Transmitter beamforming and artificial noise with delayed feedback: Secrecy rate and power allocation

Utilizing artificial noise (AN) is a good means to guarantee security against eavesdropping in a multi-input multi-output system, where the AN is designed to lie in the null space of the legitimate receiver´s channel direction information (CDI). However, imperfect CDI will lead to noise leakage at the legitimate receiver and cause significant loss in the achievable secrecy rate. In this paper, we consider a delayed feedback system, and investigate the impact of delayed CDI on security by using a transmit beamforming and AN scheme. By exploiting the Gauss-Markov fading spectrum to model the feedback delay, we derive a closed-form expression of the upper bound on the secrecy rate loss, where N_t = 2. For a moderate number of antennas where N_t >; 2, two special cases, based on the first-order statistics of the noise leakage and large number theory, are explored to approximate the respective upper bounds. In addition, to maintain a constant signal-to-interference-plus-noise ratio degradation, we analyze the corresponding delay constraint. Furthermore, based on the obtained closed-form expression of the lower bound on the achievable secrecy rate, we investigate an optimal power allocation strategy between the information signal and the AN. The analytical and numerical results obtained based on first-order statistics can be regarded as a good approximation of the capacity that can be achieved at the legitimate receiver with a certain number of antennas, N_t. In addition, for a given delay, we show that optimal power allocation is not sensitive to the number of antennas in a high signal-to-noise ratio regime. The simulation results further indicate that the achievable secrecy rate with optimal power allocation can be improved significantly as compared to that with fixed power allocation. In addition, as the delay increases, the ratio of power allocated to the AN should be decreased to reduce the secrecy rate degradation.