ECE 495 — Electromagnetic Warfare and Survivability

Block 1: Radar Fundamentals
Lesson 8: Detection Theory

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

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Block 1 · Lesson 8 · Where We Left Off

Where We Left Off

L3 gave us $R_{\max}$. We plugged in some $S_{\min}$ from the data sheet.

But where does $S_{\min}$ come from? It is not a constant — it is a contract between detection probability and false-alarm rate.

Today

Noise, SNR, $P_d$, $P_{fa}$, ROC curves, and integration — the math that decides whether a return becomes a track.

$S_{\min}$ is a contract, not a constant.

Block 1 · Lesson 8 · Objectives

Lesson Objectives

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

Compute the noise floor for a given bandwidth and noise figure
Explain the detection threshold and the $P_d / P_{fa}$ tradeoff
Read a ROC curve and identify the SNR that matches a mission requirement
Quantify the SNR gain from coherent and non-coherent integration

Detection is an SNR business.

Block 1 · Lesson 8 · Part One

The Threshold

One comparison decides whether a return becomes a track. It has exactly two ways to be wrong, and you do not get to make both of them small.

Block 1 · Lesson 8 · The Decision

The Detection Problem

Every radar return is a signal sample buried in noise. Compare the sample to a threshold:

The decision

Above threshold $\Rightarrow$ declare target
Below threshold $\Rightarrow$ declare noise

Two error modes

Miss — target present, threshold not crossed
False alarm — noise alone crosses the threshold

Every detection is a thresholded coin flip.

Block 1 · Lesson 8 · Noise Floor

The Noise Floor: $N = kTB$

Thermal noise sets the floor below which we cannot see:

Noise floor in dBm

\[ N_{\text{dBm}} = -174 + 10\log_{10}(B_{\text{Hz}}) + \text{NF}_{\text{dB}} \]

$-174$ dBm/Hz is $kT_0$, the thermal floor at the $290$ K reference temperature
Bandwidth $B$ adds $10\log_{10}B$
Noise figure NF is the receiver's own contribution

Quick example: $B = 10$ MHz, NF $= 3$ dB $\;\Rightarrow\; N \approx -101$ dBm.

Block 1 · Lesson 8 · SNR and the Threshold

SNR and the Detection Threshold

SNR $= S/N$ at the detector input. Set the threshold low and almost every target crosses it — along with a great many noise spikes.

threshold

Threshold

low

$P_{fa}$

0.159

$P_d$

0.995

Block 1 · Lesson 8 · SNR and the Threshold

Raise It, and Both Tails Go

Slide it right: $P_{fa}$ falls by a factor of 118 — and $P_d$ walks down with it.

threshold

Threshold

high

$P_{fa}$

0.0014

$P_d$

0.726

No threshold gives high $P_d$ and low $P_{fa}$ without enough SNR — $P_d = 0.9$ at $P_{fa} = 10^{-6}$ needs about 13 dB on a steady target.

Block 1 · Lesson 8 · Part Two

Buying Your Way Out

The threshold only slides you along one curve. SNR moves you to a better curve — and integration is how you pay for it, in time.

Block 1 · Lesson 8 · ROC

The Receiver Operating Characteristic

Plot $P_d$ vs $P_{fa}$ as the threshold sweeps. Each SNR value gives a different curve.

Higher SNR pushes the curve toward the top-left corner (perfect detector)
Pick an operating $P_{fa}$, read off the achievable $P_d$
Or fix a mission $P_d$ and read off the required SNR

The ROC turns "good detector" into one number: SNR at the detector input.

Block 1 · Lesson 8 · Integration

Integration Gain: Spending Time for SNR

Combine $N$ pulses on the same target. SNR improves:

Coherent

Sum complex returns — phase preserved
Gain $= N$ linear, i.e. $+10\log_{10}(N)$ dB
100 pulses buy 20 dB

Non-coherent

Sum magnitudes — phase discarded
Gain $\approx \sqrt{N}$, i.e. $\approx +5\log_{10}(N)$ dB
100 pulses buy $\sim$10 dB

Roughly $\sqrt{N}$ at low per-pulse SNR — the true non-coherent gain lies between $\sqrt{N}$ and $N$.

Coherent integration requires phase coherence across the dwell. Modern AESAs lean on this hard.

Integration buys SNR — coherently with $N$, non-coherently with $\sqrt{N}$.

Block 1 · Lesson 8 · Closing the Loop

Closing the Loop on $R_{\max}$

Recall from L3

\[ R_{\max} = \left[\frac{P_t \, G_t \, G_r \, \lambda^{2} \, \sigma}{(4\pi)^{3} \, S_{\min}}\right]^{1/4} \]

We treated $S_{\min}$ as a constant. It is actually

\[ S_{\min} = \frac{kT B \cdot \text{NF} \cdot \text{SNR}_{\text{req}}}{G_{\text{int}}} \]

Tighter mission $P_d \Rightarrow$ higher SNR$_{\text{req}} \Rightarrow$ larger $S_{\min}\Rightarrow$ shorter $R_{\max}$
More integration $\Rightarrow$ smaller $S_{\min} \Rightarrow$ longer $R_{\max}$
This is how a "longer dwell" becomes a "longer range"

Today's Interactive Demo: ROC Explorer

Companion script: L8_PdPfaAndROC.m
L9 is a Project 1 work day — we put L1–L8 to use

Live demo

ROC Explorer

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

Demo hand-off; drive it from here. Drag the threshold and watch the operating point slide along one curve while the two shaded tails in the lower panel grow and shrink together — those tails are the false alarm and detection probabilities, read off directly. Raise the single-pulse signal-to-noise slider and the whole curve lifts toward the top-left corner; that is a different curve, not a different point on the same one. The pulse selector and the coherent checkbox do the same thing by spending time instead of power. Watch the required signal-to-noise readout: it answers the mission question directly, so set the threshold for a false-alarm probability of one in a million and it reads twelve point six decibels — the reading's thirteen, to within half a decibel. Note that it is the EFFECTIVE signal-to-noise needed, so it does not move when you integrate; what moves is the per-pulse figure on the line underneath. Companion script runs on their own.

Block 1 · Lesson 8 · Type-along

Your Turn

Laptops open, MATLAB up. Ten lines: set one threshold from a false-alarm contract, then count what crosses it.

Block 1 · Lesson 8 · Quick Exercise

Quick Exercise / Type-along: Set the Threshold

Type this with me. Bare MATLAB — no toolboxes.

rng(495); M = 1e6;                    % a million noise-only cells
Pfa_want = 1e-3;                      % the false-alarm contract
T = sqrt(2)*erfcinv(2*Pfa_want);      % threshold that honors it
fprintf('T = %.3f  Pfa = %.2e\n', T, mean(randn(M,1) > T));
for snr_dB = [6 10 13]
    d = sqrt(2*10^(snr_dB/10));       % separation, in noise sigmas
    Pd_count = mean((d + randn(M,1)) > T);
    Pd_calc  = 0.5*erfc((T-d)/sqrt(2));
    fprintf('%2d dB: Pd = %.3f (calc %.3f)\n', snr_dB, Pd_count, Pd_calc);
end

$T$ never moved. $P_{fa}$ never moved. Only $P_d$ moved. As the B-21, which number can you attack?

Why contract for $P_{fa} = 10^{-3}$ and not the $10^{-6}$ from the reading?

Block 1 · Lesson 8 · Wrap-Up

Wrap-Up

Key ideas

Detection is a thresholded coin flip set by SNR
ROC curves trade $P_d$ for $P_{fa}$ at fixed SNR
Coherent integration: $+10\log N$ dB; non-coherent: $+5\log N$ dB
$S_{\min}$ in the radar range equation is a $P_d / P_{fa}$ contract, not a constant

Next lesson

L9: Project 1 work day — B-21 threat detection range analysis
HW1 due at the start of L10
Block 1 is now complete: L1–L8 are the toolkit the project uses