PE1-3.4 Detection: Correlation & Matched Filter Receivers

What this is and why it exists

The subject now applies its machinery to the problem it exists for. A known signal was sent, noise was added, and something must decide what arrived.

The answer is one of the most useful results in engineering. To detect a known signal in white noise, filter with a time-reversed copy of that signal. No other linear filter does better.

The vocabulary

• Detection — deciding which of a set of known signals was transmitted.
• Simple binary detection — two candidate signals, symmetric about the origin.
• General binary detection — two arbitrary candidate signals.
• M-ary detection — more than two candidates.
• Decision region — the part of signal space in which a given candidate is chosen.
• Correlation receiver — multiplies the received signal by each candidate and integrates.
• Matched filter — a filter whose impulse response is the time-reversed candidate signal.
• Sampling instant — the moment at which the filter output is read.

The mental model

Start from the previous topic's result. The signals differ in mean and the noise is the same under each hypothesis, so the detector is linear. What that means physically is that the receiver forms a weighted sum of the received samples, with the weights given by the candidate signal.

Forming a weighted sum of a received signal using another signal as the weights is exactly correlation. So the optimum receiver multiplies the incoming signal by each candidate, integrates over the symbol period, and picks the largest. That is the correlation receiver, and it is the direct implementation of the mathematics.

The simple binary case reproduces the results the digital modulation topic reached from signal space, which is a good consistency check between two subjects. The general binary case allows arbitrary signals rather than a symmetric pair, and adds a term accounting for their different energies. M-ary detection extends it to any number of candidates, and the decision regions become a partition of signal space, one region per candidate.

Now the second implementation. Consider a filter whose impulse response is the candidate signal reversed in time. Convolution flips one of its arguments. So convolving the received signal with a time-reversed candidate produces, at the right instant, exactly the correlation integral. Sample the output at the end of the symbol and you have the same number the correlation receiver computed.

That is the equivalence, and it is worth working through once. They look like different machines. One multiplies and integrates against a stored copy. The other is a filter with a sampler. They compute the same thing, so you can reason with whichever is more convenient and build whichever is cheaper.

The optimality is the part that makes this the central result. Among all linear filters, the matched filter maximises the ratio of signal to noise at the sampling instant. The reason is worth stating in words. The filter weights each part of the incoming signal by how much signal is there. Where the signal is strong it listens hard; where the signal is weak it listens less, so it takes in less noise from those parts. No other weighting does better, and any other weighting either misses signal or admits extra noise.

One result then underlies a great deal. Digital receivers use it for every symbol. Radar uses it to pull an echo out of noise. Every synchronisation scheme uses it to find a known pattern in a stream. Learning it once pays repeatedly.

What you should now be able to explain or do

Derive the correlation receiver from the linear detector result. Handle general binary detection with unequal signal energies, and describe M-ary decision regions. Show that a matched filter sampled at the right instant computes the correlation integral. State what the matched filter maximises, and explain in words why no other linear filter beats it. Name three places the result is used.

Check yourself