IEEE TRANSACTIONS ON SIGNAL PROCESSING, ACCEPTED FOR PUBLICATION, AUGUST 2017 1 On Linear Precoding for the Two-User MISO Broadcast Channel with Confidential Messages and Per-Antenna Constraints Ayman Mostafa, Member, IEEE, and Lutz Lampe, Senior Member, IEEE Abstract—We study the design of linear precoders for secure transmission in the two-user multiple-input single-output (MISO) broadcast channel with confidential messages (BC-CM). The transmitter has multiple antennas, and each user has a single receive antenna. Two independent messages are simultaneously transmitted, one intended for each user, and each message should be kept confidential from the other user. Assuming real- valued transmitted signals, we design the linear precoders subject to total and per-antenna average power constraints, and also subject to amplitude constraints. In both cases, we tackle the design problem via weighted secrecy sum rate maximization. The resulting problem, however, involves a fractional objective, making it nonconvex and difficult to solve. Nevertheless, we show that this difficult problem can be transformed into a more tractable problem, for which a solution can be obtained by an iterative search algorithm. In addition, we characterize a condition under which the obtained solution is guaranteed to be optimal. Furthermore, we show that the problem formulation and solution approach can be easily extended to handle the robust version of the design problem with uncertain channel information. We provide numerical examples to demonstrate the performance of the proposed precoder in terms of the achievable secrecy rate regions subject to the aforementioned constraints. We also demonstrate the performance of the robust precoder under different channel uncertainty levels. Index Terms—Amplitude constraint, MISO broadcast channel with confidential messages, per-antenna power constraint, robust linear precoding, secrecy rate region. I. I NTRODUCTION T HE foundations of information-theoretic security were laid down by Wyner in his seminal paper [1] that stud- ied the problem of secret communication over the degraded broadcast channel. In that paper, Wyner introduced the so- called wiretap channel model to describe the scenario in which the transmitter has a secret message intended for one receiver, while the other receiver, whose channel is degraded, acts only as an eavesdropper. Wyner also proposed the notion of secrecy capacity as a performance measure that specifies the maximum communication rate that guarantees reliable reception of the secret message by the intended receiver, and entire hiddenness from the eavesdropper. This motivated the Manuscript received January 9, 2017; revised June 15, 2017 and July 28, 2017; accepted August 5, 2017. The associate editor coordinating the review of this manuscript and approving it for publication was Dr. Yue Rong. This work was supported by the Natural Sciences and Engineering Research Council of Canada (NSERC). 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). characterization of the secrecy capacity of the scalar Gaussian wiretap channel [2]. Wyner’s model was then extended to the (nondegraded) broadcast channel in which the eavesdropper’s channel need not be degraded [3]. Such an extension has ultimately led to the characterization of the secrecy capacity of the multiple-input single-output (MISO) and multiple-input multiple-output (MIMO) Gaussian wiretap channels [4], [5]. The wiretap channel model was further extended in [6] to the two-user broadcast channel with confidential messages (BC-CM), and the secrecy capacity region was adopted as the performance measure. This model captures the practically relevant scenario in which the transmitter has two independent secret messages, one intended for each receiver, and each message should be kept confidential from the other receiver. Achievability of the secrecy capacity region of the two-user MISO BC-CM was established in [7] using the so-called secret dirty-paper coding (S-DPC) scheme under the total (average) power constraint. This coding scheme was then extended in [8] to the MIMO BC-CM, and it was shown that the secrecy capacity region is rectangular under the matrix power (or input covariance matrix) constraint. Under the total power constraint, however, the secrecy capacity region can be only found by performing an exhaustive search over the set of all input covariance matrices that satisfy the total power constraint. Due to the complexity of S-DPC and the lack of a simple solution to the practical case of total power constraint, the authors in [9] proposed a low-complexity linear precoding scheme for the two-user MIMO BC-CM based on generalized singular value decomposition. The work in [10] also characterized a secrecy rate region for the two-user MIMO BC-CM under the total power constraint via formulating a nonconvex weighted secrecy sum rate maximization problem. An iterative algorithm based on a block successive lower- bound maximization method was proposed to solve such a nonconvex problem. In practical multiple-antenna systems, each antenna element is equipped with a separate power amplifier. Therefore, per- antenna power constraints are often necessary to model hard- ware limitations in practical systems. With such constraints, however, the design problem may become more difficult to handle. Nonetheless, several works in the literature have considered the transmitter optimization problem subject to per-antenna power constraints [11]–[15]. In [11], the authors considered the design of zero-forcing (ZF) linear precoders for weighted sum rate maximization in the multi-user MISO broadcast channel. For the multi-user MIMO case, the authors 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.2017.2745454 Copyright (c) 2017 IEEE. Personal use is permitted. For any other purposes, permission must be obtained from the IEEE by emailing pubs-permissions@ieee.org.