Dispersed beamforming approach for secure communication in UDN

Abstract

Wireless communication systems are prone to many security breaches due to open nature of the medium and exponential a rise in subscribers. Hence, physical layer security (PLS) has emerged as one of the dominant low complexity alternatives to overcome the impact of eavesdropping by managing the physical characteristics of the medium. In this paper, we ensure PLS to moving users which tends to experience rise in handover, as a result of proximity between users and base station. This study is based on ultra-dense network (UDN). To tackle this challenge, novel secure beamforming named as beam broadening and beam merging have been proposed. Besides, we propose a synchronization approach called synchronized mobility clustering for UDN to reduce the overheads generated due to the exchange of information about moving users. More specifically, we derive an analytical expression for secrecy outage probability—an important security metric. The effect of proposed approaches have been validated through numerical results and the results show the effectiveness of the proposed approaches against eavesdropping. Finally, the performance of the proposed scheme is evaluated and compared with the conventional beamforming approach. However, this proposed approach works well for a varied density of users and location to be targeted.

Appendix

1.1 Proof of Lemma 1

As defined earlier, it can defined as the probability that the secrecy rate falls below a desired threshold. It is an important physical layer security parameter and used widely to determine the performance of the network for a proposed security algorithm. Rewriting (11) to derive the generalized expression for SOP as

$$S_{r} = \hbox{max} \left[ {C_{i} - C_{e} } \right]^{ + } = \hbox{max} \left[ {log_{2} \left( {1 + \gamma_{i} } \right) - log_{2} \left( {1 + \gamma_{e} } \right)} \right]^{ + } ,$$

(26)

$$S_{r} = log_{2} \left( {\frac{{1 + \gamma_{i} }}{{1 + \gamma_{e} }}} \right),$$

(27)

We then express the cumulative distribution function (CDF) of $\gamma_{i}$ as

$$F\left( {\gamma_{i} } \right) = 1 - { \exp }\left( {\frac{{ - \gamma_{i} }}{{\bar{\gamma }_{i} }}} \right),\gamma_{i} > 0$$

(28)

As such, the corresponding PDF of $\gamma_{i}$ can be written as

$$f\left( {\gamma_{i} } \right) = \frac{1}{{\bar{\gamma }_{i} }}\exp \left( {\frac{{ - \gamma_{i} }}{{\bar{\gamma }_{i} }}} \right),$$

(29)

where $\bar{\gamma }_{i}$ is the average SINR of the ith user at a distance of $d_{i,b}$ meters from bth serving BS. The expected $\bar{\gamma }_{i}$ can be expressed as

$$\bar{\gamma }_{i} = \frac{{P_{u} E\left\{ {\left\| {\varvec{h}_{i,b} } \right\|} \right\}}}{{\sigma_{i}^{2} }},$$

(30)

Similarly, the CDF $\left( {F\left( {\gamma_{e} } \right)} \right)$ and PDF $\left( {f\left( {\gamma_{e} } \right)} \right)$ of eve can be written as

$$F\left( {\gamma_{e} } \right) = 1 - { \exp }\left( {\frac{{ - \gamma_{e} }}{{\bar{\gamma }_{e} }}} \right),\gamma_{e} > 0,$$

(31)

$$f\left( {\gamma_{i} } \right) = \frac{1}{{\bar{\gamma }_{e} }}\exp \left( {\frac{{ - \gamma_{e} }}{{\bar{\gamma }_{e} }}} \right),$$

(32)

The variables $\gamma_{i}$ and $\gamma_{e}$ are independent and identically distributed random variable and the joint probability distribution function of $\gamma_{i}$ and $\gamma_{e}$ can represented as

$$f\left( {\gamma_{i} ,\gamma_{e} } \right) = f\left( {\gamma_{i} } \right).f\left( {\gamma_{e} } \right) = \frac{1}{{\bar{\gamma }_{i} \bar{\gamma }_{e} }}\exp \left( { - \frac{{\gamma_{e} }}{{\bar{\gamma }_{e} }} - \frac{{\gamma_{i} }}{{\bar{\gamma }_{i} }}} \right),$$

(33)

From the basic definition of SOP, we have

$$P_{o} \left( {R_{t} } \right) = Pr\left( {S_{r} < \left. {R_{t} } \right|\gamma_{i} } \right),$$

(34)

Putting (27) in above equation, we get

$$= Pr\left( {log_{2} \left( {\frac{{1 + \gamma_{i} }}{{1 + \gamma_{e} }}} \right) < \left. {R_{t} } \right|\gamma_{i} < \gamma_{e} } \right),$$

(35)

On rearranging the above equation, we have

$$= Pr(\gamma_{i} < \left. {2^{{R_{t} }} \left( {1 + \gamma_{e} } \right) - 1} \right|\gamma_{i} < \gamma_{e} ),$$

(36)

$$P_{o} \left( {R_{t} } \right) = 1 - \mathop {\iint }\limits_{{\gamma_{i} < 2^{{R_{t} }} \left( {1 + \gamma_{e} } \right) - 1}}^{{}} f\left( {\gamma_{i} ,\gamma_{e} } \right) d\gamma_{i } d\gamma_{e} ,$$

(37)

$$= \mathop \smallint \limits_{0}^{\infty } \begin{array}{*{20}c} {\frac{1}{{\bar{\gamma }_{i} }}\exp \left( {\frac{{ - \gamma_{i} }}{{\bar{\gamma }_{i} }}} \right)d\gamma_{i} } & {\mathop \smallint \limits_{{\gamma_{i} }}^{{2^{{R_{t} }} \left( {1 + \gamma_{e} } \right) - 1}} \frac{1}{{\bar{\gamma }_{e} }}\exp \left( {\frac{{ - \gamma_{e} }}{{\bar{\gamma }_{e} }}} \right)d\gamma_{e} } \\ \end{array} ,$$

(38)

$$= 1 - \frac{{\bar{\gamma }_{i} }}{{\bar{\gamma }_{i} + \bar{\gamma }_{e} 2^{{R_{t} }} }}\exp \left( { - \frac{{2^{{R_{t} }} - 1}}{{\bar{\gamma }_{i} }}} \right),$$

(39)

Hence, for a given set of values for $\bar{\gamma }_{i}$ and $\bar{\gamma }_{e}$, the resultant expression for secrecy outage is presented as (39).