DSP 101
M10 · L02
Module 10: Correlation & Detection

Signal Detection in Noise

Is the signal there, or is it just noise? This is the detection problem — and it underlies radar, communications, medical diagnostics, and spectrum sensing.

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DSP 101
M10 · L02
The Problem

Two Hypotheses

Every detector chooses between two scenarios. H0: only noise is present. H1: signal plus noise. It computes a test statistic and compares to a threshold.

Observation Model
H0: y[n] = w[n]  |  H1: y[n] = s[n] + w[n]
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DSP 101
M10 · L02
Error Types

False Alarm vs. Missed Detection

  • False Alarm (PFA): noise triggered the detector
  • Missed Detection (1−PD): signal present but undetected
  • Raise threshold → fewer false alarms, more misses
  • Lower threshold → fewer misses, more false alarms
  • These cannot both be minimized simultaneously
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DSP 101
M10 · L02
Optimal Criterion

Neyman–Pearson

Fix PFA = α. Among all detectors that meet this constraint, pick the one with maximum PD. The answer is the likelihood ratio test.

Likelihood Ratio Test
\Lambda(\mathbf{y})=\frac{p(\mathbf{y}|H_1)}{p(\mathbf{y}|H_0)}\underset{H_0}{\overset{H_1}{\gtrless}}\gamma
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DSP 101
M10 · L02
In AWGN

Correlation is Optimal

For a known signal in white Gaussian noise, the NP optimal test statistic simplifies to the cross-correlation of the received signal with the known template.

Optimal Test Statistic
T(\mathbf{y})=\sum_{n=0}^{N-1}y[n]\,s[n]\underset{H_0}{\overset{H_1}{\gtrless}}\gamma
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DSP 101
Interactive
ROC Curve

Detection vs. False Alarm

Drag the SNR slider. The ROC curve bows toward the ideal top-left corner as SNR increases.

SNR10 dB
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DSP 101
M10 · L02
Reading the ROC

What the Curve Tells You

  • Diagonal = random guesser (no better than chance)
  • Top-left corner = perfect detector
  • AUC (area under curve) → 1 is perfect, 0.5 is random
  • Higher SNR → curve bows further toward top-left
  • Each point on curve = one threshold setting
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DSP 101
M10 · L02
Improving Detection

Integration Gain

Average N independent samples: noise variance shrinks by N, so effective SNR improves by 10 log10 N dB. GPS uses 1023 chips to gain ~30 dB — signals hidden below the noise floor become detectable.

10×
+10 dB gain
100×
+20 dB gain
1023×
~30 dB (GPS)
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DSP 101
Applications
Real-World Use

Where It Appears

  • Radar — CFAR processing, PFA ~ 10−6
  • Digital comms — bit decision = binary detection
  • Medical tests — sensitivity & specificity on ROC curve
  • Cognitive radio — detect primary user before transmitting
  • GPS — correlate PRN code below noise floor
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DSP 101
Quick check

Check what stuck

Four questions from this lesson. Answer to see why — the explanation shows whether you got it right or wrong. Nothing is scored or saved.

Question 1 of 0
Score 0/0

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DSP 101
Up Next
Coming Up

Matched Filtering

We proved correlation is optimal for detection — next we design the filter that implements it. The matched filter maximizes output SNR at the sampling instant and is the engine behind pulse compression in radar.

Next Lesson
Matched Filtering →
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