Matched Filtering
The optimal linear filter for detecting a known signal in noise. It maximizes output SNR at the sampling instant — and turns out to be the correlator receiver in disguise.
Maximize Output SNR
Given y[n] = s[n] + w[n], find the linear filter h[n] whose output, sampled at n = N−1, gives the largest possible SNR. The answer is derived using the Cauchy–Schwarz inequality.
Energy Bound
The maximum achievable SNR equals the signal energy divided by noise variance. No linear filter can beat this bound — only more energy or less noise can improve detection.
Filter = Correlator
Sampling the matched filter output at n = N−1 is exactly the cross-correlation of the received signal with the template. The two are mathematically identical.
Boost Where the Signal Is
The matched filter magnitude response equals the signal’s magnitude spectrum: |H| = |S|. It emphasizes frequency bands where the signal is strong, letting white noise pass equally everywhere.
- High signal energy at frequency ω → large |H(ejω)|
- Low signal energy at frequency ω → small |H(ejω)|
- Result: maximum coherent integration of signal energy
Long Pulse, Sharp Spike
Transmit a long LFM chirp for energy; match-filter on receive to compress it. Processing gain = 10 log(T·B) dB. Resolution stays 1/B regardless of pulse length.
Minimum BER Receiver
The correlator/matched-filter receiver achieves the minimum bit error rate for a given Eb/N0 in AWGN. Modern standards use RRC pulse shaping — one RRC at TX, matched RRC at RX.
- BPSK, QPSK, QAM — all use matched-filter sampling
- Root Raised Cosine — splits the Nyquist filter into TX/RX pair
- Zero ISI at sampling instants when cascade forms raised cosine
Invisible to Spectrum Analyzers
GPS signals arrive at −130 dBm — 20 dB below thermal noise. The receiver correlates with a 1023-chip PRN code for 1 ms, gaining ~30 dB of processing gain. The signal appears out of the noise.
The Matched Filter
- Impulse response: h[n] = s*[N−1−n]
- Max SNR = Es/σ² — only energy matters
- Sampling output = cross-correlation with template
- Frequency response: |H| = |S|
- Used in: radar, comms, GPS, sonar
Radar, Sonar, Comms
We now have all the tools — correlation, detection theory, and matched filtering. Next we see how they combine in real systems: radar range measurement, sonar imaging, GPS acquisition, and digital demodulation.