Applications — Radar, Sonar & Communications

Bringing It All Together

Throughout Module 10 we have built a complete theoretical toolkit: autocorrelation and cross-correlation for measuring signal similarity, the Neyman–Pearson detection framework for making binary decisions in noise, and the matched filter for extracting the maximum signal-to-noise ratio from a noisy observation. Now we deploy these tools in three of the most important application domains in electrical engineering: radar, sonar, and digital communications.

Each domain poses the same fundamental problem — detecting and characterizing a signal of known shape embedded in noise — but with different physical channels, different time and frequency scales, and different figures of merit. Seeing how the same mathematical machinery operates across domains deepens intuition and shows why the theory developed in this module is not academic abstraction but engineering bedrock.

Radar: Range, Velocity, and Resolution

Radar (Radio Detection And Ranging) transmits a radio-frequency pulse and listens for the echo reflected from a target. The round-trip propagation delay τ determines the range R = cτ/2, where c is the speed of light. Detecting the echo is exactly a binary hypothesis test: H₀ (no target, noise only) versus H₁ (target present, signal + noise). The NP-optimal test statistic is the cross-correlation of the received waveform with the transmitted waveform — i.e., the matched filter output.

The matched filter does two things simultaneously in radar. First, it maximizes the peak SNR of the echo, enabling detection at the greatest possible range. Second, when the transmitted waveform is a chirp (LFM), the matched filter compresses the echo into a narrow spike whose width is approximately 1/B, where B is the swept bandwidth. This pulse compression decouples range resolution (controlled by B) from transmitted energy (controlled by pulse duration T), allowing modern radar to achieve both long detection range and fine range resolution.

A moving target introduces a Doppler frequency shift f_d = 2v/λ, where v is the radial velocity and λ is the wavelength. This shifts the echo spectrum away from the transmitted spectrum, reducing the matched filter output. To measure velocity, modern radars use a bank of matched filters each tuned to a different Doppler shift, or equivalently, a 2-D correlation over both delay and Doppler — the ambiguity function, which is the cross-correlation of the transmitted signal with a Doppler-shifted copy of itself.

Sonar: Underwater Acoustics

Sonar (Sound Navigation And Ranging) is the acoustic analogue of radar. Instead of electromagnetic waves it uses sound waves propagating through water at approximately 1500 m/s — about 200,000 times slower than light, but with far lower attenuation at the frequencies used. The mathematics is identical: transmit a known waveform, apply a matched filter to the received echo, and compare the output to a threshold.

Active sonar transmits a ping and listens for the return. Passive sonar only listens, using cross-correlation between sensors to determine the direction and range of a sound source. In passive sonar, the “known signal” is replaced by the cross-correlation between two spatially separated hydrophones: a peak in the cross-correlation at lag τ indicates that the same wavefront arrived at the two sensors with a time difference of τ, giving the angle of arrival.

The key challenge in sonar is multipath propagation. Sound bounces off the ocean surface, the seafloor, and thermal layers, creating multiple delayed and attenuated copies of the transmitted signal at the receiver. The matched filter output shows multiple peaks — one for each propagation path — and the receiver must identify the direct-path return to avoid range errors. This is closely related to the channel equalization problem in digital communications.

Digital Communications: Demodulation and Channel Estimation

In a digital communications system, the transmitter sends a stream of symbols, each represented by a pulse shape g[n]. The channel adds noise, and possibly multipath reflections. The receiver must decide which symbol was sent at each time instant, minimizing the probability of a wrong decision. This is again a hypothesis test: for binary signaling, H₀ (bit 0, pulse −g[n]) versus H₁ (bit 1, pulse +g[n]).

The matched filter for this problem has impulse response h[n] = g*[N−1−n], the time-reversed conjugate of the pulse shape. Sampling the output at the end of each symbol period gives the sufficient statistic for the optimal decision. For binary antipodal signaling in AWGN, the resulting bit error probability is:

When the channel introduces multipath (inter-symbol interference, ISI), the matched filter alone is insufficient. The received signal is the convolution of the transmitted sequence with the channel impulse response, and overlapping pulses cause each symbol to interfere with its neighbors. Modern receivers combine matched filtering with an equalizer (e.g., a minimum mean square error (MMSE) equalizer or a Viterbi decoder on a trellis) to remove ISI before making decisions.

Channel estimation uses cross-correlation directly. The transmitter sends a known pilot sequence p[n] whose autocorrelation is impulsive (like a PN sequence). The receiver correlates the received signal with the known pilot to estimate the channel impulse response h[n]: the cross-correlation peak at delay k gives the channel coefficient at tap k. This is exactly the matched-filter approach applied to the identification problem rather than the detection problem.

GPS Acquisition: Correlation in Two Dimensions

GPS acquisition is a striking case where correlation must be performed over two unknowns simultaneously: the code phase (which chip offset to use) and the carrier Doppler offset (the satellite is moving relative to the receiver). The receiver maintains a 2-D search grid and computes the correlation of the incoming signal with every combination of code phase and Doppler hypothesis.

Modern GPS receivers use the FFT to accelerate this search. For a fixed Doppler hypothesis, the correlation over all code phases is a circular cross-correlation and can be computed in O(N log N) operations using the FFT, instead of O(N²) for a direct search. Once the correct cell (code phase, Doppler) is identified by a peak exceeding the detection threshold, the receiver enters tracking mode, using phase-locked loops (PLLs) and delay-locked loops (DLLs) to maintain synchronization.

Medical Ultrasound: Correlation Imaging

Medical ultrasound is a close cousin of sonar. A transducer emits a short acoustic pulse (typically 2–15 MHz) and records the echoes returning from tissue boundaries. The time-of-flight gives the depth of each reflector, and the amplitude of the matched-filter output gives the reflection coefficient. Scanning the beam electronically produces a 2-D cross-sectional image (B-mode scan) in real time.

Doppler ultrasound measures blood flow velocity by tracking the Doppler shift of echoes from moving red blood cells. This is exactly the radar Doppler problem scaled down to acoustic frequencies and centimeter ranges. The same ambiguity function trade-off applies: a long pulse resolves velocity well but provides poor axial (depth) resolution; a short pulse gives sharp depth resolution but poor velocity resolution.

The Correlation Engine: A Common Architecture

Stepping back, every application in this lesson shares an identical processing chain:

1. Transmit (or observe) a known waveform s[n].
2. Receive r[n] = s[n − τ] * h[n] + w[n] (delayed, convolved with channel, plus noise).
3. Correlate: compute R_rs[k] = Σ r[n] s*[n−k].
4. Threshold: declare detection if max|R_rs[k]| > γ.
5. Estimate: read off delay, Doppler, or symbol from the peak location.

This architecture is the matched filter / correlator receiver, and it is optimal in the Neyman–Pearson and minimum-BER senses. The only degrees of freedom are the choice of waveform s[n] — which shapes the ambiguity function and the autocorrelation sidelobes — and the threshold γ, which trades off P_fa against P_d.