3GPP Antenna Beamforming Reference Notebook

1.1 Global Coordinate System (GCS)

The 3GPP Global Coordinate System, defined in TR 38.901 §7.1, adopts a spherical coordinate triplet $(r, \theta, \phi)$ anchored to a right-hand Cartesian frame $(x, y, z)$. The two angular quantities have precise physical interpretations that differ from many engineering conventions:

Relationship to Cartesian Coordinates

A point $\mathbf{r} = (x,y,z)$ in the GCS maps to spherical coordinates via:

The two orthonormal tangent vectors at a field point $(\theta,\phi)$ are:

Together with $\hat{e}_r = (\sin\theta\cos\phi,\;\sin\theta\sin\phi,\;\cos\theta)^T$, the triple $\{\hat{e}_r, \hat{e}_\theta, \hat{e}_\phi\}$ forms a right-hand orthonormal basis at every point. The Cartesian $+x$ unit vector expressed in this basis is:

Broadside Direction and Downtilt

For a vertical panel (face pointing along $+x$), broadside corresponds to $\phi = 0°$ and $\theta = 90°$, which is the positive $x$-axis direction. Mechanical downtilt $\psi_{DT}$ rotates the panel about the horizontal $y$-axis so the broadside direction becomes $\theta = 90° + \psi_{DT}$ — i.e., $\psi_{DT}$ degrees below the horizon. A typical macro base station uses $\psi_{DT} \in [2°, 10°]$.

Fig. 1.1 — 3GPP GCS axes. Zenith angle $\theta$ measured from $+z$; azimuth $\phi$ in the $xy$-plane from $+x$.

1.2 Local Coordinate System (LCS) TR 38.901 §7.1.3

Each antenna panel (or individual array element) has its own Local Coordinate System (LCS). The LCS is obtained from the GCS by three successive right-hand rotations parameterized by:

The overall rotation matrix mapping GCS to LCS is:

Expanding the product explicitly:

GCS-to-LCS Angle Transformation

Given a direction $\hat{r}_{GCS} = (\sin\theta\cos\phi,\;\sin\theta\sin\phi,\;\cos\theta)^T$ in the GCS, the corresponding LCS angles $(\theta_{LCS},\phi_{LCS})$ are obtained by:

Since $\mathbf{R}$ is orthogonal, its inverse equals its transpose — a useful computational shortcut.

Polarization Basis Transformation

The polarization basis vectors also rotate under the LCS transformation. The field-component transformation is governed by the polarization rotation matrix $\boldsymbol{\Phi}$:

where $\boldsymbol{\Phi}$ is a $2\times2$ rotation matrix dependent on all five angles and is evaluated at each $(θ,φ)$ sample.

1.3 Polarization Basis TR 38.901 Table 7.3-2

The electric-field vector at any far-field point is decomposed into two orthogonal polarization components in the tangent plane at $(\theta,\phi)$:

$\hat{e}_\theta$ is the vertical-like polarization (varies with $\theta$); $\hat{e}_\phi$ is the horizontal-like polarization (perpendicular to the meridian plane).

Cross-Polarized Elements in 5G Massive MIMO

Practical 5G NR antenna elements use slant ±45° dipoles rather than pure vertical/horizontal. A ±45° pair spans the same polarization space as $(\hat{e}_\theta, \hat{e}_\phi)$ but with lower mutual coupling and improved angular decorrelation. The slant-45° patterns are obtained by setting $\gamma = \pm 45°$ in the LCS rotation, yielding:

Cross-Polarization Discrimination (XPD)

XPD quantifies the isolation between the two polarization ports. Higher XPD indicates less cross-contamination between co-pol and cross-pol signal paths:

2.1 Horizontal (Azimuth) Cut — $A_{EH}(\phi)$

The azimuth element pattern describes gain variation in the horizontal plane (the $\theta = 90°$ slice). TR 38.901 Table 7.3-1 specifies a squared-cosine model with a front-to-back floor:

Key Sample Points

Roll-off Slope

The quadratic-in-dB model corresponds to a Gaussian beam in linear power. The slope at broadside is zero (smooth peak), and the effective directivity in the azimuth plane is $10\log_{10}(2\pi/\Delta\Omega_{az})$ where $\Delta\Omega_{az}$ is the solid-angle fraction subtended by the main lobe. For $\phi_{3\mathrm{dB}} = 65°$ and a Gaussian model, this contributes roughly $10\log_{10}(360°/65°) \approx 7.4\,\mathrm{dB}$ to azimuth directivity.

2.2 Vertical (Elevation/Zenith) Cut — $A_{EV}(\theta)$

The vertical element pattern is defined as a function of the zenith angle $\theta$. Because zenith $= 90°$ is broadside, the argument is $(\theta - 90°)$, which equals the elevation angle with sign inverted:

Key Sample Points

Floor Saturation Angle

The sidelobe floor $SLA_v = 30\,\mathrm{dB}$ is reached when:

So for $\theta < 90° - 102.8° = -12.8°$ (non-physical) or $\theta > 192.8°$ (also non-physical given $\theta \in [0°,180°]$), the floor is never actually hit via the pure quadratic formula — the $-23.0\,\mathrm{dB}$ at nadir ($\theta = 180°$) is well above the $-30\,\mathrm{dB}$ floor. This means $SLA_v$ only matters when a mechanical or electronic tilt is applied that shifts the pattern reference.

2.3 Combined 3D Element Pattern — $A_E(\theta,\phi)$ 3D

The full 3D element gain is constructed from the two cuts using a conservative combination rule that caps the total front-to-back attenuation at $A_m$:

Step-by-Step Derivation of the Formula

1. Both $A_{EH}(\phi) \leq 0\,\mathrm{dB}$ and $A_{EV}(\theta) \leq 0\,\mathrm{dB}$ because they represent attenuation relative to broadside (always non-positive).
2. Their sum $A_{EH} + A_{EV} \leq 0$ also represents total directional loss. If both cuts are at $-15\,\mathrm{dB}$, naively the total would be $-30\,\mathrm{dB}$ — but this cannot exceed the physical front-to-back limit.
3. The negation $-(A_{EH}+A_{EV}) \geq 0$ converts the loss (negative dB) to a positive attenuation quantity.
4. $\min[-(A_{EH}+A_{EV}),\; A_m]$ caps total attenuation at $A_m = 30\,\mathrm{dB}$. Combined off-axis losses cannot exceed the isotropic front-to-back floor.
5. Finally, $G_{\max}$ minus this capped attenuation gives the absolute element gain $A_E$ in dBi, normalized to isotropic.

Numerical Examples

2.4 Element Radiation Efficiency and Polarization

The element pattern model above describes the directivity shape. Real antenna elements also exhibit non-unity radiation efficiency, and when packed in a panel, suffer mutual coupling:

Radiation Efficiency

Radiation efficiency relates radiated power to accepted (incident) power:

where $D_E$ is the directivity and $G_E$ is the realized gain. For a well-matched element in a large panel, $\eta_r \approx 85\%\text{–}95\%$, contributing $-0.2$ to $-0.7\,\mathrm{dB}$ of gain loss. This is typically absorbed into the $G_{\max} = 8\,\mathrm{dBi}$ specification in TR 38.901.

Mutual Coupling

Envelope Correlation Coefficient (ECC)

ECC $\rho_e$ measures how correlated two antenna ports' radiation patterns are, which determines the effective MIMO diversity gain. It is computed from the element field patterns:

For well-designed ±45° cross-pol elements with $0.5\lambda$ spacing in 5G NR panels, $\rho_e < 0.1$ is routinely achieved, ensuring near-independent spatial streams and maximum MIMO capacity benefit.

2.5 Element Pattern Polar Plot

The canvas below renders the azimuth cut $A_{EH}(\phi)$ (blue) and the elevation cut $A_{EV}(\theta)$ (orange) of the single-element pattern from TR 38.901 Table 7.3-1. Both curves are plotted on a polar axis spanning $-30\,\mathrm{dB}$ (center) to $0\,\mathrm{dB}$ (outer rim), with dB value mapped linearly to radius.

Interpreting the Asymmetry

The azimuth cut (blue) shows a smooth roll-off out to $\pm 90°$, then an abrupt floor. The elevation cut (orange) reaches only $-23.0\,\mathrm{dB}$ at nadir because the floor is not hit within the physical $[0°, 180°]$ zenith range (as derived in §2.2). Both cuts have identical HPBW of $65°$ (full $-3\,\mathrm{dB}$ beamwidth = $65°$, half-angle $= 32.5°$).