Ferkans — Interactive Telecom Tutor

From Theory to Hardware

The preceding sections derived array patterns assuming isotropic elements with no interaction. Real arrays deviate from this ideal: elements couple electromagnetically, each element has its own radiation pattern, polarization must be managed, and the beamforming architecture dictates cost, power, and flexibility. This section addresses the practical considerations that bridge textbook array theory and deployed antenna systems.

Definition:
Mutual Coupling

Mutual coupling is the electromagnetic interaction between closely spaced antenna elements. When element $n$ is excited, some energy couples into neighbouring elements $n \pm 1, n \pm 2, \ldots$ , altering their currents and thus the radiation pattern.

The coupling is modelled by the mutual impedance matrix $\mathbf{Z} \in \mathbb{C}^{N \times N}$ , where $Z_{mn}$ is the voltage induced on element $m$ by unit current on element $n$ . The actual element currents are

$\mathbf{i} = \mathbf{Z}^{-1}\,\mathbf{v}$

where $\mathbf{v}$ is the applied voltage vector. The effective steering vector becomes

$\mathbf{a}_{\text{eff}}(\theta) = \mathbf{C}\,\mathbf{a}(\theta)$

where $\mathbf{C} = \mathbf{Z}^{-1}$ (or a normalised version thereof) is the coupling matrix. Coupling is strongest for $d < \lambda/2$ and decays roughly as $1/(kd)$ for parallel dipoles.

,

Definition:
Element Pattern vs Array Pattern

The element pattern $F_e(\theta)$ is the radiation pattern of a single element in isolation. The array pattern is the product of the element pattern and the array factor:

$F_{\text{total}}(\theta) = F_e(\theta) \cdot |\mathrm{AF}(\theta)|^2$

This is the pattern multiplication principle. In practice:

The element pattern provides a natural envelope that suppresses grating lobes outside the element's main beam
In the 3GPP model, the element pattern is a truncated $\cos^2$ with 65 $^\circ$ HPBW and 30 dB front-to-back ratio
The embedded element pattern (measured when all other elements are terminated in matched loads) differs from the isolated pattern due to mutual coupling

,

Array Factor with Element Pattern

Compare the array pattern with and without the element pattern. The isotropic element shows grating lobes at full strength; the 3GPP element pattern suppresses them. This illustrates why the element pattern acts as a spatial anti-aliasing filter.

Parameters

Number of elements

N

8

Spacing

d/\lambda

0.5

Element pattern type

Grating Lobe Onset Animation

Watch what happens as the element spacing

d/\lambda

increases from 0.3 to 1.5. Below

\lambda/2

, only the main lobe is visible. As spacing exceeds

\lambda

, grating lobes — full-amplitude replicas of the main beam — enter the visible region, causing spatial ambiguity.

Grating lobe appearance as element spacing sweeps from

0.3\lambda

to

1.5\lambda

for an 8-element ULA. The

d = \lambda/2

Nyquist boundary is the safe upper limit for general scanning.

Grating Lobe Onset

Observe how grating lobes appear as element spacing $d/\lambda$ increases beyond 0.5. For $d = \lambda$ , grating lobes reach full main-lobe amplitude at $\pm 90^\circ$ . The plot shows the power pattern in dB with grating lobe positions marked.

Parameters

Number of elements

N

8

Spacing

d/\lambda

0.5

Definition:
Polarization Diversity

Polarization diversity exploits orthogonal polarization states to create independent (or weakly correlated) channels. Two co-located antennas with orthogonal polarizations (e.g., $\pm 45^\circ$ slant, or V/H) experience largely independent fading because:

Cross-polarized scattered components are weakly correlated
The cross-polar discrimination (XPD) is typically 8-15 dB in urban environments

This effectively doubles the number of independent spatial channels without increasing the physical array size. The 3GPP standard mandates dual-polarized elements at the base station, and modern UE designs use 2 or 4 cross-polarized antennas.

Beamforming Architecture Comparison — Block diagrams of the three beamforming architectures: (a) **Analog** — one RF chain drives all $N$ elements via phase shifters, forming a single beam. (b) **Digital** — each element has its own RF chain, enabling $N$ simultaneous beams and full spatial multiplexing. (c) **Hybrid** — $N_{\text{RF}} < N$ RF chains each drive a subarray, balancing cost and multiplexing capability.

Analog vs Digital vs Hybrid Beamforming

The beamforming architecture determines how weights $\mathbf{w} \in \mathbb{C}^N$ are applied:

Analog beamforming: phase shifters adjust the phase of each element's signal in the RF domain. Only one beam can be formed per RF chain. Weights are constrained to constant modulus ( $|w_n| = 1/\sqrt{N}$ ).

Digital beamforming: each element has its own RF chain, ADC/DAC, and baseband processing. Arbitrary complex weights, multiple simultaneous beams, full spatial multiplexing.

Hybrid beamforming: a compromise. $N_{\text{RF}} < N$ RF chains each drive a subarray via analog phase shifters. Digital precoding operates across RF chains.

The choice is driven by cost, power, and the carrier frequency: at mmWave, digital beamforming with hundreds of elements is prohibitively expensive, making hybrid architectures dominant.

Beamforming Architecture Comparison

Property	Analog	Digital	Hybrid
RF chains	1	$N$	$N_{\text{RF}} < N$
Weight constraint	Constant modulus	Arbitrary complex	Mixed
Simultaneous beams	1	$N$	$N_{\text{RF}}$
Spatial multiplexing	No	Full	Partial
Power consumption	Low	Very high	Medium
Hardware cost	Low	Very high	Medium
Typical use	mmWave single-user	Sub-6 GHz massive MIMO	mmWave multi-user
5G NR example	FR2 UE	FR1 64T64R	FR2 gNB

Historical Note: Phased Arrays in Radar

Electronically steered phased arrays were first developed for military radar in the 1950s-60s. The AN/SPY-1 Aegis radar (deployed 1983) used a 4,480-element planar array to simultaneously track hundreds of targets — demonstrating the power of electronic beam steering decades before it entered telecommunications. The transition from radar to communications accelerated with 5G NR (2018), which adopted massive MIMO phased arrays as a core technology for both sub-6 GHz and mmWave bands. Many of the beamforming algorithms used in 5G (MVDR, LCMV, DFT codebooks) originated in the radar and sonar communities.

Quick Check

A hybrid beamforming architecture has $N = 64$ antenna elements and $N_{\text{RF}} = 4$ RF chains. What is the maximum number of independent data streams it can spatially multiplex?

64

4

16

1

Correction:

4

Correct. The number of independent data streams is limited by $N_{\text{RF}} = 4$ . Each RF chain processes one spatial stream. The 64 elements provide beamforming gain per stream but the multiplexing order is at most $N_{\text{RF}}$ .

⚠️Engineering Note

Hybrid Beamforming Constraints at mmWave

Hybrid beamforming is the dominant architecture for 5G NR FR2 (24.25-52.6 GHz) base stations due to practical constraints:

Power consumption: Each RF chain (ADC + mixer + LNA) draws $\sim 250$ mW at 28 GHz. A 256-element fully digital array would consume $\sim 64$ W in RF chains alone — exceeding the thermal budget of a compact panel.
Phase shifter resolution: Commercial mmWave phase shifters provide 5-7 bit resolution. Below 5 bits, the quantization loss ( $\sim 0.2$ dB at 5 bits) and spurious sidelobes become noticeable.
Subarray architectures: Most commercial 5G mmWave products (e.g., Qualcomm QTM545) use 4-8 RF chains driving 16-32 elements each, for a total of 128-256 elements.
Calibration: The analog phase shifters must be calibrated to within $\pm 5^\circ$ to maintain beam pointing accuracy. Temperature drift of $\sim 1^\circ$ /10°C requires periodic over-the-air calibration.

Practical Constraints

•
RF chain power budget: ~250 mW per chain at 28 GHz
•
Phase shifter resolution: 5-7 bits in commercial products
•
Calibration accuracy: ±5° phase, ±0.5 dB amplitude

Key Takeaway

The choice between analog, digital, and hybrid beamforming is ultimately a tradeoff between three quantities: spatial multiplexing capability ( $N_{\text{RF}}$ independent streams), power consumption ( $\propto N_{\text{RF}}$ ), and hardware cost ( $\propto N_{\text{RF}}$ ). At sub-6 GHz, digital wins; at mmWave, hybrid is the pragmatic choice; analog suffices only for single-user fixed links.

Mutual Coupling

Electromagnetic interaction between adjacent antenna elements that alters their impedance and radiation patterns. Modelled by the mutual impedance matrix $\mathbf{Z}$ .

Polarization Diversity

Using orthogonally polarized antennas (e.g., $\pm 45^\circ$ ) to create independent fading channels without additional spatial separation. Effectively doubles the MIMO rank.

Hybrid Beamforming

A beamforming architecture combining analog phase shifters (per element) with digital precoding (per RF chain), using $N_{\text{RF}} < N$ RF chains to balance cost, power, and multiplexing capability.

Practical Antenna Considerations

From Theory to Hardware

Definition: Mutual Coupling

Definition: Element Pattern vs Array Pattern

Array Factor with Element Pattern

Parameters

Grating Lobe Onset Animation

Grating Lobe Onset

Parameters

Definition: Polarization Diversity

Beamforming Architecture Comparison

Analog vs Digital vs Hybrid Beamforming

Beamforming Architecture Comparison

Historical Note: Phased Arrays in Radar

Quick Check

Hybrid Beamforming Constraints at mmWave

Key Takeaway

Mutual Coupling

Polarization Diversity

Hybrid Beamforming

Definition:
Mutual Coupling

Definition:
Element Pattern vs Array Pattern

Definition:
Polarization Diversity