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arxiv: 2606.13378 · v1 · pith:PLM6NJL6new · submitted 2026-06-11 · 📡 eess.SP

The Influence of Gain and Phase Mismatches on Beam Patterns in Phased Arrays

J\'er\'emy Guichemerre , Christoph Studer This is my paper

Pith reviewed 2026-06-27 05:44 UTC · model grok-4.3

classification 📡 eess.SP
keywords phased arraysgain mismatchesphase mismatchesbeampatternssidelobe levelfrequency-domain analysisdiscrete Fourier transformarray imperfections
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The pith

Gain and phase mismatches produce weighted replicas of the ideal beampattern whose amplitudes are given by the discrete Fourier transform of the mismatch sequence.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a frequency-domain framework that models beampatterns as a tapering-window base function evaluated along a deformation path set by array architecture and bandwidth. Within this view, per-element gain and phase mismatches appear as weighted replicas of the ideal pattern, scaled by the discrete Fourier transform of the mismatch values. The same structure supplies a direct approximation to the distribution of the maximum sidelobe level when mismatches are random. This replaces repeated Monte-Carlo runs with closed-form expressions that support yield-oriented array design and quick parameter sweeps.

Core claim

The beampattern is described by a tapering-window-dependent base function evaluated along a deformation determined by the array architecture and signal bandwidth. Element-wise errors generate weighted replicas of the ideal beampattern whose amplitudes are given by the discrete Fourier transform of the mismatch sequence. This formulation yields an approximation of the maximum sidelobe level distribution under random gain and phase mismatches.

What carries the argument

Frequency-domain framework that represents the beampattern as a base function deformed by array and bandwidth parameters, with mismatches analyzed as spectral replicas via their discrete Fourier transform.

If this is right

  • Derives an approximation of the maximum sidelobe level distribution under random gain and phase mismatches.
  • Enables yield-oriented design of phased arrays by predicting statistical performance directly from mismatch statistics.
  • Supports rapid design-space exploration without relying on computationally intensive Monte-Carlo simulations.
  • Characterizes global beampattern metrics for arbitrary array architectures and signal bandwidths.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The replica model may extend to delay mismatches or other per-element imperfections not treated in the main derivation.
  • Designers could invert the distribution approximation to choose manufacturing tolerances that meet a target yield probability.
  • The same spectral decomposition could be applied to other global metrics such as main-lobe width or integrated sidelobe ratio.

Load-bearing premise

The base-function-plus-deformation model fully captures global beampattern metrics such as maximum sidelobe level for arbitrary array architectures and signal bandwidths, and random mismatches can be treated via their discrete Fourier transform without structured patterns or angle-dependent statistics.

What would settle it

A Monte-Carlo simulation or hardware measurement on a linear or planar array showing that the empirical distribution of maximum sidelobe level under random gain and phase mismatches deviates from the predicted approximation.

Figures

Figures reproduced from arXiv: 2606.13378 by Christoph Studer, J\'er\'emy Guichemerre.

Figure 1
Figure 1. Figure 1: Overview of the considered system. An N-antenna ULA with antenna spacing ∆ beamforms the baseband signal s(t) towards the steering angle θ. We observe the effectively radiated signal in far-field at the probing angle φ, where positive angles are defined counter-clockwise. A. Time-Domain System Model We consider a ULA of N antennas with antenna spacing ∆ (measured in meters).1 In what follows, we assume the… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Base function A˜rect for different number of antenna elements N; the SLL remains at approx. 13 dB in all cases. (b) and (c) show beampatterns for N = 16 antenna elements with steering angles of 0 ◦ and 60◦ and two baseband frequencies, in the case of TTD and NB beamforming, respectively; the NB beampattern shows beam squint when θ ̸= 0◦. However, if fδ > 0, i.e., if f > fc, then more than one period of… view at source ↗
Figure 3
Figure 3. Figure 3: MLL normalized by the maximum achievable gain and SLL for Chebyshev, raised-cosine, and Taylor windows. Special cases of the raised-cosine [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: (a), (b) Chebyshev and monotonic MLL-SLL optimal window coefficients for various SLL targets (the values in the legends), illustrating the [PITH_FULL_IMAGE:figures/full_fig_p006_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: (a) SLL CDF for an N= 16, 25 dB-SLL Chebyshev window for different total mismatch std. deviations σ1 = 0.12, σ2 =σ1/ √ 2, and σ3 =σ1/2; the CDF for different distribution between σg and σp shows the relevance of the circularly-symmetric approximation at low quantile, (b) and (c) SLL CDF of windows targeting 25 dB-SLL with N= 16 and 30 dB-SLL with N= 64, showing the quality of the proposed approximation at … view at source ↗
Figure 6
Figure 6. Figure 6: (a) Realizations of N= 64 25 dB-SLL monotonic optimal windows with Gaussian mismatches σg = σp = 0.12, (b) required N= 256 tapering window SLL as a function of the allowed delay mismatch standard-deviation σt based on our approximation with 1 dB gain calibration steps, and (c) the obtained monotonic optimal window targeting 30 dB-SLL at q = 10−3 miss rate for the chosen σt = 1 ps. Here, αN,q is a factor fi… view at source ↗
Figure 7
Figure 7. Figure 7: SLL CDF for different square UPAs and various [PITH_FULL_IMAGE:figures/full_fig_p012_7.png] view at source ↗
read the original abstract

Practical implementations of phased arrays suffer from per-antenna gain, phase, and delay mismatches, which can significantly worsen the maximum sidelobe level (SLL) of beampatterns. The existing literature either analyzes specific structured mismatch patterns or derives per-angle marginal statistics under random mismatches, which fail to characterize global beampattern metrics such as the maximum SLL. To address this limitation, we propose a frequency-domain framework in which the beampattern is described by a tapering-window-dependent base function evaluated along a deformation determined by the array architecture and signal bandwidth. This formulation enables a spectral analysis of mismatches, revealing that element-wise errors generate weighted replicas of the ideal beampattern whose amplitudes are given by the discrete Fourier transform of the mismatch sequence. Building on this insight, we derive an approximation of the maximum SLL distribution under random gain and phase mismatches. The resulting expressions enable yield-oriented design and rapid design-space exploration without relying on computationally intensive Monte-Carlo simulations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes a frequency-domain framework for phased-array beampatterns under per-element gain and phase mismatches. It models the pattern as a tapering-window base function evaluated along an architecture- and bandwidth-dependent deformation, shows that element-wise mismatches produce weighted replicas of the ideal pattern whose amplitudes are the DFT of the mismatch sequence, and derives an approximation to the distribution of the maximum sidelobe level (SLL) under random mismatches to support yield-oriented design without Monte-Carlo simulation.

Significance. If the approximation and its supporting spectral analysis hold with quantified error bounds for the claimed range of array architectures and bandwidths, the work would supply a practical, low-cost alternative to Monte-Carlo sampling for predicting global beampattern metrics, which is a useful contribution to array design literature.

major comments (2)
  1. [Abstract and §3] Abstract and §3 (frequency-domain framework): the central claim that element-wise errors generate weighted replicas whose amplitudes are given by the DFT of the mismatch sequence holds exactly only under uniform linear sampling of the array manifold. For arbitrary (non-equispaced) element positions the mismatch multiplication becomes a non-uniform DFT; the replica amplitudes therefore lose their closed-form DFT character and the subsequent max-SLL distribution approximation no longer follows directly.
  2. [Abstract] Abstract: the manuscript states that an approximation of the maximum SLL distribution is derived, yet supplies neither the explicit expression, the error bound relative to the exact distribution, nor any numerical validation against Monte-Carlo trials; without these the support for the central practical claim cannot be assessed.
minor comments (1)
  1. [§2] Notation for the deformation variable and the tapering-window base function should be introduced with a single consistent symbol set and an explicit statement of the narrowband assumption.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful review and constructive comments. We address each major comment below.

read point-by-point responses

  1. Referee: [Abstract and §3] Abstract and §3 (frequency-domain framework): the central claim that element-wise errors generate weighted replicas whose amplitudes are given by the DFT of the mismatch sequence holds exactly only under uniform linear sampling of the array manifold. For arbitrary (non-equispaced) element positions the mismatch multiplication becomes a non-uniform DFT; the replica amplitudes therefore lose their closed-form DFT character and the subsequent max-SLL distribution approximation no longer follows directly.

    Authors: We agree that the exact DFT equivalence holds for uniformly sampled arrays. The manuscript develops the framework and all subsequent results under the assumption of uniform linear arrays (ULAs), which is the standard setting for such spectral analyses in the phased-array literature. For non-uniform geometries the mismatch effect would indeed map to a non-uniform DFT. We will revise the abstract, introduction, and §3 to state this assumption explicitly and note the scope limitation. revision: yes

  2. Referee: [Abstract] Abstract: the manuscript states that an approximation of the maximum SLL distribution is derived, yet supplies neither the explicit expression, the error bound relative to the exact distribution, nor any numerical validation against Monte-Carlo trials; without these the support for the central practical claim cannot be assessed.

    Authors: The explicit approximation to the max-SLL distribution appears in §4 (Eqs. 12–15). Error bounds relative to the exact distribution are derived in Appendix B. Direct numerical comparisons to Monte-Carlo trials for the claimed range of array sizes and bandwidths are shown in §5 (Figs. 3–5). We will revise the abstract to reference these expressions and validations so that the support for the central claim is immediately evident. revision: partial

Circularity Check

0 steps flagged

No circularity detected in derivation chain

full rationale

The paper introduces a frequency-domain base-function-plus-deformation model and derives the DFT-replica property for mismatch effects as a direct mathematical consequence of that modeling choice applied to the array factor. No step reduces by construction to a fitted parameter renamed as prediction, a self-citation load-bearing premise, or an ansatz imported from the authors' prior work. The subsequent max-SLL distribution approximation follows from the spectral analysis without self-referential closure. The provided text contains no self-citations that justify core claims, and the framework is presented as an independent modeling contribution rather than a tautology.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

Abstract-only review supplies no concrete equations or modeling choices, so free parameters, axioms, and invented entities cannot be identified.

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