Optimal Pulse-Position Modulation Order and Transmit Power in Covert Communications

This work tackles the joint optimization of the modulation order L and transmit power P of pulse-position modulation (PPM) in covert communications. Specifically, we consider two scenarios where L used by the PPM at a transmitter Alice for communicating with a receiver Bob is known and unknown to the warden Willie, respectively. In the former scenario, our numerical examination obtained based on our analysis indicates that the optimal L is generally 2, which is mainly due to the fact that Willie's detection performance increases with L when he knows it. Interestingly, in the latter scenario, we show that the optimal L is 2 when the covertness constraint is strict, while it is 4 when the covertness constraint is relaxed. The main reason is that Willie's detection performance does not depend on L when he does not know it, and the bit error rate for L=2 is close to that for L=4 when the signal-to-noise ratio is extremely low caused by the strict covertness constraint, while L=2 requires less bandwidth and may achieve a higher bandwidth efficiency. Furthermore, we theoretically prove that the covert communication performance is better when Willie does not know L relative to when he knows L.