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arxiv: 2512.08887 · v2 · submitted 2025-12-09 · 📡 eess.SP

A Fast Broadband Beamspace Transformation

Nakul Singh , Coleman DeLude , Mark Davenport , Justin Romberg This is my paper

Pith reviewed 2026-05-16 23:11 UTC · model grok-4.3

classification 📡 eess.SP
keywords beamformingbroadband signalsbeamspace transformationnon-uniform Fourier transformToeplitz systemssensor arrayscomputational complexity
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The pith

A fast beamspace transformation forms B beams from M sensors in O(M log N + B log N) operations per sample.

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

The paper introduces a method for multi-beamforming on broadband array data that achieves essentially linear scaling in the number of sensors M, comparable to narrowband FFT methods. It encodes N samples from each of M sensors into coefficients via a non-uniform spaced Fourier transform, then forms each of B beams by solving a small Toeplitz-structured linear system. This replaces the O(MB) cost of delay-and-sum beamformers while supporting high accuracy and enabling efficient beamspace tasks such as off-grid angle interpolation and interferer nulling. A reader would care because it makes real-time broadband processing feasible for large sensor arrays without sacrificing performance.

Core claim

Our algorithm works by taking N samples off of each of M sensors and encoding the sensor outputs into a set of coefficients using a special non-uniform spaced Fourier transform. From these coefficients, each beam is formed by solving a small system of equations that has Toeplitz structure. The total runtime complexity is O(M log N + B log N) operations per sample.

What carries the argument

non-uniform spaced Fourier transform that encodes the broadband sensor outputs into coefficients, followed by solving a small Toeplitz-structured linear system per beam

Load-bearing premise

The non-uniform spaced Fourier transform accurately captures the broadband sensor outputs for the chosen N and the resulting small Toeplitz systems remain well-conditioned and solvable to high precision for the array geometries and bandwidths of interest.

What would settle it

Run the algorithm on broadband signals with increasing fractional bandwidth or irregular array spacing and measure whether beamforming accuracy falls below a set threshold or the observed runtime deviates from the predicted O(M log N + B log N) scaling.

Figures

Figures reproduced from arXiv: 2512.08887 by Coleman DeLude, Justin Romberg, Mark Davenport, Nakul Singh.

Figure 1
Figure 1. Figure 1: The axes above are in the 2D discrete-time Fourier transform domain; the horizontal ω1 axis is the spatial frequency (Fourier transform across array elements m), the vertical ω2 axis is the frequency in time (Fourier transform across samples n). (a) The array measurements {ym,n} for a signal bandlimited to Ω at carrier frequency fc are in the linear span of discrete-time sinusoids with frequencies along th… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Beamformed SNR vs. Nominal SNR for a 128-element ULA with 64 snapshots. (b) Beamformed SNR vs. Nominal SNR for a 16×16-element UPA with 64 snapshots. In contrast, delay and sum beamformers experience saturation at higher SNRs due to the truncation of the interpolating filter. This induces a bias that begins to dominate the error as the noise variance decreases. This truncation effect is even more prono… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Runtime vs. array size for a ULA. The number of beams scales as B ≈ M for all algorithms. Two filter lengths R = 16, 32 are used for measuring the DS, FDS runtime. (b) Runtime as a function of array size for a UPA, where the sensors are arranged in a square grid of dimensions √ M × √ M. For all algorithms, the number of beams scales approximately linearly with the number of sensors, i.e., B ≈ M. Two fi… view at source ↗
Figure 4
Figure 4. Figure 4: (a) Variance vs. number of snapshots N plotted for five values of L. As N → ∞, the variance approaches the ideal array gain. (b) Bias vs. γ plotted for Fourier extension and Fourier series. Fourier extension has negligible bias once we go over the bound given in (4) by a modest amount. The variance will be the trace of this kernel matrix, as E[||F∗ (F HF + δI) −1Fn||2 L2 ] = Z T 0 r(t, t)dt = σ 2X k,l Qk,l… view at source ↗
Figure 5
Figure 5. Figure 5: (a) Zoomed-in beam pattern for interpolated and true beams on a 128-element ULA. A total of 256 beams were formed, out of which only 8 were used for interpolating to an off-grid angle θ = π 3 . Beam patterns for 20 frequencies in [fc − Ω, fc + Ω] are superimposed to get the frequency response over the entire band. (b) Interference bias for a 128-element ULA with 64 snapshots. The interference bias is small… view at source ↗
Figure 6
Figure 6. Figure 6: (a) Runtime for interference cancellation vs. number of snapshots. The experiment was done for single and multi-interference cancellation. In both cases, we observe that the beamspace-based nulling is much faster than computing the projection in array space. (b) Beam pattern for a 128-element ULA after interference cancellation with 64 snapshots. R=16 tap length is used for DS and FDS. Beam patterns for 20… view at source ↗
Figure 7
Figure 7. Figure 7: (a) Runtime vs. array size for a Linear array with non-uniform array spacing. The elements were perturbed with random offsets in their position. The number of beams scales as B ≈ M for all algorithms. Two filter lengths R = 16, 32 are used for measuring the DS runtime. (b) Runtime as a function of array size for a planar array where the sensors are arranged in a square grid of dimensions √ M × √ M. The ele… view at source ↗
Figure 8
Figure 8. Figure 8: A pseudo polar sampling grid in the 3D Fourier domain. The samples are equispaced along the ω3 axis and in slope along each slanted line. where ω1(ℓ) = − π sin φ sin ϕ ˜fc  fc + ϵ LTs · (ℓ − L 2 )  ω2(ℓ) = − π sin φ cos ϕ ˜fc  fc + ϵ LTs · (l − L 2 )  ω3(ℓ) = 2πϵ L ·  ℓ − L 2  Analogous to the linear case, this basically means that F H θ y is a collection of L equally spaced samples along a slanted l… view at source ↗
read the original abstract

We present a new computationally efficient method for multi-beamforming in the broadband setting. Our "fast beamspace transformation" forms $B$ beams from $M$ sensor outputs using a number of operations per sample that scales linearly (to within logarithmic factors) with $M$ when $B\sim M$. While the narrowband version of this transformation can be performed efficiently with a spatial fast Fourier transform, the broadband setting requires coherent processing of multiple array snapshots simultaneously. Our algorithm works by taking $N$ samples off of each of $M$ sensors and encoding the sensor outputs into a set of coefficients using a special non-uniform spaced Fourier transform. From these coefficients, each beam is formed by solving a small system of equations that has Toeplitz structure. The total runtime complexity is $\mathcal{O}(M\log N+B\log N)$ operations per sample, exhibiting essentially the same scaling as in the narrowband case and vastly outperforming broadband beamformers based on delay and sum whose computations scale as $\mathcal{O}(MB)$. Alongside a careful mathematical formulation and analysis of our fast broadband beamspace transform, we provide a host of numerical experiments demonstrating the algorithm's favorable computational scaling and high accuracy. Finally, we demonstrate how tasks such as interpolating to ``off-grid" angles and nulling an interferer are more computationally efficient when performed directly in beamspace.

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 / 2 minor

Summary. The manuscript presents a fast beamspace transformation for broadband multi-beamforming from M sensors to B beams. It encodes N samples per sensor via a non-uniform spaced Fourier transform into coefficients, then forms each beam by solving a small Toeplitz-structured linear system. The claimed per-sample complexity is O(M log N + B log N), matching narrowband FFT scaling and outperforming O(MB) delay-and-sum methods. A mathematical formulation, complexity analysis, numerical experiments on scaling and accuracy, and beamspace applications (off-grid interpolation, nulling) are provided.

Significance. If the accuracy and conditioning claims hold with rigorous support, the method would enable efficient coherent broadband beamforming at large array sizes, with practical benefits for sonar, radar, and communications where delay-and-sum is too slow. The beamspace view also simplifies downstream tasks. The numerical scaling results and explicit complexity derivation are strengths.

major comments (2)
  1. [Mathematical formulation and analysis] Mathematical formulation and analysis section: The O(M log N + B log N) complexity and high-accuracy claim rest on the non-uniform spaced Fourier transform exactly (or to machine precision) encoding the broadband sensor outputs and the resulting per-beam Toeplitz systems remaining well-conditioned. No explicit error bounds on the transform approximation for wide fractional bandwidths or irregular geometries, nor conditioning analysis (e.g., condition-number bounds or stabilization requirements), are provided; this is load-bearing for the precision guarantee without extra operations that would alter the stated complexity.
  2. [Numerical experiments] Numerical experiments section: The reported accuracy and scaling results lack details on the specific array geometries, fractional bandwidths, and signal types tested, as well as quantitative error metrics (e.g., beam pattern deviation or MSE versus exact delay-and-sum) across the full parameter range; without these, the claim of 'high accuracy' for arbitrary broadband cases remains only moderately supported.
minor comments (2)
  1. [Mathematical formulation] Notation for the non-uniform Fourier transform coefficients and the Toeplitz matrix entries should be defined more explicitly with respect to the sensor positions and frequency support to improve readability.
  2. [Numerical experiments] Figure captions for the scaling plots should include the exact values of M, N, B, and bandwidth used in each experiment for reproducibility.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their positive assessment and constructive comments on our manuscript. We agree that the points raised warrant minor revisions to strengthen the mathematical analysis and experimental details, and we address each major comment below.

read point-by-point responses

  1. Referee: [Mathematical formulation and analysis] Mathematical formulation and analysis section: The O(M log N + B log N) complexity and high-accuracy claim rest on the non-uniform spaced Fourier transform exactly (or to machine precision) encoding the broadband sensor outputs and the resulting per-beam Toeplitz systems remaining well-conditioned. No explicit error bounds on the transform approximation for wide fractional bandwidths or irregular geometries, nor conditioning analysis (e.g., condition-number bounds or stabilization requirements), are provided; this is load-bearing for the precision guarantee without extra operations that would alter the stated complexity.

    Authors: We appreciate the referee's identification of this foundational aspect. The non-uniform Fourier transform in our formulation is constructed to encode the broadband signals exactly for the given sampling, but we agree that explicit error bounds for wide fractional bandwidths and irregular geometries, along with conditioning analysis of the Toeplitz systems, would provide stronger support. In the revised manuscript, we will add a dedicated subsection deriving error bounds for the transform under these conditions and providing condition-number bounds with stabilization requirements where applicable. These additions will preserve the stated O(M log N + B log N) complexity. revision: yes

  2. Referee: [Numerical experiments] Numerical experiments section: The reported accuracy and scaling results lack details on the specific array geometries, fractional bandwidths, and signal types tested, as well as quantitative error metrics (e.g., beam pattern deviation or MSE versus exact delay-and-sum) across the full parameter range; without these, the claim of 'high accuracy' for arbitrary broadband cases remains only moderately supported.

    Authors: We agree that expanded details on the experimental parameters and quantitative metrics would better substantiate the accuracy claims. In the revised numerical experiments section, we will explicitly describe the array geometries (e.g., uniform linear arrays with specified inter-element spacing), the fractional bandwidth ranges tested (including wideband cases up to 100%), and the signal types (e.g., broadband noise and chirp signals). We will also include quantitative metrics such as beam pattern deviation and MSE compared to exact delay-and-sum beamforming, reported across the full parameter space. revision: yes

Circularity Check

0 steps flagged

No circularity; derivation uses standard non-uniform FFT and Toeplitz properties

full rationale

The claimed O(M log N + B log N) complexity follows directly from applying a non-uniform spaced Fourier transform to N samples per sensor followed by solving small Toeplitz systems per beam. These operations rest on well-known properties of Fourier bases and structured linear algebra solvers; no equation or complexity claim reduces by construction to a fitted parameter, self-defined quantity, or load-bearing self-citation. The paper's analysis and experiments are self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The approach relies on standard signal-processing assumptions about array data and transform properties rather than introducing many new fitted parameters or invented entities.

free parameters (1)
  • N (samples per sensor)
    Chosen to balance accuracy and cost for given bandwidth; not derived from first principles.
axioms (1)
  • domain assumption Sensor outputs can be accurately encoded by a non-uniform spaced Fourier transform for broadband signals
    Invoked in the encoding step of the algorithm.

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