International Journal of Antennas and Propagation

Volume 2015 (2015), Article ID 594791, 12 pages

http://dx.doi.org/10.1155/2015/594791

## Aligned Space Time Block Codes for the 2-User X Channel with Secrecy Constraints

Department of EEC Engineering, Korea Tech, Cheonan 330-708, Republic of Korea

Received 18 March 2015; Revised 29 June 2015; Accepted 1 July 2015

Academic Editor: Junping Geng

Copyright © 2015 Manar Mohaisen. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Interference alignment (IA) is a technique used to reduce the dimension of the interference, where consequently the multiplexing rate is increased. In the 2-user X channel, combining IA with space-time block codes increases the diversity gain. These gains are achieved with the cost of leaked information at unintended receivers, where this leaked information can be used to decode other receiver’s signals. In this paper, we consider each of the two two-antenna receivers as an eavesdropper with 1 or 2 additional eavesdropping antennas. As such, we suggest receiver structures to answer the question: “Is the leaked information sufficient to properly decode the unintended signals?” besides quantifying the leaked information in terms of secrecy sum rates (SSR). Interestingly, we show that the SSR is negative, indicating that the quality of the eavesdropped signals is superior to that of the intended signals. To assure confidentiality, we propose an interleaved multiple rotation-based transformation scheme that neutralizes any a priori knowledge about the structure of the eavesdropped information and rotates the transmitted symbols using orthogonal matrices, preserving both the power and the distance between symbols.

#### 1. Introduction

In wireless communications systems, interference plays a major role in defining the achievable performance and capacity [1]. In conventional receivers, in multiuser scenarios, the interference is either ignored, hence considered as an additional noise, or jointly decoded via employing successive interference cancellation (SIC) detectors [2–4]. In both cases, the dimensions of the interference remain the same, leading to degraded performance and diversity gain in the first case, while powerful algorithms should be employed in the case of SIC algorithms so as to avoid degradation in the performance due to interference.

Interference alignment is a transmission technique used to reduce the dimensions of the interference while maintaining the useful signals discernible at the intended receivers. This is achievable by precoding the transmitter signals such that the interference is aligned at unintended receivers [5]. As such, interference is removed at the intended receivers using simple mathematical operations leading to an interference-free system, where appropriate decoding algorithms can then be used to decode the useful signals. In [5], Jafar and Shamai proposed a linear alignment algorithm for the two-user X channel, which achieves the maximum data rate of () symbols/channel use and a diversity gain of 1, with as the number of transmit antennas.

In addition to the multiplexing gain, quantified by the unit symbols/channel use, the diversity gain is an important measure of the system performance. When the channel is in deep fading, systems with unity diversity gain suffer from low signal-to-noise ratio (SNR) at the receiver side, leading to degradation in the bit-error rate (BER). Several diversity techniques have been proposed in the literature to explore further diversity gain [6–8]. In [9], a technique that combines interference alignment in X channel and Alamouti diversity scheme with two transmit antennas has been proposed to achieve the maximum multiplexing gain of () and the full diversity gain of 2, which is equal to the number of antennas at each of the four nodes. Furthermore, the proposed scheme inherits the space-time orthogonality of the Alamouti algorithm, and hence a simple linear receiver, that avoids computationally complex matrix inversion is required to achieve the aforementioned gains.

In analogy to other multiuser communication systems with interuser interference [10–12], keeping confidentiality arises as one of the main challenges in the two-user X channel system with interference alignment. In such a system, each receiver can be seen as an internal eavesdropper that, besides decoding its intended symbols, it uses the leaked information to decode other receiver’s intended symbols. The accuracy of decoding the unintended symbols depends on the number of additional spatial resources available at the eavesdropper.

In [13], the eavesdropper is an external agent and the system is modeled as a wiretap channel. However, in the X-channel with interference alignment system considered in this paper, the eavesdropper is the other intended receiver in the X-channel system. While in [13] it is possible to design the precoding matrices in order to deprive the eavesdropper of the capability of decoding the unintended symbols, and it is impossible to do so in the case of the X-channel system since both transmitters are employing joint precoding as will be explained later. We conclude therefore that the work introduced in [13], though very solid, cannot be applied to the case of the X-channel with interference alignment.

Another related work was introduced in [14], where authors proposed a secrecy algorithm which can be only applied in time-division duplex (TDD) systems because authors make use of the channel reciprocity principle. Another shortcoming of the proposed algorithm in [14] is that it is mainly based on the received signal strength indication (RSSI) which is inaccurate and insecure. The RSSI of users that have totally independent channels might be the same especially in indoor pico- or microcells scenarios, where they are so common in the long-term evolution (LTE) system. The drawbacks of using the RSSI in communication systems are outlined in [15] based on experimental study.

The merits of this paper are summarized as follows:(1)Unlike in conventional works [11, 12] where only the amount of leaked information is examined and therefore is given in terms of secrecy sum rate (SSR) values, we go beyond this first stage by answering the question: “Is the leaked information sufficient to decode the unintended signals?” To this end, we investigate the receiver structures in the case of a single and two additional eavesdropping antennas. The BER performance is then evaluated for both the intended and unintended signals.(2)Based on the obtained receiver structures, the mutual information and the SSRs are derived taking into consideration the information used in the decoding stage.(3)To render useless the leaked information about other receiver’s signals, we propose an interleaved multiple rotation-based transformation (IMRBT) algorithm that consists of two stages, namely, interleaving stage and rotation stage. In the interleaving stage, symbols are interleaved so that any a priori information about the structure of the eavesdropped signals becomes useless. Then, interleaved symbols are rotated using orthonormal matrices such that both the power and the distance between symbols in the Euclidean space are kept intact.

The rest of the paper is organized as follows. In Section 2, we introduce the system model and review related works. In Section 3, we investigate the decoding capabilities of the unintended signals at the eavesdropper, in the cases of no additional, single additional, and two additional eavesdropping antennas. In Section 4, we derive the SSRs of the intended symbols and the unintended symbols. We introduce the proposed IMRBT scheme in Section 5 and present simulation results in Section 6. Finally, we draw conclusions in Section 7.

We briefly introduce the notations used in this paper. We employ boldface uppercase letters for matrices and boldface lowercase letters for vectors. The superscripts , , and denote transpose, conjugate transpose, and conjugate, respectively. is a circular symmetric complex Gaussian random variable with mean and variance . Finally, is the probability of .

#### 2. System Model and Previous Work

##### 2.1. System Model

Consider a two-user X channel with eavesdropping as depicted in Figure 1. Each transmitter has independent and* confidential* symbols for each of the receivers. These symbols are drawn independently from a finite modulation set . Transmitter 1 has and intended for receiver 1 and receiver 2, respectively. In , the superscript denotes the index of the symbol, the first subscript denotes the index of the transmitter, and the second subscript denotes the index of the intended receiver. Likewise, transmitter 2 has and intended for receiver 1 and receiver 2, respectively. Vectors , for , are encoded using the space-time block coder (STBC) block to generate the matrices , for . Finally, encoded symbols are beamformed and linearly combined to generate block codes , for , with denoting the number of channel uses. In the deployed scenario, each receiver is equipped with * legal* receive antennas and * eavesdropping* receive antennas. To denote the channels between the transmitters and the legal receive antennas, we use , , , and to denote the matrices coupling transmitter 1 and receiver 1, transmitter 2 and receiver 1, transmitter 1 and receiver 2, and transmitter 2 and receiver 2, respectively. While employing receive antennas at each receiver is sufficient to recover its intended symbols, extra eavesdropping antennas are required to leak more information about other receiver’s symbols, so that efficient decoding is achieved. To denote the channels between the transmitters and the eavesdropping receive antennas, we use , , , and to denote the channels between transmitter 1 and receiver 1, transmitter 2 and receiver 1, transmitter 1 and receiver 2, and transmitter 2 and receiver 2, respectively. The elements in the channel matrices in Figure 1 are independently and identically distributed (i.i.d.) circular Gaussian random variables, . These matrices were pseudorandomly generated following the aforementioned characteristics. signal matrices received at the legal antennas of receiver 1 and receiver 2, respectively, are given bySimilarly, the received signal matrices at the eavesdropping antennas of receiver 1 and receiver 2 are given byEntries in the additive white Gaussian noise (AWGN) matrices, , , , and , are i.i.d. , where and denotes the SNR.