---
title: Direction-Shift Keying (DSK)
url: https://www.emergentmind.com/topics/direction-shift-keying-dsk
type: topic
---

# Direction-Shift Keying (DSK)

Searching arXiv for recent papers on Direction-Shift Keying and closely related usage of the acronym DSK.
Direction-Shift Keying (DSK) is a spatial-modulation variant in which information is encoded in the Direction-of-Arrival (DoA) of the received signal rather than in its amplitude or phase. In the formulation developed for mmWave systems, a distributed antenna system (DAS) at the base station activates exactly one remote radio unit or transmit antenna per symbol interval, and a multi-antenna mobile device identifies the active transmitter by matching the observed spatial signature—specifically, inter-antenna time differences (TDoA)—to a pre-known library of direction templates [2509.01103]. A second, conceptually related realization appears in dual-RIS indoor routing, where source bits select a receive direction or zone through RIS phase control; in that setting, DSK is effectively realized as index modulation over spatial directions and is operationally equivalent to SSK-based detection at the receiver [2511.18610]. The term therefore denotes a class of schemes in which the conveyed information is the selected propagation direction, beam, or directional index.

## 1. Conceptual definition and scope

DSK encodes bits in a directional index. In the mmWave DAS formulation, the directional index is the active transmit site whose geometry induces a distinct DoA/TDoA fingerprint at the receiver; in the dual-RIS formulation, the directional index is the selected receive antenna or indoor zone toward which the signal is coherently steered by the second RIS [2509.01103]. In both cases, information is carried by “which direction” is activated rather than by conventional amplitude/phase modulation at the transmitter [2511.18610].

This places DSK within the broader family of index modulation. The dual-RIS paper explicitly interprets the method as “bit-dependent direction selection at the physical layer,” with the selected receive antenna acting as the index and with \(\log_2(N_r)\) bits encoded through the choice of one among \(N_r\) receive directions [2511.18610]. A plausible implication is that DSK can be viewed as a directional specialization of spatial or beam-index modulation whenever a controllable physical mechanism maps bits to a propagation direction.

A recurrent source of confusion is acronym overlap. In [2603.02721], “DSK” denotes Doppler Shift Keying rather than Direction-Shift Keying. That work concerns delay–Doppler domain modulation and keys information over Doppler bins, not over physical propagation directions. The shared acronym does not indicate a shared modulation principle beyond the generic idea of index selection.

## 2. Canonical mmWave DSK with distributed antenna systems

In the mmWave formulation, the base station comprises \(M_t\) spatially distributed transmit antennas at positions \(b_m = [b_{x,m}, b_{y,m}]\), while the mobile device has \(M\) receive antennas at known offsets \(a_n\) from its geometric center [2509.01103]. During each symbol period, exactly one base-station antenna is active, so the transmitted waveform is

\[
x_m(t) = \begin{cases}
s(t), & m = v,\\
0, & \text{otherwise.}
\end{cases}
\]

where \(v \in \{1,\dots,M_t\}\) is the active index and \(s(t)\) is a known, unmodulated pulse of duration \(T\) and power \(P\) [2509.01103].

The received complex baseband signal at the \(n\)-th mobile-device antenna is modeled as

\[
r_n(t) = \rho\, e^{-j 2\pi f_c \tau_n^v}\, e^{-j 2\pi \Delta f\, t}\, s\big(t - \tau_n^v\big) + w_n(t),
\]

with \(w_n(t)\sim\mathcal{CN}(0,\sigma^2)\), \(\tau_n^v\) the propagation delay from the active transmitter to receive antenna \(n\), and \(\rho\) the path gain [2509.01103]. Defining

\[
\alpha_n^v \triangleq \rho\, e^{-j 2\pi f_c \tau_n^v}\, e^{-j 2\pi \Delta f\, t},
\]

the model becomes

\[
r_n(t) = \alpha_n^v\, s(t-\tau_n^v) + w_n(t).
\]

The receiver does not estimate full complex channel coefficients in the usual CSI sense. Instead, it uses precomputed TDoA vectors

\[
\boldsymbol{\Delta \tau}^m = \big[\tau_2^m-\tau_1^m,\, \dots,\, \tau_M^m-\tau_1^m\big],
\]

from which all pairwise delays \(\Delta\tau_{i,j}^m = \tau_i^m - \tau_j^m\) are determined [2509.01103]. These vectors serve as direction templates or DoA fingerprints. The far-field assumption connects geometry and delay structure, and the strongest path governs the arrival delays.

The same paper gives a conventional steering-vector form for a mobile-device ULA,

\[
a_r(\theta) = \big[1,\ e^{-j k d_r \sin\theta},\ e^{-j 2 k d_r \sin\theta},\ \dots,\ e^{-j (M-1) k d_r \sin\theta}\big]^T,
\]

with \(k=2\pi/\lambda\), while emphasizing that DSK uses the equivalent time-domain delay representation

\[
\frac{\alpha_i^m}{\alpha_j^m} = e^{-j 2\pi f_c\, \Delta\tau_{i,j}^m}.
\]

Under LoS, \(\tau_n^m \approx \|b_m - (ā+a_n)\|/c\); under single-bounce NLoS, the image method is used to define the path length and hence the delay [2509.01103].

## 3. Detection principle and phase-noise invariance

For hypothesis \(H_m\), stating that the \(m\)-th transmitter is active, the received signals satisfy \(r_n(t) \sim \mathcal{CN}\big(\alpha_n^m s(t - \tau_n^m),\sigma^2\big)\). The optimal detector derived for an \(M\)-antenna mobile device reduces to a weighted sum of TDoA-aligned pairwise cross-correlations [2509.01103]:

\[
\tilde{m} = \arg\max_{m\in\{1,\dots,M_t\}} \sum_{i=2}^{M} \frac{1}{i-1}\,\sum_{j=1}^{i-1} \mathrm{Re}\left\{ \int \frac{\alpha_i^m}{\alpha_j^m}\, R_{i,j}^m(t)\, dt \right\},
\]

where

\[
R_{i,j}^m(t) \triangleq r_j\big(t - \Delta \tau_{i,j}^m\big)\, r_i^*(t), \quad \frac{\alpha_i^m}{\alpha_j^m} = e^{-j 2\pi f_c\, \Delta\tau_{i,j}^m}.
\]

Under the true hypothesis and in the noiseless case, the aligned correlation term becomes

\[
\mathrm{Re}\left\{\int \frac{\alpha_i^m}{\alpha_j^m}\, R_{i,j}^m(t)\, dt\right\} = \rho^2\, E_s,\quad E_s=\int |s(t)|^2\, dt,
\]

whereas under a false hypothesis the TDoA mismatch drives the inner product toward zero as the mismatch grows [2509.01103]. The detector therefore operates by measuring consistency between the received spatial signature and the candidate directional template.

A central analytical result is that DSK is invariant to common phase rotation. The received signal includes the factor \(e^{-j 2\pi \Delta f\, t}\), yet in the ratio

\[
\frac{\alpha_i^m}{\alpha_j^m}
= \frac{\rho\, e^{-j 2\pi f_c \tau_i^m}\, e^{-j 2\pi \Delta f\, t}}
{\rho\, e^{-j 2\pi f_c \tau_j^m}\, e^{-j 2\pi \Delta f\, t}}
= e^{-j 2\pi f_c\, (\tau_i^m - \tau_j^m)},
\]

the phase-noise and frequency-offset term cancels exactly [2509.01103]. The decision statistic thus depends only on relative delays or relative spatial phase implied by geometry, not on common oscillator phase. The paper consequently states that DSK “inherently cancels the phase noise, requiring no additional compensation” [2509.01103].

This detector structure clarifies the distinction between DSK and conventional SSK. SSK also conveys information via an index, but the cited mmWave analysis argues that SSK still requires reliable CSI per antenna, whereas DSK uses only the spatial signature or TDoA and thus avoids explicit channel equalization and phase tracking [2509.01103].

## 4. Direction Coherence Time and mobility scaling

The principal analytical contribution of the mmWave study is the introduction of Direction Coherence Time (DCT), defined as the temporal duration over which the DoA/TDoA vector remains approximately invariant [2509.01103]. This is contrasted with the conventional Channel Coherence Time (CCT), which measures persistence of the complex channel itself.

For CCT, the coherence function under phase noise is given as

\[
J_{\mathrm{CCT}}(t_c) = \frac{B - |\Delta f - \Delta f'|}{B}\, \left| J_0\!\left(2\pi\, \frac{t_c v}{\lambda}\right)\right|,
\]

and, using the threshold \(|J|^2 \ge 0.5\), one obtains

\[
t_{\mathrm{CCT}} \simeq \frac{1}{2\pi}\,\frac{\lambda}{v}\; J_0^{-1}\!\left(\frac{1}{\sqrt{2}\, \sqrt{\frac{B}{B - |\Delta f - \Delta f'|}}}\right)
\;\le\; \frac{9}{16\pi}\,\frac{\lambda}{v}.
\]

Hence CCT scales linearly with wavelength \(\lambda\) and inversely with velocity \(v\) [2509.01103].

For DCT, the paper derives an exact direction-coherence function and a lower bound. The lower bound is

\[
J_{\mathrm{DCT}}(t_c) \;\ge\; \Big|J_0\!\big(2\pi\, \frac{l B}{c}\, \frac{t_c v}{d}\big)\Big|,
\]

which leads to

\[
t_{\mathrm{DCT}} \;\ge\; \frac{9}{16\pi}\,\frac{d}{v}\,\frac{c}{l B}.
\]

The governing scaling law is therefore \(t_{\mathrm{DCT}} \propto d/v\), whereas \(t_{\mathrm{CCT}} \propto \lambda/v\) [2509.01103]. The resulting coherence-time gain ratio is expressed as

\[
\frac{t_{\mathrm{DCT}}}{t_{\mathrm{CCT}}} \approx \frac{d}{\lambda}\,\frac{c}{l B}.
\]

The paper states that this reveals “a coherence time gain on the order of \(d/\lambda\) (equivalent to more than four orders of magnitude.)” and gives an example at \(f=60\) GHz, \(d=100\) m, \(l=0.1\) m, and \(B=100\) MHz, for which the ratio is approximately \(6\times 10^5\) [2509.01103].

This suggests that DSK’s robustness in high-frequency mobile settings is fundamentally geometric: directional signatures evolve with motion at a rate governed by the transmitter–receiver distance, while complex carrier phase evolves at the wavelength scale.

## 5. RIS-based directional realization and relation to SSK

A second realization of DSK appears in the dual-RIS indoor architecture of [2511.18610]. The setup consists of a single-antenna transmitter near RIS\(_1\), a second RIS\(_2\) equipped with a low-power controller, and a receiver with \(N_r\) antennas. There is no LoS between transmitter and receiver, while a LoS is assumed between RIS\(_1\) and RIS\(_2\). Both RISs have \(N\) passive reflecting elements [2511.18610].

The paper’s central mechanism is “bit-controlled intelligent signal routing.” RIS\(_1\) passively reflects the incident signal, whereas RIS\(_2\) is dynamically configured “based on source data bits” to steer the signal toward a specific receiver or indoor zone. In the stated implementation, RIS\(_2\) “performs antenna selection by adjusting its phase profile such that the signal is constructively directed to one of the \(N_r\) receive antennas, as determined by incoming information bits” [2511.18610]. This is described as a direct instantiation of DSK, since bits select a transmission direction and information is carried by the activated direction.

The channel matrices are

\[
\mathbf{G} \in \mathbb{C}^{N \times N}, \qquad \mathbf{H} \in \mathbb{C}^{N_r \times N},
\]

with element-wise forms

\[
g_{k,i}=\alpha_{k,i}e^{-j \psi_{k,i}}, \qquad h_{i,p} = \beta_{i,p}e^{-j \theta_{i,p}},
\]

and Rician fading on both hops [2511.18610]. For a selected antenna \(m \in \{1,\ldots,N_r\}\),

\[
\mathbf{h}_{m} = \begin{bmatrix} h_{1,m}, & \cdots, & h_{N,m} \end{bmatrix}.
\]

RIS phases are designed to maximize SNR at the selected direction via coherent alignment across the cascaded channel, using the effective channel

\[
\mathbf{h}_{\text{eff}} = \mathbf{G} \mathbf{h}_m^{\mathrm{T}},
\]

and the closed-form phase rule

\[
\phi_{i}^{\text{opt}} = - \left( \psi_{k,i} + \theta_{i,m} \right), \quad \forall i \in \{ 1, \dots, N \}.
\]

The received signal and per-direction SNR are then

\[
y_m = \sqrt{E_s} \zeta\mathbf{h}_m \mathbf{G} \boldsymbol{\Theta}_i^{\text{opt}} + n_{m},
\]

\[
\gamma_m = \frac{\zeta^2E_s\left| \sum_{k=1}^{N} \sum_{i=1}^{N} \alpha_{k,i} \beta_{i,m} \right|^2}{N_0},
\]

with

\[
\zeta=\left( \frac{\lambda_c}{4 \pi d_1d_2} \right)^2.
\]

Detection follows SSK principles: the natural decision rule implied by the model is

\[
\hat{m} = \argmax_{m \in \{1,\dots,N_r\}} \gamma_m = \argmax_m |y_m|^2.
\]

The paper explicitly states that “SSK modulation through RIS control” encodes \(\log_2(N_r)\) bits by selecting one of the receiver antennas, so here DSK and SSK are equivalent realizations of index modulation with the selected receive direction as the index [2511.18610]. Unlike conventional directional modulation driven at the transmitter RF chain, this realization is achieved by passive beamforming at RIS\(_2\) with extremely low energy consumption.

## 6. Performance characteristics, comparisons, and limitations

The mmWave DSK study validates several robustness claims through simulations. In static regimes without phase noise and with perfect CSI available to SSK, DSK’s SER versus SNR is reported to be close to SSK for \(M_t=4\) and nearly indistinguishable for \(M_t=16\) in the circular DAS geometry [2509.01103]. Under mobility, SSK degrades sharply once the update time exceeds the CCT, whereas DSK remains flat until the DCT. For the 30 GHz case, the paper reports \( \mathrm{CCT} \approx 1.9\times 10^{-4}\,\mathrm{s}\) at \(v=30\) km/h, while DCT occurs near \(0.3\) s, corresponding to roughly three orders of magnitude longer persistence [2509.01103]. Under phase noise, DSK is described as essentially flat across five decades of phase-noise standard deviation, while SSK exhibits a severe SER floor.

The same work also links DSK to pilot overhead. Conventional CSI-dependent schemes tied to CCT require frequent updates; by contrast, DSK can refresh at the DCT scale. In the roadside-unit scenario, DSK maintains SER \(\approx 10^{-2}\) while overhead is reduced to \(\approx 0.02\%\), whereas SSK maintains low SER only when overhead is high, around \(\approx 20\%\)–\(44\%\) [2509.01103].

In the dual-RIS realization, the performance metrics are outage probability and capacity. The outage probability is

\[
P_{\text{out}} = \Pr[\gamma_m \leq \gamma_{\text{out}}],
\]

and the single-stream capacity is

\[
C(\gamma_m) = \log_2\!\left(1 + \gamma_m \right), \qquad
C_{\text{avg}} = \mathbb{E}_{\gamma_m}\!\left[ \log_2\!\left(1 + \gamma_m \right) \right].
\]

For large \(N\), the deterministic equivalent is

\[
C_{\text{avg}} \approx \log_2\!\left(1 + \bar{\gamma}_m \right) = \log_2\!\left(1 + \frac{\zeta^2 E_s \mu_Z^2}{N_0} \right).
\]

The reported simulation trends are specific. With \(N = 64, 128, 256\), \(f_c = 3\) GHz, \(d=10\) m, \(N_r=2\), and \(\gamma_{\text{out}} = 10\) dB, “the outage probability decreases significantly as the number of reflecting elements increases,” with \(N=256\) best [2511.18610]. With \(N = 64\), \(N_r=2\), and \(f_c = 3\) GHz versus \(10\) GHz, “the 3 GHz frequency consistently exhibits the lowest outage probability,” while higher frequencies yield higher outage due to greater path loss and absorption. Capacity similarly improves with increasing \(N\), and declines with distance for \(d = 5,10,20\) m. However, in the reported comparison between dual-RIS and single-RIS outage for \(N = 128\), \(N_r=2\), \(d=10\) m, \(f_c=3\) GHz, and \(\gamma_{\text{out}} = 10\) dB, “single-RIS consistently exhibits the lowest outage probability” under the tested conditions [2511.18610].

Several limitations are explicit across the two formulations. The dual-RIS analysis assumes perfect CSI, continuous ideal phase shifts, and a quasi-static indoor environment; discrete quantization, mutual coupling, calibration, switching speed, signaling overhead, and channel estimation mechanisms are not quantified [2511.18610]. The mmWave DSK analysis assumes a single dominant ray, far-field geometry, known or precomputed TDoA templates, and timing synchronization plus geometric calibration [2509.01103]. Both studies therefore characterize favorable operating regimes rather than universally resolved deployment conditions.

A common misconception is that DSK necessarily dominates SSK. The available evidence is more qualified. In the mmWave DAS setting, DSK is robust to short coherence time and phase noise relative to CSI-dependent SSK [2509.01103]. In the dual-RIS indoor setting, the DSK/SSK-like routing mechanism offers flexible direction control, but a single-RIS benchmark shows lower outage probability than dual-RIS in the reported tests [2511.18610]. The comparative outcome depends on which impairment is dominant: CSI aging and phase noise in one case, or multi-hop coordination and path loss in the other.

## 7. Terminological extensions and open research directions

Within the supplied literature, DSK denotes distinct concepts in different subfields. The mmWave work defines DSK as Direction-Shift Keying and grounds it in DoA/TDoA-based spatial detection with DAS geometry [2509.01103]. The dual-RIS work does not title the method DSK, but its description of “bit-controlled intelligent signal routing” and “data-dependent direction selection at the physical layer” makes it an explicit directional index-modulation realization [2511.18610]. By contrast, [2603.02721] uses “DSK” for Doppler Shift Keying, where bits are mapped to Doppler-bin indices in the delay–Doppler domain. That paper is relevant primarily as a terminological caution: the acronym alone does not identify the directional modulation family.

The open problems identified in the directional literature are concrete. The mmWave study notes that prior DSK results were “largely simulation-based” and limited in geometry and receive-array size, and it addresses the optimal detector, coherence law, and phase-noise resilience for general \(M\)-antenna mobile devices [2509.01103]. It also indicates dependence on resolvable TDoA/DoA signatures, dominant-path stability, and accurate geometry or calibration. The dual-RIS study identifies future work on “further optimizing the system model, exploring real-world implementation challenges, and extending the proposed framework to support emerging technologies in wireless communication,” while highlighting multi-RIS systems and integration with next-generation networks [2511.18610].

A plausible implication is that future Direction-Shift Keying research will separate into at least two branches. One branch will continue the mmWave/sub-THz DAS line, emphasizing DCT-aware reference updating, phase-noise immunity, and geometry-based detection. The other will pursue environment-controlled realizations in which RIS phase profiles instantiate the direction index directly. Both branches retain the same core principle: data are mapped to a directional choice whose spatial signature is more stable, or more controllable, than conventional complex channel coefficients.

Source: https://www.emergentmind.com/topics/direction-shift-keying-dsk