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Analysis and Approximation of Spatially Wideband Array Factor of Thinned Antenna Arrays

Marcin Wachowiak, André Bourdoux, Sofie Pollin

arXiv:2609.39480v1eess.SP

TL;DR

Thinned arrays can provide larger apertures with fewer elements, but their operation can enter the spatially wideband regime, where bandwidth modifies the array factor. This work extends a convolution-based SWB AF approximation and sidelobe analysis to thinned arrays; expected sidelobe levels decrease with distance from the mainlobe.

  • Problem

    Thinned arrays can increase aperture enough to violate the SNB criterion, making accurate characterization of their SWB array factor necessary.

    Source: A. Problem Statement · p. 1

  • Method

    The paper expresses the SWB AF as a spatially variant convolution and extends expected SL and PSL analysis from SNB thinned arrays to the SWB regime.

    Source: C. Contributions · pp. 1–2, E. Generalization of the SWB AF… · p. 4

  • Results

    Expected SWB SL and PSL are approximately inversely proportional to DλBf|∆u|, with suppression more effective farther from the mainlobe.

    Source: §I · p. 10

  • Takeaways & Limitations

    For thinned arrays, noise-like sidelobes away from the mainlobe are particularly susceptible to averaging by the SWB kernel.

    Source: C. Contributions · p. 2, §III · p. 10

    The closed-form SWB SL and PSL expressions are for uniformly thinned arrays with a uniform spectrum.

    Source: §III · p. 10

Abstract

from arXiv · show

This work investigates the spatially wideband (SWB) array factor (AF) of linear thinned antenna arrays. The SWB AF is formulated as a matched filter and expressed as a spatially variant convolution of a spatially narrowband (SNB) AF and a SWB kernel. Originally derived for uniform arrays, the SWB AF approximation is revisited and extended to thinned arrays. The approximation is shown to remain accurate across a wide range of parameters. Representing the SWB AF as a convolution offers an intuitive interpretation of how the bandwidth-aperture product modifies the SNB AF. Specifically, the SWB kernel acts as an averaging window for the SNB AF, suppressing the sidelobe levels (SL) of the SNB AF. In thinned arrays, the sidelobes away from the mainlobe can be treated as random, noise-like, making averaging and suppression particularly effective. Next, the analytical formulas for the expected SL and peak-to-sidelobe level (PSL) in SNB thinned arrays are first introduced and subsequently extended to the SWB regime. Finally, simple closed-form expressions for the expected SL and PSL levels of the uniformly thinned arrays with a uniform spectrum are provided. The expected SWB SL and PSL are formulated as functions of the bandwidth-aperture product and are shown to steadily decrease with distance from the mainlobe.

I. INTRODUCTION

Thinning enables larger apertures with fewer antenna elements, but can push arrays beyond the spatially narrowband regime and require accurate spatially wideband modeling. The paper extends the SWB array-factor approximation and sidelobe analysis to thinned arrays.

A. Problem Statement

  • Thinning increases aperture and randomizes element spacing to reduce antenna count, but the resulting aperture can push arrays into the SWB regime.Accurate modeling then requires characterizing the SWB array factor.Source: A. Problem Statement · p. 1

B. Relevant works

  • Earlier thinned-array studies focused on SNB analysis, while included bandwidth in PSL analysis but lacked closed-form expressions and detailed suppression insight.Source: B. Relevant works · p. 1

C. Contributions

  • The paper extends the SWB AF to thinned arrays, using a spatially variant convolution to separate SWB and SNB contributions.It extends expected SL and PSL analysis to SWB arrays and evaluates suppression across a wide range of parameters.Source: C. Contributions · pp. 1–2

A. Spatially narrowband array factor (SNB AF)

The SNB array factor is a weighted sum of elementwise relative phase shifts evaluated at a single frequency. Its accuracy requires the propagation delay across the aperture to be much smaller than the signal coherence time.

  • The SNB AF sums weighted relative phase shifts across antenna elements for a single frequency.The antenna weights specify each element’s amplitude contribution.Source: A. Spatially narrowband array factor… · p. 2
  • The SNB criterion is DλBf ≪ 1, requiring aperture propagation delay to be much smaller than signal coherence time.Here D is aperture, B is bandwidth, and the criterion can become restrictive for large apertures or moderate fractional bandwidths.Source: A. Spatially narrowband array factor… · p. 2
  • The paper uses “spatially” to distinguish regimes jointly determined by array extent and signal bandwidth, not bandwidth alone.Source: A. Spatially narrowband array factor… · p. 2

B. Spatially wideband array factor (SWB AF)

The SWB AF matches the transmitted or received signal jointly across spatial and frequency domains, integrating the SNB AF over the signal bandwidth. Fixing the evaluation and beamforming points at the same distance preserves an angular AF interpretation.

  • The SWB AF integrates the SNB AF across the signal bandwidth, weighted by the signal energy spectrum, as a matched filter.The frequency-domain matching uses the signal spectrum’s squared magnitude.Source: B. Spatially wideband array factor… · p. 2
  • Fixing the observation and beamforming points at the same distance keeps the SWB AF as an angular cross-cut rather than a range-dependent measure.For a far-field linear array, the point of interest is described by distance and angle, and the angular variable is direction-sine difference.Source: B. Spatially wideband array factor… · p. 2
  • The SWB AF can also be expressed as a product of the spatial window and the signal autocorrelation function.The direct SWB AF expression does not readily show how bandwidth-aperture product modifies the SNB AF, motivating approximations.Source: B. Spatially wideband array factor… · p. 3

C. Approximation of spatially wideband array factor

The approximation represents the SWB AF as a spatially variant convolution of the SNB AF and an SWB kernel. A continuous, aperiodic approximation is valid under stated sampling and aliasing conditions; uniform spectrum gives the strongest sidelobe-averaging case considered.

  • The SWB AF formulation is interpreted as a spatially variant convolution, with the SWB kernel weighting the SNB AF.The convolution separates the SNB AF contribution from the bandwidth-dependent kernel.Source: C. Approximation of spatially wideband… · p. 3
  • For the SWB kernel in (11), the antenna-element sum is approximated by a continuous aperture integral, accurate when Bf ≪ 1/(2daλ), with daλ = da/λ denoting spacing normalized to wavelength.Source: C. Approximation of spatially wideband… · p. 3
  • The aperiodic convolution approximates the periodic one when the SNB AF mainlobe is much narrower than alias spacing and M ≫ 1.The SNB AF acts as an anti-aliasing filter, so the kernel need not decay strictly before the first alias position.Source: C. Approximation of spatially wideband… · p. 3
  • A uniform amplitude spectrum yields the narrowest SWB kernel and strongest sidelobe averaging among the spectra considered.Spectrum tapering concentrates the kernel peak, narrowing its width and reducing sidelobe averaging and suppression.Source: C. Approximation of spatially wideband… · p. 3

D. Thinned antenna arrays

Thinning selects antenna slots according to a spatial density profile, with the fill parameter controlling the expected number of active antennas. The maximum achievable fill depends on that profile.

D. Thinned antenna arrays

  • The fill parameter η controls the expected number of antennas as a fraction of the fully populated array size.The number of active antennas is random under density-based thinning.Source: D. Thinned antenna arrays · p. 4
  • Each slot is represented by a binary occupation variable, with activation probability set by the scaled absolute spatial density profile.The scaling parameter α is chosen based on η to maintain the desired antenna count.Source: D. Thinned antenna arrays · p. 4
  • As the array grows, the maximum achievable fill becomes a function of the spatial density profile; Table I reports η_max by profile.The supplied passage does not provide the table’s individual values.Source: D. Thinned antenna arrays · p. 4, Table I · p. 4

E. Generalization of the SWB AF approximation for thinned arrays

The SWB AF approximation extends to thinned arrays by separating the SNB AF from the SWB kernel. Its accuracy conditions carry over, while kernel averaging suppresses thinned-array sidelobes.

E. Generalization of the SWB AF approximation for thinned arrays

  • The universal approximation extends to thinned arrays without substantial changes to the signal model by separating the SNB AF and SWB kernel contributions.The separation gives a spatially variant convolution interpretation.Source: E. Generalization of the SWB AF… · p. 4
  • An arbitrary spatial window extends the signal model to arbitrary array factors, while negative density-taper values correspond to phase-flipped elements.The window scaling normalizes precoder power to unity.Source: E. Generalization of the SWB AF… · p. 4
  • The smoothness condition Bf ≪ 1/(2daλ) is unchanged, and accuracy also requires Mth ≫ 1.The SNB AF acts as a narrow spatial filter that reduces approximation error from aliasing.Source: E. Generalization of the SWB AF… · pp. 4–5
  • The SWB kernel averages noise-like sidelobes across the spatial domain, substantially reducing their power relative to the SNB AF.The components and averaging effect are illustrated for a uniformly thinned array.Source: E. Generalization of the SWB AF… · p. 5, Figure 2 · p. 5
  • Increasing fractional bandwidth widens the SWB kernel and reduces sidelobes; increasing aperture leaves kernel width fixed but enables stronger averaging of variable sidelobes.The cited figures compare exact and approximate SWB AFs across selected bandwidths and aperture sizes.Source: E. Generalization of the SWB AF… · p. 5, Figure 3 · p. 5, Figure 4 · p. 6
  • The approximation also extends to arbitrarily spaced nonuniform arrays by taking a dense-grid limit as daλ → 0 and M → ∞.The aperture is held fixed in the limit.Source: E. Generalization of the SWB AF… · p. 5

F. Statistical sidelobe level (SL) analysis of SNB AF

The section models sidelobes in density-tapered thinned arrays through their random spatial window and derives expected sidelobe behavior from the resulting array-factor statistics. Away from the mainlobe, the fluctuation term supports an average-SL estimate, with a simpler expression for uniform spatial profiles.

F. Statistical sidelobe level (SL) analysis of SNB AF

  • The expected AF corresponds to an equivalent deterministic amplitude-tapered array, while the sidelobe fluctuation contributes a noise-like term to expected power.Independent occupation variables allow the expected product for distinct elements to be written as a product of expectations.Source: F. Statistical sidelobe level (SL)… · p. 6
  • Far enough from the mainlobe and its first sidelobes, the deterministic AF sidelobes are treated as negligible relative to the uniform noise-like fluctuation term.This far-sidelobe assumption is the basis for approximating average SL from the statistical component.Source: F. Statistical sidelobe level (SL)… · p. 6
  • Average SL compares average sidelobe power with the deterministic AF peak, which occurs at Δu=0 for an even, non-negative spatial profile.Source: F. Statistical sidelobe level (SL)… · p. 7
  • For a uniform spatial profile p_m=η, the average-SL expression simplifies to a familiar result.Source: F. Statistical sidelobe level (SL)… · p. 7

G. Statistical peak-to-sidelobe level (PSL) analysis of SNB AF

The section derives expected PSL from far-sidelobe statistics using level-crossing and extreme-value approximations. For a uniform spatial profile, analytical SL and PSL predictions match Monte Carlo results and are uniform across angles.

G. Statistical peak-to-sidelobe level (PSL) analysis of SNB AF

H. Approximation of statistical sidelobe suppression in SWB regime

For a uniform spectrum, the SWB kernel can be approximated by a rectangular window, making the SWB AF a variable-width moving average of the SNB AF. The resulting statistical SL and PSL predictions agree with simulations, while the suppression varies with angle.

H. Approximation of statistical sidelobe suppression in SWB regime

  • For large DλBf|∆u|, approximating the sine integral by a sign function yields a unit-area rectangular SWB kernel of width Bf|∆u|.This gives the SWB AF an interpretation as a variable-width moving average of the SNB AF.Source: H. Approximation of statistical… · p. 8
  • SWB sidelobe predictions use the complex fluctuation term for sufficiently large ∆u and derive its variance by integrating the sidelobe autocorrelation against the kernel.For a rectangular kernel, its autocorrelation is triangular, whose Fourier transform is a squared sinc function.Source: H. Approximation of statistical… · pp. 8–9
  • The expected SWB SL and PSL are obtained by substituting the wideband variance and second central moment into their corresponding statistical expressions.The PSL calculation updates the SNB second central moment using the SWB per-element variance.Source: H. Approximation of statistical… · p. 9
  • Predicted SWB sidelobe levels agree with Monte Carlo simulations, while kernel averaging reduces SL and PSL versus SNB with angle-varying suppression.The SWB result is shown for Bf = 0.1; suppression varies because the kernel depends on angle.Source: H. Approximation of statistical… · p. 9

I. Approximation of SL and PSL of uniformly thinned arrays

For uniformly thinned arrays with a uniform spectrum, the expected SWB SL and PSL admit closed-form approximations. Their suppression strengthens with the bandwidth-aperture product and distance from the mainlobe, though accuracy degrades near the mainlobe.

I. Approximation of SL and PSL of uniformly thinned arrays

  • For a uniform spatial profile, activation probability is η and per-element variance is 1 − η, enabling closed-form expected SL and PSL expressions for uniformly thinned arrays.The derivation uses a uniform spectrum.Source: §I · pp. 9–10
  • The closed-form derivation approximates discrete sums by integrals, requiring Bf ≪ 1/(2daλ) for discretized uniformly thinned arrays.The Riemann-sum approximation is accurate when the function varies smoothly and slowly across elements.Source: §I · pp. 9–10
  • In the SWB regime, expected SL scales relative to the SNB expression by DλBf|∆u|.Source: §I · p. 10
  • Expected SWB SL and PSL are approximately inversely proportional to DλBf|∆u|, with suppression more effective farther from the mainlobe.Approximation accuracy degrades near the mainlobe, where ν is small.Source: Figure 9 · p. 10, §I · p. 10

III. CONCLUSION

The work extends SWB array-factor analysis and statistical sidelobe formulas to thinned arrays. For uniformly thinned arrays with a uniform spectrum, expected SL and PSL decrease with the bandwidth-aperture product and angle difference, with suppression increasingly effective away from the mainlobe.

III. CONCLUSION

  • The work extends the SWB array factor to thinned arrays and represents it as a spatially variant convolution of the SNB array factor.Source: §III · p. 10
  • The convolution interpretation separates the SWB contribution and explains how the bandwidth-aperture product averages and smooths the SNB array factor.Source: §III · p. 10
  • Analytical expected SL and PSL formulas are extended to SWB arrays, including arbitrary element density profiles.Source: §III · p. 10
  • For uniformly thinned arrays with a uniform spectrum, expected SWB SL and PSL are inversely proportional to the bandwidth-aperture product and angle difference.Suppression becomes increasingly effective away from the mainlobe.Source: §III · p. 10
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