Uniform Linear Array (ULA)

• ULA is a linear arrangement of uniformly spaced antenna elements providing analytically tractable models and optimal far-field and near-field performance.
• Its design supports advanced beam synthesis techniques, including Bessel and curving beams, for enhanced spatial multiplexing and blockage avoidance.
• Practical implementations leverage efficient calibration and FFT-based precoding to achieve high spectral efficiency and directional gain in diverse environments.

A uniform linear array (ULA) is a fundamental architecture in array signal processing and multi-antenna communications, consisting of $N$ antenna elements placed at uniform spacing $\Delta$ along a straight line, usually aligned with the $x$ or $y$ axis. ULAs are preferred for their analytically tractable array manifold, optimal far-field spatial discrimination, and highly regular physical structure. Their theoretical properties underpin a range of applications from direction-of-arrival (DOA) estimation to massive MIMO and near-field 6G systems. Recent research comprehensively characterizes their performance in near- and far-field regimes, optimal beamforming, codebook design, and calibration.

1. Physical and Mathematical Model of the ULA

A canonical ULA of $N$ elements is described by antenna locations

$$\mathbf p_{t,n} = [x_{t,n},\, 0]^T, \qquad x_{t,n} = \frac{-N + 2n - 1}{2}\,\Delta,\quad n=1,\dots,N,$$

where $\Delta$ is the inter-element spacing, and array aperture is $2R$ with $R = \frac{N-1}{2}\Delta$ (Uchimura et al., 18 Mar 2025). The array response (steering vector) for a plane wave from azimuth $\theta$ is

$$a(\theta) = [1,\, e^{-j k \Delta \sin \theta},\,\dotsc,\,e^{-j k (N-1)\Delta \sin \theta}]^T,$$

with $k = 2\pi/\lambda$ the wavenumber (Tominaga et al., 2024). For arbitrary user position $\mathbf p_u = [x_u, y_u]^T$ in the 2-D plane, the field from each element includes the spherical wave phase term $e^{-j k d_n}$, where $d_n$ is the distance between $\mathbf p_{t,n}$ and the wavefront.

2. Far-Field and Near-Field Regimes

The transition from far-field (Fraunhofer) to near-field (Fresnel) is governed by the Rayleigh distance

$r_{\mathrm{Rayleigh}} = \frac{2 D^2}{\lambda},$

with $D = (N-1)\Delta$ being the array aperture (Zheng et al., 2023). For $r \gg r_{\mathrm{Rayleigh}}$, plane-wave steering vectors suffice. For $r \lesssim r_{\mathrm{Rayleigh}}$, full spherical-wave steering is required. Near-field effects, such as focusing in both angle and range, become significant at large-aperture and millimeter-wave/THz frequencies, enabling spatial multiplexing of users at the same angle but different ranges (Kosasih et al., 2024).

The beam pattern (array factor) of a ULA in the far field is

$$AF_{\mathrm{ULA}}(\Delta \theta) = \frac{1}{N} \sum_{n=0}^{N-1} e^{j k n \Delta \sin \Delta \theta} = \frac{\sin(N k \Delta \sin \Delta \theta / 2)}{N \sin(k \Delta \sin \Delta \theta / 2)}$$

(Kosasih et al., 2024). In the near-field, focusing capability is quantified by the SNR level-set ellipsoid around a focal position; feasibility requires an aperture at least $L \gtrsim 4.4\lambda$, with $L$ half the array length (Mestre et al., 11 Feb 2025).

3. Beamforming and Advanced Near-Field Beam Synthesis

A ULA enables standard beamsteering, maximum ratio transmission (MRT) for focusing, and advanced wavefront synthesis:

• Bessel beams and curving beams: Closed-form ULA phase laws enable Bessel (“conical”) and curving beams for enhanced near-field control. For a Bessel beam steered to azimuth $\theta_A$ with cone angle $\alpha$, the per-element phase is (Uchimura et al., 18 Mar 2025)

$$\phi_n = \begin{cases} k\,|\sin(\alpha-\theta_A)|\,x_{t,n}, & x_{t,n} \ge 0, \ -k\,|\sin(\alpha+\theta_A)|\,x_{t,n}, & x_{t,n} < 0, \end{cases}$$

subject to steering constraints $|\theta_A| \le \alpha < \frac{\pi}{2} - |\theta_A|$.

• Steering and sampling limits: The maximum “non-diffracting” range

$$d_{\max} = \frac{R\,\cos(\alpha+|\theta_A|)}{\sin\alpha}$$

and the anti-aliasing constraint $$\Delta < \lambda/\left[2 \sin(\alpha+|\theta_A|)\right]$$ govern array design.

• Curving beam design: Lagrangian optimization yields parabolic beam envelopes, enabling beams to route around obstacles, with per-element phase derived from the tangent-ray parabola construction. Optimal parameters are found by linear programming with closed-form KKT candidate solutions (Uchimura et al., 18 Mar 2025).
• Comparative Robustness: Bessel beams exhibit self-healing and non-diffraction over $d_{\max}$; beamfocusing (spherical/MRT) attains higher peak SNR but is less robust to blockage; curving beams route around obstructions at the cost of reduced intensity.

4. Capacity, Spectral Efficiency, and Array Optimization

For LOS MIMO, ULAs (with SNR-dependent physical or electronic rotation) can approach the optimal spatial-multiplexing capacity for any SNR regime. With “Rayleigh spacing” $d = \sqrt{\lambda D / N}$, and rotation angle $\varphi^* = \arccos(\rho(SNR)/N)$, ULAs reach the tight upper bound

$$C(SNR) \leq \rho(SNR) \log_2\left(1 + \frac{N_t N_r}{\rho(SNR)^2} SNR\right)$$

across SNR, where $\rho(SNR)$ is the integer channel rank dictated by eigenmode thresholds (Do et al., 2020). Hardware-efficient reconfigurable architectures use a small number of radially oriented ULAs to electronically switch the optimal angle, with three arrays capturing 96% of capacity for all relevant SNR.

In multiuser Rician settings, rotationally optimized ULAs (RULA) maximize average spectral efficiency $\bar{R}$ by aligning the array toward the spatial centroid of active users. Gains of up to $0.4$ bps/Hz per user are observed at high Rician $K$-factor and moderate array size $M$ (Tominaga et al., 2024).

Reduced-complexity FFT-based (DFT) precoding and combining algorithms exploit the near-circulant structure of ULA channels, scaling as $O(N\log N)$ and yielding the same asymptotic rate as full SVD-based schemes (Do et al., 2020).

5. Codebook Design and Quantization in the Near Field

In extremely large-scale ULAs ($M \gg 1$), near-field spherical-wave effects render plane-wave codebooks suboptimal. The absolute correlation between two ULA focusing vectors $\mathbf{b}(r, \theta)$ and $\mathbf{b}(r', \theta')$ in the ($\alpha$, $\beta$) parameter space, where $\alpha = \frac{\lambda \cos^2\theta}{4r}$ and $\beta = \sin\theta$, admits an elliptical approximation: $$\tau(\Delta\alpha, \Delta\beta) \approx 1 + p_\alpha (\Delta\alpha M^2)^2 + p_\beta (\Delta\beta M)^2,$$ with fitted constants. For a minimum target correlation $c$, the codebook support is partitioned using rectangular or hexagonal grids in ($\alpha, \beta$), with optimal spacings $\Delta\alpha$, $\Delta\beta$ providing guaranteed correlation (Zheng et al., 2023). The hexagonal grid reduces feedback overhead by $\sim$25%. Near-field codebook design crucially requires angular oversampling ($\sim O(M)$ grid points) relative to distance ($\sim O(\sqrt{M})$), as near-field beams are more sensitive to angle than range (Zheng et al., 2023).

6. Calibration, DOA Estimation, and Underlying Array Signal Processing

• Calibration via Toeplitz Inverse Eigenvalue Problem: A ULA’s ideal array covariance is Toeplitz. Given the eigenvalues and sub-diagonal moduli of the uncalibrated (possibly phase-perturbed) covariance matrix, ULA calibration reduces to reconstructing the Toeplitz covariance with prescribed modulus and spectrum. In the real-symmetric case, only two solutions exist, easily identified by the maximum-entropy criterion (physical solution: main lobe at boresight). For full complex Hermitian matrices, physical calibration imposes a rank-one constraint on the Hadamard–quotient of the observed and reconstructed matrices; the required phase profile is computed via Newton’s method (Abramovich et al., 2023). Reliable calibration in practical scenarios requires snapshot support $T \gg N\log N$ for strong elementwise convergence.
• Underdetermined DOA and KR Subspace Approaches: For $N$ ULA elements, the real-valued Khatri–Rao subspace method achieves $2N-2$ degrees of freedom for DOA estimation (resolving up to $2N-2$ sources), while reducing computational cost by a factor of four over conventional complex methods. It removes noise contributions exactly by specific real-valued transforms, enabling efficient real-domain subspace decomposition and spectral search (Duan et al., 2015).

7. Practical Implementation, Limitations, and Trade-Offs

Practical ULA deployments, as studied in mobile mmWave arrays, demonstrate the following key empirical characteristics:

• Analog beamforming (e.g., maximal ratio combining) with an 8-element ULA at 60 GHz achieves up to $15$ dB array gain in free space. However, gain varies $\pm15$ dB with orientation and exhibits deep outages ($\sim-14$ dB worst-case) when user hand or body blocks the main lobe—a consequence of the ULA’s directional response and collinear geometry. Distributed array topologies sacrifice peak gain for improved robustness (Haneda et al., 2018).
• Near-field ULA designs (e.g., with focused beamforming or multi-array modular configurations) can multiplex users at different ranges in the same angular direction, providing a path toward efficient massive MIMO/6G user scheduling (Kosasih et al., 2024).
• For 3D near-field beamfocusing, closed-form SNR ellipsoid criteria delineate the spatial region where focusing is feasible. Aperture lengths $L \ge 4.4\lambda$ are essential to achieve any local SNR maximum (Mestre et al., 11 Feb 2025).

Design Choice	Impact	Trade-Offs/Notes
Aperture size ($N$, $\Delta$)	Higher gain, range, angular resolution	Increases cost, hardware, PAPR
Beam type (Bessel/curving/focused)	Controls robustness, blockage, range	Bessel: self-healing; Focused: high peak, low reliability; Curving: obstacle avoidance
Codebook tiling (rectangular/hexagonal)	Overhead and quantization accuracy	Hexagonal: $\sim$25% fewer codewords for same correlation
Spacing ($\Delta$)	Grating lobe avoidance, steering flexibility	$\Delta < \lambda/[2\sin(\alpha+|\theta_A|)]$

In summary, the ULA remains the canonical design for analytically optimal beamforming, robust array processing, and emerging near-field functionality in massive MIMO systems. Modern research provides closed-form array design, sampling, and codebook formulas; advanced beam synthesis for blockage and trajectory control; and practical guidelines for calibration and robust operation in real-world hardware and environments (Uchimura et al., 18 Mar 2025, Do et al., 2020, Zheng et al., 2023, Tominaga et al., 2024, Mestre et al., 11 Feb 2025, Kosasih et al., 2024, Haneda et al., 2018, Abramovich et al., 2023, Duan et al., 2015, Miller et al., 2020).