DSP 101
M10 · L04
Module 10: Correlation & Detection

Applications — Radar, Sonar & Comms

The same matched-filter algorithm powers five different technologies: radar, sonar, GPS, digital communications, and medical ultrasound. One theory, many domains.

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DSP 101
M10 · L04
Common Architecture

One Algorithm, Five Domains

Every application follows the same chain: transmit known waveform → receive noisy echo → cross-correlate → threshold → estimate delay.

  • Radar — radio echoes, range & velocity
  • Sonar — acoustic echoes, underwater depth
  • GPS — PRN code, position fix
  • Comms — symbol pulses, bit decisions
  • Ultrasound — tissue echoes, medical imaging
02 / 12
DSP 101
Radar
Radar: Range Measurement

Time Delay → Distance

The matched filter peak at delay τ gives range R = cτ/2. After LFM pulse compression, range resolution is c/(2B), independent of pulse duration.

cτ/2
Range
c/2B
Resolution
2v/λ
Doppler Shift
03 / 12
DSP 101
Radar
Range & Velocity Together

The Ambiguity Function

A moving target shifts the echo spectrum. To measure both delay and Doppler, compute the 2-D cross-correlation over all (τ, fd) pairs — the ambiguity function.

Ambiguity Function
\chi(\tau,f_d)=\int s(t)\,s^*(t-\tau)\,e^{j2\pi f_d t}\,dt
04 / 12
DSP 101
Sonar
Underwater Acoustics

Sound at 1500 m/s

Same algorithm as radar, but with acoustic waves in water. Passive sonar cross-correlates two hydrophones: the TDOA peak gives bearing angle without transmitting.

TDOA Bearing
τ = d sinθ / c  →  θ = arcsin(cτ/d)
05 / 12
DSP 101
Communications
Digital Communications

Minimum BER Receiver

The correlator/matched-filter receiver achieves the minimum bit error rate for a given Eb/N0. No other linear receiver can do better in AWGN.

Bit Error Probability
P_b=Q\!\left(\sqrt{\dfrac{2E_b}{N_0}}\right)
06 / 12
DSP 101
Channel Est.
Pilot-Based Estimation

Correlation Identifies the Channel

Send a pilot with impulsive autocorrelation. Correlate the received signal with the pilot — the cross-correlation peak at delay k equals the channel coefficient h[k].

  • LTE/5G NR use pilot tones in OFDM resource grids
  • WiFi uses preamble sequences for channel estimation
  • GPS uses the PRN code as both pilot and data carrier
07 / 12
DSP 101
GPS
2-D Correlation Search

Code Phase × Doppler

GPS acquisition searches a grid of code delays and Doppler frequencies. FFT-based methods cut complexity from O(N²) to O(N log N) per Doppler bin.

1023
Code phases
~100×
FFT speedup
<1 s
Cold acquisition
08 / 12
DSP 101
Medical
Medical Ultrasound

Sonar Inside the Body

Acoustic pulses (2–15 MHz) echo off tissue boundaries. Matched-filter depth profiling builds B-mode images. Doppler ultrasound measures blood flow by tracking frequency shift of echoes from red blood cells.

Same Trade-off
Long pulse → good velocity, poor depth resolution. Short pulse → sharp depth, poor velocity.
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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

10 / 12
DSP 101
Summary
Module 10 Complete

The Correlation Engine

  • Radar: range = cτ/2, resolution = c/(2B) after pulse compression
  • Sonar TDOA → bearing; multipath creates multiple peaks
  • Comms: matched filter → min BER = Q(√(2Eb/N0))
  • Channel est.: pilot cross-correlation recovers h[k]
  • GPS: 2-D (delay × Doppler) FFT-accelerated search
11 / 12
DSP 101
Module 10
Module 10 Complete

Correlation & Detection

You’ve now seen how a single mathematical idea — the cross-correlation — underpins every system that must find a known signal in noise. From GPS satellites 20,000 km overhead to the ultrasound probe in a hospital, matched filtering is the common thread.

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