Doppler Squint Effect in Wideband Systems

• Doppler Squint Effect is the frequency-dependent coupling between Doppler shift, propagation direction, and array geometry, thus breaking the narrowband approximation.
• It appears in diverse domains such as OTFS, ODDM, massive MIMO, and SAR, where its impact requires phase-consistent processing to mitigate performance penalties.
• Proper modeling of DSE is critical to reduce delay-spread extension, inter-symbol interference, and beam squint in wideband, high-mobility systems.

Doppler Squint Effect (DSE) denotes, in current wave-propagation, radar, and wireless-communication literature, a set of closely related couplings between Doppler shift, propagation direction, frequency, and array geometry that become explicit when narrowband or purely collinear descriptions are insufficient. In wideband delay–Doppler modulation, DSE is the frequency dependence of Doppler and the associated time–frequency phase coupling; in wideband array processing it is intertwined with beam squint and with the joint time–frequency–space signature of moving paths; in synthetic aperture radar (SAR), squint appears through a non-zero Doppler centroid frequency; and, in a complementary classical wave treatment, Doppler shift and aberration arise from the same phase field, so the direction from which a wave arrives need not coincide with the instantaneous line of sight to the emitter (Alejos et al., 2023, Wang et al., 2023, Wang et al., 2 Aug 2025, Hamidi, 2020).

1. Terminological scope and domain-specific meanings

The term is not used with a single discipline-independent formula. In the cited literature, it appears in several related but non-identical senses, all centered on the breakdown of a naive identification between “the Doppler of a path” and a single geometry-independent scalar.

Domain	Meaning used in the literature	Hallmark expression or observation
OTFS	Frequency-dependent Doppler in a wideband time-variant channel	$$H_i(t,f)=\beta_i e^{j2\pi \frac{\nu_i}{f_c}(f_c+f)t} e^{-j2\pi f\tau_i}$$
ODDM	Wideband interactive dispersion in the delay–Doppler plane	$e^{j2\pi b_i f t}$, $\tau_i(t)=\tau_i-b_i t$, $\nu_i(f)=\nu_i+b_i f$
Massive MIMO / mmWave	Joint Doppler and beam-squint dependence of the spatial signature	$\mathbf a(\theta,f)$ depends explicitly on $f$
SAR	Non-zero Doppler centroid caused by non-zero squint angle	Azimuth spectrum is shifted from baseband

In OTFS, the defining statement is that “the Doppler shift brought by the high mobility is frequency-dependent,” and this frequency dependence is named the Doppler Squint Effect (Wang et al., 2023). In ODDM, the same phenomenon is called “wideband interactive dispersion,” caused by the term $e^{j2\pi b_i f t}$ in the time-variant frequency response $H(t,f)$ (Wang et al., 2 Aug 2025). In multi-UAV wideband massive MIMO, the cited work does not always use the exact label “Doppler squint,” but it explicitly develops a channel model in which Doppler and beam squint are jointly present and the effective spatial signature depends on Doppler, frequency, and array geometry (Zhao et al., 2019). In strip-map SAR, the non-zero Doppler centroid frequency is “the result of non-zero squint angle,” and, if uncompensated, it de-focuses the image (Hamidi, 2020).

A plausible implication is that DSE is best understood as a family of Doppler–direction–frequency couplings rather than as a single narrow phenomenon. The common thread is that propagation, motion, and observation are no longer separable by a narrowband or purely line-of-sight approximation.

2. Classical phase-function foundation: Doppler shift, aberration, and angular mismatch

A rigorous classical foundation for DSE-like coupling is provided by the phase-function treatment of the Doppler effect. In the frame where the propagating medium is at rest, the wave is described by a phase field $\varphi(\mathbf r,t)$, and both observed frequency and propagation direction are derived from this same object: $$f'=\frac{1}{2\pi}\frac{d\varphi}{dt}, \qquad \mathbf k'=-\nabla\varphi.$$ For a moving observer, the derivative is convective,

$e^{j2\pi b_i f t}$0

For a moving emitter, the phase function derived from the emission geometry yields the general classical Doppler expression

$e^{j2\pi b_i f t}$1

where $e^{j2\pi b_i f t}$2 is the propagation direction of the wave at the observer, not the emitter-position angle. The same analysis gives the aberration relation between the emitter-position angle $e^{j2\pi b_i f t}$3 and the propagation angle $e^{j2\pi b_i f t}$4: $e^{j2\pi b_i f t}$5

$e^{j2\pi b_i f t}$6

This explicitly separates the direction in which the observer sees the emitter from the direction from which the wave actually propagates (Alejos et al., 2023).

That separation is the classical core of a squint interpretation. The paper states that “the sound comes from one direction, but the emitter is seen in a different direction,” and it further shows that the Doppler shift vanishes when $e^{j2\pi b_i f t}$7, not when $e^{j2\pi b_i f t}$8. The corresponding emitter viewing angle is

$e^{j2\pi b_i f t}$9

This directly contradicts the common simplification that zero Doppler occurs at closest approach. In the general moving-emitter–moving-observer case, the frequency depends on the emitter velocity relative to the medium, the observer velocity components relative to the medium, and the relative geometry; the result can նաև be rewritten in terms of the observer velocity relative to the emitter, which the paper emphasizes is valid in any reference frame (Alejos et al., 2023).

3. Wideband delay–Doppler systems: frequency-dependent Doppler as the canonical DSE

In modern OTFS literature, DSE is the failure of the narrowband assumption that each path has a single Doppler $\tau_i(t)=\tau_i-b_i t$0 independent of frequency. The wideband time-varying frequency response is

$\tau_i(t)=\tau_i-b_i t$1

so the Doppler at frequency $\tau_i(t)=\tau_i-b_i t$2 is

$\tau_i(t)=\tau_i-b_i t$3

In the time–frequency grid of OTFS, this induces the additional phase coupling

$\tau_i(t)=\tau_i-b_i t$4

so the channel is no longer separable in time index $\tau_i(t)=\tau_i-b_i t$5 and frequency index $\tau_i(t)=\tau_i-b_i t$6. In the delay–Doppler domain, a nonzero-Doppler path is no longer a Dirac impulse but a constant-modulus 2D exponential, and each discrete path contribution becomes nonzero across all $\tau_i(t)=\tau_i-b_i t$7, albeit concentrated around its nominal indices (Wang et al., 2023).

The same structural idea appears in ODDM under the label “wideband interactive dispersion.” There the time-variant frequency response is

$\tau_i(t)=\tau_i-b_i t$8

with

$\tau_i(t)=\tau_i-b_i t$9

The effect is an extra delay–Doppler spread and more complicated power leakage outside the peak region. The equivalent discrete delay support expands to

$\nu_i(f)=\nu_i+b_i f$0

so the extra delay spread is about $\nu_i(f)=\nu_i+b_i f$1. RCP-ODDM preserves a circular structure but introduces an additional phase factor $\nu_i(f)=\nu_i+b_i f$2, whereas ZP-ODDM can recover a cleaner convolutional form if the zero-padding region absorbs the DSE-induced extended delay spread (Wang et al., 2 Aug 2025).

A practical CP-OTFS formulation makes the accumulation mechanism explicit over a long OTFS frame. The baseband channel is

$\nu_i(f)=\nu_i+b_i f$3

and the per-symbol discrete kernel contains the term

$\nu_i(f)=\nu_i+b_i f$4

The resulting DSE causes both delay-spread extension and an extra phase shift. To suppress inter-symbol interference under this model, the sufficient CP condition becomes

$\nu_i(f)=\nu_i+b_i f$5

This is stricter than the usual DSE-free condition and reflects the time-varying effective delay $\nu_i(f)=\nu_i+b_i f$6 (Wang et al., 2023).

4. Array processing, beam squint, and the “doubly squint” regime

In wideband massive-MIMO channels, DSE is naturally coupled to beam squint because the steering vector itself depends on frequency. For the multi-UAV mmWave model, the wideband single-path channel is

$\nu_i(f)=\nu_i+b_i f$7

with

$\nu_i(f)=\nu_i+b_i f$8

After stacking across blocks, each path is represented by the rank-1 spatio–temporal–frequency atom

$\nu_i(f)=\nu_i+b_i f$9

This makes the path signature simultaneously angle-selective, time-selective, and frequency-selective. The cited work treats this structure with gridless compressed sensing, estimating direction of arrival, Doppler shift, and complex gain jointly, and then exploits angular reciprocity and Doppler shift reciprocity for downlink reconstruction (Zhao et al., 2019).

In massive MIMO-OTFS, the literature explicitly distinguishes beam squint from Doppler squint and calls their simultaneous presence the “doubly squint effect.” The simplified equivalent channel contains two characteristic factors: $\mathbf a(\theta,f)$0 for beam squint, and

$\mathbf a(\theta,f)$1

for Doppler squint. The second term means that each subcarrier experiences a slightly different effective Doppler and that the phase distortion accumulates jointly over time index $\mathbf a(\theta,f)$2 and subcarrier index $\mathbf a(\theta,f)$3. The paper characterizes the maximum DSE phase as approximately

$\mathbf a(\theta,f)$4

or, equivalently, through $\mathbf a(\theta,f)$5, and then designs chirp-based channel estimation together with analog and digital precoding to compensate both squints (Duan et al., 11 Apr 2025).

The distinction matters. Beam squint is a space–frequency–angle coupling; Doppler squint is a time–frequency–Doppler coupling. In wideband large-scale arrays, they are not interchangeable, but they can be inseparable at the system level.

5. SAR usage: squint angle and Doppler centroid

In strip-map SAR, DSE appears in a geometrically older but operationally central form: the non-zero Doppler centroid caused by non-zero squint angle. For a point target with slant range $\mathbf a(\theta,f)$6, the baseband echo is modeled as

$\mathbf a(\theta,f)$7

A Taylor expansion of $\mathbf a(\theta,f)$8 around closest approach produces a linear azimuth phase term whose coefficient is the Doppler centroid frequency. The paper gives

$\mathbf a(\theta,f)$9

and states that the azimuth spectrum is shifted from baseband when the squint angle is nonzero (Hamidi, 2020).

In the Range–Doppler chain, this centroid appears explicitly after azimuth compression: $f$0 If $f$1 is not accurately estimated and compensated, azimuth compression is mismatched and the image de-focuses. The cited work therefore estimates the fractional Doppler centroid by minimizing image entropy,

$f$2

The operational principle is that a focused image has minimum entropy, so the correct centroid is selected by an autofocus-like search over candidate $f$3 values (Hamidi, 2020).

This SAR usage is narrower than the wideband OTFS or ODDM meaning, but the conceptual thread is the same: geometry shifts the relation between propagation and the Doppler variable that the processor assumes to be centered or stationary.

6. Modeling consequences, common misconceptions, and nonstandard extensions

A recurrent misconception is that DSE is negligible whenever mobility is modest. The cited literature instead shows that severity is governed by joint scale parameters: bandwidth, frame duration, array aperture, and block size, in addition to velocity. In OTFS, the accumulated DSE phase is proportional to $f$4, so larger $f$5 and $f$6 can make the effect material even when $f$7 is numerically small (Wang et al., 2023). In ODDM, the rule-of-thumb parameter is $f$8, and the paper states that one may neglect DSE only when $f$9 (Wang et al., 2 Aug 2025). In massive-MIMO OTFS, the comparable quantity is $e^{j2\pi b_i f t}$0 (Duan et al., 11 Apr 2025).

Another misconception is that DSE is merely beam squint under a different name. The literature distinguishes them sharply. Beam squint comes from frequency-dependent array response; Doppler squint comes from frequency-dependent Doppler or time-varying delay. Their coexistence is the “doubly squint effect,” not a terminological redundancy (Duan et al., 11 Apr 2025).

The performance penalties of ignoring DSE are concrete in the cited studies. In OTFS, with $e^{j2\pi b_i f t}$1 and $e^{j2\pi b_i f t}$2 km/h, ignoring DSE causes DD-channel NMSE above $e^{j2\pi b_i f t}$3, and a BER floor appears around $e^{j2\pi b_i f t}$4 under model mismatch (Wang et al., 2023). In CP-OTFS, at $e^{j2\pi b_i f t}$5 km/h and $e^{j2\pi b_i f t}$6, model NMSE is about $e^{j2\pi b_i f t}$7 when DSE is ignored, and the conventional model exhibits a BER floor of approximately $e^{j2\pi b_i f t}$8 for SNR $e^{j2\pi b_i f t}$9 dB (Wang et al., 2023). In ODDM, the RF scenario with $H(t,f)$0 and $H(t,f)$1 km/h yields NMSE exceeding $H(t,f)$2, while in the UWA scenario the NMSE can exceed $H(t,f)$3, implying that the narrowband model is completely inadequate (Wang et al., 2 Aug 2025).

A separate and nonstandard extension appears in a paper on tracking anomalies, which interprets discrepancies in flyby data through chirp d’Alembertian travelling wave solutions, clock acceleration, and distance proportional shifts, and explicitly frames the effect as a kind of spectral squint in time/range space rather than as the frequency-dependent Doppler or beam-squint models used in OTFS, ODDM, massive MIMO, or SAR (Guruprasad, 2015). This suggests a distinct usage rather than a direct continuation of the mainstream communications and radar formulations.

Across these literatures, the durable lesson is methodological: once the phase evolution depends jointly on time, frequency, and geometry, correct processing must be phase-consistent across those dimensions. The corresponding remedies are likewise domain-specific but structurally similar: phase-function formulations for non-collinear motion, DSE-aware dictionaries and sparse recovery in delay–Doppler systems, joint angle–Doppler tracking in wideband arrays, hybrid precoding under doubly squint, and centroid-aware autofocus in SAR (Alejos et al., 2023, Zhao et al., 2019, Duan et al., 11 Apr 2025, Hamidi, 2020).