# Wireless Covert Communications
Aided by Distributed Cooperative Jamming
over Slow Fading Channels

Tong-Xing Zheng, , Ziteng Yang, Chao Wang, , Zan Li, , Jinhong Yuan, , and
Xiaohong Guan T.-X. Zheng and Z. Yang are with the School of Information and
Communications Engineering, Xi’an Jiaotong University, Xi’an 710049, China.
T.-X. Zheng is also with the National Mobile Communications Research
Laboratory, Southeast University, Nanjing 210096, China. T.-X. Zheng and X.
Guan are also with the Ministry of Education Key Laboratory for Intelligent
Networks and Network Security, Xi’an Jiaotong University, Xi’an 710049, China.
(e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS>xhguan@mail.xjtu.edu.cn).C. Wang and Z. Li are with the State Key Laboratory
of Integrated Services Networks, Xidian University, Xi’an 710071, China. C.
Wang is also with the National Mobile Communications Research Laboratory,
Southeast University, Nanjing 210096, China. (e-mail<EMAIL_ADDRESS>zanli@xidian.edu.cn).J. Yuan is with the School of Electrical Engineering and
Telecommunications, University of New South Wales, Sydney, NSW 2052, Australia
(e-mail: j.yuan@unsw.edu.au).

###### Abstract

In this paper, we study covert communications between a pair of legitimate
transmitter-receiver against a watchful warden over slow fading channels.
There coexist multiple friendly helper nodes who are willing to protect the
covert communication from being detected by the warden. We propose an
uncoordinated jammer selection scheme where those helpers whose instantaneous
channel gains to the legitimate receiver fall below a pre-established
selection threshold will be chosen as jammers radiating jamming signals to
defeat the warden. By doing so, the detection accuracy of the warden is
expected to be severely degraded while the desired covert communication is
rarely affected. We then jointly design the optimal selection threshold and
message transmission rate for maximizing covert throughput under the premise
that the detection error of the warden exceeds a certain level. Numerical
results are presented to validate our theoretical analyses. It is shown that
the multi-jammer assisted covert communication outperforms the conventional
single-jammer method in terms of covert throughput, and the maximal covert
throughput improves significantly as the total number of helpers increases,
which demonstrates the validity and superiority of our proposed scheme.

###### Index Terms:

Covert communications, cooperative jamming, jammer selection, outage
probability, covert throughput.

##  I Introduction

With the advent of the 5G information age, the dependence of human being on
wireless networks has grown rapidly. An unprecedentedly large amount of
confidential and sensitive information, e.g., ID and credit card details, etc,
is being delivered through wireless media. Security and privacy have become
ever-increasingly crucial for wireless networks since wireless transmission is
particularly prone to security attacks such as eavesdropping and surveillance,
etc, due to the broadcast nature of electromagnetic propagation. In tradition,
the information security issue is addressed at the upper layers of a
communication system leveraging the key-based cryptographic encryption [1],
[2]. However, even the encryption algorithms recognized complicated enough
currently are still at risk of being broken by supercomputers and massively
parallel computing. On the other hand, information-theoretic security has
appeared as an appealing physical-layer alternative to complement conventional
upper-layer security protocols, which aims to achieve wireless secrecy by
adequately exploiting the intrinsic characteristics of wireless channels, thus
circumventing the employment of secrecy keys as well as sophisticated
encryption algorithms [3, 4, 5, 6]. It should be pointed out that both upper-
and physical-layer security mechanisms have mainly been designed for
protecting confidential content from being intercepted by unauthorized third
parties, whereas few has been concentrated on shielding the communication
itself to escape monitoring from watchful adversaries.

In fact, to guarantee the concealment of message delivery is urgently required
for the purpose of a high level of security for various wireless applications,
e.g., secret military operations, authoritarian government monitoring, and
internal interaction of the Internet of Things (IoTs) [7], etc. The exposure
of communication process will cause suspicion at the monitor and hence attract
further signal demodulation and information decipher. In order to overcome the
above problem and to provide a more direct and powerful safeguard on security
and privacy for wireless communications systems, covert communications has
emerged as a new paradigm for wireless security and it has attracted
significant research interest recently [8].

###  I-A Previous Works

Covert communications aims to hide the very existence of communication
behaviors from a vigilant warden, i.e., guaranteeing a very low probability of
being detected by the warden, while attaining a certain level of decoding
performance for the target receiver. Although spread-spectrum techniques have
been widely used to obscure military communications against adversaries in the
early 20 century [9], many fundamental issues have not been well tackled,
e.g., it is still challenging to answer for spread-spectrum techniques that
_how much information can be successfully delivered subject to a certain level
of covertness requirement?_ It is until recently that the fundamental limits
of covert communications have been characterized from an information-theoretic
viewpoint. In [10], a square root law was first established for covert
communications in additive Gaussian white noise (AWGN) channels, which states
that at most $\mathcal{O}(\sqrt{n})$ information bits can be delivered
reliably and covertly simultaneously to the intended receiver in $n$ channel
uses under the surveillance of a warden. This pioneering work has laid an
information-theoretic foundation for the research of covert communications and
has opened up two research directions: 1) explore fundamental limits of covert
communications based on an information-theoretic framework and devise
practical code structures to realize covert communications; 2) excavate the
feasibility of covert communications for realistic circumstances and exploit
advanced techniques to improve covert communications.

Along the first research direction mentioned above, researchers have
generalized the work of [10] into various wireless channel models, such as
binary symmetrical channels (BSC) [11], discrete memoryless channels (DMC)
[12, 13], multiple access channels [14], parallel AWGN channels [15], and
multi-input multi-output (MIMO) AWGN channels [16], etc. These works have been
focused on capturing the scaling law of the amount of covertly delivered
information on $\sqrt{n}$ and examining the feasibility of the square root law
proposed in [10]. It is not difficult to find that, according to the square
root law, the achievable covert rate would approach zero as
$n\rightarrow\infty$, which is definitely undesired for practical
applications. In fact, such a square root law is overly pessimistic as the
intrinsic uncertainty of wireless channels and noise has not been incorporated
therein, which then has stimulated the second research direction as will be
detailed below.

Recently, extensive efforts have been devoted to exploring the opportunity and
condition for attaining a positive covert rate. Specifically, the authors in
[17] and [18] proved that a positive covert rate can be achieved when the
background noise power is unknown to the warden. The authors in [19] analyzed
the maximal achievable covert rate under both bounded and unbounded noise
uncertainty models. The authors in [20, 21, 22, 23] evaluated the influence of
imperfect channel state information (CSI) and finite blocklength on covert
communications, respectively. In particular, the authors in [23] proved that
increasing the blocklength is always rewarding for covert throughput for a
delay-sensitive system, and adopting a random transmit power can provide
additional performance gain. The authors in [24] exploited ambient
interference signals to improve the rate of covert communications. The authors
in [25] further explored the benefits of both centralized and distributed
multi-antenna transmitters on covert communications against randomly located
wardens. The authors in a recent work [26] considered a multi-antenna warden
scenario, showing that even a slight increase in the antenna number at the
warden side can result in a dramatic deterioration in covert throughput.

Node cooperation has also been leveraged to enable and enhance covert
communications recently. For instance, the authors in [27] investigated the
dual-hop covert communication under fading channels. It was shown that with
the aid of a relay and the exploitation of channel uncertainty,
$\mathcal{O}(n)$ information bits can be conveyed reliably and covertly in $n$
channel uses, which broke through the square root law proposed in [10]. The
authors in [28] examined multi-hop routing in covert wireless networks under
AWGN channels. The authors in [29] further studied secrecy and covert
communications via a multi-hop networks under the surveillance of an unmanned
aerial vehicle (UAV). It is worth mentioning that, instead of assisting the
covert communication, relay nodes might also privately send messages to the
destination node by greedily utilizing the resources provided by the source
node. In this case, the source node becomes a warden, whose mission is to
monitor in real-time the covert communication from the untrusted relay. The
interested readers are referred to [30, 31] for more details in this
direction.

###  I-B Motivation and Our Contributions

The aforementioned endeavors on covert communications have mainly focused on
taking advantage of already-existed favorable environmental uncertainty, such
as noise, channel uncertainty, and auxiliary relays, etc. Different from
these, recent studies have revealed that intentionally introducing
interference, e.g., via a friendly jamming node using random transmit power
[32] or via a full-duplex receiver [33, 34, 35] with either fixed or random
jamming power, could also be beneficial for covert communications, since
additional uncertainty and thus more confusion can be created to the warden
when the warden is analyzing the statistics of perceived energy. Inspired by
this idea, in this paper we, for the first time, investigate covert
communications assisted by multiple friendly jammers over slow fading
channels. We aim to excavate more potential of cooperative jamming in terms of
improving covert communications compared with that of a single helper. We
propose to intelligently select jammers based on their individual channel
status in order to balance covertness and throughput performance. To the best
of our knowledge, this idea has not yet been reported by the literature on
covert communications. The main contributions of our work are summarized as
follows.

  * •

We put forward an uncoordinated opportunistic jammer selection scheme to
facilitate the covert communication against a warden over slow fading
channels. To be specific, in order to degrade the detection performance of the
warden while not leaving severe interference to the desired receiver, each
cooperative node is prudently activated as jammer only when its instantaneous
channel gain with respect to the destination receiver becomes smaller than a
certain threshold. By doing so, even the jammers use fixed transmit power, the
warden is still uncertain of the aggregate interference power due to the
random number of actually activated jammers. In addition, the covertness and
reliability performance can be well balanced by properly adjusting the
selection threshold.

  * •

Under the proposed jammer selection scheme, we first examine the worst-case
scenario for covert communications where the warden is always capable of
employing the optimal detection parameter to minimize detection errors. In
this scenario, we derive tractable expressions for both the detection error
probability at the warden and the transmission outage probability at the
desired receiver. Subject to a covertness constraint in which the average
detection error probability is restricted to exceed a certain value, we then
jointly design the optimal selection threshold and transmission rate to
maximize the covert throughput.

  * •

We present adequate numerical results to validate our theoretical findings. By
comparing with the single-jammer aided covert communications, we demonstrate
the superiority of our proposed multi-jammer scheme in terms of covert
throughput enhancement. We also provide various useful insights into the
multi-jammer aided covert communication. In particular, we show that, as the
number of cooperative nodes increases, the covert throughput first
significantly improves but then reaches saturation. It is interesting to find
that, in order to meet a certain level of covertness requirement, there exists
a minimum number of cooperative nodes that should be deployed, and the minimum
number increases as the jamming power becomes smaller.

###  I-C Organization and Notations

The remainder of this paper is organized as follows. Section II details the
system model and relevant hypotheses. Section III analyzes the detection error
probability at the warden considering a worst-case scenario for covert
communications. In Section IV, we first derive analytical expressions for the
average detection error probability and the transmission outage probability,
based on which we then optimize the covert throughput subject to a covertness
constraint. Numerical results are presented in Section V, followed by some
insights into parameter designs. Section VI concludes the paper.

_Notations_ : Bold lowercase letters denote column vectors. $|\cdot|$,
$(\cdot)^{{\dagger}}$, $\mathbb{P}[\cdot]$, and $\mathbb{E}_{v}[\cdot]$ denote
the absolute value, conjugate, probability, and mathematical expectation taken
over a random variable $v$, respectively. $\mathcal{CN}(u,\sigma^{2})$ denotes
the circularly symmetric complex Gaussian distribution with mean $u$ and
variance $\sigma^{2}$. $\binom{N}{m}\triangleq\frac{N!}{m!(N-m)!}$ denotes the
combinatorial number. $\Gamma(a)\triangleq\int_{0}^{\infty}e^{-t}t^{a-1}dt$,
$\Gamma\left(a,x\right)\triangleq\int_{x}^{\infty}{e^{-t}t^{a-1}}dt$, and
$\gamma\left(a,x\right)\triangleq\int_{0}^{x}{e^{-t}t^{a-1}}dt$ denote the
gamma function, the upper incomplete gamma function, and the lower incomplete
gamma function, respectively.

##  II System Model

###  II-A Communication Scenario and Assumptions

Figure 1: Illustration of covert communications in a multi-node cooperative
network, where multiple Cooperative nodes help to achieve covert
communications between Transmitter and Receiver in the presence of Warden.

We consider a cooperative covert communication network, where transmitter T
aims to covertly deliver messages to receiver R under the surveillance of
warden W, as depicted in Fig. 1. There coexist a set of $N$ friendly nodes
${\rm J}_{k}$, for $k=1,\cdots,N$, who collaborate to shield the ongoing
covert communication from T to R to make it as invisible as possible at W.
Specifically, a portion of the cooperative nodes will be judiciously selected
acting as jammers for sending interference signals to confuse W, as will be
described in the next subsection. We assume that each node in the network is
equipped with a single antenna.

To model the wireless channel between two nodes, we adopt a combined channel
model incorporating a standard distance-based path loss governed by the
exponent $\alpha$ along with Rayleigh fading. We consider channel reciprocity,
and the channel gains between T and R, T and W, ${\rm J}_{k}$ and R, and ${\rm
J}_{k}$ and W can be expressed as $h_{t,r}d_{t,r}^{-{\alpha}/{2}}$,
$h_{t,w}d_{t,w}^{-{\alpha}/{2}}$, $h_{j_{k},r}d_{j_{k},r}^{-{\alpha}/{2}}$ ,
and $h_{j_{k},w}d_{j_{k},w}^{-{\alpha}/{2}}$, respectively, where $h$ denotes
the independent and identically distributed (i.i.d.) fading coefficient
obeying $\mathcal{CN}(0,1)$, and $d$ denotes the corresponding path distance.
The network topology, including the locations of nodes and the distances
between each other, is considered to be overt to all the nodes in the
network.111This assumption can be tenable for some real-life circumstances.
Typical examples include that W is a government regulatory agency whose
location is public to all, or communication entities are deployed on the top
of conspicuous buildings such that they are visible and the distance can be
easily measured. Moreover, it is a worst case that assuming W knows the
location information of all nodes, which is preferred for a robust design.

We consider a two-phase channel estimation scheme. Specifically, in the first
phase, R broadcasts a public pilot sequence to enable all the $N$ friendly
nodes ${\rm J}_{k}$ to know the instantaneous CSI of their channels to R,
i.e., $h_{j_{k},r}$, which is required for the jammer selection scheme as will
be described in the next subsection;222We assume that Willie only focuses on
monitoring the covert communication from T to R while R can be overt to
Willie, and hence there is no covertness requirement for the pilot delivery of
R.  in the second phase, T covertly sends a secret pilot sequence shared
between itself and R to facilitate R to estimate their channel $h_{t,r}$ for
decoding signals, and meanwhile the selected jammers send different pseudo-
random pilot sequences to conceal T’s pilot sequence against Willie.333As
noted in [10], T and R possess a common secret randomness resource in order to
enable covert communications. In our channel estimation scheme, such a
resource is a secret pseudo-random pilot sequence shared between T and R prior
to communication which is unknown to Willie. As long as the sequence length is
sufficiently large, it will become nearly orthogonal to those pilot sequences
sent by jammers. In this way, R can easily match T’s pilot sequence and
extract the channel $h_{t,r}$, while Willie cannot detect the pilot delivery
of T due to the cover of jamming pilot sequences. We also examine two
situations where W can and cannot acquire the instantaneous CSI of the channel
between itself and T, i.e., $h_{t,w}$, as will be detailed in In Sec.
III-B.444By assuming the availability of the instantaneous value of $h_{t,w}$
at W, we expect to explore the worst performance of covert communications.

###  II-B Jammer Selection Scheme

In order to conceal the covert communication against W, an uncoordinated
jammer selection scheme is employed, where those cooperative nodes with their
individual power gains related to R being smaller than a certain threshold
$\tau$ will be activated to transmit jamming signals independently; whilst the
others remain silent.555Our scheme requires only a low-level collaboration.
This is fundamentally different from those higher-level collaboration schemes
such as distributed zero-forcing jamming, which require to collect global CSI
and optimize the collaborative beamformer weights coordinately, thus resulting
in a high overhead and implementation complexity due to signal sharing and
synchronization. Mathematically, the threshold-based jammer selection
criterion can be described by the following indicator function

$\displaystyle\mathcal{I}_{j_{k}}=\begin{cases}1,&{\left|h_{j_{k},r}\right|^{2}d_{j_{k},r}^{-\alpha}}/{\sigma_{r}^{2}}<\tau,\\\
0,&{\left|h_{j_{k},r}\right|^{2}d_{j_{k},r}^{-\alpha}}/{\sigma_{r}^{2}}\geq\tau,\end{cases}$
(1)

where $\mathcal{I}_{j_{k}}=1$ indicates that $\rm J_{k}$ is selected as jammer
to send jamming signals, and $\mathcal{I}_{j_{k}}=0$ corresponds to a silent
node. Note that the proposed jammer selection criterion takes both small-scale
channel fading and large-scale path loss into consideration. In addition, the
covert communication without jammers assistance or with all jammers being
selected are two special cases of our scheme with $\tau=0$ and
$\tau\rightarrow\infty$, respectively.

We should emphasize that, in order to confuse W, all the cooperative nodes
will continuously choose to send jamming signals or keep silent according to
the jammer selection criterion given in (1), no matter whether T is
transmitting or not. The reason behind such a threshold-based jammer selection
criterion is that there is a trade-off between the covertness and reliability
for covert communications. It is obvious that, setting a large selection
threshold $\tau$ helps to activate a large number of jammers, which is
beneficial for achieving covertness since in this way W will be severely
confused about whether his perceived increased energy comes from T or the
jammers. In this sense, for the purpose of covertness, it is preferred to
choose the value of $\tau$ as large as possible to dramatically degrade the
detection performance of W. As a coin has two sides, an overly high selection
threshold $\tau$ will definitely cause severe interference to the desired
receiver, thus impairing transmission reliability. As will be discussed later
for parameter optimization, only by properly designing the selection
threshold, can we strike a good balance between transmission covertness and
reliability so as to achieve a high level of covert throughput.

Considering quasi-static Rayleigh fading, the cumulative distribution function
(CDF) of $|h_{j_{k},r}|^{2}$ in (1) is denoted as
$F_{|h_{j_{k},r}|^{2}}(x)=1-e^{-x}$, and accordingly the probability of
selecting ${\rm J}_{k}$ as jammer can be given by

$p_{j_{k}}=1-e^{-d_{j_{k},r}^{\alpha}{\sigma_{r}^{2}}\tau}.$ (2)

If T transmits messages in a certain time slot $i$, the signal received at R
can be expressed as follows,

$\bm{y}_{r}\left[i\right]=\sqrt{P_{t}}\frac{h_{t,r}}{d_{t,r}^{\alpha/2}}\bm{x}_{t}\left[i\right]+\sum_{k=1}^{N}{\mathcal{I}_{j_{k}}\sqrt{P_{j}}\frac{h_{j_{k},r}}{d_{j_{k},r}^{\alpha/2}}\bm{v}_{j_{k}}\left[i\right]}+\bm{n}_{r}\left[i\right],$
(3)

where $\bm{x}_{t}[i]$ is the Gaussian signal transmitted by T satisfying
$\mathbb{E}[\bm{x}_{t}[i]\bm{x}_{t}^{{\dagger}}[i]]=1$, $i=1,2,\cdots,n$, with
$n$ being the number of channel uses for delivering a whole codeword, and
$P_{t}$ denotes the transmit power. $\bm{v}_{j_{k}}[i]$ is the Gaussian
jamming signal radiated by ${\rm J}_{k}$ in the current time slot satisfying
$\mathbb{E}\left[\bm{v}_{j_{k}}[i]\bm{v}_{j_{k}}^{{\dagger}}[i]\right]=1$, and
$P_{j}$ denotes the jamming power. $\bm{n}_{r}[i]$ is the AWGN at R with
variance $\sigma_{r}^{2}$, i.e.,
$\bm{n}_{r}[i]\sim\mathcal{CN}(0,\sigma_{r}^{2})$.

It is worth noting that, compared with conventional jammer-assisted covert
communications where the jammer uses random transmit power to confuse W, in
this paper we simply consider the same fixed jamming power for all the helper
nodes. Nevertheless, we can still create uncertainty for W in terms of his
received power with the aid of the proposed opportunistic jammer selection
scheme. Since W is unaware of the instantaneous channel gains between ${\rm
J}_{k}$ and R, even the value of selection threshold $\tau$ is known by W, W
still cannot determine the exact number of jammers, let alone the aggregate
interference power. As a consequence, W is not able to judge whether the
increased received power is due to the start of covert communications or just
the activation of new jammers.

###  II-C Detection Strategy at Warden

In this subsection, we will detail the detection strategy for W. Note that W
attempts to detect whether the covert communication between T and R takes
place or not in a certain time slot. Thus, W faces a binary hypothesis testing
problem, where the null hypothesis $\mathcal{H}_{0}$ states that T is not
transmitting, while the alternative hypothesis $\mathcal{H}_{1}$ states that T
is communicating to R. Under the above two hypotheses, the signal received at
W can be expressed as below

$\displaystyle\bm{y}_{w}[i]=\begin{cases}\sum\limits_{k=1}^{N}{\mathcal{I}_{j_{k}}\sqrt{P_{j}}\frac{h_{j_{k},w}}{d_{j_{k},w}^{\alpha/2}}\bm{v}_{j_{k}}[i]}+\bm{n}_{w}[i],&\mathcal{H}_{0},\\\
\sqrt{P_{t}}\frac{h_{t,w}}{d_{t,w}^{\alpha/2}}\bm{x}_{t}[i]+\sum\limits_{k=1}^{N}{\mathcal{I}_{j_{k}}\sqrt{P_{j}}\frac{h_{j_{k},w}}{d_{j_{k},w}^{\alpha/2}}\bm{v}_{j_{k}}[i]}+\bm{n}_{w}[i],&\mathcal{H}_{1},\end{cases}$
(4)

where $\bm{n}_{w}[i]$ is the AWGN at W with variance $\sigma_{w}^{2}$, i.e.,
$\bm{n}_{w}[i]\sim\mathcal{CN}(0,\sigma_{w}^{2})$.

We follow a commonly used assumption in literature on covert communications
that W adopts a radiometer to detect the covert communication from T to R [19,
24, 25, 26, 20]. This is not only because the radiometer is easy to implement
and is of low complexity in practice, but also because of its sufficiency in
detecting covert communications for the scenario under investigation as
justified in Appendix -A.  Specifically, W employs the average received power
$T_{w}$ with $n$ channel uses as the test statistic, which is given below

$T_{w}=\frac{1}{n}\sum\limits_{i=1}^{n}\left|\bm{y}_{w}[i]\right|^{2}.$ (5)

The ultimate goal of W is to determine whether $\bm{y}_{w}$ is under
$\mathcal{H}_{0}$ or under $\mathcal{H}_{1}$ by analyzing $T_{w}$. According
to the mechanism of radiometer, the decision criterion is characterized as
follows

$T_{w}\mathop{\gtrless}\limits_{\mathcal{D}_{0}}^{\mathcal{D}_{1}}\mu,$ (6)

where $\mu$ is a predetermined detection threshold, and $\mathcal{D}_{0}$ and
$\mathcal{D}_{1}$ indicate that W makes decisions in favor of
$\mathcal{H}_{0}$ and $\mathcal{H}_{1}$, respectively. We also adopt a common
assumption that W is capable of using an infinite number of signal
observations for ease of detection [19, 24, 25, 26, 20], i.e.,
$n\rightarrow\infty$, which leads to

$\displaystyle
T_{w}=\begin{cases}\sigma_{j}^{2}+\sigma_{w}^{2},&\mathcal{H}_{0},\\\
\sigma_{c}^{2}+\sigma_{j}^{2}+\sigma_{w}^{2},&\mathcal{H}_{1}.\end{cases}$ (7)

where
$\sigma_{c}^{2}\\!\\!\triangleq\\!\\!P_{t}{\left|h_{t,w}\right|^{2}}{d_{t,w}^{-\alpha}}$
and
$\sigma_{j}^{2}\\!\\!\triangleq\\!\\!\sum_{k=1}^{N}{{\mathcal{I}_{j_{k}}}P_{j}{\left|h_{j_{k},w}\right|^{2}}{d_{j_{k},w}^{-\alpha}}}$
denote the power of the covert signal and the aggregate jamming signals,
respectively.

In this paper, we employ the detection error rate to measure the detection
performance of W, which consists of the false alarm rate and the miss
detection rate. Specifically, the false alarm rate is defined as the
probability that W makes a decision in favor of $\mathcal{D}_{1}$ while
$\mathcal{H}_{0}$ is true, which is denoted by
$\mathcal{P}_{FA}\triangleq\mathcal{P}(\mathcal{D}_{1}|\mathcal{H}_{0})$; the
miss detection rate is defined as the probability that W makes a decision in
favor of $\mathcal{D}_{0}$ while $\mathcal{H}_{1}$ is true, and we denote it
as $\mathcal{P}_{MD}\triangleq\mathcal{P}(\mathcal{D}_{0}|\mathcal{H}_{1})$.
We assume that the priori probabilities of hypotheses $\mathcal{H}_{0}$ and
$\mathcal{H}_{1}$ are equal. Under this assumption, the detection error
probability of W is defined as

$\xi\triangleq\mathcal{P}_{FA}+\mathcal{P}_{MD}.$ (8)

In practice, it is impossible, or at least difficult, to know the exact value
of the detection threshold $\mu$ set by W. For the purpose of a robust design,
we examine a worst-case scenario for covert communications where W can always
use the optimal detection threshold to achieve a minimum detection error
probability, denoted as ${\xi}^{*}$. However, T cannot acquire the
instantaneous CSI of the channel between itself and W $|h_{t,w}|^{2}$, it is
impossible to calculate the exact detection error probability. Instead, we
employ the average detection error probability ${\bar{\xi}}$ to quantify the
covertness performance of the system, which is averaged over $|h_{t,w}|^{2}$.

###  II-D Optimization Problem

In this paper, we focus on the core metric termed covert throughput, which is
defined as [25]

$\Omega\triangleq R\left(1-\delta\right),$ (9)

where $R$ denotes the transmission rate of messages adopted at T, and $\delta$
denotes the transmission outage probability of the covert communication from T
to R, which will be detailed in Sec. IV-B, quantifying the likelihood that the
reliability of the covert communication with rate $R$ cannot be guaranteed
over fading channels.

The goal of this work is to achieve the maximal covet throughput under the
premise that a certain level of covertness can be promised. Specifically, we
aim at maximizing the covert throughput $\Omega$ subject to a covertness
constraint ${\bar{\xi}}^{*}\geq 1-\varepsilon$ by jointly designing the
optimal transmission rate $R$ and the jammer selection threshold $\tau$. Here,
${\bar{\xi}}^{*}$ denotes the minimum average detection error probability at W
for the worst-case scenario of covert communications, and
$\varepsilon\in[0,1]$ is a prescribed threshold corresponding to the maximal
acceptable probability of being detected by W. Mathematically, the problem of
maximizing the covert throughput can be formulated as follows

$\max\limits_{R,\tau}\Omega=R\left(1-\delta\right),\quad\rm{s.t.}\quad{\bar{\xi}}^{*}\geq\rm{1}-\varepsilon.$
(10)

##  III Analysis of Covertness Performance

In this section, we first derive the expression for the detection error
probability as defined in (8) by calculating the false alarm rate and miss
detection rate, respectively, considering a given value of $h_{t,w}$.
Afterwards, we consider a worst-case covert communication scenario where W can
always adjust the detection threshold $\mu$ to fading channels and network
geometry for improving its detection accuracy, and then design the optimal
detection threshold to minimize the average detection error probability, which
is averaged over $h_{t,w}$.

###  III-A Detection Error Probability

Revisiting the detection strategy at W described in Sec. II-C, once given a
detection threshold $\mu$, we need to derive the false alarm rate
$\mathcal{P}_{FA}$ and miss detection rate $\mathcal{P}_{MD}$, respectively.
One of the major obstacles to the calculation of $\mathcal{P}_{FA}$ and
$\mathcal{P}_{MD}$ lies in the term, i.e., the aggregate jamming power
$\sigma_{j}^{2}$ in (7). First, the indicator $\mathcal{I}_{j_{k}}$ is closely
related to the proposed opportunistic jammer selection scheme (1) such that it
is a random variable strongly coupled with another random herein, i.e., the
channel gain $|h_{j_{k},w}|^{2}$. Moreover, the distribution of the summation
$\sigma_{j}^{2}$ is intractable to derive, since the terms therein are
independent but non-identically distributed (i.n.i.d.) due to different
distances $d_{j_{k},w}$.

In order to overcome the above issues, we note that the combinations of
selected jammers might vary in different time slots, and introduce the
notation $\phi_{m}^{s}$ to represent the $s$-th subset among $\binom{N}{m}$
subsets of different combinations of selected jammers with cardinality $m$
($m=0,1,2,\ldots,N$), and the corresponding probability is denoted as
$\mathcal{P}_{\phi_{m}^{s}}$. For completeness, we denote $\phi_{0}^{1}$ as
null set indicating no jammer is selected. In the following lemma, we
calculate the probability $\mathcal{P}_{\phi_{m}^{s}}$.

###### Lemma 1

The probability that the set of the selected jammers is $\phi_{m}^{s}$ is
given by

$\displaystyle\mathcal{P}_{\phi_{m}^{s}}=\begin{cases}\prod\limits_{k=1}^{N}\left(1-p_{j_{k}}\right),&m=0\\\
\prod\limits_{j_{q}\in\phi_{m}^{s}}p_{j_{q}}\prod\limits_{j_{l}\notin\phi_{m}^{s}}\left(1-p_{j_{l}}\right),&m=1,\cdots,N-1\\\
\prod\limits_{k=1}^{N}p_{j_{k}},&m=N\\\ \end{cases}$ (11)

where $p_{j_{k}}$ denotes the probability that ${\rm J}_{k}$ is chosen as
jammer, which has been derived in (2).

###### Proof 1

The result follows directly by noting the fact that all the jammers are
activated independently.

By introducing $\phi_{m}^{s}$, the aggregate jamming power $\sigma_{j}^{2}$
perceived at W for a certain time slot with $m$ jammers being activated can be
rewritten as
$\sum_{j_{q}\in\phi_{m}^{s}}P_{j}{\left|h_{j_{q},w}\right|^{2}d_{j_{q},w}^{-\alpha}}$.
Further, we define
$X_{m,s}\triangleq\sum_{j_{q}\in\phi_{m}^{s}}{\left|h_{j_{q},w}\right|^{2}d_{j_{q},w}^{-\alpha}}$,
and leverage a common gamma approximation method [36] to derive its
distribution by treating it as a gamma random variable.

###### Lemma 2

By regarding
$X_{m,s}\triangleq\sum_{j_{q}\in\phi_{m}^{s}}{\left|h_{j_{q},w}\right|^{2}d_{j_{q},w}^{-\alpha}}$
as a gamma random variable, its probability density function (PDF) can be
approximated as

$f_{X_{m,s}}\left(x_{m,s};v_{m,s},\omega_{m,s}\right)\approx\frac{x_{m,s}^{v_{m,s}-1}e^{-\frac{x_{m,s}}{\omega_{m,s}}}}{\omega_{m,s}^{v_{m,s}}\Gamma\left(v_{m,s}\right)},$
(12)

where $v_{m,s}$ and $\omega_{m,s}$ are respectively defined as below

$\displaystyle
v_{m,s}\triangleq\frac{\left(\sum_{j_{q}\in\phi_{m}^{s}}d_{j_{q},w}^{-\alpha}\right)^{2}}{\sum_{j_{q}\in\phi_{m}^{s}}d_{j_{q},w}^{-2\alpha}},\quad\omega_{m,s}\triangleq\frac{\sum_{j_{q}\in\phi_{m}^{s}}d_{j_{q},w}^{-2\alpha}}{\sum_{j_{q}\in\phi_{m}^{s}}d_{j_{q},w}^{-\alpha}}.$
(13)

###### Proof 2

The values of $v_{m,s}$ and $\omega_{m,s}$ can be obtained by matching the
first and second moments of $X_{m,s}$, and a detailed proof is relegated to
Appendix -B.

The accuracy of the approximation (12) will be confirmed later by simulation
results, and hence in the following analysis we will replace the exact PDF of
the random variable $X_{m,s}$ with (12) for mathematical tractability.
Resorting to Lemmas 1 and 2, we can calculate the false alarm rate and miss
detection rate in the following theorem.

###### Theorem 1

The false alarm rate $\mathcal{P}_{FA}$ and the miss detection rate
$\mathcal{P}_{MD}$ at W are respectively given by

$\displaystyle\mathcal{P}_{FA}$
$\displaystyle=\begin{cases}1,&\mu\leq\sigma_{w}^{2},\\\
\sum\limits_{m=1}^{N}\sum\limits_{s=1}^{\binom{N}{m}}\mathcal{P}_{\phi_{m}^{s}}\frac{\Gamma\left(v_{m,s},\frac{\mu-\sigma_{w}^{2}}{\omega_{m,s}P_{j}}\right)}{\Gamma\left(v_{m,s}\right)},&\mu>\sigma_{w}^{2},\\\
\end{cases}$ (14) $\displaystyle\mathcal{P}_{MD}$
$\displaystyle=\begin{cases}0,&\mu\leq\rho_{1},\\\
\mathcal{P}_{\phi_{0}^{1}}+\sum\limits_{m=1}^{N}\sum\limits_{s=1}^{\binom{N}{m}}\mathcal{P}_{\phi_{m}^{s}}\frac{\gamma\left(v_{m,s},\frac{\mu-\rho_{1}}{{\omega_{m,s}P}_{j}}\right)}{\Gamma\left(v_{m,s}\right)},&\mu>\rho_{1},\end{cases}$
(15)

where $\mathcal{P}_{\phi_{m}^{s}}$ has been provided by Lemma 1 and
$\rho_{1}\triangleq\sigma_{c}^{2}+\sigma_{w}^{2}$.

###### Proof 3

Recalling the detection strategy described in (6), the false alarm rate can be
calculated from (7) as

$\displaystyle\mathcal{P}_{FA}=\mathbb{P}\left[\sigma_{j}^{2}+\sigma_{w}^{2}>\mu\right]$
$\displaystyle\stackrel{{\scriptstyle\mathrm{(a)}}}{{=}}\begin{cases}1,&\mu\leq\sigma_{w}^{2},\\\
\sum\limits_{m=0}^{N}\sum\limits_{s=1}^{\binom{N}{m}}\mathcal{P}_{\phi_{m}^{s}}\mathbb{P}\left[\sum\limits_{j_{q}\in\phi_{m}^{s}}{\frac{\left|h_{j_{q},w}\right|^{2}}{d_{j_{q},w}^{\alpha}}}>\frac{\mu-\sigma_{w}^{2}}{P_{j}}\right],&\mu>\sigma_{w}^{2}.\end{cases}$
(16)

where (a) follows from treating the combination of the selected jammers as a
random variable and applying the total probability theorem by traversing all
possible combinations.

Similarly, the miss detection rate is computed as

$\displaystyle\mathcal{P}_{MD}=\left[\sigma_{c}^{2}+\sigma_{j}^{2}+\sigma_{w}^{2}<\mu\right]$
$\displaystyle=\begin{cases}0,&\mu\leq\rho_{1},\\\
\sum\limits_{m=0}^{N}\sum\limits_{s=1}^{\binom{N}{m}}\mathcal{P}_{\phi_{m}^{s}}\mathbb{P}\left[\sum\limits_{j_{q}\in\phi_{m}^{s}}\frac{\left|h_{j_{q},w}\right|^{2}}{d_{j_{q},w}^{\alpha}}<\frac{\mu-\rho_{1}}{P_{j}}\right],&\mu>\rho_{1}.\end{cases}$
(17)

The proof can be completed by calculating the inner probabilities in (3) and
(3) resorting to Lemma 2.

Note that $\rho_{1}$ in Theorem 1 is the sum of received signal power and
noise power at W. Theorem 1 shows that the noise power $\sigma_{w}^{2}$ and
the signal-plus-noise power $\rho_{1}$ are two important boundaries for the
detection at W. Specifically, a too small detection threshold, e.g.,
$\mu\leq\sigma_{w}^{2}$, will lead to a complete false alarm; where if W
choose $\mu$ lower than $\rho_{1}$, a miss detection issue can be avoid. For
other situations, the detection accuracy at W heavily depends on the aggregate
jamming power level. In particular, if no jammer is activated when T is
communicating to R, a miss detection event will occur if W set $\mu>\rho_{1}$.
That is why there exists an additional term $\mathcal{P}_{\phi_{0}^{1}}$ for
$\mathcal{P}_{MD}$ in (15) compared with $\mathcal{P}_{FA}$ in (14).

After obtaining the false alarm rate $\mathcal{P}_{FA}$ and miss detection
rate $\mathcal{P}_{MD}$, the detection error probability can be calculated as
(8), which is provided by the following corollary.

###### Corollary 1

The detection error probability $\xi$ at W for a prescribed detection
threshold $\mu$ is given by

$\displaystyle\xi=\begin{cases}1,&\mu\leq\sigma_{w}^{2},\\\
\sum\limits_{m=1}^{N}\sum\limits_{s=1}^{\binom{N}{m}}\mathcal{P}_{\phi_{m}^{s}}\frac{\Gamma\left(v_{m,s},\frac{\mu-\sigma_{w}^{2}}{\omega_{m,s}P_{j}}\right)}{\Gamma\left(v_{m,s}\right)},&\sigma_{w}^{2}<\mu\leq\rho_{1},\\\
\mathcal{P}_{\phi_{0}^{1}}+\sum\limits_{m=1}^{N}\sum\limits_{s=1}^{\binom{N}{m}}\mathcal{P}_{\phi_{m}^{s}}\frac{\Gamma\left(v_{m,s},\frac{\mu-\sigma_{w}^{2}}{\omega_{m,s}P_{j}}\right)}{\Gamma\left(v_{m,s}\right)}+\\\
\quad\quad\sum\limits_{m=1}^{N}\sum\limits_{s=1}^{\binom{N}{m}}\mathcal{P}_{\phi_{m}^{s}}\frac{\gamma\left(v_{m,s},\frac{\mu-\rho_{1}}{{\omega_{m,s}P}_{j}}\right)}{\Gamma\left(v_{m,s}\right)},&\mu>\rho_{1}.\end{cases}$
(18)

Corollary 1 shows that $\xi=1$ when $\mu\leq\sigma_{w}^{2}$, which indicates a
completely incorrect detection. Some observations can be further made on how
the detection performance at W is impacted by the selection threshold $\tau$
of the proposed jammer selection scheme and the jamming power $P_{j}$.

1) As $\tau\rightarrow\infty$, the detection error probability $\xi$ in (18)
can be simplified as

$\displaystyle\lim\limits_{\tau\rightarrow\infty}{\xi}=\begin{cases}1,&\mu\leq\sigma_{w}^{2},\\\
\frac{\Gamma\left(v_{N},\frac{\mu-\sigma_{w}^{2}}{\omega_{N}P_{j}}\right)}{\Gamma\left(v_{N}\right)},&\sigma_{w}^{2}<\mu\leq\rho_{1},\\\
1-\frac{\Gamma\left(v_{N},\frac{\mu-\rho_{1}}{\omega_{N}P_{j}}\right)-\Gamma\left(v_{N},\frac{\mu-\sigma_{w}^{2}}{{\omega_{N}P}_{j}}\right)}{\Gamma\left(v_{N}\right)},&\mu>\rho_{1}.\end{cases}$
(19)

where

$v_{N}=\frac{\left(\sum_{k=1}^{N}d_{j_{k},w}^{-\alpha}\right)^{2}}{\sum_{k=1}^{N}d_{j_{k},w}^{-2\alpha}},\quad\omega_{N}=\frac{\sum_{k=1}^{N}d_{j_{k},w}^{-2\alpha}}{\sum_{k=1}^{N}d_{j_{k},w}^{-\alpha}}.$

We notice that $\tau\rightarrow\infty$ corresponds to the case where all the
cooperative nodes are chosen to send jamming signals to facilitate the covert
communication such that the uncertainty imposed at W by jammers vanishes.
Then, we can see from (19) that $\lim_{\tau\rightarrow\infty}{\xi}$ is solely
determined on the detection threshold $\mu$ at W for a given value of
$h_{t,w}$, but is independent of the selection threshold $\tau$.

2) As $P_{j}\rightarrow\infty$, it is easy to find that
$\frac{\mu-\sigma_{w}^{2}}{\omega_{m,s}P_{j}}\rightarrow 0$ and
$\frac{\mu-\rho_{1}}{\omega_{m,s}P_{j}}\rightarrow 0$. Accordingly, we can
obtain $\lim_{P_{j}\rightarrow\infty}\xi\rightarrow
1-\mathcal{P}_{\phi_{0}^{1}}$ when $\sigma_{w}^{2}<\mu\leq\rho_{1}$ and
$\lim_{P_{j}\rightarrow\infty}\xi\rightarrow 1$ when $\mu>\rho_{1}$. This
implies that the minimum detection error probability that W can achieve
approaches $\xi^{*}=1-\mathcal{P}_{\phi_{0}^{1}}$ when a large enough jamming
power is employed against surveillance. Moreover, it is as expected that
$\xi\rightarrow 1$ as $\tau,P_{j}\rightarrow\infty$, which can also be
verified from (19).

3) As $\tau,P_{j}\rightarrow 0$, which is equivalent to the case without the
assistance of jammers, we can readily derive that $\xi\rightarrow 0$ when
$\sigma_{w}^{2}<\mu\leq\rho_{1}$ and $\xi\rightarrow 1$ otherwise. This
implies, if only W chooses the value of $\mu$ within
$(\sigma_{w}^{2},\rho_{1}]$, it can successfully detect the covert
communication between T and R.

The above analyses clearly demonstrate the superiority of our proposed
threshold-based jammer selection scheme.

###  III-B Minimum Average Detection Error Probability

Since T cannot acquire the instantaneous CSI related to W $h_{t,w}$, it is
impossible to calculate the exact detection error probability at W. Instead,
we focus on the average detection error probability over $h_{t,w}$.666As
discussed in [19, 24], the average detection error probability captures the
average covertness performance from a Bayesian statistics perspective. There
exists another approach for evaluating the covertness performance, which
captures the probability that covert communication fails from an outage
perspective, e.g., covert outage probability. Following [19, 24], we can
easily prove that the two kinds of metrics are equivalent in assessing the
covertness performance of the system under investigation.  We consider a
worst-case scenario for covert communications where the optimal detection
threshold $\mu$ is designed from W’s viewpoint which results in a minimum
average detection error probability. In particular, the worst-case covert
communication should consider that, for improving detection accuracy, W can
adjust the detection threshold based on not only the distances between himself
to T and jammers, but also the CSI of $h_{t,w}$. In the following, we
distinguish two situations depending on whether W has the knowledge of the
instantaneous CSI $h_{t,w}$ or not.

_Case 1: W knows the instantaneous CSI of $h_{t,w}$._ In this situation, W is
able to adjust the detection threshold $\mu$ to the change of the detection
channel $h_{t,w}$ in real-time so as to achieve a minimum detection error
probability. Revisiting $\xi$ in (18), the minimum detection error probability
under a given value of $h_{t,w}$ can be derived by
$\xi^{*}\left(h_{t,w}\right)=\min_{\mu>0}\xi$, which turns out to be a
function of a single random variable $h_{t,w}$. Specifically, when
$\sigma_{w}^{2}<\mu\leq\rho_{1}$, we can easily prove that $\xi$ monotonically
decreases from $\xi=1$ as $\mu$ increases, since
$\Gamma\left(v_{m,s},\frac{\mu-\sigma_{w}^{2}}{\omega_{m,s}P_{j}}\right)$
decreases with $\mu$. We can also confirm that $\xi\rightarrow 1$ when
$\mu\rightarrow\infty$. This indicates that, the minimum detection error
probability can be achieved at either $\mu=\rho_{1}$ or within the interval
$\mu\in(\rho_{1},\infty)$.

Based on the above discussion, the average minimum detection error probability
can be calculated as

${\bar{\xi}}_{1}^{*}=\mathbb{E}_{h_{t,w}}\left[\min_{\mu\in[\rho_{1},\infty)}\xi(h_{t,w})\right].$
(20)

_Case 2: W does not know the instantaneous CSI of $h_{t,w}$._ In this
situation, W can only design the optimal $\mu$ to minimize the average
detection error probability based on the statistics of $h_{t,w}$. Before
proceeding to the calculation of the minimum average detection error
probability, we first derive the expression for the average detection error
probability based on (18) by the following theorem.

###### Theorem 2

The average detection error probability at W for a given detection threshold
$\mu$ is given by

$\displaystyle\bar{\xi}=\begin{cases}1,&\mu\leq\sigma_{w}^{2},\\\
\sum\limits_{m=1}^{N}\sum\limits_{s=1}^{\binom{N}{m}}\mathcal{P}_{\phi_{m}^{s}}\left(\frac{\Gamma\left(v_{m,s},\frac{\mu-\sigma_{w}^{2}}{\omega_{m,s}P_{j}}\right)}{\Gamma\left(v_{m,s}\right)}-\int_{0}^{\rho_{2}}e^{-x}\times\right.\\\
\quad\left.\frac{\Gamma\left(v_{m,s},\frac{\mu-\rho_{1}(x)}{{\omega_{m,s}P}_{j}}\right)}{\Gamma\left(v_{m,s}\right)}dx\right)+1-e^{-\rho_{2}},&\mu>\sigma_{w}^{2},\end{cases}$
(21)

where $\rho_{1}(x)\triangleq P_{t}d_{t,w}^{-\alpha}x+\sigma_{w}^{2}$ and
$\rho_{2}\triangleq{\left(\mu-\sigma_{w}^{2}\right)d_{t,w}^{\alpha}}/{P_{t}}$.

###### Proof 4

We see from (15) that the miss detection event only occurs when
$\mu>\rho_{1}$, i.e., $\left|h_{t,w}\right|^{2}<\rho_{2}$. Let
$x\triangleq\left|h_{t,w}\right|^{2}$, then the average detection error
probability for a given $\mu$ can be calculated according to (14), (15), and
(18) as

$\displaystyle\bar{\xi}$
$\displaystyle=\mathbb{E}_{h_{t,w}}\left[\xi(h_{t,w})\right]=\mathcal{P}_{FA}+\mathbb{E}_{h_{t,w}}\left[\mathcal{P}_{MD}\right]$
$\displaystyle=\begin{cases}1,&\mu\leq\sigma_{w}^{2},\\\
\mathcal{P}_{FA}+\int_{0}^{\rho_{2}}{\mathcal{P}_{MD}f_{|h_{t,w}|^{2}}\left(x\right)dx},&\mu>\sigma_{w}^{2},\end{cases}$
(22)

where $f_{|h_{t,w}|^{2}}\left(x\right)=e^{-x}$ is the PDF of $|h_{t,w}|^{2}$.
Substituting (14) and (15) into (4), and the proof can be completed after some
algebraic operations.

Note that $\bar{\xi}$ in (4) for $\mu>\sigma_{w}^{2}$ contains two parts,
where the first part reflects the negative impact of the opportunistic jammer
selection on the detection accuracy at W, and the second one arises due to the
absence of the instantaneous CSI of $h_{t,w}$.

In the following corollary, we provide more concise expressions for
$\bar{\xi}$ in (21) under the large and small selection threshold regimes,
i.e., considering $\tau\rightarrow\infty$ and $\tau\rightarrow 0$,
respectively.

###### Corollary 2

As $\tau\rightarrow\infty$ and $\tau\rightarrow 0$, the average detection
error probability $\bar{\xi}$ for $\mu>\sigma_{w}^{2}$ approaches

$\lim\limits_{\tau\rightarrow\infty}\bar{\xi}=\frac{\Gamma\left(v_{N},\frac{\mu-\sigma_{w}^{2}}{\omega_{N}P_{j}}\right)}{\Gamma\left(v_{N}\right)}+\int_{0}^{\rho_{2}}{e^{-x}\frac{\gamma\left(v_{N},\frac{\mu-\rho_{1}(x)}{{\omega_{N}P}_{j}}\right)}{\Gamma\left(v_{N}\right)}dx},$
(23) $\lim\limits_{\tau\rightarrow 0}\bar{\xi}=1-e^{-\rho_{2}},$ (24)

respectively, where $v_{N}$ and $\omega_{N}$ have been defined in (19).

Corollary 2 shows that as either $\tau\rightarrow\infty$ or $\tau\rightarrow
0$, $\bar{\xi}$ converges to a constant value independent of $\tau$, which
implies that the uncertainty of the number of selected jammers vanishes at
this time.

Recalling (21), just similar to the analysis with respect to (18), we can
prove that the optimal detection threshold $\mu$ that minimizes the average
detection error probability $\bar{\xi}$ in (21) exists within the interval
$\mu\in[\sigma_{w}^{2},\infty)$.

According to the above analysis, the minimum average detection error
probability can be derived by

${\bar{\xi}}_{2}^{*}=\min_{\mu\in[\sigma_{w}^{2},\infty)}\mathbb{E}_{h_{t,w}}\left[\xi(h_{t,w})\right].$
(25)

Combined with Corollary 2, some observations can be made on the asymptotic
behavior of ${\bar{\xi}}_{2}^{*}$. To be specific, we can derive that
$\lim_{\tau\rightarrow\infty}{\bar{\xi}}_{2}^{*}\rightarrow 1$ as
$P_{j}\rightarrow\infty$ and
$\lim_{\tau\rightarrow\infty}{\bar{\xi}}_{2}^{*}\rightarrow 1-e^{-\rho_{2}}$
as $P_{j}\rightarrow 0$. Besides, we have $\lim_{\tau\rightarrow
0}{\bar{\xi}}_{2}^{*}\rightarrow 0$, which is achieved when W sets
$\mu=\sigma_{w}^{2}$. Note that although adopting sufficiently large values of
$\tau$ and $P_{j}$ can result in a quite poor detection performance, it will
bring severe interference to R to degrade the reliability. These results
reflect the necessary of choosing a proper value of $\tau$ in order to balance
well between covertness and reliability.

##  IV Covert Throughput Maximization

In this section, we first calculate the transmission outage probability
$\delta$ of the covert communication from T to R, based on which we then
design the optimal selection threshold $\tau$ and transmission rate $R$ to
maximize the covert throughput subject to a covertness constraint.

###  IV-A Transmission Outage Probability

According to (3), the signal-to-interference-plus-noise ratio (SINR) at R is
given by

$\Lambda_{r}=\frac{P_{t}\left|h_{t,r}\right|^{2}}{\sum_{k=1}^{N}{\mathcal{I}_{j_{k}}P_{j}\left|h_{j_{k},r}\right|^{2}+\sigma_{r}^{2}}}.$
(26)

Since $\left|h_{j_{k},r}\right|^{2}$ is a random variable to T, a transmission
outage event occurs when the channel capacity
$C={\rm{log}}_{2}{\left(1+\Lambda_{r}\right)}$ falls below the transmission
rate $R$, i.e., $C<R$. The corresponding probability that this event happens
is provided by the following lemma.

###### Lemma 3

The transmission outage probability from T to R is given by

$\begin{split}\delta=&\left(1-\lambda\right)\mathcal{P}_{\phi_{0}^{1}}+\sum_{m=1}^{N}\sum_{s=1}^{\binom{N}{m}}\mathcal{P}_{\phi_{m}^{s}}\times\\\
&\quad\left(1-\prod_{j_{k}\in\phi_{m}^{s}}\frac{\lambda\left(1-e^{-\left(\varphi
P_{j}d_{t,r}^{\alpha}+d_{j_{k},r}^{\alpha}\right)\sigma_{r}^{2}\tau}\right)}{\left(\varphi
P_{j}d_{j_{k},r}^{-\alpha}d_{t,r}^{\alpha}+1\right)\left(1-e^{-d_{j_{k},r}^{\alpha}\sigma_{r}^{2}\tau}\right)}\right),\end{split}$
(27)

where $\varphi\triangleq{\left(2^{R}-1\right)}/{P_{t}}$ and $\lambda\triangleq
e^{-d_{t,r}^{\alpha}\varphi\sigma_{r}^{2}}$.

###### Proof 5

Based on the definition of a transmission outage event, the transmission
outage probability can be calculated as

$\displaystyle\delta=\mathbb{P}\left[{\rm{log}}_{2}{\left(1+\Lambda_{r}\right)}<R\right]$
$\displaystyle\stackrel{{\scriptstyle\mathrm{(a)}}}{{=}}\sum_{m=0}^{N}\sum_{s=1}^{\binom{N}{m}}\mathcal{P}_{\phi_{m}^{s}}\mathbb{P}\left[\frac{\left|h_{t,r}\right|^{2}}{d_{t,r}^{\alpha}}<\varphi\left(\sum_{j_{k}\in\phi_{m}^{s}}\frac{\left|h_{j_{k},r}\right|^{2}}{d_{j_{k},r}^{\alpha}}+\sigma_{r}^{2}\right)\right]$
$\displaystyle\stackrel{{\scriptstyle\mathrm{(b)}}}{{=}}\left(1-\lambda\right)\mathcal{P}_{\phi_{0}^{1}}+\sum_{m=1}^{N}\sum_{s=1}^{\binom{N}{m}}\mathcal{P}_{\phi_{m}^{s}}\times$
$\displaystyle\qquad\left(1-\lambda\prod_{j_{k}\in\phi_{m}^{s}}{\mathbb{E}_{\left|h_{j_{k},r}\right|^{2}}\left[e^{-\varphi
P_{j}\left|h_{j_{k},r}\right|^{2}d_{j_{k},r}^{-\alpha}d_{t,r}^{\alpha}}\right]}\right)$
$\displaystyle=\left(1-\lambda\right)\mathcal{P}_{\phi_{0}^{1}}+\sum_{m=1}^{N}\sum_{s=1}^{\binom{N}{m}}\mathcal{P}_{\phi_{m}^{s}}\Bigg{(}1-\lambda\prod_{j_{k}\in\phi_{m}^{s}}\times$
$\displaystyle\qquad\int_{0}^{{\sigma_{r}^{2}\tau}{d_{j_{k},r}^{\alpha}}}{\check{f}}_{\left|h_{j_{k},r}\right|^{2}}\left(y\right)e^{-\varphi
P_{j}\left|h_{j_{k},r}\right|^{2}d_{j_{k},r}^{-\alpha}d_{t,r}^{\alpha}}dy\Bigg{)},$
(28)

where (a) follows from the total probability theorem and (b) holds by using
the CDF of ${|h_{t,r}|}^{2}$, i.e.,
$F_{\left|h_{t,r}\right|^{2}}\left(x\right)=1-e^{-x}$. In addition, since the
selected jammers satisfy
$\left|h_{j_{k},r}\right|^{2}d_{j_{k},r}^{-\alpha}/\sigma_{r}^{2}<\tau$, the
PDF of $\left|h_{j_{k},r}\right|^{2}$ can be derived as a conditional PDF
given below,

$\displaystyle{\check{f}}_{\left|h_{j_{k},r}\right|^{2}}\left(y\right)=\begin{cases}\frac{e^{-y}}{1-e^{-d_{j_{k},r}^{\alpha}\sigma_{r}^{2}\tau}},&0\leq
y\leq d_{j_{k},r}^{\alpha}\sigma_{r}^{2}\tau,\\\ 0,&otherwise.\end{cases}$
(29)

Substituting (29) into (5) and solving the integral complete the proof.

The first part in (27) is actually the transmission outage probability without
jammers and the second part reflects the increased risk of causing
transmission outage due to the introduction of jammers. The following
corollary further provides the transmission outage probability for the
asymptotic cases with $\tau\rightarrow 0$ and $\tau\rightarrow\infty$,
respectively.

###### Corollary 3

As the selection threshold $\tau\rightarrow 0$ and $\tau\rightarrow\infty$,
the transmission outage probability $\delta$ respectively approaches

$\lim\limits_{\tau\rightarrow 0}\delta=1-\lambda,$ (30)
$\lim\limits_{\tau\rightarrow\infty}\delta=1-\lambda\prod_{k=1}^{N}\frac{1}{\varphi
P_{j}d_{j_{k},r}^{-\alpha}d_{t,r}^{\alpha}+1}.$ (31)

###### Proof 6

The results follow directly by substituting the results that $p_{k}\rightarrow
0$ as $\tau\rightarrow 0$, and $p_{k}\rightarrow 1$ and $e^{-\left(\varphi
P_{j}d_{t,r}^{\alpha}+d_{j_{k},r}^{\alpha}\right)\tau}\rightarrow 0$ as
$\tau\rightarrow\infty$. into (27).

Corollary 3 shows that the transmission outage probability tends to be
constant as either $\tau\rightarrow 0$ or $\tau\rightarrow\infty$, which will
be confirmed in Fig. 5.

###  IV-B Optimization of Covert Throughput

Note that the transmission outage probability $\delta$ is a function of both
the transmission rate $R$ and the selection threshold $\tau$. The problem of
maximizing the covert throughput $\Omega$ subject to a covertness constraint
${\bar{\xi}}_{\iota}^{*}\geq\rm{1}-\varepsilon$ for $\iota=1,2$, as shown in
(10), can be solved by first designing the optimal $\tau$ to minimize $\delta$
for a fixed $R$ and then deriving the optimal $R$ to achieve the globally
maximum $\Omega^{*}$. To begin with, the subproblem of minimizing $\delta$ can
be formulated as

$\min_{\tau>0}\delta(\tau),\quad\rm{s.t.}\quad{\bar{\xi}}_{\iota}^{*}\geq\rm{1}-\varepsilon,\forall\iota=1,2,$
(32)

with the optimal $\tau^{*}$ characterized by the following proposition.

###### Proposition 1

For any given transmit power $P_{t}$, jamming power $P_{j}$, total number of
cooperative nodes $N$, and maximum tolerable detection probability
$\varepsilon$, the optimal selection threshold $\tau^{*}$ for minimizing the
transmission outage probability $\delta$ satisfies

$\left.{{\bar{\xi}}_{\iota}^{*}}\right|_{\tau=\tau^{*}}=1-\varepsilon,\quad\iota=1,2.$
(33)

###### Proof 7

According to the jammer selection criteria described in (1), it is obvious
that using a larger value of selection threshold $\tau$ activates more jammers
and causes more severe interference to R, thus resulting in a larger
transmission outage probability $\delta$. In that sense, the optimal
$\tau^{*}$ that minimizes $\delta$ should be the minimum achievable value of
$\tau$ that meets the covertness constraint
${\bar{\xi}}_{\iota}^{*}\geq\rm{1}-\varepsilon$ for $\iota=1,2$. Note that the
confusion on the detection at W created by jammers weakens as $\tau$
decreases, which yields a smaller minimum average detection error probability
${\bar{\xi}}_{\iota}^{*}$. In other words, ${\bar{\xi}}_{\iota}^{*}$ is a
monotonically increasing function of $\tau$. Based on the above discussion, it
is not difficult to conclude that the optimal $\tau^{*}$ satisfies
${\bar{\xi}}_{\iota}^{*}=\rm{1}-\varepsilon$, and the proof is completed.

From (33), we can conveniently obtain the optimal $\tau^{*}$ by calculating
the inverse of ${\bar{\xi}}_{\iota}^{*}$ at $1-\varepsilon$, e.g., using a
simple bisection method. Afterwards, we proceed to the optimal transmission
rate $R$ by solving the following subproblem.

$\max_{R>0}\Omega(R)=R(1-\delta(R)),$ (34)

where $\delta(R)$ is an increasing function of $R$. We can show that
$\Omega\rightarrow 0$ as either $R\rightarrow 0$ or $R\rightarrow\infty$ due
to $\delta(R)\rightarrow 1$ for the latter. This indicates that neither a too
large nor a too small covert rate can produce a high covert throughput, and we
should carefully design the covert rate to improve the covert throughput.
Moreover, as will be verified in the numerical results, the covert throughput
$\Omega(R)$ is a first-increasing-then-deceasing function of $R$. As a
consequence, the optimal $R^{*}$ that maximizes $\Omega(R)$ can be efficiently
calculated using the bisection method.

By now, we have obtained the optimal selection threshold $\tau^{*}$ and covert
rate $R^{*}$, substituting which into (9) yields the maximum covert throughput
$\Omega^{*}$.

##  V Numerical Results

In this section, we present numerical results to validate our theoretical
analyses and to study the achievable performance of covert communications for
the multi-jammer cooperative networks. For simplicity, $N$ cooperative nodes
are randomly generated in a circle centered at T with a radius $2d_{t,r}$,
where $d_{t,r}$ is the distance between T and R. Unless specified otherwise,
we assume that W does not know the instantaneous CSI of the detection channel
$h_{t,w}$ between itself and T, which is rational due to the covert
transmission from T. We define a reference distance $d_{0}=1000$m, and set
$d_{t,r}=d_{0}$, $\alpha=4$, $P_{t}=10$dBm, $\sigma_{w}^{2}=-120$dBm, and
$\sigma_{r}^{2}=-120$dBm. In what follows, we will first evaluate the minimum
average detection error probability at W under the two scenarios where W does
and does not know $h_{t,w}$, and consider the transmission outage probability
of the covert communication between T and R. We will then examine the
influence of various key arguments on the covert throughput focusing on the
case without knowing the channel $h_{t,w}$ at W.

Figure 2: ${\bar{\xi}}_{1}^{*}$ and ${\bar{\xi}}_{2}^{*}$ vs. lg$\tau$ for
different values of $N$, with $P_{j}=10$dBm and $d_{t,w}=1.2d_{0}$.

In Fig. 2, we aim to examine the detection performance with and without the
knowledge of the instantaneous CSI of $h_{t,w}$ at W, and we illustrate the
corresponding minimum average detection error probabilities
${\bar{\xi}}_{1}^{*}$ and ${\bar{\xi}}_{2}^{*}$ versus lg$\tau$ for different
values of $N$. It is expected that both ${\bar{\xi}}_{1}^{*}$ and
${\bar{\xi}}_{2}^{*}$ increase with $N$, since the increase of $N$ creates
more randomness of the aggregate jamming power to W, which hinders its
detection. It can be seen that ${\bar{\xi}}_{1}^{*}$ is always lower than
${\bar{\xi}}_{2}^{*}$, and the gap enlarges for a smaller $N$. This is as
expected, meaning that once the instantaneous CSI of the detection channel is
acquired by W, the covertness will be further compromised.

Figure 3: ${\bar{\xi}}_{2}^{*}$ vs. lg$\tau$ for different values of $P_{j}$
and $d_{t,w}$, with $N=10$.

Fig. 3 plots the minimum average detection error probability
${\bar{\xi}}_{2}^{*}$ versus lg$\tau$ for different values of $P_{j}$ and
$d_{t,w}$. We note that when $\tau\rightarrow\infty$, ${\bar{\xi}}_{2}^{*}$
approaches a fixed value, which verifies Corollary 2. We find that
${\bar{\xi}}_{2}^{*}$ increases with $P_{j}$, since large jamming power
shields the covert communication more effectively from being detected by W. We
also show that ${\bar{\xi}}_{2}^{*}$ decreases significantly as $d_{t,w}$
decreases. The reason behind is that the path loss between T and W becomes
smaller as $d_{t,w}$ decreases, which makes the power of covert communications
much more dominate compared with the jamming power, thus remarkably improving
the detection accuracy for W.

Figure 4: ${\bar{\xi}}_{2}^{*}$ and $\delta$ vs. lg$\tau$ for different values
of $P_{j}$, with $N=4$ and $R=1$bits/s/Hz.

Fig. 4 shows the minimum average detection error probability
${\bar{\xi}}_{2}^{*}$ and the transmission outage probability $\delta$ versus
lg$\tau$ for different values of jamming power $P_{j}$. It can be seen that
the Monte-Carlo simulation results match well with the theoretical values from
(21) and (27), respectively, which verifies the correctness of the theoretical
analyses and the accuracy of the approximation method given in (12). We can
also observe that both ${\bar{\xi}}_{2}^{*}$ and $\delta$ are monotonically
increasing functions of lg$\tau$, which is consistent with what is concluded
in Proposition 1.

Figure 5: $\delta$ vs. lg$\tau$ for different values of $P_{j}$ and $R$, with
$N=10$ and $d_{t,w}=1.2d_{0}$.

Fig. 5 depicts the transmission outage probability $\delta$ versus lg$\tau$
for different jamming power $P_{j}$ and covert rate $R$. We observe that
$\delta$ monotonically increases with lg$\tau$, $R$, and $P_{j}$, and
approaches a constant value as either $\tau\rightarrow 0$ or
$\tau\rightarrow\infty$, which is consistent with Corollary 3. In addition,
the convergence value of $\delta$ as $\tau\rightarrow 0$ is solely determined
by $R$ and is independent of $P_{j}$, since no jammer is likely to be
activated for a small enough $\tau$.

Figure 6: $\tau^{*}$ vs. $P_{j}$ for different values of $N$, with
$\varepsilon=0.1$.

Fig. 6 shows the optimal selection threshold $\tau^{*}$ versus $P_{j}$ for
different values of $N$. We show that $\tau^{*}$ is a monotonically decreasing
function of $N$ and $P_{j}$. This is because the uncertainty of
differentiating desired signal and jamming power increases with $N$ and
$P_{j}$, which produces a larger detection error probability at W. As a
result, smaller $\tau$ is allowed to satisfy a certain covertness constraint
for larger $N$ and $P_{j}$.

Figure 7: $\Omega$ vs. $R$ for different values of $N$, with $P_{j}=10$dBm,
$d_{t,w}=1.2d_{0}$, and $\varepsilon=0.1$.

Fig. 7 illustrates the covert throughput $\Omega$ versus the covert rate $R$
for different numbers of cooperative nodes $N$, considering the situations
with and without the instantaneous CSI of the detection channel $h_{t,w}$. We
see that $\Omega$ increases first and then decreases as $R$ increases, and
approaches zero as either $R\rightarrow 0$ or $R\rightarrow\infty$. It can
also be observed that $\Omega$ increases as $N$ becomes larger. This is
because the increase of $N$ will undoubtedly create more randomness of total
jamming power and degrade the detection accuracy at W, whereas the
transmission outage probability between T and R will not increase
significantly owing to the jammer selection criterion. We also find that
$\Omega$ with ${\bar{\xi}}_{2}^{*}$ is larger than that with
${\bar{\xi}}_{1}^{*}$ particularly for the small $N$ regime, since a higher
level of covertness can be promised when W is unaware of the instantaneous CSI
of the channel from T to itself.

Figure 8: $\Omega^{*}$ vs. $1-\varepsilon$ for different values of $N$, with
$P_{j}=10$dBm and $d_{t,w}=1.2d_{0}$.

Fig. 8 plots the maximum covert throughput $\Omega^{*}$ versus $1-\varepsilon$
for different numbers of cooperative nodes $N$. We also adopt the single-
jammer scheme adopted in [32] as the benchmark scheme, where variable jamming
power is exploited to confuse the detector. Comparing the two schemes, we show
that $\Omega^{*}$ for the multi-jammer scheme significantly outperforms that
for the single-jammer scheme for a quite wide range of $\varepsilon$. This
demonstrates the superiority of our proposed multi-jammer scheme in terms of
covert communications. Nevertheless, we can observe that the single-jammer
scheme appears to be superior for an extremely small value of $\varepsilon$
which corresponds to a high level of covertness requirement. The underlying
reason is that the randomness of total jamming power created by a certain
number of cooperative nodes might not be sufficient to satisfy a stringent
enough covert constraint, whereas the single-jammer scheme can always
guarantee uncertainty at W by enlarging the jamming power budget and adopting
random power continuously.

Figure 9: $\Omega^{*}$ vs. $N$ for different values of $P_{j}$, with
$d_{t,w}=1.2d_{0}$ and $\varepsilon=0.1$.

Fig. 9 depicts the maximum covert throughput $\Omega^{*}$ versus $N$ for
different values of $P_{j}$. It is as expected that $\Omega^{*}$ increases
with $N$, and we can see that as $N$ continuously increases, the improvement
of covert throughput becomes less remarkable. This suggests that a moderate
number of cooperative nodes should be favorable for assisting the covert
communication. In addition, there exists a minimum number $N_{min}$ of
cooperative nodes below which the covertness constraint cannot be satisfied
under a certain level of jamming power, and the value of $N_{min}$ becomes
smaller as $P_{j}$ increases, e.g., $N_{min}=5$ for $P_{j}=10$dBm and
$N_{min}=8$ for $P_{j}=5$dBm. Interestingly, we also observe that a smaller
(larger) $P_{j}$ can produce a higher $\Omega^{*}$ when $N$ is large (small)
enough, which is attributed to the trade-off between covertness and
reliability.

##  VI Conclusion

This paper investigated covert communications assisted by multiple friendly
cooperative nodes over slow fading channels, in which transmitter T aims to
covertly deliver messages to receiver R under the surveillance of warden W,
while a portion of the cooperative nodes will be intelligently selected as
jammers to send interference signals to confuse W. We proposed an
opportunistic jammer selection scheme, based on which we analyzed the average
detection error probability at W and the transmission outage probability of
the covert communication between T and R. We then jointly designed the optimal
selection threshold and the covert rate to maximize the covert throughput
subject to a certain covertness constraint. By comparing with the single-
jammer case, numerical results demonstrated the superiority of our proposed
multi-jammer scheme in terms of covert throughput. We showed that there exists
an optimal jammer selection threshold for maximizing the covert throughput,
and the optimal value decreases as either the number of cooperative nodes or
the jamming power increases. We also revealed that a minimum number of
cooperative nodes should be deployed in order to meet a certain level of
covertness requirement; as the number of cooperative nodes continuously
increases, the covert throughput first significantly improves but then reaches
saturation.

###  -A Proof of the Sufficiency of Radiometer

To justify the use of a radiometer as the detector, we prove that the test
statistic $T_{w}$ in (7) is sufficient for W’s detection.

The received signal at W in (4) can be rewritten as

$\bm{y}_{w}[i]=\sqrt{P_{t}}\frac{h_{t,w}}{d_{t,w}^{\alpha/2}}\bm{x}_{t}[i]\theta+\sum\limits_{k=1}^{N}{\mathcal{I}_{j_{k}}\sqrt{P_{j}}\frac{h_{j_{k},w}}{d_{j_{k},w}^{\alpha/2}}\bm{v}_{j_{k}}[i]}+\bm{n}_{w}[i],$
(35)

where $\theta$ represents the indicator of whether T is transmitting or not,
with $\theta=1$ meaning that T is transmitting while $\theta=0$ otherwise.
Since we assume that $\bm{x}_{t}[i],\bm{v}_{j_{k}}[i]\sim\mathcal{CN}(0,1)$,
$\bm{n}_{w}[i]\sim\mathcal{CN}(0,\sigma_{w}^{2})$, we have
$\bm{y}_{w}[i]\sim\mathcal{CN}(0,\sigma_{\theta}^{2})$, where
$\sigma_{\theta}^{2}=\sigma_{c}^{2}\theta+\sigma_{j}^{2}+\sigma_{w}^{2}$ with
$\sigma_{c}^{2}$ and $\sigma_{j}^{2}$ defined in (7). Since W is unaware of
the instantaneous CSI between ${\rm J}_{k}$ and R, the number of selected
jammers is not available at W such that $\sigma_{j}^{2}$ becomes a random
variable to W. However, the PDF $f_{\sigma_{j}^{2}}(x)$ of $\sigma_{j}^{2}$ is
assumed known at W.

According to the Fisher-Neyman factorization theorem, W’s test statistic
$T_{w}=(1/n)\sum_{i=1}^{n}\left|\bm{y}_{w}[i]\right|^{2}$ is sufficient for
the indicator $\theta$ if and only if the conditioned probability
$\mathcal{P}(\bm{y}_{w}|\theta)$ depends on $\bm{y}_{w}$ only through $T_{w}$.

From (35), $\mathcal{P}(\bm{y}_{w}|\theta)$ can be calculated as

$\displaystyle\mathcal{P}(\bm{y}_{w}|\theta)$
$\displaystyle=\int_{0}^{\infty}\prod_{i=1}^{n}\left\\{\frac{1}{\sqrt{2\pi\sigma_{\theta}^{2}}}e^{-\frac{\left|\bm{y}_{w}[i]\right|^{2}}{2\sigma_{\theta}^{2}}}\right\\}f_{\sigma_{j}^{2}}\left(x\right)dx$
$\displaystyle=\int_{0}^{\infty}\left(2\pi\sigma_{\theta}^{2}\right)^{-\frac{n}{2}}e^{-\frac{n}{2\sigma_{\theta}^{2}}T_{w}}f_{\sigma_{j}^{2}}\left(x\right)dx.$
(36)

We clearly show that $\mathcal{P}(\bm{y}_{w}|\theta)$ relies on $\bm{y}_{w}$
only via $T_{w}$ regardless of the form of $f_{\sigma_{j}^{2}}\left(x\right)$.
This completes the proof.

###  -B Proof of Lemma 2

The $y$-th cumulant of a random variable $X_{m,s}$ is defined as

$Q_{X_{m,s}}^{\left(y\right)}=\left.\frac{d^{y}\mathbb{E}_{X_{m,s}}\left[e^{\omega
X_{m,s}}\right]}{d\omega^{y}}\right|_{\omega=0}.$ (37)

The mean and variance of $X_{m,s}$ are
$\mu_{X_{m,s}}=Q_{X_{m,s}}^{\left(1\right)}$ and
$\sigma_{X_{m,s}}^{2}=Q_{X_{m,s}}^{\left(2\right)}-\left(Q_{X_{m,s}}^{\left(1\right)}\right)^{2}$,
respectively. Hence, the parameters $v_{m,s}$ and $\omega_{m,s}$ can be
calculated as $v_{m,s}={\mu_{X_{m,s}}^{2}}/{\sigma_{X_{m,s}}^{2}}$ and
$\omega_{m,s}={\sigma_{X_{m,s}}^{2}}/{\mu_{X_{m,s}}}$. Recall that
$X_{m,s}=\sum_{j_{k}\in\phi_{m}^{s}}{\left|h_{j_{k},w}\right|^{2}d_{j_{k},w}^{-\alpha}}$.
Due to the mutual independence among $\left\\{h_{j_{k},w}\right\\}$ for
$j_{k}\in\phi_{m}^{s}$, we can calculate $\mathbb{E}_{X_{m,s}}\left[e^{\omega
X_{m,s}}\right]$ for a deterministic $\phi_{m}^{s}$ as follows,

$\mathbb{E}_{X_{m,s}}\left[e^{\omega
X_{m,s}}\right]=\prod_{j_{k}\in\phi_{m}^{s}}{\mathbb{E}_{\left|h_{j_{k},w}\right|^{2}}\left[e^{-\omega\left|h_{j_{k},w}\right|^{2}d_{j_{k},w}^{-\alpha}}\right]}.$
(38)

Denote
$g_{j_{k},w}\left(\omega\right)=\mathbb{E}_{\left|h_{j_{k},w}\right|^{2}}\left[e^{-\omega\left|h_{j_{k},w}\right|^{2}d_{j_{k},w}^{-\alpha}}\right]$
such that $\mathbb{E}_{X_{m,s}}\left[e^{\omega
X_{m,s}}\right]=\prod_{j_{k}\in\phi_{m}^{s}}{g_{j_{k},w}\left(\omega\right)}$
and
$Q_{X_{m,s}}^{\left(y\right)}=\left.\left({d^{y}\prod_{j_{k}\in\phi_{m}^{s}}{g_{j_{k},w}\left(\omega\right)}}/{d\omega^{y}}\right)\right|_{\omega=0}$.
Then, we have $\left.g_{j_{k},w}\left(\omega\right)\right|_{\omega=0}=1$,
$\left.\left({dg_{j_{k},w}\left(\omega\right)}/{d\omega}\right)\right|_{\omega=0}=d_{j_{k},w}^{-\alpha}$,
and
$\left.\left({d^{2}g_{j_{k},w}\left(\omega\right)}/{d\omega^{2}}\right)\right|_{\omega=0}=2d_{j_{k},w}^{-2\alpha}$.
Based on these results, the first and second cumulants of $X_{m,s}$ can be
respectively calculated as

$\begin{split}Q_{X_{m,s}}^{\left(1\right)}&=\sum_{j_{k}\in\phi_{m}^{s}}\prod_{j_{p}\in\phi_{m}^{s}\backslash
j_{k}}\left.\left(g_{j_{p},w}\left(\omega\right)\frac{dg_{j_{k},w}\left(\omega\right)}{d\omega}\right)\right|_{\omega=0}\\\
&=\sum_{j_{k}\in\phi_{m}^{s}}d_{j_{k},w}^{-\alpha},\end{split}$ (39)
$\displaystyle Q_{X_{m,s}}^{\left(2\right)}$
$\displaystyle=\sum_{j_{k}\in\phi_{m}^{s}}\left(\prod_{j_{p}\in\phi_{m}^{s}\backslash
j_{k}}g_{j_{p},w}\left(\omega\right)\frac{d^{2}g_{j_{k},w}\left(\omega\right)}{d\omega^{2}}+\frac{dg_{j_{k},w}\left(\omega\right)}{d\omega}\right.$
$\displaystyle\quad\left.\left.\times\sum_{j_{p}\in\phi_{m}^{s}\backslash
j_{k}}\prod_{j_{q}\in\phi_{m}^{s}\backslash\left\\{j_{k},j_{p}\right\\}}g_{j_{q},w}\left(\omega\right)\frac{dg_{j_{p},w}\left(\omega\right)}{d\omega}\right)\right|_{\omega=0}$
$\displaystyle=\sum_{j_{k}\in\phi_{m}^{s}}\left(2d_{j_{k},w}^{-2\alpha}+d_{j_{k},w}^{-\alpha}\sum_{j_{p}\in\phi_{m}^{s}\backslash
j_{k}}d_{j_{p},w}^{-\alpha}\right).$ (40)

After some algebraic manipulations, we can obtain
$\mu_{X_{m,s}}=\sum_{j_{k}\in\phi_{m}^{s}}d_{j_{k},w}^{-\alpha}$ and
$\sigma_{X_{m,s}}^{2}=\sum_{j_{k}\in\phi_{m}^{s}}d_{j_{k},w}^{-2\alpha}$,
substituting which into $v_{m,s}$ and $\omega_{m,s}$ completes the proof.

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