On the Resilience of Direction-Shift Keying Against Phase Noise and Short Channel Coherence Time at mmWave Frequencies

The rapid variation of the wireless channel (short channel coherence time) and the phase noise are two prominent concerns in Millimeter-wave (mmWave) and sub-Terahertz systems communication systems. Equalizing the channel effect and tracking the phase noise necessitate dense pilot insertion. Direction-Shift Keying (DSK), a recent variant of Spatial Modulation (SM), addresses these challenges by encoding information in the Direction-of-Arrival (DoA) using a distributed antenna system (DAS), rather than relying on amplitude or phase. DSK has been shown to extend coherence time by up to four orders of magnitude. Despite its promise, existing DSK studies are largely simulation-based and limited to simplified roadside unit scenarios and mobile device (MD) equipped with only two antennas. DSK's performance in general settings, along with the fundamental laws governing its behavior, such as coherence time and resilience to phase noise, remain open problems. In this paper, we derive the structure of the optimal detector for the case of $M$-antenna MD. Then, we establish the governing law for DSK's coherence time, termed the Direction Coherence Time (DCT), defining the the temporal duration over which the DoA remains approximately invariant. We analytically establish that DCT scales with $d/v$ (transmitter-receiver distance over velocity), while the Channel Coherence Time (CCT) scales with $\lambda/v$, revealing a coherence time gain on the order of $d/\lambda$ (equivalent to more than four orders of magnitude.) Furthermore, we prove that DSK inherently cancels the phase noise, requiring no additional compensation. Analytical predictions are validated through simulations, confirming the robustness and scalability of DSK in high-frequency mobile environments.