Advanced DSP - FFT windows: matched filters

A filter which passes all the signal frequency components while suppressing any frequency components where there is only noise allows to pass the maximum amount of signal power:

Unfortunately, there is still significant noise power in each of the filter bands. So this frequency components also allows more noise to pass through. To make matters worse, some signal frequency components are rather small - so they contribute relatively little extra signal power, while still allowing through all the noise from that filter band.

To get the maximum signal to noise ratio, we want to allow through all the signal frequency components - but to take more notice of signal frequency components that are large and so contribute more to improving the overall signal to noise ratio. We can do this by weighting the contributions from each filter band proportionally to the signal power:

The diagram shows a filter whose frequency response is designed to exactly match the frequency spectrum of the signal. Such a filter gives the best possible improvement in signal to noise ratio, and is called a matched filter.

The diagram shows a noisy square wave. Passing this through a single band passfilter improves the signal to noise ratio: but passing it through a matched filter gives the maximum possible improvement in signal to noise ratio.

A matched filter is in fact the same as correlating a signal with a copy of itself.