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Knowledge Hubwireless communication basics4. Array Geometry: ULA, URA, and RIS

wireless communication basics learning note

4. Array Geometry: ULA, URA, and RIS

Compare uniform linear and rectangular arrays, derive their steering behaviour, and connect array geometry to 2D/3D beam control and RIS design.

Why Geometry Matters

An antenna array is a set of radiating or reflecting elements placed at known positions. The element positions determine which spatial phase patterns the array can produce and distinguish.

Two common geometries are:

  • Uniform linear array (ULA): elements lie along one line with equal spacing.
  • Uniform rectangular array (URA): elements form a two-dimensional grid with equal row and column spacing.

The same geometries appear in active antenna arrays and reconfigurable intelligent surfaces. In an active array, each element may transmit or receive through an RF chain or subarray network. In an RIS, programmable cells change the phase and sometimes amplitude of reflected or transmitted waves.

Original comparison of uniform linear and uniform rectangular array geometries
Original diagram: a ULA samples one spatial axis, whereas a URA samples two axes for azimuth-elevation steering and 3D focusing.
References for this section3

Uniform Linear Array

Consider NN elements separated by distance dd along the x-axis. For a far-field plane wave arriving from angle θ\theta, adjacent elements observe a path-length difference related to dsin⁡θd\sin\theta. The corresponding phase difference is

ψ=kdsin⁡θ,k=2πλ\psi = kd\sin\theta, \qquad k=\frac{2\pi}{\lambda}

A steering vector can be written as

a(θ)=[1ejψej2ψ⋯ej(N−1)ψ]T\mathbf{a}(\theta) = \begin{bmatrix} 1 & e^{j\psi} & e^{j2\psi} & \cdots & e^{j(N-1)\psi} \end{bmatrix}^{\mathsf{T}}

The exact sine or cosine form changes with the angle convention and array orientation. The important idea is that the phase progresses linearly across the elements.

A ULA is computationally simple and useful when the dominant steering problem is one-dimensional. It cannot independently resolve both azimuth and elevation unless other geometry, movement, or prior information supplies the missing dimension.

References for this section3

Uniform Rectangular Array

An M×NM\times N URA extends the array into two dimensions. For element indices (m,n)(m,n), the steering phase depends on both coordinates:

ψm,n=k(mdxux+ndyuy)\psi_{m,n} = k\left(md_xu_x+nd_yu_y\right)

where dxd_x and dyd_y are element spacings and uxu_x, uyu_y are direction-cosine components determined by azimuth and elevation.

For an ideal separable grid, the URA steering vector can be expressed as a Kronecker product of two linear-array steering vectors:

aURA(φ,θ)=ay(φ,θ)⊗ax(φ,θ)\mathbf{a}_{\mathrm{URA}}(\varphi,\theta) = \mathbf{a}_{y}(\varphi,\theta)\otimes\mathbf{a}_{x}(\varphi,\theta)

This structure allows two-dimensional beam steering, narrower beams, and better angular discrimination. It also increases the number of control coefficients and the difficulty of channel estimation and calibration.

References for this section3

ULA and URA Comparison

PropertyULAURA
Element layoutOne-dimensional lineTwo-dimensional grid
Principal angular controlOne planeAzimuth and elevation
Number of elementsNNM×NM\times N
Hardware/control complexityLowerHigher
3D beamformingLimitedNatural
Typical useSimple arrays, low-cost sensing, one-plane scansBase stations, imaging, 3D localization, large RIS panels
References for this section3

Element Spacing and Grating Lobes

Element spacing is a central design parameter. If spacing is too large, different directions can produce indistinguishable phase progressions, creating grating lobes. A widely used safe rule for broad scanning is

d≤λ2d\leq\frac{\lambda}{2}

Half-wavelength spacing is not a universal optimum. Tighter spacing increases mutual coupling, while larger spacing may be acceptable for a restricted scan sector. The final design depends on scan range, element pattern, bandwidth, array size, and fabrication constraints.

At 28 GHz, λ≈10.7 mm\lambda\approx10.7\ \mathrm{mm}, so half-wavelength spacing is approximately 5.35 mm. A 20 × 20 half-wavelength URA therefore has an aperture width near 10 cm before edge spacing and packaging are considered.

References for this section3

Aperture and Beamwidth

Beamwidth is governed mainly by electrical aperture size rather than element count alone. A larger aperture measured in wavelengths produces a narrower main lobe and higher directivity. Adding more elements within the same fixed aperture improves sampling and control but does not increase aperture indefinitely.

This distinction matters when comparing RIS sizes. A 20 × 20 RIS with half-wavelength spacing has a physically larger aperture than a 10 × 10 RIS at the same frequency, so it can form a narrower spatial response. Narrow beams improve angular selectivity but demand more accurate alignment and channel knowledge.

References for this section3

Array Factor and Element Pattern

The total radiation pattern is commonly interpreted as

Ftotal(θ,φ)=Felement(θ,φ) AF(θ,φ)F_{\mathrm{total}}(\theta,\varphi) = F_{\mathrm{element}}(\theta,\varphi)\, AF(\theta,\varphi)

The element pattern describes one antenna or RIS cell. The array factor describes interference created by the element positions and complex weights. Ideal array-factor calculations can overestimate performance if they ignore element directivity, coupling, blockage, phase-dependent loss, and the surface mounting structure.

References for this section3

RIS Geometry

For a far-field RIS model, each cell is often assigned a complex reflection coefficient

Γm,n=βm,nejϕm,n\Gamma_{m,n} = \beta_{m,n}e^{j\phi_{m,n}}

where βm,n\beta_{m,n} is reflection amplitude and ϕm,n\phi_{m,n} is programmed phase. A desired phase gradient across the grid steers the reflected beam.

Real RIS hardware may offer only a few discrete phase states. A 1-bit RIS typically switches between two nominal states separated by approximately 180 degrees, while a 2-bit design offers four states. The realized phase difference may vary with frequency, incident angle, bias network, and unit-cell coupling.

References for this section3

Far Field and Near Field

In the far field, an arriving wave is approximated as planar and direction is the main spatial variable. Near a large aperture, wavefront curvature becomes visible. The phase across the array then depends on the distance from every element to the user:

ϕm,n∝−2πλrm,n\phi_{m,n} \propto -\frac{2\pi}{\lambda}r_{m,n}

This spherical-wave structure allows range-angle focusing and can support 3D localization from a single large array or RIS. It also means far-field steering vectors can become inaccurate in near-field scenarios.

References for this section3

Choosing a Geometry

Use a ULA when the problem is predominantly one-dimensional, cost and control simplicity matter, or a compact teaching/test setup is needed. Use a URA when the system needs azimuth-elevation control, narrow 3D beams, spatial focusing, or robust localization in two or three dimensions.

References for this section3

Takeaway

Array geometry determines the spatial information available to a wireless system. A ULA offers simple one-plane control. A URA offers richer 3D control through a two-dimensional aperture. Spacing, aperture size, element response, phase resolution, and near-field behaviour decide how much of the theoretical advantage appears in real hardware.

References for this section3

Complete references and further reading