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Characterization of Beam-Squint and Beam-Split Effects in RIS-assisted Multi-Frequency Networks

Mohammad Amin Saeidi, Hina Tabassum

arXiv:2609.07995v1eess.SP

TL;DR

Frequency mismatch can shift RIS beams and introduce additional maximum-gain directions, with phase quantization adding harmonic beams. The paper derives closed-form characterizations for continuous and quantized UPA-RISs and a correction for peaks of the superposed pattern. Simulations validate the analysis and report sum-rate improvement from beam-split-aware codeword selection in a two-RIS, two-user, multi-band network.

  • Problem

    Frequency mismatch can shift the main beam or produce multiple maximum-gain directions, potentially reducing array gain in directional RIS links.

    Source: §I · p. 1

  • Method

    The paper derives continuous-phase peak conditions and uses Fourier harmonic decomposition plus perturbation-based correction to characterize quantized UPA-RIS peaks.

    Source: §VIII · p. 14

  • Results

    Numerical results validate the peak and split characterizations and show that beam-split-aware codeword selection improves sum-rate in a two-RIS, two-user, multi-band network.

    Source: §VIII · p. 14

  • Takeaways & Limitations

    Closed-form peak predictions can form compact candidate codeword sets for cross-band selection without exhaustive angular sweeps.

    Source: B. Beam-split-aware Codeword Selection · p. 11

    The study excludes TTD-based wideband architectures, which modify RIS hardware with frequency-dependent delay elements, and uses exhaustive finite-codebook search as an upper bound.

    Source: §VIII · p. 14

Abstract

from arXiv · show

Multi-frequency operation in beyond-5G and 6G systems renders the beam directions of large-aperture arrays frequency-dependent, giving rise to beam misalignment effects critical for reconfigurable intelligent surfaces (RISs). When an RIS phase profile is configured at one frequency but illuminated at another, the reflected field may exhibit main-lobe deviation and additional dominant lobes, degrading beamforming performance. In this paper, we develop a unified analytical framework to characterize these effects for uniform planar array-based RISs under both continuous and finite-resolution phase control. We formalize beam-squint as the deviation of the maximum-gain beam from the desired elevation-azimuth angle pair and beam-split as the emergence of additional maximum-gain beams under frequency mismatch and quantization. For continuous-phase RISs, we derive necessary and sufficient peak conditions, obtain closed-form elevation-azimuth peak families, and establish feasibility conditions as functions of the frequency ratio, incidence and configuration angles, and element spacing. Extending the analysis to b-bit phase quantization, we express the quantized beampattern as a superposition of harmonic array responses via Fourier expansion of the quantization-error phasor, leading to closed-form per-harmonic peak families, dominance ordering, and feasibility conditions. To capture peak displacement induced by harmonic superposition, we further develop a perturbation-based correction leveraging dominant-harmonic curvature and residual-harmonic gradients, improving peak-location accuracy without exhaustive two-dimensional searches. Numerical simulations validate the analysis, quantify peak deviations and beam-split probabilities, and demonstrate sum-rate gains by including analytically predicted squinted/split and harmonic candidates in multi-RIS, multi-user, multi-band networks.

I. INTRODUCTION

Frequency mismatch can shift RIS beams or create multiple maximum-gain directions, while practical phase quantization adds further harmonic beams. The paper develops a characterization framework for these effects and specifies the RIS model and peak-finding objective.

I. INTRODUCTION

  • Frequency mismatch shifts the main beam and can create additional dominant beams, with angular deviations potentially reducing array gain in directional links.The paper motivates analysis of these effects for wideband and multi-band operation.Source: §I · p. 1

A. Related Work

  • Prior work addresses frequency-dependent beams through hardware or deployment changes, and also studies their relevance to sensing, localization, and channel estimation.The paper distinguishes its focus on characterizing passive RIS beams from optimization methods targeting active and passive beamforming variables.Source: A. Related Work · pp. 1–2

B. Motivation and Contributions

  • The proposed framework predicts squinted, split, and quantization-induced beam locations and existence conditions for continuous and finite-resolution RIS phases.For quantized phases, it uses harmonic decomposition and a correction procedure; predicted directions are proposed for frequency-aware beam management.Source: B. Motivation and Contributions · p. 2

II. SYSTEM MODEL AND PROBLEM STATEMENT

  • The model fixes the RIS phase profile at a design frequency and maximizes normalized beamforming gain over observation angles under continuous or nearest-neighbor-quantized phases.Elements are assumed frequency-independent and unit-modulus to isolate array-level effects; maximum-gain directions define squint and split.Source: §II · p. 3

III. BEAM-SQUINT AND BEAM-SPLIT EXISTENCE AND CHARACTERIZATION: CONTINUOUS RIS PHASE PROFILES

For continuous-phase RISs, the analysis characterizes maximum-gain directions under a mismatch between the design and incident frequencies. Feasible integer phase-slope pairs distinguish a squinted beam from additional split lobes.

A. Beam-squint and Beam-split Characterization

  • The RIS profile remains fixed at its design-frequency setting while the incident signal has a different frequency, producing the phase pattern analyzed for maximum gain.The zero integer pair identifies beam-squint; other feasible pairs identify additional maximum-gain beam-split lobes, not sidelobes.Source: A. Beam-squint and Beam-split… · pp. 3–4
  • The analysis covers conventional single-layer passive UPA-RISs with diagonal phase responses; stacked or beyond-diagonal architectures are left for future work.Those extensions require different effective-aperture models.Source: A. Beam-squint and Beam-split… · p. 3
  • Neighboring element phases must differ by integer multiples of 2π in both array directions for the beamforming-gain upper bound to be attained.The resulting continuous-phase peak locations are obtained by solving the elevation and azimuth relations for integer index pairs.Source: A. Beam-squint and Beam-split… · p. 4

B. Beam-Squint/Split Existence Conditions (Elevation Angles)

Feasibility of continuous-phase peak indices depends on interval lengths set by element spacing, frequency ratio, and angles. The elevation and azimuth conditions differ, with azimuth feasibility also depending on the elevation peak.

B. Beam-Squint/Split Existence Conditions (Elevation Angles)

  • When 2αzρ < 1, at most one elevation index is feasible, so elevation beam-split cannot occur and a peak may be absent.For 1 ≤ 2αzρ < 2, a squint peak is guaranteed, while split depends on the interval center; for 2αzρ ≥ 2, both occur.Source: B. Beam-Squint/Split Existence… · p. 4
  • For ρ < 1, the elevation squint index mz = 0 is feasible for all incidence and design elevation angles; for ρ > 1, feasibility depends on those angles.The elevation feasibility test limits the arcsine argument to [−1, 1].Source: B. Beam-Squint/Split Existence… · p. 4

C. Beam-Squint/Split Existence Conditions (Azimuth Angles)

  • Azimuth feasibility is assessed only after a feasible elevation peak exists, and its interval length depends on 2αyρAO.At 2αyρAO ≥ 2, at least two azimuth indices are feasible and both squint and split occur.Source: C. Beam-Squint/Split Existence… · pp. 4–5
  • Unlike elevation, azimuth squint is not guaranteed when ρ ≤ 1 because AO can be less than 1; for 1 ≤ 2αyρAO < 2, split depends on the interval center.The framework uses the elevation and azimuth feasibility sets to limit the integer-pair search illustrated for the continuous RIS case.Source: C. Beam-Squint/Split Existence… · p. 5

IV. BEAM-SQUINT AND BEAM-SPLIT EXISTENCE AND CHARACTERIZATION: b-BIT RIS PHASE PROFILES

Finite-resolution phase control adds a non-affine quantization error that prevents direct use of the continuous-phase peak condition. The paper uses a Fourier expansion to analyze the quantized beampattern as harmonic responses.

IV. BEAM-SQUINT AND BEAM-SPLIT EXISTENCE AND CHARACTERIZATION: b-BIT RIS PHASE PROFILES

  • Under b-bit quantization, the element-wise phase error depends on position and is no longer affine, so the continuous-phase peak condition cannot be applied directly.The analysis considers nearest-neighbor quantization of the continuous phase profile.Source: §IV · p. 5
  • The framework derives closed-form peak families and existence conditions per harmonic, then analyzes harmonic dominance to characterize quantized beam-squint and beam-split.The Fourier series of the periodic quantization-error phasor is established in the supplied lemmas.Source: §IV · p. 5, A. Beam Pattern with b-bit Quantized RIS · p. 6

A. Beam Pattern with b-bit Quantized RIS

  • Fourier expansion of the quantization-error phasor expresses the beampattern as a weighted sum of array responses with affine phase slopes.Each harmonic can then be analyzed separately using the continuous-phase peak result.Source: A. Beam Pattern with b-bit Quantized RIS · pp. 5–6

B. Beam Pattern Transformation and Per-harmonic Peaks

The quantized beampattern is transformed into harmonic responses, whose peak locations and feasibility can be analyzed harmonic by harmonic. A strength threshold limits feasibility checks to selected harmonics.

B. Beam Pattern Transformation and Per-harmonic Peaks

  • Fourier substitution expresses the quantized beampattern through harmonic terms, each with an affine phase progression that yields per-harmonic maximum-gain angle pairs.The ℓ = 0, (mz, my) = (0, 0) peak corresponds to beam-squint; nonzero harmonics correspond to quantization-induced beams counted as beam-split.Source: B. Beam Pattern Transformation and… · p. 6

C. Existence of Per Harmonic Beam-Squint/Split

  • Feasibility analysis is restricted to harmonics whose relative strength compared with ℓ = 0 is at least threshold η.For ℓ < 0, the selected set follows by solving κℓ ≥ η.Source: C. Existence of Per Harmonic… · p. 6
  • For 1-bit quantization, harmonics ℓ = 0 and ℓ = −1 have equal relative strength and exceed all others, while for b ≥ 2, ℓ = 0 is strongest.The dominance ordering informs which quantization-induced beams may need consideration.Source: C. Existence of Per Harmonic… · p. 6
  • An elevation or azimuth peak is feasible when its corresponding integer-index interval contains at least one integer.The supplied conditions use ceiling of the lower bound no greater than floor of the upper bound for mz and my, respectively.Source: C. Existence of Per Harmonic… · p. 7

V. PER-HARMONIC BEAM-SQUINT/SPLIT ANGLE CORRECTION MECHANISM

The quantized pattern superposes infinitely many harmonics, so aggregate peaks can shift from isolated per-harmonic predictions. The correction approach treats other harmonics as a local perturbation around a nominal harmonic peak.

V. PER-HARMONIC BEAM-SQUINT/SPLIT ANGLE CORRECTION MECHANISM

  • Superposition of infinitely many harmonics can displace isolated per-harmonic peak positions from the peaks of the aggregate pattern.The correction method refines predicted locations by accounting for interference from other harmonics.Source: §V · p. 7

A. Decomposition of Harmonic of Interest and Perturbation

  • The aggregate pattern is decomposed into a harmonic of interest and a residual containing the contributions of all other harmonics.The correction assumes the residual is locally smaller than the harmonic of interest.Source: A. Decomposition of Harmonic of… · p. 7
  • The actual peak is estimated near the nominal harmonic peak by linearizing the aggregate stationarity condition around that nominal location.The estimate assumes the actual peak lies close to the isolated harmonic peak.Source: A. Decomposition of Harmonic of… · p. 7
  • A Newton step using the full aggregate Hessian can move the estimate toward sidelobes or null directions because the aggregate pattern is highly nonconvex and has many extrema.The passage identifies this as a reason to avoid using the full Hessian directly for the angle shift.Source: A. Decomposition of Harmonic of… · p. 7

B. Local Surrogate Function-based Optimization

The correction uses dominant-harmonic curvature with residual-harmonic gradients to estimate peak shifts and assess local validity. The framework also bounds candidate counts and gives alternative costs for truncated-harmonic and exact-field correction.

B. Local Surrogate Function-based Optimization

  • The local surrogate retains the dominant harmonic's negative curvature and treats the residual as a linear perturbation, yielding a unique shift estimate.This uses the dominant harmonic's curvature rather than the full aggregate Hessian.Source: B. Local Surrogate Function-based… · p. 8
  • Correction accuracy depends on the residual remaining a local perturbation and on sufficiently strong curvature; a large residual gradient or weak curvature can produce a shift that violates the local approximation.The validity metric χ(p) < 1 indicates residual variation over the predicted shift is smaller than the isolated harmonic peak power; smaller values indicate a more reliable approximation.Source: B. Local Surrogate Function-based… · p. 8
  • Increasing Ny and Nz strengthens isolated-lobe curvature and reduces sensitivity to residual harmonics, whereas increasing ρ weakens curvature through a 1/ρ^2 factor.Weaker curvature can produce a larger correction step.Source: B. Local Surrogate Function-based… · p. 8
  • Comparing Fig. 3 and Fig. 4, increasing the number of RIS elements improves the accuracy of both per-harmonic and corrected peak locations.The passage attributes the improvement to increased curvature at per-harmonic peak locations.Source: B. Local Surrogate Function-based… · p. 8

1) Per-harmonic peak complexity:

  • Feasible per-harmonic candidate counts scale as O(αz/ρ + 1) in elevation and O(αy/ρ + 1) in azimuth.Each candidate angle pair is obtained in constant time.Source: 1) Per-harmonic peak complexity · p. 8

2) Perturbation-based correction complexity:

  • With a truncated harmonic set Kc, correction costs O(|Kc|(Ny + Nz)) per candidate; exact quantized-field gradients require summing over all NyNz RIS elements.The exact-gradient cost dominates the separable isolated-harmonic Hessian evaluation.Source: 2) Perturbation-based correction… · pp. 8–9

VI. USE-CASE STUDY: BEAM-SPLIT AWARE SCHEDULING IN MULTI-RIS, MULTI-USER, MULTI-BAND NETWORK

The use case evaluates two RISs serving two UEs on separate frequency bands, with each RIS also considered for cross-band assistance under frequency mismatch. Its codebooks separately represent home and cross incidence conditions.

VI. USE-CASE STUDY: BEAM-SPLIT AWARE SCHEDULING IN MULTI-RIS, MULTI-USER, MULTI-BAND NETWORK

  • The two-BS, two-RIS, two-UE scenario evaluates cross-band assistance, with direct BS–UE paths blocked to isolate RIS-reflected frequency-mismatch effects.Achievable rate is computed using the complete multipath cascaded channel, although candidate phase profiles are generated from LoS geometry.Source: §VI · pp. 9–10

A. Quantized codebooks at ρ = 1 (home and cross incidence)

  • Each RIS codebook samples design directions on a uniform direction-cosine grid, applies the phase-alignment rule, and quantizes each resulting profile to b bits.The grid resolution Qmin trades angular snapping error against the number of codewords evaluated.Source: A. Quantized codebooks at ρ = 1 (home… · p. 10, §VI · p. 10
  • Separate home- and cross-incidence codebooks are required because changing the BS incidence changes the quantized phase pattern for a given design direction.The two codebooks reflect the two-RIS, two-BS geometry.Source: A. Quantized codebooks at ρ = 1 (home… · p. 10

B. Beam-split-aware Codeword Selection

The section uses closed-form peak families to build compact cross-assistance codeword shortlists, then validates peak predictions and split-feasibility criteria. Continuous peak predictions match exhaustive search, while quantized estimates benefit from correction for harmonic interference.

B. Beam-split-aware Codeword Selection

  • The method solves for design directions whose mismatched harmonic peaks align with a cross-user, maps them to codewords, and prunes candidates using cross- and home-steering distances.It retains the home-maximizing codeword as a fallback; the resulting candidate sets are jointly evaluated by sum-rate without exhaustive angular sweeps.Source: B. Beam-split-aware Codeword Selection · pp. 10–11

VII. NUMERICAL RESULTS AND DISCUSSIONS

  • The numerical evaluation averages 5 × 10^3 trials per point, using direction-cosine sampling and dominant harmonics for quantized peak characterization.The benchmarks distinguish continuous profiles from b-bit profiles with and without the proposed correction.Source: §VII · p. 11

A. Beam-Squint/Split Analytical Validation

  • Continuous peak families coincide with exhaustive-search maximizers, while the correction consistently reduces quantized peak-location errors.For quantized patterns, residual-harmonic gradients shift the predicted peak using dominant-harmonic curvature; larger element spacing also reduces error.Source: A. Beam-Squint/Split Analytical… · p. 11, Figure 6 · p. 11

B. Beam-split Existence (Elevation and Azimuth)

  • For the continuous case, elevation split probability is essentially zero below 2αzρ = 1, becomes one above 2αzρ = 2, and transitions smoothly for 1 ≤ 2αzρ < 2 depending on the random interval center.The criterion is |Mz| ≥ 2, where Mz is the feasible-integer set.Source: B. Beam-split Existence (Elevation and… · p. 11
  • Conditioned on a valid elevation peak, azimuth split occurs when My contains at least two integers; elevation projection shortens its feasibility interval and shifts high split probability to larger αy.The Fig. 8 contours mark thresholds involving 2αyρA_O.Source: B. Beam-split Existence (Elevation and… · pp. 11–12, Figure 8 · p. 12

C. Per-frequency Gain and Beam-split Probability

The section relates frequency mismatch and quantization to peak displacement and beam splitting, then evaluates correction accuracy and beam-split-aware codeword selection. Its simulations show that including predicted split/harmonic candidates improves sum-rate over simpler selection baselines in the considered two-RIS, two-user, multi-band network.

C. Per-frequency Gain and Beam-split Probability

  • For continuous-phase RISs, design-direction degradation is zero at ρ = 1 and increases as the maximum-gain lobe shifts away from the design direction.Beam-split is more likely mainly for ρ < 1; for ρ > 1, degradation is primarily due to beam-squint.Source: C. Per-frequency Gain and Beam-split… · p. 12
  • For 1-bit RISs, quantization adds harmonic peak families, producing higher beam-split probability than for continuous-phase RISs.At some mismatched frequencies, harmonic components can preserve more design-direction gain than the shifted continuous-phase beams.Source: C. Per-frequency Gain and Beam-split… · p. 12

D. Number of RIS Elements in b-bit Quantization

  • Mean peak-location error decreases with RIS size, and the proposed correction is most beneficial for small and moderate sizes.The correction benefit is more visible for 1-bit quantization, while larger ρ can increase residual displacement by weakening angle-domain curvature.Source: D. Number of RIS Elements in b-bit… · p. 13
  • The perturbation validity metric χ(p) and angular correction norm ∥∆p∥2 capture different aspects of correction size: lobe-relative locality and physical angular displacement.Both measures decrease with RIS size; χ(p) is largest for coarse quantization, especially 1-bit, while increasing ρ can enlarge the angular correction step.Source: D. Number of RIS Elements in b-bit… · p. 13

E. Beam-split-aware Codeword Selection

  • The proposed beam-split-aware selection improves average sum-rate over home-link-only and squint-aware split-agnostic selection, while exhaustive finite-codebook search remains highest.The proposed method uses a small set of analytically predicted split/harmonic candidates; exhaustive search evaluates a much larger candidate space.Source: E. Beam-split-aware Codeword Selection · p. 14, Figure 12 · p. 14
  • Increasing Qmin from 8 to 16 improves all selection methods, while the proposed method remains above the home-link-only and squint-aware baselines.With Lkeep = 5, the proposed method evaluates at most 36 phase-profile pairs across two RISs, versus up to 16384 or 262144 for exhaustive search.Source: E. Beam-split-aware Codeword Selection · p. 14

VIII. CONCLUSION

  • The paper characterizes continuous and b-bit quantized UPA-RIS peak families and develops a perturbation-based correction for peaks displaced by harmonic superposition.The correction uses dominant-harmonic curvature and residual-harmonic gradients; numerical results include a two-RIS, two-UE multi-band network.Source: §VIII · p. 14
  • TTD-based wideband architectures are outside the evaluated passive finite-resolution RIS scope, so exhaustive finite-codebook search serves as the upper bound.Source: §VIII · p. 14

APPENDIX A: PROOF OF LEMMA 1

The appendices prove a phase-increment criterion by accumulating unit-step changes across the array, then establish periodicity and Fourier-series convergence for the quantization-error phasor.

APPENDIX A: PROOF OF LEMMA 1

  • The sufficiency proof accumulates fixed 2π phase increments along the y and z directions to show each element phase is congruent to the reference phase modulo 2π.Source: Appendix A · p. 14

APPENDIX B: PROOF OF LEMMA 2

  • The proof establishes that the quantization-error phasor g(x) is 1-periodic by applying the rounding-function shift identity.Source: Appendix B · p. 15
  • Bounded variation over one period lets the Dirichlet–Jordan test establish pointwise Fourier-series convergence, including the stated midpoint value at jumps.Source: Appendix B · p. 15

APPENDIX C: PROOF OF LEMMA 3

The appendices derive a Fourier representation for the quantization-error phasor and show that feasible isolated per-harmonic peaks have strictly negative angular curvature under the stated conditions.

APPENDIX C: PROOF OF LEMMA 3

  • The Fourier-series proof fixes a harmonic index ℓ and sets a = ∆ + 2πℓ to obtain the coefficient representation of g(x).Source: Appendix C · p. 15

APPENDIX D: NEGATIVE DEFINITENESS OF HJℓ(p)

  • The proof establishes the required full rank of G for the stated positive spacing and frequency-ratio conditions and observation-angle range.Source: Appendix D · p. 15
  • At a feasible per-harmonic peak, the phase-slope derivatives vanish, reducing the angular Hessian to GᵀHβG.The appendix concludes that this Hessian is strictly negative definite under the stated conditions.Source: Appendix D · p. 15

APPENDIX E: GRADIENT AND HESSIAN OF J USED IN (42)

The appendix supplies derivative identities used to form the angular gradient and Hessian of the squared array response, including the aggregate-gradient relation at a per-harmonic nominal peak.

APPENDIX E: GRADIENT AND HESSIAN OF J USED IN (42)

  • Complex-exponential derivative identities provide the component formulas for the Hessian of the squared response J = |f|2.Source: Appendix E · p. 16
  • At a per-harmonic nominal peak, the aggregate residual gradient equals the gradient of J because the isolated-harmonic gradient is zero there.Source: Appendix E · p. 16
  • The appendix lists first- and second-order derivatives of the phase slopes with respect to the observation angles.Source: Appendix E · p. 16
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