Modular Linear Arrays (MLAs)

• MLAs are modular array architectures that combine multiple sub-arrays with large inter-module gaps to form extended apertures for applications in communications, radar, and sensing.
• They exploit non-uniform spherical wave propagation to enable precise beam focusing, narrow main lobes, and high spatial resolution compared to conventional arrays.
• MLAs offer flexible deployment with reduced active elements, improving hardware efficiency and spectral performance in XL-MIMO, near-field beamforming, and multi-user scheduling.

A Modular Linear Array (MLA) is a general class of array architectures in which a long, sparse, and typically quasi-one-dimensional aperture is formed by physically and electrically combining multiple regular or irregular sub-arrays (modules), each typically a short uniform or sparse linear array, separated by inter-module gaps much larger than the inter-element spacing within the modules. MLAs underpin a range of applications in extremely large-scale MIMO communications, near-field wireless beamforming, radar, and sensing by offering the ability to synthesize large apertures while providing deployment flexibility and significant hardware efficiency. The key technical innovation is the exploitation of the radiative near field and non-uniform spherical wave (NUSW) propagation, enabling main-lobe narrowing, depth-domain (range) focusing, high spatial resolution, and spatial multiplexing in settings that challenge conventional uniform, fully-filled linear arrays.

1. Modular Linear Array Architectures

The canonical MLA consists of $N$ modules, each an array of $M$ elements with inter-element spacing $d$ (typically $d\approx\lambda/2$), regularly arranged along a common axis but separated by much larger inter-module gaps ($\Gamma d$, $D$, or analogous, where $\Gamma\gg M$). The total aperture is therefore

$L_{\text{MLA}} = [(N-1)\Gamma + (M-1)]d,$

where module centers are separated by $\Gamma d$, and the total element count is $N M$ (Li et al., 2023, Li et al., 2022, Li et al., 2022). In various domains, "module" may refer to physical 'planks' on a conical surface (as in radar (Rocca et al., 2021)), widely spaced uniform linear arrays (ULAs) treated as sub-arrays (Kosasih et al., 12 May 2025, Kosasih et al., 2024), or more general arrangements. MLA design is motivated by the need to deploy large apertures around physical constraints (e.g., windows, pillars), while reducing the number of active elements and associated RF chains.

2. Near-Field Modelling and System Representation

MLAs require near-field propagation models that account for amplitude and phase variation across large, sparsely filled arrays:

• Non-Uniform Spherical Wave (NUSW) Model: The baseband channel coefficient from a transmitter at $M$0 to element $M$1 located at $M$2 is

$M$3

with array response

$M$4

and amplitude taper as $M$5 (Li et al., 2023, Li et al., 2022, Li et al., 2022, Kosasih et al., 12 May 2025).

• Sub-Array Based Uniform Spherical Wave (USW) Models: For $M$6 beyond the Rayleigh distance of sub-arrays ($M$7), intra-module amplitude variation is negligible. The array response can be factorized as a sum or Kronecker product over module and sub-module steering vectors with either module-dependent or common angle assumptions (Li et al., 2023, Li et al., 2023).
• Far-Field (Uniform Plane Wave, UPW) Limit: For $M$8 the array operates in the plane-wave regime, recovering the classical, linear phase progression across the entire aperture (Li et al., 2022).

The explicit distinction between NUSW and UPW regimes is critical. In MLAs, in contrast to collocated ULAs, array-scale near-field effects dominate unless the user is far outside the array’s radiative near field (González-Coma et al., 10 Jan 2025, Li et al., 2023).

3. Beam Focusing, Grating Lobes, and Spatial Resolution

MLAs leverage widely spaced modules for enhanced spatial focusing, at the expense of characteristic grating lobes:

• Beam Focusing Patterns: The normalized array gain when steering to and observing from $M$9 and $d$0 is given by the normalized inner product

$d$1

which, under common-angle USW, can be factored into an inter-module (slow) and intra-module (fast) Dirichlet kernel. The main-lobe width along $d$2 is narrowed by the effective aperture as

$d$3

compared to the collocated ULA $d$4, with $d$5 (Li et al., 2023, Li et al., 2023).

• Grating Lobe Structure: Large inter-module spacing ($d$6) induces grating lobes at fixed angular spacings,

$d$7

causing side-beams in both far-field and, less coherently, in near-field patterns (Li et al., 2023, Li et al., 2023, González-Coma et al., 10 Jan 2025).

• Near-Field Lobe Suppression: In the near field, non-linear phase terms across widely spaced modules prevent full coherence at grating-lobe directions, suppressing their amplitude compared to far field (Li et al., 2023). The intra-module array factor further envelopes and can suppress sidelobes if module parameters are co-designed.

The following table summarizes key dependencies:

Parameter	Effect on MLA Beampattern	Design Implication
$d$8 (module size)	Nulls/envelope for grating lobes	Larger $d$9 suppresses sidelobes
$d\approx\lambda/2$0	Main-lobe width, grating lobe period	Tradeoff: resolution vs. lobes
$d\approx\lambda/2$1 (modules)	Angular resolution	More $d\approx\lambda/2$2 → narrower main lobe

4. Performance Analysis and Asymptotic Behavior

MLA performance in communications and sensing is governed by the interaction of array geometry and near-field propagation:

• SNR and Scaling Laws: The maximum achievable SNR in the near field under NUSW saturates with increasing aperture, converging as $d\approx\lambda/2$3 to

$d\approx\lambda/2$4

reflecting diminishing returns from adding distant modules as path loss dominates. In contrast, the UPW model (inappropriate for MLA regime) predicts unbounded linear SNR growth in $d\approx\lambda/2$5 (Li et al., 2022, Li et al., 2022).

• Antenna-Efficiency: By exploiting aperture extension, an MLA can match the near-field beamfocusing and spatial multiplexing capability of a fully filled ULA using substantially fewer antennas. For example, two subarrays can achieve the same focal resolution with up to 36% antenna count reduction if the spacing criterion $d\approx\lambda/2$6 is met (Kosasih et al., 2024, Kosasih et al., 12 May 2025, Alshumayri et al., 9 Nov 2025).
• Velocity Estimation: The Cramér–Rao bound for transverse velocity estimation is inversely proportional to the square of the 'stretched' aperture (inter-module spacing); increasing module separation provides direct gains in transverse velocity accuracy, with only mild penalty for radial velocity (Alshumayri et al., 9 Nov 2025).
• Interference and Multi-User Throughput: MLAs, by virtue of finer spatial resolution, offer up to 70% higher sum spectral efficiency in dense multi-user downlink as compared to collocated ULAs, provided user grouping/scheduling avoids mutual grating-lobe overlap (González-Coma et al., 10 Jan 2025, Li et al., 2023).

5. Multi-User Scheduling and Beam Management Strategies

MLAs in large-scale multi-user MIMO require specialized algorithms for user resource allocation and interference management:

• Interference Characterization: The overlap of beam patterns, particularly at grating-lobe angles, can produce strong inter-user interference. The array response cross-correlation can be bounded using Fresnel integrals or Dirichlet kernel analyses (González-Coma et al., 10 Jan 2025, Li et al., 2023).
• User Grouping and Resource Assignment: Greedy grouping methods assign users to orthogonal resource blocks such that users lying in each other’s grating lobe patterns are separated, optimizing per-user SINR and net sum-rate (Li et al., 2023).
• Low-Complexity Scheduling: Algorithms such as rectangular search scheduler (RSS) and front-line scheduler (FLS) exploit unique MLA spatial signatures to provide near-optimal performance with substantially reduced complexity versus brute-force selection (González-Coma et al., 10 Jan 2025).

6. Implementation Considerations and Design Guidelines

MLA realizations involve key practical trade-offs:

• Module Synchronization: Coherent operation of modules, especially under near-field focusing and subarray-based beamforming, requires precise relative phase calibration (typically coaxial cabling, distributed oscillator synchronization) (Kosasih et al., 2024).
• Sparse/Flexible Mounting: MLAs accommodate physical constraints (building facades, radome structure) that preclude fully filled arrays, with design freedom in module count, size, and spacing to optimize for site-specific deployment (Rocca et al., 2021, Li et al., 2022).
• Hardware Efficiency: Digital architectures may process module outputs in parallel and combine digitally, enabling complexity $d\approx\lambda/2$7 (for $d\approx\lambda/2$8 modules of $d\approx\lambda/2$9 elements each) (Kosasih et al., 12 May 2025).
• Antenna Savings: Proper design of aperture filling fraction (antenna-to-aperture ratio) allows for substantial reduction in the number of antenna elements while satisfying target focusing or velocity estimation accuracy (see (Kosasih et al., 2024, Alshumayri et al., 9 Nov 2025) for quantitative expressions).
• Parameter Selection: Optimal spatial resolution and sidelobe suppression are realized by choosing $\Gamma d$0–$\Gamma d$1, module size $\Gamma d$2–8, and module spacing $\Gamma d$3 large enough to reach the desired Rayleigh distance while avoiding excessive lobes (Li et al., 2023, González-Coma et al., 10 Jan 2025).

7. Representative MLA Applications and Performance Summaries

MLAs have been investigated across several domains:

• XL-MIMO Communications: Modular arrays provide high spatial and range resolution for near-field beamfocusing, depth multiplexing (serving multiple users at the same angle but different distances), and sum-rate enhancement in next-generation gMIMO systems (Kosasih et al., 12 May 2025, Kosasih et al., 2024, González-Coma et al., 10 Jan 2025, Li et al., 2023).
• Predictive Beamforming: Near-field velocity estimation and predictive tracking are enhanced by inter-module separation, with fundamentally lower CRB for transverse velocity and robust performance under target motion (Alshumayri et al., 9 Nov 2025).
• Sparse Phased Array Radar: Modular conical arrangements with compressive sensing-optimized plank (sub-array) geometry have been validated for multi-beam radar, achieving order-27% reduction in element count with negligible degradation in beam pattern fidelity (Rocca et al., 2021).
• Parametric Channel Estimation: MLAs enable low-complexity, triangulation-based 2D localization and accurate parametric channel recovery, facilitating practical near-field multi-parameter estimation (Kosasih et al., 12 May 2025).

In all cases, extensive numerical and experimental evidence corroborates that modular sparsity, properly harnessed, yields high-efficiency, high-resolution near-field array systems with application-driven flexibility (González-Coma et al., 10 Jan 2025, Li et al., 2023, Li et al., 2022).

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Key references:

(Li et al., 2023, Kosasih et al., 2024, Li et al., 2022, Rocca et al., 2021, Alshumayri et al., 9 Nov 2025, González-Coma et al., 10 Jan 2025, Li et al., 2022, Kosasih et al., 12 May 2025, Li et al., 2023)