Direction Coherence Time (DCT)

• Direction Coherence Time (DCT) is defined as the duration during which the received signal’s direction-of-arrival (DoA) or timing signature (TDoA) remains invariant, dictated by the geometric relationship between transmitter and receiver.
• DCT scales as d/v (transmitter–receiver distance over velocity) compared to classical channel coherence time's λ/v, offering up to 10,000× longer stability in high-frequency, mobile environments.
• The extended DCT enables significant reductions in pilot overhead and robust phase-noise immunity in Direction-Shift Keying systems, enhancing reliable detection under mobility.

Searching arXiv for papers on Direction Coherence Time and closely related Direction-Shift Keying work. Direction Coherence Time (DCT) is the temporal duration over which the Direction-of-Arrival (DoA) of a received signal remains approximately invariant, or equivalently the interval during which the spatial signature, or TDoA, associated with a given transmitter remains reliably stable at the receiver despite receiver motion. In the context of Direction-Shift Keying (DSK), DCT is introduced as the governing coherence scale for direction-based detection in mmWave and sub-Terahertz systems, where classical Channel Coherence Time (CCT) is short and phase noise is a major impairment. The central analytical result is that DCT scales with transmitter–receiver distance over velocity, $d/v$, whereas CCT scales with wavelength over velocity, $\lambda/v$, implying a coherence-time gain on the order of $d/\lambda$ (Chraiti et al., 1 Sep 2025).

1. Definition and problem setting

DCT is defined as the temporal duration over which the DoA remains approximately invariant. The same definition is expressed in terms of the spatial signature or TDoA associated with a transmitter: as long as that signature changes only slightly, it remains valid for reliable DSK detection. Physically, DCT quantifies how long a mobile device’s TDoA or DoA references for a particular transmitter remain valid before they are significantly perturbed by mobility (Chraiti et al., 1 Sep 2025).

The concept is introduced in a setting where rapid variation of the wireless channel and phase noise are identified as two prominent concerns in mmWave and sub-Terahertz systems. In conventional channel-based schemes, equalizing the channel effect and tracking phase noise necessitate dense pilot insertion. DSK addresses these challenges by encoding information in the DoA using a distributed antenna system (DAS), rather than relying on amplitude or phase. Existing DSK studies are described as largely simulation-based and limited to simplified roadside unit scenarios and mobile devices equipped with only two antennas; the analytical formulation of DCT is presented as addressing the open problems of coherence time and resilience to phase noise in more general settings (Chraiti et al., 1 Sep 2025).

For direction-based signaling, this shifts the relevant invariance notion away from the channel impulse response and toward geometric stability. A plausible implication is that DCT is not merely a renamed CCT, but a distinct timescale attached to direction-preserving propagation geometry.

2. Analytical characterization and scaling laws

The main analytical characterization contrasts DCT with the classical CCT. CCT is the metric that quantifies how long the channel impulse response remains stable, and its scaling is stated as

$t_{\mathrm{CCT}} \propto \frac{\lambda}{v},$

where $\lambda$ is the wavelength and $v$ is the receiver velocity. DCT, by contrast, scales as

$t_{\mathrm{DCT}} \propto \frac{d}{v},$

where $d$ is the distance between transmitter and receiver and $v$ is again the receiver velocity (Chraiti et al., 1 Sep 2025).

The paper also gives explicit bounds:

$$t_{\mathrm{DCT}} \geq \frac{1}{2\pi} J_0^{-1}\left(\frac{1}{\sqrt{2}}\right)\frac{d}{v}\frac{c}{lB},$$

and

$\lambda/v$0

where $\lambda/v$1 is the zeroth-order Bessel function, $\lambda/v$2 is the speed of light, $\lambda/v$3 is the receive antenna element spacing, and $\lambda/v$4 is the signal bandwidth. The corresponding ratio is given as

$\lambda/v$5

This is reported to reveal a coherence-time gain on the order of $\lambda/v$6, equivalent to more than four orders of magnitude, and the paper states that DCT can be up to $\lambda/v$7 longer than CCT (Chraiti et al., 1 Sep 2025).

These expressions make DCT a geometry-dependent coherence measure rather than a wavelength-limited one. In mmWave regimes, where $\lambda/v$8 is small, that distinction becomes especially pronounced.

3. Physical interpretation and comparison with channel coherence time

The physical intuition given for CCT is that it changes as the user moves by a fraction of the wavelength, which at mmWave is only a few millimeters. As a result, vehicular motion can cause the channel to fluctuate at sub-millisecond scales. DCT behaves differently: it changes only when the geometric angle subtended from transmitter to receiver changes significantly, that is, after the user has moved a distance comparable to the link range $\lambda/v$9, which is much larger than $d/\lambda$0 (Chraiti et al., 1 Sep 2025).

The paper attributes the extension of DCT over CCT to the fact that the angular geometry between two distant nodes is much less sensitive to small positional changes than the phase of a multipath channel. This gives DCT a macroscopic geometric basis, whereas CCT is dominated by microscopic phase evolution.

Aspect	Channel Coherence Time (CCT)	Direction Coherence Time (DCT)
Definition	Time channel remains invariant	Time DoA/TDoA remains invariant
Governs	Validity of channel knowledge for SSK/QAM/OFDM	Validity of DoA knowledge for DSK
Scales as	$d/\lambda$1	$d/\lambda$2
Implication	Frequent pilot updates	Far less frequent pilot updates
Impacted by Phase Noise?	Yes	No
Order-of-magnitude	Microseconds at mmWave/car speed	Milliseconds or longer

The paper gives an example in which CCT is short in mmWave, specifically $d/\lambda$3 or 6 microseconds at 30 GHz and 100 km/h, while DCT can be hundreds of milliseconds or longer for comparable settings. This contrast explains why a direction-based signaling architecture can remain operational in high-mobility regimes where channel-based tracking becomes pilot-dominated (Chraiti et al., 1 Sep 2025).

4. Role in Direction-Shift Keying

DSK is described as a recent variant of Spatial Modulation (SM). Its operating principle is to encode information in the direction of signal arrival by switching between spatially distributed antennas at the base station, rather than in amplitude or phase. The receiver then uses TDoA or DoA matching to determine which transmitter was active. In this framework, DCT is the interval during which the relevant TDoA or DoA signatures remain valid for detection (Chraiti et al., 1 Sep 2025).

For the case of an $d/\lambda$4-antenna mobile device, the paper derives the structure of the optimal detector. The summary states that DSK detection is based on maximizing cross-correlations over possible TDoA alignments and is fundamentally independent of channel phase and phase noise. A simplified detector expression is reported as

$d/\lambda$5

Only the stability of the direction-related signature over the DCT interval is required; the detection rule does not rely on coherent channel equalization in the conventional sense (Chraiti et al., 1 Sep 2025).

This ties DCT directly to the signaling architecture. In effect, DCT is the operational validity horizon of the reference geometry used by DSK.

5. Phase-noise resilience, pilot overhead, and performance implications

The paper states that DSK inherently cancels the phase noise and requires no additional compensation. More specifically, the detection metric does not depend on the instantaneous channel phase, which is corrupted by phase noise, but only on timing differences, i.e., TDoA. It is further stated that no CSI or phase tracking is required for DSK, unlike SSK or QAM (Chraiti et al., 1 Sep 2025).

This has direct consequences for overhead. With short CCT, amplitude/phase modulations and classical spatial modulation require dense pilot insertion to track the channel, which is associated with overhead, reduced spectral efficiency, complexity, and vulnerability to phase noise. With long DCT, DSK allows very sparse reference updates, and pilot or reference updates are needed only when DCT expires, which is stated to be 3–4 orders of magnitude less frequent than CCT expiration. The data further reports that SSK may require 20–30% overhead at mmWave, while DSK operates at below 0.1% overhead without performance degradation (Chraiti et al., 1 Sep 2025).

Simulation findings summarized in the source indicate that DSK’s symbol error rate is flat as a function of phase-noise standard deviation, in sharp contrast to SSK, where phase noise drastically degrades performance. The same summary reports that DSK’s SER does not degrade for long intervals between reference updates, nor with increased phase noise or mobility. Taken together, these results identify DCT as the key resource enabling robustness to both mobility and oscillator impairments in the reported system model (Chraiti et al., 1 Sep 2025).

6. Relation to other coherence-time notions

DCT should be distinguished from other coherence-time concepts used in communication theory and physics. In interference networks, “network coherence time” refers to a model in which all channel links share the same coherence pattern, and in the main result of that line of work the case $d/\lambda$6 means that channel coefficients change every symbol time. That notion governs degrees of freedom under perfect CSIR and instantaneous but finite-precision CSIT; it is a network-wide channel-variation parameter rather than a direction-invariance timescale (Davoodi et al., 2017).

A separate use of “coherence time” appears in discrete time crystal experiments, where coherence is assessed through decay and recovery of driven many-body dynamics. There, a DTC echo sequence is used to determine whether apparent signal decay reflects irreversible decoherence or hidden coherences recoverable by echo techniques. In that context, coherence time refers to recoverable phase coherence in a periodically driven spin system, not to the invariance of DoA or TDoA under mobility (Rovny et al., 2018).

This suggests that DCT occupies a specific conceptual niche: it is neither a generic channel stationarity measure nor a many-body lifetime metric, but a geometry-driven coherence measure tailored to direction-based signaling.

7. Significance and open analytical perspective

Within the reported framework, DCT provides the governing law for how long direction references remain valid in a mobile high-frequency link. Its significance lies in the replacement of a wavelength-limited coherence constraint by a distance-limited one: DCT scales with $d/\lambda$7, while CCT scales with $d/\lambda$8. Because $d/\lambda$9 can be $t_{\mathrm{CCT}} \propto \frac{\lambda}{v},$0–$t_{\mathrm{CCT}} \propto \frac{\lambda}{v},$1 in typical mmWave systems, the resulting coherence-time advantage is reported as more than four orders of magnitude, with demonstrations up to $t_{\mathrm{CCT}} \propto \frac{\lambda}{v},$2 (Chraiti et al., 1 Sep 2025).

The same source also frames DCT as answering open problems left by prior DSK literature, which had been largely simulation-based and confined to simplified settings. By deriving the detector structure for the $t_{\mathrm{CCT}} \propto \frac{\lambda}{v},$3-antenna mobile device case, analytically establishing the DCT scaling law, and proving phase-noise immunity, the work positions DCT as the fundamental timescale for direction/angle-based wireless schemes in high-frequency mobile environments (Chraiti et al., 1 Sep 2025).

A plausible implication is that future direction-based system design will treat DCT as the primary refresh interval for geometry-dependent references, in contrast to CCT-driven pilot scheduling in conventional channel-based architectures.