Impact of Discrete Phase Shifts on RIS-Enabled Covert Communications

This work examines covert communication performance gain achieved by deploying a reconfigurable intelligent surface (RIS) with discrete phase shifts under the scenarios with and without direct transmission links. To this end, we first derive the average receive signal power at the receiver Bob and the warden Willie as functions of the RIS number of elements <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> and the number of bits <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d$</tex-math></inline-formula> for its discrete phase shift quantization level. Our analytical results reveal that Bob's average power is proportional to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N^{2}4^{d}$</tex-math></inline-formula>, while Willie's average power is proportional to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> and does not depend on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d$</tex-math></inline-formula>. These findings provide clear design guidelines for optimizing RIS configurations to improve covert performance via increasing <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$N$</tex-math></inline-formula> or <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d$</tex-math></inline-formula>. To explicitly quantify the performance gain, we analyze the transmission outage probability from the transmitter Alice to Bob and Willie's minimum total detection error probability, which allows us to optimally design Alice's transmit power via minimizing the transmission outage probability subject to the covertness constraint established based on Willie's detection performance, offering practical insights for covert system implementation. Our results also demonstrate that a low-resolution RIS (e.g., <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$d=3$</tex-math></inline-formula>) can achieve covert communication performance that is very close to that of an ideal RIS with infinite phase resolution, highlighting the practical value of deploying low-resolution RIS in covert communication systems.