Phase-Only Zero Forcing for Secure Communication With Multiple Antennas

Artificial noise (AN) transmission can enhance the secrecy in multiantenna wireless channels by superimposing judiciously constructed synthetic noise signals over information signals during transmission. Motivated by the large-scale multiple-input multiple-output systems, we study AN transmission under a constraint that all elements (weights) in each beamforming vector must have a constant magnitude, but can have arbitrary phases. For the special case of one AN beamforming vector and one legitimate receiver, we derive a necessary and sufficient condition for finding a beamforming vector in the null space of the legitimate receiver's channel and provide a geometric interpretation. For the independent identically distributed Rayleigh fading channel, we derive an approximate expression for the probability of failing to find such a beamforming vector, and show that it decreases exponentially in the square of the number of transmit antennas. For the general case, we propose a numerical algorithm for obtaining a set of mutually orthogonal AN beamforming vectors in the null space of all the receivers. Our approach involves reducing the problem to an unconstrained nonlinear programming problem, which is then solved using the Gauss-Newton method. We show numerically that our proposed algorithm performs significantly better than a heuristic relaxation approach. Finally, for a multiantenna system with multiple RF chains, we show numerically that the secrecy rate achieved by our proposed approach is close to that achieved by the AN transmission with unconstrained beamforming, when the number of transmit antennas is sufficiently large.