# Coverage and Capacity of Joint Communication and Sensing in Wireless
Networks

Nicholas R. Olson, Jeffrey G. Andrews, and Robert W. Heath, Jr N. R. Olson and
J. G. Andrews are with 6G@UT and WNCG at The University of Texas at Austin,
Austin, TX, USA (email<EMAIL_ADDRESS>jandrews@ece.utexas.edu). R. W.
Heath Jr. is with North Carolina State University, Raleigh, North Carolina,
USA (email: rwheathjr@ncsu.edu).

###### Abstract

From an information theoretic perspective, joint communication and sensing
(JCAS) represents a natural generalization of communication network
functionality. However, it requires the reevaluation of network performance
from a multi-objective perspective. We develop a novel mathematical framework
for characterizing the sensing and communication coverage probability and
ergodic capacity in JCAS networks. We employ an information theoretic
formulation of radar tracking to extend the notions of coverage probability
and ergodic capacity to the radar setting. Using this framework, we analyze
the downlink sensing and communication coverage and capacity of a JCAS network
employing a shared multicarrier waveform and analog beamforming. Leveraging
tools from stochastic geometry, we derive upper and lower bounds for these
quantities. We also develop several general technical results including: i) a
method for obtaining closed form bounds on the Laplace Transform of a shot
noise process, ii) an analog of Hölder’s Inequality to the setting of harmonic
means, and iii) a relation between the Laplace and Mellin Transforms of a non-
negative random variable. We use the derived bounds to investigate the
performance of JCAS networks under varying base station and blockage density.
Among several insights, our analysis indicates that network densification
improves sensing performance - in contrast to communications.

###### Index Terms:

Joint Communication and Sensing, Stochastic Geometry, Coverage Probability,
Ergodic Capacity, Estimation Rate

##  I Introduction

There is increasing interest in leveraging future communication networks and
waveforms to provide the additional services of user localization and
environmental sensing. Such networks may be termed _joint communication and
sensing networks_ or JCAS networks. Armed with these additional services, JCAS
networks could enable precision navigation in urban environments, monitor
activity in a given coverage area, provide collision avoidance services to
autonomous vehicles, enhance remote automation, and facilitate AR/VR
applications [1]. Moreover, the environmental information obtained through
sensing could improve communication performance and reliability by
facilitating channel estimation, beam alignment, and user tracking [2].
Indeed, one may view these important communication network functions as forms
of sensing themselves. While JCAS as a concept encompasses a variety of
methodologies, cooperative or jointly designed strategies – in which a common
waveform is used for simultaneous communication and radar sensing – seem to be
the most efficient usage of network resources. Hence, we restrict our
attention to this setting.

At a high level, the addition of radar sensing to communication networks is
not necessarily exotic: sensing targets represent additional, albeit non-
cooperative, sources of information with which the network interacts [2], [3].
However, the constraints and objectives of the communication and sensing
functions may not necessarily be in alignment. In particular, regimes in which
the communication function performs well may not be suitable for radar sensing
and vice versa. Hence, while sensing provides a natural generalization of
communication network functionality, the addition of this function requires
the re-evaluation of network performance from a multi-objective perspective.

While there is an array of dynamics at play in the performance of JCAS
networks, of particular concern is the impact of interference and spatial
attributes - which is the focus of our work. With respect to the communication
function, ascertaining the impact of these phenomena on network performance is
well now understood. One may employ a stochastic geometry approach (for
instance as in [4]) to characterize the coverage probability and ergodic
capacity of the network. In the radar case, the determination of suitable
performance metrics and analytical approaches is less established. We propose
generalizations of coverage probability and ergodic capacity to the radar case
based on an information theoretic framework, detailed in the following
section. Formally, we consider sensing coverage probability to be defined as
the probability that the rate of information about the parameters of interest
extracted from a typical radar target being tracked by the network exceeds
some threshold, and the sensing ergodic capacity as the expectation of this
quantity (or equivalently the average rate of information extracted from all
sensing targets being tracked by the network).

###  I-A Prior Work

Joint communication and sensing, sometimes referred to as joint radar and
communication (JRC) or integrated sensing and communication (ISAC), has
emerged as a promising potential function for future cellular networks [1],
[5]. In depth surveys of prior work, implementation approaches, and network
integration issues may be found in [2], [6], [7]. While much of the prior work
in this area has focused on signal processing and waveform design issues, for
instance in [8], our focus is on network wide performance analysis. Stochastic
geometry provides a powerful analytical framework for the analysis of wireless
networks, and has been widely used in a variety of settings. Notably in [4]
for the analysis of coverage and capacity in cellular networks. This was
extended in [9], for the analysis of mmWave cellular networks upon which we
base some of our system model.

With respect to the analysis of JCAS in wireless networks, there have been
relatively few works. In [10], a stochastic geometry approach is used to
characterize the performance of radar range detection and communication
coverage probability using a time multiplexed system in an ad hoc network. The
authors extend their analysis of this setting in [11]. In the setting of an
indoor network, [12] characterizes the detection performance of a radar system
amidst clutter that is time multiplexed with a communication system. In [13],
the radar detection performance in a vehicular network employing a shared
waveform for communication and sensing is characterized. In [14], a radar
network is considered which leverages sensing waveforms for communication. The
detection probability is characterized when multiple terminals share
information in addition to the communication coverage probability. Focusing on
exclusively on radar performance, in [15] a stochastic geometry approach is
employed to characterize the impact of network interference on radar detection
and false alarm rate in an ad hoc network. In a similar vein, the performance
of a sensor network to detect blockages is characterized in [16], also using a
stochastic geometry framework. Finally, in [17] the detection performance of a
heterogenous cellular network is considered in which the network access is
split between radar and communications functions using a stochastic geometry
framework. Performance analysis is conducted for a variety of cooperative
methods in which the individual detection hypotheses of multiple receivers are
fused according to some hard decision rules.

In a related but distinct setting from what we consider in this paper, there
have been a few stochastic geometry analyses of the performance of
localization using uplink signals from UEs in cellular networks. This was
considered, for instance, in [18].

###  I-B Contributions and Outline

To the best of our knowledge, our work is the first to present a rigorous
analytical framework for coverage and capacity in JCAS networks. In Sec. II,
we develop a conceptual framework for JCAS networks which motivates an
information theoretic formulation of sensing and communication coverage and
capacity. Focusing on the setting of a shared multicarrier waveform, we
characterize these metrics as

$\displaystyle\mathbb{P}^{0}({\text{SINR}_{\text{com}}}\geq\tau)$
$\displaystyle\mathbb{P}^{0}({\text{SINR}_{\text{rad}}}\geq\tau)$ (1)

for communication and sensing coverage probability, and

$\displaystyle\mathbb{E}^{0}\left[\log(1+{\text{SINR}_{\text{com}}})\right]$
$\displaystyle\mathbb{E}^{0}\left[\log(1+G\cdot{\text{SINR}_{\text{rad}}})\right]$
(2)

for ergodic communication and sensing capacity. While
${\text{SINR}_{\text{com}}}$ is an actual SINR, ${\text{SINR}_{\text{rad}}}$
is only an SINR by analogy: it is a more general construct which captures the
effectiveness of the signal for estimating the targets’ parameters of
interest. Crucially, this metric depends on the i) assumptions regarding the
target parameters to be estimated and ii) the variation/structure of the
excitation waveform over time and frequency. Next, in Sec. III, we construct a
network model using stochastic geometry with which we induce stochastic
expressions for ${\text{SINR}_{\text{com}}}$ and ${\text{SINR}_{\text{rad}}}$
with respect to the typical communication user and sensing target. Following
this, we establish a series of novel results that outline an approach to
obtain integral closed form upper and lower bounds for the expression in (1)
and (2) in Sections IV – VI. We detail preliminary results pertaining the
development of these bounds in Sec. IV. We then detail their derivations in
Sections V and VI, respectively. Finally, we present a numerical case study of
a JCAS network in Sec. VII. We conclude the paper in Sec. VIII. Our analysis
requires many intermediate results, the proofs of which we relegate to
Appendices A through I at the end of the paper.

Though we focus specifically on the setting of a shared mulitcarrier waveform
in a cellular context, our framework may be readily adapted to other waveforms
and network topologies. Moreover, through the course of our analysis, we
develop a few results that are of independent interest. These include: i) a
generic method for obtaining closed form upper and lower bounds on the Laplace
Transform of a shot noise process, ii) a new analog of Hölder’s Inequality to
the setting of harmonic means, and iii) a relation between the Laplace and
Mellin Transforms of a non-negative random variable.

###  I-C Notation

We employ the following notation throughout the paper. Deterministic scalars
or vectors are denoted by lower case letters and random variables or matrices
by upper case letters. Scalar objects are denoted in non-bold font, while
vectors and matrices are bolded. For a random variable $X$, we denote its
Laplace Transform as $\mathcal{L}_{X}(s)=\mathbb{E}[e^{-sX}]$ and its Mellin
Transform as $\mathcal{M}_{X}(p)=\mathbb{E}[X^{p-1}]$. Similarly, for a
measure $\nu$ over $\mathbb{G}$, its Laplace Transform is denoted as
$\mathcal{L}_{\nu}(s)=\int_{\mathbb{G}}e^{-sx}\nu(dx)$ and its Mellin
Transform $\mathcal{M}_{\nu}(p)=\int_{\mathbb{G}}x^{p-1}\nu(dx)$. We denote
the normal distribution with mean $\bm{\mu}$ and covariance $\mathbb{R}$ as
$\mathcal{N}(\bm{\mu},\mathbb{R})$, and the circularly symmetric complex
normal distribution with the same parameters as
$\mathcal{CN}(\bm{\mu},\mathbb{R})$. The gamma distribution with shape
parameter $\alpha$ and scale parameter $\beta$ is denoted as
$\text{Gamma}(\alpha,\beta)$. For $k\in\mathbb{N}$, $[k]$ denotes the interval
$\\{1,\dots,k\\}$. The symbol $\leq_{\text{s.t.}}$ denotes stochastic
dominance: $X\leq_{\text{s.t.}}Y$ if
$\mathbb{P}(X\geq\tau)\leq\mathbb{P}(Y\geq\tau)$ for all $\tau\in\mathbb{R}$.

##  II Characterization of Joint Communication and Sensing in Cellular
Networks

In constructing a modeling framework for JCAS networks, there are two
fundamental questions:

  1. Q1)

What is an appropriate conceptual framework for JCAS networks?

  2. Q2)

How can the performance of communication and sensing be characterized within
this conceptual framework?

In this section, we first address these questions, arguing for a conceptual
model in which sensing targets are treated as non-cooperative communication
users with an analogous notion of channel capacity to the communication case.
We then use the resulting framework to characterize a notion of sensing
coverage and capacity in the setting of a JCAS network employing a common
multicarrier waveform.

###  II-A Conceptual View of JCAS Networks

We deal with the Q1 in this subsection. Namely, we seek to address, at a high
level, the principal objects considered in our framework, their interaction,
and their utility with respect to the network. Throughout this section we take
an explicitly cellular view, although these concepts may be generalized to
other topologies.

One may view JCAS networks as involving the interaction among three types of
objects: physical network resources or base stations (BSs), communication
users or user equipment (UEs), and sensing targets or sensed objects (SOs).
BSs are responsible for the transmission of downlink communication data to
UEs, the reception of uplink communication data from UEs, and (potentially)
the transmission of sensing waveforms used to monitor SOs. UEs are responsible
for the transmission of uplink communication data to BSs and the reception of
downlink communication data from BSs. They may also participate in sensing SOs
in the network. To distinguish BS sensing from UE sensing we shall refer to
the transmission of radar waveforms by the base station as downlink sensing,
and the transmission of radar waveforms by the UEs as uplink sensing following
[19]. In general, any combination of uplink or downlink sensing may take place
in a JCAS network. One may generally consider SOs as passive agents or non-
cooperative sources of information. However, the set of UEs and SOs need not
be disjoint. For example, a UE that is also a vehicle may benefit from radar
sensing by the network.

The interaction between BS and UEs is, or course, well understood in the
cellular context. This paradigm may be extended, by analogy, to the SOs to
conceptualize their interaction with the network. We argue that this motivates
the treatment of SOs as another type of user. Such a notion requires that SOs
be discoverable by the network in some sense, and, once they are discovered,
that they have a quantifiable, well defined notion of utility from the
network. With respect to the discoverability requirement, one could generalize
or adapt initial access methods for UEs to estimate the presence of unknown
SOs. In radar systems this corresponds with the split between target detection
and target tracking. Once this procedure has been performed, SOs could then be
scheduled for tracking and associated with (a) serving BS(s) in a manner
similar to UEs – of course subject to detection uncertainty.

With respect to the notion of utility, one may exploit the level of certainty
that the network has about the parameters of interest associated with a given
SO. This concept is in some ways similar to the notion of a data queue for a
UE. Over time, the uncertainty about a given SO increases as it evolves, just
as uplink or downlink data continues to arrive for UEs. The uncertainty about
a given SO may be reduced by observing it through radar sensing, just as the
data queues for a UE may be reduced by uplink or downlink communication.
Consequently, the notions of throughput, delay, latency, reliability, and
coverage apply here as well. As was the case with UEs, these quantities are
fundamentally related to the sensing information rate available to the SOs.

###  II-B An Information Theoretic Perspective of Communication and Sensing

With the conceptual model in mind, we now address Q2 and turn to the problem
of performance characterization. The performance of both network functions is
inherently tied to the communication and sensing information rates experienced
by the UEs and SOs. These rates may be characterized by the capacities of
their respective channels.

For a given link occupying bandwidth $B$ with a signal-to-intererence-and-
noise-ratio ${\text{SINR}_{\text{com}}}$, the capacity, or communication rate,
of the link over an AWGN channel with interference treated as noise is given
by the well known Shannon Formula [20]. If one assumes that sinc-like pulses
are employed, letting $T_{\text{s}}$ denote the symbol rate of the system,
this may be expressed as

$\displaystyle
C_{\text{com}}=\frac{1}{T_{\text{s}}}\log_{2}(1+{\text{SINR}_{\text{com}}})$
$\displaystyle\text{[bits/s]}.$ (3)

Note that $T_{\text{s}}C_{\text{com}}$ is the spectral efficiency of the link,
which we refer to as communication efficiency hereafter.

In a similar vein, one may take an analogous, information theoretic view of
radar systems. Such an approach was first proposed by [21] and later extended
by [22]. Recently, it has been used to study waveform design and rate bounds
for joint radar and communication in [3, 23, 24] and termed estimation rate.
The estimation rate should be interpreted as the marginal information gained
about a target through a single sensing procedure [24]. As a BS in the network
tracks an SO in its vicinity it forms a belief about the associated parameters
of interest, $\bm{\theta}$, in the form of a distribution, $P$. At the same
time, the SO’s parameters of interest evolve according to some stochastic
dynamical process and increase the network’s uncertainty with respect to these
parameters. Hence, at a given sensing interval the BS must estimate the
residual uncertainty about the SO with respect to the prediction based on $P$.
Viewing this setting as an non-cooperative communication channel, one may
characterize the sensing quality via the mutual information between the radar
return and the parameters of interest, $\bm{\theta}$.

In general, an explicit characterization of this expression is difficult to
obtain. Hence, we employ a metric proposed in [23] based on the mutual
information between the processed return and the parameters of interest. Note
that this is a lower bound for the exact expression by the Data Processing
Inequality. For a radar system operating with a coherent processing interval
of $T_{\text{CPI}}$, attempting to estimate $\bm{\theta}$ with effective noise
following estimation $\mathbb{N_{\text{est}}}$ (independent of $\bm{\theta}$),
the estimation rate is defined as

$\displaystyle
C_{\text{rad}}=\frac{I(\bm{\theta}+\mathbb{N_{\text{est}}};\bm{\theta})}{T_{\text{CPI}}}.$
(4)

Where $I(\cdot;\cdot)$ denotes mutual information. In keeping with the
communication case, we refer to the quantity $T_{\text{CPI}}C_{\text{rad}}$ as
the sensing efficiency.

Under the assumption that the prior is Gaussian,
$P=\mathcal{N}\left(\bm{\overline{\theta}},\mathbb{Q}\right)$, Gaussian
estimation noise minimizes the mutual information for a fixed covariance, $R$,
[25]. Thus, taking $N_{est}\sim\mathcal{N}\left(\mathbb{0},\mathbb{R}\right)$,
we may formulate a pessimistic expression for $C_{\text{rad}}$ as

$\displaystyle
C_{\text{rad}}=\frac{1}{2T_{\text{CPI}}}\log_{2}\left(\left\lvert\mathbb{I}+\mathbb{Q}^{\frac{1}{2}}\mathbb{R}^{-1}\mathbb{Q}^{\frac{1}{2}}\right\rvert\right)$
$\displaystyle\text{[bits/s]}.$ (5)

###  II-C Characterization of the Estimation Rate and Definition of Sensing
Performance Metrics

The usage of multicarrier waveforms for radar (most commonly OFDM) has gained
interest over the past decade [5], [26], [27]. Such waveforms are an
attractive option for JCAS networks in that they provide excellent
communication capability while providing sufficient flexibility for sensing
purposes [28]. Moreover, the usage of a multicarrier waveform allows for radar
estimation to be performed at the symbol level in the frequency domain, rather
than directly through the baseband waveform [27]. This allows for some of the
benefits of multicarrier methods for communication purposes, such as reduction
of ISI, to carry over for sensing. Motivated by [27], we consider sensing to
take place in the frequency domain (that is, estimation is performed using the
output of an multicarrier receiver) as this has the added benefit of
facilitating comparison between sensing and communication.

We now characterize the estimation rate in the setting where a multicarrier
waveform used to sense a single target in the presence of interference from
the JCAS network. The setting is as follows:

  1. i)

The target is modeled as a point source and characterized by its relative
range, $r$, and relative velocity $v_{\text{rel}}$. Letting $c_{0}$ denotes
the speed of light and $f_{\text{c}}$ denote the center frequency, these
translate to inducing a delay, $\nu=2r/c_{0}$, and Doppler shift,
$f_{\text{D}}=2v_{\text{rel}}f_{\text{c}}/c_{0}$ to the transmitted waveform.
We assume that $f_{\text{c}}\gg B$, which implies that $f_{D}$ is constant
over the sub-carriers. Then the unknown target parameters are
$\bm{\theta}=(r,v_{\text{rel}})$ is assumed to come from a Gaussian prior with
covaraince, $\mathbb{Q}$.

  2. ii)

The waveform consists of $N_{\text{c}}$ subcarriers, and the maximum burst
duration of the radar excitation signal is $N_{\text{s}}$ multicarrier
symbols. A subset, $\mathbb{S}_{\text{rad}}\in\\{0,1\\}^{N_{\text{s}}\times
N_{\text{c}}}$, of these resource elements are used for sensing.

  3. iii)

The total length of each multicarrier symbol is
$T_{\text{MC}}=T_{\text{s}}+T_{\text{g}}$. $T_{\text{s}}$ denotes the symbol
duration and $T_{\text{g}}$ denotes the guard interval. This results in a a
sub-carrier spacing of $\Delta f=1/T_{\text{s}}$. It is assumed that these
parameters are such that $\nu<T_{\text{g}}$ and $f_{\text{D}}\ll\Delta f$.
This simplifies the estimation problem and nearly always holds in practice
[5], [26].

  4. iv)

The waveform may be shared by the communication function. Hence, arbitrary
message bearing symbols are transmitted on the scheduled resource elements.
Let the frequency domain symbols be denoted as $X_{m,n}$ for
$(m,n)\in\mathbb{S}_{\text{rad}}$.

Under these assumptions, the frequency domain representation of the
multicarrier radar return is given by the matrix
$\mathbb{F}\in\mathbb{C}^{N_{s}\times N_{c}}$ with support over
$\mathbb{S}_{\text{rad}}$ (following [26] [27])

$\displaystyle(\mathbb{F})_{m,n}=e^{-j2\pi\Delta f\tau n}e^{j2\pi
T_{\text{MC}}f_{D}m}+Z_{m,n}.$
$\displaystyle\mathbb{Z}\sim\mathcal{CN}(0,\text{diag}\\{\lvert
X_{m,n}\rvert^{-2}{\text{SINR}}_{m,n}^{-1}\\}_{(m,n)\in\mathbb{S}_{\text{rad}}}),$
(6)

where ${\text{SINR}}_{m,n}$ denotes the SINR at the $(m,n)^{th}$ resource
element. For simplicity, we have discarded the global random phase offset.
This results in a minor improvement to the Cramer Rao Bound (upon which our
metric for sensing rate is based), and simplifies the derivation.

We may obtain tractable bounds on the estimation rate in this setting by way
of the following theorem.

Theorem 1: Estimation Rate Bounds in the Multicarrier Setting The following
statements are true

  1. i)

Let $\mathbb{J}$ denote the Fisher Information of the estimation problem
stated in (6). Txhen the estimation rate as stated in (5) is upper bounded as

$\displaystyle C_{\text{rad}}\leq
C_{\text{rad}}^{UB}=\frac{1}{2T_{\text{CPI}}}\log_{2}\left(\left\lvert\mathbb{I}+\mathbb{Q}^{\frac{1}{2}}\mathbb{J}\mathbb{Q}^{\frac{1}{2}}\right\rvert\right).$
(7)

  2. ii)

Let $k_{1}=2\Delta f/c_{0}$ and $k_{2}=2T_{\text{MC}}f_{\text{c}}/c_{0}$, and
define $G$ and $\eta_{m,n}$ as

$\displaystyle G$
$\displaystyle=\sum_{(m,n)\in\mathbb{S}_{\text{rad}}}8\pi^{2}\left(\left(k_{1}(\mathbb{Q}^{\frac{1}{2}})_{1,1}n-k_{2}(\mathbb{Q}^{\frac{1}{2}})_{1,2}m\right)^{2}+\left(k_{1}(\mathbb{Q}^{\frac{1}{2}})_{1,2}n-k_{2}(\mathbb{Q}^{\frac{1}{2}})_{2,2}m\right)^{2}\right),$
(8) $\displaystyle\eta_{m,n}$
$\displaystyle=G^{-1}\left(\left(k_{1}(\mathbb{Q}^{\frac{1}{2}})_{1,1}n-k_{2}(\mathbb{Q}^{\frac{1}{2}})_{1,2}m\right)^{2}+\left(k_{1}(\mathbb{Q}^{\frac{1}{2}})_{1,2}n-k_{2}(\mathbb{Q}^{\frac{1}{2}})_{2,2}m\right)^{2}\right).$
(9)

Further, define ${\text{SINR}_{\text{rad}}}$ as

$\displaystyle{\text{SINR}_{\text{rad}}}=\sum_{(m,n)\in\mathbb{S}_{\text{rad}}}\eta_{m,n}\lvert
X_{m,n}\rvert^{2}{\text{SINR}}_{m,n}.$ (10)

Then the upper bound for the $C_{\text{rad}}^{UB}$ in (7) can itself be
bounded as

$\displaystyle\frac{1}{2}\log_{2}\left(1+G\cdot{\text{SINR}_{\text{rad}}}\right)\leq
T_{\text{CPI}}C_{\text{rad}}^{UB}\leq\log_{2}\left(1+\frac{1}{2}G\cdot{\text{SINR}_{\text{rad}}}\right).$
(11)

Proof: See Appendix A.

Theorem 1 forms the basis for our characterization of sensing coverage
probability and ergodic capacity. Part i) is a straightforward application of
the Cramer Rao Lower Bound that establishes an upper bound for the estimation
rate that is agnostic to the choice of estimator (so long as it is unbiased).
Part ii) is the main result from Theorem 1 and states that the estimation rate
upper bound in part i) may be upper and lower bounded by logarithmic functions
of ${\text{SINR}_{\text{rad}}}$ \- a weighted average of the per-resource-
element SINRs modulated by the transmit symbols. The weights are derived from
the Fisher Information Matrix, $J$, and capture the relative importance of
each resource element in estimating $\bm{\theta}$. The scale factor, $G$, is
an analog to the processing gain from observing $\bm{\theta}$ over
$\mathbb{S}_{\text{rad}}$.

The fact that ${\text{SINR}_{\text{rad}}}$ forms upper and lower bounds for
$T_{\text{CPI}}C_{\text{rad}}^{UB}$ in (11) motivate its usage as a metric to
characterize sensing coverage and capacity. Namely, we may say that an SO is
covered by the JCAS network if the estimation rate for its associated link is
above some threshold, $\tau$. Hence, in light of these bounds a necessary
condition for this to occur is
${\text{SINR}_{\text{rad}}}\geq\frac{2^{\tau+1}-2}{G}$ and a sufficient
condition is ${\text{SINR}_{\text{rad}}}\geq\frac{2^{2\tau}-1}{G}$. Therefore,
by characterizing the distribution of ${\text{SINR}_{\text{rad}}}$ we may
characterize necessary and sufficient conditions on sensing coverage. In light
of this, in the following sections we shall define the sensing coverage
probability as

$\displaystyle
P_{\text{c}}^{\text{rad}}(\tau)=\mathbb{P}^{0}\left({\text{SINR}_{\text{rad}}}\geq\tau\right).$
(12)

Where $\mathbb{P}^{0}$ denotes the Palm measure associated with the process of
SOs. Moreover, we shall take upper and lower bounds for the ergodic sensing
efficiency to be

$\displaystyle\mathbb{E}^{0}\left[\frac{1}{2}\log_{2}\left(1+G\cdot{\text{SINR}_{\text{rad}}}\right)\right]\leq
T_{\text{CPI}}\mathbb{E}^{0}[C_{\text{rad}}^{UB}]\leq\mathbb{E}^{0}\left[\log_{2}\left(1+\frac{1}{2}G\cdot{\text{SINR}_{\text{rad}}}\right)\right].$
(13)

Finally, while we have assumed multicarrier radar in the derivation of these
metrics, we note that these quantities may be applicable to other settings.
For instance, under similar assumptions FMCW waveforms have a similar form to
that of (6), and hence a corresponding notion of ${\text{SINR}_{\text{rad}}}$
could be defined along the lines of Theorem 1.

###  II-D Characterization of the Communication Rate and Definition of
Communication Performance Metrics

Unlike the sensing function, the communication function may employ any
arbitrary ensemble of resource elements. Moreover, the overall communication
rate is determined by the achievable rates on each resource element. Let such
an allocation for a given user over $N_{\text{s}}$ multicarrier symbols be
denoted as
$\mathbb{S}_{\text{com}}\in\\{0,1\\}^{N_{\text{s}}\times N_{\text{c}}}$.
111Note that $\mathbb{S}_{\text{com}}$ may partially or fully overlap with
$\mathbb{S}_{\text{rad}}$ due to the assumption that a common waveform is used
for both network functions. Then, the ergodic communication efficiency may be
expressed as

$\displaystyle
T_{\text{s}}\sum_{(m,n)\in\mathbb{S}_{\text{com}}}\mathbb{E}^{0}\left[C_{\text{com}}^{(m,n)}\right]=\sum_{(m,n)\in\mathbb{S}_{\text{com}}}\mathbb{E}^{0}\left[\log_{2}(1+{\text{SINR}}_{m,n})\right].$
(14)

Where $\mathbb{E}^{0}$ denotes expectation taken with respect to the Palm
measure associated with the process of UEs. Given that
$\\{{\text{SINR}}_{m,n}\\}_{(m,n)\in\mathbb{S}_{\text{com}}}$ are identically
distributed, it suffices to characterize the above expectation for an
arbitrary resource element. Letting
${\text{SINR}_{\text{com}}}={\text{SINR}}_{m,n}$, for some
$(m,n)\in\mathbb{S}_{\text{com}}$, the above formulation for the ergodic
communication efficiency reduces to

$\displaystyle\left\lvert\mathbb{S}_{\text{com}}\right\rvert\mathbb{E}^{0}\left[\log_{2}(1+{\text{SINR}_{\text{com}}})\right].$
(15)

In light of this formulation,x we define the communication coverage
probability as

$\displaystyle
P_{\text{c}}^{\text{com}}(\tau)=\mathbb{P}^{0}\left({\text{SINR}_{\text{com}}}\geq\tau\right).$
(16)

Note that, in either case, the characterization of the above
probability/expectation does not depend on $\mathbb{S}_{\text{com}}$ – in
contrast to the radar setting. Thus, in this sense, the scheduling dynamics
involved in determining $\mathbb{S}_{\text{com}}$ may be decoupled from the
process of characterizing the coverage and capacity. This is not the case for
the radar setting as ${\text{SINR}_{\text{rad}}}$ is a weighted averaged
average over the per-resource-element SINRs in $\mathbb{S}_{\text{rad}}$.
Consequently, the ergodic sensing efficiency requires the joint consideration
of the per-resource-element SINRs and the scheduling dynamics involved in the
determination of $\mathbb{S}_{\text{rad}}$

While the temporal dynamics involved in scheduling sensing and communication
are an important aspect in the performance of JCAS networks, they are outside
the scope of this work. Hence, we shall consider $\mathbb{S}_{\text{rad}}$ and
$\mathbb{S}_{\text{com}}$ to be fixed in the following sections. That is, we
effectively consider a "snapshot" model of the network and characterize the
JCAS coverage and capacity at a representative instant in time. (This
obfuscates the impacts of mobility and traffic dynamics, and thus our analysis
may apply to networks in which these attributes are prominent).

##  III System Model

Having presented our conceptual framework for a JCAS network and characterized
the sensing and communication performance metrics in the multicarrier setting,
we now present a detailed system model upon which ${\text{SINR}_{\text{rad}}}$
and ${\text{SINR}_{\text{com}}}$ are constructed. The key assumptions of the
model are enumerated as An) with n being the assumption number.

###  III-A Spatial Attributes

In general, the spatial components of a JCAS network may be represented by the
tuple,

$\displaystyle\\{\Phi_{\text{B}},\Phi_{\text{U}},\Phi_{\text{S}},\Theta_{\text{block}}\\}$
(17)

Where, $\Phi_{\text{B}}$ is a point process on $\mathbb{R}^{2}$ modeling the
locations of BSs in the JCAS network, $\Phi_{\text{U}}$ is a point process on
$\mathbb{R}^{2}$ modeling the locations of UEs in the JCAS network, and
$\Phi_{\text{S}}$ is a point process on $\mathbb{R}^{2}$ modeling the
locations of SOs in the JCAS network. $\Theta_{\text{block}}$ is a set process
on $\mathbb{R}^{2}$ representing the locations and shapes of blockages in the
network. We impose the following constraints.

  1. A1)

$\Phi_{\text{B}}$, $\Phi_{\text{U}}$, and $\Phi_{\text{S}}$ are stationary,
mutually independent PPPs with intensities $\lambda_{\text{B}}$,
$\lambda_{\text{U}}$, and $\lambda_{\text{S}}$, respectively. We assume that
$\lambda_{\text{U}}\gg\lambda_{\text{B}}$ and
$\lambda_{\text{S}}\gg\lambda_{\text{B}}$.

Note that the independence of $\Phi_{\text{U}}$ and $\Phi_{\text{S}}$ implies
that the SOs and UEs are distinct.

Using the notation we have just introduced and the notions of sensing and
communication coverage probability defined in Sec. II, we may characterize the
JCAS coverage probability,
$P_{\text{c}}^{\text{JCAS}}(\tau_{\text{com}},\tau_{\text{rad}})$, as the
expected fraction of users (UEs or SOs) which are covered by the network.
Formally, this may be stated as follows. Let $\\{A_{n}\\}_{n\in\mathbb{N}}$ be
a convex averaging sequence on $\mathbb{R}^{2}$, and ${\text{SINR}}(X_{k})$
denote that radar or communication SINR of a user at $X_{k}$, depending on
whether or not it is a UE or SO. Similarly, define $\tau(X_{k})$ as the target
threshold, say $\tau_{\text{com}}$ if $X_{k}\in\Phi_{\text{U}}$ or
$\tau_{\text{rad}}$ if $X_{k}\in\Phi_{\text{S}}$. Then,

$\displaystyle P_{\text{c}}^{\text{JCAS}}(\tau_{com},\tau_{rad})$
$\displaystyle=\mathbb{E}\left[\lim_{n\rightarrow\infty}\frac{1}{\Phi_{\text{U}}(A_{n})+\Phi_{\text{S}}(A_{n})}\sum_{X_{k}\in(\Phi_{\text{U}}+\Phi_{\text{S}})\cap
A_{n}}\mathds{1}\\{{\text{SINR}}(X_{k})\geq\tau(X_{k})\\}\right]$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\mbox{(a)}}}}{{=}}\mathbb{P}_{\Phi_{\text{U}}+\Phi_{\text{S}}}^{0}\left({\text{SINR}}^{0}\geq\tau^{0}\right)$
$\displaystyle\stackrel{{\scriptstyle\mathclap{\mbox{(b)}}}}{{=}}\frac{\lambda_{\text{U}}}{\lambda_{\text{U}}+\lambda_{\text{S}}}\mathbb{P}_{\Phi_{\text{U}}}^{0}\left({\text{SINR}_{\text{com}}}^{0}\geq\tau_{\text{com}}\right)+\frac{\lambda_{\text{S}}}{\lambda_{\text{U}}+\lambda_{\text{S}}}\mathbb{P}_{\Phi_{\text{S}}}^{0}\left({\text{SINR}_{\text{rad}}}^{0}\geq\tau_{\text{rad}}\right),$
(18)

where (a) follows from the Ergodic Theorem of Random Measures [29, Theorem
8.3.4] and (b) follows from the independence of $\Phi_{\text{U}}$ and
$\Phi_{\text{S}}$ and the Superposition Theorem. Hence, by characterizing the
sensing and communication coverage probabilities separately, we may
immediately obtain the JCAS coverage probability. A corresponding definition
and result holds for the JCAS ergodic capacity.

Finally, by Slivnyak’s Theorem, the Palm measures associated with
$\Phi_{\text{U}}$ and $\Phi_{\text{S}}$ reduce to their nominal measures.
Thus, without loss of generality, we assume the typical UE and SO to be
located at the origin.

  1. A2)

We shall take $\Theta_{\text{block}}$ to be a Boolean line process. Moreover,
we simplify its impact by approximating the induced LoS regions using the
independent, exponential blockage model. That is, a given link of length $r$
is LoS or NLoS (independent of the others) with probability [30]

$\displaystyle p_{\text{LoS}}(r)=\exp(-\beta r).$ (19)

Where $\beta$ is a parameter related to the density of blockages in
$\Theta_{\text{block}}$ their corresponding geometry.

  2. A3)

Communication may occur over either LoS and NLoS links, but sensing is LoS
only. This is motivated by the fact that NLoS range/velocity estimation is
significantly more challenging due to the greater inherent ambiguity, and
typically requires multiple receivers to resolve the location of the SO.

In light of these assumptions, we employ the following notation. Let $X_{k}$
denote the canonical enumeration of the atoms of $\Phi_{\text{B}}$. Then let
$M_{k}$ be a mark associated with $X_{k}$ denoting whether or not the link
from $X_{k}$ to the origin is LoS. Then $\\{M_{k}\\}_{k\in\mathbb{N}}$ are
independent and $M_{k}\sim\text{Bern}(p_{\text{LoS}}\left(\left\lVert
X_{k}\right\rVert_{2}\right))$. Let $\tilde{\Phi}_{\text{B}}$ denote the
marked PPP $\\{(X_{k},M_{k})\\}_{k\in\mathbb{N}}$. Let $\Phi_{\text{L}}$
denote the set of points in $\Phi_{\text{B}}$ that are LoS and
$\Phi_{\text{N}}$ denote the set of points in $\Phi_{\text{B}}$ that are NLoS.

###  III-B Channel and Radar Cross Section Models

  1. A4)

The LoS/NLoS path loss functions are

$\displaystyle
g_{\text{L}}(r)=K_{\text{L}}r^{-\alpha_{\text{L}}}e^{-\gamma_{\text{L}}r}\text{\quad
when LoS}$ $\displaystyle
g_{\text{N}}(r)=K_{\text{N}}r^{-\alpha_{\text{N}}}e^{-\gamma_{\text{N}}r}\text{\quad
when NLoS.}$ (20)

Thus the path loss to the origin of $X_{k}\in\tilde{\Phi}_{\text{B}}$ is

$\displaystyle L(\left\lVert X_{k}\right\rVert_{2})=g_{\text{L}}(\left\lVert
X_{k}\right\rVert_{2})M_{k}+g_{\text{N}}(\left\lVert
X_{k}\right\rVert_{2})(1-M_{k}).$ (21)

  2. A5)

In a similar manner the path loss of the LoS radar return is given by the two
way monostatic path loss model [31]

$\displaystyle
g_{\text{L,ret}}(r)=\frac{K_{\text{L}}}{4\pi}r^{-2\alpha_{\text{L}}}e^{-2\gamma_{\text{L}}r}.$
(22)

To avoid aberrant behavior of the path loss function for small $r$, we further
assume that

  1. A6)

The path loss functions are zero for distances such that they would exceed 1.
Thus the actual LoS path loss is
$g_{\text{L}}(r)\mathds{1}\\{g_{\text{L}}(r)\leq 1\\}$, and similar forms hold
for the other pathloss models.

This effectively removes interferers that are unrealistically close to the
typical UE or serving BS for the typical SO.

Recall that ${\text{SINR}_{\text{rad}}}$ is a weighted average over a set of
scheduled resource elements, $\mathbb{S}_{\text{rad}}$. Thus, to model this
expression, we require the distribution of the fading terms over each of the
$(m,n)\in\mathbb{S}_{\text{rad}}$ resource elements for each BS
$X_{k}\in\Phi_{\text{B}}$, $\\{|H^{k}_{m,n}|^{2}\\}$.

  1. A7)

We assume that the sensing procedure occurs within a coherence interval, and
thus that $|H^{k}_{m,n}|^{2}$ do not vary in time. Hence we drop the $m$
subscript.

  2. A8)

For interfering links, the fading terms are distributed as follows

$\displaystyle|H_{n}^{k}|^{2}\sim\text{Gamma}\left(N_{\text{L}},N_{\text{L}}\right)\text{\quad
when $X_{k}$ is LoS to the receiver}$ (23)
$\displaystyle|H_{n}^{k}|^{2}\sim\text{Gamma}\left(N_{\text{N}},N_{\text{N}}\right)\text{\quad
when $X_{k}$ is NLoS to the receiver}.$ (24)

Here, $N_{L},N_{N}\geq 1$ denotes the order of the fading model. Note that
$N_{L},N_{N}=1$ reduces to Rayleigh fading. We further assume that
$\\{|H^{k}_{n}|^{2}\\}$ are IID.

  3. A9)

The typical UE faces Rayleigh fading on the desired signal. That is $\lvert
H^{0}\rvert^{2}\sim\text{Exp}(1)$. This assumption is to facilitate analytical
tractability, and has a minor impact on the overall coverage probability in
the event of no fading, or other fading, or shadowing distributions [32].

  4. A10)

The radar cross section of the typical SO, $\kappa_{\text{CS}}$, is constant
over the excitation duration. We assume that this term is exponentially
distributed (denoted as the Swerling III model in radar settings). That is,
$\kappa_{\text{CS}}\sim\text{Exp}(1)$.

###  III-C Beamforming Models

As mentioned in the introduction, we consider that the BSs are capable of
analog beamforming only. Hence, the main utility of the antenna arrays is to
increase transmit or receive gain in a particular direction. Thus,

  1. A11)

the BSs and UEs perform directional beamforming for both communication and
sensing. That is, for a communication link, the BS and UE select the beam
directions which maximize the received power. Since sensing is LoS only, the
BS points its transmit and receive beams in the direction of the desired
SO(s).

Here it is worth noting explicitly, that we consider a monostatic scenario for
sensing in which the serving BS is both the transmitter and receiver. This
requires a full duplex transceiver at the base station in addition to antenna
separation [5]. Prior work in the area of full duplex has demonstrated its
potential feasibility [33], [34], [35]. This is especially true at mmWave, and
such duplexing forms are expected

  1. A12)

We model the antenna patterns of BSs/ UEs using the sectored model. The
antenna patterns for BSs and UEs are parameterized by the terms in the
following table.

Type |  Main lobe Gain |  3 dB Beam Beam width |  Front to Back Ratio
---|---|---|---
BS, Tx |  $G_{\text{B,Tx}}$ |  $\theta_{\text{B,Tx}}$ |  $\xi_{\text{B,Tx}}$
BS, Rx |  $G_{\text{B,Rx}}$ |  $\theta_{\text{B,Tx}}$ |  $\xi_{\text{B,Rx}}$
UE, Rx |  $G_{\text{U,Rx}}$ |  $\theta_{\text{U,Rx}}$ |  $\xi_{\text{U,Rx}}$

Thus, the total antenna gain for the desired signal of the typical UE is
$G_{\text{B,Tx}}G_{\text{U,Rx}}$, and the total antenna gain for the radar
return is $G_{\text{B,Tx}}G_{\text{B,Rx}}$

  2. A13)

The antenna gain from interfering links are independent random variables. For
the $k^{th}$ interferer, the normalized antenna gain over the $m^{th}$
multicarrier symbol is modeled as

$\displaystyle B_{m}^{k}\sim\begin{cases}1&\text{ with prob.
}p_{\text{B}}=\frac{\theta_{\text{B,Tx}}}{2\pi}\\\ \xi_{\text{B,Tx}}&\text{
with prob. }1-\frac{\theta_{\text{B,Tx}}}{2\pi}.\end{cases}$ (25)

  3. A14)

In a similar manner, since the BS process is isotropic, the receive gain
between the typical UE and $k^{th}$ interferer is modeled as

$\displaystyle Z_{U}^{k}\sim\begin{cases}1&\text{ with prob.
}\frac{\theta_{\text{U,Rx}}}{2\pi}\\\ \xi_{\text{U,Rx}}&\text{ with prob.
}1-\frac{\theta_{\text{U,Rx}}}{2\pi}.\end{cases}$ (26)

These models are justified under the assumption that the UE and SO densities
are much greater than the BS density. Additionally, note that one may define
analogous random variable for the overall antenna gain between the serving BS
for the typical SO and the $k^{th}$ interfering BS, $Z_{B}^{k}$. However, as a
consequence of the SO association policy, the process of interfering BSs in
non-isotropic. Hence, it does not admit an expression in the same form
$Z_{U}^{k}$. We shall develop a distribution for $Z_{B}^{k}$ in the following
section.

  1. A15)

Transmit beam directions are to fixed within a time slot (composed of multiple
multicarrier symbols) to facilitate communication. As a simplifying
assumption, $B^{k}_{m}\perp\\!\\!\\!\perp B^{k}_{s}$ if $m$ and $s$ fall in
different time slots. $B^{k}_{m}=B^{k}_{s}$ otherwise.

Using the notation developed thus far, the total antenna gain over the
$k^{th}$ interfering link for the typical UE is
$G_{\text{B,Tx}}G_{\text{U,Rx}}B^{k}Z_{U}^{k}$, and total antenna gain over
the $k^{th}$ interfering link on the $m^{th}$ multicarrier symbol for the
typical SO is $G_{\text{B,Tx}}G_{\text{B,Rx}}B_{m}^{k}Z_{B}^{k}$.

###  III-D UE and SO Association Policies

As we are considering a cellular network, UE and SO association plays an
important role in coverage and capacity.

  1. A16)

The typical UE is associated with BS that has strongest path loss. Let $X_{0}$
denote the location of the serving BS associated with the typical UE:

$\displaystyle X_{0}=\arg\sup\\{L(\left\lVert
X_{k}\right\rVert_{2}):X_{k}\in\tilde{\Phi}_{\text{B}}\\}.$ (27)

  2. A17)

The typical SO is associated with the nearest LoS BS. Let $X_{0}$ denote the
location of the serving BS associated with the typical SO. Given
$\Phi_{\text{L}}\neq\emptyset$,

$\displaystyle X_{0}=\arg\inf\\{\left\lVert
X_{k}\right\rVert_{2}:X_{k}\in\Phi_{\text{L}}\\}$ (28)

Such an association process represents a naive form of network sensing. In
practice, there would likely be some errors in the SO association process due
to detection uncertainty and imperfections in the fusion of the the detection
pictures of different BSs. We leave relaxations of this assumption that
account for these non-idealities to future work.

###  III-E Communication SINR Model

Let the normalized noise power at the typical UE be defined as

$\displaystyle\nu_{\text{com}}=\frac{\sigma_{\text{N}}}{G_{\text{B,Tx}}G_{\text{U,Rx}}}$
(29)

Then, we may express ${\text{SINR}_{\text{com}}}$ as

$\displaystyle{\text{SINR}_{\text{com}}}=\frac{\lvert
H^{0}\rvert^{2}L(\left\lVert
X_{0}\right\rVert_{2})}{\sum_{X_{k}\in\tilde{\Phi}_{\text{B}}\setminus\\{X_{0}\\}}\lvert
H^{k}\rvert^{2}B^{k}Z_{U}^{k}L(\left\lVert
X_{k}\right\rVert)+\nu_{\text{com}}}.$ (30)

In the subsequent section we shall take $F_{L}=\lvert
H^{k}\rvert^{2}B^{k}Z_{U}^{k}$ for interfering LoS BSs and $F_{N}=\lvert
H^{k}\rvert^{2}B^{k}Z_{U}^{k}$ for interfering NLoS BS.

###  III-F Sensing SINR Model

Before stating ${\text{SINR}_{\text{rad}}}$, we make the following further
assumptions.

  1. A18)

The transmitted symbols over the radar excitation signal are constant modulus.
That is $\lvert X_{m,n}\rvert=1$ for all $(m,n)\in\mathbb{S}_{\text{rad}}.$

This assumption is made to facilitate tractability. We note that, though the
variability of transmit symbols is an important factor in practical JCAS
waveform design, their impact is on the overall sensing coverage probability
defined with respect to ${\text{SINR}_{\text{rad}}}$ is limited.

In light of A15 and A18, the terms involved in the expression for
${\text{SINR}_{\text{rad}}}$ vary only over time slots rather than
multicarrier symbols. Hence, we may simplify the expression somewhat by
reducing the dimensionality of $\mathbb{S}_{\text{rad}}$ to the time slot
level. Specifically, let the radar excitation take place over $K$ time slots,
and define

$\displaystyle\mathbb{S}_{\text{rad}}^{t}=\\{(m,n)\in\mathbb{S}_{\text{rad}}:m\in\text{$t^{th}$
time slot}\\}$ $\displaystyle\forall t\in[K].$ (31)

Then, we may define the reduced radar resource allocation matrix over time
slots and sub-carriers as

$\displaystyle\mathbb{\tilde{S}}_{\text{rad}}=\\{(t,n)\in\\{0,1\\}^{T\times
N_{c}}:(\cdot,n)\in\mathbb{S}_{\text{rad}}^{t}\\}.$ (32)

The corresponding weights for $(t,n)\in\mathbb{\tilde{S}}_{\text{rad}}$ are
defined as

$\displaystyle\theta_{t,n}=\sum_{(m,n)\in\mathbb{S}_{\text{rad}}^{t}}\eta_{m,n}.$
(33)

In the following analysis, we shall further employ the notation

$\displaystyle
w_{t}=\sum_{(\cdot,n)\in\mathbb{\tilde{S}}_{\text{rad}}}\theta_{t,n},$
$\displaystyle
q_{n}=\sum_{(t,\cdot)\in\mathbb{\tilde{S}}_{\text{rad}}}\theta_{t,n}.$ (34)

Moreover, we shall take $\text{support}(\mathbb{w})=T\leq K$ and
$\text{support}(\mathbb{q})=N\leq N_{c}$.

Finally, we make the following assumption regarding the impact of blockages on
the interfering BSs from the perspective of the serving BS for the typical SO.

  1. A19)

The LoS/NLoS status of interfering BSs with respect to the serving BS is
independent of their LoS/NLoS status with respect to the typical SO at the
origin.

We may now state the model for ${\text{SINR}_{\text{rad}}}$. Let the
normalized noise variances at the serving BS for the typical SO be defined as

$\displaystyle\nu_{\text{rad}}=\frac{\sigma_{\text{N}}}{G_{\text{B,Tx}}G_{\text{B,Rx}}}.$
(35)

Then, let the product of the fading term on the $n^{th}$ sub carrier and
normalized transmit antenna gain at the $t^{th}$ time slot for the $k^{th}$
interfering BS be denoted as $F_{t,n}^{k}=\lvert
H^{k}_{n}\rvert^{2}B^{k}_{t}$. ${\text{SINR}_{\text{rad}}}$ may be stated as

$\displaystyle{\text{SINR}_{\text{rad}}}=\mathds{1}\\{\Phi_{\text{L}}(\mathbb{R}^{2})>0\\}\sum_{(t,n)\in\mathbb{\tilde{S}}_{\text{rad}}}\theta_{t,n}\frac{\kappa_{\text{CS}}g_{\text{L,ret}}(\left\lVert
X_{0}\right\rVert)}{\sum_{X_{k}\in\tilde{\Phi}_{\text{B}}\setminus\\{X_{0}\\}}F_{t,n}^{k}Z_{B}^{k}L(\left\lVert
X_{k}-X_{0}\right\rVert)+\nu_{\text{rad}}}.$ (36)

Note that $L(\left\lVert X_{k}-X_{0}\right\rVert)$ is defined with respect to
the LoS marks of associated with $X_{k}$ with respect to the serving BS rather
than the origin. The indicator term,
$\mathds{1}\\{\Phi_{\text{L}}(\mathbb{R}^{2})>0\\}$, denotes that fact that
sensing is LoS only. Hence, if there are no LoS BSs with respect to the
typical SO, its SINR (and consequently its estimation rate) is zero.

##  IV Preliminary Results

Before proceeding with the analysis of these SINR models, we take a brief
detour and establish some necessary preliminary results. First, we argue for
an ensemble of alternate radar SINR models, as characterizing the distribution
of ${\text{SINR}_{\text{rad}}}$, as stated, is analytically prohibitive. We
establish upper and lower bounding models as well as an approximate SINR
model. These alternate models share a common form with
${\text{SINR}_{\text{com}}}$, and, hence, we additionally derive generic upper
and lower bounds for Laplace Transforms of shot noise processes, based on
results from [36], which lead to bounds on the distributions of the SINR
models which we consider.

###  IV-A Alternative Radar SINR Models and their Relationship to
${\text{SINR}_{\text{rad}}}$

We begin by defining alternate radar SINR models, and establishing their
relationship to ${\text{SINR}_{\text{rad}}}$. Characterizing the distribution
of ${\text{SINR}_{\text{rad}}}$ is challenging due to the fact that it is
expressed as a mean of the inverse interference experienced over the allocated
resource elements in $\mathbb{S}_{\text{rad}}$. Indeed one may recognize the
denominator as a harmonic mean. Our alternate SINR models make use of the
relationship between different types of means, which are defined as follows.

Definition: (Arithmetic, Geometric, and Harmonic Means) Let $X$ be a non-
negative random variable over some countable state space $\mathcal{X}$ with
PMF $p$. The arithmetic mean of $X$ with respect to $p$ is

$\displaystyle{\text{AM}}(X,p)=\sum_{x\in\mathcal{X}}p_{X}(x)\cdot
x=\mathbb{E}[X].$ (37)

The geometric mean of $X$ with respect to $p$ is

$\displaystyle{\text{GM}}(X,p)=\prod_{x\in\mathcal{X}}x^{p_{X}(x)}=\exp(\mathbb{E}[\log(X)]).$
(38)

The harmonic mean of $X$ with respect to $p$ is

$\displaystyle{\text{HM}}(X,p)=\left(\sum_{x\in\mathcal{X}}p_{X}(x)\cdot
x^{-1}\right)^{-1}=\mathbb{E}[X^{-1}]^{-1}.$ (39)

Moreover, Jensens Inequality implies the following inequalities (the AM-GM-HM
Inequality):

$\displaystyle{\text{HM}}(X,p)\leq{\text{GM}}(X,p)\leq{\text{AM}}(X,p).$ (40)

Using this notation and letting $\bm{\theta}$ denote the ensemble of weights
over $\mathbb{\tilde{S}}_{\text{rad}}$ we may express
${\text{SINR}_{\text{rad}}}$ as

$\displaystyle{\text{SINR}_{\text{rad}}}=\mathds{1}\\{\Phi_{\text{L}}(\mathbb{R}^{2})>0\\}\frac{\kappa_{\text{CS}}g_{\text{L,ret}}(\left\lVert
X_{0}\right\rVert)}{HM\left(\left\\{\sum_{X_{k}\in\tilde{\Phi}_{\text{B}}\setminus\\{X_{0}\\}}F_{t,n}^{k}Z_{B}^{k}L(\left\lVert
X_{k}-X_{0}\right\rVert)+\nu_{\text{rad}}\right\\}_{(t,n)\in\mathbb{\tilde{S}}_{\text{rad}}},\bm{\theta}\right)}.$
(41)

Rather than dealing with this expression directly, we consider the following
three SINR models: ${\text{SINR}_{\text{AM}}}$, ${\text{SINR}_{\text{GM}}}$,
and ${\text{SINR}_{\text{HM}}}$. Letting $\mathbf{F_{k}}$ denote the ensemble
of fading terms $\\{F_{t,n}^{k}\\}_{(t,n)\in\mathbb{\tilde{S}}_{\text{rad}}}$,
these are defined as

$\displaystyle{\text{SINR}_{\text{AM}}}=\mathds{1}\\{\Phi_{\text{L}}(\mathbb{R}^{2})>0\\}\frac{\kappa_{\text{CS}}g_{\text{L,ret}}(\left\lVert
X_{0}\right\rVert)}{\sum_{X_{k}\in\tilde{\Phi}_{\text{B}}\setminus\\{X_{0}\\}}{\text{AM}}\left(\mathbf{F_{k}},\bm{\theta}\right)Z_{B}^{k}L(\left\lVert
X_{k}-X_{0}\right\rVert)+\nu_{\text{rad}}},$ (42)
$\displaystyle{\text{SINR}_{\text{GM}}}=\mathds{1}\\{\Phi_{\text{L}}(\mathbb{R}^{2})>0\\}\frac{\kappa_{\text{CS}}g_{\text{L,ret}}(\left\lVert
X_{0}\right\rVert)}{\sum_{X_{k}\in\tilde{\Phi}_{\text{B}}\setminus\\{X_{0}\\}}{\text{GM}}\left(\mathbf{F_{k}},\bm{\theta}\right)Z_{B}^{k}L(\left\lVert
X_{k}-X_{0}\right\rVert)+\nu_{\text{rad}}},$ (43)
$\displaystyle{\text{SINR}_{\text{HM}}}=\mathds{1}\\{\Phi_{\text{L}}(\mathbb{R}^{2})>0\\}\frac{\kappa_{\text{CS}}g_{\text{L,ret}}(\left\lVert
X_{0}\right\rVert)}{\sum_{X_{k}\in\tilde{\Phi}_{\text{B}}\setminus\\{X_{0}\\}}{\text{HM}}\left(\mathbf{F_{k}},\bm{\theta}\right)Z_{B}^{k}L(\left\lVert
X_{k}-X_{0}\right\rVert)+\nu_{\text{rad}}}.$ (44)

We relate these to the exact model, ${\text{SINR}_{\text{rad}}}$, by the
following proposition.

Proposition 1: (Relation Between Proposed SINR Models) The following
stochastic orderings hold:

$\displaystyle{\text{SINR}_{\text{AM}}}\leq_{\text{s.t.}}{\text{SINR}_{\text{rad}}}\leq_{\text{s.t.}}{\text{SINR}_{\text{HM}}},$
(45)

and

$\displaystyle{\text{SINR}_{\text{AM}}}\leq_{\text{s.t.}}{\text{SINR}_{\text{GM}}}\leq_{\text{s.t.}}{\text{SINR}_{\text{HM}}},$
(46)

Proof: See Appendix B.

These orderings arise from the AM-GM-HM Inequality, Hölder’s Inequality, and a
generalization of Hölder’s Inequality to harmonic means (proved in Appendix
I). Overall, these orderings indicate that both ${\text{SINR}_{\text{rad}}}$
and ${\text{SINR}_{\text{GM}}}$ are lower bounded by
${\text{SINR}_{\text{AM}}}$ and upper bounded by ${\text{SINR}_{\text{HM}}}$.
This motivates ${\text{SINR}_{\text{GM}}}$ for use as an approximation of
${\text{SINR}_{\text{rad}}}$, and establishes ${\text{SINR}_{\text{AM}}}$ and
${\text{SINR}_{\text{HM}}}$ as lower and upper bounding models, respectively.

By the assumption that the radar cross section $\kappa_{\text{CS}}$ is
exponentially distributed we may characterize the distributions of the SINR
models via the Laplace Transform of their corresponding interference
processes. Each of the three alternate models differ only in the interference
fading terms – ${\text{AM}}\left(\mathbf{F_{k}},\bm{\theta}\right)$,
${\text{GM}}\left(\mathbf{F_{k}},\bm{\theta}\right)$, or
${\text{HM}}\left(\mathbf{F_{k}},\bm{\theta}\right)$ – and, thus their
interference Laplace Transforms differ only in the Laplace Transforms of these
terms. Consequently, we shall present characterizations of the radar coverage
probability in terms of a generic model,

$\displaystyle{\text{SINR}_{\text{gen}}}=\mathds{1}\\{\Phi_{\text{L}}(\mathbb{R}^{2})>0\\}\frac{\kappa_{\text{CS}}g_{\text{L,ret}}(\left\lVert
X_{0}\right\rVert)}{\sum_{X_{k}\in\tilde{\Phi}_{\text{B}}\setminus\\{X_{0}\\}}F_{k}Z_{B}^{k}L(\left\lVert
X_{k}-X_{0}\right\rVert)+\nu_{\text{rad}}},$ (47)

where, within each subset of LoS/NLoS points, $\\{F_{k}\\}_{k\in\mathbb{N}}$
are arbitrary IID non-negative random variables with finite first moment. We
shall take $F_{L}$ to denote a LoS fading term and $F_{N}$ to denote an NLoS
fading term. Explicit expressions for the CCDFs of the aforementioned radar
SINR models may then be obtained using expressions for the Laplace Transforms
for their corresponding interference fading terms, derived in the subsequent
section.

###  IV-B A Generic Method For Obtaining Upper and Lower Bounds on
Interference Laplace Transforms

${\text{SINR}_{\text{gen}}}$ and ${\text{SINR}_{\text{com}}}$ belong to a
common class of SINR models in that

  1. i)

the fading term on the desired signal is exponentially distributed,

  2. ii)

conditioned on the location of the serving base station, the interference is
independent of the desired signal and follows a Poisson shot noise process,

  3. iii)

and the interference fading terms are IID (per each class of LoS/NLoS points).

As a consequence of i) the distribution of an SINR model from this class may
be characterized in terms of the Laplace Transform of its interference
process. By ii) and iii), the conditional interference Laplace Transform
admits an integral closed form expression in terms of Campbell’s Formula.

While a powerful framework, the usage of Campbell’s Formula often leads to
results expressed in terms of an incomputable integral. This is especially
true in the case of ${\text{SINR}_{\text{gen}}}$. In light of this, we develop
a general method for obtaining upper and lower bounds on the positive real
branch of the Laplace Transform of a shot noise process (i.e. Poisson
interference process). We consider the following generic interference process

$\displaystyle I=\sum_{R_{k}\in\Phi}F_{k}g(R_{k}).$ (48)

Where $\Phi$ is an arbitrary PPP with intensity measure $\Lambda$,
$\\{F_{k}\\}$ are arbitrary fading terms, and $g$ is an arbitrary path loss
function. Of particular interest is the Laplace Transform of $I$, which by
Campbell’s Formula may be expressed as

$\displaystyle\mathcal{L}_{I}(s)=\exp\left(-\int_{\mathbb{R}_{+}}\left(1-\mathcal{L}_{F}(s\cdot
g(r))\right)\Lambda(dr)\right).$ (49)

We obtain upper and lower bounds on $\mathcal{L}_{I}(s)$, for
$s\in\mathbb{R}_{+}$, using the theorem to follow. The theorem requires some
notation, which we define first before its statement.

Our general approach to obtaining upper and lower bounds on
$\mathcal{L}_{I}(s)$ for $s\in\mathbb{R}_{+}$ is to obtain lower and upper
bounds on the integral term in (49). We obtain these bounds by approximating
the intensity measure using lower and upper bounding atomic measures with a
finite number of atoms. These approximate measures are such that the moments
of the path loss with respect to $\Lambda$ are matched over arbitrary
intervals. To that end, we make use of the Mellin Transform of the path loss
with respect to $\Lambda$, restricted to a set, $A\subseteq\mathbb{R}_{+}$,
which we denote as 222In many practical scenarios – such as our case – these
may be obtained closed form.

$\displaystyle\mathcal{M}_{(\Lambda\circ
g^{-1})}(p;A)=\int_{A}g(r)^{p-1}\Lambda(dr).$ (50)

Our bounds utilize the following two functionals, which have the
interpretation as finite support approximations of $\Lambda$. The first,
$H_{\text{LB}}(s,A;1-\mathcal{L}_{F},\mathcal{M}_{(\Lambda\circ g^{-1})})$, we
define as

$\displaystyle H_{\text{LB}}(s,A;1-\mathcal{L}_{F},\mathcal{M}_{(\Lambda\circ
g^{-1})})=\mathcal{M}_{(\Lambda\circ
g^{-1})}(1;A)\left(p\left(1-\mathcal{L}_{F}(sx_{1})\right)+(1-p)\left(1-\mathcal{L}_{F}(sx_{2})\right)\right),$
(51) $\displaystyle\alpha$ $\displaystyle=\frac{\mathcal{M}_{(\Lambda\circ
g^{-1})}(1;A)\mathcal{M}_{(\Lambda\circ
g^{-1})}(4;A)-\mathcal{M}_{(\Lambda\circ
g^{-1})}(2;A)\mathcal{M}_{(\Lambda\circ
g^{-1})}(3;A)}{\mathcal{M}_{(\Lambda\circ
g^{-1})}(1;A)\mathcal{M}_{(\Lambda\circ
g^{-1})}(3;A)-\mathcal{M}_{(\Lambda\circ g^{-1})}(2;A)^{2}}$ (52)
$\displaystyle\beta$ $\displaystyle=\frac{\mathcal{M}_{(\Lambda\circ
g^{-1})}(3;A)^{2}-\mathcal{M}_{(\Lambda\circ
g^{-1})}(2;A)\mathcal{M}_{(\Lambda\circ
g^{-1})}(4;A)}{\mathcal{M}_{(\Lambda\circ
g^{-1})}(1;A)\mathcal{M}_{(\Lambda\circ
g^{-1})}(3;A)-\mathcal{M}_{(\Lambda\circ g^{-1})}(2;A)^{2}}$ (53)
$\displaystyle\gamma$ $\displaystyle=\sqrt{\alpha^{2}+4\beta}$ (54)
$\displaystyle x_{1}$ $\displaystyle=\frac{\alpha-\gamma}{2}$ (55)
$\displaystyle x_{2}$ $\displaystyle=\frac{\alpha+\gamma}{2}$ (56)
$\displaystyle p$ $\displaystyle=\frac{\mathcal{M}_{(\Lambda\circ
g^{-1})}(1;A)(\alpha+\gamma)-2\mathcal{M}_{(\Lambda\circ
g^{-1})}(2;A)}{2\gamma\mathcal{M}_{(\Lambda\circ g^{-1})}(1;A)}.$ (57)

Next, we define
$H_{\text{UB}}(s,A;1-\mathcal{L}_{F},\mathcal{M}_{(\Lambda\circ g^{-1})})$ as

$\displaystyle\begin{split}&H_{\text{UB}}(s,A;1-\mathcal{L}_{F},\mathcal{M}_{(\Lambda\circ
g^{-1})})=\\\ &\mathcal{M}_{(\Lambda\circ
g^{-1})}(1;A)\left((1-p^{\prime}-q^{\prime})\left(1-\mathcal{L}_{F}(sz_{1})\right)+p^{\prime}\left(1-\mathcal{L}_{F}(sz_{2})\right)+q^{\prime}\left(1-\mathcal{L}_{F}(sz_{3})\right)\right),\end{split}$
(58) $\displaystyle z_{1}$ $\displaystyle=\inf_{r\in A}g(r)$ (59)
$\displaystyle z_{3}$ $\displaystyle=\sup_{r\in A}g(r)$ (60)
$\displaystyle\alpha$ $\displaystyle=\frac{\mathcal{M}_{(\Lambda\circ
g^{-1})}(2;A)}{\mathcal{M}_{(\Lambda\circ g^{-1})}(1;A)}-z_{1}$ (61)
$\displaystyle\beta$ $\displaystyle=\frac{\mathcal{M}_{(\Lambda\circ
g^{-1})}(3;A)-2z_{1}\mathcal{M}_{(\Lambda\circ
g^{-1})}(2;A)}{\mathcal{M}_{(\Lambda\circ g^{-1})}(1;A)}-z_{1}^{2}$ (62)
$\displaystyle\gamma$ $\displaystyle=\frac{\mathcal{M}_{(\Lambda\circ
g^{-1})}(4;A)-3z_{1}\mathcal{M}_{(\Lambda\circ
g^{-1})}(3;A)-3z_{1}^{2}\mathcal{M}_{(\Lambda\circ
g^{-1})}(2;A)}{\mathcal{M}_{(\Lambda\circ g^{-1})}(1;A)}-z_{1}^{3}$ (63)
$\displaystyle q^{\prime}$
$\displaystyle=\frac{\alpha\gamma-\beta^{2}}{\alpha(z_{3}-z_{1})^{3}-2\beta(z_{3}-z_{1})^{2}+\gamma(z_{3}-z_{1})}$
(64) $\displaystyle p^{\prime}$
$\displaystyle=\frac{\left(\alpha-q^{\prime}z_{3}\right)^{2}}{\beta-q^{\prime}z_{3}^{2}}$
(65) $\displaystyle z_{2}$
$\displaystyle=\frac{\beta-q^{\prime}z_{3}^{2}}{\alpha-q^{\prime}z_{3}}.$ (66)

Using these functionals, we now state the theorem.

Theorem 2: (Bounds on the Laplace Transform of a Poisson shot noise process)
Let $\Phi$ be a PPP on $\mathbb{R}_{+}$ with intensity measure $\Lambda$,
$\\{F_{k}\\}_{k\in\Phi}$ be a series of IID marks in $\mathbb{R}_{+}$
associated with $\Phi$ such that their first moment is finite, and
$g:\mathbb{R}_{+}\rightarrow\mathbb{R}_{+}$ be an arbitrary path loss function
such that $\Lambda(g)$ is finite. Let $I$ denote the shot noise process as
given in (48), and $\mathcal{L}_{I}(s)$ denote its Laplace Transform.

Moreover, let $N_{w}\in\mathbb{N}$ and $\\{d_{i}\\}_{i\in[N_{w}]}$ be a
partitioning of $\mathbb{R}_{+}$. Without loss of generality assume $d_{i}\leq
d_{i+1}$ and $d_{0}=0$. Then, for $s\in R_{+}$ the following hold

  1. i)

If $\Lambda(\mathbb{R}_{+})=\infty$

$\displaystyle\mathcal{L}_{I}(s)\geq\exp\left(-\sum_{i=1}^{N_{w}}H_{\text{LB}}\left(s,[d_{i-1},d_{i}];1-\mathcal{L}_{F},\mathcal{M}_{(\Lambda\circ
g^{-1})}\right)+s\mathbb{E}[F]\mathcal{M}_{(\Lambda\circ
g^{-1})}\left(s;[d_{N_{w}},\infty)\right)\right),$ (67)

and

$\displaystyle\mathcal{L}_{I}(s)\leq\exp\left(-\sum_{i=1}^{N_{w}}H_{\text{UB}}\left(s,[d_{i-1},d_{i}];1-\mathcal{L}_{F},\mathcal{M}_{(\Lambda\circ
g^{-1})}\right)\right).$ (68)

  2. ii)

Otherwise, if $\Lambda(\mathbb{R}_{+})<\infty$,

$\displaystyle\mathcal{L}_{I}(s)\geq\exp\left(-\sum_{i=1}^{N_{w}}H_{\text{LB}}\left(s,[d_{i-1},d_{i}];1-\mathcal{L}_{F},\mathcal{M}_{(\Lambda\circ
g^{-1})}\right)+H_{\text{LB}}\left(s,[d_{N_{w}},\infty);1-\mathcal{L}_{F},\mathcal{M}_{(\Lambda\circ
g^{-1})}\right)\right),$ (69)

and

$\displaystyle\mathcal{L}_{I}(s)\leq\exp\left(-\sum_{i=1}^{N_{w}}H_{\text{UB}}\left(s,[d_{i-1},d_{i}];1-\mathcal{L}_{F},\mathcal{M}_{(\Lambda\circ
g^{-1})}\right)+H_{\text{UB}}\left(s,[d_{N_{w}},\infty);1-\mathcal{L}_{F},\mathcal{M}_{(\Lambda\circ
g^{-1})}\right)\right).$ (70)

Proof: See Appendix C.

In these expressions,
$H_{\text{LB}}\left(\cdot,[d_{1},d_{2}];(\cdot),\mathcal{M}_{(\Lambda\circ
g^{-1})}\right)$ and
$H_{\text{UB}}\left(\cdot,[d_{1},d_{2}];(\cdot),\mathcal{M}_{(\Lambda\circ
g^{-1})}\right)$ may be interpreted as finite support approximations of
$\Lambda$ over the interval $[d_{1},d_{2}]$. Hence the summation of these
functionals over the partitions represents a finite order approximation of
$\Lambda$ over $\mathbb{R}_{+}$. These functionals arise as generalizations of
results in [36], which enable us to conclude that they result in upper and
lower bounds for the complementary Laplace Transform, $1-\mathcal{L}_{F}$, in
the argument of of exponential function in (49).

##  V Characterization of Sensing Coverage Probability and Ergodic Estimation
Sensing Efficiency

We now characterize bounds on the CCDF of ${\text{SINR}_{\text{gen}}}$, which,
in tandem with expressions for the corresponding Laplace Transforms of the
interference fading terms, lead to bounds on the CCDFs of the alternate SINR
models, ${\text{SINR}_{\text{HM}}}$, ${\text{SINR}_{\text{GM}}}$, and
${\text{SINR}_{\text{AM}}}$. These, in turn lead to bounds and approximations
for the CCDF of ${\text{SINR}_{\text{rad}}}$.

In light of Theorem 2, in order to characterize the CCDF of
${\text{SINR}_{\text{gen}}}$, we require a characterization of the
interference process conditioned on $X_{0}$. Due to the asymmetry of the
interferers induced by the SO association policy, the intensity measure of
this process lacks a closed form expression. We therefore derive closed form
upper and lower bounds for this quantity. These bounds admit closed form
expressions for their associated path loss Mellin Transforms, and hence allow
for the application of Theorem 2.

We begin by characterizing the interference process. Using these results, we
derive upper and lower bounds on the CCDF of ${\text{SINR}_{\text{gen}}}$.
Finally, we present results pertaining to the Laplace Transforms of the
interference fading terms of the alternate SINR models. Using the bounds on
${\text{SINR}_{\text{gen}}}$, these immediately imply bounds on the alternate
SINR models, which are stated at the end of this section. As a corollary of
these bounds and the bounds on the estimation rate in Theorem 1, we further
obtain bounds on the ergodic sensing efficiency.

###  V-A Characterization of the Distribution of the Desired Signal and
Conditional Interference PPP

Here, we derive the distribution of the serving BS location,
$X_{0}=(R_{0},\theta_{0})$, and the reduced Palm distribution of the point
process of interferers, $\Phi_{\text{B}}^{!X_{0}}$. We will then use this to
characterize the point process of the distances of interfering BSs with
respect to the serving BS conditioned on $R_{0}$:

$\displaystyle\Pi_{\text{B}}^{0}=\left\\{\left\lVert
X_{k}-X_{0}\right\rVert_{2}:X_{k}\in\Phi_{\text{B}}^{!X_{0}}\right\\}.$ (71)

In so doing, we further characterize the distribution of the receive antenna
gain variables, $\\{Z_{B}^{k}\\}_{k\in\Pi_{\text{B}}^{0}}$, as marks with
respect to $\Pi_{\text{B}}^{0}$.

The following lemma establishes the form of the distribution of $X_{0}$.

Lemma 1: (Distribution of the Location of the Serving Base Station) $R_{0}$
and $\theta_{0}$ are independent with $\theta_{0}\sim\text{Uniform}([0,2\pi))$
and $R_{0}$ distributed as

$\displaystyle
R_{0}\sim\begin{cases}\frac{f_{R_{0}}(r)}{1-\exp(\frac{-2\pi\lambda_{\text{B}}}{\beta^{2}})}&\text{w.p.
}1-\exp(\frac{-2\pi\lambda_{\text{B}}}{\beta^{2}})\\\ \infty&\text{w.p.
}\exp(\frac{-2\pi\lambda_{\text{B}}}{\beta^{2}})\end{cases}$ (72)

where

$\displaystyle
f_{R_{0}}(r)=2\pi\lambda_{\text{B}}\exp\left(\frac{-2\pi\lambda_{\text{B}}}{\beta^{2}}(1-e^{-\beta
r}(\beta r+1))\right).$ (73)

Proof: This may be proved in a manner similar to that of [30, Theorem 10].
$\hfill\Box$

Note that $R_{0}$ takes the value of $\infty$ when
$\Phi_{\text{L}}(\mathbb{R}^{2})=0$. Since ${\text{SINR}_{\text{gen}}}$ is $0$
when $\Phi_{\text{L}}(\mathbb{R}^{2})=0$, we shall restrict our attention to
the case where $R_{0}<\infty$. We now characterize $\Pi_{\text{B}}^{0}$.

Lemma 2: (Characterization of the Palm Interference Process) Condition on
$X_{0}=(r_{0},\phi_{0})$. Define the function $J(r;r_{0},z)$ as

$\displaystyle
J(r;r_{0},z)=\int_{z}^{1}(1-u^{2})^{-1/2}\exp\left(-\beta\sqrt{r^{2}-2rr_{0}u+r_{0}^{2}}\right)du.$
(74)

Then, for all $\phi_{0}$, $\Pi_{\text{B}}^{0}$ is a PPP on $\mathbb{R}_{+}$
with intensity function

$\displaystyle\lambda_{\text{B}}^{0}(r;r_{0})=2\lambda_{\text{B}}r\left(\pi-J\left(r;r_{0},\frac{r}{2r_{0}}\right)\mathds{1}\\{r\leq
2r_{0}\\}\right).$ (75)

Moreover, the normalized receive antenna gain variables
$\\{Z_{B}^{k}\\}_{k\in\Pi_{\text{B}}^{0}}$ may be treated as independent marks
of $\Pi_{\text{B}}^{0}$. $\\{Z_{B}^{k}\\}_{k\in\Pi_{\text{B}}^{0}}$ take
values in $\\{\xi_{\text{B,Rx}},1\\}$, and the probability that $Z_{B}^{k}=1$
is given by $p_{\text{B,Rx}}(R_{k})$ where

$\displaystyle
p_{\text{B,Rx}}(R_{k})=\frac{\frac{\theta_{\text{B,Rx}}}{2}-J\left(R_{k};r_{0},\max\left\\{\cos\left(\frac{\theta_{\text{B,Rx}}}{2}\right),\frac{R_{k}}{2r_{0}}\right\\}\right)\mathds{1}\\{R_{k}\leq
2r_{0})\\}}{\pi-J\left(r;R_{0},\frac{R_{k}}{2r_{0}}\right)\mathds{1}\\{R_{k}\leq
2r_{0}\\}}$ (76)

Proof: See Appendix D.

Note that $\Pi_{\text{B}}^{0}$ represents the point process of the distances
of interfering base stations with respect to the serving base station located
at a distance $R_{0}$ from the origin. In contrast to typical stochastic
geometry based analyses of cellular networks, there is now a non-zero
probability of interfering base stations being closer to the radar receiver
(which is the serving base station in this case) than the typical SO. This
"exclusion region" (that is the region $B(0,R_{0})$ about the typical SO) is
an important aspect of this process which has a significant impact on the
sensing coverage probability of the network.

Note also that we have discarded the marks for the LoS and NLoS base stations
with respect to the origin. This is due to the fact that the serving base
station is also the receiver, and hence observes different LoS/NLoS base
stations. As a simplification, we have assumed that the LoS/NLoS status of
interfering base stations with respect to the serving base station are
independent of the LoS and NLoS status of the base stations observed by the
typical SO following A2.

Thus, under this assumption, we represent the interfering BSs as a marked
point process

$\displaystyle\tilde{\Pi}_{\text{B}}^{0}=\\{(Z_{B}^{k},M_{k},R_{k}):R_{k}\in\Pi_{\text{B}},M_{k}\sim\text{Bern}(p_{\text{LoS}}(R_{k})),Z_{B}^{k}\sim(1-\xi_{\text{B,Rx}})\text{
Bern}(p_{\text{B,Rx}}(R_{k}))+\xi_{\text{B,Rx}}\text{ IID}\\}.$ (77)

Equivalently, by the Independent Thinning Theorem the receive antenna gain
marks partition this process into two independent classes of points: those
that inside/outside of the receive beam. These are denoted as
$\Pi_{\text{B,1}}^{0}$ and $\Pi_{\text{B,2}}^{0}$, respectively. Their
respective intensity functions are $\lambda_{\text{B,1}}^{0}(r;R_{0})$ and
$\lambda_{\text{B,2}}^{0}(r;R_{0})$.

In light of Lemma 2, the corresponding expressions for these terms are not
amenable to the characterization of their associated path loss Mellin
Transforms. Hence, we derive the following upper and lower bounds on
$\lambda_{\text{B,1}}^{0}$ and $\lambda_{\text{B,2}}^{0}(r;R_{0})$. These
bounds are defined with respect to a collection of parameters (dependent on
$R_{0}$), which for clarity of exposition we relegate to Appendix E.

Lemma 8: (Bounds on the Intensity Measures of the Palm Interference Process)
Let $k(\theta_{\text{B,Rx}})$, $k^{\ell}(\theta_{\text{B,Rx}})$,
$k^{u}(\theta_{\text{B,Rx}})$,
$\\{(r_{i},c_{i},m_{i})\\}_{i\in[k(\theta_{\text{B,Rx}})]}$,
$\\{(r^{\ell}_{i},c^{\ell}_{i},m^{\ell}_{i})\\}_{i\in[k^{\ell}(\theta_{\text{B,Rx}})]}$,
$\\{(r^{u}_{i},c^{u}_{i},m^{u}_{i})\\}_{i\in[k^{u}(\theta_{\text{B,Rx}})]}$,
and $\\{\gamma_{k}\\}_{k\in[M_{a}+1]}$ (where $M_{a}\in\mathbb{N}$) be as
defined in Appendix E. Then

  1. i)

The following upper bounds hold:
$\lambda_{\text{B,1}}^{0}(r;R_{0})\leq\rho^{0}_{\text{B,1}}(r;R_{0})$ and
$\lambda_{\text{B,2}}^{0}(r;R_{0})\leq\rho^{0}_{\text{B,2}}(r;R_{0})$, where

$\displaystyle\rho^{0}_{\text{B,1}}(r;R_{0})=2\lambda_{\text{B}}r\left(\frac{\theta_{\text{B,Rx}}}{2}\right.-\frac{\theta_{\text{B,Rx}}}{2}\sum_{i=1}^{k(\theta_{\text{B,Rx}})}\mathds{1}\left\\{r\in\left[r_{i-1},\min\left\\{r_{i},2R_{0}\cos\left(\frac{\theta_{\text{B,Rx}}}{2}\right)\right\\}\right]\right\\}e^{-\beta(c_{i}+m_{i}r)}$
$\displaystyle-\sum_{i=1}^{k^{u}(\theta_{\text{B,Rx}})}\frac{\pi}{2}\mathds{1}\left\\{r\in\left[\max\left\\{r_{i-1}^{u},2R_{0}\cos\left(\frac{\theta_{\text{B,Rx}}}{2}\right)\right\\},\min\left\\{r_{i}^{u},2R_{0}\right\\}\right]\right\\}e^{-\beta(c_{i}^{u}+m_{i}^{u}r)}$
$\displaystyle\left.+\sum_{k=0}^{M_{a}+1}\frac{\gamma_{k}}{(2R_{0})^{2k+1}}\mathds{1}\left\\{r\in\left[\max\left\\{r_{i-1}^{u},2R_{0}\cos\left(\frac{\theta_{\text{B,Rx}}}{2}\right)\right\\},\min\left\\{r_{i}^{u},2R_{0}\right\\}\right]\right\\}r^{2k+1}e^{-\beta(c_{i}^{u}+m_{i}^{u}r)}\right),$
(78)

and

$\displaystyle\rho^{0}_{\text{B,2}}(r;R_{0})=2\lambda_{\text{B}}r\left(\frac{2\pi-\theta_{\text{B,Rx}}}{2}\right.-\sum_{i=1}^{k^{\ell}(\theta_{\text{B,Rx}})}\frac{\pi}{2}\mathds{1}\left\\{r\in\left[r_{i-1}^{\ell},\min\left\\{r_{i}^{\ell},2R_{0}\cos\left(\frac{\theta_{\text{B,Rx}}}{2}\right)\right\\}\right]\right\\}e^{-\beta(c_{i}^{\ell}+m_{i}^{\ell}r)}$
$\displaystyle+\sum_{k=0}^{M_{a}+1}\frac{\gamma_{k}}{(2R_{0})^{2k+1}}\mathds{1}\left\\{r\in\left[r_{i-1}^{\ell},\min\left\\{r_{i}^{\ell},2R_{0}\cos\left(\frac{\theta_{\text{B,Rx}}}{2}\right)\right\\}\right]\right\\}r^{2k+1}e^{-\beta(c_{i}^{\ell}+m_{i}^{\ell}r)}$
$\displaystyle+\mathds{1}\left\\{r\in\left[0,\min\left\\{R_{0},2R_{0}\cos\left(\frac{\theta_{\text{B,Rx}}}{2}\right)\right\\}\right]\right\\}\frac{\theta_{\text{B,Rx}}}{2}e^{-\beta(R_{0}-r)}$
$\displaystyle\left.+\mathds{1}\left\\{r\in\left[R_{0},2R_{0}\cos\left(\frac{\theta_{\text{B,Rx}}}{2}\right)\right]\right\\}\frac{\theta_{\text{B,Rx}}}{2}e^{-\beta(r-R_{0})}\right).$
(79)

  2. ii)

The following lower bounds hold:
$\lambda_{\text{B,1}}^{0}(r;R_{0})\geq\nu^{0}_{\text{B,1}}(r;R_{0})$ and
$\lambda_{\text{B,2}}^{0}(r;R_{0})\geq\nu^{0}_{\text{B,2}}(r;R_{0})$, where

$\displaystyle\nu^{0}_{\text{B,1}}(r;R_{0})=2\lambda_{\text{B}}r\left(\frac{\theta_{\text{B,Rx}}}{2}-\mathds{1}\left\\{r\in\left[0,2R_{0}\cos\left(\frac{\theta_{\text{B,Rx}}}{2}\right)\right]\right\\}\frac{\theta_{\text{B,Rx}}}{2}e^{-\beta\lvert
R_{0}-r\rvert}\right.$
$\displaystyle-\frac{\pi}{2}\mathds{1}\left\\{r\in\left[2R_{0}\cos\left(\frac{\theta_{\text{B,Rx}}}{2}\right),2R_{0}\right]\right\\}e^{-\beta\lvert
R_{0}-r\rvert}$
$\displaystyle\left.+\sum_{k=0}^{M_{a}}\frac{\gamma_{k}}{(2R_{0})^{2k+1}}\mathds{1}\left\\{r\in\left[2R_{0}\cos\left(\frac{\theta_{\text{B,Rx}}}{2}\right),2R_{0}\right]\right\\}r^{2k+1}e^{-\beta\lvert
R_{0}-r\rvert}\right),$ (80)

and

$\displaystyle\nu^{0}_{\text{B,2}}(r;R_{0})=2\lambda_{\text{B}}r\left(\frac{2\pi-\theta_{\text{B,Rx}}}{2}-\frac{\pi}{2}\mathds{1}\left\\{r\in\left[0,2R_{0}\cos\left(\frac{\theta_{\text{B,Rx}}}{2}\right)\right]\right\\}e^{-\beta\lvert
R_{0}-r\rvert}\right.$
$\displaystyle+\sum_{k=0}^{M_{a}}\frac{\gamma_{k}}{(2R_{0})^{2k+1}}\mathds{1}\left\\{r\in\left[0,2R_{0}\cos\left(\frac{\theta_{\text{B,Rx}}}{2}\right)\right]\right\\}r^{2k+1}e^{-\beta\lvert
R_{0}-r\rvert}$
$\displaystyle\left.+\frac{\theta_{\text{B,Rx}}}{2}\sum_{i=1}^{k(\theta_{\text{B,Rx}})}\mathds{1}\left\\{r\in\left[r_{i-1},\min\left\\{r_{i},2R_{0}\cos\left(\frac{\theta_{\text{B,Rx}}}{2}\right)\right\\}\right]\right\\}e^{-\beta(c_{i}+m_{i}r)}\right).$
(81)

Proof: See Appendix E.

Though somewhat tedious, these bounds are useful in that they are expressed in
terms of piecewise poly-exponential functions – which arise from upper/lower
bounds on the $J(\cdot)$ function in the definition of
$\lambda_{\text{B}}^{0}$ and correspond to the so called exclusion region.
These, in turn, allow us to characterize the corresponding path loss Mellin
Transforms in closed form. Note that $M_{a}$ and
$\\{\gamma_{k}\\}_{k\in[N_{a}]}$ correspond to constants in a finite order
polynomial approximation of the $\arccos$ function, which is necessary to
characterize upper and lower bounds on $\lambda_{\text{B,1}}^{0}$ and
$\lambda_{\text{B,2}}^{0}$ in the manner above.

By the Superposition Theorem, using
$\rho^{0}_{\text{B,1}}(r;R_{0})$/$\rho^{0}_{\text{B,2}}(r;R_{0})$ instead of
$\lambda_{\text{B,1}}^{0}(r;R_{0})$/$\lambda_{\text{B,2}}^{0}(r;R_{0})$ in the
definition of $\Pi_{\text{B,1}}^{0}$/$\Pi_{\text{B,2}}^{0}$ lead to upper
bounds on the interference term of ${\text{SINR}_{\text{gen}}}$, and hence
lower bounds on the overall SINR model. Similarly using
$\nu^{0}_{\text{B,1}}(r;R_{0})$/$\nu^{0}_{\text{B,2}}(r;R_{0})$ leads to upper
bounds on the SINR model. We conclude our analysis of the Palm interference
process by deriving the path loss Mellin Transforms associated with the
LoS/NLoS points in the upper and lower bounding interference processes defined
with respect to
$\rho^{0}_{\text{B,1}}(r;R_{0})$/$\rho^{0}_{\text{B,2}}(r;R_{0})$ and
$\nu^{0}_{\text{B,1}}(r;R_{0})$/$\nu^{0}_{\text{B,2}}(r;R_{0})$.

Applying the Independent Thinning Theorem again to $\Pi_{\text{B,1}}^{0}$ and
$\Pi_{\text{B,2}}^{0}$ with respect to the LoS Marks, results in four
independent processes: $\Pi_{\text{L,1}}^{0}$, $\Pi_{\text{N,1}}^{0}$,
$\Pi_{\text{L,2}}^{0}$, and $\Pi_{\text{N,2}}^{0}$. These denote the set of
points that are LoS/NLoS with respect to the serving BS and inside the receive
beam and those that are LoS/NLoS with respect to the serving BS and outside
the receive beam, respectively. Let their associated intensity measures be
denoted as $\lambda_{\text{L,1}}^{0}$, $\lambda_{\text{N,1}}^{0}$,
$\lambda_{\text{L,2}}^{0}$, and $\lambda_{\text{N,2}}^{0}$, respectively. As
an immediate consequence of Lemma 8 and the definition of $p_{\text{LoS}}(r)$,
we have

$\displaystyle\nu^{0}_{\text{L,1}}(r;R_{0})=e^{-\beta
r}\nu^{0}_{\text{B,1}}(r;R_{0})\leq\lambda_{\text{L,1}}^{0}(r;R0)\leq
e^{-\beta r}\rho^{0}_{\text{B,1}}(r;R_{0})=\rho^{0}_{\text{L,1}}(r;R_{0}),$
(82)

and

$\displaystyle\nu^{0}_{\text{N,1}}(r;R_{0})=(1-e^{-\beta
r})\nu^{0}_{\text{B,1}}(r;R_{0})\leq\lambda_{\text{N,1}}^{0}(r;R0)\leq(1-e^{-\beta
r})\rho^{0}_{\text{B,1}}(r;R_{0})=\rho^{0}_{\text{N,1}}(r;R_{0}).$ (83)

Similar inequalities and definitions hold for
$\nu^{0}_{\text{L,2}}$/$\rho^{0}_{\text{L,2}}$ and
$\nu^{0}_{\text{N,2}}$/$\rho^{0}_{\text{N,2}}$.

Using these quantities define

$\displaystyle\mathcal{M}_{\nu^{0}_{\text{a,k}}\circ
g_{a}^{-1}}(p;A,R_{0})=\int_{A}g_{a}(r)^{{}_{p}-1}\nu^{0}_{\text{a,k}}(r;R_{0})dr$
$\displaystyle a\in\\{L,N\\},k\in\\{1,2\\},$ (84)

and

$\displaystyle\mathcal{M}_{\rho^{0}_{\text{a,k}}\circ
g_{a}^{-1}}(p;A,R_{0})=\int_{A}g_{a}(r)^{{}_{p}-1}\rho^{0}_{\text{a,k}}(r;R_{0})dr$
$\displaystyle a\in\\{L,N\\},k\in\\{1,2\\}.$ (85)

These admit closed form expressions in terms of the generalized incomplete
Gamma function, which are listed in Lemma 9, found in Appendix F.

###  V-B Bounds on the CCDF of the Generic Radar SINR Model

Using the results we have developed thus far, we now obtain integral closed
form bounds on the CCDF of ${\text{SINR}_{\text{gen}}}$. This is the primary
vehicle used to obtain bounds and approximations for the radar coverage
probability.

Theorem 3: (Bounds on the CCDF of ${\text{SINR}_{\text{gen}}}$) Let
$N_{w}\in\mathbb{N}$ denote an arbitrary number of windows employed in the
bounds for interference Laplace Transform, and $M_{a}\in\mathbb{N}$ denote an
arbitrary order for the $\arccos$ approximation used in the bounds on the
conditional interference intensity measures.

Additionally, let $\\{d^{L}_{i}\\}_{i=0}^{N_{w}+1}$ denote arbitrary, ordered
boundaries for the windows used to partition $\mathbb{R}_{+}$ for the LoS
interference terms and $\\{d^{N}_{i}\\}_{i=0}^{N_{w}+1}$ denote arbitrary,
ordered boundaries for the windows used to partition $\mathbb{R}_{+}$ for the
NLoS interference terms. Without loss of generality, take
$d^{L}_{0},d^{N}_{0}=0$ and $d^{L}_{N_{w}},d^{N}_{N_{w}+1}=\infty$. For
conciseness, let $W^{L}_{i}=[d^{L}_{i-1},d^{L}_{i}]$ and
$W^{N}_{i}=[d^{N}_{i-1},d^{N}_{i}]$.

Finally, define $\xi_{1}=1$ and $\xi_{2}=\xi_{\text{B,Rx}}$. The following
bounds hold.

  1. i)

The CCDF of ${\text{SINR}_{\text{gen}}}$ is upper bounded as
$\mathbb{P}({\text{SINR}_{\text{gen}}}\geq\tau)\leq P_{\text{c,
rad}}^{\text{UB}}\left(\tau;\mathcal{L}_{F_{L}},\mathcal{L}_{F_{N}}\right)$,
where

$\displaystyle P_{\text{c, rad}}^{\text{UB}}(\tau;$
$\displaystyle\mathcal{L}_{F_{L}},\mathcal{L}_{F_{N}})=\left(1-\exp\left(\frac{-2\pi\lambda_{\text{B}}}{\beta^{2}}\right)\right)\int_{\mathbb{R}_{+}}f_{R_{0}}(u)\exp\left({\frac{-\tau\nu_{\text{rad}}}{g_{\text{L,ret}}(u)}}\right)$
$\displaystyle\prod_{k=1}^{2}\exp\left(-\sum_{i=1}^{N_{w}+1}H_{\text{UB}}\left(\frac{\tau\xi_{k}}{g_{\text{L,ret}}(u)},W^{L}_{i};1-\mathcal{L}_{F_{L}},\mathcal{M}_{\nu^{0}_{\text{L,k}}\circ
g_{\text{L}}^{-1}}(\cdot;\cdot,u)\right)\right)$
$\displaystyle\prod_{k=1}^{2}\exp\left(-\sum_{i=1}^{N_{w}}H_{\text{UB}}\left(\frac{\tau\xi_{k}}{g_{\text{L,ret}}(u)},W^{N}_{i};1-\mathcal{L}_{F_{N}},\mathcal{M}_{\nu^{0}_{\text{N,k}}\circ
g_{\text{N}}^{-1}}(\cdot;\cdot,u)\right)\right)du.$ (86)

  2. ii)

The CCDF of ${\text{SINR}_{\text{gen}}}$ is lower bounded as
$\mathbb{P}({\text{SINR}_{\text{gen}}}\geq\tau)\geq P_{\text{c,
rad}}^{\text{LB}}\left(\tau;\mathcal{L}_{F}\right)$, where

$\displaystyle P_{\text{c, rad}}^{\text{LB}}(\tau;$
$\displaystyle\mathcal{L}_{F_{L}},\mathcal{L}_{F_{N}})=\left(1-\exp\left(\frac{-2\pi\lambda_{\text{B}}}{\beta^{2}}\right)\right)\int_{\mathbb{R}_{+}}f_{R_{0}}(u)\exp\left({\frac{-\tau\nu_{\text{rad}}}{g_{\text{L,ret}}(u)}}\right)$
$\displaystyle\prod_{k=1}^{2}\exp\left(-\sum_{i=1}^{N_{w}+1}H_{\text{LB}}\left(\frac{\tau\xi_{k}}{g_{\text{L,ret}}(u)},W^{L}_{i};1-\mathcal{L}_{F_{L}},\mathcal{M}_{\rho^{0}_{\text{L,k}}\circ
g_{\text{L}}^{-1}}(\cdot;\cdot,u)\right)\right)$
$\displaystyle\prod_{k=1}^{2}\exp\left(-\sum_{i=1}^{N_{w}}H_{\text{LB}}\left(\frac{\tau\xi_{k}}{g_{\text{L,ret}}(u)},W^{N}_{i};1-\mathcal{L}_{F_{N}},\mathcal{M}_{\rho^{0}_{\text{N,k}}\circ
g_{\text{N}}^{-1}}(\cdot;\cdot,u)\right)\right.$ $\displaystyle\hskip
108.405pt\left.+\frac{\tau\xi_{k}}{g_{\text{L,ret}}(u)}\mathbb{E}[F_{N}]\mathcal{M}_{\rho^{0}_{\text{N,k}}\circ
g_{\text{N}}^{-1}}(2;W^{N}_{N_{w}+1},u)\right)du.$ (87)

Proof: See Appendix G.

While we have established Theorem 3 in the specific setting of the
${\text{SINR}_{\text{gen}}}$, it may be readily extended to other settings.
Generally, when the path loss Mellin Transforms of the interference process
may be obtained in closed form, Theorem 3 provides a robust framework for
obtaining integral closed form upper and lower bounds for SINR models with
Rayleigh fading on the desired signal. Using an approach similar to [9], this
framework may be exploited for Nakagami fading as well.

Up to now, we have left the partitions used in the bound for the interference
Laplace Transforms as arbitrary parameters. Choosing these terms correctly is
important for obtaining tight bounds. The problem of finding optimal
partitions is generally difficult (it is akin to a generalization of weighted
k-means clustering to a continuum), hence we propose a heuristic method base
on uniformly sampling the reciprocal of the path loss functions.

Definition 3: (Uniform Path Gain Windowing) Let $N_{w}\in\mathbb{N}$ and
choose a desired end point, $d_{N_{w}}\in\mathbb{R}_{+}$. Further, for
$a\in\\{L,N\\}$ define $f_{\text{a}}(r)=1/g_{\text{a}}(r)$ and define
$d_{min}=g_{\text{a}}^{-1}(1)$. Then the partitions in Theorem 3 may be
constructed as follows

$\displaystyle\Delta_{a}=\frac{f_{\text{a}}^{-1}(d_{N_{w}})-f_{\text{a}}^{-1}(d_{min})}{N_{w}}$
(88) $\displaystyle
d^{a}_{i}=\begin{cases}f_{\text{a}}\left(f_{\text{a}}^{-1}(d_{min})+\Delta_{a}i\right)&i\leq
N_{w}\\\ \infty&\text{otherwise}\end{cases}$ (89) $\displaystyle
W^{a}_{i}=[d^{a}_{i-1},d^{a}_{i}].$ (90)

While suboptimal, this method attempts to capture the tradeoff between the
relative strength of closer interferers with the fact that the density of
interferers (on $\mathbb{R}_{+}$) increases with distance. Indeed, as we
demonstrate in Section VII, this windowing method leads to sharp bounds on
coverage and capacity.

###  V-C Bounds on Ergodic Sensing Efficiency Using the Generic Model

Before proceeding, it is worth mentioning that the upper and lower bounds for
the sensing coverage probability with respect to the generic model in Theorem
3, in tandem with Theorem 1, imply the following bounds on the ergodic sensing
efficiency.

Corollary 3: (Bounds on the Ergodic Sensing Efficiency) The ergodic sensing
efficiency defined with respect to the generic SINR model,
${\text{SINR}_{\text{gen}}}$, is lower bounded as

$\displaystyle
T_{\text{CPI}}\mathbb{E}[C_{\text{rad}}^{UB}]\geq\mathbb{E}\left[\frac{1}{2}\log_{2}\left(1+G\cdot{\text{SINR}_{\text{rad}}}\right)\right]\geq\frac{G}{2\ln(2)}\int_{\mathbb{R}_{+}}P_{\text{c,
rad}}^{\text{LB}}\left(\tau;\mathcal{L}_{F_{L}},\mathcal{L}_{F_{N}}\right)\left(1+G\tau\right)^{-1}d\tau.$
(91)

Similarly, it is upper bounded as

$\displaystyle
T_{\text{CPI}}\mathbb{E}[C_{\text{rad}}^{UB}]\leq\mathbb{E}\left[\log_{2}\left(1+\frac{1}{2}G\cdot{\text{SINR}_{\text{rad}}}\right)\right]\leq\frac{G}{2\ln(2)}\int_{\mathbb{R}_{+}}P_{\text{c,
rad}}^{\text{UB}}\left(\tau;\mathcal{L}_{F_{L}},\mathcal{L}_{F_{N}}\right)\left(1+\frac{G}{2}\tau\right)^{-1}d\tau.$
(92)

Proof: The terms inside the expectations follow from Theorem 1. Using that
bounds from Theorem 3, one may show that the expectations are lower bounded by
the integral terms in a manner similar to [37, Theorem 2]. $\hfill\Box$

###  V-D Characterizing the Laplace Transform of the Interference Fading Terms

Having bounded the CCDF of ${\text{SINR}_{\text{gen}}}$, all that remains to
characterize the alternate SINR models is to determine the Laplace Transforms
of their fading terms: ${\text{AM}}(\mathbb{F},\bm{\theta})$,
${\text{GM}}(\mathbb{F},\bm{\theta})$, and
${\text{HM}}(\mathbb{F},\bm{\theta})$. Due to the structure of the means
involved in their definitions, these are non-trivial. We provide bounds and
approximations for these quantities in the following series of lemmas.

Since the fading terms are dependent on the LoS/NLoS status of the BS in
question, wherever necessary we use $a\in\\{L,N\\}$ as a proxy for subscripts
denoting LoS/NLoS parameters. Specifically, we shall denote the dependence of
the LoS/NLoS fading parameters as
$\mathcal{L}_{{\text{AM}}(\mathbb{F},\bm{\theta})}(s;N_{a})$,
$\mathcal{L}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{a})$, and
$\mathcal{L}_{{\text{HM}}(\mathbb{F},\bm{\theta})}(s;N_{a})$

We begin with the Laplace Transform of ${\text{GM}}(\mathbb{F},\bm{\theta})$.
Notably, we make use of the follow relation between Laplace and Mellin
Transforms.

Proposition 2: (Relation between Mellin and Laplace Transforms of Nonnegative
Random Variables) Let $X$ be a non-negative random variable and recall that
$\Gamma(p)=\int_{\mathbb{R}_{+}}x^{p-1}e^{-x}dx$. Then

$\displaystyle\mathcal{L}_{X}(s)=\mathcal{M}^{-1}\\{\mathcal{M}_{X}(1-p)\Gamma(p)\\}(s).$
(93)

Proof: See Appendix I.

Lemma 10: (Exact Form and Approximation for the Laplace Transform of
${\text{GM}}(\mathbb{F},\bm{\theta})$) The following hold.

  1. i)

$\mathcal{L}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{a})$ may be expressed
as

$\displaystyle\mathcal{L}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{a})=\sum_{r=0}^{T}p_{\text{B}}^{T-r}(1-p_{\text{B}})^{r}\sum_{j=1}^{T\choose
r}\mathcal{L}_{{\text{GM}}(\mathbb{H},\mathbb{q})}\left(s\prod_{i\in
T_{j}^{r}}\xi_{\text{B,Tx}}^{w_{t_{i}}};N_{a}\right),$ (94)

where $T^{r}_{j}$ denotes the $j^{th}$ possible permutation of the indices of
$T$ time slots such that $r$ of them have $B_{t}=\xi_{\text{B,Tx}}$, and

$\displaystyle\mathcal{L}_{{\text{GM}}(\mathbb{H},\mathbb{q})}\left(s;N_{a}\right)=\frac{1}{\Gamma(N_{\text{a}})^{N}}H^{1,N_{c}}_{N_{c},1}\left(\frac{s}{N_{\text{a}}}\Bigg{|}\begin{matrix}&(1-N_{a},q_{1})\dots(1-N_{a},q_{N_{c}})\\\
&(0,1)\end{matrix}\right)$ (95)

with $H^{m,n}_{p,q}(\cdot|\cdot)$ denoting Fox’s H function.

  2. ii)

${\text{GM}}(\mathbb{H},\mathbb{q})$ is closely approximated by a gamma random
variable,
$H^{\prime}\sim\text{Gamma}(\alpha_{0}(\mathbb{q},N_{\text{a}}),\beta_{0}(\mathbb{q},N_{\text{a}}))$,
where $\alpha_{0}(\mathbb{q},N_{\text{a}})$ and
$\beta_{0}(\mathbb{q},N_{\text{a}})$ are such that the moments of $H^{\prime}$
and matched to $GM(\mathbb{H},\mathbb{q})$:

$\displaystyle\frac{\alpha_{0}(\mathbb{q},N_{\text{a}})}{\beta_{0}(\mathbb{q},N_{\text{a}})}=N_{\text{a}}^{-1}\prod_{n=1}^{N}\frac{\Gamma(q_{n}+N_{\text{a}})}{\Gamma(N_{\text{a}})}$
(96)
$\displaystyle\frac{\alpha_{0}(\mathbb{q},N_{\text{a}})(\alpha_{0}(\mathbb{q},N_{\text{a}})+1)}{\beta_{0}(\mathbb{q},N_{\text{a}})^{2}}=N_{\text{a}}^{-2}\prod_{n=1}^{N}\frac{\Gamma(2q_{n}+N_{\text{a}})}{\Gamma(N_{\text{a}})}.$
(97)

Thus,

$\displaystyle\mathcal{L}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{a})\approx\sum_{r=0}^{T}p_{\text{B}}^{T-r}(1-p_{\text{B}})^{r}\sum_{j=1}^{T\choose
r}\left(1+\frac{s\xi_{\text{B,Tx}}^{\sum_{i\in
T_{j}^{r}}w_{t_{i}}}}{\beta_{0}(\mathbb{q},N_{\text{a}})}\right)^{-\alpha_{0}(\mathbb{q},N_{\text{a}})}.$
(98)

Proof: See Appendix H.

Though exact, part i) is difficult to use as Fox’s H function is challenging
to implement in a numerically efficient manner. Hence we employ the gamma
approximation in part ii). The gamma approximation is justified in that, for
large $N$, ${\text{GM}}(\mathbb{H},\mathbb{q})$ is closely approximated by a
log-normal random variable (a consequence of the Central Limit Theorem). A
Log-normal random variable is, in turn, is closely approximated by a gamma
random variable.

While (98) may be evaluated efficiently for small $T$, for large $T$ its
evaluation becomes cumbersome. Hence, we derive upper and lower bounds in the
following lemma.

Lemma 11: (Bounds on the Laplace Transform of
${\text{GM}}(\mathbb{F},\bm{\theta})$) For $s\in\mathbb{R}_{+}$, the
approximation of $\mathcal{L}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{a})$
in (98) is upper bounded by
$\mathcal{L}^{\text{UB}}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{a})$,
where

$\displaystyle\mathcal{L}^{\text{UB}}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{a})=\prod_{t=1}^{T}\left(p_{\text{B}}+(1-p_{\text{B}})\left(1+\frac{s\xi_{\text{B,Tx}}}{\beta_{0}(\mathbb{q},N_{\text{a}})}\right)^{-\alpha_{0}(\mathbb{q},N_{\text{a}}){w_{t}}}\right).$
(99)

It is lower bounded by
$\mathcal{L}^{\text{UB}}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{a})$

$\displaystyle\mathcal{L}^{\text{LB}}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{a})=\sum_{r=0}^{T}{T\choose
r}p_{\text{B}}^{T-r}(1-p_{\text{B}})^{r}\left(1+\frac{s{T\choose
r}^{-1}\sum_{j=1}^{T\choose r}\xi_{\text{B,Tx}}^{\sum_{i\in
T_{j}^{r}}w_{t_{i}}}}{\beta_{0}(\mathbb{q},N_{\text{a}})}\right)^{-\alpha_{0}(\mathbb{q},N_{\text{a}})}.$
(100)

Proof: See Appendix H.

Finally, we now consider the Laplace Transforms of
${\text{AM}}(\mathbb{F},\bm{\theta})$ and
${\text{HM}}(\mathbb{F},\bm{\theta})$. Since these correspond to lower and
upper bounding models, we derive lower and upper bounds, respectively.

The Laplace Transform of ${\text{AM}}(\mathbb{F},\bm{\theta})$ may be lower
bounded as follows.

Lemma 12: (Lower Bound on the Laplace Transform of
${\text{AM}}(\mathbb{F},\bm{\theta})$) For $s\in\mathbb{R}_{+}$, the Laplace
Transform of ${\text{AM}}(\mathbb{F},\bm{\theta})$ is lower bounded by
$\mathcal{L}^{\text{LB}}_{{\text{AM}}(\mathbb{F},\bm{\theta})}(s;N_{a})$,
where

$\displaystyle\mathcal{L}^{\text{LB}}_{{\text{AM}}(\mathbb{F},\bm{\theta})}(s;N_{a})=\sum_{r=0}^{T}{T\choose
r}p_{\text{B}}^{T-r}(1-p_{\text{B}})^{r}\left(1+\frac{s{T\choose
r}^{-1}\sum_{j=1}^{T\choose r}\left((\xi_{\text{B,Tx}}-1){\sum_{i\in
T_{j}^{r}}w_{t_{i}}}+1\right)}{NN_{\text{a}}}\right)^{-NN_{\text{a}}}.$ (101)

Proof: See Appendix H.

The Laplace Transform of ${\text{HM}}(\mathbb{F},\bm{\theta})$ may be upper
bounded as follows.

Lemma 13: (Lower Bound on the Laplace Transform of
${\text{HM}}(\mathbb{F},\bm{\theta})$) For $s\in\mathbb{R}_{+}$, the Laplace
Transform of ${\text{HM}}(\mathbb{F},\bm{\theta})$ is upper bounded by
$\mathcal{L}^{\text{UB}}_{{\text{HM}}(\mathbb{F},\bm{\theta})}(s;N_{a})$.
Depending on $N_{a}$ this may be expressed as follows.

  1. i)

If $N_{a}=1$,

$\displaystyle\mathcal{L}^{\text{UB}}_{{\text{HM}}(\mathbb{F},\bm{\theta})}(s;N_{a})=p_{\text{B}}^{T}\left(1+\frac{s}{\left(\sum_{n=1}^{N}q_{n}\right)^{2}}\right)^{-1}+(1-p_{\text{B}}^{T})\left(1+\frac{\xi_{\text{B,Tx}}s}{\left(\sum_{n=1}^{N}q_{n}\right)^{2}}\right)^{-1}.$
(102)

  2. ii)

Otherwise, for $N_{a}>1$, define

$\displaystyle
m_{1}=\left(\left(p_{\text{B}}+(1-p_{\text{B}})\xi_{\text{B,Tx}}^{-1}\right)\frac{N_{\text{a}}}{N_{\text{a}}-1}\right)^{-1},$
(103) $\displaystyle
m_{2}=\left(\sum_{r=0}^{T}p_{\text{B}}^{T-r}(1-p_{\text{B}})^{r}\sum_{j=1}^{T\choose
r}\left(1-\left(\sum_{i\in T_{j}^{r}}w_{t_{i}}\right)+\left(\sum_{i\in
T_{j}^{r}}w_{t_{i}}\right)\xi_{\text{B,Tx}}^{-1}\right)^{-2}\right)N_{\text{a}}^{-2}\prod_{n=1}^{N}\frac{\Gamma(2q_{n}+N_{\text{a}})}{\Gamma(N_{\text{a}})}$
(104)

Then,

$\displaystyle\mathcal{L}^{\text{UB}}_{{\text{HM}}(\mathbb{F},\bm{\theta})}(s;N_{a})=\frac{m_{2}-m_{1}^{2}}{m_{2}}+\frac{m_{1}^{2}}{m_{2}}\exp\left(-s\frac{m_{2}}{m_{1}}\right).$
(105)

Proof: See Appendix H.

In closing, recall that the lower bound in Theorem 3 requires the mean of the
NLoS fading term. Hence, we further require
$\mathbb{E}\left[{\text{GM}}(\mathbb{F},\bm{\theta})\right]$ and
$\mathbb{E}\left[{\text{AM}}(\mathbb{F},\bm{\theta})\right]$. For
${\text{GM}}(\mathbb{F},\bm{\theta})$ this follows from part ii) of Lemma 10.
The mean of ${\text{AM}}(\mathbb{F},\bm{\theta})$ may be readily obtained as
$\mathbb{E}\left[{\text{AM}}(\mathbb{F},\bm{\theta})\right]=\mathbb{E}\left[\lvert
H\rvert^{2}B\right]$ noting linearity of the arithmetic mean.

###  V-E Statements for Radar Coverage Probability

Having characterized the Laplace Transforms of the interference fading terms
of the alternate SINR models, we arrive at our main results: bounds and
approximations for the sensing coverage probability.

Theorem 4: (Characterization of Sensing Coverage Probability) The following
bounds and approximations hold on the sensing coverage probability,
$\mathbb{P}({\text{SINR}_{\text{rad}}}\geq\tau)$.

  1. i)

The sensing coverage probability is lower bounded as

$\displaystyle\mathbb{P}({\text{SINR}_{\text{rad}}}\geq\tau)\geq P_{\text{c,
rad}}^{\text{LB}}\left(\tau;\mathcal{L}^{\text{LB}}_{{\text{AM}}(\mathbb{F},\bm{\theta})}(s;N_{\text{L}}),\mathcal{L}^{\text{LB}}_{{\text{AM}}(\mathbb{F},\bm{\theta})}(s;N_{\text{N}})\right).$
(106)

  2. ii)

The sensing coverage probability is upper bounded as

$\displaystyle\mathbb{P}({\text{SINR}_{\text{rad}}}\geq\tau)\leq P_{\text{c,
rad}}^{\text{UB}}\left(\tau;\mathcal{L}^{\text{UB}}_{{\text{HM}}(\mathbb{F},\bm{\theta})}(s;N_{\text{L}}),\mathcal{L}^{\text{UB}}_{{\text{HM}}(\mathbb{F},\bm{\theta})}(s;N_{\text{N}})\right).$
(107)

  3. iii)

The sensing coverage probability is approximately lower bounded as

$\displaystyle\mathbb{P}({\text{SINR}_{\text{rad}}}\geq\tau)\approx\mathbb{P}({\text{SINR}_{\text{GM}}}\geq\tau)\geq
P_{\text{c,
rad}}^{\text{LB}}\left(\tau;\mathcal{L}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{\text{L}}),\mathcal{L}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{\text{N}})\right).$
(108)

  4. iv)

The sensing coverage probability is approximately upper bounded as

$\displaystyle\mathbb{P}({\text{SINR}_{\text{rad}}}\geq\tau)\approx\mathbb{P}({\text{SINR}_{\text{GM}}}\geq\tau)\leq
P_{\text{c,
rad}}^{\text{UB}}\left(\tau;\mathcal{L}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{\text{L}}),\mathcal{L}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{\text{N}})\right).$
(109)

Proof: Using Proposition 1, we may bound and approximate the CCDF of
${\text{SINR}_{\text{rad}}}$ using the alternate SINR models,
${\text{SINR}_{\text{AM}}}$, ${\text{SINR}_{\text{GM}}}$, and
${\text{SINR}_{\text{HM}}}$. We may then employ Theorem 3 to further bound the
CCDFs of these respective models. The fact that the bounds for the
interference Laplace Transform in Theorem 2 are increasing in the Laplace
Transform of the fading terms concludes the proof, leveraging Lemmas 10
through 13. $\hfill\Box$

With respect to iii) and iv), looser bounds may be obtained using
$\mathcal{L}^{\text{LB}}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{\text{L}})$
and
$\mathcal{L}^{\text{LB}}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{\text{N}})$,
and
$\mathcal{L}^{\text{UB}}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{\text{L}})$
and
$\mathcal{L}^{\text{UB}}_{{\text{GM}}(\mathbb{F},\bm{\theta})}(s;N_{\text{N}})$,
respectively. Further, Theorem 4 may be used in tandem with Corollary 3 to
obtain approximate/exact upper and lower bounds on the ergodic sensing
efficiency. Overall, these results offer a highly expressive, yet tractable
means to analytically characterize the performance of radar tracking in a JCAS
network. In addition to capturing the effects of non-isotropic interference,
blockages, and varying antenna gains, our model further accounts for the
structure of the excitation waveform over time and frequency. This is done in
a rigorous manner by leveraging the Fisher Information of the induced
estimation problem via Theorem 1.

##  VI Characterization of Communication Coverage Probability and Ergodic
Communication Efficiency

We now conclude the presentation of our analytical results by characterizing
the communication coverage probability and ergodic communication efficiency.
As there is little novel in the expression for communication coverage
probability (aside from a usage of Theorem 2), we present that main result
without proof. Indeed the analysis of ${\text{SINR}_{\text{com}}}$ is largely
similar to that of ${\text{SINR}_{\text{gen}}}$.

Theorem 5: (Characterization of Communication Coverage Probability) The path
loss Mellin Transform of the LoS interference process may be expressed with
respect to the $G_{L}$ function defined in Appendix F,

$\displaystyle\mathcal{M}_{\lambda_{\text{L}}^{0}\circ
g_{\text{L}}^{-1}}(p;A,R_{0})=2\pi\lambda_{\text{B}}K_{\text{L}}^{p-1}G_{L}\left(p-1,A;[R_{0},\infty),\alpha_{\text{L}},\gamma_{\text{L}},1,\beta\right),$
(110)

where $\lambda_{\text{L}}^{0}(r;R_{0})=2\pi\lambda_{\text{B}}re^{-\beta
r}\mathds{1}\\{r\geq R_{0}\\}$.

Similarly, for the NLoS interference process, using the $G_{N}$ function
defined in Appendix F, we have

$\displaystyle\mathcal{M}_{\lambda_{\text{N}}^{0}\circ
g_{\text{N}}^{-1}}(p;A,R_{0})=2\pi\lambda_{\text{B}}K_{\text{N}}^{p-1}G_{N}\left(p-1,A;[R_{0},\infty),\alpha_{\text{N}},\gamma_{\text{N}},1,0,\beta\right),$
(111)

where $\lambda_{\text{N}}^{0}(r;R_{0})=2\pi\lambda_{\text{B}}r(1-e^{-\beta
r})\mathds{1}\\{r\geq R_{0}\\}$.

The LoS equivalent distribution of the serving base station (see [37, Lemma
5]) may be expressed as

$\displaystyle
f_{R_{0}}(r)=2\pi\lambda_{\text{B}}\lambda_{\text{EQ}}^{*}(r)e^{-2\pi\lambda_{\text{B}}\Lambda_{\text{EQ}}^{*}([0,r])},$
(112)

where

$\displaystyle\lambda_{\text{EQ}}^{*}(r)=re^{-\beta
r}+\frac{\psi^{-1}(r)(\alpha
r^{-1}+\gamma_{\text{L}})}{\gamma_{\text{N}}\psi^{-1}(r)+\alpha}\psi^{-1}(r)\left(1-e^{-\beta\psi^{-1}(r)}\right),$
(113)

and

$\displaystyle\Lambda_{\text{EQ}}^{*}([0,r])$
$\displaystyle=\beta^{-2}\left(\frac{\beta^{2}}{2}[\psi^{-1}(r)]^{2}+e^{-\beta\psi^{-1}(r)}\left(\beta\psi^{-1}(r)+1\right)-e^{-\beta
r}(\beta r+1)\right).$ (114)

Finally, the Laplace Transform for the LoS/NLoS interference fading terms may
be expressed as

$\displaystyle\mathcal{L}_{F_{a}}(s)=\mathbb{E}_{B,Z_{U}}\left[\left(1+\frac{sBZ_{U}}{N_{\text{a}}}\right)^{-N_{\text{a}}}\right].$
(115)

Then, letting $N_{w}\in\mathbb{N}$ and defining
$\\{W_{i}^{L}\\}_{i\in[N_{w}+1]}$ and $\\{W_{i}^{N}\\}_{i\in[N_{w}+1]}$ as in
Definition 3, the following hold

  1. i)

The communication coverage probability is lower bounded as
$\mathbb{P}({\text{SINR}_{\text{com}}}\geq\tau)\geq P_{\text{c,
com}}^{\text{LB}}\left(\tau;\mathcal{L}_{F_{L}},\mathcal{L}_{F_{N}}\right)$
where

$\displaystyle P_{\text{c,
com}}^{\text{LB}}(\tau;\mathcal{L}_{F_{L}},\mathcal{L}_{F_{N}})=\int_{\mathbb{R}_{+}}f_{R_{0}}(u)\exp\left({\frac{-\tau\nu_{\text{rad}}}{g_{\text{L}}(u)}}\right)$
$\displaystyle\exp\left(-\sum_{i=1}^{N_{w}+1}H_{\text{LB}}\left(\frac{\tau}{g_{\text{L}}(u)},W^{L}_{i};1-\mathcal{L}_{F_{L}},\mathcal{M}_{\lambda_{\text{L}}^{0}\circ
g_{\text{L}}^{-1}}(\cdot;\cdot,u)\right)\right)$
$\displaystyle\exp\left(-\sum_{i=1}^{N_{w}}H_{\text{LB}}\left(\frac{\tau}{g_{\text{L}}(u)},W^{N}_{i};1-\mathcal{L}_{F_{N}},\mathcal{M}_{\lambda_{\text{N}}^{0}\circ
g_{\text{N}}^{-1}}(\cdot;\cdot,u)\right)+\frac{\tau}{g_{\text{L}}(u)}\mathbb{E}[F_{N}]\mathcal{M}_{\lambda_{\text{N}}^{0}\circ
g_{\text{N}}^{-1}}(2;W^{N}_{N_{w}+1},u)\right)du.$ (116)

  2. ii)

The communication coverage probability is upper bounded as
$\mathbb{P}({\text{SINR}_{\text{com}}}\geq\tau)\leq P_{\text{c,
com}}^{\text{UB}}\left(\tau;\mathcal{L}_{F_{L}},\mathcal{L}_{F_{N}}\right)$
where

$\displaystyle P_{\text{c, com}}^{\text{UB}}$
$\displaystyle(\tau;\mathcal{L}_{F_{L}},\mathcal{L}_{F_{N}})=\int_{\mathbb{R}_{+}}f_{R_{0}}(u)\exp\left({\frac{-\tau\nu_{\text{rad}}}{g_{\text{L}}(u)}}\right)$
$\displaystyle\exp\left(-\sum_{i=1}^{N_{w}+1}H_{\text{UB}}\left(\frac{\tau}{g_{\text{L}}(u)},W^{L}_{i};1-\mathcal{L}_{F_{L}},\mathcal{M}_{\lambda_{\text{L}}^{0}\circ
g_{\text{L}}^{-1}}(\cdot;\cdot,u)\right)\right)$
$\displaystyle\exp\left(-\sum_{i=1}^{N_{w}}H_{\text{UB}}\left(\frac{\tau}{g_{\text{L}}(u)},W^{N}_{i};1-\mathcal{L}_{F_{N}},\mathcal{M}_{\lambda_{\text{N}}^{0}\circ
g_{\text{N}}^{-1}}(\cdot;\cdot,u)\right)\right)du.$ (117)

In a similar manner to Corollary 3 we may bound the ergodic communication
efficiency as follows

Corollary 5: (Bounds on the Ergodic Communication Efficiency) The ergodic
communication efficiency may be bounded as follow.

$\displaystyle
T_{\text{s}}\mathbb{E}\left[C_{\text{com}}^{(m,n)}\right]=\mathbb{E}\left[\log_{2}(1+{\text{SINR}_{\text{com}}})\right]\geq\frac{1}{\ln(2)}\int_{\mathbb{R}_{+}}P_{\text{c,
rad}}^{\text{LB}}\left(x;\mathcal{L}_{F_{L}},\mathcal{L}_{F_{N}}\right)\left(1+\tau\right)^{-1}d\tau,$
(118)

and

$\displaystyle
T_{\text{s}}\mathbb{E}\left[C_{\text{com}}^{(m,n)}\right]=\mathbb{E}\left[\log_{2}(1+{\text{SINR}_{\text{com}}})\right]\leq\frac{1}{\ln(2)}\int_{\mathbb{R}_{+}}P_{\text{c,
rad}}^{\text{UB}}\left(x;\mathcal{L}_{F_{L}},\mathcal{L}_{F_{N}}\right)\left(1+\tau\right)^{-1}d\tau.$
(119)

##  VII Numerical Analysis

We now present numerical analysis of an mmWave JCAS network using a
multicarrier waveform leveraging the analytical results from Sec. V and Sec.
VI. and discuss the insights they provide from a system design perspective.

Unless otherwise stated, the network parameters are set as follows. As we are
concerned with 5G-Adv./6G mmWave networks, we consider the center frequency to
be $f_{c}=75$ GHz. Using the Friis transmission model the pathloss intercept
for the LoS case is set to $K_{\text{L}}=\left(\frac{c_{0}}{4\pi
f_{c}}\right)^{2}$, and the NLoS intercept is set to be 15 dB lower. Based on
[38], we set the LoS path loss exponent to $\alpha_{\text{L}}=2$ and the NLoS
path loss exponent to $\alpha_{\text{N}}=3.2$. Similarly, from [39] the LoS
absorption coefficient is set to $\gamma_{\text{L}}=5\text{e}^{-6}$ and the
NLoS absorption coefficient to $\gamma_{\text{N}}=5\text{e}^{-3}$. The beam
widths are set to $\theta_{\text{B,Tx}}=5$ degrees at the BS transmitter,
$\theta_{\text{B,Rx}}=5$ degrees at the BS receiver, and
$\theta_{\text{U,Rx}}=30$ degrees at the UE. The front to back ratios are
$\xi_{\text{B,Tx}}=-35$ dB at the BS transmitter, $\xi_{\text{B,Rx}}=-20$ dB
at the BS receiver, and $\xi_{\text{U,Rx}}=-15$ dB at the UE. We set main lobe
gain to a physically realizable value given the beam width and front to back
ratios following [40]. This results in antenna gains of $G_{\text{B,Tx}}=31$
dB, $G_{\text{B,Rx}}=19.8$ dB, and $G_{\text{U,Rx}}=13.2$ dB. Thus, assuming a
transmit power of $P_{\text{t}}=15$ dBm and noise power of
$P_{\text{n}}=-123.2$ dBm, the normalized SNR for communication case is
$\nu_{\text{com}}^{-1}=182.4$ dB and $\nu_{\text{rad}}^{-1}=189.02$ dB.
Finally, we assume that the LoS interference fading terms are of the order
$N_{\text{L}}=3$, and the NLoS terms are $N_{\text{N}}=2$.

Regarding the parametrization of the multicarrier waveform and the composition
of $\mathbb{S}_{\text{rad}}$, we follow [5] and [41]. Using numerology 3 in
the 5G NR specification, we set the subcarrier spacing to $\Delta f=120$ kHz
and the guard interval to $T_{g}=570$ ns. Thus, the constants $k_{1}$ and
$k_{2}$ from Theorem 1 are $k_{1}=8\text{e}^{-4}$ and
$k_{2}=8.903\text{e}^{-3}$. Moreover, $\mathbb{S}_{\text{rad}}$ is composed in
a manner following [5]. Under the objectives that our range resolution is
$\Delta r=1$ m, our velocity resolution is $\Delta v=1.33$ m/s, our maximum
unambiguous range is $r_{\text{max}}=300$ m, and our max velocity is
$v_{\text{max}}=200$ km/h we use one in every $\Delta N_{\text{s}}=3$
multicarrier symbols and one in every $\Delta N_{\text{c}}=14$ subcarriers. In
total, the excitation signal covers $N_{\text{s}}=264$ multicarrier symbols
and uses the total bandwidth, corresponding to $N_{\text{c}}=3168$
subcarriers. We set the prior covariance for the range/velocity estimates to
the identity matrix.

Finally, we set the blockage density to $\beta^{-1}=140$ (which corresponds to
an urban environment) and effective cell radius
$r_{c}=\sqrt{1/(\pi\lambda_{\text{B}})}=100$ m. The key network parameters are
summarized in the following table.

Parameter |  Value |  Parameter |  Value |  Parameter |  Value
---|---|---|---|---|---
$\bm{f_{c}}$ |  $75$ GHz |  $\bm{G_{\text{B,Rx}}}$ |  $19.8$ dB |  $\bm{\Delta r}$ |  $1$ m
$\bm{K_{\text{L}}}$ |  $-75.96$ dB |  $\bm{\xi_{\text{B,Rx}}}$ |  $-20$ dB |  $\bm{\Delta v}$ |  $1.33$ m/s
$\bm{K_{\text{N}}}$ |  $-90.96$ dB |  $\bm{\theta_{\text{B,Rx}}}$ |  $5$ deg. |  $\bm{r_{\text{max}}}$ |  $300$ m
$\bm{\alpha_{\text{L}}}$ |  $2$ |  $\bm{G_{\text{U,Rx}}}$ |  $13.2$ dB |  $\bm{v_{\text{max}}}$ |  $200$ km/h
$\bm{\alpha_{\text{N}}}$ |  $3.2$ |  $\bm{\xi_{\text{U,Rx}}}$ |  $-15$ dB |  $\bm{N_{\text{s}}}$ |  $264$
$\bm{\gamma_{\text{L}}}$ |  $5e^{-6}$ |  $\bm{\theta_{\text{U,Rx}}}$ |  $30$ deg. |  $\bm{N_{\text{c}}}$ |  $3168$
$\bm{\gamma_{\text{N}}}$ |  $5e^{-3}$ |  $\bm{N_{\text{L}}}$ |  $3$ deg. |  $\bm{\Delta N_{\text{s}}}$ |  $3$
$\bm{P_{\text{t}}}$ |  $15$ dBm |  $\bm{N_{\text{N}}}$ |  $2$ deg. |  $\bm{\Delta N_{\text{c}}}$ |  $14$
$\bm{P_{\text{n}}}$ |  $123.2$ dBm |  $\bm{\Delta f}$ |  $120$ kHz |  $\bm{Q}$ |  $I$
$\bm{G_{\text{B,Tx}}}$ |  $31$ dB |  $\bm{T_{g}}$ |  $570$ ns |  $\bm{\beta^{-1}}$ |  $140$
$\bm{\xi_{\text{B,Tx}}}$ |  $-35$ dB |  $\bm{k_{1}}$ |  $8e^{-4}$ |  $\bm{r_{c}}$ |  $100$ m
$\bm{\theta_{\text{B,Tx}}}$ |  $5$ deg. |  $\bm{k_{2}}$ |  $8.903e^{-3}$ |  $\cdot$ |  $\cdot$

###  VII-A Sensing and Communication Coverage Probability

Figure 1: This figure depicts analytical upper and lower bounds and simulated
CCDFs for various radar SINR models. These indicate that a) the analytical
bounds are tight in most cases, and b) the approximate radar SINR model,
${\text{SINR}_{\text{GM}}}$, is a close fit to the true model,
${\text{SINR}_{\text{rad}}}$. Figure 2: This figure depicts analytical upper
and lower bounds and simulated CCDFs for the communication SINR models. Like
the radar case, these indicate that the analytical bounds are tight.

Figs. 1 and 2 depict the sensing and communication coverage probabilities for
the JCAS network detailed above. For the sensing case both analytical bounds
and simulated CCDFs are depicted for ${\text{SINR}_{\text{AM}}}$,
${\text{SINR}_{\text{GM}}}$, ${\text{SINR}_{\text{HM}}}$, and the radar SNR.
For the sake of comparison, simulated curves are depicted for
${\text{SINR}_{\text{rad}}}$. For the communication case, analytical bounds
and simulated curves are depicted for ${\text{SINR}_{\text{com}}}$. The
analytical bounds were obtained using Theorems 4 and 5, and the simulated
curves were obtained using Monte Carlo simulations of the JCAS network. An
immediate takeaway is that the analytical upper and lower bounds are generally
tight in all cases.

Figs. 1 indicates that ${\text{SINR}_{\text{GM}}}$ is a good approximation of
${\text{SINR}_{\text{rad}}}$. For the aforementioned parametrization of the
network, their respective CCDFs are empirically quite close. The usage of
${\text{SINR}_{\text{GM}}}$ is further motivated by the upper and lower
bounding models ${\text{SINR}_{\text{HM}}}$ and ${\text{SINR}_{\text{AM}}}$
from Proposition 1. The upper bounding model, ${\text{SINR}_{\text{HM}}}$,
lies above the CCDFs for ${\text{SINR}_{\text{GM}}}$ and
${\text{SINR}_{\text{rad}}}$ by a few dB. The closeness of
${\text{SINR}_{\text{GM}}}$ as an approximation for
${\text{SINR}_{\text{rad}}}$ is further underscored by the analytical upper
bound. The lower bounding model differs by 5 to 10 dB for low SINR thresholds,
$\tau$. It is expected to be somewhat looser, as its derivation involve two
applications of Jensen’s Inequality with respect to
${\text{SINR}_{\text{rad}}}$ in comparison to only one for
${\text{SINR}_{\text{HM}}}$.

These figures also demonstrate the utility of Theorem 2 in obtaining tractable
bounds on the distributions of SINR models, when their distributions may be
expressed in terms of the Laplace Transform of their interference process. In
both figures, matching upper and lower bounds are shown for
${\text{SINR}_{\text{GM}}}$ and ${\text{SINR}_{\text{com}}}$. These bounds are
close – especially for the case of ${\text{SINR}_{\text{com}}}$ – and differ
by only a few dB in the sensing case and less than a dB in the communication
case.

From a system design perspective, these figures demonstrate that interference
and blockages play a greater role in the link quality of a typical sensing
target than that of a typical communication user. While the overall average
SINR is lower in the sensing case due to the decreased path loss, this is made
up for in part by the distribution of the excitation signal over time and
frequency (which increases the processing gain analog, $G$). Looking at
relative features, then, we see a prominent downward shift in the sensing SINR
curves due to the probability that a typical SO may have no LoS BSs. Moreover,
in comparison to the communication case, we see that the gap between the SNR
and SINR is slightly wider. This is partially explained by the SO association
procedure, which leaves open the possibility of interferers being closer than
the sensing target.

###  VII-B Ergodic Estimation and Communication Efficiency with Varying
Network Density

Figure 3: This figure depicts lower (a) and upper (b) bounds for the ergodic
estimation efficiency of the JCAS network with respect to various SINR models
as well as bounds on the ergodic communication efficiency (c) in the high
blockage regime with $\beta^{-1}=360.67$. Figure 4: This figure depicts lower
(a) and upper (b) bounds for the ergodic estimation efficiency of the JCAS
network with respect to various SINR models as well as bounds on the ergodic
communication efficiency (c) in the low blockage regime with
$\beta^{-1}=72.13$. Figure 5: This figure depicts how the sensing and
communication ergodic efficiencies vary jointly with the network density and
the LoS pathloss exponent, $\alpha_{\text{L}}$, in the high blockage regime.
Lower bounds for the ergodic efficiencies are depicted for sensing with
respect to the lower bounding model, ${\text{SINR}_{\text{AM}}}$, (a) and
communications (b). Similarly, upper bounds for the ergodic efficiencies are
depicted for sensing with respect to the upper bounding model,
${\text{SINR}_{\text{HM}}}$, (c) and communications (d). A few representative
level sets for these surfaces are shown as block lines to improve clarity.

In Figs. 3 and 4 we consider the impact of base station and blockage density
on the ergodic sensing and communication efficiency of the JCAS network. To
this end, we sweep the average cell radius at different levels of blockage
density – as captured by $\beta$. We consider a low blockage regime
$\beta^{-1}=360.67$ and a high blockage regime, $\beta^{-1}=72.13$. These are
set such that the median LoS probability occurs at $r=50$ and $r=250$ meters,
respectively. Note that we also adjust the maximum unambiguous range for
sensing to be $3r_{c}$ to track with varying network density. This changes the
structure of $\mathbb{S}_{\text{rad}}$.

The curves depicted in the figures were obtained using Corollaries 3 and 5.
For the sake of discussion, we have included new radar SINR model for the
sensing case, ${\text{SINR}_{\text{typ}}}$. This model is simply the SINR of
the radar return at an arbitrary resource element, and is meant to highlight
the impact of the distribution of the excitation signal over time and
frequency. Formally, it is defined as

$\displaystyle{\text{SINR}_{\text{typ}}}=\mathds{1}\\{\Phi_{\text{L}}(\mathbb{R}^{2})>0\\}\frac{\kappa_{\text{CS}}g_{\text{L,ret}}(\left\lVert
X_{0}\right\rVert)}{\sum_{X_{k}\in\tilde{\Phi}_{\text{B}}\setminus\\{X_{0}\\}}\lvert
H^{k}\rvert^{2}BZ_{B}^{k}L(\left\lVert
X_{k}-X_{0}\right\rVert)+\nu_{\text{rad}}}.$ (120)

Note that bounds on its CCDF may be obtained using the bounds on
${\text{SINR}_{\text{GM}}}$ from Theorem 4 with $\mathbb{S}_{\text{rad}}$
having support over a single resource element.

In each of the figures, the bounds for the radar case depict the ergodic
efficiency with respect to the approximate model, ${\text{SINR}_{\text{GM}}}$,
the typical resource element, ${\text{SINR}_{\text{typ}}}$, the SIR with
respect to the approximate model, ${\text{SIR}_{\text{GM}}}$, and the SNR,
${\text{SNR}_{\text{rad}}}$. The lower bounds additionally include the sensing
efficiency with respect to the lower bounding SINR model,
${\text{SINR}_{\text{AM}}}$, and the upper bounds include the estimation
efficiency with respect to the upper bounding SINR model,
${\text{SINR}_{\text{HM}}}$. The communication ergodic efficiency is depicted
with respect to the SINR ${\text{SINR}_{\text{com}}}$, SIR,
${\text{SIR}_{\text{com}}}$, and SNR, ${\text{SNR}_{\text{com}}}$. Upper and
lower bounds are obtained using the corresponding bounds in Theorems 4 and 5
along with Corollaries 3 and 5. While the upper and lower bounds on ergodic
sensing efficiency differ by around a factor of 2, they exhibit the same
trends in all cases.

These figures reveal several insights into the performance trends of
communication and sensing in JCAS networks. Blockage density plays a
significant role in the performance of both functions, though its impact on
sensing is more pronounced. Generally, increasing blockage density improves
JCAS network performance at high to moderate base station densities, and
degrades performance at lower base station densities. The improvement is due
to the reduction in overall interference power, while the reduction is due to
the degradation of the desired signal power resulting from the scarcity of LoS
base stations. This effect is compounded on sensing, as it requires a LoS base
station, while communication can leverage strong NLoS base stations. Hence,
the faster decreasing of ergodic sensing efficiency with increasing cell
radius at moderate and high blockage densities is due to increasing
probability that the target is invisible to the network (i.e. the probability
that $R_{0}=\infty$).

More interesting trends are observed with respect to the variation of the
performance of communication and sensing with respect to base station density.
These trends are similar at different blockage densities, so we focus
primarily on Fig. 4. At moderate to high base station densities, in the range
of $r_{c}\leq 75$ m, the sensing efficiency is relatively insensitive or
increases to changes in base station density. While the strength of the
desired signal improves, as indicated by the sensing efficiency with respect
to ${\text{SNR}_{\text{rad}}}$, these improvements are mostly offset by the
increased interference power. Interestingly, this is not the case for
communication efficiency, which decreases with increasing base station density
in this range – a well known phenomena in the cellular networks [42].

To provide further insight into these trends, we investigate the variation of
communication and sensing ergodic efficiency with respect to both the LoS
pathloss exponent, $\alpha_{\text{L}}$, and network density in Fig. 5. Note
that the upper band lower bounds for the sensing case are computed with
respect to ${\text{SINR}_{\text{HM}}}$ and ${\text{SINR}_{\text{AM}}}$,
respectively, as these models provide true upper and lower bounds. This figure
confirms the trends observed in Figs. 3 and 4: over the range of pathloss
exponents considered, sensing ergodic efficiency is non-decreasing with
increasing network density, while communication efficiency is maximized for
some value of $r_{c}$ and then decreases thereafter. Two phenomena play into
this. First, the serving BS for the typical SO and the the typical UE face
different interference processes. In the sensing case, by the assumption that
the typical SO is served by the nearest LoS BS, the serving BS observes a
return from an object inside its Voronoi cell, and interference outside its
Voronoi cell. While this leaves open the possibility, that an interferer is
nearer than the desired BS, this has a generally repulsive effect on
interferers due to the Voronoi structure. By contrast, in the communication
case, the typical UE is always associated with the BS having the lowest path
loss, but observers a homogenous processes of interferers further than the
serving BS. We hypothesize that the sensing association policy leads to
greater variability in ${\text{SINR}_{\text{rad}}}$, with a bias toward
favorable SINR values, in comparison to the sensing case.

In addition to this, the sensing densification trends are further driven by
the difference between the pathloss functions. In particular, the difference
between the two way radar LoS path loss model and the regular LoS path loss
model. While the additional distance induced by the reflection reduces overall
radar pathloss, the derivative of the radar path loss is steeper due to the
factor of two difference in the path loss exponent and absorption factor.
Viewing the increase in base station density as decreasing the distances of
all links, the marginal improvement on the desired signal, which is based on
the radar path loss model, outstrips the marginal increase in interference
power, which is based on either the LoS or NLoS path loss models. Hence, we
observe the aforementioned insensitivity of ergodic sensing efficiency at
moderate base station densities, and the increase thereof at high densities.
Overall, these results highlight a tension between optimal regimes of base
station density for communication and sensing: the sensing function performs
best at high base station densities, while the communication performance is
optimized at moderate base station densities.

Finally, we comment on a few remaining features of interest. First, note that
the sensing efficiency with respect to ${\text{SINR}_{\text{GM}}}$ and
${\text{SINR}_{\text{HM}}}$ dominates the sensing efficiency with respect to
${\text{SINR}_{\text{typ}}}$. While the this is not the case for the
efficiency with respect to ${\text{SINR}_{\text{AM}}}$, this is likely due to
the model being more pessimistic, as mentioned previously. This indicates that
the distribution of the excitation signal over time likely reduces the impact
of worst case interference scenarios in which a BS in the main lobe of the
receive beam is also pointing its transmit beam in the direction of the
serving BS. Additionally, at low base station densities, the ergodic
efficiency with respect to the ${\text{SIR}_{\text{GM}}}$ tracks more closely
with ${\text{SNR}_{\text{rad}}}$ than does the ergodic efficiency of
${\text{SIR}_{\text{com}}}$ with respect to ${\text{SNR}_{\text{com}}}$. This
indicates that, even at low base station densities, interference remains a
more prominent feature on the performance of the sensing function than for
communications.

##  VIII Conclusion

We have presented a novel analytical framework for characterizing the coverage
probability and ergodic capacity of JCAS in cellular networks. Using an
information theoretic framework based on the estimation rate metric developed
in [3], we have extended the notion of coverage probability to the radar
setting, defining it as the probability that the rate of information extracted
about the parameters of interest associated with a typical sensing target
being tracked by the network exceeds some threshold. Similarly, we define
ergodic sensing capacity as the expected value of this quantity. Focusing on
the multicarrier setting, we have established upper and lower bounds on the
estimation rate in terms of an aggregate SINR induced by taking a weighted
average of per-resource element SINRs experienced by the radar receiver over
the excitation signal. Using this model, we developed a stochastic geometry
framework for characterizing the sensing and communication coverage
probabilities and ergodic capacities in a mmWave JCAS network employing a
shared multicarrier waveform and directional beamforming. As a first step in
our analysis, we derived a generic method for obtaining closed form upper and
lower bounds on the Laplace Transform of a shot noise process. We leveraged
this result to establish bounds on the CCDFs of a series of bounding and
approximate radar SINR models and the communication SINR model. We further
established expressions for the ergodic sensing and communication efficiencies
as corollaries with respect to these bounds. Over the course of our analysis,
we developed some noteworthy, independent results: an analog to Hölder’s
inequality in the setting of harmonic, rather than geometric, means of
functions in a measure space, and a result linking the Laplace Transform of a
non-negative random variable to its Mellin Transform via an inverse Mellin
Transform. Using our bounds for JCAS coverage and capacity, we have
investigated performance trends in a numerical case study. Among several
insights, our numerical analysis indicates the possibility that network
densification improves sensing performance – in contrast to the communication
function.

Though we have focused on the setting of a cellular network employing a shared
multicarrier waveform, our framework may be readily extended to other
settings. Future work in this area could address the impact of imperfect SO
association with the serving BS due to detection uncertainty or account for
heterogeneity in BS capability. It would also be interesting to investigate
other forms of BS cooperation or sensing policies among BSs, perhaps
accounting for more structured deployments using non-Poisson spatial models.
Interference mitigation techniques for sensing would be of particular concern
here. Our framework may also be adapted to the analysis of sensing meta-
distributions, which could shed some light on the variation of coverage among
the SOs. Finally, extensions of our analysis to incorporate more fine-grained
angular estimation, potentially considering adaptive beamforming, would be of
interest as well.

##  Appendix A Proof of Theorem 1

###  A-A Proof of Theorem 1

We first prove part i). For any unbiased estimator of $\theta$, by the Cramer
Rao Lower Bound we have that $Cov(N_{est})=R\gtrsim J^{-1}$ Hence it follows
that

$\displaystyle C_{\text{rad}}$
$\displaystyle=\frac{1}{2T_{\text{CPI}}}\log_{2}\left(\left\lvert\mathbb{I}+\mathbb{Q}^{\frac{1}{2}}\mathbb{R}^{-1}\mathbb{Q}^{\frac{1}{2}}\right\rvert\right)\leq\frac{1}{2T_{\text{CPI}}}\log_{2}\left(\left\lvert\mathbb{I}+\mathbb{Q}^{\frac{1}{2}}\mathbb{J}\mathbb{Q}^{\frac{1}{2}}\right\rvert\right)=C_{\text{rad}}^{UB}.$
(121)

Part ii). the FIM for the estimation problem stated in (6) may be expressed as
[43]

$\displaystyle\begin{split}\mathbb{J}&=\sum_{(m,n)\in\mathbb{S}_{\text{rad}}}8\pi^{2}|X_{m,n}|^{2}{\text{SINR}}_{m,n}\begin{bmatrix}&\Delta
f^{2}\left(\frac{2}{C_{0}}\right)^{2}n^{2}&-T_{\text{MC}}\frac{2f_{c}}{c_{0}}\Delta
f\frac{2}{C_{0}}mn\\\ &-T_{\text{MC}}\frac{2f_{c}}{c_{0}}\Delta
f\frac{2}{C_{0}}mn&T_{\text{MC}}^{2}\left(\frac{2f_{c}}{c_{0}}\right)^{2}m^{2}\end{bmatrix}.\end{split}$
(122)

Moreover, the following inequalities hold.

$\displaystyle
1+Tr\left(\mathbb{Q}^{\frac{1}{2}}\mathbb{J}\mathbb{Q}^{\frac{1}{2}}\right)\leq\left\lvert\mathbb{I}+\mathbb{Q}^{\frac{1}{2}}\mathbb{J}\mathbb{Q}^{\frac{1}{2}}\right\rvert\leq\frac{1}{4}Tr\left(\mathbb{I}+\mathbb{Q}^{\frac{1}{2}}\mathbb{J}\mathbb{Q}^{\frac{1}{2}}\right)^{2}.$

The lower bound follows from the positive semi-definiteness of
$\mathbb{Q}^{\frac{1}{2}}\mathbb{J}\mathbb{Q}^{\frac{1}{2}}$, and the upper
bound follows from the AM-GM inequality. Hence, the proof follows noting that
$G\cdot{\text{SINR}_{\text{rad}}}=Tr\left(\mathbb{Q}^{\frac{1}{2}}\mathbb{J}\mathbb{Q}^{\frac{1}{2}}\right)$.
$\hfill\Box$

##  Appendix B Proof of Proposition 1

This proposition inherently relies on the fact that
${\text{SINR}_{\text{rad}}}$ is a function of a certain type of empirical mean
of the interference over $\mathbb{\tilde{S}}_{\text{rad}}$ (the harmonic mean
specifically). One can then obtain upper and lower bounds using inequalities
for different types of means, and a Hölder-like in Inequality for obtaining
lower bounds on a mean of measures using the measure of the means.

Hence, before proving this proposition, we first summarize some key
inequalities. Of note, we prove a new Hölder-like inequality for measures of
harmonic means.

###  B-A Summary of Intermediate Lemmas

To prove Proposition 1, we shall exploit the following extension of Hölder’s
Inequality, which allows one to bound the geometric mean of ensembles of
measures of functions.

Lemma A: (Hölder’s Inequality for Geometric Means of Sets of Functions) Let
$p$ be a PMF on $[n]$ for some $n\in\mathbb{N}$, and let
$(\mathbb{G},\mathcal{G},\mu)$ be a measure space. Let $\\{f_{k}\\}_{k\in[n]}$
be a sequence measurable functions from $\mathbb{G}$ to $\mathbb{C}$. Then

$\displaystyle\left\lVert\prod_{k=1}^{n}f_{k}^{p_{k}}\right\rVert_{1}\leq\prod_{k=1}^{n}\left\lVert
f_{k}\right\rVert_{1}^{p_{k}}.$ (123)

Or, alternatively

$\displaystyle\left\lVert{\text{GM}}(\\{f_{k}\\}_{k\in[n]},p)\right\rVert_{1}\leq{\text{GM}}(\\{\left\lVert
f_{k}\right\rVert_{1}\\}_{k\in[n]},p).$ (124)

Note that the above norms are taken with respect to $\mu$.

We now generalize Hölder’s Inequality to the setting of Harmonic means.

Lemma B: (A Hölder-Like Inequality for Harmonic Means of Sets of Functions)
Let $p$ be a PMF on $[n]$ for some $n\in\mathbb{N}$, and let
$(\mathbb{G},\mathcal{G},\mu)$ be a measure space. Let $\\{f_{k}\\}_{k\in[n]}$
be a sequence measurable functions from $\mathbb{G}$ to $\mathbb{R}_{+}$. Then

$\displaystyle\left\lVert\left(\sum_{k=1}^{n}p_{k}f_{k}^{-1}\right)^{-1}\right\rVert_{1}\leq\left(\sum_{k=1}^{n}p_{k}\left\lVert
f_{k}\right\rVert_{1}^{-1}\right)^{-1}.$ (125)

Or, alternatively

$\displaystyle\left\lVert{\text{HM}}(\\{f_{k}\\}_{k\in[n]},p)\right\rVert_{1}\leq{\text{HM}}(\\{\left\lVert
f_{k}\right\rVert_{1}\\}_{k\in[n]},p).$ (126)

Note that the above norms are taken with respect to $\mu$.

Moreover, equality is obtained iff $\\{f_{k}\\}_{k\in[n]}$ are linearly
dependent.

Proof: See Appendix I.

###  B-B Proof of Proposition 1

Note that the SINR models differ only in the interference terms. Thus, it
suffices to establish bounds on the these terms only. For conciseness, let

$\displaystyle
I_{t,n}=\sum_{R_{k}\in\tilde{\Pi}_{\text{B}}^{0}}L(R_{k})Z_{B}^{k}F_{t,n}^{k}+\nu_{\text{rad}},$

and defeine
$\mathbf{I}=\\{I_{t,n}\\}_{(t,n)\in\mathbb{\tilde{S}}_{\text{rad}}}$. By the
AM-GM-HM Inequality, we have that

$\displaystyle{\text{HM}}(\mathbf{I}+\nu_{\text{rad}},\bm{\theta})\leq{\text{GM}}(\mathbf{I}+\nu_{\text{rad}},\bm{\theta})\leq{\text{AM}}(\mathbf{I}+\nu_{\text{rad}},\bm{\theta}).$
(127)

We now show that we may lower bound
${\text{HM}}(\mathbf{I}+\nu_{\text{rad}},\bm{\theta})$ and
${\text{GM}}(\mathbf{I}+\nu_{\text{rad}},\bm{\theta})$ using the Hölder-like
Inequalities in Lemmas A and B.
${\text{GM}}(\mathbf{I}+\nu_{\text{rad}},\bm{\theta})$ may be expressed as

$\displaystyle{\text{GM}}(\mathbf{I}+\nu_{\text{rad}},\bm{\theta})=\prod_{(t,n)\in\mathbb{\tilde{S}}_{\text{rad}}}\left(\sum_{R_{k}\in\tilde{\Pi}_{\text{B}}^{0}}L(R_{k})Z_{B}^{k}F_{t,n}^{k}+\nu_{\text{rad}}\right)^{\theta_{t,n}}.$
(128)

Note that the fading and beam alignment terms inside the sum may be
interpreted as marks associated with $R_{k}\in\tilde{\Pi}_{\text{B}}^{0}$,
which is itself a marked process. More precisely, we shall let

$\displaystyle\tilde{\Pi}_{\text{B}}^{\prime}=\\{(R_{k},\mathbf{F}_{k},Z_{B}^{k},M_{k}):R_{k}\in\Pi_{\text{B}}^{0}\\}$
(129)

denote the corresponding marked process. Since
$\tilde{\Pi}_{\text{B}}^{\prime}$ is defined with respect to a Palm process
which is itself defined with respect to the stationary process
$\Phi_{\text{B}}$, by [29, Prop-7.1.8] we may conclude the existence of the
random fields $\mathbf{F}(r)$, $Z_{B}(r)$, and $M(r)$ over the index set
$\mathbb{R}_{+}$ such that $\mathbf{F}(R_{k})=\mathbf{F}_{k}$,
$Z_{b}(R_{k})=Z_{B}^{k}$, and $M(R_{k})=M_{k}$ almost surely. Then, we may
express (128) as

$\displaystyle{\text{GM}}(\mathbf{I}+\nu_{\text{rad}},\bm{\theta})=\prod_{(t,n)\in\mathbb{\tilde{S}}_{\text{rad}}}\left(\int_{\mathbb{R}_{+}}L(r,M(r))F_{t,n}(r)Z_{B}(r)\Pi_{\text{B}}^{0}(dr)+\nu_{\text{rad}}\right)^{\theta_{t,n}}.$
(130)

We argue Lemma A (Hölder’s Inequality) may be applied to (130). This requires