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.

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, and it is a sampling problem. The elements sample the aperture at spacing \(d\), so the pattern can alias — repeat itself — in exactly the way a coarsely sampled signal does. The aliased copies are grating lobes: full-strength duplicates of the main lobe pointing somewhere you never asked for. Whether one appears depends on the spacing and the scan angle together:

\[ d \le \frac{\lambda}{1+|\sin\theta_0|}\quad\Rightarrow\quad\text{no grating lobe in the visible region.} \]

Read the bound before you memorize a number. At broadside (\(\theta_0 = 0\)) it permits a full \(d \le \lambda\) — spacing alone never creates a grating lobe there. It is steering that tightens the bound: scan to \(30^\circ\) and it drops to \(0.67\lambda\), scan to \(60^\circ\) and it drops to \(0.54\lambda\). An array with \(d = 0.9\lambda\) is perfectly clean staring straight ahead and grows a second full-strength beam the moment it scans off boresight. This is why designers pick \(d \le \lambda/2\): it is the one choice that satisfies the bound at every scan angle, out to \(\theta_0 = 90^\circ\).

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.

Type-along

Type this in MATLAB. It builds the array factor the honest way — by adding up the \(N\) element phasors — for an 8-element array with wide spacing, \(d = 0.9\lambda\).

N = 8;  dlam = 0.9;  th0 = 0;          % elements, d/lambda, steer (deg)
th = -90:0.25:90;                      % look angles (deg)
n  = (0:N-1).';                        % element index, as a column
beta = -2*pi*dlam*sind(th0);           % programmed phase per element
AF = sum(exp(1j*n*(2*pi*dlam*sind(th) + beta)), 1)/N;
plot(th, 20*log10(max(abs(AF),1e-3)), 'LineWidth',1.5);
hold on;  grid on;  ylim([-40 0]);
xlabel('theta (deg)');  ylabel('|AF| (dB)');
fprintf('theta0=%2d | beta=%7.1f deg/elem | limit %.2f | used %.2f\n', ...
        th0, rad2deg(beta), 1/(1+abs(sind(th0))), dlam);

Run it. Then change one thing — th0 = 30 on line 1 — and run it again. The second curve lands on top of the first.

1. At \(\theta_0 = 0\) the spacing is nearly double the \(\lambda/2\) rule. How many full-strength lobes do you see?

2. At \(\theta_0 = 30^\circ\), how many? Where is the new one?

3. Compare the two limit values the script printed. Which run violated its own limit, and why?

Solution

1. One, at \(0^\circ\), about \(7^\circ\) wide. Wide spacing alone did not produce a grating lobe — at broadside the bound is \(d \le \lambda\), and \(0.9\lambda\) clears it.

2. Two: the steered main lobe at \(+30^\circ\) and a full-strength grating lobe near \(-38^\circ\). Predicted by \(\sin\theta_g = \sin\theta_0 - \lambda/d = 0.5 - 1.111 = -0.611\), so \(\theta_g = -37.7^\circ\).

3. The printed limit dropped from \(1.00\) to \(0.67\) while \(d/\lambda\) stayed at \(0.90\). Steering broke the array, not spacing by itself — that is exactly what \(d \le \lambda/(1+|\sin\theta_0|)\) says. The programmed phase moved with it: \(\beta = 0\) to \(\beta = -162^\circ\) per element.

The EW reading: a threat array spaced this way radiates — and listens — at full gain in a direction its operator never chose, and that direction moves as the beam scans.

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.