---
title: Uniform Linear Array (ULA)
url: https://www.emergentmind.com/topics/uniform-linear-array-ula
type: topic
---

# Uniform Linear Array (ULA)

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 $R = \frac{N-1}{2}\Delta$ [2503.13890]. 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 [2402.05583]. 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 [2306.01458]. 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 [2412.03232].

The beam pattern (array factor) of a ULA in the far field is
\[
AF_{\mathrm{ULA}}(\Delta \theta) = \frac{1}{N} \sum_{n=0}^{N-1} e^{j k n \Delta \sin \Delta \theta}
= \frac{\sin(N k \Delta \sin \Delta \theta / 2)}{N \sin(k \Delta \sin \Delta \theta / 2)}
\]
[2412.03232]. 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 [2502.07318].

## 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 [2503.13890]
  \[
  \phi_n =
  \begin{cases}
    k\,|\sin(\alpha-\theta_A)|\,x_{t,n}, & x_{t,n} \ge 0, \\
    -k\,|\sin(\alpha+\theta_A)|\,x_{t,n}, & x_{t,n} < 0,
  \end{cases}
  \]
  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 [2503.13890].

- **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 [2004.12039]. 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 $M$ [2402.05583].

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 [2004.12039].

## 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 [2306.01458]. 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 [2306.01458].

## 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 [2305.13394]. 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 [1511.06828].

## 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 [1802.08591].
- 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 [2412.03232].
- 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 [2502.07318].

| 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 [2503.13890, 2004.12039, 2306.01458, 2402.05583, 2502.07318, 2412.03232, 1802.08591, 2305.13394, 1511.06828, 2006.14204].

Source: https://www.emergentmind.com/topics/uniform-linear-array-ula