1 Coverage and Rate of Joint Communication and Parameter Estimation in Wireless Networks

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Coverage and Rate of Joint Communication and

Parameter Estimation in Wireless Networks

Nicholas R. Olson, Jeffrey G. Andrews, and Robert W. Heath, Jr.

Abstract

From an information theoretic perspective, joint communication and sensing (JCAS) represents a natural gen-

eralization of communication network functionality. However, it requires the re-evaluation of network performance

from a multi-objective perspective. We develop a novel mathematical framework for characterizing the sensing and

communication coverage probability and ergodic rate in JCAS networks. We employ a formulation of sensing

parameter estimation based on mutual information to extend the notions of coverage probability and ergodic

rate to the radar setting. We deﬁne sensing coverage probability as the probability that the rate of information

extracted about the parameters of interest associated with a typical radar target exceeds some threshold, and

sensing ergodic rate as the spatial average of the aforementioned rate of information. Using this framework, we

analyze the downlink sensing and communication coverage and rate of a mmWave JCAS network employing a

shared waveform, directional beamforming, and monostatic sensing. Leveraging tools from stochastic geometry,

we derive upper and lower bounds for these quantities. We also develop several general technical results including:

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. We use the derived bounds to numerically

investigate the performance of JCAS networks under varying base station and blockage density. Among several

insights, our numerical analysis indicates that network densiﬁcation improves sensing SINR performance – in

contrast to communications.

Index Terms

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

N. R. Olson and J. G. Andrews are with 6G@UT and WNCG at The University of Texas at Austin, Austin, TX, USA (email:

nolson@utexas.edu, jandrews@ece.utexas.edu). R. W. Heath Jr. is with North Carolina State University, Raleigh, North Carolina, USA

(email: rwheathjr@ncsu.edu).

arXiv:2210.02289v2 [cs.IT] 15 Jan 2024

I. INTRODUCTION

There is increasing interest in leveraging communication networks to provide the additional services

of user localization and radar sensing – a concept termed joint communication and sensing or JCAS. The

deﬁning feature of JCAS networks is that the network is co-designed to perform the dual functions of

communication and sensing: network transceivers, spectrum, and even waveforms are used, potentially

simultaneously, to communicate with user equipment (UEs) and to detect, locate, and track objects of

interest (which we refer to as sensed objects or SOs). 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]. One may view these important communication network functions as forms of sensing

themselves.

Through the introduction of a parallel sensing objective, JCAS networks require the reconsideration of

nearly every network layer from a multi-objective perspective. Waveforms and antenna array codebooks

must now be designed to efﬁciently convey data for communication and to provide sufﬁcient detection

and tracking performance for sensing [3]. Scheduling at the MAC layer must now not only consider

tradeoffs with respect to trafﬁc ﬂows among UEs, but also tradeoffs with respect to sensing coverage for

SOs [4]. Likewise, network deployments and protocols must be designed with the performance of both

functions in mind. One of the key challenges inherent in this design problem is jointly accounting for and

mitigating the effects of intercell interference. Developing tractable models which capture the impact of

this phenomena on both functions and allow for insight into tradeoffs with respect to each is an important

step in addressing this issue.

This phenomena has been adeptly addressed in the setting of wireless communication networks by

characterizing notions of network coverage probability and ergodic rate using stochastic geometry. Inspired

by this, to address the intercell interference issue in the JCAS setting, we take a macroscopic view and

seek to quantify JCAS performance through the lens of coverage and rate. That is, we seek to address

the questions “What fraction of UEs and SOs achieve satisfactory performance?” (i.e. the coverage

probability), and “What is the average performance of all UEs and SOs?” (i.e. the ergodic performance).

In particular we focus on downlink communication and parameter estimation via monostatic radar

sensing, and employ performance characterizations thereof based on information theoretic objects. Our

restriction to parameter estimation, the objective of radar tracking, as opposed to detection is motivated

by its lack of study in related prior work, and that, in our view, it is the more challenging task. For

communication, we quantify performance via the Shannon rate of the links for each UE. Thus, we consider

a UE to be covered if its rate is satisfactorily high, and deﬁne the ergodic performance as the spatial

average of these rates. For the parameter estimation objective, we exploit a metric based on the mutual

information between the measured returns of the SOs and their associated parameters of interest. Dividing

this quantity by the time taken to perform the measurement, one obtains an analogous sensing rate: the

rate of information gained about the parameters of interest of an SO via a measurement procedure. This

sensing rate metric allows for the natural generalization of the concepts of the coverage probability and

ergodic rate to the sensing objective. These, in tandem with the communication metrics, lead to a precise

notion of JCAS coverage probability and ergodic rate.

A. Prior Work

Joint communication and sensing, sometimes referred to as joint radar and communication (JRC),

integrated sensing and communication (ISAC), or dual function radar and communication (DFRC), 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 has been widely

used for the analysis of wireless networks in a variety of settings. Notably in [9] for the analysis of

coverage and rate in cellular networks. This was extended in [10], 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 within a stochastic geometry framework,

there have been relatively few works. In [11], the authors characterize the performance of radar range

detection and communication coverage probability using a time multiplexed system in an ad hoc network.

Certain aspects of their analysis of this setting are extended in [12]. In the setting of an indoor network,

[13] characterizes the detection performance of a radar system amidst clutter, but without considering in-

terference, that is time multiplexed with a communication system. In [14], the radar detection performance

in a vehicular network employing a shared waveform for communication and sensing is characterized. In

[15], 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 [16] the authors 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 [17]. Finally, in [18]

the detection performance of a heterogenous cellular network is considered in which the network access

is split between radar and communications functions. 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.

While these works offer some insight into JCAS performance, they employ somewhat simpliﬁed and/or

limited models of radar detection and do not address the performance of the parameter estimation problem

inherent in radar tracking. Indeed, many of these models fail to account directly for the impact of the

radar waveform on performance and instead simply study a narrowband SINR model at a speciﬁc time

slot as a proxy for sensing performance.

B. Contributions and Summary

To the best of our knowledge, our work is the ﬁrst to present a rigorous analytical framework with which

to characterize the joint performance of communication and parameter estimation in JCAS networks. In

Sec. II, we develop a notion of sensing performance based on the mutual information between the radar

return, Y, and the SO’s parameters of interest, Θ. As summarized earlier, this allows for the natural

generalization of the concepts of coverage probability and ergodic rate to be applied to sensing, while still

maintaining close correspondence with the more traditional estimation theoretic metric of error covariance.

We note further, that the application of mutual information to radar has been widely employed in prior

work. Such an approach was ﬁrst proposed by [19] and later extended by [20]. Recently, it has been used

to study waveform design and rate bounds for joint radar and communication in [21], [22], [23], [24].

Unlike communications, the sensing mutual information is typically intractable. To address this issue,

we establish that the sensing mutual information is approximately lower bounded in terms of the Fisher

Information Matrix (FIM), J(Θ), as

I(Y; Θ) ≳1

2log I+Q1

2EΘ[J(Θ)] Q1

2−c. (1)

Where Qis the covariance of Θunder the reference prior, and cis a non-negative constant depending only

on the differential entropy of the prior for Θscaled to have identity covariance. Focusing on the setting

of a shared multi-carrier waveform, we further establish that the log-determinant term admits upper and

lower bounds of the form

2log (1 + GSINRrad)≤1

2log I+Q1

2EΘ[J(Θ)] Q1

2≤log 1 + G

2SINRrad.(2)

Where Gis a constant derived from the FIM, and SINRrad is an average of the SINRs over the resource

elements employed for sensing. We argue that these bounds imply that SINRrad may be used as one

would the communication SINR, SINRcom, to characterize the coverage and rate performance of parameter

estimation. Therefore, we equivalently characterize the JCAS coverage probability as the joint fraction of

UEs and SOs whose corresponding SINR is above some corresponding threshold and the JCAS ergodic

rate as the joint spatial average of the corresponding rate functions.

Leveraging the communication and sensing rate functions, we ﬁnally establish that, when the UEs

follow a stationary, ergodic point process, ΦU, with intensity λU, and the SOs follow an independent

stationary, ergodic point process, ΦS, with intensity, λS, the JCAS coverage probability may be expressed

Pc, JCAS(τcom, τrad) = λU

λU+λS

ΦU(SINRcom ≥τcom) + λS

λU+λS

ΦS(SINRrad ≥τrad),(3)

where P0

ΦUand P0

ΦSdenote the Palm measures associated with ΦUand ΦS. Similarly the JCAS ergodic

rate may be expressed as

Ec, JCAS =λU

λU+λS

ΦU[log(1 + SINRcom)] + λS

λU+λS

ΦSk

2log 1 + G

kSINRrad k∈ {1,2}.(4)

Hence, even though the SINR models arise from a network in which communication and sensing are

performed simultaneously (and thereby strongly coupled), it sufﬁces to analyze the performance of each

function separately in characterizing network-wide JCAS coverage and rate performance.

Having formally developed our notion of JCAS coverage probability and ergodic rate, in Sec. III we

detail a system model for a mmWave JCAS network performing downlink communication and monostatic

sensing of doppler and delay using a shared multi-carrier waveform. From this model, we induce stochastic

expressions for SINRcom and SINRrad with respect to the typical UE and SO. Using these, in Sec. IV

through Sec. VI we establish a series of novel results that outline an approach to obtain integral closed

form upper and lower bounds and approximations for the JCAS coverage and rate of the network. We

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1CoverageandRateofJointCommunicationandParameterEstimationinWirelessNetworksNicholasR.Olson,JeffreyG.Andrews,andRobertW.Heath,Jr.AbstractFromaninformationtheoreticperspective,jointcommunicationandsensing(JCAS)representsanaturalgen-eralizationofcommunicationnetworkfunctionality.However,itrequiresther...

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