11institutetext: Queen’s University Belfast, Belfast, United Kingdom, BT9 5AH
11email<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>
# Secrecy Outage Probability of Cognitive Small-Cell Network with Unreliable
Backhaul Connections††thanks: This work was supported in part by the Royal
Society-SERB Newton International Fellowship under Grant NF151345

Jinghua Zhang  Chinmoy Kundu  Emi Garcia-Palacios

###### Abstract

In this paper, we investigate the secrecy performance of underlay cognitive
small-cell radio network with unreliable backhaul connections. The secondary
cognitive small-cell transmitters are connected to macro base station by
wireless backhaul links. The small-cell network is sharing the same spectrum
with the primary network ensuring that a desired outage probability constraint
in the primary network is always satisfied. We propose an optimal transmitter
selection (OTS) scheme for small-cell network to transfer information to the
destination. The closed-form expression of secrecy outage probability are
derived. Our result shows that increasing the primary transmitter’s transmit
power and the number of small-cell transmitter can improve the system
performance. The backhaul reliability of secondary and the desired outage
probability of the primary also have significant impact on the system.

###### Keywords:

Unreliable Backhaul Cognitive Radio Network Small-Cell Network Physical Layer
Security Secrecy Outage Probability.

##  1 Introduction

Due to the explosion of data-intensive applications and wireless systems such
as the Internet of Things (IoT) and smart cities, the deployment of wireless
infrastructure is expected to get more dense and heterogeneous in the near
future [1]. To reach such high data rate, the backhaul links connecting the
macro-cell and many small-cells in the heterogeneous networks (HetNets) are
also expected to become dense. In the conventional wired backhaul network,
high reliability wired links and high data rate can be expected, however, the
deploying and sustaining the large-scale wired links requir excessive capital
investment for all the connections [2, 3]. This leads wireless backhaul as
alternative solution since it has been proven cost-effective and flexible in
practical systems. However, wireless backhaul is unlikely reliable as wired
backhaul due to non-line-of-sight (n-LOS) propagation and fading of wireless
channels [4].

The aforementioned rapid development in wireless devices and services is also
pushing the demand for spectrum while most of licensed spectrum bands are
occupied [5]. In recent years, the investigation on cognitive radio (CR)
techniques [6] has attracted many experts’ attention. CR optimises the current
spectrum usage, which allows unlicensed secondary users to share the same
spectrum with the licensed primary users in an opportunistic manner. The
authors in [7] analysed the impact of the primary network on the secondary
network. In [8], the authors optimizing the time and power allocation in the
secondary network. To improve the CR or noncognitive network performance, user
selection is always among the secondary users and relays in the literature [9,
10, 11, 12].

For a complete study, we also consider the challenges of security in the
wireless communication network. Due to the broadcasting nature of wireless
channels, the confidential information in wireless network is vulnerable to
eavesdropping and security attacks. In reality, CR networks are easily
susceptable to eavesdropping. The conventional way from upper layer security
is deploying data encryption for secure communication, on the other hand,
physical layer security (PLS) obtain the advantage from the randomness of the
wireless channels for information security extensively. PLS has become
increasingly popular to deal with wiretapping and possible loss of
confidentiality. Some research has investigated the secrecy performance using
PLS [9, 13, 14, 15, 16].

Nevertheless, all the aforementioned work did not take into account the impact
of unreliable backhaul on PLS of CR network. Some literatures in CR network
only consider the interference on the primary network. In some literature on
backhaul CR networks [4, 17, 18, 19], authors investigated the secrecy
performance but not consider the system with secondary user selection schemes.
The impact of guaranteeing outage as a quality-of-service (QoS) in the primary
networks also not considered in the aforementioned paper. Our research address
these key issues in CR network with backhaul. We investigate the secrecy
performance of CR network with unreliable backhaul connections. Based on those
considerations, our contribution of this paper is summarise as follows:

  1. 1.

We take into account the backhaul unreliability in secrecy performance. We
develop the close-form expression of the secrecy outage probability.

  2. 2.

We consider interference both in primary and secondary receiver.

  3. 3.

We consider primary QoS constraint metric as outage probability, which is
different from other works in CR.

  4. 4.

Our model investigates a small-cell transmitter selection schemes, namely,
optimal transmitter selection (OTS) which prioritizes the maximum channel gain
S–D, and also assesses the influences of varying the number of small-cell
transmitter.

The rest of the paper is organised as follow. In section 2, the system can
channel models are described. Section 3 demonstrates the secrecy outage
probability of propose system. Numerical results from monte-carlo simulations
are showcased in section 4. Finally, the paper is concluded in section 5.

##  2 System and channel models

As illustrated in Fig. 1, the system is consisting of a primary network with
one primary transmitter, T, one primary receiver, R and a secondary network
consisting of $K$ small-cells transmitters,
$\\{\text{S}_{1},...,\text{S}_{k},...,\text{S}_{K}\\}$ which are connected to
a macro-base station, BS, by unreliable backhaul links, one secondary
destination, D, and one eavesdropper, E. All nodes are equipped with single
antenna. We assume all nodes are sufficiently separated from each other so
that $\text{T}-\text{R}$, $\text{T}-\text{D}$, $\text{T}-\text{E}$,
$\text{S}-\text{R}$, $\text{S}-\text{D}$ and $\text{S}-\text{E}$ experience
independent and identically distributed Rayleigh fading. The channel between
nodes are denoted by $h_{X}$ where X= $\\{$ $TR$, $TD$, $TE$, $SR$, $SD$, $SE$
$\\}$ channel power gains are exponential distributed with parameters
$\lambda_{x}$ for x=$\\{$ $tr$, $td$, $te$, $sr$, $sd$, $se$ $\\}$,
respectively. The noise at R , E and D is modelled as the additive white
Gaussian noise (AWGN) with zero mean and variances $N_{0}$. One best
transmitter will be selected among K small-cell transmitters, to transfer
information to D. While the message is sent from the BS to small-cell
transmitters, the backhaul link might have certain probability of failure.
Backhaul reliability is modelled as Bernoulli process with success
probability, $\mathbb{P}(\mathbb{I}_{k}=1)=\Lambda$, and failure probability
is $\mathbb{P}(\mathbb{I}_{k}=0)=1-\Lambda$ for each $k=1,\cdots,K$. We
investigate the secrecy outage probability (SOP) of the secondary network.

Figure 1: Underlay cognitive radio network with unreliable backhaul
connections.

###  2.1 Interference at Primary and Secondary Power Constraint

The primary network is interfered from the selected secondary transmitter,
$S_{k}$, for $k=\\{1,...,K\\}$ via interference channels $h_{S_{k}R}$ during
the secondary network transmission. The signal-to-interference-plus-noise-
ratio (SINR) at R is given as

$\displaystyle\Gamma_{R}=\frac{P_{T}|h_{TR}|^{2}}{P_{S}|h_{S_{k}R}|^{2}+N_{0}},$
(1)

where $P_{S}$ is the maximum allowed transmit power of small-cell transmitter
which satisfies the primary network QoS constraint, $h_{TR}$ is the channel
coefficient of the $\text{T}-\text{R}$ link, and $h_{S_{k}R}$ is the channel
coefficient of the $\text{S}-\text{R}$ link. To protect the primary network,
the secondary network transmitters must adapt their transmit power. Moreover,
the secondary network transmit power must be limitedly the QoS of the primary
network which characterized by its desired outage probability. The primary
network outage probability should be below a desired level, $\Phi$. The
desired outage probability constraint is defined as follows

$\displaystyle\mathbb{P}\left[{\Gamma_{\text{R}}}<\Gamma_{0}\right]\leq\Phi,$
(2)

where $\Gamma_{0}=2^{\beta}-1$, $\beta$ is the target rate of the primary
network, and $0<\Phi<1$. From (1) and (2), output power of secondary
transmitter can be derived from the desired outage probability at the primary
network

$\displaystyle P_{S}=\left\\{\begin{array}[]{ll}P_{T}\lambda_{sr}\xi,\ \ \
\textnormal{if}\ \ \ \xi>0\\\ 0,\ \ \ \ \ \ \ \ \ \ \ \
\textnormal{otherwise}.\\\ \end{array}\right.$ (5)

where

$\displaystyle\xi=\frac{1}{\lambda_{tr}\Gamma_{0}}\left[\frac{\textnormal{exp}\left(\frac{-\lambda_{tr}\Gamma_{0}}{\Gamma_{T}}\right)}{1-\Phi}-1\right].$
(6)

Here, we used CDF of $\Gamma_{\text{R}}(x)$ to find $P_{S}$ in close-form. The
CDF of $\Gamma_{\text{R}}(x)$ can be derived from the definition of CDF as

$\displaystyle
F_{{\text{R}}}(x)=1-\frac{\frac{\lambda_{sr}\Gamma_{T}}{\lambda_{tr}\Gamma_{S}}}{x+\frac{\lambda_{sr}\Gamma_{T}}{\lambda_{tr}\Gamma_{S}}}\textnormal{exp}\left(\frac{-\lambda_{tr}x}{\Gamma_{T}}\right),$
(7)

where $\Gamma_{T}=\frac{P_{T}}{N_{0}}$ and $\Gamma_{S}=\frac{P_{S}}{N_{0}}$.

###  2.2 Proposed Source Selection and Interference at the Secondary

To mitigate the eavesdropping, OTS scheme is proposed where a source is
selected to forward the message such that it maximize $\text{S}_{k}-\text{D}$
link power gain as

$\displaystyle k^{*}=\arg\max_{1\leq k\leq K}P_{S}|h_{S_{k}D}|^{2}.$ (8)

Due to the unreliability of the backhaul, the selected link may not be active.
To consider backhaul reliability into the performance analysis, we model
backhaul reliability using Bernoulli random variable $\mathbb{I}$. The SINR at
D can be given as

$\displaystyle\Gamma_{SD}=\mathbb{I}{\tilde{\Gamma}}_{SD},$ (9)

where

$\displaystyle{\tilde{\Gamma}}_{SD}=\frac{P_{S}\max[|h_{S_{k}D}|^{2}]}{P_{T}|h_{TD}|^{2}+N_{0}}.$
(10)

SINR at E can be similarly expressed as

$\displaystyle\Gamma_{SE}=\dfrac{P_{S}|h_{S_{k}E}|^{2}}{P_{T}|h_{TE}|^{2}+N_{0}}.$
(11)

Conditioned on the source has already been selected, E always experience its
intercepted signal power as independent exponentially distributed, hence,
while finding the distribution of $\Gamma_{SE}$ no backhaul reliability
parameter comes into play in (11). However, that is not true for $\Gamma_{SD}$
in (9). The distribution of $\Gamma_{SD}$ will be the mixture distribution of
$\mathbb{I}$ and ${\tilde{\Gamma}}_{SD}$. Now the distribution of
$\Gamma_{SD}$ can be obtained from the mixture distribution due to backhaul
reliability as

$\displaystyle f_{SD}(x)=(1-\Lambda)\delta(x)+\Lambda{\tilde{f}}_{SD},$ (12)

where $f_{SD}(x)$, ${\tilde{f}}_{SD}(x)$ are the PDFs of $\Gamma_{SD}$ and
${\tilde{\Gamma}}_{SD}$, respectively and $\delta(x)$ is delta function. CDF
of $f_{SD}(x)$ can be obtained just by integrating it and finding the CDF of
${\tilde{\Gamma}}_{SD}$.

The CDF of ${\tilde{\Gamma}}_{SD}$ can be evaluate from the definition of CDF
with the help of the CDF of $\max|h_{S_{k}D}|^{2}$ and the PDF of
$|h_{TD}|^{2}$ as

$\displaystyle{\tilde{F}}_{{SD}}(x)$
$\displaystyle=\mathbb{P}\left[\frac{P_{S}\max|h_{S_{k}D}|^{2}}{P_{T}|h_{TD}|^{2}+N_{0}}<x\right]$
$\displaystyle=\mathbb{P}\left[P_{S}\max_{k=1,...,K}|h_{S_{k}D}|^{2}<(P_{T}|h_{TD}|^{2}+N_{0})x\right]$
$\displaystyle=1-\sum_{k=1}^{K}\binom{K}{k}\frac{(-1)^{k+1}\frac{\lambda_{td}\Gamma_{S}}{k\lambda_{sd}\Gamma_{T}}}{x+\frac{\lambda_{td}\Gamma_{S}}{k\lambda_{sd}\Gamma_{T}}}\exp\left(\frac{-k\lambda_{sd}x}{\Gamma_{S}}\right).$
(13)

The CDF of ${\Gamma}_{SD}$ then can be evaluate with the help of (12) as

$\displaystyle{F}_{SD}(x)=1-\Lambda\sum_{k=1}^{K}\binom{K}{k}\frac{(-1)^{k+1}\frac{\lambda_{td}\Gamma_{S}}{k\lambda_{sd}\Gamma_{T}}}{x+\frac{\lambda_{td}\Gamma_{S}}{k\lambda_{sd}\Gamma_{T}}}\exp\left(\frac{-k\lambda_{sd}x}{\Gamma_{S}}\right).$
(14)

The CDF of $\Gamma_{S_{k}E}$ can be obtained from the definition of CDF
similar to ${\tilde{F}}_{SD}$ as

$\displaystyle
F_{{SE}}(x)=1-\frac{\frac{\lambda_{te}\Gamma_{S}}{\lambda_{se}\Gamma_{T}}}{x+\frac{\lambda_{te}\Gamma_{S}}{\lambda_{se}\Gamma_{T}}}\exp\left(\frac{-\lambda_{se}x}{\Gamma_{S}}\right),$
(15)

and the PDF of $\Gamma_{SE}$ can be expressed after differentiating (15) as

$\displaystyle
f_{{SE}}(x)=\frac{\frac{\lambda_{te}}{\Gamma_{T}}\exp\left(\frac{-\lambda_{se}x}{\Gamma_{S}}\right)}{x+\frac{\lambda_{te}\Gamma_{S}}{\lambda_{se}\Gamma_{T}}}+\frac{\frac{\lambda_{te}\Gamma_{S}}{\lambda_{se}\Gamma_{T}}\exp\left(\frac{-\lambda_{se}x}{\Gamma_{S}}\right)}{\left(x+\frac{\lambda_{te}\Gamma_{S}}{\lambda_{se}\Gamma_{T}}\right)^{2}}.$
(16)

##  3 Secrecy Outage Probability

In this section, we investigate the SOP of the secondary network. Towards
deriving those performances, the secrecy capacity is required to be defined
first. The secrecy capacity can be expressed for as

$\displaystyle
C_{S}=\left[\log_{2}(1+\Gamma_{SD})-\log_{2}(1+\Gamma_{E})\right]^{+},$ (17)

where $\log_{2}(1+\Gamma_{SD})$ is the instantaneous capacity at D,
$\log_{2}(1+\Gamma_{E})$ is the instantaneous capacity of the wiretap channel
at E, $\Gamma_{E}=\Gamma_{SE}$ and $[x]^{+}=\max(x,0)$. The SOP is defined as
the probability that the secrecy rate is lower than a certain threshold,
$R_{th}$. The SOP is given as

$\displaystyle\mathcal{P}_{out}(R_{th})$ $\displaystyle=Pr(C_{S}<R_{th})$
$\displaystyle=\mathbb{P}\left[\Gamma_{SD}<R_{th}(1+\Gamma_{E})-1\right]$
$\displaystyle=\int_{0}^{\infty}F_{SD}(\rho(x+1)-1)f_{SE}(x)dx,$ (18)

where $\rho=2^{R_{th}}-1$, $R_{th}$ is the target rate of the secondary
network. Substituting (2.2) and (16) into (3), outage secrecy probability can
be evaluated as

$\displaystyle\mathcal{P}_{out}(R_{th})=1-A\cdot I_{1}-B\cdot I_{2},$ (19)

where $I_{1}$ and $I_{2}$ can be expressed respectively as

$\displaystyle I_{1}=$
$\displaystyle\int_{0}^{\infty}\frac{1}{(x+a)(x+b)}\exp\left(-cx\right)dx,$
(20) $\displaystyle I_{2}=$
$\displaystyle\int_{0}^{\infty}\frac{1}{\left(x+a\right)\left(x+b\right)^{2}}\exp\left(-cx\right)dx,$
(21)

with
$a=\frac{\lambda_{td}\Gamma_{S}+k\rho\lambda_{sd}\Gamma_{T}-k\lambda_{sd}\Gamma_{T}}{k\rho\lambda_{sd}\Gamma_{T}}$,
$b=\frac{\lambda_{te}\Gamma_{S}}{\lambda_{se}\Gamma_{T}}$ and
$c=\frac{k\rho\lambda_{sd}+\lambda_{se}}{\Gamma_{S}}$

$\displaystyle A=$
$\displaystyle\sum_{k=1}^{K}\binom{K}{k}\Lambda(-1)^{k+1}\frac{\lambda_{te}\lambda_{td}\Gamma_{S}}{k\rho\lambda_{sd}\Gamma_{T}^{2}}\exp\left(\frac{-k\lambda_{sd}\left(\rho-1\right)}{\Gamma_{S}}\right),$
(22) $\displaystyle B=$
$\displaystyle\sum_{k=1}^{K}\binom{K}{k}\Lambda(-1)^{k+1}\frac{\lambda_{te}\lambda_{td}\Gamma_{S}^{2}}{k\rho\lambda_{se}\lambda_{sd}\Gamma_{T}^{2}}\exp\left(\frac{-k\lambda_{sd}\left(\rho-1\right)}{\Gamma_{S}}\right).$
(23)

We can utilize the partial fraction to transform multiplication into summation
and solve (20) and (21) using

$\displaystyle\frac{1}{(x+a)(x+b)}=-\frac{1}{(a-b)(x+a)}+\frac{1}{(a-b)(x+b)},$
(24) $\displaystyle\frac{1}{(x+a)(x+b)^{2}}$
$\displaystyle=\frac{1}{(a-b)^{2}(x+a)}-\frac{1}{(a-b)^{2}(x+b)}+\frac{1}{(a-b)(x+b)^{2}}.$
(25)

For the final solution, we have used the integral solution of the form [20],
eq.(3.352.4) and [20], eq.(3.353.3) to get

$\displaystyle I_{1}=$
$\displaystyle\frac{1}{a-b}\exp\left(ac\right)\mathrm{Ei}\left(-ac\right)-\frac{1}{a-b}\exp\left(bc\right)\mathrm{Ei}\left(-bc\right),$
(26) $\displaystyle I_{2}=$
$\displaystyle-\frac{1}{\left(a-b\right)^{2}}\exp\left(ac\right)\mathrm{Ei}\left(-ac\right)+\frac{1}{\left(a-b\right)^{2}}\exp\left(bc\right)\mathrm{Ei}\left(-bc\right)$
$\displaystyle+$
$\displaystyle\frac{1}{a-b}\left(c\exp\left(bc\right)\mathrm{Ei}\left(-bc\right)+\frac{1}{b}\right).$
(27)

##  4 Numerical Results and Discussions

Figure 2:  SOP versus $P_{T}$(dB) for different numbers of secondary users.

In this section, Monte Carlo simulations are provide to validate the
theoretical analyses. Without loss of generality, we assume all nodes are
affected by the same noise power $N_{0}$, and the following parameters are
set: $\beta$=0.5 bits/s/Hz, $R_{th}=0.5$ bits/s/Hz,
$\lambda_{tr}=3,\lambda_{td}=-6,\lambda_{sd}=3,\lambda_{sr}=-3,\lambda_{te}=6,\lambda_{se}=-3$
dB.

Fig. 2 shows the SOP versus $P_{T}$ for different number of small-cell
transmitters, K=2, K=4, and K=6. The network parameters are set as $\Phi=0.1$
and $\Lambda=0.99$. It shows that the analysis match with simulation. It can
be observed that the number of small-cell transmitters strongly affects the
SOP. As the number of smell-cell transmitter increase, SOP improves. However,
the increase of transmitter i.e. K=2 to K=4 has more improvement compared to
k=4 to K=6. As we increase $P_{T}$ SOP decreases first and converges to its
floor after certain values.

Figure 3:  SOP versus $P_{T}$(dB) for different values of $\Lambda$ .

Fig. 3 plots the SOP versus $P_{T}$ for different value of backhaul
reliability, $\Lambda=0.95$ , $\Lambda=0.97$ and $\Lambda=0.99$ with K=6,
$\Phi=0.1$. We observed that the SOP reduces when the $\Lambda$ increases.
This is intuitive that as the reliability of the backhaul link improve of
secrecy also improves.

Figure 4:  SOP versus $P_{T}$(dB) for different values of $\Phi$.

In Fig. 4, the SOP is investigated versus $P_{T}$ with three different value
of primary QoS constraint, $\Phi=0.01$, $\Phi=0.05$ and $\Phi=0.1$ with K=6,
$\Lambda=0.99$. We observed that increasing $\Phi$ result in a reduction in
the SOP. This is because the secondary network are allowed to have higher
transmit power by relaxing the QoS requirement of the primary network.

##  5 Conclusion

In this paper, we have taken into account the backhaul connection reliability
of studying the SOP of underlay cognitive radio network. A proposed selection
scheme enhance the system’s secrecy performance. The small-cell transmitter
power met the desired outage probability. The results have proved that
increasing the primary transmitter’s power and the number of small-cell
transmitter enhance the system’s secrecy performance. In addition, our results
shows that the backhaul reliability and the desired outage probability of the
primary network are important parameter relative to the scaling of the secrecy
performance. Increasing backhaul reliability will result in base station
having higher success rate to connect with small-cell transmitters, and
relaxing the QoS requirement of the primary network will allow small-cell
transmitter to have higher transmit power, which will improve over all secrecy
of the system.

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