Uniform Linear Arrays (ULA)

• Uniform Linear Arrays are configurations of equally spaced antennas that yield tractable analytical structures for beamforming and spectral analysis.
• They enable high-rank LoS channel modeling, near-field beamfocusing, and efficient spatial multiplexing in modern MIMO and wireless communication systems.
• Advanced designs address calibration via Toeplitz methods, inter-element spacing trade-offs, and modular or sparse architectures to reduce hardware costs.

A uniform linear array (ULA) is an arrangement of $N$ antenna elements spaced equally along a straight line. ULAs are foundational in array processing, wireless communications, and electromagnetic sensing due to their tractable algebraic structure, capacity for high-rank line-of-sight (LoS) channels, analytical beampatterns, and direct connection to Toeplitz and Vandermonde matrices. The performance, limitations, and advanced architectures built from ULAs—such as modular or sparse arrays—are central in contemporary MIMO systems, near-field communication, massive MIMO, calibration, and multipath management.

1. Array Geometry, Channel Models, and Analytical Structure

A ULA consists of $N$ equally spaced sensor elements with inter-element spacing $d$, position vector $x_n = (n-1)d$, $n=1,\dots,N$. The far-field steering vector to angle $\theta$ is

$$\mathbf{a}(\theta) = [1,\,e^{-j2\pi d\sin\theta/\lambda},\,\ldots,\,e^{-j2\pi (N-1)d\sin\theta/\lambda}]^T.$$

In LoS MIMO, the baseband channel with range $D$ and array spacings $d_t$, $d_r$ yields a matrix $N$0, which for parallel ULAs reduces (under $N$1 aperture) to a Vandermonde structure $N$2, with the antenna separation product $N$3 (Do et al., 2020). This simple form allows for direct analysis of singular-value distributions, array rank, and ultimately spatial multiplexing capacity.

In more general electromagnetic channels, the full dyadic Green’s function model expresses the channel as

$N$4

where $N$5 for each element $N$6 (Mestre et al., 11 Feb 2025).

The output spatial covariance matrix for an ideal (calibrated) ULA is Hermitian Toeplitz, $N$7, facilitating both spectral and calibration algorithms via eigenvalue analysis and the Toeplitz inverse eigenvalue problem (Abramovich et al., 2023).

2. Capacity, Multiplexing, and Rotational or Reconfigurable ULAs

ULA geometry alone, even in absence of multipath, can induce high-rank LoS channels by exploiting spatially varying path-length differences, especially in the near-field or for large apertures. For two parallel ULAs (transmit, receive; min size $N$8), the asymptotic channel admits up to $N$9 spatial streams, depending on the chosen $d$0 (Do et al., 2020).

A derived capacity upper bound for any antenna arrangement (given SNR) is

$d$1

with $d$2 set by singular value structure. For a ULA, SNR-dependent rotation of the receiver/transmitter can make $d$3 (or equivalently, $d$4 if $d$5), permitting asymptotic attainment of $d$6 at low and high SNR, and for $d$7 (Do et al., 2020).

To circumvent mechanical rotation, a bank of $d$8 radially oriented ULAs can be deployed, each activated electronically at the desired angle. Using only $d$9 arrays, each angled appropriately, $x_n = (n-1)d$0 of the channel capacity can be achieved over the entire SNR range of practical interest, with further $x_n = (n-1)d$1 producing diminishing returns (Do et al., 2020).

3. Beamforming, Beamfocusing, and Near-Field Effects

Classical ULA beamforming forms directional beams in angle (far field); the half-power beamwidth is $x_n = (n-1)d$2, reducing to $x_n = (n-1)d$3 radians for $x_n = (n-1)d$4 (Kosasih et al., 12 May 2025). In the near field, phased arrays can create beams focusable in both range and angle, termed “beamfocusing.” The focal width (beamwidth) and “beamdepth” (along $x_n = (n-1)d$5-axis) can be expressed in closed form via Fresnel integrals. The normalized array gain at focal range $x_n = (n-1)d$6 and observation point $x_n = (n-1)d$7 is

$x_n = (n-1)d$8

Beamfocusing feasibility in the holographic regime demands array length $x_n = (n-1)d$9; the region in which beamfocusing can be achieved is determined by closed-form criteria involving the position and elevation angle of the receiver (Mestre et al., 11 Feb 2025).

In the holographic (continuous-aperture) ULA, spatial multiplexing and high-rank LoS channels can be achieved at almost any range if three orthogonal polarizations are used, and two parallel ULAs suffice for angle-depth spatial multiplexing in the radiative near field (Mestre et al., 2024, Kosasih et al., 2024).

Design Parameter	Value/Formula	Purpose
Far-field beamwidth	$n=1,\dots,N$0 (rad) ($n=1,\dots,N$1)	Angular discrimination
Beamfocusing size	$n=1,\dots,N$2	Min. for feasible focus
Aperture for MLA	$n=1,\dots,N$3 as in (Kosasih et al., 2024)	MLA vs. single ULA perf
Modular ULA, 2 subarrays	$n=1,\dots,N$436% fewer antennas	Equal focus with savings

4. Direction-of-Arrival (DOA) Estimation and Identifiability

ULAs, when spaced with $n=1,\dots,N$5, yield unambiguous (non-wrapped) phase increments for single-frequency DOA estimation. The identifiability condition requires that no two directions $n=1,\dots,N$6 yield identical patterns of wrapped phases at all sensor pairs,

$n=1,\dots,N$7

with $n=1,\dots,N$8 the normalized pairwise spacing, $n=1,\dots,N$9 integer. For a ULA, this reduces to the half-wavelength criterion: only $\theta$0 avoids wrap-induced ambiguities (Chen et al., 2020). Exceeding this limit leaks ambiguity, unless non-uniform spacings or multifrequency techniques are used. Practical design typically strictly enforces $\theta$1.

5. Calibration and Toeplitz Inverse Eigenvalue Problems

The array covariance matrix of a perfect ULA is Hermitian Toeplitz; phase calibration errors create a diagonal phase distortion $\theta$2 that preserves covariance eigenvalues and entry-wise magnitudes. The Toeplitz inverse eigenvalue problem (ToIEP) for ULAs is: reconstruct $\theta$3 from eigenvalues and magnitudes $\theta$4. For the real case, only two Toeplitz solutions exist—one physical (correct beam pointing), one non-physical (spatially reversed); a sign-flipping algorithm (complexity $\theta$5) selects the physical one (Abramovich et al., 2023). The complex case requires Newton iteration with rank-one constraints and alternating projections. Accurate calibration demands $\theta$6 samples for robust “strong convergence.” Fully augmentable MRAs inherit the Toeplitz structure and can feed ULA ToIEP if $\theta$7 signal subspace dimension.

6. Sparse Arrays, Modular Architectures, and Hardware Reductions

Sparse arrays often concatenate multiple ULAs (“ULA fitting”), providing large virtual coarrays for super-resolution DOA estimation with reduced mutual coupling (Shi et al., 2021). Array polynomials encode sub-ULA positions, and the difference coarray is their convolution. Parametric designs such as “UF-3BL/4BL” use base and transfer layers with explicit formulas to ensure hole-free, low-coupling configurations. For N large, spatial efficiency exceeds 90%; these arrays achieve robust DOA estimation in strong coupling regimes.

Modular linear arrays (MLAs) built from two or more widely separated ULAs maintain the beamfocusing properties of a single full ULA, achieving identical beampatterns and beamdepth with approximately 36% fewer antennas. Analytical expressions for beampatterns and MLA beamdepth (and ripple design) facilitate array design (Kosasih et al., 2024, Kosasih et al., 12 May 2025).

7. Performance Trade-offs, Space-Constrained Design, and Extended Architectures

In massive MIMO under physical aperture constraints (array size $\theta$8 fixed), increasing ULA antenna count reduces $\theta$9, thereby degrading angular resolution and causing the normalized mean interference (NMI) to saturate at a non-zero floor; in unconstrained arrays with $$\mathbf{a}(\theta) = [1,\,e^{-j2\pi d\sin\theta/\lambda},\,\ldots,\,e^{-j2\pi (N-1)d\sin\theta/\lambda}]^T.$$0 fixed, NMI decays as $$\mathbf{a}(\theta) = [1,\,e^{-j2\pi d\sin\theta/\lambda},\,\ldots,\,e^{-j2\pi (N-1)d\sin\theta/\lambda}]^T.$$1 (Miller et al., 2020). Uplink SINR with MMSE processing follows suit: for finite aperture, SINR saturates; for unconstrained, SINR scales. ULA design for massive MIMO therefore trades off inter-element spacing, aperture length, mutual coupling, and spatial discrimination.

Additional ULA concepts include:

• Dielectric loading: Interposing dielectric elements between antennas reduces the orthodox $$\mathbf{a}(\theta) = [1,\,e^{-j2\pi d\sin\theta/\lambda},\,\ldots,\,e^{-j2\pi (N-1)d\sin\theta/\lambda}]^T.$$2 antenna spacing for LoS orthogonality, enabling compact full-rank ULAs at mmWave (Hälsig et al., 2015).
• Rotary ULA: Actively rotating the ULA is a low-cost degree of freedom to exploit LoS geometry, yielding substantial spectral efficiency increases in Rician LoS-dominated MU-MIMO networks (Tominaga et al., 2024).
• Near-field Bessel and curving beams: Closed-form ULA phase laws produce robust quasi-non-diffractive or “self-healing” beams for blockage mitigation, with Nyquist-type spatial sampling criteria and analytical bounds on steering and range (Uchimura et al., 18 Mar 2025).
• Polarization-multiplexed ULAs: Tri-orthogonal dipole elements guarantee, in the large-aperture limit, two spatial streams universally and up to three in favorable LoS, providing holographic multiplexing capacity (Mestre et al., 2024).

In sum, the ULA remains a keystone of high-dimensional signal processing and radio system engineering, offering deep analytic tractability, versatile implementation, and extensibility to next-generation array architectures (Do et al., 2020, Mestre et al., 2024, Mestre et al., 11 Feb 2025, Abramovich et al., 2023, Kosasih et al., 12 May 2025, Kosasih et al., 2024, Tominaga et al., 2024, Chen et al., 2020, Miller et al., 2020, Shi et al., 2021, Hälsig et al., 2015, Uchimura et al., 18 Mar 2025).