ECE 495 — Electromagnetic Warfare and Survivability

Block 2: RF Electronic Warfare
Lesson 13: Angle of Arrival

Juan Jurado, Col, USAF, Ph.D.
Permanent Professor and Head
Electrical and Computer Engineering
Fall 2026

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Block 2 · Lesson 13 · Where We Left Off

Where We Left Off

L12 built the RWR's PDW: frequency, pulse width, TOA, power — and one field we measured but never explained. AoA.

AoA is the only spatial parameter in the word. It is how the processor tells two emitters apart when their pulses overlap, and how the cockpit knows which way to look.

Today

How an antenna with no moving parts answers where is it?

Block 2 · Lesson 13 · Objectives

Lesson Objectives

By the end of this lesson, you will be able to:

Explain why AoA is the hardest field in the PDW to measure well
Describe amplitude comparison and where its accuracy runs out
Derive the interferometer relation $\Delta\phi = \tfrac{2\pi d}{\lambda}\sin\theta$
Trace the accuracy–ambiguity trade that sets baseline length

Block 2 · Lesson 13 · Part One

Two Handles on Bearing

Every other field in the word is a stopwatch reading on a single antenna. Bearing is geometry — and a receiver that only listens has exactly two physical quantities it can turn into an angle.

Block 2 · Lesson 13 · The Hard Parameter

Why AoA Is the Hard Parameter

Frequency, PW, and PRI are all timing measurements on a single antenna. AoA is different — it is geometry, and the RWR is on a moving aircraft.

The field the processor deinterleaves on — it splits emitters that share a band
It cues the aircrew and points countermeasures at the right threat
One bearing is a line, not a fix — a position needs two (L14)

Every other PDW field says what. AoA says where.

Block 2 · Lesson 13 · Two Methods

Two Ways to Find the Angle

A passive receiver has two physical handles on bearing:

Amplitude — how loud the signal is in beams that point different ways
Phase — how the wavefront arrives at antennas spaced apart

Method	Reads	Character
Amplitude comparison	Power ratio across beams	Coarse, cheap, instant
Phase interferometry	Phase across a baseline	Fine, but can wrap

Most RWRs do amplitude comparison; precision systems add interferometry.

Block 2 · Lesson 13 · Amplitude Comparison

Amplitude Comparison: Squinted Beams

Point several broad antennas in different directions so their patterns overlap. A signal off boresight is louder in the beam aimed nearer to it.

The classic RWR: four antennas, one per quadrant; the power ratio between adjacent beams gives bearing within the overlap
Single pulse — no waiting, no second emitter needed
Wideband and cheap; degrades gracefully as signals weaken

Accuracy is limited by how well you know each antenna pattern. Typical four-quadrant bearing lands within roughly $10$–$30^\circ$ — enough to label a quadrant, not to target.

Block 2 · Lesson 13 · Part Two

Phase, and Its Price

Phase can be read to a small fraction of a cycle, and that is where sub-degree bearing comes from. It is also where the trap is.

Block 2 · Lesson 13 · Geometry

Phase Interferometry: The Geometry

The extra leg

A plane wave from off-boresight angle $\theta$ reaches the far element after traveling an extra $d\sin\theta$ — the leg of a right triangle with hypotenuse $d$.

At boresight the wave hits both elements at once and the extra path is zero. That extra path is a measurable phase difference between the two channels.

Block 2 · Lesson 13 · Phase to Bearing

From Phase to Angle

The extra path $d\sin\theta$ becomes phase by dividing by wavelength — one full wavelength of path is $2\pi$ of phase:

Interferometer relation

\[ \Delta\phi = \frac{2\pi d}{\lambda}\,\sin\theta \qquad\Longrightarrow\qquad \theta = \arcsin\!\left(\frac{\lambda\,\Delta\phi}{2\pi d}\right) \]

Measure one number, $\Delta\phi$; invert for the bearing $\theta$
A fixed phase error maps to a smaller angle error as $d/\lambda$ grows
Long baselines give sub-degree accuracy — far past amplitude comparison

Stretch the baseline, sharpen the bearing.

Block 2 · Lesson 13 · Phase to Bearing

The Catch: Phase Wraps

A phase meter cannot tell $\Delta\phi$ from $\Delta\phi + 2\pi$ — the reading folds into $[-\pi,\pi]$. Staying unambiguous across $\pm 90^\circ$ means $|\Delta\phi|\le\pi$, and $\sin\theta = 1$ turns that into:

Interferometer relation

\[ \Delta\phi = \frac{2\pi d}{\lambda}\,\sin\theta \qquad\Longrightarrow\qquad \theta = \arcsin\!\left(\frac{\lambda\,\Delta\phi}{2\pi d}\right) \]

The ambiguity limit

\[ d \le \lambda/2 \]

But $d \le \lambda/2$ is a short baseline — back to coarse accuracy. Make $d$ large and the phase wraps: a fine but wrong bearing. Accuracy wants a long baseline; unambiguity wants a short one, and you cannot have both with a single pair.

Block 2 · Lesson 13 · Resolving It

Resolving It: Multi-Baseline

Use more than two elements — a short baseline and a long one:

Short pairhalf a wavelength · coarse, no wrap

→

Which cyclethe coarse bearing predicts it

→

Long pairunwrap, invert · fine bearing

Short pair picks the cycle; long pair sets the precision
Real systems use several baselines and calibrate each across the band

Block 2 · Lesson 13 · Head to Head

Amplitude vs Interferometry

	Amplitude comparison	Interferometry
Reads	Power ratio across beams	Phase across a baseline
Accuracy	Coarse ($\sim 10$–$30^\circ$)	Fine ($<1^\circ$ possible)
Pulses	Single pulse	Single pulse
Cost / size	Low; a few antennas	Higher; spaced array + cal
Failure mode	Pattern calibration error	Phase wrap / ambiguity

Many fielded RWRs amplitude-compare for the quadrant, then interferometer-refine when precision matters
Both are passive and instantaneous — no scan, no transmit

Today's Interactive Demo: AoA Accuracy Explorer

Companion script: L13_InterferometricAoA.m
In L14 we take these bearings to two receivers and cross them into a fix

Live demo

AoA Accuracy Explorer

usafa-ece.github.io/ece-495-ew

Demo hand-off; drive it from here. This is a two-baseline array: a short pair fixed at half a wavelength, and a long pair on the slider. Four controls — true bearing, frequency, long baseline, and SNR. Read the upper plot first: the navy curve is the long pair's true unwrapped phase against bearing, the blue curve is the short pair's, and the two dashed grey lines are the plus and minus pi limits the receiver can actually read. Notice the blue curve never leaves that band and the navy one does — that contrast is the whole ambiguity problem. The panel on the right lists three bearings: the short pair's coarse answer, the long pair's naive answer, and the resolved answer. Start on the d equals lambda over two chip: green card, all three agree, all three coarse. Then take the stretch chip, which puts the long pair at two and a half wavelengths. The card turns red, the naive row jumps tens of degrees, and the resolved row does not — the cycle counter steps to plus one, a whole turn of phase is added back, and the fine bearing lands within a few tenths. Say out loud that the red card flags the guarantee for the whole plus or minus ninety degree field of regard, so it can read ambiguous while the resolved answer is perfectly good. The polar plot is the same story as rays: green riding just inside navy, red swung away, blue visibly off truth. If there is time, drop the SNR to zero on a very long baseline and let the short pair pick the wrong cycle — the unwrap is only ever as good as the coarse bearing feeding it. The companion script prints this same pipeline as numbers.

Block 2 · Lesson 13 · Quick Exercise

Quick Exercise / Type-along: Pick the Baseline

Type this with me — one X-band threat, four candidate baselines.

lambda = 0.03;                      % X-band, 3 cm
th     = 35;                        % true bearing, deg off boresight
d      = [0.5 1 3 8]*lambda;        % four candidate baselines
phi    = 2*pi*d/lambda*sind(th);    % true phase difference, rad
phiw   = mod(phi+pi, 2*pi) - pi;    % what the phase meter reads
est    = asind(phiw*lambda./(2*pi*d));      % naive inversion
sig    = 20*lambda./(2*pi*d*cosd(th));      % 20 deg phase err, deg
fprintf('  d/lam  phi(rad)  wrapped  est(deg)  sig(deg)\n');
fprintf('%7.1f %9.2f %8.2f %9.2f %9.2f\n', ...
        [d/lambda; phi; phiw; est; sig]);
fprintf('\nTruth = %g deg.  Unambiguous needs d <= lambda/2.\n', th);

Only one row recovers 35°. Which column told you the rest would fail?

Then: the 8λ row reads 0.49° — HW2's σ_θ. What buys the right cycle?

Block 2 · Lesson 13 · Wrap-Up

Wrap-Up

Key ideas

AoA is the only spatial PDW field — it deinterleaves, cues, and warns
Amplitude comparison: power ratio across beams — coarse, cheap, instant
Interferometry: $\Delta\phi = \tfrac{2\pi d}{\lambda}\sin\theta$ — fine bearing from one phase measurement
Long baseline sharpens accuracy but wraps; multi-baseline buys both

Next lesson

L14: Triangulation and fusion — crossing bearings into a fix
Pre-flight quiz for L14 due before class