# Reading — Antenna Patterns and Beamforming

By the end of this lesson you should be able to:

1. Identify the **main lobe, side lobes, and back lobes** in an antenna pattern.
2. Read a **polar plot** and extract half-power beamwidth (HPBW) and side-lobe level (SLL).
3. Use the gain / aperture / beamwidth rule of thumb to size an antenna.
4. Explain how an **electronically scanned array (ESA)** steers a beam, and what that buys an EW operator.

## The antenna is the radar's eye

The range equation in L3 hid a lot inside the gain terms $G_t$ and $G_r$. A radar only sees energy that arrives through its antenna, and the antenna's **pattern** decides what gets in and from where. Two radars with identical transmitters can behave completely differently because their patterns differ. For EW, the pattern is not a detail — it is the map of where the radar is strong, where it is weak, and where it can be fooled.

## Anatomy of a pattern

A pattern is the antenna's gain as a function of direction. It has three features worth naming:

- **Main lobe** — the direction of peak gain, where the radar wants to look. Pointing the main lobe is "pointing the radar."
- **Side lobes** — smaller lobes of unintended sensitivity off boresight. Energy arriving through a side lobe still reaches the receiver. These are **the back door for EW**: a jammer that is nowhere near the main beam can still inject energy through a side lobe.
- **Back lobe** — a lobe pointing roughly opposite the main lobe. Usually small, but never exactly zero.

:::{admonition} Key Concept
:class: key-concept

Side lobes are where an antenna listens when it thinks it isn't. A jammer off the main beam, entering through a $-18$ dB side lobe, is attenuated by 18 dB — but if it is loud enough, it still gets in. Much of electronic protection (EP) is the art of closing this back door.
:::

## Reading a polar plot

Patterns are drawn in polar coordinates: gain in dB versus angle. Two numbers do most of the work:

- **HPBW** (half-power beamwidth) — the angular width between the two points where the gain falls 3 dB below the peak. It measures how tightly the beam is focused.
- **SLL** (side-lobe level) — the peak side lobe relative to the main lobe, in dB (always negative). A **uniformly illuminated** aperture has its first side lobe at about $-13.2$ dB — a textbook constant worth memorizing.

You can lower the side lobes by **tapering** the illumination (feeding the edges of the aperture less than the center). Tapering trades SLL for HPBW: lower side lobes, but a wider main beam. There is no free lunch. A single polar plot is one cut (azimuth *or* elevation); two orthogonal cuts together describe the full 3D pattern.

## Gain, aperture, and beamwidth

For an aperture of size $D$ at wavelength $\lambda$, two rules of thumb tie everything together:

$$
\theta_{\text{HPBW}} \approx \frac{70\,\lambda}{D}\ \text{(degrees)}, \qquad
G \approx \frac{30{,}000}{\theta_{\text{az}}\cdot\theta_{\text{el}}}.
$$

A bigger aperture (in wavelengths) means a narrower beam and higher gain. Work a quick X-band example: $\lambda = 3$ cm and a $0.5 \times 0.5$ m antenna.

$$
\theta_{\text{HPBW}} \approx \frac{70\cdot0.03}{0.5} \approx 4.2^\circ \ \text{in each plane}, \qquad
G \approx \frac{30{,}000}{4.2\cdot4.2} \approx 1700 \approx 32\ \text{dBi}.
$$

The tighter the beam, the more gain — but the less sky you cover per scan position. That tension between **coverage** and **gain** is exactly why an IADS uses different radars for different jobs, which is the subject of L7.

## Beamforming: steering without moving

Replace the dish with $N$ small radiators arranged in a line — an **array**. Each element transmits the same signal, but with a programmed phase $\phi_n$. By sloping the phase across the array, you tilt the wavefront, and the main lobe points wherever you choose. The phase step needed to steer to angle $\theta_s$ is

$$
\Delta\phi = \frac{2\pi}{\lambda}\,d\,\sin\theta_s,
$$

where $d$ is the element spacing. At the steering angle the element contributions add in phase (constructive interference), so the beam forms there. Nothing physically moves — the beam is steered **electronically**, in the time it takes to load new phase values. More elements give a narrower beam and, with tapering, lower side lobes.

There is a catch. If the spacing $d$ grows beyond about $\lambda/2$, the array produces full-strength copies of the main lobe in unwanted directions — **grating lobes**. They are the angular-domain version of the Nyquist sampling limit: sample the aperture too coarsely and the pattern aliases. A well-designed array keeps $d \le \lambda/2$ to suppress them at broadside.

## ESA and AESA

An **ESA** steers its beam with phase shifters. An **AESA** (active ESA) goes further: every element has its own transmit/receive module. This architecture is what makes modern threats hard:

- **Microsecond beam pointing** — track many targets and interleave search and track within a single dwell.
- **Graceful degradation** — a few dead modules cost a few percent of gain, not the whole radar.
- **Multiple simultaneous beams** — different waveforms in different directions at once.
- **Low-probability-of-intercept waveforms** — precise phase control enables emissions that are hard to detect.

Physics still wins at the edges, though: when an array steers far off boresight, its effective aperture shrinks as $\cos\theta_s$ and the beam broadens as $1/\cos\theta_s$. Steering to $60^\circ$ doubles the HPBW and drops gain by about 3 dB. Most new fire-control and IADS surveillance radars are AESA.

## Why this matters for EW

- **Side-lobe leak.** Off-axis jamming enters through side lobes. The EP responses — side-lobe cancellation (an auxiliary antenna subtracts the side-lobe signal), side-lobe blanking (an omni reference rejects strong pulses outside the main beam), and low-SLL antenna design — all exist to shut that door.
- **Main-beam pointing.** Knowing where the threat is looking tells you what it is *doing* — searching, tracking, or about to engage.
- **ESA agility.** Microsecond re-pointing compresses the threat's kill chain and defeats classical deception that assumed a slow mechanical scan.
- **Polarization purity.** Cross-polarization leakage shows up in the pattern and matters for both jamming and low-observable design.

::::{admonition} Quick Exercise
:class: quick-exercise

1. A polar plot shows a $3^\circ$ main beam, $-20$ dB side lobes, and the beam re-points in microseconds. Mechanical dish or AESA?
2. An off-axis jammer at $15^\circ$ enters through a $-18$ dB side lobe. List two EP options.
3. An AESA steers from boresight to $60^\circ$. What happens to its HPBW and effective aperture?

:::{admonition} Solution
:class: dropdown

1. AESA — a mechanical dish cannot re-point in microseconds, and the low side lobes suggest a tapered electronic array.
2. Any two of: side-lobe cancellation with an auxiliary antenna, a side-lobe blanker, a low-SLL (tapered) antenna, polarization filtering, or spatial nulling with multiple beams.
3. HPBW broadens by $1/\cos 60^\circ = 2\times$; the effective aperture drops by $\cos 60^\circ = 0.5$, costing about 3 dB of gain. You always pay at large scan angles.
:::

::::

## Wrap-Up

A pattern has a main lobe, side lobes, and a back lobe; the side lobes are the EW back door. HPBW scales as $\lambda/D$ and gain as aperture $/\lambda^2$, so focus and coverage trade against each other. Arrays steer the beam by sloping the phase across their elements, and AESAs add microsecond agility, multi-beam operation, and graceful degradation — at the cost of broadening as $1/\cos\theta_s$ off boresight. Next, **L7** drops these antennas into a real integrated air-defense system and sorts out which radar does which job.