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Energy efficiency optimization for MISO wiretap channels with statistical and imperfect CSI

This paper considers an energy-efficient transmit design in a multiple-input single-output (MISO) wiretap channel. In particular, a transmitter sends one confidential message to a legitimate receiver, which must be kept perfectly secure from multiple external eavesdroppers. Assuming partial eavesdroppers' channel state information (ECSI) at the transmitter, we aim to design the transmit beamformer such that the outage secrecy energy efficiency (SEE) is maximized. Two ECSI uncertainty scenarios are studied, i.e., statistical ECSI and imperfect ECSI. For the statistical ECSI case, only the distribution of ECSI is available at the transmitter. The SEE outage constraint admits a closed-form expression in this case, and an approximate solution can be obtained by solving a sequence of convex optimization problems. However, this does not apply to the imperfect ECSI case, where the channel lies in some uncertainty set centered at the true channel. We show that a conservative transmit scheme can be derived by invoking the Bernstein-type inequality. It is proved that our proposed methods always yield a solution of rank one, with either statistical or imperfect CSI. Numerical results are finally presented to confirm the efficacy of our proposed methods.