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Wideband Large-Array Processing and Sparse Design for Angle Imaging

Ziyu Zhou, Wei Dai

arXiv:2608.13005v1eess.SP

TL;DR

Wideband virtual arrays offer dense angle imaging with sparse physical arrays, but irregular element spacing can undermine stable recovery. This paper introduces a geometry-based coverage criterion with deterministic condition-number guarantees and uses it to design sparse arrays for stable full-field imaging, validated by numerical results.

  • Problem

    Frequency-induced virtual elements increase spatial sampling, but their irregular gaps can make imaging matrices ill-conditioned and prevent stable angle recovery.

  • Method

    The paper links virtual-aperture coverage to conditioning, derives deterministic condition-number bounds, and designs non-uniform sparse arrays using the coverage criterion.

  • Results

    Numerical results show that coverage-criterion designs remain stable under noise and achieve moderate condition numbers across reported configurations.

  • Takeaways & Limitations

    Geometric coverage provides practical guidance for designing sparse arrays that maintain stable full-field angle imaging with fewer physical antennas.

Abstract

from arXiv · show

This paper shows that wideband large-array processing can recover a large number of angle pixels with far fewer antenna elements. The key advantage of wideband signaling is that different frequencies induce different virtual arrays, whose union forms a virtual array with a substantially increased number of effective virtual elements. Thus, a sparse physical array can support far more spatial samples than physical antennas. Motivated by this capability, we study the recovery of angular responses across the full field of view $[-90^\circ, 90^\circ)$, discretized according to the improved angular resolution, and refer to this sensing regime as angle imaging. However, the resulting virtual array is inherently irregular, clustered, and does not automatically guarantee stable recovery. To address this challenge, we introduce a coverage criterion that estimates the number of stably recoverable angle pixels, without computationally intensive singular-value-based conditioning tests over candidate image dimensions. For systems satisfying this criterion, we theoretically establish deterministic condition-number bounds that characterize stable angle imaging. Building on this criterion, we derive non-uniform sparse array designs that minimize the number of physical antennas while maintaining recovery over the full field of view. Simulation results show that the proposed criterion provides practical guidance for stable system design, and that the resulting sparse arrays can recover substantially more angle pixels than the number of physical antennas, with representative designs supporting over ten times as many angle pixels as physical antennas.

I. INTRODUCTION … C. Received Signal Model

The paper develops stable, hardware-efficient wideband angle imaging by exploiting frequency-induced virtual arrays while addressing their irregular geometry and frequency-dependent channels. It formulates the received-signal model, introduces a coverage-based stability framework, and derives sparse-array designs for dense full-FoV recovery.

  • I. INTRODUCTION: Wideband frequency diversity creates virtual spatial samples whose union can greatly exceed the number of physical antennas, enabling angle imaging without angular sparsity assumptions.The approach targets dense angular responses over a continuum of directions, with angular resolution, numerical stability, and hardware cost as key requirements.
  • I. INTRODUCTION: Irregular, clustered virtual arrays can contain large gaps that correlate imaging-matrix columns and destabilize recovery, especially with frequency-dependent scattering responses.The paper therefore treats virtual-aperture geometry as central to stable angle imaging rather than relying only on total virtual-element count.
  • A. Our Contributions: The coverage criterion tests whether a physical array and signal band stably support a prescribed number of contiguous angle pixels and yields deterministic condition-number bounds.This replaces computationally intensive conditioning evaluations with a geometric test based on the largest uncovered virtual-aperture gap.
  • A. Our Contributions: A closed-form non-uniform sparse-array design derives antenna positions directly from virtual-aperture coverage requirements, avoiding iterative or combinatorial array search.Numerical results verify that the designed arrays are well conditioned and support substantially more angle pixels than physical antennas.
  • A. Our Contributions: The paper models dense angular-response reconstruction over a prescribed FoV with frequency-dependent steering, where each frequency-antenna pair is treated as a virtual spatial sample.Its nominal angular resolution is governed by the highest operating frequency rather than the center frequency, and the model incorporates frequency-dependent scattering responses.
  • B. Notation and Organization: The paper defines notation for matrices, vectors, scalars, ordered sets, eigenvalues, singular values, and identity matrices, then organizes the remaining sections around modeling, stability, design, results, and conclusions.The organization proceeds from the wideband signal model through virtual-aperture properties, coverage analysis, sparse-array design, numerical results, and conclusions.
  • A. Transmitted Signal Model: The transmitted wideband probing signal occupies [fL, fH], has bandwidth B = fH − fL, and is sampled at Mf uniformly spaced frequencies with spacing ∆f = B/(Mf −1).The signal is transmitted using one transmitting antenna, with tones indexed over the sampled frequency set.
  • B. Frequency-Dependent Wireless Channel: Frequency-dependent channel variation is modeled through total phase excursion ∆Φ = 2πr(Mf −1) and a finite-dimensional Fourier-series expansion, with r ∈[0, 1] controlling variation.The coefficient γ(fmf, θ) incorporates path loss, phase delay, reflection, scattering, and target radar cross section, while Nb = 1 represents a flat channel.

D. Inverse problem for imaging · III. OPPORTUNITIES AND CHALLENGES OF WIDEBAND ANGLE IMAGING · A. Virtual-Array Interpretation

Wideband angle imaging is posed as an inverse problem and interpreted through a frequency-induced virtual array. This virtual array increases effective spatial sampling but can be irregular, causing ill-conditioning and unstable recovery.

  • D. Inverse problem for imaging: Angle imaging recovers G from noisy measurements y through a linear inverse system.The angular response γ(f_mf, n_θ) is reconstructed from G⋆ using (3).
  • III. OPPORTUNITIES AND CHALLENGES OF WIDEBAND ANGLE IMAGING: Each physical antenna-frequency sample pair acts as a virtual spatial sample, allowing sparse arrays to support more angle pixels.Additional virtual samples enlarge the effective spatial sampling set and mitigate spatial aliasing found in sparse narrowband arrays.
  • III. OPPORTUNITIES AND CHALLENGES OF WIDEBAND ANGLE IMAGING: Non-uniform virtual samples can create large aperture gaps, making the system matrix ill-conditioned and recovery unstable.The virtual-array geometry therefore creates both an opportunity for expanded sampling and a challenge for stable imaging.
  • III. OPPORTUNITIES AND CHALLENGES OF WIDEBAND ANGLE IMAGING: The section formalizes the virtual-array interpretation before discussing wideband angle imaging’s opportunities and challenges.This structure connects the frequency-induced sampling mechanism to its recovery implications.
  • A. Virtual-Array Interpretation: Given a frequency sample set F and physical linear array P_a, the virtual array is defined as their collection of virtual elements.This definition establishes the frequency- and geometry-dependent sampling set used for wideband imaging.
  • A. Virtual-Array Interpretation: Repeated virtual-element values are retained as separate virtual samples.Multiplicity is preserved in the virtual sampling representation.
  • A. Virtual-Array Interpretation: The virtual array has the physical array’s width, with its largest location attained at f = f_H.Its sampling locations are jointly induced by frequency and spatial samples.
  • A. Virtual-Array Interpretation: Even with uniformly spaced antennas and frequency samples, virtual sampling locations are generally denser and non-uniform.The construction is illustrated in Fig. 1.

B. Opportunities … A. Coverage Criterion

Wideband signaling improves angular resolution and mitigates spatial aliasing, enabling sparse arrays with larger inter-element spacings. Because irregular virtual-element gaps can still destabilize recovery, the paper introduces a geometry-driven coverage criterion that estimates stably recoverable angle pixels and provides condition-number guarantees.

  • B. Opportunities: Wideband signaling improves theoretical angular resolution by replacing the narrowband carrier frequency fc with the larger frequency fH.The wideband resolution is governed by fHW/c through the effective window width.
  • B. Opportunities: Wideband frequency diversity relaxes the half-wavelength spacing constraint, allowing much larger inter-element spacings for sparse-array design.Across frequencies, spatially aliased responses become distinguishable.
  • B. Opportunities: Under d = 1.2c/fc, the narrowband response exhibits prominent aliasing peaks, whereas the wideband response remains free of such ambiguities.The comparison uses the same sparse ULA configuration in both cases.
  • C. Challenges: More virtual elements do not ensure stable recovery because irregular virtual-array sampling can create large gaps and highly correlated system-matrix columns.The resulting virtual aperture is clustered and may be under-sampled despite having many elements.
  • C. Challenges: Stable imaging is characterized by the maximum number Nθ of contiguous angle pixels supported with a controlled spectral condition-number bound.The associated supported FoV is a contiguous u-domain interval with width approximately Nθures; full-FoV imaging occurs when it covers the entire angular domain.
  • IV. COVERAGE CRITERION FOR STABLE WIDEBAND ANGLE IMAGING: The coverage criterion is an SVD-free, geometry-driven rule that uses the maximum gap of the effective virtual array instead of testing each candidate system-matrix condition number.Under the criterion, the associated system matrices admit explicit condition-number upper bounds.
  • A. Coverage Criterion: Supporting Nθ angle pixels requires adjacent virtual-element gaps no larger than W/Nθ, ensuring coverage of the virtual aperture.For each effective virtual array, full coverage is achieved when the coverage length is at least its maximum wrap-around inter-element spacing, and this condition must hold for all i.
  • A. Coverage Criterion: The threshold ϵ controls effective virtual-array size: larger ϵ reduces active frequency samples and supported Nθ, while smaller ϵ may increase supported Nθ.The criterion’s full-FoV pixel count is denoted Nθ,max ≜⌊2WfH/c⌋.

B. Conditioning Analysis under Flat Channels

For flat channels, the coverage criterion (CC) is given a deterministic conditioning interpretation through a Voronoi-weighted system matrix. The maximum virtual-array gap controls both the supportable angular-grid size and an explicit condition-number upper bound, enabling SVD-free stability characterization.

  • Conditioning criterion: Stability is quantified by the spectral condition number κ(A) = σmax(A)/σmin(A), with controlled upper bounds defining a stable system.The analysis applies this definition to full-column-rank flat-channel system matrices.
  • Voronoi weighting: Voronoi weights convert the non-uniform virtual-array geometry into a weighted flat-channel matrix through deterministic row scaling of the original observations.The weighting requires no additional measurements and preserves the underlying observation model through row weighting.
  • Condition-number bound: Theorem 4 establishes an explicit upper bound on the condition number of the Voronoi-weighted flat-channel matrix when the angular grid is selected according to the CC.The bound depends on the maximum circular inter-element gap of the virtual array.
  • Condition-number interpretation: The maximum virtual-array gap determines the CC-based angular-grid size and yields the corresponding condition-number bound for the weighted system.The weighted system recovers the same angular response as the original system because it is obtained by applying a scaling matrix to the observations.
  • Implication: The CC provides an SVD-free route to characterize supportable angular-grid size, while stability is quantified through an explicit condition-number upper bound.This establishes a directly controlled weighted inverse problem for flat-channel angle imaging.

C. Conditioning analysis under frequency-dependent channels

For frequency-dependent channels, stability is analyzed through weighted basis-wise blocks and their concatenation. Stable angle imaging follows when basis-wise spectral floors control conditioning and inter-basis leakage remains sufficiently small.

  • Conditioning analysis: The frequency-dependent analysis defines a Voronoi-weighted system matrix and studies stability without changing the angular-response coefficients to be recovered.The analysis therefore focuses on conditioning of the weighted frequency-dependent system matrix.
  • Conditioning analysis: The conditioning analysis first bounds each basis-wise block’s self-correlation energy, then controls concatenation through cross-correlation among different basis-wise components.This two-step structure yields condition-number bounds for the full system from block-level properties and inter-block coupling.
  • Basis-wise conditioning: Each active-frequency row-submatrix is full-column rank, while retaining all virtual elements provides controlled singular-value and condition-number bounds for each basis-wise block.The lower bound on the minimum singular value depends on the coverage threshold ϵ.
  • Basis-wise conditioning: A smaller ϵ may support a larger N_ccθ for fixed measurements but weakens each block’s spectral floor, whereas a larger ϵ is more conservative and improves conditioning.The threshold therefore trades recoverable angle-pixel count against the strength of the spectral floor.
  • Full-system conditioning: Theorem 6 controls the full condition number when inter-basis leakage δ remains below the diagonal-block spectral floor.Diagonal blocks accumulate dominant energy constructively, while off-diagonal blocks are cancellation-dominated and typically weak, producing approximate block diagonal dominance.

V. NON-UNIFORM ARRAY DESIGN

This section develops a non-uniform sparse array design that enables all effective virtual arrays to satisfy the coverage criterion on a common full-FoV angular grid. The design derives the finest common angle spacing and antenna placements, yielding arrays substantially sparser than conventional ULAs while linking required density to channel variation.

  • Design objective: The design enforces the coverage criterion across all effective virtual arrays on a common full-FoV angular grid.It first determines the finest common angle spacing supported by all effective sub-bands, then derives the corresponding physical-antenna placement.
  • Common angular grid: The resulting array supports full-FoV angle imaging with spacing ∆u = c/(Wf⋆), where f⋆ ≜ fL + B/Nb.The common spacing is selected to account for the maximum virtual extent available from the lowest-frequency effective sub-band.
  • Common angular grid: A terminal gap is unavoidable for higher-frequency effective sub-bands at the finest spacing, and adding antennas within [0, W] cannot remove it.This limitation motivates reducing the common cyclic-aperture length before deriving the finest shared spacing.
  • Antenna placement: The proposed antenna locations are obtained by placing each antenna as sparsely as permitted by all active frequency sets and solving the resulting recursion.The closed-form locations are expressed as the conventional spacing c/(2fH) multiplied by a dimensionless factor that can substantially exceed one.
  • Effect of channel variation: A smaller Nb requires fewer physical antennas for a prescribed width W, whereas increasing Nb requires a denser array to maintain the same imaging capability.Smaller Nb gives wider effective sub-bands and more frequency diversity; larger Nb uses more diversity to accommodate channel variation.

VI. NUMERICAL RESULTS

The numerical results examine four representative frequency bands using a fixed 40 MHz frequency spacing and evaluate both unweighted and weighted system-matrix condition numbers.

  • VI. NUMERICAL RESULTS: Four frequency bands—C-band, X-band, K-band, and W-band—are evaluated with frequency spacing Δf = 40 MHz.The bands span 4–8 GHz, 8–12 GHz, 21–26 GHz, and 77–81 GHz, respectively.
  • VI. NUMERICAL RESULTS: Condition-number evaluations report both unweighted and weighted system matrices.Weighted condition numbers follow the deterministic stability analysis, while unweighted values show empirical behavior of the original observation model.

A. Coverage Criterion Performance

The proposed coverage criterion is validated on imaging systems in both flat-channel and frequency-dependent cases.

  • Coverage Criterion Performance: Validation first considers the flat-channel case with N_b = 1.
  • Coverage Criterion Performance: Validation then proceeds to the frequency-dependent case with N_b > 1.

1) Flat channel case: … VII. CONCLUSION

The proposed coverage criterion (CC) yields stable wideband angle imaging with sparse arrays in flat and frequency-dependent channels, while guiding non-uniform designs across the full field of view. Simulations show accurate recovery, robustness to noise and channel variation, and substantially more recoverable angle pixels than physical antennas.

  • 1) Flat channel case:: Three X-band arrays occupying the same width, including a 25-antenna random array with fixed endpoints, were generated to test CC stability across geometries.All arrays use width 50 × c/(2fH).
  • 1) Flat channel case:: Condition numbers stayed below 5 unweighted and below 2 weighted for all three arrays, indicating stable angular-response recovery.The CC-designed systems accurately recovered prescribed-grid responses, including a non-sparse complex-valued ground truth.
  • 1) Flat channel case:: 100-trial RMSE evaluations across listed SNR levels found the CC-generated imaging systems numerically stable under noise perturbations.RMSE was measured between recovered and ground-truth γ for representative Array 2 configurations.
  • 2) Frequency-dependent channel case:: With frequency-dependent responses modeled using Nb = 4 and ϵ = 0.25, the CC supported Nccθ = 27 angle pixels, with weighted and unweighted condition numbers of 89.14 and 69.65.Recovered responses at three representative frequency samples remained in close agreement with ground truth despite fewer supported pixels than in the flat-channel case.
  • 2) Frequency-dependent channel case:: As Nb increases, Nccθ generally decreases, showing that stronger frequency variation reduces supportable imaging capability.Bands with smaller α(1) perform better when channel variation is mild, whereas larger-α(1) bands degrade more slowly under stronger selectivity.
  • B. Non-uniform array design: The non-uniform design targets approximately 100 full-FoV angle pixels using as few physical antennas as possible within width W = 100 × c/(2fH).Because antenna counts are discrete, each band uses the realizable design closest to the 100-pixel target.
  • 2) Frequency-dependent channel case:: For frequency-dependent channels, supported pixels increased with physical antennas but decreased with Nb, while reconstruction RMSE remained below 10^-2 for all shown configurations.Larger Nb creates more active frequency sets, leaving fewer elements in each effective virtual array.

APPENDIX · A. Proof of Theorem 4

The proof models the wideband response as a bandlimited trigonometric polynomial sampled at generally nonuniform virtual-element locations. It then applies irregular-sampling energy bounds and matrix-output identities to establish the theorem’s condition-number result.

  • A. Proof of Theorem 4: The virtual elements are discrete, generally nonuniform sampling locations along a continuous spatial coordinate.The received signal at each virtual element is treated as a sample of the spatial-domain response.
  • A. Proof of Theorem 4: Known unit-modulus modulation centers the angular-index range around zero.After centering, the spatial response is treated as a bandlimited trigonometric polynomial in the spatial coordinate.
  • A. Proof of Theorem 4: The proof defines continuous L2 energy over the aperture interval to connect the continuous response with discrete samples.This energy provides the continuous quantity used in the subsequent sampling bound.
  • A. Proof of Theorem 4: The adaptive-weight irregular sampling bound relates continuous energy to Voronoi-weighted sample energy.For the circular array, the weights come from (31), and the sampling-gap parameter is the maximum wrap-around inter-element gap dmax.
  • A. Proof of Theorem 4: The middle term in the energy inequality is expressed as the energy of the weighted matrix output.This rewrite follows directly from (61).
  • A. Proof of Theorem 4: Orthogonality of complex exponentials over [0, W) supplies the next energy identity in the proof.The identity converts the relevant expression into the form needed for bounding the weighted matrix output.
  • A. Proof of Theorem 4: Taking the infimum and supremum over all unit-norm γ yields lower and upper energy bounds.These extremal bounds are then used to obtain the theorem’s conditioning relation.
  • A. Proof of Theorem 4: Monotonicity of (1+ρ)/(1−ρ) for 0 ≤ρ < 1 transfers the sampling bound to the condition-number bound.The proof concludes after applying this monotonicity step.

B. Proof of Lemma 5

The proof establishes that the active row submatrix is full column rank using the coverage condition and Theorem 4, then combines active/inactive row bounds to prove Lemma 5.

  • Full-column-rank argument: The coverage condition yields θ consecutive angle pixels, so Theorem 4 gives a positive lower singular-value bound and establishes that e A_i is full column rank.Positive row scalings preserve column rank, transferring the result to the active row submatrix.
  • Active/inactive row decomposition: Active rows satisfy |β_i,mf| > ϵ, while inactive rows satisfy |β_i,mf| ≤ ϵ, enabling separate bounds for the corresponding row contributions.These active and inactive components are used in the Gram-matrix analysis of D_i e A_fl.
  • Completion of the proof: The resulting inequalities, together with the full-column-rank property, establish the lemma.Substitution of the derived bound into the preceding expression completes the proof.

C. Proof of Theorem 6

The proof bounds the Gram matrix through the block-diagonal structure of QD and Weyl’s inequality. Under δ < ηmin, the Gram matrix is positive definite, while the scaled-unitary DFT factor preserves the condition number.

  • Eigenvalue bounds: The block-diagonal structure of QD determines its extreme eigenvalues.These eigenvalue bounds provide the starting point for the proof.
  • Eigenvalue bounds: Weyl’s inequality transfers the eigenvalue bounds to the full Gram matrix.This step supports the subsequent definiteness conclusion.
  • Positive definiteness: Under δ < ηmin, the Gram matrix is positive definite.This establishes the required definiteness condition.
  • Condition-number preservation: The scaled-unitary DFT-row factor scales all singular values equally and therefore does not change the condition number.The relation EEH = NbINb implies scaling by √Nb.
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