IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. XX, NO.YY, MONTH YEAR 1 Optimal and Robust Beamforming for Secure Transmission in MISO Visible-Light Communication Links Ayman Mostafa, Student Member, IEEE, and Lutz Lampe, Senior Member, IEEE Abstract—This work considers secure downlink transmission in indoor multiple-input, single-output (MISO) visible-light com- munication (VLC) links. In particular, we study the design of transmit beamformers that maximize the achievable secrecy rate subject to amplitude constraints imposed by the limited dynamic range of the light-emitting diodes (LEDs). Such constraints render the design problem nonconvex and difficult to solve. We show, however, that this nonconvex problem can be transformed into a solvable quasiconvex line search problem. We also con- sider the more realistic case of imperfect channel information regarding the receiver’s and eavesdropper’s links. We tackle the worst-case secrecy rate maximization problem, again subject to amplitude constraints. In our treatment, uncertainty in the receiver’s channel is due to limited feedback, and is modelled by spherical sets. On the other hand, there is no feedback from the eavesdropper, and the transmitter shall utilize the line-of-sight (LoS) channel gain equation to map the eavesdropper’s nominal location and orientation into an estimate of the channel gain. Thus, we derive uncertainty sets based on inaccurate information regarding the eavesdropper’s location and orientation, as well as the emission pattern of the LEDs. We also consider channel mismatches caused by the uncertain non-line-of-sight (NLoS) components. We provide numerical examples to demonstrate the performance gain of the optimal beamformer compared to suboptimal schemes, and the robust beamformer compared to its non-robust counterparts. We also evaluate the worst-case secrecy rate performance of the robust beamformer in a typical VLC scenario along with the aforementioned uncertainty sources. Index Terms—physical-layer security, visible-light communi- cations, amplitude constraints, robust beamforming, worst-case secrecy rate. I. I NTRODUCTION I NFORMATION-THEORETIC security was pioneered by Wyner back in the mid-1970s with his seminal work [1] that introduced the wiretap channel model and the notion of secrecy capacity as a performance metric for reliable and secure communications. Almost three decades after, inter- est in physical-layer security has been revived by the need for additional secrecy measures that do not jeopardize low complexity at the receiver. In physical-layer security systems, Manuscript received March 7, 2016; revised July 12, 2016; accepted August 12, 2016. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). A preliminary version of a portion of this material was presented at the 2015 ICC Workshop in Visible Light Communications and Networking (VLCN), London, UK, June 2015. c  2016 IEEE. Personal use of this material is permitted. However, permission to use this material for any other purposes must be obtained from the IEEE by sending a request to pubs-permissions@ieee.org. A. Mostafa and L. Lampe are with the Department of Electrical and Computer Engineering, The University of British Colombia, Vancouver, BC, V6T 1Z4, Canada (emails: amostafa@ece.ubc.ca; lampe@ece.ubc.ca). the transmitter exploits dissimilarities among the channels of different receivers and adopts signaling and/or coding schemes that ensure reliable reception by the intended receivers and, at the same time, hinder unintended or unauthorized receivers from inferring the transmitted messages [2]–[5]. Nevertheless, the secrecy performance of physical-layer security systems can severely deteriorate by inaccurate channel information, especially if the unintended receiver is a malicious user or passive eavesdropper attempting to hide its presence from the transmitter. Performance sensitivity, however, can be alleviated by adopting transmission schemes that explicitly take channel uncertainty into account. Such schemes are typically referred to as robust transmission schemes. Visible-light communication (VLC) is a wireless transmis- sion technology that exploits illumination devices, mostly high-brightness light-emitting diodes (LEDs), for short-range data connectivity [6]. In VLC systems, information is relayed by the means of modulating the output intensity of the LEDs, whereas at the receiver side, the data signal is recovered using simple photodetectors. Typical lighting systems utilize multiple LEDs to provide uniform illumination. Thus, VLC systems can readily benefit from multiple-antenna techniques to enhance the reliability and/or security of VLC networks. Being a broadcast channel, data transmitted over VLC links are inherently vulnerable to overhearing by unintended receivers or eavesdroppers existing in the service area illuminated by the transmit LEDs. Therefore, secure transmission in VLC systems using physical-layer security techniques has been proposed in [7]. In this paper, we consider the design of transmit beamform- ers for secure downlink transmission in indoor multiple-input, single-output (MISO) VLC links in the presence of a passive eavesdropper attempting to overhear information conveyed by light waves to the legitimate receiver. Assuming uniform input distribution, our performance measure is the secrecy rate expression derived in [7] for amplitude-constrained wiretap channels. Under the premise of perfect channel information, we first consider the design of optimal beamformers that maximize the achievable secrecy rate subject to amplitude constraints. Such constraints render the optimization problem nonconvex and difficult to solve. Nevertheless, we show that this nonconvex problem can be recast as a solvable quasicon- vex line search problem. Next, we consider the more general and more realistic case in which the transmitter has uncertain information regarding the receiver’s and eavesdropper’s chan- nels. We study the design of robust beamformers that maxi- This is the author's version of an article that has been published in this journal. Changes were made to this version by the publisher prior to publication. The final version of record is available at http://dx.doi.org/10.1109/TSP.2016.2603964 Copyright (c) 2016 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.