# On The Robustness of Channel Allocation in Joint Radar And Communication
Systems:
An Auction Approach

Ismail Lotfi, Hongyang Du, Dusit Niyato , Sumei Sun , and Dong In Kim Ismail
Lotfi, Hongyang Du and Dusit Niyato are with the School of Computer Science
and Engineering, Nanyang Technological University, Singapore 639798, Singapore
E-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>Sumei Sun is with the Institute for Infocomm Research, A*STAR, Singapore
138632, Singapore
E-mail<EMAIL_ADDRESS>Dong In Kim is with the Department of
Electrical and Computer Engineering, Sungkyunkwan University (SKKU), Suwon
16419, South Korea
E-mail<EMAIL_ADDRESS>
###### Abstract

Joint radar and communication (JRC) is a promising technique for spectrum re-
utilization, which enables radar sensing and data transmission to operate on
the same frequencies and the same devices. However, due to the multi-objective
property of JRC systems, channel allocation to JRC nodes should be carefully
designed to maximize system performance. Additionally, because of the
broadcast nature of wireless signals, a watchful adversary, i.e., a warden,
can detect ongoing transmissions and attack the system. Thus, we develop a
covert JRC system that minimizes the detection probability by wardens, in
which friendly jammers are deployed to improve the covertness of the JRC nodes
during radar sensing and data transmission operations. Furthermore, we propose
a robust multi-item auction design for channel allocation for such a JRC
system that considers the uncertainty in bids. The proposed auction mechanism
achieves the properties of truthfulness, individual rationality, budget
feasibility, and computational efficiency. The simulations clearly show the
benefits of our design to support covert JRC systems and to provide incentive
to the JRC nodes in obtaining spectrum, in which the auction-based channel
allocation mechanism is robust against perturbations in the bids, which is
highly effective for JRC nodes working in uncertain environments.

###### Index Terms:

Robust optimization, multi-item auction, joint radar and communication,
friendly jammers, covert communication.

##  1 Introduction

With the growing number of mobile users in cellular networks and given the
limited available bandwidth, several approaches have been proposed at
different levels of the network design to accommodate these users and
efficiently allocate the spectrum. In addition, different technologies are
being integrated into these network devices to perform a variety of tasks such
as communication, radar sensing, and power transfer [1, 2]. This allows
efficient spectrum re-utilization and minimizes hardware costs. Recently,
spectrum sharing between radar and communication systems has led to the
emergence of joint radar and communication (JRC) systems [1]. In JRC systems,
simultaneous radar sensing and data transmission can be performed by the same
device on the same frequency. However, JRC systems are vulnerable to jamming
and eavesdropping attacks, and only a few works address this issue.

In our previous work [2], we have developed a framework for JRC systems that
enables continuous operation of the system under jamming attacks. We developed
an intelligent deception strategy based on ambient backscatter communication
and deep reinforcement learning. However, the proposed model faces some
scalability issues. First, if the network has multiple JRC nodes that are
communicating with a receiver, it becomes harder for the receiver to decode
the modulated ambient backscatter signals. Second, a warden can still detect
ongoing radar sensing signals and knows exactly which JRC node is trying to
sense the environment, hence making the JRC node potentially vulnerable to
attackers. The warden can eavesdrop the transmitted signals (communication
signals and radar signals) to infer sensitive information about the users.
Therefore, minimizing the detection probability by a warden is crucial for
preventing such attacks as the warden will be unable to distinguish between
noise and real ongoing transmission and sensing activities. As such, hiding
the transmitted signals can both minimize the leakage of sensitive information
and prevent jamming attacks from occurrence.

Recently, covert communication has been proposed as an effective solution to
hide the ongoing signal transmission from discovery by a third party [3, 4].
Specifically, the idea consists of deploying friendly jammers to transmit fake
signals to disturb the warden about the ongoing transmission. Since the warden
has a predefined energy threshold to detect ongoing communication, the
addition of jamming signals will increase the false alarm and misdetection
probabilities at the warden. As a result, the warden is forced to increase its
detection threshold and allows the legitimate transmitter to transmit silently
and covertly thanks to the jamming signals.

Since the covert communication technique can be applicable and compatible with
any wireless systems without changing the system operation, it is a low-cost
and efficient solution, for example, compared with the frequency hopping (FH)
technique which requires the transmitters and receivers to switch among
subcarriers frequently. As such, the FH technique increases the complexity to
the circuit design and requires a sophisticated synchronization method.
Moreover, the warden could scan a wide range of frequencies, overcoming the FH
technique. Therefore, covert communications can avoid other external attacks
as the transmission is hidden. With a variety of modulation schemes developed
for JRC systems such as orthogonal frequency division multiplexing (OFDM) [5,
6], the idea of covert communication can be extended to enable covert radar
sensing and hence, a covert JRC system. However, radio resource management
issues arise when there are multiple JRC nodes in the network. Specifically,
these JRC nodes can have different spectrum requirements to achieve their
targeted data rates and sensing operations. The JRC nodes need to carefully
design their valuation functions based on their private information, e.g.,
channel state information (CSI) and importance of radar sensing over
communication and vice-versa, to ensure that the obtained spectrum from the
SSP is worth its price, i.e., positive utility.

Even though several works have studied at the signal processing level a
coexistence of radar sensing and data transmission on the same waves and/or on
the same devices [1], optimal channel allocation for JRC nodes has received
far less attention. In particular, a major part of existing works focused on
finding the optimal scheduling and modulation scheme for radar and
communication signals, see for example the comprehensive review in [7].
However, fewer works addressed the problem of allocating the available
bandwidth to the JRC nodes so that the utilities of both the SSP and the JRC
nodes are maximized [8]. With the significant changes at the lower layers of
the network to accommodate JRC systems, and due to their multi-objective
nature, deriving an optimal channel allocation strategy by the spectrum
service provider (SSP) to the JRC nodes becomes more complex and a significant
problem of interest. In addition, when demands from the JRC nodes exceed the
available spectrum, the SSP needs to strategically allocate the available
channels to maximize its revenue.

Several auction mechanisms have been developed to address the problem of
overloaded requests for the limited bandwidth in wireless networks [9, 10,
11]. Auction mechanisms are efficient tools to allocate the channels to the
users that value them the most. In such mechanisms, the objective can be to
maximize the social welfare of the system which is defined as the sum of all
users’ utilities. For instance, the authors in [8] designed a simple VCG-like
auction mechanism for channel allocation for JRC nodes. However, traditional
auction mechanisms do not consider the uncertainty of the submitted bids of
the wireless nodes which can make the derived optimal auction solution
infeasible in practice [12]. Here, the bids are derived from the values of
channels perceived by the wireless nodes that depend on each device’s
detection sensitivity and capability. For instance, when a JRC node calculates
the valuation of each channel to derive the bids, it can incorporate some
uncertain parameters, e.g., the location of the warden or the channel gain,
and therefore its realized utility can be less than expected. Furthermore, the
utilities of some users can become negative and according to prospect theory,
users’ response to losses is much stronger than that to the corresponding
gains [13]. Therefore, a mechanism that considers the uncertainty in bids is
needed when deriving the optimal auction solution.

Motivated by the above mentioned issues, in this paper, we develop a robust
and efficient multi-item auction mechanism for channel allocation in covert
JRC systems under uncertainty of bids. The objective is to develop a risk-
averse algorithm for channel allocation that maximizes the social welfare of
the system under uncertainty of JRC nodes’ valuations. Different from the
traditional Vickrey–Clarke–Groves (VCG) mechanism in which the auctioneer is
risk-neutral, in the proposed robust auction mechanism, the auctioneer is
risk-averse, i.e., the auctioneer accepts to obtain a lower revenue but with
higher confidence, which is practical for several safety critical
applications. To the best of our knowledge, this is the first work that
considers the problem of robust channel allocation for covert JRC systems.

The main contributions of our work are as follows:

  1. 1.

We design a novel covert JRC system in which friendly jammers are deployed to
transmit artificial noise and prevent wardens from detecting ongoing
transmissions of the JRC nodes. The proposed design is shown to enable the JRC
nodes to perform covertly their radar sensing and data transmission operations
with low detection probability.

  2. 2.

As the task of designing a reliable channel allocation system by the SSP
remains challenging, i.e., immune against uncertainty, we develop a robust
multi-item auction mechanism to allocate the channels to the JRC nodes. Unlike
previous works on covert communication, we consider the uncertainty about the
warden, such as its location, in the design of the auction mechanism and show
how the auction outcomes are affected by the uncertainty range of the warden.

  3. 3.

The proposed auction based-model guarantees the properties of individual
rationality (IR), incentive compatibility (IC), and budget feasibility (BF).
This makes the system resilient both against intentional market manipulation
attacks and perturbations in the submitted spectrum bids.

  4. 4.

We conduct extensive simulations to validate the proposed covert JRC system
and derive important properties about the proposed robust auction mechanism
compared to deterministic auction mechanisms over different scenarios. The JRC
nodes are able to covertly perform their radar sensing and data transmission
operations while the SSP is able to derive an optimal allocation strategy that
reflects its risk-aversion.

The rest of the paper is organized as follows. Section II reviews related
works. Section III and IV describe our system model and the proposed robust
auction mechanism, respectively. The evaluation results are then presented in
section V. Section VI concludes the paper.

##  2 Related Works

Since our work is a novel design that incorporates several techniques, we
review related works which study the problem of spectrum allocation in JRC
systems, covert communication and robust optimization in wireless networks.

###  2.1 Covert Communication

Different from upper-layer security techniques that try to protect the content
from being intercepted by a third party, covert communication has the goal of
hiding the ongoing communication itself from wardens. In [3], covert
communication was first addressed from an information theory perspective where
a square root limit on the number of covertly transmitted bits was derived. In
[14], a zero-sum game was formulated between the transmitter and the jammers
as the first player and a set of wardens as the second player. The authors
studied how the warden adjusts its detection threshold to increase its
detection probability while the transmitter and the jammers vary their
transmission power to increase the detection error probability (DEP) at the
wardens. In [15], the authors proposed an uncoordinated jammer selection
scheme where a set of friendly jammers cooperate to hide an ongoing
transmission in a distributed fashion, which is based on their channel gains
to the legitimate receiver. However, they only considered the covertness of
only one communication channel. If the network has multiple simultaneous
transmitters, their proposed jammer selection scheme becomes inefficient as
different receivers would have different channel gain thresholds. Furthermore,
the additional cost to the transmitter of deploying multiple jammers was not
investigated. Finally, we should note that, to the best of our knowledge, the
covertness of JRC systems was not addressed before in the literature.

###  2.2 Spectrum Allocation for JRC Systems

Spectrum allocation by JRC nodes for both radar sensing and data transmission
has received a significant interest in the literature [7]. For instance, in
[16], a resource management and scheduling (RMS) process is developed to
quantify JRC system’s performance where different quality-of-service (QoS)
metrics are derived for different use cases of JRC nodes. In [17], authors
used deep reinforcement learning to optimally schedule the spectrum resources
by JRC nodes. In [18], a pricing-based mechanism was proposed for JRC systems
but with a focus on handling the interference between the radar and
communication functions instead of the market model. However, less focus was
given to the layer where the SSP allocates the spectrum to the JRC nodes. In
[19], a hierarchical game model was proposed to allocate spectrum resources
for OFDM-based JRC systems. A set of SSPs compete with each other to attract
more JRC nodes to be their clients while the JRC nodes determine their optimal
spectrum demands to maximize their utilities. In [8], a VCG-like auction
mechanism was developed to allocate channels for autonomous vehicles (AVs).
However, the authors considered all the channels to have the same valuation,
reducing the problem to a single-item auction. In addition, existing works on
mechanism design for spectrum allocation did not consider the security of the
wireless system into their design, e.g., wardens, which can limit the
application of the proposed mechanisms in practical scenarios. Finally, both
works in [19] and [8] did not consider the uncertainty about users’ valuations
which can negatively impact the utility of the users. In fact, a quite small
perturbation in the submitted bids can make the solution to the auction model
infeasible and thus practically meaningless [12].

###  2.3 Robust Optimization in Wireless Networks

Most of the existing works on optimization problems are based on the
assumption that the data defining the constraints and the objective function
of the problem are obtained precisely. However, in many practical scenarios,
this assumption does not hold and several sources can perturb the obtained
data. For instance, wireless networks have different sources of uncertainties
that can significantly impact the network performance and valuation metrics
such as distance between nodes, CSI and mobile users’ dynamic spectrum
demands. To tackle this problem, robust optimization has been proposed in
which the optimization is performed over an uncertainty set and the objective
is to optimize a worst-case function.

Even though robust optimization is being explored for more than two decades
[12, 20], very few works adopted robustness in wireless network optimization
problems. In [21], the problem of uncertainty about wireless sensors’
locations was considered to design a robust solution that maximizes the
extracted data, minimizes the consumed energy, and maximizes the network
lifetime. The robust solution is defined as the one with the best worst-case
objective over the uncertainty set. The authors showed that as the uncertainty
set increases in size, the robust solution provides a significant improvement
in the worst-case but with the expense of some loss in optimality, known as
the price of robustness [20]. A distributed robust optimization problem was
developed in [22] to solve the problem of power and rate control in wireless
communication networks under the uncertainty of CSI. In a recent work [23], a
robust optimization approach was considered for mobile data offloading
problem, which is formulated as a multi-item auction where the spectrum for
mobile users is auctioned by the SSP to offload from the main base station to
other access points. The proposed auction mechanism uses historical data from
previous bids to determine the winners and payments. However, the proposed
model relies only on previous bids and does not consider the realized new bids
during the auction process. This makes the derived solution suboptimal as
users might change their submitted bids over time. In addition, a limited
discussion was provided about the valuation function of the mobile users,
which significantly impacts the derived uncertainty set. Finally, existing
works on mechanism design for spectrum allocation did not consider the
security of the wireless system in their design, which can limit the
application of the proposed mechanisms in practical scenarios.

Compared with existing works, the proposed auction mechanism in this paper has
the following characteristics: (1) we consider the uncertainty in the
submitted bids to derive a robust solution that prevents violation of optimal
auction properties and is always feasible in practice; (2) we provide examples
of relevant uncertainty sets to our problem while the proposed mechanism still
works for general uncertainty sets; and (3) we show that our robust mechanism
can model the risk level of the SSP towards the solution based on the size of
the uncertainty set. Even though the obtained total social welfare can be
lower than the one obtained by deterministic auctions, i.e., which do not
consider the uncertainty of bids, it is less risky and more robust in terms of
feasibility.

##  3 System Model

In this section, we first describe our proposed covert JRC system and then
present the metrics used by the JRC nodes to evaluate the spectrum and derive
their bids. Finally, we present the auction market model where we define the
utilities of the SSP and the JRC nodes, followed by the properties of the
desired optimal auction solution.

###  3.1 Covert JRC System

Figure 1: An illustration of the proposed channel allocation auction model and
the covert JRC system with the friendly jammer, where a warden is trying to
detect the ongoing signals between the JRC node and a receiver/obstacle.

Figure 1 presents the network model under consideration. We consider a set
$\mathcal{N}=\\{1,\dots,N\\}$ of JRC nodes. These JRC nodes are under the
coverage area of an SSP that has a set $\mathcal{M}=\\{1,\dots,M\\}$ of
channels for allocation, using time-division multiple access (TDMA) to the JRC
nodes. Therefore, each channel can be used only by one JRC node at a time,
which minimizes mutual interference between neighboring JRC nodes. Each
channel has $M_{c}$ orthogonal sub-carriers that are used by each JRC node to
modulate OFDM symbols to transmit simultaneously data and sense the
environment [5, 24]. However, due to the nature of the wireless signal
transmission, a warden can detect ongoing data and radar transmissions. To
overcome this security issue, we consider that the SSP deploys friendly
jammers to lower the probability of warden’s success detection to enable the
covert JRC system. Friendly jammers help by transmitting random signals to
increase the uncertainty at the warden about the ongoing transmission when
analyzing the received energy111This can be further extended to have multiple
jammers assisting each JRC node similar to the work presented in [15], which
adds more uncertainty to the warden about the aggregate interference power..
Without loss of generality, we consider that each JRC node is assisted with
one jammer and has one warden trying to detect its data transmission and radar
sensing. Similar to [15], the jamming power is considered fixed for all the
jammers to an optimal value that balances between increasing the DEP at the
warden and the outage of the legitimate receiver. Note here that the focus of
the paper is on channel allocation through the auction mechanism. The
optimization of jamming and transmit power of data and radar transmissions can
be done, e.g., as in [15]. The covertness of the radar signals is meant to
prevent the warden from identifying the exact JRC node that is trying to sense
the environment.

It is interesting to note that on one hand, allocating one jammer for each JRC
node might be a waste of resources and one jammer can offer covertness to
several JRC nodes. On the other hand, if we use one jammer for more than one
JRC node, the jammer will have to transmit at an average power of the covered
JRC nodes. This makes some of the JRC nodes much weaker as the used jamming
power is far from the optimal jamming power that would be used in the case of
single jammer single JRC node scenario, and hence, their transmission can be
easily detected by wardens. We should also highlight that existing works,
e.g., [25, 15], suggest that increasing the number of jammers per transmitter
brings much more covertness compared to the scenario of single-jammer single-
transmitter architecture. Nevertheless, associating one jammer for each JRC
node is an important concern in large scale applications, e.g., device-to-
device (D2D) communications for extended coverage. In that scenario, it is
possible to use existing JRC nodes as friendly jammers. In this case, how to
motivate these nodes to act as friendly jammers would be an interesting issue
for further study.

###  3.2 Valuation Metrics

Different from traditional resource allocation problems, both the radar
sensing and data transmission functions need to be jointly optimized in JRC
systems. The JRC nodes, therefore, have to consider radar sensing and data
transmission performance simultaneously to evaluate the valuation of the
spectrum to be acquired through the auction from the SSP. Furthermore, too
high transmit power makes the transmitter’s sensitive information (e.g.,
location) more detectable to the warden [26]. Therefore, we first analyze the
DEP of the warden under generalized fading channels and obtain the closed-form
expression of DEP, with arbitrary transmit and jamming power. With the help of
the friendly jammer, the transmitter transmits JRC signals while ensuring that
the warden’s DEP is close to $1$. Only then we can ensure that the
communication is covert [27]. With such a precondition, we further analyze the
radar and communication performance of the system. We consider the mutual
information (MI) between the received signal and the target impulse response
to be an important valuation metric for radar systems [24, 5]. The accuracy of
the estimated target parameters increases with an increase of the MI [5]. In
addition, channel capacity (CC) enables the computation of the highest data
rate that can be reached via a communication channel and is an important
metric for communication systems. It has been shown in [28] and [5] that
minimizing the minimum mean square error (MMSE) in estimating the target
impulse response is equivalent to maximizing the MI and that careful
adjustment of the transmit power according to the channel state information
(CSI) increases the data rate. Therefore, in this work, we adopt MI and CC as
the major performance metrics used by the JRC nodes to evaluate the spectrum
resources. Because we ensure that the transmission is covert (DEP $\to$ 1), we
can call MI and CC as covert MI and covert CC, respectively.

####  A. Channel Model

First, we adopt a three-dimensional Cartesian coordinate system to represent
locations. The locations of a JRC node $i$, a jammer $g$, a receiver $b$, and
a warden $w$ are denoted by
${\mathbf{q}}_{i}=\left[{{x_{i}},{y_{i}},{z_{i}}}\right]^{T}$ ,
${{\mathbf{q}}_{g}}={\left[{{x_{g}},{y_{g}},{z_{g}}}\right]^{T}}$,
${\mathbf{q}}_{b}=\left[x_{b},y_{b},z_{b}\right]^{T}$ and
${\mathbf{q}}_{w}={\left[{{x_{w}},{y_{w}},{z_{w}}}\right]^{T}}$, respectively.
The distance between two devices $d_{1}$ and $d_{2}$ is expressed as
$D_{d_{1}d_{2}}={\|{\mathbf{q}}_{d_{1}}-{\mathbf{q}}_{d_{2}}\|}$, and
$\alpha_{d_{1}d_{2}}$ is the corresponding path loss exponents. We then use
the $\alpha-\mu$ distribution to model the small-scale fading, which is a
general fading model that includes several important other distributions, such
as the Weibull, One-Sided Gaussian, Rayleigh, and Nakagami. The probability
density function (PDF) and the cumulative distribution function (CDF)
expressions of a squared $\alpha-\mu$ random variable $\Upsilon$ are given by
[29]:

${f_{\Upsilon}}\left(\gamma\right)=\frac{{\alpha{\gamma^{\frac{{\alpha\mu}}{2}-1}}}}{{2{\beta^{\frac{{\alpha\mu}}{2}}}\Gamma\left(\mu\right)}}\exp\left({-{{\left({\frac{\gamma}{\beta}}\right)}^{\frac{\alpha}{2}}}}\right),$
(1)

and

${F_{\Upsilon}}\left(\gamma\right)=\frac{{\gamma\left({\mu,{\gamma^{\frac{\alpha}{2}}}{\beta^{-\frac{\alpha}{2}}}}\right)}}{{\Gamma\left(\mu\right)}},$
(2)

respectively, where $\Gamma\left(\cdot\right)$ is the gamma function [30, eq.
(8.310.1)],
$\beta=\frac{{\bar{\Upsilon}\Gamma\left(\mu\right)}}{{\Gamma\left({\mu+\frac{2}{\alpha}}\right)}}$,
${\bar{\gamma}}=E\left(\gamma\right)$, and $\gamma\left(\cdot\right)$ is the
incomplete gamma function [30, eq. (8.35)].

####  B. Detection Error Probability at Warden

The warden’s objective is to minimize the DEP of the ongoing signal
transmission, i.e., data and radar signals. For JRC node $i$’s sub-carrier $m$
of channel $j$, the DEP is defined as [31]:

${\xi_{m}^{(ij)}}=\mathcal{P}_{FA}+\mathcal{P}_{MD},$ (3)

where $\mathcal{P}_{FA}$ is the probability of false alarm, which is defined
as
$\Pr\left({\sigma_{c}^{2}+D_{gw}^{-{\alpha_{gw}}}{p_{g}^{(J)}}h_{wg}^{2}>{\varepsilon_{m}}}\right)$,
$\mathcal{P}_{MD}$ is the probability of miss detection, which is defined as
$\Pr\left({D_{iw}^{-{\alpha_{iw}}}{p_{m}^{(T)}}h_{wm}^{2}+D_{gw}^{-{\alpha_{gw}}}{p_{g}^{(J)}}h_{wg}^{2}+{\sigma_{c}^{2}}<{\varepsilon_{m}}}\right)$,
${\sigma_{c}^{2}}$ is the noise power, ${\varepsilon_{m}}$ is the detection
threshold, $p_{m}^{(T)}$ is the transmit power, $p_{g}^{(J)}$ is the jamming
power,
$h_{wm}^{2}\sim\alpha-\mu\left({{\alpha_{wm}^{(ij)}},{\mu_{wm}^{(ij)}},{{\bar{\gamma}}_{wm}^{(ij)}}}\right)$,
and
$h_{wg}^{2}\sim\alpha-\mu\left({{\alpha_{wg}^{(ij)}},{\mu_{wg}^{(ij)}},{{\bar{\gamma}}_{wg}^{(ij)}}}\right)$.
As there are $M_{c}$ sub-carriers for each channel, we consider the DEP for
channel $j$ to be the minimum over all DEP for each sub-carrier, i.e.:

$\xi_{w}^{(ij)}=\min_{m\in M_{c}}\xi_{m}^{(ij)}.$ (4)

###### Theorem 1

The closed-form DEP can be derived as (B.), where ${C_{1w}}\triangleq
D_{iw}^{-{\alpha_{iw}}}{p_{m}^{(T)}}$, and ${C_{2w}}\triangleq
D_{gw}^{-{\alpha_{gw}}}{p_{g}^{(J)}}$.

###### Proof:

Let ${Y_{1}}\triangleq\sigma_{c}^{2}+{C_{2w}}h_{wg}^{2}$ and
${Y_{2}}\triangleq{C_{1w}}h_{wm}^{2}+{C_{2w}}h_{wg}^{2}+\sigma_{c}^{2}={C_{1w}}h_{wm}^{2}+{Y_{1}}$.
According to the definition of ${\xi_{m}^{(ij)}}$, we have

${\xi_{m}^{(ij)}}=1-{F_{{Y_{1}}}}\left({{\varepsilon_{m}}}\right)+{F_{{Y_{2}}}}\left({{\varepsilon_{m}}}\right).$
(A-1)

In the following, we derive the CDF expressions of $Y_{1}$ and $Y_{2}$. With
the help of definition of CDF, we have

$\displaystyle{F_{{Y_{1}}}}\\!\left(y\right)\\!=\\!{F_{h_{wg}^{2}}}\\!\\!\left(\\!{\frac{{y\\!-\\!\sigma_{c}^{2}}}{{{C_{2w}}}}}\\!\right)\\!=\\!\frac{{\gamma\\!\left(\\!{{\mu_{wg}^{(ij)}},{{\left({\frac{{y-\sigma_{c}^{2}}}{{{C_{2w}}}}}\right)}^{\frac{{{\alpha_{wg}^{(ij)}}}}{2}}}\\!{\beta_{wg}^{(ij)}}^{-\frac{{{\alpha_{wg}^{(ij)}}}}{2}}}\right)}}{{\Gamma\left({{\mu_{wg}^{(ij)}}}\right)}}.$
(A-2)

The CDF of $Y_{2}$ can be expressed as [32]

${F_{{Y_{2}}}}\left(y\right)=\int_{0}^{\infty}{{F_{{Y_{1}}}}\left({y-t}\right)\frac{1}{{{C_{1w}}}}{f_{h_{wm}^{2}}}\left({\frac{t}{{{C_{1w}}}}}\right){\text{d}}t}.$
(A-3)

Substituting the CDF and PDF expressions in (A-3), with the help of [33, eq.
(06.06.07.0002.01)], [33, eq. (01.03.07.0001.01)], and [30, eq. (3.194.3)], we
can express ${F_{{Y_{2}}}}\left(y\right)$ as (B.), shown at the top of the
next page, which can be re-written in closed-form with the help of the
definition of multivariate Fox’s $H$-function [34, eq. (A-1)].

$\displaystyle{F_{{Y_{2}}}}\left(y\right)=\frac{{{\alpha_{wm}^{(ij)}}}}{{2{{\left({{C_{1w}}{\beta_{wm}^{(ij)}}}\right)}^{\frac{{{\alpha_{wm}^{(ij)}}{\mu_{wm}^{(ij)}}}}{2}}}\Gamma\left({{\mu_{wm}^{(ij)}}}\right)\Gamma\left({{\mu_{wg}^{(ij)}}}\right)}}{\left({y-\sigma_{c}^{2}}\right)^{\frac{{{\alpha_{wm}^{(ij)}}{\mu_{wm}^{(ij)}}}}{2}}}{\left({\frac{1}{{2\pi
i}}}\right)^{2}}$ $\displaystyle\times\int_{{{\cal L}_{1}}}{\int_{{{\cal
L}_{2}}}{\frac{{\Gamma\\!\left({1-\frac{{{s_{1}}{\alpha_{wg}^{(ij)}}}}{2}}\right)\Gamma\\!\left({{s_{1}}+{\mu_{wg}^{(ij)}}}\right)\Gamma\\!\left({-{s_{1}}}\right)\Gamma\\!\left({\frac{{{\alpha_{wm}^{(ij)}}{\mu_{wm}^{(ij)}}}}{2}+\frac{{{s_{2}}{\alpha_{wm}^{(ij)}}}}{2}}\right)}}{{\Gamma\\!\left({1+\frac{{{\alpha_{wm}^{(ij)}}{\mu_{wm}^{(ij)}}}}{2}+\frac{{{s_{2}}{\alpha_{wm}^{(ij)}}}}{2}-\frac{{{s_{1}}{\alpha_{wg}^{(ij)}}}}{2}}\right)\Gamma\\!\left({1-{s_{1}}}\right)\Gamma^{-1}\\!\left({-{s_{2}}}\right)}}}}{\left({\frac{{y-\sigma_{c}^{2}}}{{{C_{1w}}{\beta_{wm}^{(ij)}}}}}\right)^{\frac{{{s_{2}}{\alpha_{wm}^{(ij)}}}}{2}}}{\left({\frac{{y-\sigma_{c}^{2}}}{{{C_{2w}}{\beta_{wg}^{(ij)}}}}}\right)^{\frac{{-{s_{1}}{\alpha_{wg}^{(ij)}}}}{2}}}d{s_{2}}d{s_{1}}$
(A-4)

Thus, by substituting CDF expressions of $Y_{1}$ and $Y_{2}$ into (A-1), the
DEP can be derived as (B.), which completes the proof. ∎

$\displaystyle\xi_{m}^{(ij)}=$ $\displaystyle
1-\frac{{\gamma\left({{\mu_{wg}^{(ij)}},{{\left({\frac{{{\varepsilon_{m}}-\sigma_{c}^{2}}}{{{C_{2w}}}}}\right)}^{\frac{{{\alpha_{wg}^{(ij)}}}}{2}}}{\beta_{wg}^{(ij)}}^{-\frac{{{\alpha_{wg}^{(ij)}}}}{2}}}\right)}}{{\Gamma\left({{\mu_{wg}^{(ij)}}}\right)}}-\frac{{{\alpha_{wm}^{(ij)}}{{\left({{\varepsilon_{m}}-\sigma_{c}^{2}}\right)}^{\frac{{{\alpha_{wm}^{(ij)}}{\mu_{wm}^{(ij)}}}}{2}}}}}{{2{{\left({{C_{1w}}{\beta_{wm}^{(ij)}}}\right)}^{\frac{{{\alpha_{wm}^{(ij)}}{\mu_{wm}^{(ij)}}}}{2}}}\Gamma\left({{\mu_{wm}^{(ij)}}}\right)\Gamma\left({{\mu_{wg}^{(ij)}}}\right)}}$
$\displaystyle\times
H_{1,0:1,2:0,0}^{0,1:2,1:1,1}\left(\\!\\!{\left.{\begin{array}[]{*{20}{c}}{\frac{{{C_{1w}}{\beta_{wm}^{(ij)}}}}{{{\varepsilon_{m}}-\sigma_{c}^{2}}}}\\\
{\frac{{{C_{2w}}{\beta_{wg}^{(ij)}}}}{{{\varepsilon_{m}}-\sigma_{c}^{2}}}}\end{array}}\\!\\!\right|\\!\\!\begin{array}[]{*{20}{c}}{\left({1+\frac{{{\alpha_{wm}^{(ij)}}{\mu_{wm}^{(ij)}}}}{2}:-\frac{{{\alpha_{wg}^{(ij)}}}}{2},\frac{{{\alpha_{wm}^{(ij)}}}}{2}}\right):\left({1-{\mu_{wg}^{(ij)}},1}\right)\left({1,1}\right);\left({1-\frac{{{\alpha_{wm}^{(ij)}}{\mu_{wm}^{(ij)}}}}{2},\frac{{{\alpha_{wm}^{(ij)}}}}{2}}\right)}\\\
{-:\left({1,-\frac{{{\alpha_{wg}^{(ij)}}}}{2}}\right)\left({0,1}\right);\left({0,1}\right)}\end{array}}\\!\\!\right)$
(9)

Note that although the warden’s estimate of the channel state is imperfect
(including the JRC node’s transmit power and the jamming power, which are
factors in (B.)), to verify the robustness of the proposed covert system
design, we consider that the warden knows the perfect information (as the
worst-case scenario). This assumption is actually common in the literature,
e.g., in [25, 35]. If we can still guarantee that the DEP is arbitrarily close
to 1 under the worst-case scenario, covert communication is successfully
achieved. This is also clarified later in the results section (Figure 2).

####  C. Covert Channel Capacity

The covert CC obtained under the precondition that DEP $\to$ 1, which reflects
the covert communication rate. For the channel $j$ of JRC node $i$, the CC is
defined as [5, 19]:

${C_{ij}}=\sum\limits_{m=1}^{{M_{c}}}\Delta
f{\log_{2}}\left({1+\frac{{D_{bi}^{-{\alpha_{bi}}}{p_{m}^{(T)}}{{\left|{{h_{m}^{(ij)}}}\right|}^{2}}}}{{\sigma_{c}^{2}+D_{gi}^{-{\alpha_{gi}}}{p_{g}^{(J)}}{{\left|{{h_{g}^{(ij)}}}\right|}^{2}}}}}\right),$
(6)

where
${\left|{{h_{m}^{(ij)}}}\right|^{2}}\sim\alpha-\mu\left({{\alpha_{m}^{(ij)}},{\mu_{m}^{(ij)}},{{\bar{\gamma}}_{m}^{(ij)}}}\right)$
and
${\left|{{h_{g}^{(ij)}}}\right|^{2}}\sim\alpha-\mu\left({{\alpha_{g}^{(ij)}},{\mu_{g}^{(ij)}},{{\bar{\gamma}}_{g}^{(ij)}}}\right)$
represent the small scale fading of each sub-carrier $m$ and the jammer,
respectively [29]. ${D_{bi}^{-{\alpha_{bi}}}}$ and ${D_{gi}^{-{\alpha_{gi}}}}$
denote the large scale fading between the receiver and the JRC node $i$ and
between the jammer $j$ and the JRC node $i$, respectively. $p_{m}^{(T)}$ and
$p_{g}^{(J)}$ are the transmit power of the $m$-th sub-carrier and the jammer,
respectively. $\Delta f=\frac{1}{T}$ is the sub-carrier interval with the
duration of elementary OFDM symbol $T$ and $\sigma_{c}^{2}$ is the noise
variance.  The jammer’s location and its jamming power are publicly shared by
the SSP to enable the JRC nodes to calculate the covert channel capacity
defined in (6).

###### Theorem 2

The closed-form expression of CC can be derived as (C.), where
$H_{\cdot}^{\cdot}\left(\cdot\right)$ is the multivariate Fox’s $H$-function
[34, eq. (A-1)], ${C_{1}}\triangleq D_{bi}^{-{\alpha_{bi}}}{p_{m}^{(T)}}$, and
${C_{2}}\triangleq D_{gi}^{-{\alpha_{gi}}}{p_{g}^{(J)}}$.

$\displaystyle{C_{ij}}$
$\displaystyle=\sum\limits_{m=1}^{{M_{c}}}\frac{2\Delta f}{{\ln
2}}\frac{{{{\left({2{{\left({{C_{2}}{\beta_{g}^{(ij)}}}\right)}^{\frac{{{\alpha_{g}^{(ij)}}{\mu_{g}^{(ij)}}}}{2}}}\Gamma\left({{\mu_{g}^{(ij)}}}\right)}\right)}^{-1}}}}{{{{\left({{C_{1}}{\beta_{m}^{(ij)}}}\right)}^{\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2}}}\Gamma\left({{\mu_{m}^{(ij)}}}\right)}}{\left({\sigma_{c}^{2}}\right)^{\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2}+\frac{{{\alpha_{g}^{(ij)}}{\mu_{g}^{(ij)}}}}{2}}}$
$\displaystyle\times
H_{1,0:3,3;1,1}^{0,1:2,2;1,1}\left(\\!\\!{\left.{\begin{array}[]{*{20}{c}}{\frac{{{C_{1}}{\beta_{m}^{(ij)}}}}{{\sigma_{c}^{2}}}}\\\
{\frac{{{C_{2}}{\beta_{g}^{(ij)}}}}{{\sigma_{c}^{2}}}}\end{array}}\\!\\!\right|\\!\\!\begin{array}[]{*{20}{c}}{\left({1+\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2}+\frac{{{\alpha_{g}^{(ij)}}{\mu_{g}^{(ij)}}}}{2}:1,1}\right):\left({1,\frac{2}{{{\alpha_{m}^{(ij)}}}}}\right)\left({\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2},1}\right)\left({1\\!+\\!\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2},1}\right);\left({1,\frac{2}{{{\alpha_{g}^{(ij)}}}}}\right)}\\\
{-:\left({\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2},1}\right)\left({\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2},1}\right)\left({1+\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2},1}\right);\left({\frac{{{\alpha_{g}^{(ij)}}{\mu_{g}^{(ij)}}}}{2},1}\right)}\end{array}}\\!\\!\\!\right)$
(11)

###### Proof:

Let ${C_{i}}=\sum\limits_{m=1}^{{M_{c}}}\Delta f{C_{m}}$. The $C_{m}$ can be
expressed as

${C_{m}}=\int_{0}^{\infty}{\log\left({1+\gamma}\right){f_{X}}\left(\gamma\right){\text{d}}\gamma},$
(B-1)

where
$X\buildrel\Delta\over{=}\frac{{{C_{1}}{{\left|{{h_{m}^{(ij)}}}\right|}^{2}}}}{{\sigma_{c}^{2}+{C_{2}}{{\left|{{h_{g}^{(ij)}}}\right|}^{2}}}}$.
Next, we first derive ${f_{X}}\left(\gamma\right)$. Let
${X_{1}}={C_{1}}{\left|{{h_{m}^{(ij)}}}\right|^{2}}$ and
${X_{2}}=\sigma_{c}^{2}+{C_{2}}{\left|{{h_{g}^{(ij)}}}\right|^{2}}$, we have
${f_{{X_{1}}}}\left(x\right)=\frac{1}{{{C_{1}}}}{f_{{{\left|{{h_{m}^{(ij)}}}\right|}^{2}}}}\left({\frac{x}{{{C_{1}}}}}\right)$
and
${f_{{X_{2}}}}\left(x\right)=\frac{1}{{{C_{2}}}}{f_{{{\left|{{h_{g}^{(ij)}}}\right|}^{2}}}}\left({\frac{{x-\sigma_{c}^{2}}}{{{C_{2}}}}}\right)$.
The PDF of $X$ can be expressed as [36]
${f_{X}}(x)=\int_{0}^{\infty}y{f_{{X_{1}}}}(xy){f_{{X_{2}}}}(y){\text{dy}}$.
With the help of PDF expressions of ${{{X_{1}}}}$ and ${{X_{2}}}$, we have

${f_{X}}(x)\\!=\\!\frac{{{x^{\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2}-1}}{{\left(\\!{2{{\left(\\!{{C_{2}}{\beta_{g}^{(ij)}}}\\!\right)}^{\frac{{{\alpha_{g}^{(ij)}}{\mu_{g}^{(ij)}}}}{2}}}\Gamma\\!\left(\\!{{\mu_{g}^{(ij)}}}\right)}\\!\right)}^{-1}}}}{{2{{\left(\\!{{\alpha_{g}^{(ij)}}{\alpha_{m}^{(ij)}}}\\!\right)}^{-1}}{{\left(\\!{{C_{1}}{\beta_{m}^{(ij)}}}\\!\right)}^{\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2}}}\Gamma\\!\left(\\!{{\mu_{m}^{(ij)}}}\\!\right)}}{I_{{A_{1}}}},$
(B-2)

where

$\displaystyle{I_{{A_{1}}}}=$
$\displaystyle\int_{0}^{\infty}{{{\left({y-\sigma_{c}^{2}}\right)}^{\frac{{{\alpha_{g}^{(ij)}}{\mu_{g}^{(ij)}}}}{2}-1}}\exp\left({-{{\left({\frac{{xy}}{{{C_{1}}{\beta_{m}^{(ij)}}}}}\right)}^{\frac{{{\alpha_{m}^{(ij)}}}}{2}}}}\right)}$
$\displaystyle\times{y^{\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2}}}\exp\left({-{{\left({\frac{{y-\sigma_{c}^{2}}}{{{\beta_{g}^{(ij)}}{C_{2}}}}}\right)}^{\frac{{{\alpha_{g}^{(ij)}}}}{2}}}}\right){\text{dy}}.$
(B-3)

With the help of [33, eq. (01.03.07.0001.01)], we can re-write ${f_{X}}(x)$ as

$\displaystyle{f_{X}}\\!(x)\\!=\\!\\!\frac{{{x^{\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2}\\!-\\!1}}{{\left({2{{\left(\\!{{C_{2}}{\beta_{g}^{(ij)}}}\\!\right)}^{\frac{{{\alpha_{g}^{(ij)}}{\mu_{g}^{(ij)}}}}{2}}}\Gamma\\!\left({{\mu_{g}^{(ij)}}}\right)}\right)}^{-\\!1}}}}{{2{{\left(\\!{{\alpha_{g}^{(ij)}}{\alpha_{m}^{(ij)}}}\\!\right)}^{-\\!1}}{{\left({2\pi
i}\right)}^{2}}{{\left(\\!{{C_{1}}{\beta_{m}^{(ij)}}}\\!\right)}^{\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2}}}\Gamma\\!\left({{\mu_{m}^{(ij)}}}\right)}}$
$\displaystyle\times\int_{{\mathcal{L}_{1}}}{\int_{{\mathcal{L}_{2}}}\\!\\!\\!{\Gamma\\!\left({-\\!{s_{1}}}\\!\right)}}\\!{\left({\frac{1}{{{\beta_{g}^{(ij)}}{C_{2}}}}}\right)^{\frac{{{s_{2}}{\alpha_{g}^{(ij)}}}}{2}}}{\left({\frac{x}{{{C_{1}}{\beta_{m}^{(ij)}}}}}\right)^{\frac{{{s_{1}}{\alpha_{m}^{(ij)}}}}{2}}}$
$\displaystyle\times\Gamma\left({-{s_{2}}}\right){I_{A1}}d{s_{1}}d{s_{2}},$
(B-4)

where the integration path of $\mathcal{L}_{1}$ and $\mathcal{L}_{2}$ goes
from $\sigma_{1}-i\infty$ to $\sigma_{1}+i\infty$ and $\sigma_{2}-i\infty$ to
$\sigma_{2}+i\infty$, respectively, and $\sigma_{1},\sigma_{2}\in\mathbb{R}$,
${I_{A1}}=\int_{0}^{\infty}{{y^{\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2}+\frac{{{s_{1}}{\alpha_{m}^{(ij)}}}}{2}}}{{\left({y-\sigma_{c}^{2}}\right)}^{\frac{{{\alpha_{g}^{(ij)}}{\mu_{g}^{(ij)}}}}{2}+\frac{{{s_{2}}{\alpha_{g}^{(ij)}}}}{2}-1}}}{\text{dy}}$,
which can be solved with the help of [30, eq. (3.194.3)]. Let
${t_{1}}=\frac{{{s_{1}}{\alpha_{m}^{(ij)}}}}{2}$ and
${t_{2}}=\frac{{{s_{2}}{\alpha_{g}^{(ij)}}}}{2}$. By substituting ${f_{X}}(x)$
into (B-1), the $C_{m}$ can be expressed as (C.), shown at the top of the next
page, where
${I_{B}}=\int_{0}^{\infty}{\log\left({1+\gamma}\right){\gamma^{{t_{1}}+\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2}-1}}{\text{d}}\gamma}$.

$\displaystyle{C_{m}}=$ $\displaystyle\frac{4}{{{{\left({2\pi
i}\right)}^{2}}}}\frac{{{{\left({2{{\left({{C_{2}}{\beta_{g}^{(ij)}}}\right)}^{\frac{{{\alpha_{g}^{(ij)}}{\mu_{g}^{(ij)}}}}{2}}}\Gamma\\!\left({{\mu_{g}^{(ij)}}}\right)}\right)}^{-1}}}}{{2{{\left({{C_{1}}{\beta_{m}^{(ij)}}}\right)}^{\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2}}}\Gamma\\!\left({{\mu_{m}^{(ij)}}}\right)}}{\left({\sigma_{c}^{2}}\right)^{\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2}+\frac{{{\alpha_{g}^{(ij)}}{\mu_{g}^{(ij)}}}}{2}}}$
$\displaystyle\times\int_{{{\cal L}_{1}}}{\int_{{{\cal
L}_{2}}}{\frac{{\Gamma\\!\left({-\frac{{2{t_{1}}}}{{{\alpha_{m}^{(ij)}}}}}\right)\Gamma\\!\left({\frac{{{\alpha_{g}^{(ij)}}{\mu_{g}^{(ij)}}}}{2}+{t_{2}}}\right)\Gamma\\!\left({-\frac{{2{t_{2}}}}{{{\alpha_{g}^{(ij)}}}}}\right)}}{\Gamma^{-1}\\!\left({-\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2}-\frac{{{\alpha_{g}^{(ij)}}{\mu_{g}^{(ij)}}}}{2}-{t_{1}}-{t_{2}}}\right){\Gamma\\!\left({-\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2}-{t_{1}}}\right)}}}}{\left({\frac{{\sigma_{c}^{2}}}{{{C_{1}}{\beta_{m}^{(ij)}}}}}\right)^{{t_{1}}}}{\left({\frac{{\sigma_{c}^{2}}}{{{C_{2}}{\beta_{g}^{(ij)}}}}}\right)^{{t_{2}}}}{I_{B}}d{t_{1}}d{t_{2}}$
(B-5)

With the help of [37, eq. (2.6.9.21)] and [30, eq. (8.334.3)], $I_{B}$ can be
solved. Substituting $I_{B}$ into (C.), using the definition of multivariate
Fox’s $H$-function [34, eq. (A-1)], we can obtain (C.) to complete the proof.
∎

Note here that the so called ”covert channel capacity” is the same as standard
channel capacity with the condition that the DEP is close to 1. If the DEP is
far lower than 1, the communication is no longer covert but the capacity of
the channel to transmit data is not affected. In other words, the DEP can be
less than 0.9 but still the derived channel capacity is correct. However, it
is not common in the literature to refer to such system as “covert“. See for
example references [15, 4, 25], where the communication is said to be covert
only when DEP is between 0.9 and 1.

####  D. Covert Radar Mutual Information

When we ensure that (B.) is arbitrarily close to $1$ by adjusting the transmit
and jamming power, we can guarantee that the signals will not be detected
[27]. Now the MI of the JRC system is called _covert radar MI_. For the
channel $j$ of JRC node $i$, the MI is defined as [5, 19]:

$\displaystyle{I_{ij}}=\\!\frac{\Delta
f{T_{\rm{pri}}}}{2}\sum\limits_{m=1}^{{M_{c}}}{{{\log}_{2}}}\\!\\!\left(\\!{1\\!+\\!\frac{{T_{\rm{pri}}D_{bi}^{-{\alpha_{bi}}}{p_{m}^{(T)}}{{\left|{G\left({{f_{m}}}\right)}\right|}^{2}}}}{{\Psi\left({{f_{m}}}\right)\\!+\\!D_{gi}^{-{\alpha_{gi}}}{p_{g}^{(J)}}{{\left|{J\left({{f_{m}}}\right)}\right|}^{2}}}}}\\!\right),$
(8)

where ${T_{{\rm{pri}}}}={T_{\rm pulse}}/\delta$ is the pulse repetition
interval of the radar system, ${T_{\rm pulse}}$ is the radar pulse duration,
$\delta$ is the radar duty factor, ${G\left({{f_{m}}}\right)}$,
${J\left({{f_{m}}}\right)}$ and ${\Psi\left({{f_{m}}}\right)}$ are energy
spectral densities (ESDs) of the transmitted signal, jamming signal and noise,
respectively. According to [38, eq. (5)], ESD is viewed as uniform in each
sub-channel, and we can consider that
${\left|{G\left({{f_{m}}}\right)}\right|^{2}}\sim\alpha-\mu\left({{\alpha_{rm}^{(ij)}},{\mu_{rm}^{(ij)}},{{\bar{\gamma}}_{rm}^{(ij)}}}\right)$,
${\left|{J\left({{f_{m}}}\right)}\right|^{2}}\sim\alpha-\mu\left({{\alpha_{rg}^{(ij)}},{\mu_{rg}^{(ij)}},{{\bar{\gamma}}_{rg}^{(ij)}}}\right)$,
and ${\left|{\Psi\left({{f_{m}}}\right)}\right|^{2}}={\sigma_{r}^{2}}$,
${\sigma_{r}^{2}}$ is the noise power, $T_{\rm{pri}}$ is the signal duration,
and $f_{m}=f_{c}+m\Delta f$ is the $m$-th subcarrier frequency with $f_{c}$
the central frequency. Note that it is possible for the warden to detect
reflected signals during radar sensing. However, with the help of the jamming
signals, the warden will not be able to know which JRC node has initiated the
radar sensing, which is the objective of the covertness for radar sensing.

###### Theorem 3

The covert radar MI rate can be expressed in closed-form as (D.), where
${C_{3}}\triangleq{T_{\rm{pri}}}D_{bi}^{-{\alpha_{bi}}}{p_{m}^{(T)}}$, and
${C_{4}}\triangleq D_{gi}^{-{\alpha_{gi}}}{p_{g}^{(J)}}$.

###### Proof:

Following the similar steps to Theorem 2, we can derive (D.) to complete the
proof. ∎

$\displaystyle{I_{ij}}$
$\displaystyle=\mathop{\sum}\limits_{m=1}^{{M_{c}}}\frac{{\Delta f{T_{\rm
pri}}}}{{\ln
2}}\frac{{{{\left({2{{\left({{C_{4}}{\beta_{rg}^{(ij)}}}\right)}^{\frac{{{\alpha_{rg}^{(ij)}}{\mu_{rg}^{(ij)}}}}{2}}}\Gamma\left({{\mu_{rg}^{(ij)}}}\right)}\right)}^{-1}}}}{{{{\left({{C_{3}}{\beta_{rm}^{(ij)}}}\right)}^{\frac{{{\alpha_{m}^{(ij)}}{\mu_{m}^{(ij)}}}}{2}}}\Gamma\left({{\mu_{rm}^{(ij)}}}\right)}}{\left({\sigma_{r}^{2}}\right)^{\frac{{{\alpha_{rm}^{(ij)}}{\mu_{rm}^{(ij)}}}}{2}+\frac{{{\alpha_{rg}^{(ij)}}{\mu_{rg}^{(ij)}}}}{2}}}$
$\displaystyle\times
H_{1,0:3,3;1,1}^{0,1:2,2;1,1}\\!\left(\\!\\!\\!{\left.{\begin{array}[]{*{20}{c}}{\frac{{{C_{3}}{\beta_{rm}^{(ij)}}}}{{\sigma_{r}^{2}}}}\\\
{\frac{{{C_{4}}{\beta_{rg}^{(ij)}}}}{{\sigma_{r}^{2}}}}\end{array}}\\!\\!\\!\right|\\!\\!\\!\begin{array}[]{*{20}{c}}{\left({1\\!+\\!\frac{{{\alpha_{rm}^{(ij)}}{\mu_{rm}^{(ij)}}}}{2}+\frac{{{\alpha_{rg}^{(ij)}}{\mu_{rg}^{(ij)}}}}{2}:1,1}\right):\left({1,\frac{2}{{{\alpha_{rm}^{(ij)}}}}}\right)\left({\frac{{{\alpha_{rm}^{(ij)}}{\mu_{rm}^{(ij)}}}}{2},1}\right)\left({1+\frac{{{\alpha_{rm}^{(ij)}}{\mu_{rm}^{(ij)}}}}{2},1}\right);\left({1,\frac{2}{{{\alpha_{rg}^{(ij)}}}}}\right)}\\\
{-:\left({\frac{{{\alpha_{rm}^{(ij)}}{\mu_{rm}^{(ij)}}}}{2},1}\right)\left({\frac{{{\alpha_{rm}^{(ij)}}{\mu_{rm}^{(ij)}}}}{2},1}\right)\left({1+\frac{{{\alpha_{rm}^{(ij)}}{\mu_{rm}^{(ij)}}}}{2},1}\right);\left({\frac{{{\alpha_{rg}^{(ij)}}{\mu_{rg}^{(ij)}}}}{2},1}\right)}\end{array}}\\!\\!\\!\right)$
(13)

###  3.3 Auction Model

Figure 1 presents the proposed auction model. We consider that the SSP, as the
auctioneer, is offering a unit bundle that consists of a set of channels and
friendly jammers to enable covertness for the JRC nodes as the bidders. The
SSP conducts an auction by broadcasting its available spectrum resources to
the JRC nodes at every time period $T_{b}$ (e.g., every 10 seconds). The JRC
nodes buy spectrum resources from the SSP and use them for radar sensing and
data transmission. Each JRC node $i$ submits its bid vector
$\mathbf{b}_{i}=(b_{1},b_{2},\dots,b_{M})$ to the SSP. Each element of the
vector $\mathbf{b}_{i}$ represents the bid that JRC node $i$ is willing to pay
for channel $j$. Setting $b_{ij}=0$ means that the JRC node is not interested
in channel $j$. Before the auction starts, the SSP first calculates the
nominal allocation and reservation prices (defined later in Section 4).  The
calculation of the reservation prices prevents market manipulation by setting
a lower bound on acceptable amounts of bids for any JRC node in order to be
included in the winner list. After receiving the bids from the JRC nodes, the
SSP (as the auctioneer) runs the winner selection algorithm to derive the
final allocation vector $\mathbf{a}_{i}=(a_{1},a_{2},\dots,a_{M})$ and the
payment ${p_{i}}$ for each JRC node $i$. The winning JRC nodes are then
allowed to use the channels according to their allocation vectors
$\mathbf{a}_{i},\forall i\in\mathcal{N}$. In the following, we define the
utility functions of the JRC nodes and the SSP and the social welfare
maximization problem.

####  A. Utility Functions

The utility of the SSP is defined as the difference between the payment that
it receives from all JRC nodes and the total cost to maintain the channels:

$u_{SSP}=\sum\limits_{i\in\mathcal{N}}p_{i}-c(\mathbf{x}),$ (10)

where $p_{i}$ is the payment given by JRC node $i$ and
$c(\mathbf{x})=\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}}c_{j}x_{ij}$
is the total channel cost for the allocation vector
$\mathbf{x}=\\{x_{ij}\\}_{i\in N,j\in M}$ and $c_{j}$ is the per channel cost
for the SSP. The channel cost includes the required computing resources to
maintain the channel and the cost of friendly jammers for ensuring the
covertness of the JRC system. The cost of channel $j$ is expressed as follows:

$\begin{multlined}c_{j}=\kappa_{1,j}p_{\rm{FJ},j}+\kappa_{2,j},\end{multlined}c_{j}=\kappa_{1,j}p_{\rm{FJ},j}+\kappa_{2,j},$
(11)

where $p_{\rm{FJ},j}$ is the total jamming power used to covert channel $j$,
$\kappa_{1,j}$ is the per unit cost of the jamming power, and $\kappa_{2,j}$
is a constant that reflects the licensing fees for channel $j$.

We consider TDMA for the radar and communication functions by the JRC system.
Specifically, for some time slots, the allocated channel will be used for
radar sensing and then for data transmission in the other time slots. Each JRC
node $i$ has a private valuation of channel $j$ denoted by $v_{ij}$ which is
unknown to the SSP. The valuation for each JRC node can vary because of the
hardware specific design for each JRC node, e.g., supported wireless
technologies that operate on different bandwidths. Also, the valuations given
by a JRC node $i$ can differ from one channel to another channel because each
channel provided by the SSP can have different transmission characteristics
and channel fading parameters. We define the valuation as follows:

$\begin{multlined}v_{ij}=\mathcal{I}_{ij}(\eta_{1}I_{ij}+\eta_{2}C_{ij})\xi_{w},\end{multlined}v_{ij}=\mathcal{I}_{ij}(\eta_{1}I_{ij}+\eta_{2}C_{ij})\xi_{w},$
(12)

where $\mathcal{I}_{ij}$ is an indicator function in the form of a binary
matrix that reflects the ability of JRC node $i$ to use channel $j$ or not,
and is known to the SSP. $\eta_{1}$ and $\eta_{2}$ are scaling factors, and
$\xi_{w}$ is the DEP at warden $w$. The DEP in (12) reflects the discount in
the valuation due to the probabilities that the warden detects the ongoing
transmission by the JRC node.  The DEP is chosen to get multiplied into the
weighted sum of the covert channel capacity and the covert MI in (12) because
as the DEP decreases, the output of the valuation function in (12) needs to
decrease linearly. If the DEP was just an addition term, the change in the
final valuation output would be less apparent222further mathematical
explanation can be found in [39].. In other words, in (12), we are counting
the percentage that we are able to protect against the warden, which is
reflected using the DEP value (between 0 and 1). For instance, if the DEP is
high (close to 1), this would imply a meaningful allocation to the JRC node,
i.e., the performed communication and sensing are successfully covert.
Otherwise, if the DEP is low, that indicates a wasted resource allocation.

Note that the impact of the two scaling factors cannot be observed beyond the
JRC node itself. Specifically, the output of (12) is just a number which will
be used later during the auction process. Changing the weighting factors will
only increase or decrease the submitted bids by the JRC node, i.e., its
chances to be among the winners. The form of (12) has the objective to help
the JRC node to determine the best price to submit so as it maximizes its
benefit from getting the resource. The impact of changes of the value computed
by (12) is explored later in the results section. When the JRC node obtains
the spectrum, then based on those coefficients it will allocate the spectrum
proportionally for both functionalities based on a TDMA scheme, as discussed
in the paper. Furthermore, the scaling factors can vary dynamically over time
based on the JRC node’s demand for data transmission or target sensing to
assert a certain trade-off as we demonstrated in our previous work [8].

The JRC node $i$’s utility is then defined as the difference between its
valuation for all the channels and its payment $p_{i}$, which is expressed by
the following quasilinear preference function:

$u_{i}=\begin{cases}\sum\limits_{j\in\mathcal{M}}v_{ij}x_{ij}-p_{i},&\text{if
JRC node $i$ wins},\\\ 0,&\text{otherwise}.\end{cases}$ (13)

####  B. Social Welfare Maximization

The solution to the auction mechanism is the maximization of the social
welfare function which is defined as the sum of all the utilities, i.e., the
utility of the SSP and all the utilities of the JRC nodes. Formally, the
social welfare function is defined as:

$SW=u_{SSP}+\sum\limits_{i\in\mathcal{N}}u_{i}$
$=\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}}v_{ij}x_{ij}-\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}}c_{j}x_{ij}$
$=\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}}(v_{ij}-c_{j})x_{ij}.$
(14)

####  C. Properties of The Auction Mechanism

Before solving the maximization problem (14), the following properties need to
be satisfied for an auction to be optimal and efficient:

  * •

Incentive compatibility (IC): The JRC node $i$ has no incentive to submit a
false bid as for every other bid $v^{\prime}$, the obtained utility is lower
than the utility the JRC node gets by submitting its true valuation $v$.
Formally,

$\sum\limits_{j\in\mathcal{M}}v^{\prime}_{ij}x_{ij}-p^{(v^{\prime})}_{i}\leq\sum\limits_{j\in\mathcal{M}}v_{ij}x_{ij}-p^{(v)}_{i},\quad\forall
i\in\mathcal{N},$ (15)

where $p^{(v)}_{i}$ and $p^{(v^{\prime})}_{i}$ are the obtained payments for
the true valuation $v$ and any other valuation $v^{\prime}$, respectively.

  * •

Budget feasibility (BF): The payment vector is budget feasible. Formally,

$p^{(v)}_{i}\leq B_{i},\quad\forall i\in\mathcal{N},$ (16)

where $B_{i}$ is the the maximum budget for JRC node $i$ and is assumed to be
publicly known. The property of BF is of crucial importance in multi-item
auctions. This is because in real systems, the buyers always have a limited
budget that they need not to exceed. For example, in an auction mechanism that
does not consider BF, if a bidder is selected amongst the winners for several
items but he/she cannot pay for all the items, the solution becomes
infeasible. One of the main problems in multi-item auctions is that the bidder
does not know in advance how many items he/she will win and hence, its budget
needs to be incorporated into the optimization problem.

  * •

Individual rationality (IR): The utilities must be non-negative for all JRC
nodes, i.e., $u_{i}\geq 0,\forall i\in\mathcal{N}$

  * •

Computational efficiency (CE): The proposed solution to the optimization
problem should be computed in a polynomial time.

Therefore, to derive an optimal auction that satisfies the above mentioned
properties, problem (14) is rewritten as:

$\displaystyle\begin{split}\max\limits_{\mathbf{x}}SW=\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}}(v_{ij}-c_{j})x_{ij}\end{split}$
(17a) $\displaystyle\begin{split}\hskip
14.22636pts.t.\sum\limits_{j\in\mathcal{M}}v_{ij}x_{ij}\leq B_{i},\quad\forall
i\in\mathcal{N},\end{split}$ (17b) $\displaystyle\begin{split}\hskip
14.22636ptp^{(v)}_{i}\leq\sum\limits_{j\in\mathcal{M}}v_{ij}x_{ij},\quad\forall
i\in\mathcal{N},\end{split}$ (17c) $\displaystyle\begin{split}\hskip
14.22636pt\sum\limits_{j\in\mathcal{M}}v^{\prime}_{ij}x_{ij}-p^{(v^{\prime})}_{i}\leq\sum\limits_{j\in\mathcal{M}}v_{ij}x_{ij}-p^{(v})_{i},\quad\forall
i\in\mathcal{N},\end{split}$ (17d) $\displaystyle\begin{split}\hskip
28.45274pt\sum\limits_{i\in\mathcal{N}}x_{ij}\leq 1,\quad\forall
j\in\mathcal{M},\end{split}$ (17e) $\displaystyle\begin{split}\hskip
28.45274pt\mathbf{x}\geq 0,\end{split}$ (17f)

where $x_{ij}$ is the probability that channel $j$ is allocated to the JRC
node $i$, and $c_{j}$ is the cost for each channel $j$. The constraints (17b),
(17c) and (17d), refers to BF, IR and IC, respectively [40]. The constraints
(17e) and (17f) ensure that the vector of allocation probabilities sums to 1.
Note that channels are indivisible items, i.e., each channel is allocated to
only one JRC node at a time. Therefore, we are restricted to integral values
for the allocation vector $\mathbf{x}$, which we explain later in Algorithm 2.

Finally, the SSP needs to take into consideration the uncertainty in bids when
deriving the solution to the auction mechanism. The SSP is able to construct
an uncertainty set for these valuations based on the previously submitted bids
by the JRC nodes. It can then use the constructed uncertainty set during the
channel allocation phase to derive an optimal allocation strategy that
reflects its risk-aversion attitude about uncertain parameters in the system,
e.g., the warden’s location. The size of the uncertainty set determines the
risk-aversion level of the SSP, i.e., how robust we want to be. If the SSP has
a high level of risk-aversion, it will consider a large uncertainty set and
vice-versa. This has also been validated by other existing works, e.g., [41],
in which the authors showed that the knowledge of a large number of historical
data, i.e., a larger uncertainty set, gives an exhaustive set of scenarios,
and guarantees the reliability of the derived solution, i.e., high risk-
aversion level. Furthermore, in [42], under the assumption of normal
distribution, the authors were able to derive an expression that links the
size of the uncertainty set to the risk level. Therefore, in the following
section, we develop a robust multi-item auction mechanism that takes into
consideration the uncertainty of bids by all the JRC nodes for each channel.

##  4 Auction-based Mechanism for Channel Allocation

In this section, we formulate the multi-item auction based JRC resource
allocation as a robust optimization problem. The objective is to maximize the
social welfare of the system for all valuations by the JRC nodes in the
constructed uncertainty set. Unlike previous works that consider the network
geometry to be overt to all the nodes [15, 26], the uncertain parameters are
typically not known to the transmitters and the SSP, in which case we consider
the location of the warden to be the uncertain parameter, while the other
uncertain parameters can be adopted in the auction mechanism. The uncertain
parameters significantly impact the channel gain equations and the spectrum
valuation, and hence reduce the expected social welfare and violate optimal
auction properties. To overcome these challenges, we develop a robust auction
mechanism that considers the uncertainty in the bidders’ valuations [40].

###  4.1 Construction of the Uncertainty Set

The SSP can create the uncertainty set for the bids based on the type of
information it has access to. In the following, we describe two different ways
of creating the uncertainty set $\mathcal{U}$ from which the valuation vectors
are derived.

####  A. Interval Uncertainty Set

In these settings, the belief of the SSP about the valuations of the JRC nodes
is modeled based on the lowest and highest possible valuations for each JRC
node for each channel. Specifically, the SSP has geometrical information about
the transmitter, receiver, and friendly jammer. However, the exact location of
the warden is unknown either to the JRC node or to the SSP, affecting the
submitted bids by the JRC nodes for each channel. Therefore, the JRC nodes
consider that the warden is located in a cube instead of a point in the three-
dimensional Cartesian domain. Then, the JRC nodes can calculate the smallest
and largest possible values of DEP at the warden while varying its location
inside the cube. It is also possible to adopt other forms of uncertainty
intervals, such as when the warden is located on a sphere. Note that the
calculation of the DEPs can be done by methods such as the particle swarm
optimization (PSO) [43] algorithm. Then they substitutes these values in the
valuation function presented in (12) and derive the lowest and highest
valuations for each JRC node and each channel. The uncertainty set for channel
$j$ with respect to the JRC node $i$ is then defined as follows:

$\mathcal{U}_{ij}=\\{\mu_{ij}\pm\varsigma_{ij}\\},$ (18)

where $\mu_{ij}$ is the mean value for the valuation of channel $j$ by the JRC
node $i$, and $-\varsigma_{ij}$ and $+\varsigma_{ij}$ reflects the minimum and
maximum valuations normalized to zero, respectively. If the SSP has more than
one uncertain parameter, it can adopt more generalized techniques for creating
the uncertainty set, e.g., correlated historical data technique, presented in
the following.

####  B. Correlated Historical Data

If the uncertainty of the SSP about the bids is not limited to the warden’s
location, i.e., multiple or unknown factors, the belief of the SSP about the
valuations of the JRC nodes can be modeled using historical data of previous
bids. Specifically, the uncertainty set for channel $j$ is defined as:

$\mathcal{U}_{j}=\left\\{\\!\\!\\!\\!\begin{array}[]{c|c}(v_{1j},\ldots,v_{Nj})&\begin{array}[]{c}v_{ij}=f_{j}+y_{ij},\quad\forall
i\in\mathcal{N},\\\ \underline{F_{j}}\leq f_{j}\leq\overline{F_{j}},\\\
-\vartheta\leq\frac{\sum_{i=1}^{N}y_{ij}-N\cdot\mu_{g}}{\sqrt{N}\cdot\delta_{j}}\leq\vartheta,\end{array}\end{array}\\!\\!\\!\\!\right\\},$
(19)

where $f_{j}$ is a common factor between valuations that reflects the
correlation between valuations, $y_{ij}$ are independent components with mean
$\mu_{j}$ and standard deviation $\delta_{j}$. $\vartheta$ is the parameter
that controls the conservativeness of the historical data.

The robustness is then incorporated in the original problem (17) as follows:

$(\mathbf{z},\mathbf{x}^{*})\\!=\\!\operatorname*{argmax}\limits_{\mathbf{v}\in\mathcal{U}}\left\\{\\!\\!\\!\\!\begin{array}[]{c}\\!\vspace{0.3cm}\max\limits_{\mathbf{x}}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}}(v_{ij}-c_{j})x_{ij}\\\
\vspace{0.3cm}\hskip 14.22636pts.t.\sum\limits_{i\in\mathcal{N}}x_{ij}\leq
1,\quad\forall j\in\mathcal{M},\\\
\vspace{0.3cm}\sum\limits_{j\in\mathcal{M}}v_{ij}x_{ij}\leq B_{i},\quad\forall
i\in\mathcal{N},\\\
\vspace{0.3cm}\sum\limits_{j\in\mathcal{M}}v_{ij}x_{ij}\leq\sum\limits_{j\in\mathcal{M}}u_{ij}x_{ij},\\\
\quad\forall\mathbf{u}\in\mathcal{U},\forall i\in\mathcal{N},\\\ \hskip
28.45274pt\mathbf{x}\geq 0,\end{array}\\!\\!\\!\right\\},$ (20)

where $\mathbf{z}=\\{z_{ij}\\}_{i\in N,j\in M}$ is the optimal valuation
vector and the objective is to maximize the worst-case social welfare over all
the possible valuation vectors in the uncertainty set $\mathcal{U}$. By
setting
$\bar{u}^{i}_{j}=\operatorname*{arg\,min}\limits_{\mathbf{u}\in\mathcal{U}}\sum\limits_{j\in\mathcal{M}}x^{*}_{ij}u_{ij},\forall
i\in\mathcal{N}$, the problem (20) is reformulated as follows:

$(z,x^{*})=\operatorname*{argmax}\limits_{\mathbf{v}\in\mathcal{U}}\left\\{\begin{array}[]{c}\vspace{0.3cm}\max\limits_{x}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}}(v_{ij}-c_{j})x_{ij}\\\
\vspace{0.3cm}\hskip 14.22636pts.t.\sum\limits_{i\in\mathcal{N}}x_{ij}\leq
1,\quad\forall j\in\mathcal{M},\\\
\vspace{0.3cm}\sum\limits_{j\in\mathcal{M}}v_{ij}x_{ij}\leq B_{i},\quad\forall
i\in\mathcal{N},\\\
\vspace{0.3cm}\sum\limits_{j\in\mathcal{M}}v_{ij}x_{ij}\leq\sum\limits_{j\in\mathcal{M}}\bar{u}^{i}_{j}x_{ij},\quad\forall
i\in\mathcal{N},\\\ \hskip 28.45274pt\mathbf{x}\geq 0.\end{array}\right\\}.$
(21)

The dual of the inner problem (21) is as follows:

$\begin{array}[]{c}\vspace{0.3cm}\min\limits_{\boldsymbol{\omega},\boldsymbol{\phi},\boldsymbol{\psi}}\sum\limits_{j\in\mathcal{M}}\omega_{j}+\sum\limits_{i\in\mathcal{N}}\left(\phi_{i}B_{i}+\psi_{i}\sum\limits_{j\in\mathcal{M}}x^{*}_{ij}\bar{u}^{i}_{j}\right)\\\
\vspace{0.3cm}\hskip
8.5359pts.t.\quad\omega_{j}+z_{ij}\phi_{i}+z_{ij}\psi_{i}+c_{j}\geq
z_{ij},\quad\forall i\in\mathcal{N},\forall j\in\mathcal{M}\\\
\vspace{0.3cm}\hskip 28.45274pt\phi_{i},\psi_{i}\geq 0,\quad\forall
i\in\mathcal{N},\\\ \vspace{0.3cm}\hskip 28.45274pt\omega_{j}\geq
0,\quad\forall j\in\mathcal{M},\end{array}$ (22)

where $\omega_{j}$, $\phi_{i}$ and $\psi_{i}$ are elements of
$\boldsymbol{\omega}$, $\boldsymbol{\phi}$ and $\boldsymbol{\psi}$,
respectively, and are the duals corresponding to the first, second, and third
constraints in (21).

###  4.2 Robust Mechanism for Channel Allocation (RMCA)

The proposed Robust Mechanism for Channel Allocation is executed in two
phases:

####  A. Nominal Allocation and Reservation Price Calculation

The first phase of the mechanism is executed offline before the beginning of
the auction and is presented in Algorithm 1. The SSP starts first by
constructing the uncertainty set as previously described in Section 4.1 and
then uses it as an input to the algorithm with the budgets of each JRC node.
Then problem (20), which is a bilinear optimization problem and outputs the
worst-case valuation vector $\mathbf{z}$ and the nominal allocation vector
$\mathbf{x}^{*}$, is solved using Generalized Benders Decomposition [44]. The
dual of the problem (20) is then solved to calculate the reservation prices
$\mathbf{r}^{*}=\\{r_{ij}\\}_{i\in\mathcal{N},j\in\mathcal{M}}$ in steps 1-1
of Algorithm 1. The reservation prices are defined as the minimum bids that
should be submitted by each JRC node to be admissible to the winner list. As
such, if a JRC node submits a bid lower than its reservation prices, it will
not be among the winners. Note that the rationale behind setting the
reservation prices equal to the left term of the first constraint of the dual
in (22) is as follows. Since $\mathbf{z}$ is the optimal solution of the
primal for the worst-case valuation, the price that each JRC node has to pay
is equal to that at minimum. Otherwise, the SSP (the auctioneer) will have a
negative utility.

Input : Uncertainty set $\mathcal{U}$, and budgets $B_{1},\dots,B_{N}$.

Output : Reservation prices $\mathbf{r}^{*}$, and nominal allocations
$\mathbf{x}^{*}$.

1

2 begin

3 $(z,x^{*})\leftarrow$ Solve problem (20);

4 $(\omega^{*},\phi^{*},\psi^{*})\leftarrow$ Solve problem (22);

5 // Calculate reservation prices

6 foreach  _$i\in\mathcal{N}$_ do

7 foreach  _$j\in\mathcal{M}$_ do

8 $r^{*}_{ij}=\omega^{*}_{j}+z_{ij}\phi^{*}_{i}+z_{ij}\psi^{*}_{i}+c_{j}$;

9 end foreach

10

11 end foreach

12

13 end

14

Algorithm 1 RMCA.a

####  B. Final Allocation and Payment Calculation

The second phase of the mechanism is executed after the bid vector is
realized, i.e., the JRC nodes submit their bids to the SSP, and is presented
in Algorithm 2. First, the adapted allocation vector $y^{(v)}$ is calculated
by solving the following problem (23), in which the objective is to maximize
the social welfare with consideration of the previously derived reservation
prices $\mathbf{r^{*}}$ and the realized bid vector $\mathbf{v}$:

$\displaystyle\begin{split}\max\limits_{y^{(v)}}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}}(v_{ij}-c_{j}-r^{*}_{ij})y^{(v)}_{ij}\end{split}$
(23a) $\displaystyle\begin{split}\hskip
8.5359pts.t.\sum\limits_{i\in\mathcal{N}}y^{(v)}_{ij}\leq
1-\sum\limits_{i\in\mathcal{N}}x^{*}_{ij},\quad\forall
j\in\mathcal{M}\end{split}$ (23b) $\displaystyle\begin{split}\hskip
8.5359pt\sum\limits_{j\in\mathcal{M}}y^{(v)}_{ij}u_{ij}\leq
B_{i}-\sum\limits_{j\in\mathcal{M}}x^{*}_{ij}r^{*}_{ij}+\sum\limits_{j\in\mathcal{M}}x^{*}_{ij}\psi^{*}_{i}\bar{u}^{i}_{j},\\\
\quad\forall\mathbf{u}\in\mathcal{U},\forall i\in\mathcal{N},\end{split}$
(23c) $\displaystyle\begin{split}\hskip 28.45274pt\mathbf{y}^{(v)}\geq
0.\end{split}$ (23d)

Then we calculate the adapted allocation $y^{(v)_{-k}}$ which is similar to
problem (23) with JRC node $k$ removed from the set of bidders:

$\displaystyle\begin{split}\max\limits_{y^{(v)_{-k}}}\sum\limits_{i\in\mathcal{N}\backslash\\{k\\}}\sum\limits_{j\in\mathcal{M}}(v_{ij}-c_{j}-r^{*}_{ij})y^{(v)_{-k}}_{ij}\end{split}$
(24a) $\displaystyle\begin{split}\hskip
8.5359pts.t.\sum\limits_{i\in\mathcal{N}\backslash\\{k\\}}y^{(v)_{-k}}_{ij}\leq
1-\sum\limits_{i\in\mathcal{N}}x^{*}_{ij},\quad\forall
j\in\mathcal{M}\end{split}$ (24b) $\displaystyle\begin{split}\hskip
8.5359pt\sum\limits_{j\in\mathcal{M}}y^{(v)_{-k}}_{ij}u_{ij}\leq
B_{i}-\sum\limits_{j\in\mathcal{M}}x^{*}_{ij}r^{*}_{ij},\quad\forall\mathbf{u}\in\mathcal{U},\forall
i\in\mathcal{N}\backslash\\{k\\},\end{split}$ (24c)
$\displaystyle\begin{split}\hskip 28.45274pt\mathbf{y}^{(v)_{-k}}\geq
0.\end{split}$ (24d)

The payments are then calculated by using a VCG-like method, in which the JRC
nodes are charged the lowest amount that they could have bid such that they
are in the winner list [40].

Input : Realized bid vector
$\mathbf{v}=\\{v_{ij}\\}_{i\in\mathcal{N},j\in\mathcal{M}}$, reservation
prices $\mathbf{r}^{*}=\\{r_{ij}\\}_{i\in\mathcal{N},j\in\mathcal{M}}$, and
nominal allocation
$\mathbf{x}^{*}=\\{x_{ij}\\}_{i\in\mathcal{N},j\in\mathcal{M}}$

Output : Allocation vector
$\mathbf{a}^{*}=\\{a_{ij}\\}_{i\in\mathcal{N},j\in\mathcal{M}}$, and payment
$\mathbf{p}^{*}=\\{p_{ij}\\}_{i\in\mathcal{N},j\in\mathcal{M}}$

1

2 begin

3 if  _$\mathbf{v}\notin\mathcal{U}$_ then

4 Do not allocate any channel to any JRC node and exit the auction.

5 else

6 $y^{(v)}\leftarrow$ Solve problem (23);

7 foreach  _$k\in\mathcal{N}$_ do

8 $y^{(v)_{-k}}\leftarrow$ Solve problem (24);

9

10 end foreach

11 // Calculate the final allocation vector

12 foreach  _$i\in\mathcal{N}$_ do

13 foreach  _$j\in\mathcal{M}$_ do

14 $a_{ij}^{*}=y_{ij}^{v}+x_{ij}^{*}$;

15 end foreach

16

17 end foreach

18 // Calculate the payment vector

19 foreach  _$k\in\mathcal{N}$_ do

20
$p_{k}=\sum\limits_{j\in\mathcal{M}}y^{(v)}_{kj}r^{*}_{kj}+\sum\limits_{j\in\mathcal{M}}x^{*}_{kj}r^{*}_{kj}-\sum\limits_{j\in\mathcal{M}}x^{*}_{kj}\psi^{*}_{k}\bar{u}^{k}_{j}+$

21
$\sum\limits_{i\in\mathcal{N}\backslash\\{k\\}}\sum\limits_{j\in\mathcal{M}}(v_{ij}-r^{*}_{ij})y^{(v)_{-k}}_{ij}-\sum\limits_{i\in\mathcal{N}\backslash\\{k\\}}\sum\limits_{j\in\mathcal{M}}(v_{ij}-r^{*}_{ij})y^{(v)}_{ij},\quad\forall
k\in\mathcal{N}$;

22 end foreach

23

24 Allocate the $j$th channel to the $i$th JRC node with probability
$a^{*}_{ij}$ and charge $p_{i}/\sum_{j\in\mathcal{M}}a^{*}_{ij}$ to the $i$th
JRC node;

25 end if

26

27 end

28

Algorithm 2 RMCA.b

Since the channels are indivisible items, we are restricted to binary
allocations of the channels to the JRC nodes, i.e., each channel is allocated
to only one JRC node at a time. Therefore, step 2 in Algorithm 2 consists of
allocating channels randomly based on the allocation vector $\mathbf{a^{*}}$.
Moreover, the condition $\mathbf{v}\notin\mathcal{U}$ is necessary as if the
realized bid vector $\mathbf{v}$ does not belong to the uncertainty set
$\mathcal{U}$. As such, the solution to the auction mechanism will be
suboptimal as there might be negative utilities, violating the IR property.

Note here that both the JRC nodes and the SSP deal with the uncertainty but in
different phases of the algorithm (RMCA). Specifically, the first phase of the
algorithm is executed offline, i.e., before the JRC nodes start submitting
their realized bids (RMCA.a). At this point, and as mentioned is Section
4.1.1, each JRC node calculates the DEP interval using PSO algorithm. The SSP
is considered to have collected these valuation before the beginning of the
auction. The SSP uses the constructed uncertainty set during the nominal
allocation phase (RMCA.a) to derive an optimal allocation strategy that
reflects its risk-aversion about the warden’s location. Finally, when the JRC
nodes want to submit their realized bids, they draw the DEP value from a
uniform distribution in the interval between the minimum and maximum values of
the DEP.

###### Theorem 4

The proposed RMCA has the properties of individual rationality, incentive
compatibility and budget feasibility, all in expectation.

###### Proof:

Since the channel cost $c_{j}$ in the objective function of problem (20) is
constant, we can consider $v_{ij}-c_{j}$ as one variable. Therefore, with this
change of variable, the proof follows from the one derived in [40]. ∎

###  4.3 Discussion on The IC Property

The most important property that should be satisfied by auctions is IC. The
proof of the IC property is omitted here to avoid overloading the paper. Here,
we give the intuition behind the proof.

First, we should note that IC is not impacted directly by the uncertainty of
bids. The IC is the property that guarantees that the bidders have no
incentive to misreport their bids. This is guaranteed if we can prove that any
other submitted bids will not bring additional benefit for the bidder than its
true valuation, as formulated in (15). As illustrated in Algorithm 1, the
objective of the first phase of RMCA is to derive the nominal allocation, the
worst-case valuation vector and the reservation prices. This Algorithm 1 is
executed before the beginning of the auction, i.e., before the JRC nodes
submit their bids, and use only the constructed uncertainty set. The
construction of the uncertainty set, as detailed in Section 4.1, is done by
the SSP and hence, the uncertainty set cannot be forged. The second phase of
RMCA, described in Algorithm 2, is executed after the JRC nodes submit their
bids, which might be untruthful. However, a malicious JRC node is aware that
in the first phase of RMCA, the reservation prices have been calculated and
submitting a bid lower than its associated reservation price will prevent the
JRC node from being in the winner list. Even though this rational does not
totally prevent the malicious JRC node from misreporting its valuation, it can
still help preventing the SSP from getting a negative utility, as discussed in
Section 4.2. The important instruction that prevents a malicious JRC node from
misreporting its valuation/bid, is in step-2 of Algorithm 2. Specifically, if
the submitted bid is outside the uncertainty set, which is used to derive the
optimal auction solution, the auction process will be reset again and no
channel is allocated to any JRC node. In other words, a malicious JRC node
cannot misreport its bid because the IC property defined in (15) is already
integrated as a constraint in the optimization problem 20, which shows the
capability of robust optimization.

An important point to discuss also is that the IC holds only in expectation,
as shown in Theorem 4. The output of the auction model loses the total
optimality when we move from divisible items to indivisible items, which is
executed in phase 2, i.e., Algorithm 2, step-2. This is because in the case of
indivisible items we are restricted to looking for integral allocations of the
items (the channels) to the buyers (the JRC nodes). Instead of allocating the
items proportionally according to the allocation vector $a^{*}$ (which is
optimal), the channels are allocated to users randomly where the allocation
vector $a^{*}$ is regarded as a probability vector. To the best of our
knowledge, and based on a recent report [45], there is no proven auction
design for multi-item multi-buyer scenario that guarantees total IC. In
addition, the adopted auction design in this paper is the only work that can
guarantee IC for divisible items due to the inherited properties of robust
optimization [40]. When we consider uncertainty in the auction design, the IC
will hold in expectation for indivisible items (e.g., channels), which means
that in some cases it might not hold, but the solution is feasible and solves
the problem. Moreover, even if a malicious JRC node knows that the IC might
not hold in some cases, it is hard to know exactly under which circumstances.
This makes the proposed auction design significantly useful.

###  4.4 Deterministic Mechanism for Channel Allocation

To evaluate the performance of RMCA, we propose a deterministic mechanism for
channel allocation based on RMCA. The deterministic RMCA can be regarded
simply as an instance of the original RMCA in which the uncertainty set is
considered to contain only the realized bid vector. In other words, by running
the deterministic RMCA, the SSP directly derives the solution to the multi-
item multi-buyer auction problem without consideration of any perturbation in
the submitted bids. In this settings, the warden is considered to be at a
fixed distance from the JRC node and the jammer, and then the DEP is
calculated by the JRC node as in (12) based on that location333Later in the
experiments section, we show the impact of this assumption caused by not
considering the uncertainty in the system.. We first reformulate the inner
optimization problem (20) by omitting the uncertainty of the valuations, which
results in the following linear problem:

$\displaystyle\begin{split}\max\limits_{\mathbf{x}}\sum\limits_{i\in\mathcal{N}}\sum\limits_{j\in\mathcal{M}}(v_{ij}-c_{j})x_{ij}\end{split}$
(25a) $\displaystyle\begin{split}\hskip
14.22636pts.t.\sum\limits_{i\in\mathcal{N}}x_{ij}\leq 1,\quad\forall
j\in\mathcal{M},\end{split}$ (25b) $\displaystyle\begin{split}\hskip
28.45274pt\sum\limits_{j\in\mathcal{M}}v_{ij}x_{ij}\leq B_{i},\quad\forall
i\in\mathcal{N},\end{split}$ (25c) $\displaystyle\begin{split}\hskip
28.45274pt\mathbf{x}\geq 0.\end{split}$ (25d)

The dual of problem (25) is then calculated as follows:

$\begin{array}[]{c}\vspace{0.3cm}\min\limits_{\boldsymbol{\omega},\boldsymbol{\phi}}\sum\limits_{j\in\mathcal{M}}\omega_{j}+\sum\limits_{i\in\mathcal{N}}\phi_{i}B_{i}\\\
\vspace{0.3cm}\hskip 8.5359pts.t.\quad\omega_{j}+v_{ij}\phi_{i}+c_{j}\geq
v_{ij},\quad\forall i\in\mathcal{N},\forall j\in\mathcal{M}\\\
\vspace{0.3cm}\hskip 28.45274pt\phi_{i}\geq 0,\quad\forall i\in\mathcal{N},\\\
\vspace{0.3cm}\hskip 28.45274pt\omega_{j}\geq 0,\quad\forall
j\in\mathcal{M},\end{array}$ (26)

where $\omega_{j}$ and $\phi_{i}$ are elements of $\boldsymbol{\omega}$ and
$\boldsymbol{\phi}$, respectively, and are the duals corresponding to the
first and second constraints in (25).

To derive the prices, we need to solve the following problem which is a
reduced version of problem (25) where we remove a JRC node $k$ from the set of
bidders and calculate the social welfare:

$\displaystyle\begin{split}\max\limits_{x^{-k}}\sum\limits_{i\in\mathcal{N}\backslash\\{k\\}}\sum\limits_{j\in\mathcal{M}}(v_{ij}-c_{j})x^{-k}_{ij}\end{split}$
(27a) $\displaystyle\begin{split}\hskip
14.22636pts.t.\sum\limits_{i\in\mathcal{N}\backslash\\{k\\}}x^{-k}_{ij}\leq
1,\quad\forall j\in\mathcal{M},\end{split}$ (27b)
$\displaystyle\begin{split}\hskip
28.45274pt\sum\limits_{j\in\mathcal{M}}v_{ij}x^{-k}_{ij}\leq
B_{i},\quad\forall i\in\mathcal{N}\backslash\\{k\\},\end{split}$ (27c)
$\displaystyle\begin{split}\hskip 28.45274pt\mathbf{x}^{-k}\geq 0.\end{split}$
(27d)

The proposed mechanism is presented in Algorithm 3.

Input : Realized bid vector
$\mathbf{v}=\\{v_{ij}\\}_{i\in\mathcal{N},j\in\mathcal{M}}$, and budgets
$B_{1},\dots,B_{N}$.

Output : Allocation vector
$\\{a_{ij}^{*}\\}_{i\in\mathcal{N},j\in\mathcal{M}}$, and payment
$\\{p_{ij}^{*}\\}_{i\in\mathcal{N},j\in\mathcal{M}}$

1

2 begin

3 $x^{*}\leftarrow$ Solve problem (25);

4 $(\omega^{*},\phi^{*})\leftarrow$ Solve problem (26);

5 foreach  _$k\in\mathcal{N}$_ do

6 $x^{*_{-k}}\leftarrow$ Solve problem (27);

7

8 end foreach

9 // Calculate reservation prices

10 foreach  _$i\in\mathcal{N}$_ do

11 foreach  _$j\in\mathcal{M}$_ do

12 $r^{*}_{ij}=\omega^{*}_{j}+v_{ij}\phi^{*}_{i}+c_{j}$;

13 end foreach

14

15 end foreach

16

17 // Calculate the payment vector

18 foreach  _$k\in\mathcal{N}$_ do

19
$p_{k}=\sum\limits_{j\in\mathcal{M}}x^{*}_{kj}r^{*}_{kj}+\sum\limits_{i\in\mathcal{N}\backslash\\{k\\}}\sum\limits_{j\in\mathcal{M}}(v_{ij}-r^{*}_{ij})x^{*_{-k}}_{ij},\quad\forall
k\in\mathcal{N}$;

20 end foreach

21 Allocate the $j$th channel to the $i$th JRC node with probability
$x^{*}_{ij}$ and charge $p_{i}/\sum_{j\in\mathcal{M}}x^{*}_{ij}$ to the $i$th
JRC node;

22 end

23

Algorithm 3 Deterministic Mechanism for Channel Allocation.

##  5 Numerical Results

In this section, we evaluate the proposed auction mechanisms for channel
allocation in covert JRC systems. Specifically, we are interested in analyzing
the impact of uncertainty about the warden’s location on the obtained social
welfare for both the robust and the deterministic auction mechanisms. Again,
we use the latter as a benchmark scheme to evaluate the effectiveness of the
former. We also aim to investigate the impact of the number of channels and
JRC nodes on social welfare and computation time. Our solution is implemented
using _Gurobi optimizer_ and the python library RSOME [46] for robust
optimization. Experiments are run on a computer with Intel(R) Xeon(R) CPU at
2.20GHz using 13 GB of RAM and operating on Ubuntu 18.04 system.

We consider a square area of 200 m $\times$ 200 m where a set of JRC nodes,
friendly jammers and wardens are located randomly under the coverage of the
SSP. Channel costs for the SSP are sampled from a normal distribution with
mean $2\$$ and variance $1\$$. The budgets for JRC nodes are chosen uniformly
from the interval $[1.5\$,5\$]$. As alluded before, we consider that every JRC
node has one friendly jammer and one dedicated warden. Table I lists the other
simulation parameters.

TABLE I: Simulation parameters Parameter |  Value
---|---
Frequency |  $5.9\>GHz$
Bandwidth |  $50\>MHz$
$p_{i}^{max},p_{g}^{max}$ |  10 dBm
Number of sub-carriers |  10
Time-Bandwidth Product |  100
Radar Duty Factor |  0.01

###  5.1 Impact of the jamming power on the covert rate

Figure 2: Impact of the jamming power on the channel capacity, mutual
information and detection error probability.

To validate our proposed valuation metrics for the covert JRC system, we first
conduct Monte Carlo simulations in which we consider a receiver, a JRC node, a
friendly jammer and a warden that are located at
${\mathbf{q}}_{b}=\left[7,10,19\right]$,
${\mathbf{q}}_{i}=\left[3,8,0\right]$, ${\mathbf{q}}_{j}=\left[6,21,0\right]$
and ${\mathbf{q}}_{w}=\left[3,14,4\right]$, respectively. Figure 2 depicts the
impact of the jamming power on the CC, MI and DEP. As we increase the jamming
power, the DEP at the warden starts increasing only after the jamming power is
greater than $20$ ${\rm dBW}$. However, we observe that the increase of
jamming power causes a decrease of CC and MI. Compared to the case without any
jamming signals, to achieve a $97\%$ DEP at the warden, the CC and MI decrease
by $50\%$ and $53\%$, respectively. This result suggests that there is a
trade-off between the performance and covertness of the JRC system. Finally,
to ensure the covertness of the JRC system, the jamming power must be larger
than a certain threshold, i.e., $27$ ${\rm dBW}$ as shown in Figure 2. Again,
the transmit powers of a JRC node and a friendly jammer can be optimized
accordingly, e.g., by using the method provided in [15].

###  5.2 Uncertainty About Warden’s Location

We consider, as an example, the influence of uncertainty about the warden’s
location on the performance of the system, which is illustrated in Figure 3.
Specifically, during the valuation of the spectrum, the JRC node calculates
the DEP at the warden based on its belief about the warden’s location.
However, the JRC node’s belief about the warden’s location is not accurate and
therefore the derived valuations of the JRC nodes might be higher or lower
than the real valuations444Real valuations refer to the derived valuations if
the location of the warden is precisely known.. This implies that the JRC
nodes or the SSP might experience negative utilities, violating the IR
property of the optimal auction solution.

Figure 3: An illustration of uncertainty about warden’s location.

To analyze the impact of the uncertainty interval on the social welfare of the
system, in this experiment, we set the number of channels to $\mathcal{M}=10$,
the number of JRC nodes to $\mathcal{N}=20$, and vary the uncertainty interval
about the warden’s location, represented in Figure 3 by the side of a 2D
square surrounding the warden. We observe from Figure 4 that the social
welfare obtained by the deterministic auction algorithm is not affected by the
variations in the uncertainty set, while the social welfare obtained using
RMCA decreases as the uncertainty interval,i.e., box width, increases and is
lower than that of the deterministic auction algorithm [23]. This is explained
by the fact that RMCA maximizes the social welfare for the worst-case
uncertainty set while the solution derived by the deterministic auction
mechanism does not depend on the uncertainty set and uses the realized bids
only.

Even though the social welfare obtained by the deterministic auction algorithm
is higher than that obtained by RMCA, it comes with a high risk of not being
able to be achieved in reality. For instance, if the DEP at warden calculated
by the JRC nodes is higher than that if it is in reality, the JRC nodes will
have lower utility than expected and can violate the IR property of optimal
auctions, i.e., a negative utility. However, the RMCA algorithm is more robust
for variations of the DEP at warden which guarantees the feasibility and
optimality of the derived solution. The gap between the social welfare
obtained by RMCA and the deterministic auction is the price of robustness,
i.e., the higher the conservation level about the warden’s location, the
higher the performance gap between RMCA and the deterministic auction.

Figure 4: Impact of the uncertainty interval on social welfare.

To further illustrate the robustness of RMCA against the deterministic auction
for IR violation, we change the location of the warden to a position where the
DEP is lower than expected, i.e., closer to the jammer, then calculate the
utility of one of the JRC nodes for both algorithms using (13). First, we need
to distinguish between the _expected utility_ and the _true utility_. The
expected utility is the utility of a JRC node based on its belief about the
warden’s location, while the true utility is based on the true location of the
warden. In Figure 5(a), we consider that the true location of the warden is
outside the uncertainty range defined by the SSP (2 $meters$ in our settings),
and in Figure 5(b) we consider that the warden’s location is within the
uncertainty range. From Figure 5, we observe that the expected utility of the
deterministic auction is higher than that of RMCA. However, the true utility
of the deterministic auction is negative, violating IR. For RMCA, the utility
is not negative. In fact, it is negative for RMCA only if the new location of
the warden is outside the range from which the uncertainty set is derived.
Interestingly, as observed from Figure 5(b), when the true location of the
warden is within the uncertainty range of the SSP, the true utility derived by
RMCA is much higher than the expected utility. This is explained by the fact
that RMCA maximizes the worst-case social welfare, i.e., the derived optimal
solution is based on the location of the warden that has the lowest DEP. Note
that the knowledge about the violation of the IR property is not possible in
real-world scenarios as the location of the warden is usually not known.
Therefore, with careful choice of the uncertainty set, the use of RMCA
significantly minimizes the chances of violation of the IR property.

(a)

(b)

Figure 5: Impact of the uncertainty interval on one of the JRC node’s utility
in cases where (a) the true location of the warden is outside the uncertainty
set, and (b) the true location of the warden is within the uncertainty set.

###  5.3 Impact of mutual information, channel capacity and DEP on the winner
list

(a)

(b)

Figure 6: Impact of bids of a JRC node on the winner list in terms of (a) the
allocation probability and (b) social welfare.

To understand the impact of CC, MI, and DEP at the warden on the winner list,
we consider the following scenario. We set the number of channels to
$\mathcal{M}=3$ and the number of JRC nodes to $\mathcal{N}=5$. We then allow
one JRC node (ID=5) to change its location so as its valuation for the
channels increases based on (12). Figure 6(a) shows the derived allocation
probabilities using RMCA. We observe that the JRC node 5 is allocated to
channel 3 with probability one, and zero for the other channels. However,
after its average bids, i.e., the average of the submitted bids by one JRC
node to all the channels, increases from 4.47 to 4.57, the allocation
probability for JRC node 5 to channel 3 shifts from 0 to 0.45 and then
decreases again as the bids increase. To understand this strange behavior, we
show in Table II, the submitted bids for all the channels by all the JRC nodes
before and after the updated bid values. We observe that channel 3 is
allocated to JRC node 5 with probability 1 because it has a significantly
higher bid value than the other JRC nodes for this channel. However, once the
bids from the JRC node increase for channel 1 and become the highest, the
auction mechanism allocates channel 1 to JRC node 5 with a probability of
$45\%$. If we keep increasing the submitted bids of JRC node 5, the allocation
probability to channel 1 decreases as shown in Figure 6(a) which is due to the
budget constraint, i.e., constraint (17b). Specifically, since the JRC node 5
is paying more for channel 3, its ability to pay for channel 1 decreases, and
hence, the auction mechanism decreases its probability to obtain channel 1.

TABLE II: Submitted bids |  Channel 1 |  Channel 2 |  Channel 3
---|---|---|---
JRC node 1 |  4.17 |  3.11 |  3.69
JRC node 2 |  4.77 |  2.56 |  3.09
JRC node 3 |  4.42 |  4.20 |  3.12
JRC node 4 |  4.23 |  4.33 |  3.26
JRC node 5 |  4.75 |  4.07 |  4.58
JRC node 5 (varied) |  4.85 |  4.17 |  4.68

We also observe from Figure 6(b) that as the bids from JRC node 5 increase,
the social welfare for both algorithms increases. However, the social welfare
obtained by RMCA is lower than that obtained by the deterministic one, which
is similar to the results shown in Figure 4, i.e., the price of robustness
[20]. Interestingly, our simulations reveal that for the interval-based
uncertainty set, there is no difference between RMCA and the deterministic
auction algorithm in terms of the allocation probabilities. This is explained
by the fact that for our interval-based uncertainty set, RMCA can be regarded
as a deterministic auction where the locations of the wardens are fixed at the
point with the lowest DEP in the uncertainty box. Therefore, only social
welfare is impacted by the changes in bids but the allocation probabilities
are the same for both algorithms.

###  5.4 Computation time for different number of JRC nodes and channels

Figure 7 shows the computation time for both of the algorithms while varying
the number of channels and JRC nodes. We observe that the deterministic
auction has almost a constant computation time for different combinations of
the number of channels and JRC nodes. However, the RMCA has a higher
computation time for the same combinations and increases polynomial with
increases in the number of channels and JRC nodes. This is explained by the
fact that the RMCA solves a bilinear optimization problem, i.e., problem (20),
which is NP-hard for general uncertainty sets $\mathcal{U}$. However, the
computation time is still tractable which is due to the use of the Generalized
Benders Decomposition algorithm that assures a polynomial computation time if
the uncertainty set has a polynomial number of extreme points which is the
case in our interval-based uncertainty set $\mathcal{U}$. Note that since the
first phase of RMCA, i.e., RMCA.a, is executed before bid submission, the
computation time can be further reduced if the set of participating JRC nodes
is the same as in the previous round of the auction.

Figure 7: Computation time for different number of JRC nodes and channels.

###  5.5 Discussions

From the results obtained from Figure 6, we observe that even though a JRC
node increases its bid, its chances of getting a channel decrease because of
the budget constraint. In this case, the SSP obtains much higher revenue but
the JRC node’s utility decreases with no additional benefit. In other words,
the JRC node can be allocated to the same number of channels if it bids
untruthfully, violating the IC property. This case occurs because the
developed mechanism, as earlier shown, is only IC in expectation and not
dominant strategy incentive compatible (DSIC). However, obtaining the minimum
bid value by a JRC node is practically difficult because the mechanism is
probabilistic and the JRC node’s objective is to maximize its chances of
getting the channel. Moreover, JRC nodes are not aware of other bids.
Therefore, the JRC nodes are incentivized to bid truthfully.

Another main observation from our results is the distinction between ex-ante
IR and ex-poste IR. Ex-ante IR refers to the case where the JRC node
anticipates that it has a non-negative expected utility before the winner list
and prices are determined, while ex-post IR refers to the case where the JRC
node is guaranteed to have a non-negative utility after the winner list and
prices are determined. The ex-post IR property is certainly desired in our
system. This is because it represents the true utility that the JRC nodes get.
Based on (13), getting a negative utility implies that the JRC node is paying
more that it gets in benefits, which makes the JRC node reluctant to
participate in such auctions in the future. As earlier shown in Figure 5, the
deterministic RMCA has ex-ante IR but lacks ex-post IR. However, RMCA can
guarantee ex-post IR if the uncertainty set is well defined, i.e., large
enough to include all possible realization of the bids. The deterministic RMCA
cannot have this guarantee even with large uncertainty sets.

Finally, the developed deterministic auction mechanism opens an interesting
research area to explore. Specifically, the use of robust optimization tools
has enabled us to derive optimal solutions that have the properties of IC, IR
and BF. The power of robust optimization is that these properties are smoothly
incorporated in the optimization problem, making the solution to the auction
problem significantly easier than existing complicated auction designs. This
suggests that we can use robust optimization for other auction problems where
we need to guarantee the properties of IC, IR and BF and then, we might omit
the discussion about uncertainty by simply considering the uncertainty set to
contain only one item (as we have done in our deterministic auction).
Certainly, these suggestions needs further investigations and validation as
there might appear other challenges.

##  6 Conclusion

In this paper, a covert JRC system that can operate safely in the existence of
a watchful adversary has been developed. The reliability of the channel
allocation problem by the SSP to the JRC nodes was addressed, where we
proposed a robust auction mechanism to maximize the social welfare of the
system. The proposed auction mechanism was shown to be robust against
perturbations in the submitted bids. We implemented a deterministic auction
mechanism to show the benefits of robustness. Simulation results showed that
the robust auction mechanism yields better performance compared to the
deterministic auction mechanism in terms of satisfaction of the optimal
auction solution when there is uncertainty about the submitted bids. For
future works, we would like to investigate the use of deep learning in auction
design and study what type of robustness it can provide.

##  7 Acknowledgement

The authors would like to thank Prof. Chaithanya Bandi for the valuable
suggestions that helped us significantly improve the paper.

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