USAFA DFEC

ECE 495 — Electromagnetic Warfare and Survivability

Block 4: Aircraft Survivability and Mission Planning
Lesson 33: Information Geometry Under Threat
Juan Jurado, Col, USAF, Ph.D.
Permanent Professor and Head
Electrical and Computer Engineering
Fall 2026
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Block 4 · Lesson 33 · Where We Left Off

Where We Left Off

L14's demo put two things on your screen without a name: a PDOP tile and a DOP map.

Project 2 then had you build $\mathbf{J} = \frac{1}{\sigma_\theta^2}\sum_k \mathbf{H}_k^\top\mathbf{H}_k$ and invert it — without saying what $\mathbf{J}^{-1}$ promises.

L14's reading promised the formal treatment — Fisher information and the CRLB — for Block 4. This is Block 4.

Today
Name both: Fisher information, the Cramér–Rao bound, and the seams where geometry fails — for their sensors and for ours.

Block 4 · Lesson 33 · Objectives

Lesson Objectives

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

  • Build the Fisher information $\mathbf{J}$ from bearing rows
  • State the CRLB and read the error ellipse from $\mathbf{J}$'s eigenvalues
  • Name L14's PDOP tile: $\text{DRMS} = \text{PDOP}\cdot\bar R\cdot\sigma_\theta$
  • Locate seams — in their network and in ours
Geometry is information: where their sensors can't see you well is a place you can fly.

Block 4 · Lesson 33 · Part One

Information from Bearings

A bearing measures one direction only. Stack enough directions and you know where something is — or exactly which way you don't.

Block 4 · Lesson 33 · One Bearing

One Bearing, One Row

1
The bearing from a site, clockwise from North.
$\theta = \mathrm{atan2}(\Delta E,\ \Delta N)$
2
Nudge the target: only its cross-range move turns $\theta$.
$\mathbf{h} = \dfrac{[-\Delta E,\ \Delta N]}{R^2} = \dfrac{\mathbf{n}^\top}{R}$
3
Noise $\sigma_\theta$ pins cross-range to $R\sigma_\theta$; along the line of sight it says nothing.
$\dfrac{\mathbf{n}\,\mathbf{n}^\top}{R^2\sigma_\theta^2}$
N θ site R σθ along the line of sight: no information target

Block 4 · Lesson 33 · Fisher Information

Stack the Rows: the Fisher Information $\mathbf{J}$

Information adds
$$\mathbf{J} = \mathbf{H}^\top\mathbf{R}^{-1}\mathbf{H} = \sum_{k} \frac{\mathbf{n}_k\,\mathbf{n}_k^\top}{R_k^2\,\sigma_\theta^2}$$
  • Every bearing adds: every site, every look, every pass
  • $\mathbf{J}$ is 2×2 in km⁻² — and a function of where the target is
  • One site is rank one; two sites out of line give full rank
You built exactly this in Project 2 — same $\mathbf{H}_k$ rows, same sum. Today we say what its inverse promises.

Block 4 · Lesson 33 · The Bound

The Cramér–Rao Lower Bound

No unbiased estimator does better
$$\operatorname{Cov}(\hat{\mathbf{p}}) \;\succeq\; \mathbf{J}^{-1}$$
  • Not a smarter solver, not a faster computer — the geometry sets the floor
  • Least squares on small noise reaches it: Project 2's ellipse was $\mathbf{J}^{-1}$
  • It needs only positions and $\sigma_\theta$ — map it before anyone takes a bearing

So a bound on their estimator is a statement about your route.

Block 4 · Lesson 33 · The Ellipse

Reading the Ellipse from $\mathbf{J}$

β a = 1/√λmin along emin: weak b = 1/√λmax S1 S2
Eigen-decomposition
$$\mathbf{J} = \lambda_{\max}\mathbf{e}_{\max}\mathbf{e}_{\max}^\top + \lambda_{\min}\mathbf{e}_{\min}\mathbf{e}_{\min}^\top$$
  • $\mathbf{J}^{-1}$ keeps the eigenvectors and inverts the eigenvalues
  • Semi-axes $a = 1/\sqrt{\lambda_{\min}}$ along $\mathbf{e}_{\min}$ and $b = 1/\sqrt{\lambda_{\max}}$
  • $\text{DRMS} = \sqrt{a^2+b^2} = \sqrt{\operatorname{tr}\mathbf{J}^{-1}}$
  • Condition $\kappa = \lambda_{\max}/\lambda_{\min}$: 1 is a circle, $\infty$ is a seam

Block 4 · Lesson 33 · By Hand

Eigenvalues by Hand

A network's $\mathbf{J}$ at your position, in $[N\ E]$ and $\text{km}^{-2}$
$$\mathbf{J} = \begin{bmatrix}0.05 & 0.02\\ 0.02 & 0.02\end{bmatrix}$$
1
Trace and determinant: the whole 2×2 story.
$\operatorname{tr}\mathbf{J} = 0.07,\quad \det\mathbf{J} = 0.0010 - 0.0004 = 0.0006$
2
The two eigenvalues.
$\lambda = 0.035 \pm \sqrt{0.035^2 - 0.0006} = 0.06,\ 0.01$
3
Semi-axes: small eigenvalue, long axis.
$a = 1/\sqrt{0.01} = 10\ \text{km},\quad b = 1/\sqrt{0.06} = 4.08\ \text{km}$
4
Condition and DRMS.
$\kappa = 6,\quad \text{DRMS} = \sqrt{a^2 + b^2} = 10.8\ \text{km}$
5
Which way is long? Solve $(\mathbf{J} - 0.01\,\mathbf{I})\,\mathbf{e} = \mathbf{0}$.
$\mathbf{e}_{\min} \propto [\,1,\ -2\,]$: the long axis runs 117°–297°

Block 4 · Lesson 33 · Part Two

PDOP, Named

The PDOP tile on L14's screen was this bound all along, with the range scaled out of it.

Block 4 · Lesson 33 · PDOP

The L14 Tile, Named

1
Scale row $k$ by $\bar R/R_k$, with $\bar R$ the mean range.
$\tilde{\mathbf{h}}_k = \dfrac{\bar R}{R_k}\,\mathbf{n}_k^\top$
2
Stack them: it is $\mathbf{J}$ again, times a constant.
$\tilde{\mathbf{H}}^\top\tilde{\mathbf{H}} = \bar R^2\sigma_\theta^2\,\mathbf{J}$
3
What L14's demo computes, exactly.
$\text{PDOP} = \sqrt{\operatorname{tr}\big((\tilde{\mathbf{H}}^\top\tilde{\mathbf{H}})^{-1}\big)} = \dfrac{\sqrt{\operatorname{tr}\mathbf{J}^{-1}}}{\bar R\,\sigma_\theta}$
4
So the two tiles on L14's screen always obeyed
$\text{DRMS} = \text{PDOP}\cdot\bar R\cdot\sigma_\theta$

PDOP is pure geometry — the same number a GPS receiver computes from where its satellites sit.

Block 4 · Lesson 33 · PDOP

The Symmetric Pair: $\text{PDOP} = \sqrt{2}/\sin\beta$

1
Both sites at range $R$; lines of sight $\pm\beta/2$ about the bisector (axes: along it, across it).
$\mathbf{J} = \dfrac{2}{R^2\sigma_\theta^2}\begin{bmatrix}\sin^2\frac{\beta}{2} & 0\\ 0 & \cos^2\frac{\beta}{2}\end{bmatrix}$
2
Diagonal, so read the eigenvalues off — L14's ellipse falls out.
$a = \dfrac{R\sigma_\theta}{\sqrt2\,\sin(\beta/2)},\qquad b = \dfrac{R\sigma_\theta}{\sqrt2\,\cos(\beta/2)}$
3
Add the squares.
$\text{DRMS} = \dfrac{\sqrt2\,R\,\sigma_\theta}{\sin\beta} \;\Rightarrow\; \text{PDOP} = \dfrac{\sqrt2}{\sin\beta}$
Cut → PDOP:  90° → 1.41 (the best a pair can do)  ·  60° → 1.63  ·  30° → 2.83  ·  10° → 8.14  ·  0° or 180° → $\infty$

Block 4 · Lesson 33 · Quick Exercise

Quick Exercise / Type-along: Fix the Jammer

Type this with me — two of their ES sites, one of our stand-off jammers.

sig = deg2rad(2);                % ES bearing 1-sigma, rad
S   = [0 0; 0 60];               % their two ES sites [N E], km
p   = [80 60];                   % our stand-off jammer [N E], km
d   = p - S;                     % [dN dE] from each site, km
R   = hypot(d(:,1), d(:,2));     % range from each site, km
n   = [-d(:,2) d(:,1)] ./ R;     % cross-range unit vectors
J   = (n./R)' * (n./R) / sig^2;  % Fisher information, 1/km^2
[V, L] = eig(J);                 % eigenvalues ascending: min first
ab  = 1 ./ sqrt(diag(L));        % 1-sigma semi-axes [a; b], km
Ht  = mean(R) * n ./ R;          % L14's rows (Rbar/R_k) n_k'
PDOP = sqrt(trace(inv(Ht'*Ht)));
fprintf('lambda %.4f %.4f /km^2   a %.2f  b %.2f km\n', diag(L), ab);
fprintf('DRMS %.2f km   PDOP*Rbar*sig %.2f km   PDOP %.2f\n', ...
        sqrt(trace(inv(J))), PDOP*mean(R)*sig, PDOP);

$a$ comes out three times $b$. Which eigenvalue made $a$ — and which way does the long axis point?

Then: park the jammer at p = [0 120]. What does MATLAB say — and what does it mean for the jammer?

Block 4 · Lesson 33 · Quick Problem

Quick Problem: Open the Cut

Two aircraft fix an emitter from equal range — a symmetric pair.

  • Both aircraft at $\bar R = 120$ km from the emitter
  • Their bearings cross at $\beta = 30^\circ$
  • Each RWR bearing good to $\sigma_\theta = 1.5^\circ$ (1σ)
  1. (a) The PDOP of the pair.
  2. (b) The DRMS of the fix, in km.
  3. (c) The cut $\beta$ that brings DRMS down to 5 km at the same $\bar R$ and $\sigma_\theta$.
(a) 2.83  ·  (b) 8.89 km  ·  (c) $\beta = 62.7^\circ$ (or its supplement, $117.3^\circ$)

Block 4 · Lesson 33 · Seams

Seams: Where $\mathbf{J}$ Goes Blind

κ = 3.4 cut 57° κ = 81 cut 16° κ = 56 cut 165° S1 S2 seam: λmin = 0 1σ CRLB ellipses ×4 · σθ = 1° · sites 60 km apart
  • A seam: $\lambda_{\min} \to 0$ — $\mathbf{J}$ singular, or $\kappa \gg 1$
  • Two sites: the whole line through them — beyond both ends and between
  • Three or more: no exact seam unless all in line — but ridges where $\kappa$ climbs
  • Far from everyone, every cut closes: DRMS grows like $R/\sin\beta$

Block 4 · Lesson 33 · Range

Range Fills the Gap

A range + bearing sensor ($\mathbf{u}$ = unit line of sight)
$$\mathbf{J}_{\text{radar}} = \frac{\mathbf{n}\,\mathbf{n}^\top}{R^2\sigma_\theta^2} + \frac{\mathbf{u}\,\mathbf{u}^\top}{\sigma_R^2}$$
  • The range row lies along the line of sight — exactly what bearings leave empty
  • One range + bearing radar is full rank by itself: no seam
  • Deny its range — L17's noise strobe — and it is bearing-only again

A network that crosses jam strobes is a bearing-only network, seams and all.

Block 4 · Lesson 33 · Both Directions

Both Directions

The same matrix, read from opposite ends of the line of sight.

Our receivers → their emitters
  • Project 2: our bearings, our $\mathbf{J}$
  • We want $\lambda_{\min}$ large: open the cut, shorten the range
  • Pay for it in exposure — the closer leg is inside their rings
Their ES / IRST → us, or our jammer
  • Their bearings, their $\mathbf{J}$ — evaluated along our route
  • We want their $\lambda_{\min}$ small: ride the seam, keep the range long
  • Final Project T4: their DRMS along your chosen route
Geometry is information: where their sensors can't see you well is a place you can fly.

Today's Interactive Demo: Information Map

Companion script: L33_InformationMap.m
Live demo

Information Map

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

Block 4 · Lesson 33 · Wrap-Up

Wrap-Up

Key ideas
  • Each bearing adds $\mathbf{n}\mathbf{n}^\top/(R^2\sigma_\theta^2)$: cross-range information only
  • The CRLB $\mathbf{J}^{-1}$ is the floor; its eigenvalues are the ellipse, $\sqrt{\operatorname{tr}\mathbf{J}^{-1}}$ the DRMS
  • L14's PDOP, named: $\text{DRMS} = \text{PDOP}\cdot\bar R\cdot\sigma_\theta$
  • A symmetric pair: $\text{PDOP} = \sqrt2/\sin\beta$, best $\sqrt2$ at 90°
  • Seams — $\lambda_{\min}\to 0$ — are where a route hides or a fix fails
Next lesson
  • L34: Optimal Path Planning — the threat map becomes a graph, and Dijkstra finds the lowest-risk route
  • Work the L34 pre-flight before class