PE1-3.4 Detection: Correlation & Matched Filter Receivers
You can derive simple and general binary detection and M-ary detection, and show that the correlation receiver and the matched filter receiver are equivalent implementations of the optimum detector.
The matched filter is the central result of the whole subject: to detect a known signal in white noise, filter with the time-reversed signal, and no other linear filter gives better signal-to-noise ratio at the sampling instant. Its equivalence with the correlation receiver is worth proving to yourself, because they look like different machines and are the same one. This one result underlies digital receivers, radar and every synchronisation scheme you will meet.
Work through these
Introduction to detection
Moving from hypothesis testing to detecting a known signal in noise. It is the same mathematics applied to the problem the subject exists for.
Simple binary detection
The two-signal case, worked from first principles. It reproduces the digital modulation results from the communication subject, which is a good consistency check.
General binary detection
The same with arbitrary signals rather than a symmetric pair. The general form is what the receiver structures below implement.
M-ary detection
More than two possible signals, which is what any real modulation scheme uses. The decision regions become a partition of signal space.
The correlation receiver
One way to build the detector: multiply by each candidate and integrate. It is the direct implementation of the mathematics above.
The matched filter receiver
The other way: filter with a time-reversed copy of the signal and sample. It is what receivers actually contain, and it maximises signal to noise ratio at the sampling instant.
Equivalence of correlation and matched filter receivers
The two implementations above give identical results, which is worth proving once. Knowing they are the same means you can reason with whichever is more convenient.
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