ECE 495 — Electromagnetic Warfare and Survivability

Block 2: RF Electronic Warfare
Lesson 14: Triangulation and Fusion

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

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Block 2 · Lesson 14 · Where We Left Off

Where We Left Off

L13 turned a measured phase into a bearing. But one bearing is a line — the emitter is somewhere along it, near or far, we cannot tell.

Cross a second bearing from a second receiver and the lines meet. That intersection is a fix.

Today

Crossing bearings into a position, fusing more than two, and the full chain from antenna to track.

Block 2 · Lesson 14 · Objectives

Lesson Objectives

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

Triangulate an emitter by crossing two lines of position
Explain how crossing geometry, not receiver quality, sets the fix error
Describe multi-receiver fusion and why more bearings help
Trace the full RWR processing chain from intercept to track

The receivers never change — only the geometry does.

Block 2 · Lesson 14 · Part One

Crossing Bearings

Two receivers, two bearings, one point — if the geometry cooperates. That last clause is the whole lesson.

Block 2 · Lesson 14 · Triangulation

From Bearing to Fix

Each receiver gives a line of position (LOP) — the emitter lies somewhere along its bearing. Two LOPs cross at the fix.

Two receivers, two bearings, one point — if the geometry cooperates.

Block 2 · Lesson 14 · The Cut

The Cut: Geometry Sets the Error

Each bearing carries a noise band. The fix lives where the two bands overlap.

A wide cut overlaps in a small, near-circular patch. What happens to that patch as the two bearings swing toward parallel?

Block 2 · Lesson 14 · The Cut

Close the Cut and the Error Smears

Same receivers, same bearing noise — a shorter baseline, and the bands now slide along each other.

The patch is 5× longer than it is wide, and the long axis points down-range — toward and away from the receivers. Push it all the way to collinear and you get a seam: no cut at all, and range is unobservable.

Same receivers. Different geometry. A useless fix.

Block 2 · Lesson 14 · Fusion

More Receivers: Least-Squares Fusion

With three or more bearings, the lines almost never meet at a single point — noise pulls them apart.

The problem is now overdetermined: more equations than unknowns
Solve for the position that best fits all bearings — a least-squares fix
Each added receiver brings redundancy and, if well placed, a better cut
A bad bearing stands out as a large residual — you can spot and down-weight it

Two bearings give a fix. More bearings give a fix you can trust.

Block 2 · Lesson 14 · Part Two

From Bearings to a Tracked Threat

Crossing bearings across platforms is how standoff geolocation works — and the bookkeeping that feeds it is where the whole thing can quietly go wrong.

Block 2 · Lesson 14 · Networked ESM

Networked ESM in Practice

Crossing bearings across platforms is how standoff geolocation works:

Separated receivers — wingmen, ships, ground sites — share PDWs over a data link
All measurements must be registered: common time and common grid
A wide baseline holds a usable, non-collapsing cut at long range — the B-21 standoff case
Fused bearings can locate an emitter that no single receiver could place

This is the bridge from warning (where is it, roughly) to targeting (where is it, precisely).

Block 2 · Lesson 14 · The Chain

The RWR Processing Chain, End to End

Interceptcatch a pulse

→

PDWparameterize

→

Deinterleavesort by emitter

→

AoAper-pulse angle

across receivers↓

Associatesame emitter?

→

Fuse · geolocatecross bearings

→

Trackhold over time

→

Cueto the crew

L12 built the top row — intercept to per-pulse AoA on one receiver
L13–L14 add the bottom row — across receivers, into a located, tracked threat

Block 2 · Lesson 14 · Association

Association: The Step Before You Can Cross

Two receivers each report a bearing. Crossing them assumes they heard the same emitter. In a dense scene, that is a real question.

Match candidate emitters across receivers on frequency, PRI, and PW first
Only after they associate can their bearings be crossed into a fix
Cross two different emitters and you invent a ghost — a fix on nothing

Geolocation is only as good as the association feeding it.

Today's Interactive Demo: Geolocation Explorer

Companion script: L14_TwoReceiverCrossFix.m

Live demo

Geolocation Explorer

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

Demo hand-off; drive it from here. Every marker drags: two receivers and the emitter. Tiles: cut angle, CEP, the two one-sigma semi-axes, a predicted RMS, PDOP and the mean range; each bearing is drawn as a half-degree noise wedge. Open it as it loads — the dashed ×10 ellipse is the true one at ten times the radii, so they can actually watch it breathe. Push the emitter downrange first: the cut closes, the ellipse stretches along the line of sight, CEP grows, and the receivers never changed. Seam preset next — both bearings on one line, PDOP reads infinity. Worth saying out loud: PDOP is the same geometry number a GPS receiver works out from where the satellites sit. DOP map on, and the seam is a red ridge straight through the emitter — every place this pair cannot locate, drawn before anyone takes a bearing. Tick the third receiver on and the ridge breaks. Then the punchline -- ghost fixes on: a second emitter, four crossings, two of them ghosts, and on two receivers nothing at all tells them apart. The third receiver's bearing runs through the real pair and misses each ghost by kilometres; the label says by how much. That is association, and it is the step that cannot be skipped. Fix cloud on: three hundred fixes off half-degree-noisy bearings, and the CEP tile reports the share that landed inside — about half, which is the promise CEP makes. Last, the threat ring: drag a receiver in and the ellipse tightens as its marker goes red — accuracy bought with exposure. Companion script runs on their own.

Block 2 · Lesson 14 · Quick Exercise

Quick Exercise / Type-along: Size the Fix

Type this with me — a B-21 standoff leg, RWR bearings good to 0.5°.

sig  = deg2rad(0.5);             % RWR bearing 1-sigma
wp   = [-60 -25; -60 25];        % WP1, WP2 as [N E], km
d    = [0 0] - wp;               % [dN dE] to the emitter at (0,0)
R    = hypot(d(:,1), d(:,2));    % range from each waypoint, km
th   = atan2d(d(:,2), d(:,1));   % bearing, North toward East
cut  = abs(th(1) - th(2));       % crossing ("cut") angle, deg
ecr  = R(1)*sig*1e3;             % one bearing, across the LOS, m
a    = ecr/(sqrt(2)*sind(cut/2));  % down-range semi-axis, m
b    = ecr/(sqrt(2)*cosd(cut/2));  % cross-range semi-axis, m
fprintf('R = %.1f km   bearings %+.1f / %+.1f deg\n', R(1), th);
fprintf('cut = %.1f deg   e_cr = %.0f m\n', cut, ecr);
fprintf('ellipse a = %.0f m   b = %.0f m   CEP = %.0f m\n', ...
        a, b, 0.59*(a+b));

The ellipse comes out 2.4× longer one way. Which way, and why?

Then: double the leg, or halve the standoff — which buys more?

Block 2 · Lesson 14 · Wrap-Up

Wrap-Up

Key ideas

One bearing is a line; two crossing bearings are a fix
The cut angle sets the error — shallow geometry smears it along range
More receivers turn the fix into a least-squares solution you can trust
Association comes first — cross the wrong pair and you get a ghost

Next lesson

L15: Electromagnetic Protect — LPI radar, agility, side-lobe cancellation
Work the L15 pre-flight before class — you could be the one up front