Ferkans — Interactive Telecom Tutor

From Single Antennas to Arrays

A single antenna has a fixed radiation pattern — to steer the beam or increase directivity, one must physically rotate it or use a larger aperture. An antenna array solves both problems electronically: by combining signals from $N$ elements with appropriate phase shifts, one can steer the beam to any direction and sharpen it far beyond what a single element achieves. This is the foundational concept behind MIMO, massive MIMO, and 5G beamforming.

Definition:
Uniform Linear Array (ULA)

A uniform linear array (ULA) consists of $N$ identical antenna elements placed along a line (taken as the $z$ -axis) with equal inter-element spacing $d$ :

Element positions: $z_n = n d$ , for $n = 0, 1, \ldots, N-1$
All elements have the same radiation pattern (the element pattern)
The array is characterised by $N$ and $d/\lambda$

The ULA is the simplest and most widely analysed array geometry. Its properties extend naturally to planar and circular arrays (Section 7.3).

Definition:
Steering Vector

For a ULA with spacing $d$ , the steering vector for a plane wave arriving from direction $\theta$ (measured from broadside) is

$\mathbf{a}(\theta) = \bigl[1,\; e^{jkd\sin\theta},\; e^{j2kd\sin\theta},\; \ldots,\; e^{j(N-1)kd\sin\theta}\bigr]^T \in \mathbb{C}^N$

where $k = 2\pi/\lambda$ is the wavenumber. The $n$ -th component encodes the phase shift $e^{jnkd\sin\theta}$ due to the extra path length $nd\sin\theta$ to element $n$ .

The steering vector is a spatial frequency concept: it maps angle $\theta$ to a complex exponential sequence with frequency $\psi = kd\sin\theta$ .

Definition:
Array Factor

The array factor is the beam pattern produced by the array of isotropic elements, obtained by summing the element outputs with complex weights $\mathbf{w} = [w_0, w_1, \ldots, w_{N-1}]^T$ :

$\mathrm{AF}(\theta) = \mathbf{w}^H \mathbf{a}(\theta) = \sum_{n=0}^{N-1} w_n^* \, e^{jnkd\sin\theta}$

The total radiation pattern is the product of the element pattern and the array factor:

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

This factorisation is the pattern multiplication principle — it holds exactly when mutual coupling is negligible.

,

Theorem: ULA Array Factor with Uniform Weights

For a ULA with $N$ elements, spacing $d$ , and uniform weights $w_n = \frac{1}{N} e^{-jnkd\sin\theta_0}$ (steering to $\theta_0$ ), the array factor is

$\mathrm{AF}(\theta) = \frac{1}{N} \sum_{n=0}^{N-1} e^{jnkd(\sin\theta - \sin\theta_0)} = \frac{1}{N} \cdot \frac{\sin\!\bigl(\frac{N\psi}{2}\bigr)}{\sin\!\bigl(\frac{\psi}{2}\bigr)}$

where $\psi = kd(\sin\theta - \sin\theta_0)$ is the progressive phase difference. The power pattern is

$|\mathrm{AF}(\theta)|^2 = \frac{1}{N^2}\,\frac{\sin^2\!\bigl(\frac{N\psi}{2}\bigr)}{\sin^2\!\bigl(\frac{\psi}{2}\bigr)}$

This is the spatial analogue of the Dirichlet kernel from Fourier analysis. The array samples the incoming wavefront at $N$ equally spaced points, just as a DFT samples a time signal. The $\sin(N\psi/2)/\sin(\psi/2)$ function has a main lobe of width $\sim 2/(Nd\cos\theta_0)$ and sidelobes at $-13.3$ dB — identical to the spectral leakage of a rectangular window in spectral analysis.

Proof

Geometric series

Let $u = e^{jkd(\sin\theta - \sin\theta_0)}$ . Then

$\mathrm{AF}(\theta) = \frac{1}{N}\sum_{n=0}^{N-1} u^n = \frac{1}{N} \cdot \frac{1 - u^N}{1 - u}$ .

Dirichlet kernel form

$\frac{1 - u^N}{1 - u} = \frac{e^{jN\psi/2}}{e^{j\psi/2}} \cdot \frac{e^{-jN\psi/2} - e^{jN\psi/2}}{e^{-j\psi/2} - e^{j\psi/2}} = e^{j(N-1)\psi/2} \cdot \frac{\sin(N\psi/2)}{\sin(\psi/2)}$

where $\psi = kd(\sin\theta - \sin\theta_0)$ .

Taking the magnitude squared and normalising by $1/N^2$ gives the stated power pattern. $\blacksquare$

,

Example: 8-Element ULA Beam Pattern

An 8-element ULA with half-wavelength spacing ( $d = \lambda/2$ ) is steered to broadside ( $\theta_0 = 0$ ).

(a) Find the 3 dB beamwidth.

(b) Find the direction of the first null.

(c) What is the first sidelobe level?

Solution

Beamwidth

For broadside steering with $d = \lambda/2$ :

$\text{HPBW} \approx \frac{0.886\lambda}{Nd} = \frac{0.886}{N \times 0.5} = \frac{0.886}{4} = 0.2215$ rad $= 12.7^\circ$ .

More precisely, solving $|\mathrm{AF}(\theta_{3\text{dB}})|^2 = 0.5$ numerically gives $\text{HPBW} \approx 12.8^\circ$ .

First null

The first null occurs at $N\psi/2 = \pi$ : $\psi = 2\pi/N = \pi/4$ .

$kd\sin\theta_{\text{null}} = \pi/4 \Rightarrow \pi\sin\theta_{\text{null}} = \pi/4 \Rightarrow \theta_{\text{null}} = \arcsin(1/4) = 14.5^\circ$ .

First sidelobe level

For uniform weights, the first sidelobe of the Dirichlet kernel is at $\psi = 3\pi/N$ , with level

$\frac{1}{N^2} \cdot \frac{1}{\sin^2(3\pi/(2N))} \approx \frac{2}{3\pi}$

In dB: $20\log_{10}(2/(3\pi)) = -13.3$ dB relative to the main lobe peak. This is independent of $N$ for uniform weights. $\blacksquare$

ULA Beam Steering Animation

Watch the beam pattern of a 16-element ULA sweep from

-60^\circ

to

+60^\circ

and back. Observe how the main lobe narrows at broadside and broadens at large scan angles due to the

1/\cos\theta_0

foreshortening effect.

Cinematic beam sweep of a 16-element ULA (

d = \lambda/2

). The beam broadens as it scans away from broadside, consistent with the HPBW formula

0.886\lambda/(Nd\cos\theta_0)

.

ULA Beam Pattern

Explore how the beam pattern of a ULA changes with the number of elements $N$ , element spacing $d/\lambda$ , and scan angle $\theta_0$ . Observe the main lobe narrowing as $N$ increases, grating lobes appearing when $d > \lambda/2$ , and beam broadening at large scan angles.

Parameters

Number of elements

N

8

Spacing

d/\lambda

0.5

Scan angle

\theta_0

(degrees)0

Beamwidth and Directivity vs Number of Elements

See how the 3 dB beamwidth decreases and the array directivity increases as the number of elements $N$ grows. The beamwidth scales approximately as $1/N$ for half-wavelength spacing. The theoretical approximation $\text{HPBW} \approx 0.886\lambda/(Nd)$ is overlaid for comparison.

Parameters

Spacing

d/\lambda

0.5

Maximum

N

64

Theorem: ULA 3 dB Beamwidth

For a ULA with $N$ elements, spacing $d$ , steered to angle $\theta_0$ , the approximate half-power beamwidth is

$\text{HPBW} \approx \frac{0.886\,\lambda}{Nd\cos\theta_0}$

The directivity of the ULA (for $d = \lambda/2$ ) is approximately

$D \approx N$

or in dBi: $D_{\text{dBi}} \approx 10\log_{10} N$ .

The aperture of the array is $L = (N-1)d \approx Nd$ . The beamwidth is inversely proportional to the aperture measured in wavelengths, just as the time-domain resolution of a filter is inversely proportional to its bandwidth (the space-frequency duality). The $\cos\theta_0$ factor accounts for the foreshortening of the projected aperture when the beam is steered away from broadside.

Proof

From the Dirichlet kernel

The 3 dB point satisfies $|\mathrm{AF}|^2 = 1/2$ , which for the Dirichlet kernel occurs at approximately

$N\psi_{3\text{dB}}/2 \approx 1.3916$ .

With $\psi = kd(\sin\theta - \sin\theta_0) \approx kd\cos\theta_0 \cdot \Delta\theta$ for small $\Delta\theta$ around $\theta_0$ :

$\text{HPBW} = 2\Delta\theta \approx \frac{2 \times 1.3916}{Nkd\cos\theta_0/2} = \frac{2 \times 1.3916 \times \lambda}{N\pi d\cos\theta_0} \approx \frac{0.886\,\lambda}{Nd\cos\theta_0}$ . $\blacksquare$

Common Mistake: Confusing Element Spacing with Array Aperture

Mistake:

Believing that increasing element spacing $d$ always narrows the beam without consequences.

Correction:

While increasing $d$ does increase the total aperture $L = (N-1)d$ and thus narrows the main lobe, when $d > \lambda/2$ grating lobes appear — additional main-lobe-level peaks at angles where $\psi = \pm 2\pi$ . Grating lobes satisfy

$\sin\theta_g = \sin\theta_0 \pm m\lambda/d, \quad m = 1, 2, \ldots$

For $d = \lambda/2$ , grating lobes are pushed to $\theta_g = \pm 90^\circ$ (invisible endfire). For $d > \lambda/2$ , they enter the visible region and cause ambiguity and power loss.

Quick Check

A ULA with $d = \lambda$ is steered to broadside ( $\theta_0 = 0$ ). At what angles do grating lobes appear?

$\pm 30^\circ$

$\pm 90^\circ$

No grating lobes appear

$\pm 45^\circ$

Correction:

\pm 90^\circ

Correct. $\sin\theta_g = \sin\theta_0 + m\lambda/d = 0 + m$ . For $m = \pm 1$ : $\sin\theta_g = \pm 1$ , so $\theta_g = \pm 90^\circ$ . The grating lobes are right at endfire.

Key Takeaway

The ULA is a spatial DFT: the steering vector $\mathbf{a}(\theta)$ maps physical direction $\theta$ to spatial frequency $\psi = kd\sin\theta$ . Everything from the DFT world carries over — the Dirichlet kernel shapes the beam, $d = \lambda/2$ is the spatial Nyquist rate, and exceeding it causes grating lobes (spatial aliasing).

Uniform Linear Array (ULA)

An array of $N$ identical antenna elements arranged along a line with equal spacing $d$ . The simplest and most fundamental array geometry.

Steering Vector

$\mathbf{a}(\theta) \in \mathbb{C}^N$ : the vector of phase shifts across array elements for a plane wave from direction $\theta$ . The $n$ -th entry is $e^{jnkd\sin\theta}$ .

Array Factor

$\mathrm{AF}(\theta) = \mathbf{w}^H \mathbf{a}(\theta)$ : the spatial response of an array of isotropic elements to a plane wave from direction $\theta$ , determined by the weight vector $\mathbf{w}$ .

Grating Lobes

Spurious main-lobe-level peaks in the array pattern that appear when the element spacing exceeds $\lambda/2$ . Caused by spatial aliasing, analogous to spectral aliasing in temporal sampling.

Antenna Arrays: Uniform Linear Arrays

From Single Antennas to Arrays

Definition: Uniform Linear Array (ULA)

Definition: Steering Vector

Definition: Array Factor

Theorem: ULA Array Factor with Uniform Weights

Geometric series

Dirichlet kernel form

Example: 8-Element ULA Beam Pattern

Beamwidth

First null

First sidelobe level

ULA Beam Steering Animation

ULA Beam Pattern

Parameters

Beamwidth and Directivity vs Number of Elements

Parameters

Theorem: ULA 3 dB Beamwidth

From the Dirichlet kernel

Common Mistake: Confusing Element Spacing with Array Aperture

Quick Check

Key Takeaway

Uniform Linear Array (ULA)

Steering Vector

Array Factor

Grating Lobes

Definition:
Uniform Linear Array (ULA)

Definition:
Steering Vector

Definition:
Array Factor