Double Beam Squint Effect in Wideband Arrays

• Double beam squint effect is a phenomenon in wideband arrays where frequency-invariant phase control meets frequency-dependent spatial or delay variations, causing compounded misalignment.
• It spans several domains including RIS-aided links, massive MIMO, near-field localization, and THz planar arrays, impacting beamforming gain and channel capacity.
• Mitigation strategies such as codebook redesign, true time delay elements, and hybrid beamforming are crucial for maintaining effective beam alignment in these systems.

Double beam squint effect denotes a family of wideband array phenomena in which beam squint is compounded across two coupled dimensions or two cascaded stages rather than appearing only as a single frequency-dependent pointing error. Across the literature, the compounded aspect appears in several forms: BS–RIS and RIS–UE squint in cascaded RIS links, simultaneous angle and delay dependence in massive MIMO channel models, simultaneous angle and range drift in near-field focusing, concurrent azimuth and elevation squint in planar arrays, and joint main-lobe and split-lobe frequency steering in TTD-assisted ISAC. The common mechanism is unchanged: a frequency-invariant phase profile is applied to a propagation process whose spatial or delay phase scales with frequency, so the steering or focusing condition that is exact at one frequency is no longer exact at other subcarriers (Chen et al., 2021, Wang et al., 2019, Hao et al., 2022, Luo et al., 2023).

1. Terminological scope and meanings

The phrase is not standardized. Several cited works analyze compounded squint mechanisms without explicitly using the phrase “double beam squint,” while others use closely related language to emphasize two simultaneous squint dimensions. This suggests an umbrella usage rather than a single canonical definition.

Setting	What is “double”	Representative sources
Wideband phased ULA codebooks	Two-sided band-edge misalignment of one analog beam across frequency	(Cai et al., 2016, Cai et al., 2017, Yu et al., 2021)
RIS-aided cascaded links	Frequency mismatch on both BS–RIS and RIS–UE hops under one common RIS phase matrix	(Chen et al., 2021)
Massive MIMO channel modeling	Angle squint and delay squint in the same channel representation	(Wang et al., 2019, Wang et al., 2019)
Near-field wideband/ISAC	Angle squint and range squint of the focal point	(Luo et al., 2022, Luo et al., 2023)
THz planar or XL arrays	Simultaneous azimuth and elevation squint, or compounded TX/RX squint	(Ma et al., 2023, Xie et al., 25 Mar 2026)
Beam-split sensing	Main-lobe squint together with split-lobe squint	(Xu et al., 2022)

A recurring misconception is that “double beam squint” must refer to two separately transmitted beams. One of the foundational codebook papers explicitly states that it does not analyze two simultaneous beams or interactions between multiple beams labeled “double”; its analysis concerns a single analog beam whose optimal pointing differs across frequency (Cai et al., 2016). In contrast, RIS, THz, near-field, and sensing papers use the doubled aspect to indicate two coupled squint sources, two spatial dimensions, or two physical coordinates.

2. Classical beam squint in wideband phased arrays

The foundational mechanism is most transparent for a uniform linear array. For inter-element spacing $d$ and physical angle $\theta$, the spatial phase progression is

$\psi(f,\theta)=\frac{2\pi f\,d\sin\theta}{c},$

and the array response is

$$\mathbf{a}_M(f,\theta)=\frac{1}{\sqrt{M}}\big[1,e^{j\psi(f,\theta)},\ldots,e^{j(M-1)\psi(f,\theta)}\big]^T.$$

If the beamformer is designed at the carrier frequency $f_c$ for steering angle $\theta_0$, then at a different frequency $f$ the main-beam condition becomes

$\sin\theta(f)\approx \frac{f_c}{f}\sin\theta_0.$

Thus the pointing angle is frequency dependent, and the angular error obeys the small-bandwidth approximation

$$\Delta\theta(f)\approx -\frac{\Delta f}{f_0}\tan\theta_0.$$

This shows why squint is negligible near broadside and increases for oblique steering, large bandwidth, and large aperture (Cai et al., 2016, Chen et al., 2021).

The same effect appears in the array factor of a phase-shifter array. With weights fixed at $f_0$, the array factor is

$\theta$0

and its magnitude peaks when $\theta$1 (Cai et al., 2016). The normalized power pattern therefore varies across the band even when element patterns are assumed identical, isotropic, and frequency independent. In the $\theta$2 domain, the no-squint $\theta$3 dB beamwidth is approximately $\theta$4, so increasing $\theta$5 narrows the main lobe and makes a fixed-phase beam more sensitive to a given fractional bandwidth (Cai et al., 2016).

These relations also impose hard feasibility limits on wideband codebooks. For a target $\theta$6 dB guarantee over angular coverage $\theta$7, the cited codebook analysis gives

$\theta$8

with $\theta$9 (Cai et al., 2016). The basic single-beam phenomenon is therefore already a bandwidth–aperture trade-off; the “double” variants inherit and amplify that trade-off.

3. Compounded squint mechanisms

In RIS-aided wideband mmWave systems, the compounded form is especially explicit. The RIS applies one common phase matrix,

$\psi(f,\theta)=\frac{2\pi f\,d\sin\theta}{c},$0

with each $\psi(f,\theta)=\frac{2\pi f\,d\sin\theta}{c},$1 constant across all subcarriers. Yet both BS–RIS and RIS–UE links have frequency-dependent array responses and delay terms. In the cascaded channel

$\psi(f,\theta)=\frac{2\pi f\,d\sin\theta}{c},$2

the RIS cannot tune phases per subcarrier to track those variations. The resulting compounded misalignment is the double beam squint effect in RIS-aided wideband links: one squint source arises from the BS–RIS hop and another from the RIS–UE hop, while the RIS contributes only a single frequency-invariant phase pattern (Chen et al., 2021).

A second major interpretation appears in wideband massive MIMO channel estimation. There, the channel model contains frequency-dependent steering vectors and path-delay rotations simultaneously:

$\psi(f,\theta)=\frac{2\pi f\,d\sin\theta}{c},$3

The steering vectors induce angle squint, while the factors $\psi(f,\theta)=\frac{2\pi f\,d\sin\theta}{c},$4 induce delay squint. In that sense, the channel is doubly frequency dependent in angle and delay, and the support structure becomes angle–delay block sparse rather than frequency separable (Wang et al., 2019).

Near-field wideband systems add another doubled aspect. With spherical-wave propagation, the focal point itself moves in both angle and distance. For phase-only beamforming designed at $\psi(f,\theta)=\frac{2\pi f\,d\sin\theta}{c},$5, the near-field beam-squint points obey

$\psi(f,\theta)=\frac{2\pi f\,d\sin\theta}{c},$6

The focal maximum therefore traces a frequency-dependent trajectory in the $\psi(f,\theta)=\frac{2\pi f\,d\sin\theta}{c},$7 plane, not merely on the angular axis. TTD-assisted designs can deliberately reshape that trajectory, which is why near-field localization papers treat angle squint plus range squint as a two-dimensional or “double” squint phenomenon (Luo et al., 2022, Luo et al., 2023).

In planar THz arrays, the compounded structure can be attached to two spatial dimensions. One THz UPA analysis derives the gain as a product of two Dirichlet kernels, one for the horizontal projection and one for the vertical projection,

$\psi(f,\theta)=\frac{2\pi f\,d\sin\theta}{c},$8

with $\psi(f,\theta)=\frac{2\pi f\,d\sin\theta}{c},$9, $$\mathbf{a}_M(f,\theta)=\frac{1}{\sqrt{M}}\big[1,e^{j\psi(f,\theta)},\ldots,e^{j(M-1)\psi(f,\theta)}\big]^T.$$0, and $$\mathbf{a}_M(f,\theta)=\frac{1}{\sqrt{M}}\big[1,e^{j\psi(f,\theta)},\ldots,e^{j(M-1)\psi(f,\theta)}\big]^T.$$1 (Ma et al., 2023). Another THz study emphasizes simultaneous azimuth and elevation squint through two independent spatial projections in a rotated planar array and calls this double beam squint in the sense of concurrent two-dimensional angular deviation (Xie et al., 25 Mar 2026).

ISAC papers introduce still another variant. With TTD lines and enlarged antenna spacing, different OFDM subcarriers illuminate different directions while grating lobes create additional split beams. Each split beam also squints with frequency, so the doubled aspect arises from simultaneous frequency steering of the main lobe and the split lobes (Xu et al., 2022).

4. Performance consequences and quantitative regimes

The most immediate consequence is loss of beamforming gain and effective beamwidth. In squint-aware codebook design, a ULA with $$\mathbf{a}_M(f,\theta)=\frac{1}{\sqrt{M}}\big[1,e^{j\psi(f,\theta)},\ldots,e^{j(M-1)\psi(f,\theta)}\big]^T.$$2, carrier frequency $$\mathbf{a}_M(f,\theta)=\frac{1}{\sqrt{M}}\big[1,e^{j\psi(f,\theta)},\ldots,e^{j(M-1)\psi(f,\theta)}\big]^T.$$3 GHz, bandwidth $$\mathbf{a}_M(f,\theta)=\frac{1}{\sqrt{M}}\big[1,e^{j\psi(f,\theta)},\ldots,e^{j(M-1)\psi(f,\theta)}\big]^T.$$4 GHz, and $$\mathbf{a}_M(f,\theta)=\frac{1}{\sqrt{M}}\big[1,e^{j\psi(f,\theta)},\ldots,e^{j(M-1)\psi(f,\theta)}\big]^T.$$5 requires about $$\mathbf{a}_M(f,\theta)=\frac{1}{\sqrt{M}}\big[1,e^{j\psi(f,\theta)},\ldots,e^{j(M-1)\psi(f,\theta)}\big]^T.$$6 more beams to guarantee a $$\mathbf{a}_M(f,\theta)=\frac{1}{\sqrt{M}}\big[1,e^{j\psi(f,\theta)},\ldots,e^{j(M-1)\psi(f,\theta)}\big]^T.$$7 dB minimum gain across the entire band and coverage sector; with $$\mathbf{a}_M(f,\theta)=\frac{1}{\sqrt{M}}\big[1,e^{j\psi(f,\theta)},\ldots,e^{j(M-1)\psi(f,\theta)}\big]^T.$$8 under the same $$\mathbf{a}_M(f,\theta)=\frac{1}{\sqrt{M}}\big[1,e^{j\psi(f,\theta)},\ldots,e^{j(M-1)\psi(f,\theta)}\big]^T.$$9 GHz and $f_c$0 GHz setting, the increase is $f_c$1 (Cai et al., 2016). The mechanism is that the effective beamwidth shrinks away from broadside and varies across the band, so a codebook that ignores squint undercovers the sector edges.

Beam squint also reduces channel capacity. A capacity-constrained codebook compensation study reports that its design algorithm can improve the channel capacity by $f_c$2 for a ULA with $f_c$3 antennas operating at bandwidth $f_c$4 GHz and carrier frequency $f_c$5 GHz (Cai et al., 2017). The same study gives an explicit bandwidth limit: for $f_c$6 dB and $f_c$7, the upper bound is approximately $f_c$8 (Cai et al., 2017). In other words, bandwidth expansion without delay-based compensation eventually renders a fixed-phase guarantee impossible.

RIS-aided results show the same trend in a cascaded form. With $f_c$9 GHz, bandwidth $\theta_0$0 GHz, $\theta_0$1 subcarriers, $\theta_0$2 BS antennas, and $\theta_0$3 RIS elements, the gap between the per-subcarrier “Ideal” RIS phases and common-phase designs grows with bandwidth and with RIS size $\theta_0$4, indicating that squint severity scales with aperture and fractional bandwidth (Chen et al., 2021). In NLoS settings, the MCCM-based RIS design still shows approximately a $\theta_0$5 bps/Hz gap to the “Ideal” per-subcarrier phase design at $\theta_0$6 MHz bandwidth, while baselines can lose at least $\theta_0$7 bps/Hz (Chen et al., 2021).

In THz planar arrays, the cited UPA analysis condenses severity into the beam squint ratio

$\theta_0$8

where $\theta_0$9 (Ma et al., 2023). For half-wavelength spacing and fixed $f$0, a square UPA minimizes the ratio, and the analysis states that

$f$1

This identifies array shape as a geometric control variable for compounded azimuth–elevation squint (Ma et al., 2023).

Time variation can compound the phenomenon further. In multi-UAV wideband communications, the cited channel-tracking study classifies the channel as antenna-selective when

$f$2

and as time-selective when

$f$3

When both hold, the channel is doubly selective in frequency and time, so beam squint interacts with Doppler and evolving angles (Zhao et al., 2019).

5. Mitigation methodologies

The most direct mitigation is to replace static phase shifts by true time delay. The general principle is identical across several subfields: a delay element contributes a frequency-proportional phase and can therefore align wideband propagation delays across the aperture. RIS and array papers repeatedly note that TTD can, in principle, eliminate squint, but they also emphasize higher hardware complexity, latency, insertion loss, size, power, and control overhead relative to phase shifters (Chen et al., 2021, Hao et al., 2022).

In RIS-aided systems under phase-only constraints, the dominant approach is to optimize one common phase profile across subcarriers. For the LoS case, the central-frequency solution is

$f$4

which reflects according to the central-frequency geometric difference and requires only the long-term angles $f$5 and $f$6 (Chen et al., 2021). For NLoS RIS–UE channels, the mean channel covariance matrix

$f$7

is used; the RIS receive-side phase is chosen from the dominant eigenvector of $f$8, and the BS-facing phase is matched at the central subcarrier (Chen et al., 2021). This separates low-complexity common-phase design from per-subcarrier unattainability.

In THz IRS communications, delay-adjustable metasurfaces provide an explicit wideband cure. The per-element reflection is

$f$9

and the cited DAM design gives closed-form far-field and near-field delays that cancel the $\sin\theta(f)\approx \frac{f_c}{f}\sin\theta_0.$0 mismatch in the cascaded channel (Hao et al., 2022). In the far field, the delay setting is proportional to element index and target composite direction; in the near field, it is proportional to the summed BS–IRS and IRS–user path lengths (Hao et al., 2022).

Codebook redesign is the main mitigation when the architecture remains purely phase-shifter based. Two strands appear in the cited literature. One densifies the codebook to preserve a target minimum gain or capacity across frequency (Cai et al., 2016, Cai et al., 2017). Another keeps the codebook size fixed but spreads beam coverage nonuniformly, especially toward end-fire and larger $\sin\theta(f)\approx \frac{f_c}{f}\sin\theta_0.$1, to absorb the angle dilation

$\sin\theta(f)\approx \frac{f_c}{f}\sin\theta_0.$2

seen by different subcarriers (Yu et al., 2021).

Hybrid transceiver designs offer intermediate solutions. One THz HBF study constructs a frequency-flat analog combiner from the average receive subspace across subcarriers and gives closed-form digital precoders from per-subcarrier water-filling, making the analog stage robust to UPA beam squint without TTD hardware (Ma et al., 2023). A switch-based HBF study derives

$\sin\theta(f)\approx \frac{f_c}{f}\sin\theta_0.$3

for ULAs and argues that switch-based hybrid beamforming is more robust than phase-shifter HBF because switches are phase independent and more power/cost efficient (Ma et al., 2022). Another wideband massive-MIMO design splits the UWB signal into narrowband beams by lens antenna subarrays and analog subband filters, then performs per-subband alignment by a threshold-based exhaustive search (Afeef et al., 2022).

A more recent mitigation category moves geometry instead of circuitry. Six-dimensional movable antennas jointly optimize analog weights, antenna positions in a $\sin\theta(f)\approx \frac{f_c}{f}\sin\theta_0.$4D region, and $\sin\theta(f)\approx \frac{f_c}{f}\sin\theta_0.$5D rotation angles to maximize the minimum wideband gain over angle–frequency space; for a special one-dimensional coverage case, the cited result states that rotating a ULA is sufficient to achieve global optimality and eliminate beam-squint effects (Xie et al., 25 Mar 2026). In IRS-aided THz links, movable BS antennas and movable IRS subarrays are optimized by majorization–minimization to maximize the minimal received power across subcarriers and suppress the coupled BS–IRS double squint effect (Zhu et al., 29 Aug 2025).

6. Estimation, sensing, localization, and other uses

Because double beam squint breaks the narrowband steering model, it directly alters channel estimation. In wideband mmWave massive MIMO-OFDM, a compressed-sensing formulation that explicitly models

$\sin\theta(f)\approx \frac{f_c}{f}\sin\theta_0.$6

jointly estimates frequency-insensitive angles and delays together with frequency-sensitive gains (Wang et al., 2019). A related block-sparsity estimator emphasizes the angle–delay representation

$\sin\theta(f)\approx \frac{f_c}{f}\sin\theta_0.$7

and exploits common support across subcarriers to recover off-grid angles, delays, and gains without the error floors that appear when squint is ignored (Wang et al., 2019).

In IRS-aided wideband channel estimation, beam squint can create a specific two-peak ambiguity. The cited analysis shows that the mutual correlation function between steering vectors and the cascaded channel has two peaks for a single physical path: an actual angle and a false angle. The false peak is frequency dependent, and for a single-path model the two solutions are

$\sin\theta(f)\approx \frac{f_c}{f}\sin\theta_0.$8

This motivates a twin-stage orthogonal matching pursuit algorithm that first exploits cross-subcarrier consistency to suppress the false angle and then estimates delays and gains in the delay domain (Ma et al., 2021).

When Doppler is present, squint and time variation must be tracked jointly. In multi-UAV wideband communications, the channel is parameterized by direction of arrival, Doppler shift, and complex gain, and a gridless compressed sensing tracker is combined with an EKF-based DOA tracker. The same work uses angular reciprocity and Doppler reciprocity to reconstruct the downlink channel from limited uplink-side physical parameters (Zhao et al., 2019).

A separate research line reverses the usual perspective and uses double beam squint as a sensing resource. In near-field localization with TTDs, different subcarriers are deliberately forced to focus at different angles and distances, so users at different positions observe different subcarriers of maximum power (Luo et al., 2022, Luo et al., 2023). For phase-only near-field squint, the focal trajectory already satisfies

$\sin\theta(f)\approx \frac{f_c}{f}\sin\theta_0.$9

while TTD control generalizes that trajectory into a design variable (Luo et al., 2022). One THz ISAC scheme similarly maps subcarrier frequencies to directions and uses beam split to expand the sensing range; users feed back the subcarrier frequency with the maximum array gain, and direction is then inferred from the designed frequency–angle mapping (Xu et al., 2022).

THz XL-RIS sensing brings yet another variant. A frequency-selective polar-domain redundant dictionary is built to overcome hybrid-field beam squint, and localization is obtained either by complete dictionary line intersection with off-grid angle refinement or by a partial-dictionary hyperbola construction based on time difference of arrival (Li et al., 2023). In that setting, the doubled aspect is the hybrid far–near field effect together with wideband squint, both of which distort the angle–range support across subcarriers (Li et al., 2023).

Taken together, these works establish that double beam squint is not a single narrowly defined impairment but a broad class of compounded frequency-dependent beam-misalignment phenomena. Its exact form depends on architecture and propagation regime, but the central engineering fact is stable: once frequency-independent phase control confronts frequency-dependent spatial or delay structure in more than one coupled dimension, the beam ceases to be a narrowband object.