we assume that the BSs employ imperfect
cancellation mechanisms [10, 13]. As such, we consider that the residual SI
channel coefficient follows a Nagakami distribution with parameters
$\left(\mu,\sigma^{2}_{\rm SI}\right)$, where $1/\sigma^{2}_{\rm SI}$
characterizes the SI cancellation capability of the BSs. Therefore, the power
gain of the residual SI channel follows a Gamma distribution [13] with mean
$\mu$ and variance $\sigma^{2}_{\rm SI}/\mu$ i.e., $g_{\rm
SI}\sim\mathcal{G}\left(\mu,\frac{\sigma^{2}_{\rm SI}}{\mu}\right)$.

###  2.2 Blockage and sectorized antenna model

A mmWave link can be either line-of-sight (LoS) or non-LoS, depending on
whether the transmitter is visible to the receiver or not, due to the
existence of blockages. In particular, a transmitter is considered LoS by a
receiver, if and only if their communication link of length $r$ is
unobstructed by blockages. We define the LoS probability function $p_{\rm
L}(r)$ as the probability that a link of length $r$ is LoS. To simplify the
mathematical derivation of the analysis, we consider the generalized LoS ball
model [30], where the LoS probability function can be approximated by a step
function. Specifically, a link of length $r$ is in LoS state with probability
$p_{\rm L}(r)=p_{\rm L}$ if $r\leq R$, otherwise $p_{\rm L}(r)=0$, where $R$
is the maximum length of an LoS link [30] and the constant $p_{\rm L}\in[0,1]$
is the average fraction of the LoS area in the ball of radius $R$. In this
paper, the interference effect from the non-LoS signals is ignored, as we
assume the dominant interference is caused by the LoS signals [30].

For modeling the antenna directionality of the BSs, we adopt an ideal cone
antenna pattern [20]. The antenna array gain is parameterized by two values:
$1)$ the main-lobe beamwidth $\phi\in[0,2\pi]$, and $2)$ the main-lobe gain
$G$ (dB). We assume that the main-lobe of the radar’s antenna for all BSs is
facing towards the direction of their associated PoI and that perfect beam
alignment is achieved between each user and its serving BS, using the
estimated angles of arrival. On the other hand, the beams of interfering links
are assumed to be randomly oriented with respect to each other. Therefore, an
interfering BS causes interference to another node only if the alignment of
their two randomly oriented antenna patterns overlap and their link is LoS; by
simple geometrical arguments, this event has probability,
$\left(\frac{\phi}{2\pi}\right)^{2}p_{\rm L}$.

###  2.3 Power allocation

We assume that all BSs that belong to the $k$-th tier, where
$k\in\\{1,\dots,K\\}$, are allocated with the same transmit power $P_{k}$,
where $P_{k}>P_{i}$ for $k<i$. Due to the overwhelming negative effect of the
SI at the radar’s receive antenna, the DL power control for the communication
system is of paramount importance. Hence, we assume that all BSs which
implement the FD-JCAS scheme, utilize distance-proportional fractional power
control for the DL communication with their associated user, which is located
over a circle of radius $d$ centered at the BS, where $d\leq R$. The power
control scheme aims at compensating the large-scale path-loss and maintaining
the average received signal power at their associated user equal to $\rho$
[13]. To accomplish this, a BS at distance $d$ from its associated user,
adapts its transmitted signal power to $\rho L(d)^{\varepsilon}$, where
$0\leq\varepsilon\leq 1$ is the power control fraction. Based on the general
power control mechanism, the transmission power allocated to a BS that belongs
in the $k$-th tier, can be expressed as $P_{k}(d)=\min\\{\rho
L(d)^{\varepsilon},\ P_{k}\\}$, where the BSs which are unable to fully invert
the path-loss, transmit with maximum power $P_{k}$. It is important to mention
here that, for the case where $\varepsilon=1$, the path-loss is completely
compensated if $P_{k}$ is sufficiently large, and if $\varepsilon=0$, no path-
loss inversion is performed and all BSs from the same tier transmit with the
same power. The considered distance-proportional DL power control technique
only requires the knowledge of the locations of the users, opposed to the
truncated channel-inversion power control scheme, that demands further
computational resources for the channel estimation. Several approaches have
been proposed for the acquisition of this knowledge, such as low-rate feedback
schemes [13] or pilot-based sub-$6$ GHz networks [36]. For the detection
system, we consider a fixed power transmission allocation scheme i.e., BSs
from the $k$-th tier transmit with power $P_{k}$. It is important to mention
that, for the sake of simplicity, both systems of an FD-JCAS BS have equal
transmit power.

###  2.4 $\beta$-GPP preliminaries

In this subsection, we provide some background on the $\beta$-GPP; readers are
referred to [31, 7, 25] for further details. The $\beta$-GPP represents a
repulsive point process where the parameter $\beta\in(0,1]$ can be used to
“interpolate” smoothly from the GPP to the PPP. More specifically, when
$\beta$ becomes larger, the points in the $\beta$-GPP experience strong
repulsion, and therefore the points tend to be more spread out. On the other
hand, when $\beta\rightarrow 0$, the correlation among the points disappears,
which corresponds to the case where the locations of the points are
independent i.e., the PPP case.

Let $\Phi=\\{X_{j}\\}_{j\in\mathbb{N}}$, be a $\beta$-GPP with density
$\lambda$ and repulsion parameter $\beta$ i.e., $\Phi\sim{\rm
GPP}(\lambda,\beta)$. Then, the distribution of the set
$\\{\|X_{j}\|^{2}\\}_{j\in\mathbb{N}}$ can be characterized by the
distribution of the set $\\{\widetilde{B}_{i}\\}_{i\in\mathbb{N}}$, which is
obtained by independently retaining from $\\{B_{i}\\}_{i\in\mathbb{N}}$ each
$B_{i}$ with probability $\beta$, where $\\{B_{i}\\}_{i\in\mathbb{N}}$ are
independent and Gamma distributed r.v. with a shape parameter $i$ and a scale
parameter $\beta/t$ i.e., $B_{i}\sim\mathcal{G}(i,\beta/t)$, where $t$ is the
scaling parameter used to control the intensity and is equal to $t=\pi\lambda$
[7]. Therefore, the term $\|x\|^{-a}$ can be represented as
$\widetilde{B}_{i}^{-\frac{a}{2}}=B_{i}^{-\frac{a}{2}}\Xi_{i}$, where
$\\{\Xi_{i}\\}_{i\in\mathbb{N}}$ represents the set of discrete r.v., such
that $\mathbb{P}\left(\Xi_{i}=1\right)=\beta$ and
$\mathbb{P}\left(\Xi_{i}=0\right)=1-\beta$. Based on the aforementioned, the
following proposition provides the probability density function (pdf) of
$B_{i}$ [25].

######  Proposition 1.

The pdf of a r.v. $B_{i}\sim\mathcal{G}(i,\beta/t)$, is given by

$f_{B_{i}}(y)=y^{i-1}\exp\left(-\frac{t}{\beta}y\right)\Bigg{/}\left(\left(\frac{\beta}{t}\right)^{i}\Gamma[i]\right),$
(1)

where $t>0$ and $i\in\mathbb{N}$.

$\mathcal{P}_{c}^{k}\\!\leq\\!p_{\rm
L}\\!\sum_{\xi=1}^{\nu}(-1)^{\xi+1}\binom{\nu}{\xi}\exp\\!\left(-s\sigma_{n}^{2}\right)\\!\prod_{\begin{subarray}{c}j=1\\\
j\neq
i\end{subarray}}^{\infty}\\!\left(1\\!-\\!\beta_{k}\\!+\\!\int\nolimits_{\sqrt{d}}^{\sqrt{{R}}}\frac{\beta_{k}f_{C_{j,k}}(y)}{1\\!+\\!\frac{sP_{k}(d)G^{2}}{\left(\epsilon+y^{\frac{a}{2}}\right)}}dy\right)\\!\prod_{\begin{subarray}{c}z=1\\\
z\neq
k\end{subarray}}^{K}\prod_{j=1}^{\infty}\\!\left(1\\!-\\!\beta_{z}\\!+\\!\int\nolimits_{0}^{\sqrt{{R}}}\frac{\beta_{z}f_{C_{j,z}}(y)}{1+\frac{sP_{z}(d)G^{2}}{\left(\epsilon+y^{\frac{a}{2}}\right)}}dy\right),$
(3)

In the context of the considered system model, we denote as
$\\{\Xi_{j,k}\\}_{j\in\mathbb{N}}$, the set of discrete r.v. for the $k$-th
tier, such that $\mathbb{P}\left(\Xi_{j,k}=1\right)=\beta_{k}$ and
$\mathbb{P}\left(\Xi_{j,k}=0\right)=1-\beta_{k}$. Also, for $\Phi_{k}\sim{\rm
GPP}(\lambda_{k},\beta_{k})$, we define a set of independent gamma r.v.
$\\{B_{j,k}\\}_{j\in\mathbb{N}}$, where
$B_{j,k}\sim\mathcal{G}\left(j,\beta_{k}/(\pi\lambda_{k})\right)$ [33]. Let
$\widetilde{\Phi}_{k}$ depicts the locations of the active interfering BSs.
Based on the considered network deployment, the point process
$\widetilde{\Phi}_{k}$ is a thinned version of the original $\beta$-GPP
$\Phi_{k}$, and have a density equal to
$\widetilde{\lambda}_{k}(r)=\lambda_{k}\left(\frac{\phi}{2\pi}\right)^{2}p_{\rm
L}\chi_{k}$. Therefore, for $\widetilde{\Phi}_{k}\sim{\rm
GPP}(\widetilde{\lambda}_{k}(r),\beta_{k})$, we define a set of independent
gamma r.v. $\\{C_{j,k}\\}_{j\in\mathbb{N}}$, where
$C_{j,k}\\!\sim\\!\mathcal{G}(j,\beta_{k}/\\!(\pi\widetilde{\lambda}_{k}(r)))$
[33].

###  2.5 Detection model

Figure 2: An example for the timeline of the operation of both radar and
communication system of a transmitter that employs the considered FD-JCAS
scheme.

An example timeline for the considered network deployment is depicted in Fig.
2, highlighting the radar and communication operations of the FD-JCAS BSs.
More specifically, the time is slotted, and a BS follows a regular pattern of
duration $T$ slots (or time units). Regarding the radar operation (Fig. 2
$(a)$), within the first $t_{0}$ time units, where $t_{0}\ll T$, we assume
that the BS at $x\in\Phi_{k}$ firstly broadcasts a narrow pilot sequence
towards its main-lobe direction. During the remaining $T-t_{0}$ time units,
the BS measures the reflected signal power received within its main-lobe
direction, due to the existence of the object at $y\in\mathbb{R}^{2}$. The
echo measured during the time slot $\tau$ at the BS, follows from the well-
known radar equation as [17, 18, 19]

$P^{(\tau)}_{\rm
refl}(r)=\\!\frac{P_{k}Gh_{0}(\tau)L(r)}{4\pi}\frac{AL(r)}{4\pi}A_{e}=\\!P_{k}G^{2}\frac{A\ell}{4\pi}h_{0}(\tau)L^{2}(r),$

where $r=\|x-y\|$, $G$ is the transmit antenna gain, $h_{0}(\tau)$ denotes the
channel power gain of the link between the BS and the object during time slot
$\tau$, $A$ is the radar cross-section area of the object, $A_{e}$ is the
effective area i.e., $A_{e}=G\frac{c^{2}}{4\pi f^{2}}$, where $c$ is the speed
of light and $f$ is the carrier frequency, and $\ell=(c/4\pi f)^{2}$. Then, by
focusing on a particular time slot, the signal-to-interference-plus-noise
ratio (SINR) of the reflected pilot sequence, is given by

$\gamma_{k}^{(\tau)}(r)=\frac{P^{(\tau)}_{\rm refl}(r)}{\mathcal{I}+I_{\rm
SI}\left(d,k\right)\mathds{1}_{\text{JCAS}}+\sigma_{n}^{2}},$ (2)

where
$\mathcal{I}=\sum\nolimits_{z=1}^{K}\sum\nolimits_{X_{j,z}\in\widetilde{\Phi}_{z}}P_{z}(d)G^{2}\ell{h}_{j,z}(\tau)L(\|X_{j,z}-x\|)$
is the total interfering power observed by the BS at $x\in\Phi_{k}$ caused by
all active BSs from the $i$-th tier i.e., $X_{j,z}\in\widetilde{\Phi}_{z}$,
$j\in\mathbb{N}$, $h_{j,z}(\tau)\sim\Gamma[\nu,1/\nu]$ denotes the channel
power gain at time slot $\tau$ between the active interfering BS at
$X_{j,z}\in\widetilde{\Phi}_{z}$ and the BS at $x\in\Phi_{k}$;
$\mathds{1}_{\text{JCAS}}$ represents the indicator function of the event “the
BS at $x\in\Phi_{k}$ implements the FD-JCAS scheme” and $\sigma_{n}^{2}$
denotes the variance of the thermal noise. Due to the non-orthogonal operation
of communication and radar systems (see Fig. $2$), the residual SI observed at
the receive antenna of the radar system after the SI cancellation is equal to
$I_{\rm SI}(d,k)=P_{k}\left(d\right)g_{\rm SI}=\min\\{\rho
L(d)^{\varepsilon},\ P_{k}\\}g_{\rm SI}$, where $g_{\rm SI}$ is the power gain
of the residual SI channel.

For the successful detection of an object we assume two conditions. Firstly,
the SINR of the reflected pilot signal should be larger than a specific
threshold. Note that this is a typical condition in the context of detection
problems [17]. Hence, by assuming that the corresponding BS is at a distance
$v$ from the typical PoI, the probability of this first condition is evaluated
as $\mathbb{P}[\gamma_{k}^{(\tau)}(v)>\theta]$. Second, there should be no
blockage between the typical PoI and its associated BS, as well as an object
must exists at the typical PoI. If the typical PoI is blocked by a blockage,
the BS cannot detect the typical PoI successfully. The probability of this
second condition is $\zeta p_{\rm L}$. Considering both conditions, the
successful detection probability of a BS that belongs in the $k$-th tier at
the time slot $\tau$, is given by $\zeta p_{\rm
L}\mathbb{P}[\gamma_{k}^{(\tau)}(v)>\vartheta]$. Conversely, for the case
where a blockage exists between the object at the typical PoI and the BS, and
the signal-to-interference ratio of the signal reflected at the blockage is
larger than the detection threshold, a false alarm (FA) is triggered. Hence,
the FA probability of a BS that belongs in the $k$-th tier and detect a
blockage at distance $u$, is given by $(1-p_{\rm L})\zeta\mathbb{P}[P_{\rm
refl}(u)/\mathcal{I}>\vartheta]$ [17]. Based on the above-mentioned
performance metrics and the Neyman-Pearson Lemma [19], the minimum threshold,
$\theta$, for the decision rule that ensures a tolerable pre-defined false
alarm rate is assessed in Section 4.

###  2.6 Communication model

Each BS implementing the FD-JCAS scheme, communicates with its associated user
in the DL direction, when its radar system is waiting for the object’s echo
(Fig. 2 $(b)$). We assume that the users are located over a circle of radius
$d$ around their associated BSs, where $d\leq R$. For the considered highly
directional mmWave network deployments, where there exists a form of repulsion
among the networks nodes, the interference caused by nearby cells, and even
more the interference caused by the reflected signals that travel large
distances, is significantly reduced [24, 26, 27, 32, 33]. Therefore, the main
terms that prevail in the aggregate network interference are caused by the
nearby BSs that perform communication operations, allowing to neglect the
interference caused by the reflected radar signals for simplicity. The
validity of the above-mentioned assumption will be shown in the numerical
results in Section 6. Let $\gamma^{\rm D}_{k}(d)$ represents the DL SINR
observed by a user at a distance $d$ from its serving BS, that is evaluated as

$\gamma^{\rm
D}_{k}(d)=\frac{{P}_{k}(d)G^{2}{g}_{0}(\tau)L(d)}{\sum\nolimits_{i=1}^{K}\sum\nolimits_{X_{j,i}\in\widetilde{\Phi}_{i}}P_{i}(d)G^{2}\ell{g}_{j,i}(\tau)L\left(\|X_{j,i}\|\right)+\sigma_{n}^{2}},$

where $j\in\mathbb{N}$, ${g}_{0}(\tau)\sim\Gamma[\nu,1/\nu]$ and
${g}_{j,i}(\tau)\sim\Gamma[\nu,1/\nu]$ is the channel power gain during time
slot $\tau$ of the user with its serving BS and the interfering BS at
$X_{j,i}\in\widetilde{\Phi}_{i}$, respectively, and ${P}_{k}(d)$ denotes the
transmission power allocated to a BS that belongs in the $k$-th tier, based on
the power control mechanism. Following Alzer’s Lemma [25] in order to
approximate the Gamma r.v. with a weighted sum of the cumulative distribution
functions of exponential r.v., a tight upper bound of the DL coverage
probability of a user i.e., $\mathbb{P}[\gamma^{\rm D}_{k}(d)>\eta]$, is given
in the following proposition.

######  Proposition 2.

The DL coverage probability of a user at distance $d$ from its serving BS at
$X_{i,k}\in\Phi_{k}$, is given by (3) at the top of this page where
$s=\eta\xi\varpi(\epsilon+d^{a})/(P_{k}(d)G^{2})$,
$\varpi=\nu(\nu!)^{-\frac{1}{\nu}}$ and $f_{C_{i,k}}(u)$ represents the pdf of
the $\beta$-GPP distributed interfering BSs, which is given in Proposition 1
with $c=\pi\widetilde{\lambda}_{k}(r)$.

The investigation of random distances between a BS and its associated user,
which is achieved by un-conditioning the above expression with the pdf
$f(d)=\frac{2}{R^{2}}d$, is an interesting but trivial extension, and thus is
out of the scope of this work. The DL coverage performance in the context of
$\beta$-GPP deployments, is a well-investigated metric within the literature
[25, 32, 33]. Therefore, the focus of this paper lies in evaluating the
achieved detection performance of the FD-JCAS systems for the considered
network deployments, where the communication operation is indicated by the
existence of interference at the radar operation of the BSs and residual SI at
the JCAS-enabled BSs.

##  3 Cooperative Multi-point Radar Detection Technique

In this section, we introduce the proposed CoMRD technique in the context of
mmWave HetNets. Our technique exploits the cooperation among randomly located
BSs that belong in different network tiers, aiming at enhancing the network’s
detection performance. Specifically, our proposed technique is based on a two-
level procedure: $(i)$ pre-selection phase, and $(ii)$ decision phase, which
are detailed described in the following discussion. In addition, analytical
expressions are derived which will be useful for evaluating the network’s
detection performance in Section 4.

###  3.1 Pre-selection Phase

We denote by $\mathcal{C}$, the set of cooperative BSs that are selected based
on the rules of the adopted association scheme and share their sensing
information to the FC for the collaborative detection of the presence or
absence of an object at a PoI, where $|\mathcal{C}|\in\mathbb{N}$.
Conventional distance-based (DB) association schemes assume the existence of a
centralized controller that selects the $K$ closest BSs from each PoI over the
entire HetNet. However, such approaches require continuous exchange of
information between the centralized controller and the BSs of the HetNet,
resulting in increased implementation and computational complexity. Motivated
by this, we consider a low-complexity per-tier DB (PTDB) association scheme
[34, 35], whose performance is a lower bound of the one achieved with the DB
association scheme; this is shown in the numerical results of Section 6.
Specifically, the PTDB scheme selects the closest BS to the typical PoI from
each network tier $k\in\\{1,\dots\,K\\}$, which is denoted as
$X^{*}_{k}\in{\Phi}_{k}$. In contrast to the DB association policy, the
adopted scheme requires a signaling only between the BSs within the same
network tier, reducing the signaling overhead and thus alleviating the
computational complexity of the association process. This pre-selection policy
requires an a priori knowledge of the location of the BSs, which can be
obtained by monitoring the location of the BSs through a low-rate feedback
channel or by a global positioning system mechanism [13]. Hence, the set
defined by the pre-selected BSs from all network tiers, represents the set of
cooperative BSs for the object detection, and is defined as

$\mathcal{C}=\left\\{X^{*}_{k}:X^{*}_{k}=\operatorname*{arg\,min}_{j\in\mathbb{N}}\|X_{j,k}\|,k\in\left\\{1,\dots,K\right\\}\right\\},$
(4)

where $X^{*}_{k}$ represents the location of the closest BS to the PoI from
the $k$-th tier. It is worth mentioning that, the adopted pre-selection policy
is ideal for scenarios with low mobility, since it does not consider
instantaneous fading.

###  3.2 Decision Phase

At the second-level phase, each cooperative BS takes an independent binary
decision regarding the presence of an object at the PoI based on the observed
SINR. Let $\mathcal{H}_{0}$ and $\mathcal{H}_{1}$ represent the hypotheses
made by each cooperative BS, when it senses the absence and the presence of an
object at the typical PoI, respectively. In particular, the cooperative BS at
$X^{*}_{k}\in\mathcal{C}$ from the $k$-th tier selects the hypothesis
$\mathcal{H}_{1}$, when the observed SINR exceeds a pre-defined threshold
$\vartheta$ i.e., $\gamma_{k}^{(\tau)}(\|X^{*}_{k}\|)\\!\geq\\!\vartheta$;
otherwise, the cooperative BS selects the hypothesis $\mathcal{H}_{0}$. Hence,
the binary decision of the cooperative BS at $X^{*}_{k}$, is defined as
$\delta_{k}=0$ if $\mathcal{H}_{0}\ {\rm holds}$, or $\delta_{k}=1$ if
$\mathcal{H}_{1}\ {\rm holds}$, where $k\\!\in\\!\left\\{1,\dots,K\right\\}$.
Thereafter, all local sensing informations from the cooperative BSs are shared
to the FC via an ideal report channel. However, the distance between every BS
and the typical PoI is different, thus, the quality of the sensing information
of every BS is also different. Hence, in our proposed scheme, for each sensing
information, a combining weight is assigned that reflects the quality of that
local sensing information. Specifically, the weight of the cooperative BS at
$X^{*}_{k}\in\mathcal{C}$, can be defined as,

$w_{k}=\mathds{1}\left(\|X^{*}_{k}\|^{-1}\Big{/}\max\limits_{X^{*}_{j}\in\mathcal{C}}\left(\|X^{*}_{j}\|^{-1}\right)>\varsigma\right),$
(5)

where $i\in\left\\{1,\dots,\eta\right\\}$ and $\varsigma$ represents a pre-
defined fraction, with $0<\varsigma\leq 1$. Thus, the final decision takes
into account only the cooperative BSs whose distances from the typical PoI are
relatively equal to that of the closest cooperative BS. The remaining
cooperative BSs are considered to have a lower sensing quality due to their
larger distances from the typical PoI, and therefore, have a zero impact on
the final decision. Such weighted decision rule requires an a priori knowledge
of the location of the BSs. This knowledge can be obtained by monitoring the
location of the BSs through a low-rate feedback channel or by a global
positioning system mechanism [37].

In order to enhance the overall detection capability of the considered network
deployment, the FC makes the final decision by combining all weighted sensing
information based on an adopted hard-decision combining rule. Specifically,
the FC makes its decision according to the number of BSs claiming the presence
of an object at their associated PoI. We adopt a generic $\kappa$-out-of-$K$
combining rule, where the FC decides that an object is present at the typical
PoI, if and only if the $\kappa$ or more cooperative BSs decide the hypothesis
$\mathcal{H}_{1}$. Hence, the final decision of the FC at the time slot
$\tau$, conditioned on the cooperative BSs’ locations, is given by

$\displaystyle\mathcal{P}_{d}^{(\tau)}(\vartheta,\kappa,\varsigma|{\bf
r}_{X^{*}})$
$\displaystyle\\!=\\!\sum\limits_{t=\kappa}^{K}\\!\binom{K}{t}\\!\prod\limits_{k=1}^{K}\\!w_{k}\mathcal{P}_{d,k}^{(\tau)}\\!(\vartheta|r_{X^{*}_{k}})^{\delta_{k}}\\!\left(\\!1\\!-\\!\mathcal{P}_{d,k}^{(\tau)}(\vartheta|r_{X^{*}_{k}})\\!\right)^{1-\delta_{k}},$
(6)

where $\kappa\in\mathbb{N}^{+}$, ${\bf
r}_{X^{*}}=\\{r_{X^{*}_{1}},\dots,r_{X^{*}_{K}}\\}$ represents the set of the
distances between the typical PoI and the cooperative BS from each tier, and
$\mathcal{P}_{d,k}^{(\tau)}(\vartheta|r_{X^{*}_{k}})$ is the conditional
detection probability achieved by the cooperative BS that belongs to the
$k$-th tier, which according to the discussion in Section 2.5, is evaluated as
$\mathcal{P}_{d,k}^{(\tau)}(\vartheta|r_{X^{*}_{k}})=\zeta p_{\rm
L}\mathbb{P}\left[\gamma_{k}^{(\tau)}(r_{X^{*}_{k}})\geq\vartheta|r_{X^{*}_{k}}\right]$
[18, 17]. Note that, $\delta_{k}$ is the detection decision of the cooperative
BS at $X^{*}_{k}\in\mathcal{C}$ for the considered combination. It is
important to mention here that, if
$\kappa=\left\lceil\frac{K}{2}\right\rceil$, the rule is referred as the
Majority rule, if $\kappa=1$, the rule is referred as the OR rule, and if
$\kappa=K$, the rule is referred as the AND rule.

In contrast to the non-cooperative techniques [18, 19], where the detection
decision is performed locally at each BS, our proposed cooperative technique
combines all local decisions conducted by the cooperative BSs at the FC,
aiming to determine the final sensing result [37]. Hence, the need for a FC,
makes our proposed technique more costly in terms of information exchange
overhead compared to the non-cooperative counterpart. Moreover, the adopted
association scheme for the pre-selection of the $K$ cooperative BSs offers
lower implementation complexity and reduced signaling overhead compared to the
conventional approaches.

##  4 Detection Performance with CoMRD technique

In this section, we analyze the impact of our proposed technique on the
detection performance of FD-JCAS-enabled mmWave HetNets under a constant false
alarm rate (CFAR) constraint, where the BSs from the $k$-th tier are deployed
based on a $\beta$-GPP. Initially, the detection threshold is evaluated aiming
at the achievement of the desired probability of FA $P_{\rm fa}$ for each tier
$k$, where $k\in\\{1,\dots,K\\}$. Then, the SINR complementary cumulative
distribution function (ccdf) is defined, and the Laplace transform of the
overall interference function is characterized. By using the proposed CoMRD
technique, the detection performance of each hard-decision rule is derived
analytically by using stochastic geometry tools.

###  4.1 Interference Characterization

Firstly, we investigate the received interference at a cooperative BS, where
analytical and asymptotic expressions for the Laplace transform of the
received interference are derived. The aggregate interference at a cooperative
BS located at $x\in\Phi_{k}$, can be expressed as follows

$\mathcal{I}=\sum\nolimits_{z=1}^{K}\sum\nolimits_{X_{j,z}\in\widetilde{\Phi}_{z}}P_{i}(d)G^{2}\ell{h}_{j,z}(\tau)L(\|X_{j,z}-x\|),$

where $j\in\mathbb{N}$. Due to the non-orthogonal operation of communication
and radar systems, the cooperative BS at $x\in\Phi_{k}$ may also observe a
residual SI, where after the SI cancellation, is equal to

$I_{\rm SI}(d,k)=P_{k}\left(d\right)g_{\rm SI}=\min\\{\rho
L(d)^{\varepsilon},\ P_{k}\\}g_{\rm SI},$

where $g_{\rm SI}$ represents the power gain of the residual SI channel. To
characterize the interference, in the following Lemma, we compute the Laplace
transform of the r.v. $\mathcal{I}$ and $I_{\rm SI}$ evaluated at $s$.

######  Lemma 1.

The Laplace transform of the aggregate interference during time slot $\tau$,
$\mathcal{L}^{(\tau)}_{\mathcal{I}}\left(s\right)$, is given by

$\mathcal{L}^{(\tau)}_{\mathcal{I}}\left(s\right)\\!=\\!\prod_{z=1}^{K}\\!\prod\limits_{i\in\mathbb{N}\backslash
j}\\!\left(\\!1\\!-\\!\beta\\!+\\!\int_{0}^{\sqrt{{R}}}\frac{\beta
f_{C_{i,z}}(y)}{1+\frac{sP_{z}(d)G^{2}\ell}{\left(\epsilon+y^{\frac{a}{2}}\right)}}dy\right),$
(7)

and the Laplace transform of the SI, $\mathcal{L}_{I_{\rm SI}}\left(s\right)$,
can be expressed as

$\mathcal{L}_{I_{\rm SI}}\left(s\right)=\left(\frac{\mu}{\mu+s\sigma^{2}_{\rm
SI}\min\\{\rho\left(\epsilon+d^{a}\right)^{\varepsilon},P_{k}\\}}\right)^{\mu},$
(8)

where $f_{C_{i,k}}(u)$ represents the pdf of the $\beta$-GPP distributed
interfering BSs, which is given in Proposition 1 with
$c=\pi\widetilde{\lambda}_{k}$.

###### Proof.

Firstly, we compute the Laplace transform of r.v. $\mathcal{I}$ at $s$ during
time slot $\tau$, conditioned on the location of the serving BS $X_{i,k}$,
which we denote as $\mathcal{L}^{(\tau)}_{\mathcal{I}}\left(s\right)$. The
Laplace transform definition yields

$\displaystyle\mathcal{L}^{(\tau)}_{\mathcal{I}}\left(s\right)=\mathbb{E}\left[e^{-s\left(\sum\limits_{z=1}^{K}\sum\limits_{i\in\mathbb{N}\backslash
j}P_{z}(d)G^{2}\ell
h_{i,z}(\tau)L\left({C}_{i,z}^{\frac{1}{2}}\right)\Xi_{i,z}\right)}\right]$
$\displaystyle=\\!\prod\limits_{z=1}^{K}\\!\prod\limits_{i\in\mathbb{N}\backslash
j}\\!\\!\left(\\!\beta_{z}\mathbb{E}\left[e^{-sP_{z}(d)G^{2}\ell
h_{i,z}(\tau)L\left({C}_{i,z}^{\frac{1}{2}}\right)}\right]\\!+\\!1\\!-\\!\beta_{z}\\!\right)$
(9)
$\displaystyle=\\!\prod\limits_{z=1}^{K}\\!\prod\limits_{i\in\mathbb{N}\backslash
j}\\!\left(\\!1\\!-\\!\beta_{z}\\!+\\!\mathbb{E}\left[\frac{\beta_{z}}{1\\!+\\!sP_{z}(d)G^{2}\ell
L\left({C}_{i,z}^{\frac{1}{2}}\right)}\right]\right)$ (10)

where (9) is derived based on the $\beta$-GPP properties, and (10) is due to
the moment generating function of an exponential r.v.. By using the distance
distribution of $\beta$-GPP deployed BSs, which is calculated in Proposition
1, (7) can be derived. Regarding the expressions for the Laplace transform of
the SI function, it can be expressed as

$\displaystyle\mathcal{L}_{I_{\rm
SI}}\left(s\right)=\mathbb{E}\left[\exp\left(-s\min\\{\rho\left(\epsilon+d^{a}\right)^{\varepsilon},P_{k}\\}g_{\rm
SI}\right)\right],$ (11)

where $g_{\rm SI}$ is the residual SI channel that follows Gamma distribution
i.e., $g_{\rm SI}\sim\mathcal{G}\left(\mu,\frac{\sigma_{\rm
SI}^{2}}{\mu}\right)$. By averaging over the channel fading, the final
expression for the Laplace transform of the SI can be derived ∎

We end this section by computing the Laplace transform of the aggregate
interference $\mathcal{I}$ in the case of PPP (which is obtained as the limit
as $\beta_{k}\rightarrow 0$ in the Lemma 1).

######  Corollary 1.

Let $\widetilde{\Phi}_{k}\sim$PPP($\widetilde{\lambda}_{k}(r)$), the Laplace
transform of $\mathcal{I}$ during time slot $\tau$,
$\widetilde{\mathcal{L}}^{(\tau)}_{\mathcal{I}}\left(s\right)$, is given by

$\displaystyle\\!\\!\widetilde{\mathcal{L}}^{(\tau)}_{\mathcal{I}}\left(s\right)$
$\displaystyle\\!=\\!\prod_{z=1}^{K}\\!e^{-\lambda_{z}p_{\rm
L}\frac{\phi}{2}{R}^{2}\\!\left(1-\frac{\epsilon}{Q(s,z)}\right){}_{2}F_{1}\left[1,\frac{2}{a},\frac{2+a}{a},-\frac{{R}^{a}}{Q(s,z)}\right]},$
(12)

where $Q(s,i)=G^{2}\ell P_{i}s+\epsilon$.

###  4.2 Detection Threshold Under False Alarm Constraint

The detection threshold is tuned according to the principle of the CFAR [19].
Specifically, the threshold is designed (a priori) in such a way as to achieve
a desired false alarm rate ($P_{\rm fa}$) over all network realizations.
Moreover, this design takes into account the topology and the statistical
properties of the network (e.g., interference, network density, etc).
Therefore, even though this threshold is constant, it is evaluated based on a
contextual-aware design.

In order to derive compact and insightful expressions for the FA probability,
and hence, the detection performance, the detection threshold is evaluated for
the case of PPP-distributed BSs, i.e. $\beta_{k}\rightarrow 0,\ \forall
k\in\\{0,\dots,K\\}$, with omnidirectional antenna gains, i.e. $\phi=2\pi$,
and $a=4$. These assumptions provide an approximation of the actual network’s
detection performance, which is shown to be tight from numerical results in
Section 6. Hence, we assume that the required FA rate for the $k$-th tier, is
equal to $P_{\rm fa}=(1-p_{\rm
L})\zeta\mathbb{P}[\gamma_{k}^{(\tau)}(u)>\vartheta]$, where
$\gamma_{k}^{(\tau)}(u)$ is the signal reflected at a blockage at distance $u$
from the cooperative BS that belongs in the $k$-th tier, and is given by (2).
In the following lemma, the $P_{\rm fa}$ is analytically evaluated.

######  Lemma 2.

The FA probability for the $k$-th tier of PPP-distributed omnidirectional
HetNets with $a=4$, is given by

$P_{\rm fa}=(1-p_{\rm L})\zeta\Big{/}\left(1+p_{\rm
L}\sqrt{\pi\vartheta}\frac{\lambda_{k}}{\lambda_{b}}\right).$ (13)

###### Proof.

Based on the discussion in Section 2.5, the FA probability for the $k$-th
tier, can be re-written as

$\displaystyle P_{\rm fa}=(1-p_{\rm
L})\zeta\int_{0}^{R}\mathbb{P}\left[\frac{P_{\rm
refl}^{(\tau)}(u)}{\mathcal{I}}>\vartheta\right]f_{b}(u)\mathop{}\\!du$
$\displaystyle=\\!(1\\!-\\!p_{\rm L})\zeta\int_{0}^{R}e^{-\pi\lambda_{k}p_{\rm
L}R\sqrt{P_{k}s\ell}\arctan\left[\frac{R}{\sqrt{P_{k}s\ell}}\right]}f_{b}(u)\mathop{}\\!du,$
(14)

where $f_{b}(u)$ is the probability density function (pdf) of the distance
between a cooperative BS from the $k$-th tier and its closest blockage, i.e.
$f_{b}(u)=2\pi\lambda_{b}u\exp(-\pi\lambda_{b}u^{2})$ [18, 30], and (14) is
derived by using the expression in Corollary 1 under the considered
assumptions. Finally, by assuming $R\rightarrow\infty$, the final expression
can be obtained. ∎

Leveraging the expressions derived in Lemma 2, the detection threshold needed
for achieving the desired FA rate is analytically computed in the following
lemma.

######  Lemma 3.

The detection threshold $\theta_{k}$ for the $k$-th tier, which ensures a FA
rate $P_{\rm fa}$, is equal to

$\theta_{k}=\frac{A\lambda_{b}^{2}}{\lambda_{k}^{2}p_{\rm
L}\pi^{3}R^{2}}\left(1-\frac{(1-p_{\rm L})\zeta}{P_{\rm fa}}\right)^{2}.$ (15)

###### Proof.

The expression is derived by solving the expression derived in Lemma 2 with
respect to the detection threshold. ∎

The simple closed-form expression in (15) provides guidance on how to tune the
radar receiver for network tier, pinpointing the impact of all relevant system
parameters.

###  4.3 Detection Performance under CFAR Constraint

Based on Section 2.4 and by following Slivnyak’s theorem, the effective SINR
of the detection process for the cooperative BS $X^{*}_{k}\in\Phi_{k}$, can be
expressed as

$\displaystyle\gamma_{k}^{(\tau)}\\!(\widetilde{B}_{j,k}^{\frac{1}{2}})\\!=\\!\frac{P_{k}G\frac{A\ell}{4\pi}h_{0}(\tau)L^{2}\left(\widetilde{B}_{j,k}^{\frac{1}{2}}\right)}{\sum\limits_{z=1}^{K}\sum\limits_{i\in\mathbb{N}\backslash
j}\\!\\!\frac{P_{z}(d)G^{2}\ell{h}_{i,z}(\tau)}{L\left(\widetilde{C}_{i,z}^{\frac{1}{2}}\right)^{-1}}\\!+\\!I_{\rm
SI}\\!\left(d,k\right)\mathds{1}_{\text{JCAS}}\\!+\\!\sigma_{n}^{2}},$
$\displaystyle\quad\\!=\\!\frac{P_{k}G\frac{A\ell}{4\pi}h_{0}(\tau)L^{2}\left({B}_{j,k}^{\frac{1}{2}}\right)\Xi_{j,k}}{\sum\limits_{z=1}^{K}\sum\limits_{i\in\mathbb{N}\backslash
j}\\!\\!\frac{P_{z}(d)G^{2}\ell{h}_{i,z}(\tau)\Xi_{i,z}}{L\left({C}_{i,z}^{\frac{1}{2}}\right)^{-1}}\\!+\\!I_{\rm
SI}\\!\left(d,k\right)\mathds{1}_{\text{JCAS}}\\!+\\!\sigma_{n}^{2}},$ (16)

where $\Xi_{j,k}$ denotes the discrete r.v. of the $j$-th BS from the $k$-th
tier, i.e $\Xi_{j,k}=\\{0,1\\}$. The following lemma provides the conditional
detection probability under CFAR constraint in the context of $\beta$-GPP-
distributed mmWave HetNets.

$\displaystyle\mathcal{P}_{d,k}^{(\tau)}(\theta_{k}|B_{j,k})$
$\displaystyle=\chi_{k}\zeta p_{\rm L}\mathbb{P}\left[h_{0}(\tau)\geq
s\left(\sum\limits_{z=1}^{K}\sum\limits_{i\in\mathbb{N}\backslash
j}P_{z}(d)G^{2}\ell{h}_{i,z}(\tau)L\left({C}_{i,z}^{\frac{1}{2}}\right)+I_{\rm
SI}\left(d,k\right)+\sigma_{n}^{2}\right)|\Xi_{j,k}=1\right]\mathbb{P}[\Xi_{j,k}=1]$
$\displaystyle\qquad\qquad\qquad\ +(1\\!-\\!\chi_{k})\zeta p_{\rm
L}\mathbb{P}\\!\left[\\!h_{0}(\tau)\\!\geq\\!s\\!\left(\\!\sum\limits_{z=1}^{K}\\!\sum\limits_{i\in\mathbb{N}\backslash
j}\\!P_{z}(d)G^{2}\ell{h}_{i,z}(\tau)L\\!\left({C}_{i,z}^{\frac{1}{2}}\right)\\!+\\!\sigma_{n}^{2}\right)\\!|\Xi_{j,k}\\!=\\!1\right]\\!\mathbb{P}[\Xi_{j,k}\\!=\\!1],$
(18)

$\displaystyle\mathcal{P}_{d,k}^{(\tau)}(\theta_{k}|B_{j,k})$
$\displaystyle\leq\beta_{k}\zeta p_{\rm
L}\sum_{\xi=1}^{\nu}(-1)^{\xi+1}\binom{\nu}{\xi}\Bigg{(}\chi_{k}\mathbb{E}\left[\exp\left(-s\xi\varpi\left(\sum\limits_{z=1}^{K}\sum\limits_{i\in\mathbb{N}\backslash
j}P_{z}(d)G^{2}\ell{h}_{i,z}(\tau)L\left({C}_{i,z}^{\frac{1}{2}}\right)+I_{\rm
SI}\left(d,k\right)+\sigma_{n}^{2}\right)\right)\right]$
$\displaystyle\qquad\qquad\qquad\qquad\qquad+(1-\chi_{k})\mathbb{E}\left[\exp\left(-s\xi\varpi\left(\sum\limits_{z=1}^{K}\sum\limits_{i\in\mathbb{N}\backslash
j}P_{z}(d)G^{2}\ell{h}_{i,z}(\tau)L\left({C}_{i,z}^{\frac{1}{2}}\right)+\sigma_{n}^{2}\right)\right)\right]\Bigg{)},$
(19)

######  Lemma 4.

The conditional detection probability achieved by the cooperative BS at
$X^{*}_{k}\in{\Phi}_{k}$ at the time slot $\tau$, where the BSs are
distributed according to a $\beta$-GPP, is given by

$\displaystyle\mathcal{P}_{d,k}^{(\tau)}(\theta_{k}|B_{j,k})$
$\displaystyle\leq\beta_{k}\sum_{\xi=1}^{\nu}(-1)^{\xi+1}\binom{\nu}{\xi}\exp\left(-s\xi\varpi\sigma_{n}^{2}\right)$
$\displaystyle\times\mathcal{L}_{\mathcal{I}}^{(\tau)}(s\xi\varpi)\left(\chi_{k}\mathcal{L}_{I_{\rm
SI}}\left(s\xi\varpi\right)+1-\chi_{k}\right),$ (17)

where
$s=\frac{4\pi\theta_{k}(\epsilon+B_{j,k}^{\frac{a}{2}})^{2}}{P_{k}GA\ell}$,
$\varpi=\nu(\nu!)^{-\frac{1}{\nu}}$, $f_{C_{i,k}}(u)$ represents the pdf of
the $\beta$-GPP distributed interfering BSs, which is given in Proposition 1,
and $\mathcal{L}_{\mathcal{I}}^{(\tau)}\left(\alpha\right)$ and
$\mathcal{L}_{I_{\rm SI}}\left(\alpha\right)$ represent the Laplace transform
of the overall interference function and the SI, respectively, and are given
in Lemma 1

###### Proof.

Conditioned on whether the considered cooperative BS $X^{*}_{k}$ performs the
FD-JCAS scheme or not, its achieved detection performance, conditioned on its
location, is given by (18) at the top of next page, where
$s=\frac{4\pi\theta_{k}(\epsilon+B_{j,k}^{\frac{1}{2}})^{\frac{a}{2}}}{P_{k}G^{2}A\ell}$
and $h_{0}(\tau)$ is a gamma r.v. i.e., $h_{0}(\tau)\sim\Gamma[\nu,1/\nu]$. To
overcome the difficulty on Nakagami fading, Alzer’s Lemma [25] on the ccdf of
a gamma r.v. with integer parameter can be applied. This relates the ccdf of a
gamma r.v. into a weighted sum of the ccdfs of exponential r.v.. Hence, we can
bound expression (18) as in (19) at the top of next page, where
$\varpi=\nu(\nu!)^{-\frac{1}{\nu}}$,
$\mathcal{L}_{\mathcal{I}}^{(\tau)}(\alpha)$, and $\mathcal{L}_{I_{\rm
SI}}(\alpha)$ are the Laplace transforms of the overall interference function
and SI, respectively, that are derived in Lemma 1. ∎

It is noteworthy to mention that the Laplace transform of the overall
interference function in (7) is a decreasing function of $\beta$. This can be
explained by the fact that, the distance between the cooperative BSs and their
interfering BSs becomes larger as $\beta\rightarrow 1$. Specifically, when
$\beta$ becomes larger, the BSs exhibit more repulsion and tend to be
distributed more uniformly. This leads to an increase of the distance between
the cooperative BSs and their interfering BSs, resulting in a decreased
overall interference. We now state our main result for the detection
probability achieved by our proposed CoMRD technique in the context of
$\beta$-GPP mmWave HetNets.

######  Theorem 1.

The detection performance achieved by the CoMRD technique at the time slot
$\tau$, is given by

$\displaystyle\mathcal{P}_{d}^{(\tau)}(\boldsymbol{\theta},\kappa,\varsigma)$
$\displaystyle\leq\sum\limits_{t=\kappa}^{K}\binom{K}{t}\prod\limits_{k=1}^{K}\sum\limits_{j\in\mathbb{N}}w_{k}\int_{0}^{\sqrt{{R}}}\mathcal{P}_{d,k}^{(\tau)}(\theta_{k}|u)^{\delta_{k}}$
$\displaystyle\quad\times\left(1-\mathcal{P}_{d,k}^{(\tau)}(\theta_{k}|u)\right)^{1-\delta_{k}}f_{B_{j,k}}(u)du,$
(20)

where $\boldsymbol{\theta}=\\{\theta_{1},\dots,\theta_{K}\\}$,
$s=\frac{4\pi\theta_{k}(\epsilon+B_{j,k}^{\frac{a}{2}})^{2}}{P_{k}G^{2}A\ell}$,
$\mathcal{P}_{d,k}^{(\tau)}(\theta_{k}|u)$ represents the achieved detection
performance of the cooperative BS $X_{j,k}\in{\Phi}_{k}$ at the time slot
$\tau$, and $f_{B_{j,k}}(u)$ represents the pdf of the $\beta$-GPP BSs, which
is given in Proposition 1.

###### Proof.

By substituting the expression of Lemma 4 in (3.2), and by un-conditioning the
derived expression with the pdf given by (1), the result in Theorem 1 follows.
∎

To illustrate the effect of repulsion between the BSs on the achieved
detection performance, we also evaluate our proposed technique in the context
of PPP mmWave HetNets. As we previously mentioned, the independent deployment
of BSs i.e., the BSs are spatially distributed based on a PPP, is a special
case of the considered $\beta$-GPP deployment, where $\beta\rightarrow 0$. In
the following corollary, we state the achieved detection performance of our
proposed technique in the context of PPP mmWave HetNets.

######  Corollary 2.

When $\beta\rightarrow 0$, the detection performance achieved with the CoMRD
technique at the time slot $\tau$, is given by

$\displaystyle\mathcal{P}_{d}^{(\tau)}(\theta_{k},\kappa,\varsigma)\\!\leq\\!\sum\limits_{t=\kappa}^{K}\binom{K}{t}\prod\limits_{k=1}^{K}w_{k}\int_{0}^{{R}}\widetilde{\mathcal{P}}_{d,k}^{(\tau)}(\theta_{k}|u)^{\delta_{k}}$
$\displaystyle\qquad\times\\!(1\\!-\\!\widetilde{\mathcal{P}}_{d,k}^{(\tau)}(\theta_{k}|u))^{1-\delta_{k}}2\pi\lambda_{k}u\zeta
p_{\rm L}e^{-\pi\lambda_{k}\zeta p_{\rm L}u^{2}}du,$ (21)

where $\widetilde{\mathcal{P}}_{d,k}^{(\tau)}(\theta_{k}|u)$ represents the
conditional detection probability, and is given by

$\displaystyle\widetilde{\mathcal{P}}_{d,k}^{(\tau)}(\theta_{k}|u)\leq\sum_{\xi=1}^{\nu}(-1)^{\xi+1}\nu$
$\displaystyle(\nu!)^{-\frac{1}{\nu}}\widetilde{\mathcal{L}}_{\mathcal{I}}^{(\tau)}(s\xi\varpi)\exp\left(-s\xi\varpi\sigma^{2}_{n}\right)$
$\displaystyle\times\left(\chi_{k}\mathcal{L}_{I_{\rm
SI}}\left(s\xi\varpi\right)+1-\chi_{k}\right),$

where $\widetilde{\mathcal{L}}_{\mathcal{I}}^{(\tau)}(\alpha)$ and
$\mathcal{L}_{I_{\rm SI}}(\alpha)$ are given by (12) and (8), respectively,
and $s=\frac{4\pi\theta_{k}(\epsilon+u^{a})^{2}}{P_{k}G^{2}A\ell}$.

##  5 Temporal Correlation of Radar Detection Performance

In this section, we assess the effect of the temporal interference correlation
on the detection performance achieved by our proposed technique. Even though
we assume that the channel fading between different time slots is independent,
the interference caused at a certain location is correlated across the time
for the same network realization due to the fixed locations of the BSs.
Specifically, by assuming a static PoI, a fraction of the interfering BSs at a
given time slot might also cause interference at the particular PoI in future
time slots, which introduces a temporal interference correlation that needs to
be taken into account. Hence, due to the temporal correlation of interference,
if an object is successfully detected by its cooperative BSs in a given time
slot, there is a higher probability that the object will also be detected in
the future time slots [28].

$\displaystyle\mathcal{P}_{d}^{(\tau,\widehat{\tau})}(\theta_{k},\kappa,\varsigma|B_{j,k})$
$\displaystyle\stackrel{{\scriptstyle(a)}}{{=}}\beta_{k}\mathbb{P}\left[h_{0}(\tau)\geq
s\left(\mathcal{I}(\tau)+I_{\rm
SI}(R,k)\mathds{1}_{\text{JCAS}}+\sigma_{n}^{2}\right),h_{0}({\widehat{\tau}})\geq
s\left(\mathcal{I}(\widehat{\tau})+I_{\rm
SI}(R,k)\mathds{1}_{\text{JCAS}}+\sigma_{n}^{2}\right)\right],$ (23)
$\displaystyle\stackrel{{\scriptstyle(b)}}{{\leq}}\\!\beta_{k}\\!\sum_{\xi=1}^{K}(-1)^{\xi+1}\\!\binom{\nu}{\xi}\\!\exp\left(-2s\xi\varpi\sigma_{n}^{2}\right)\\!\left(\chi_{k}\big{(}\mathcal{L}_{I_{\rm
SI}}(s\xi\varpi)\big{)}^{2}\\!+\\!1\\!-\\!\chi_{k}\right)\\!\mathbb{P}\left[h_{0}(\tau)\geq
s\mathcal{I}(\tau),h_{0}(\widehat{\tau})\geq
s\xi\varpi\mathcal{I}(\widehat{\tau})\right]$
$\displaystyle=\beta_{k}\exp\left(-2s\xi\varpi\sigma_{n}^{2}\right)\left(\chi_{k}\big{(}\mathcal{L}_{I_{\rm
SI}}(s\xi\varpi)\big{)}^{2}+1-\chi_{k}\right)\mathcal{L}_{\mathcal{I}}^{(\tau,\widehat{\tau})}(s\xi\varpi).$

Consider a static object scenario, where the object at the typical PoI remains
static at the origin for multiple time slots. For this scenario, we study the
ability of our proposed technique to successfully detect the object at the
typical PoI over different time slots. Let
$\mathcal{P}_{d}^{(\tau,\widehat{\tau})}(\theta_{k},\kappa,\varsigma|X^{*}_{k})$
denote the probability of the cooperative BS $X^{*}_{k}\in{\Phi}_{k}$ to
jointly detect its associated PoI at both time slots $\tau$ and
$\widehat{\tau}$, which can be expressed as

$\mathcal{P}_{d}^{(\tau,\widehat{\tau})}\\!(\theta_{k},\kappa,\varsigma|X^{*}_{k})\\!=\\!\mathbb{P}\\!\left[\gamma_{k}^{(\tau)}(r_{X^{*}_{k}})\\!\geq\\!\theta_{k},\gamma_{k}^{(\widehat{\tau})}(r_{X^{*}_{k}})\\!\geq\\!\theta_{k}\right],$
(22)

where $\tau\\!\neq\\!\widehat{\tau}$. In the context of $\beta$-GPP mmWave
HetNets, this probability can be expressed as in (23) at the top of next page,
where
$s=4\pi\theta_{k}\left(\epsilon+B_{j,k}^{\frac{a}{2}}\right)^{2}/(P_{k}G^{2}A\ell)$,
$\varpi=\nu(\nu!)^{-\frac{1}{\nu}}$,$\mathcal{I}(t)=\sum\nolimits_{z=1}^{K}\sum\nolimits_{i\in\mathbb{N}\backslash
j}P_{z}(d)G^{2}\ell{h}_{i,z}(t)L\left(\|C_{i,z}^{\frac{1}{2}}\|\right)$ and
$\mathcal{L}_{\mathcal{I}}^{(\tau,\widehat{\tau})}(\alpha)$ represents the
joint Laplace transform of the interference functions at the time slots $\tau$
and $\widehat{\tau}$, where $\tau\neq\widehat{\tau}$, and is equal to
$\mathcal{L}_{\mathcal{I}}^{(\tau,\widehat{\tau})}(\alpha)=\mathbb{P}[h_{0}(\tau)>\alpha\mathcal{I}(\tau),h_{0}(\widehat{\tau})>\alpha\mathcal{I}(\widehat{\tau})]$;
$(a)$ is derived by substituting (2) for different time instances $\tau$ and
$\widehat{\tau}$ and due to the independence between $h_{0}(\tau)$ and
$h_{0}(\widehat{\tau})$, and $(b)$ is based on Alzer’s lemma. In the following
lemma, the expression for the joint Laplace transform of the interference
functions at the time slots $\tau$ and $\widehat{\tau}$ is evaluated.

######  Lemma 5.

The joint Laplace transform of the interference functions at the time slots
$\tau$ and $\widehat{\tau}$, is given by

$\mathcal{L}_{\mathcal{I}}^{(\tau,\widehat{\tau})}(s)\\!=\\!\int\limits_{0}^{\sqrt{{R}}}\\!\prod\limits_{z=1}^{K}\\!\prod\limits_{i\in\mathbb{N}\backslash
j}\\!\\!\left(\\!1\\!-\\!\beta\\!+\\!\frac{\beta}{1\\!+\\!\frac{\alpha
P_{z}(d)G^{2}\ell}{\epsilon+y^{\frac{a}{2}}}}\\!\right)^{2}\\!\\!f_{C_{i,z}}\\!(y)dy,$
(24)

where $f_{C_{i,z}}(u)$ is the pdf of the $\beta$-GPP distributed interfering
BSs, which is given in Proposition 1 with $c=\pi\widetilde{\lambda}_{z}$.

###### Proof.

Initially, since $h_{0}(\tau)$ and $h_{0}({\widehat{\tau}})$ are independent
for $\tau\neq\widehat{\tau}$, the joint Laplace transform of the interference
functions at the time slots $\tau$ and $\widehat{\tau}$, is given by

$\displaystyle\mathcal{L}_{\mathcal{I}}^{(\tau,\widehat{\tau})}(\alpha)=\mathbb{E}\\!\left[\\!e^{\\!-\alpha\\!\sum\limits_{z=1}^{K}\sum\limits_{i\in\mathbb{N}\backslash
j}\\!P_{z}(d)G^{2}\ell
h_{i,z}(\tau)L\\!\left(\\!{B}_{i,z}^{\frac{1}{2}}\\!\right)\Xi_{i,z}}\right.$
$\displaystyle\quad\quad\quad\quad\times\left.e^{-\alpha\sum\limits_{z=1}^{K}\sum\limits_{i\in\mathbb{N}\backslash
j}\\!P_{z}(d)G^{2}\ell
g_{i,z}(\widehat{\tau})L\\!\left(\\!{B}_{i,z}^{\frac{1}{2}}\\!\right)\Xi_{i,z}}\right]$
$\displaystyle=\mathbb{E}\\!\left[\\!\prod\limits_{z=1}^{K}\prod\limits_{i\in\mathbb{N}\backslash
j}\\!\left(\\!1\\!-\\!\beta_{z}\\!+\\!\frac{\beta_{z}}{1\\!+\\!\alpha
P_{z}(d)G^{2}\ell L\left({B}_{i,z}^{\frac{1}{2}}\right)}\right)^{2}\right]$
(25)

where (25) follows from the moment generating functional of the exponential
r.v. $h_{0}(\tau)$ and $h_{0}(\widehat{\tau})$, and by un-conditioning the
expression with the pdf $f_{C_{i,z}}(y)$ (Proposition 4), the final expression
can be derived. ∎

By substituting (24) in (22), and the resulting expression in (3.2), and by
un-conditioning the derived expression with the pdf given by (1), the joint
detection probability for the time slots $\tau$ and $\widehat{\tau}$ can be
expressed as

$\displaystyle\mathcal{P}_{d}^{(\tau,\widehat{\tau})}$
$\displaystyle(\theta_{k},\kappa\varsigma)\\!\leq\\!\sum\limits_{t=\kappa}^{K}\\!\binom{K}{t}\\!\prod\limits_{k=1}^{K}\\!\left(\sum\limits_{j\in\mathbb{N}}w_{k}\int\limits_{0}^{\sqrt{{R}}}\\!\mathcal{P}_{d,k}^{(\tau,\widehat{\tau})}(\theta_{k}\kappa,\varsigma|u)^{\delta_{k}}\right.$
$\displaystyle\quad\ \
\times\left.\left(1\\!-\\!\mathcal{P}_{d,k}^{(\tau,\widehat{\tau})}(\theta_{k},\kappa,\varsigma|u)\right)^{1-\delta_{k}}\\!f_{B_{j,k}}(u)du\\!\right),$
(26)

where $f_{B_{j,k}}(u)$ denotes the distance distribution between the
cooperative BS $X^{*}_{k}$ and the typical PoI. The following theorem
characterizes the detection probability at the time slot $\widehat{\tau}$,
conditioned on the detection probability at the time slot $\tau$, in the
context of our proposed technique and in a $\beta$-GPP deployment.

######  Theorem 2.

The achieved detection probability with the CoMRD technique at the time slot
$\widehat{\tau}$, conditioned on the detection probability at the time slot
$\tau$, is given by

$\mathcal{P}_{d}^{(\widehat{\tau}|\tau)}(\theta_{k},\kappa,\varsigma)=\frac{\mathcal{P}_{d}^{(\tau,\widehat{\tau})}(\theta_{k},\kappa,\varsigma)}{\mathcal{P}_{d}^{(\tau)}(\theta_{k},\kappa,\varsigma)},$
(27)

where $\mathcal{P}_{d}^{(\tau)}(\theta_{k},\kappa,\varsigma)$ and
$\mathcal{P}_{d}^{(\tau,\widehat{\tau})}(\theta_{k},\kappa,\varsigma)$ are
given by (20) and (26), respectively.

In order to illustrate the effect of temporal correlation on the detection
performance, we introduce the ratio of the conditional and the unconditional
detection probability, that is given by

$\displaystyle\varrho$
$\displaystyle=\frac{\mathcal{P}_{d}^{(\widehat{\tau}|\tau)}(\theta_{k},\kappa,\varsigma)}{\mathcal{P}_{d}^{(\tau)}(\theta_{k},\kappa,\varsigma)}=\frac{\mathcal{P}_{d}^{(\tau,\widehat{\tau})}(\theta_{k},\kappa,\varsigma)}{\mathcal{P}_{d}^{(\tau)}(\theta_{k},\kappa,\varsigma)^{2}}>1.$
(28)

######  Remark 1.

From (28), if the detection process succeeds at the time slot $\tau$, there is
a higher probability that the detection process succeeds at future time slots
$\widehat{\tau}$.

######  Remark 2.

From (28), we can easily observe that
$\mathcal{P}_{d}^{(\widehat{\tau}|\tau)}(\theta_{k},\kappa,\varsigma)>\mathcal{P}_{d}^{(\tau)}(\theta_{k},\kappa,\varsigma)$.
Particularly, a failure in the object detection by the CoMRD technique at the
time slot $\tau$, results in a more likely detection failure at future time
slots $\widehat{\tau}$.

##  6 Numerical Results

In this section, we plot the analytical results derived in the previous
sections along with the simulation results obtained from Monte-Carlo trials.
We focus on the special case of a HetNet with $K=3$ tiers, where their spatial
densities are $\lambda_{1}=1\ {\rm BSs/km^{2}}$, $\lambda_{2}=2\ {\rm
BSs/km^{2}}$, and $\lambda_{3}=4\ {\rm BSs/km^{2}}$, respectively, with
transmit power equal to $P_{1}=15$ dBm, $P_{2}=10$ dBm, and $P_{3}=5$ dBm,
respectively [30]. Regarding the repulsive behavior between the BSs, we assume
that the repulsive parameter of all tiers is equal to $\beta=0.9$ i.e.,
$\beta_{k}=\beta=0.9\ \forall k$. For the large-scale path-loss, we assume
$a=4$ and $\epsilon=1$, while the Nakagami fading parameter is equal to
$\nu=2$. The power control factor is $\varepsilon=0.9$ and all BSs have the
same receive sensitivity $\rho=-40$ dB [12]. The SI capabilities of BSs are
set to $\sigma^{2}_{\rm SI}=-60$ dB and $\mu=4$ [12]. The parameters for the
sectorized antenna model are set to $G=10$ dB and $\phi=\frac{\pi}{6}$ for the
main lobe gain and the main lobe beamwidth, respectively. In terms of modeling
the blockage sensitivity of mmWave signals, we assume that within a disk of
radius ${R}=400$ m around the typical target, a fraction equal to $p_{\rm
L}=0.7$ of BSs is in LoS. Unless otherwise specified, the fraction of JCAS-
capable BSs is set to $\chi_{k}=0.8,\ \forall k\in\\{1,\dots,K\\}$. Finally,
we set $A=10$ dB, $f=30$ GHz, and $\sigma_{n}^{2}=-60$ dB [20]. It is
important to note that the selection of these parameters is for the purpose of
presenting the achieved performance of our proposed framework. Using different
values will lead to a shifted network performance, but with the same
conclusions and remarks.

Figure 3: Detection probability versus $\vartheta$ for the considered
combining rules, where $\beta=\\{0.3,0.6,0.9\\}$. Figure 4: Detection
probability versus $\varsigma$ for the considered combining rules under CFAR
rate, where $P_{\rm fa}=\\{0.3,0.5\\}$.

Fig. 3 illustrates the detection performance achieved with our proposed
technique for all three hard-decision combining rules and for different
repulsion parameter values. An important observation from this figure is that
the OR combining rule achieves a significantly higher network detection
performance, outperforming the other two. This observation is based on the
fact that, for the scenario where the FC adopts the OR rule, the object
detection is achieved when at least one of the $K$ cooperative BSs
successfully senses the existence of the particular object. On the other hand,
by applying the Majority or the AND rule, a decreased detection performance is
observed since the simultaneous successful detection of the object by the
majority or all cooperative BSs, respectively, is required. Furthermore, Fig.
3 demonstrates the impact of the repulsion parameter $\beta$ on the detection
performance achieved with our proposed technique. We can easily observe that,
the detection performance
$\mathcal{P}_{d}^{(\tau)}(\vartheta,\kappa,\varsigma)$ is an increasing
function of $\beta$. This is expected, since by increasing the repulsion
parameter, the distance between interfering BSs is increased, achieving a
reduced overall interference and an enhanced detection performance. We can
finally observe that the analytical results (solid, dashed and dotted lines)
agree with the network performance given by the simulation results (markers).

Fig. 4 shows the detection performance for the adopted distance-based weighted
decision rules for two constant false alarm rates $P_{\rm fa}=\\{0.3,0.5\\}$,
where the conventional case of equal weights is depicted at $\varsigma=0$. The
detection threshold $\theta_{k}$ is selected based on Lemma 3, where
$k\in\\{1,\dots,K\\}$. An important observation is that, the achieved
detection performance of of all decision rules reduces as the tolerable false
alarm rate decreases. This observation is based on the fact that, the
acquisition of a lower false alarm rate requires the selection of a higher
detection threshold, leading to a reduced detection performance. We can also
observe that, the weighted decision rules provide a better detection
performance compared to the equal-weighted detection rules for the AND and
Majority rules. On the other hand, the adoption of weighted decision rules
reduces the detection performance of the OR rule compared to that achieved by
the equal-weighted detection rules. By comparing the performance of the
proposed CoMRD technique with the conventional non-cooperative technique
(dashed lines), which is obtained in the case where $\varsigma=1$, we observe
that the cooperation yields relative gains in detection performance with the
OR and Majority rules.

Figure 5: Detection probability versus $p_{\rm L}$ for the OR and AND
combining rules and for different number of network tiers; $P_{\rm fa}=0.5$,
$\lambda_{4}=6\ {\rm BSs/km^{2}}$ and $P_{4}=0$ dBm. Figure 6: Detection
probability versus $p_{\rm L}$ for the OR and AND combining rules and for
different number of network tiers; $P_{\rm fa}=0.5$, $\lambda_{4}=6\ {\rm
BSs/km^{2}}$ and $P_{4}=0$ dBm.

Fig. 5 reveals the impact of both the blockage and repulsion parameter on the
detection performance achieved by applying the OR and the AND combining rules
and for different number of network tiers. As expected, by increasing the
number of network tiers, the number of cooperative BSs is also increased, and
therefore the detection performance achieved by applying the OR rule
increases, in contrast to the decreasing detection performance achieved by
applying the AND rule. The latter occurs since, as the number of cooperative
BSs increases, the probability of simultaneous successful detection of an
object by all cooperative BSs is decreased. On the other hand, by applying the
OR rule, the increased number of cooperative BSs, increases the probability of
successfully detecting the object from at least one cooperative BS. Fig. 5
also reveals the impact of blockages on the network’s detection performance.
As expected, at low LoS constant values, the existence of LoS BSs improves the
network performance, since the cooperative BSs are able to successfully detect
the object. However, by increasing the LoS constant beyond a critical point,
which describes the optimal fraction of LoS BSs that provide the maximum
detection performance, the network performance decreases. This observation is
based on the fact that, the interference from the LoS BSs becomes
significantly larger than the reflected signal from the object and thus the
detection probability significantly decreases. Another important observation
is that, the aforementioned critical point depends on the repulsion parameter
of the network tiers. Specifically, by enhancing the repulsion behavior
between the BSs, the optimal detection performance is achieved with a larger
fraction of LoS BSs. Moreover, Fig. 6 shows the detection performance achieved
with the adopted PTDB association scheme compared to the conventional DB
scheme for all three hard-decision combining rules and for different number of
network tiers. As it can be seen, the performance achieved with the PTDB
scheme is upper bounded by the achieved performance of the DB scheme. Finally,
the adoption of the PTDB scheme provides lower complexity methodology for
evaluating the system performance, without being significantly deficient in
accuracy.

Figure 7: Detection probability versus $\chi$ for the considered combining
rules, where $\sigma_{\rm SI}^{2}=\\{0,\ -10,\ -20\\}$ dB and
$\varepsilon=\\{0.6\ ,0.9\\}$; $\beta=0.9$, $P_{\rm fa}=0.5$.

Fig. 7 shows the effect of the residual SI on the network’s detection
performance for the OR and the AND combining rules. We can easily observe that
by increasing the ability of the nodes to cancel the observed SI i.e.,
$\sigma_{\rm SI}^{2}\rightarrow-\infty$, the detection performance achieved
with our proposed technique is increased for all combining rules. This
observation was expected, since by decreasing the residual SI at the BSs, the
aggregate received interference is decreased, and therefore an increased SINR
is observed. An important observation from this figure is that the increased
fraction of JCAS-enabled BSs, negatively affects the detection performance.
This again is expected, since by increasing the number of BSs that
simultaneously perform both functions i.e., DL transmission and detection, the
overall interference increases due to the existence of SI, compromising the
observed SINR. We can easily observe from the figure that, the power control
can prevent the significant degradation of the network’s detection performance
by controlling the power control parameter $\varepsilon$. Specifically, by
decreasing the power control parameter i.e., $\varepsilon\rightarrow 0$, the
BSs’ transmit power for the DL communication with their associated users is
reduced, resulting in a decreased SI and therefore in an enhanced SINR. It is
important to mention here that both the fraction of JCAS-enabled BSs and the
Nakagami-$\mu$ fading parameters, have similar effect on the detection
performance achieved with the Majority rule, thus the corresponding curves are
omitted.

Figure 8: Detection probability versus $P_{1}$ for the OR and the AND
combining rules, where $\beta=\\{0,0.6,0.9\\}$; $P_{2}=\frac{P_{1}}{2}$ dB,
$P_{3}=\frac{P_{1}}{3}$ dB and $\chi=0.3$.

Fig. 8 shows the detection performance of the OR and the AND combining rules
in terms of the BSs’ transmit power $P_{1}$, where $P_{2}=\frac{P_{1}}{2}$ and
$P_{3}=\frac{P_{1}}{3}$. We can easily observe from the figure that, by
increasing the transmit power of the BSs can significantly enhance the
detection probability of its associated object. This observation is expected
since, a larger transmission power by the cooperative BSs corresponds to
receiving a stronger reflected signal power, resulting in an enhanced
detection performance. Nevertheless, beyond a critical transmit power level,
the increase of the transmission power negatively affects the detection
performance. This is again expected, since the residual SI at a receiver
becomes larger, and therefore the observed SINR is decreased. Moreover, it can
be seen that the detection performance converges to a constant floor in all
cases for high transmission powers. This is due to the fact that as the
transmission power of the network’s BSs increases, the noise in the network
becomes negligible. Fig. 8 also illustrates the achieved detection performance
of the considered combining rules, for the scenario where the spatial
distribution of the BSs is based on a PPP. It is clear from the figure that,
the spatially distributed BSs based on a repulsive point process can attain
better detection performance compared to the randomly deployed BSs. This is
due to the fact that, the distances between a cooperative BS and its
interfering BSs in a $\beta$-GPP-based deployment are greater than the
corresponding distances in a PPP-based deployment, resulting in a reduced
overall interference.

Figure 9: Detection probability and conditional detection probability versus
$\vartheta$ for the considered combining rules; $\beta=0.9$, $\chi_{k}=0.3$.

Finally, Fig. 9 highlights the effect of the interference temporal correlation
on the network’s detection performance between different time slots.
Specifically, we plot the conditional and the unconditional detection
probabilities for the three considered combining rules. We can easily observe
that the conditional detection probability overcomes the unconditional
detection probability. This was expected since, a fraction of interfering BSs
at the time slot $\tau$, in which a static object is successfully detected,
also causes interference in future time slots $\widehat{\tau}$. Therefore, as
the static object is successfully detected at the time slot $\tau$, the
probability of successful detecting the same static object at the time slot
$\widehat{\tau}$ is increased. Note that, the aforementioned observation holds
for all the considered combining rules and for all the detection thresholds
considered.

##  7 Conclusion

In this paper, we proposed an analytical framework based on stochastic
geometry and studied the detection performance of FD-JCAS systems in the
context of heterogeneous mmWave cellular networks. In particular, we
considered the scenario where all BSs exhibit detection capabilities, a
fraction of which also exhibit DL communication capabilities by exploiting the
concept of FD radio. The developed framework takes into account the spatial
correlation between the BSs and the temporal correlation of the interference,
by modeling the BSs’ locations as a $\beta$-GPP and by analyzing the joint
detection probability at two different time slots, respectively. Aiming at
enhancing the network’s detection performance, a novel cooperative detection
technique is proposed, and the achieved network performance is evaluated in
the context of three hard-decision combining rules, namely the OR, the
Majority and the AND rule. Using stochastic geometry tools, the network’s
detection performance in the context of our proposed technique was derived in
analytical expressions and the impact of blockage characteristics, residual
SI, repulsion parameter and fraction of FD-JCAS BSs have been evaluated. Our
study reveals that the proposed technique outperforms the conventional non-
cooperative detection technique. Finally, we have shown that the repulsive
behavior of the BSs can provide significantly larger detection performance to
the network.

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|  Christodoulos Skouroumounis (S’15–M’20) received the diploma in Computer
Engineer from the Electrical and Computer Engineer Department of National
Technical University of Athens (NTUA), Greece, in 2014, and a Ph.D. in
Computer Engineer from the University of Cyprus, Cyprus in 2019. He is
currently a Post-Doctoral Research Fellow with the Department of Electrical
Engineering, Computer Engineering and Informatics, Cyprus University of
Technology. His current research interests include full-duplex radio, next-
generation communication systems and cooperative networks.
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|  Constantinos Psomas (M’15–SM’19) received the BSc in Computer Science and
Mathematics from Royal Holloway, University of London in 2007 and the MSc in
Applicable Mathematics from London School of Economics the following year. In
2011, he received a PhD in Mathematics from The Open University, UK. He is
currently a Research Fellow at the Department of Electrical and Computer
Engineering of the University of Cyprus. From 2011 to 2014, he was as a
Postdoctoral Research Fellow at the Department of Electrical Engineering,
Computer Engineering and Informatics of the Cyprus University of Technology.
Dr. Psomas serves as an Associate Editor for the IEEE Wireless Communications
Letters. He received the Exemplary Reviewer Award by the IEEE Wireless
Communications Letters in 2015 and 2018. His current research interests
include wireless powered communications, cooperative networks and full-duplex
communications.
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|  Ioannis Krikidis (S’03–M’07–SM’12–F’19) received the Diploma degree in
computer engineering from the Computer Engineering and Informatics Department
(CEID), University of Patras, Greece, in 2000, and the M.Sc. and Ph.D. degrees
from École Nationale Supérieure des Télécommunications (ENST), Paris, France,
in 2001 and 2005, respectively, all in electrical engineering. From 2006 to
2007, he worked as a PostDoctoral Researcher with ENST, Paris. From 2007 to
2010, he was a Research Fellow of the School of Engineering and Electronics,
The University of Edinburgh, Edinburgh, U.K. He also has held research
positions at the Department of Electrical Engineering, University of Notre
Dame; the Department of Electrical and Computer Engineering, University of
Maryland; the Interdisciplinary Centre for Security, Reliability and Trust,
University of Luxembourg; and the Department of Electrical and Electronic
Engineering, Niigata University, Japan. He is currently an Associate Professor
at the Department of Electrical and Computer Engineering, University of
Cyprus, Nicosia, Cyprus. His current research interests include wireless
communications, cooperative networks, 5G communication systems, wireless
powered communications, and secrecy communications. He serves as an Associate
Editor for IEEE TRANSACTIONS ON WIRELESS COMMUNICATIONS, IEEE TRANSACTIONS ON
GREEN COMMUNICATIONS AND NETWORKING, and IEEE WIRELESS COMMUNICATIONS LETTERS.
He was a recipient of the Young Researcher Award from the Research Promotion
Foundation, Cyprus, in 2013, and a recipient of IEEE ComSoc Best Young
Professional Award in Academia, 2016. He has been recognized by the Web of
Science as a Highly Cited Researcher for 2017, 2018, and 2019. He has received
the prestigious ERC Consolidator Grant.
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