Advanced DSP - FFT windows: extracting signals from noise

A common use for digital filtering is to reduce unwanted noise.

Filtering is a frequency selective operation: we seek to reduce noise by suppressing noise frequency components but passing signal frequency components:

The diagram shows a simple case where the desired signal has a narrow frequency spectrum but the noise is spread out: here a narrow band pass filter is used to pass the narrow band signal but suppress most of the broad band noise.

The example works well for a narrow band signal: but if the signal's frequency spectrum were more complicated, things would not be quite so simple. In fact it is easy to imagine that if we know the frequency spectrum of the desired signal, then we might be able to design a more intelligent filter that would pass the maximum signal power while suppressing as much of the noise power as possible.

This leads to the idea of a matched filter - a filter which increases the signal to noise ratio by as much as possible.