Matched Filter and Detection Theory Basics For students.

When a known signal arrives buried in noise, the best possible receiver is not a fancier antenna — it is a matched filter: a filter whose shape is a mirror image of the expected signal. Out of every conceivable linear filter, the matched filter squeezes the maximum signal-to-noise ratio out of white noise at the moment of decision. It is the theoretical gold standard every real receiver is measured against.

Detection theory sits beside it, answering the next question: given this filtered output, how do you decide whether a 1 or a 0 was sent, and how often will you be wrong? Together they explain why digital receivers correlate against known waveforms and why error probability falls exponentially as signal energy rises.

Key Concepts

• Matched filter: impulse response h(t) = s(T − t) — the time-reversed expected signal; its output at time T equals the signal's energy.
• Signal energy (E): the total energy in one symbol; detection performance depends on E, not peak power.
• Noise spectral density (N0): white-noise power per Hz, in watts/Hz.
• Maximum SNR: 2E/N0 — no linear filter can do better in white noise.
• Threshold detection: compare the filter output to a threshold to decide the bit.
• Bit error rate (BER): for BPSK, BER = Q(√(2Eb/N0)) — errors drop steeply as Eb/N0 grows.

Formulas

• Matched impulse response: h(t) = s(T − t), 0 ≤ t ≤ T, where s(t) is the known pulse shape.
• Peak output SNR: SNRmax = 2E/N0 (dimensionless); in dB: 10 log10(2E/N0).
• BPSK error probability: Pb = Q(√(2Eb/N0)), where Q is the Gaussian tail function.
• Energy of a rectangular pulse: E = A²T (A = amplitude in volts, T = duration in seconds, 1 Ω reference).
• Assumptions: additive white Gaussian noise, perfect knowledge of the pulse shape and timing at the receiver.

Worked Example

A BPSK receiver expects rectangular pulses of amplitude A = 1 V and duration T = 1 µs, in noise with N0 = 2×10⁻⁹ W/Hz. Step 1: bit energy E = A²T = 1 × 10⁻⁶ = 10⁻⁶ J. Step 2: maximum SNR = 2E/N0 = 2×10⁻⁶/(2×10⁻⁹) = 1000 → 30 dB — the matched filter achieves this; any other filter does worse. Step 3: BER = Q(√(2×1000)) = Q(√2000) = Q(44.7) — astronomically small (far below 10⁻⁹); this link is essentially error-free. Step 4: shrink the amplitude to 0.1 V → E = 10⁻⁸ J, SNR = 10 → 10 dB, BER = Q(√20) = Q(4.47) ≈ 4×10⁻⁶ — about 4 errors per million bits. The math shows exactly why weak signals need either more energy per bit or error-correction coding.

Common Misconceptions

• "A sharper filter always helps." Wrong: the matched filter is optimal only for white noise — against coloured noise or interference, a different (whitening) filter wins.
• "Matched filtering removes the noise." Wrong: it maximises the SNR at the sampling instant; plenty of noise remains — the threshold decision then does its probabilistic best.
• "Higher amplitude is the only way to cut errors." Not quite: BER depends on energy Eb = A²T — a longer, weaker pulse can match a short, strong one.

Real-World Applications

• Radar receivers correlate echoes against the transmitted pulse — a matched filter that finds faint targets in noise.
• GPS receivers despread satellite codes with correlators, digging signals out from below the noise floor.
• Every digital modem, from dial-up to 5G, ends in a matched filter (or its digital equivalent) before the bit slicer.
• Ultrasound and sonar imaging use matched filtering to sharpen weak reflections.

The matched filter is the provably best way to fish a known signal out of white noise — mirror the pulse, sample at the peak, and no linear receiver can beat your SNR of 2E/N0. Detection theory then tells you the price of the remaining uncertainty. Every modem you own is, at heart, a correlator plus a threshold.