Ferkans — Interactive Telecom Tutor

Beyond One Dimension

A ULA steers the beam in one plane only — it has no control over the perpendicular dimension. Practical base station antennas, radar panels, and satellite dishes require two-dimensional beam control (azimuth and elevation). This section extends the ULA to planar arrays (UPA), circular arrays (UCA), and the 3GPP antenna model used for 5G NR system-level simulations.

Definition:
Uniform Planar Array (UPA)

A uniform planar array (UPA) consists of $M \times N$ elements arranged in a rectangular grid in the $yz$ -plane:

$M$ rows along the $z$ -axis with spacing $d_z$
$N$ columns along the $y$ -axis with spacing $d_y$
Element $(m, n)$ is at position $(0,\, n d_y,\, m d_z)$

The steering vector of the UPA for direction $(\theta, \phi)$ is the Kronecker product of the row and column steering vectors:

$\mathbf{a}_{\text{UPA}}(\theta, \phi) = \mathbf{a}_z(\theta) \otimes \mathbf{a}_y(\theta, \phi)$

where

$[\mathbf{a}_z(\theta)]_m = e^{jmkd_z\cos\theta}$ and $[\mathbf{a}_y(\theta,\phi)]_n = e^{jnkd_y\sin\theta\sin\phi}$

for $m = 0, \ldots, M-1$ and $n = 0, \ldots, N-1$ .

,

Theorem: UPA Array Factor Separability

For a UPA with uniform weights steered to $(\theta_0, \phi_0)$ , the array factor separates as a product of row and column array factors:

$\mathrm{AF}_{\text{UPA}}(\theta, \phi) = \mathrm{AF}_z(\theta) \cdot \mathrm{AF}_y(\theta, \phi)$

where

$\mathrm{AF}_z(\theta) = \frac{1}{M}\,\frac{\sin\!\bigl(\frac{M\psi_z}{2}\bigr)}{\sin\!\bigl(\frac{\psi_z}{2}\bigr)}, \qquad \psi_z = kd_z(\cos\theta - \cos\theta_0)$

$\mathrm{AF}_y(\theta, \phi) = \frac{1}{N}\,\frac{\sin\!\bigl(\frac{N\psi_y}{2}\bigr)}{\sin\!\bigl(\frac{\psi_y}{2}\bigr)}, \qquad \psi_y = kd_y(\sin\theta\sin\phi - \sin\theta_0\sin\phi_0)$

The Kronecker structure of the steering vector means the 2D beam pattern is the product of two independent 1D patterns. This is analogous to a 2D DFT being the outer product of two 1D DFTs. The separability breaks down when mutual coupling or non-uniform weighting couples the row and column dimensions.

Proof

Kronecker product expansion

The weight vector for steering to $(\theta_0, \phi_0)$ is $\mathbf{w} = \frac{1}{MN}\,\mathbf{a}_{\text{UPA}}(\theta_0, \phi_0)$ .

$\mathrm{AF} = \mathbf{w}^H \mathbf{a}_{\text{UPA}}(\theta, \phi) = \frac{1}{MN}\bigl[\mathbf{a}_z(\theta_0) \otimes \mathbf{a}_y(\theta_0,\phi_0)\bigr]^H \bigl[\mathbf{a}_z(\theta) \otimes \mathbf{a}_y(\theta,\phi)\bigr]$

Using the mixed-product property of the Kronecker product:

$= \frac{1}{MN}\,\mathbf{a}_z^H(\theta_0)\mathbf{a}_z(\theta) \cdot \mathbf{a}_y^H(\theta_0,\phi_0)\mathbf{a}_y(\theta,\phi)$

$= \mathrm{AF}_z(\theta) \cdot \mathrm{AF}_y(\theta, \phi)$ . $\blacksquare$

UPA 3D Beam Pattern

Visualize the 3D beam pattern of a uniform planar array. Adjust the number of rows and columns, and steer the beam in elevation ( $\theta$ ) and azimuth ( $\phi$ ). The beam narrows in both planes as the array grows, and the pattern separability is visible in the rectangular main lobe shape.

Parameters

Rows

M

4

Columns

N

4

Elevation steer

\theta_0

(deg)0

Azimuth steer

\phi_0

(deg)0

Definition:
Uniform Circular Array (UCA)

A uniform circular array (UCA) places $N$ elements uniformly on a circle of radius $R$ :

Element $n$ is at angle $\gamma_n = 2\pi n / N$
Position: $(R\cos\gamma_n,\, R\sin\gamma_n,\, 0)$

The steering vector for direction $(\theta, \phi)$ is

$[\mathbf{a}_{\text{UCA}}(\theta, \phi)]_n = \exp\!\bigl(jkR\sin\theta\cos(\phi - \gamma_n)\bigr)$

Unlike the ULA, the UCA provides uniform azimuthal coverage — the beam pattern is identical in all azimuth directions (no broadening at endfire). This makes it attractive for omnidirectional scanning applications.

3GPP Antenna Model for 5G NR

The 3GPP standard (TR 38.901, Section 7.3) defines a dual-polarized UPA model for base station and UE antennas. Key features:

Panel composed of $M_g \times N_g$ antenna panels, each with $M \times N$ cross-polarized elements ( $\pm 45^\circ$ slant)
Element pattern: directional with 65 $^\circ$ HPBW in both planes and 30 dB front-to-back ratio
Element radiation pattern (vertical cut): $A_E^V(\theta) = -\min\!\bigl[12\bigl(\frac{\theta - 90^\circ}{\theta_{3\text{dB}}}\bigr)^2,\, \mathrm{SLA}_V\bigr]$ dB
Composite pattern: element pattern $\times$ array factor
Supports per-panel and per-element beamforming

This model is the de facto standard for system-level 5G NR simulations and is used throughout 3GPP evaluation methodologies.

ULA vs UPA vs UCA

Property	ULA	UPA	UCA
Dimensions	1D (line)	2D (rectangle)	1D (circle)
Total elements	$N$	$M \times N$	$N$
Beam control	1 plane	2 planes (az + el)	Azimuth only
Steering vector	$e^{jnkd\sin\theta}$	Kronecker $\mathbf{a}_z \otimes \mathbf{a}_y$	$e^{jkR\sin\theta\cos(\phi - \gamma_n)}$
AF separable?	N/A (1D)	Yes (rows $\times$ cols)	No
Azimuth symmetry	No (broadens at endfire)	No	Yes (uniform)
Grating lobe risk	$d > \lambda/2$	$d_y$ or $d_z > \lambda/2$	Depends on $R$
Typical use	Analysis, small UE	5G BS panels	Direction finding

Quick Check

A UPA has $M = 8$ rows and $N = 8$ columns with $d_y = d_z = \lambda/2$ . What is the approximate directivity?

9 dBi

18 dBi

12 dBi

24 dBi

Correction:

18 dBi

Correct. For a UPA with half-wavelength spacing, $D \approx MN = 64 = 18.1$ dBi. The separability means the total gain is approximately the product of row and column gains.

Uniform Planar Array (UPA)

A rectangular $M \times N$ grid of antenna elements with uniform spacing $d_y$ and $d_z$ . Provides 2D beam steering in azimuth and elevation. The standard geometry for 5G base station panels.

Uniform Circular Array (UCA)

$N$ elements uniformly spaced on a circle of radius $R$ . Provides azimuthally symmetric beam patterns, useful for omnidirectional scanning and direction finding.

Planar and Circular Arrays