Why is the matched filter's impulse response h(t) = s(T−t) — a time-reversed copy of the target signal — rather than simply h(t) = s(t)?
ATime reversal corrects for phase distortion introduced by the transmission channel
BBecause convolution involves time reversal, the output at time T equals the cross-correlation of the received signal with s(t), which maximizes the signal component at the decision point
CThe time reversal ensures causality — a non-reversed filter would require predicting future inputs
DUsing h(t) = s(t) would cancel the target signal rather than detect it
Convolution of the received signal r(t) with h(t) = s(T−t), evaluated at time T, gives y(T) = ∫r(τ)s(τ)dτ — the inner product (cross-correlation) of the received signal with the template. The time reversal in h is precisely what converts convolution into correlation at the output time T. When r(t) contains s(t), this inner product is large (signal aligns with its own template). The Cauchy-Schwarz inequality proves this inner product is maximized relative to noise when h is the time-reversed signal — which is why the matched filter is the optimal detector.
Question 2 Multiple Choice
A radar system uses a 100 μs frequency-swept chirp pulse with 1 MHz bandwidth and matched filtering at the receiver. What does the matched filter achieve that a 100 μs unmodulated pulse cannot?
AThe chirp allows multiple targets to be detected simultaneously by correlating with different frequency segments
BTime-bandwidth product compression — the matched filter collapses the long chirp into a sharp correlation peak of width ~1/B, achieving fine range resolution while transmitting the energy of the long pulse
CThe matched filter corrects for Doppler shifts introduced by moving targets before detection
DFrequency sweeping prevents jamming, and the matched filter cancels the sweep to recover the original pulse shape
This is pulse compression. A chirp of duration T and bandwidth B has a time-bandwidth product TB = 100 μs × 1 MHz = 100. The matched filter compresses this to a correlation peak of width ~1/B, giving range resolution equivalent to a 1 μs pulse — while transmitting the energy of a 100 μs pulse. This resolves the fundamental radar tradeoff: long pulses give high energy (good SNR) but poor range resolution; short pulses give fine resolution but little energy. Matched filtering with chirp achieves both simultaneously.
Question 3 True / False
The output of a matched filter at the decision time T equals the cross-correlation between the received signal and the target waveform.
TTrue
FFalse
Answer: True
Convolution of r(t) with h(t) = s(T−t), evaluated at time T, gives y(T) = ∫r(τ)s(τ)dτ — the inner product of r and s, which is the cross-correlation at zero lag. This equivalence between LTI convolution with the time-reversed template and correlation is the key mathematical insight. When r contains s, the correlation is large; when r is noise with no structure matching s, the correlation fluctuates near zero. The matched filter is therefore a correlation detector implemented as an LTI system.
Question 4 True / False
The matched filter achieves maximum SNR for detecting a known signal regardless of the noise characteristics, because its optimality depends primarily on the signal structure.
TTrue
FFalse
Answer: False
The matched filter achieves maximum SNR specifically for WHITE GAUSSIAN noise — noise with equal power spectral density at all frequencies. The Cauchy-Schwarz proof of optimality assumes this white noise structure. For colored noise (non-uniform power spectrum), the optimal filter is a modified whitened matched filter that first decorrelates the noise before applying the template. The matched filter remains highly effective in practice for many noise models, but the theoretical guarantee of optimality is specific to white Gaussian noise.
Question 5 Short Answer
Explain in your own words why the matched filter maximizes signal-to-noise ratio rather than simply maximizing signal amplitude.
Think about your answer, then reveal below.
Model answer: The matched filter maximizes the ratio of signal energy to noise power at the decision point, not just signal amplitude. At time T, the signal component equals the cross-correlation of the template with itself — a fixed quantity tied to the signal's energy. The noise component depends on the filter's bandwidth: a filter responding to more frequencies passes more noise power. By choosing h(t) = s(T−t), the filter weights its frequency response toward the frequencies where the signal has energy, avoiding amplification of frequencies containing only noise. The Cauchy-Schwarz inequality proves no other linear filter achieves a higher SNR — scaling the filter amplifies signal and noise equally, leaving the ratio unchanged.
This is why matched filtering is described as 'signal-specific projection.' It projects the received waveform onto the template, extracting the maximum possible signal component relative to noise by exploiting the known structure of the target waveform. Any other linear filter either passes frequencies where the signal is weak (reducing SNR) or rejects frequencies where the signal is strong (also reducing SNR).