Fourier and signal processing

Suppose we have a signal consisting of some gaussian peaks and noise, like a chromatogram for instance. A plot of such a signal is given in Fig. 15(a), and its power spectrum in Fig. 15(b). 

We just applied a low-pass filter (LP). The opposite would be a high-pass filter (HP). Under other circumstances, it may be useful to select a band of frequencies somewhere in the frequency range, not necessarily to the high or low extreme. In that case we would be using a band-pass filter (BP). Which filter is appropriate depends on the type of signal and the type of noise. 

A filter can be implemented in the time domain as well. It would be the convolution of the signal with the back transform of the weight function we apply to the frequencies. Vice versa, a filter designed in the time domain can be implemented in the Fourier domain as a multiplication with the Fourier transform of the impulse response of the filter. 

The hard cut-off we applied in Fig. 15 amounts to a weight function with the shape of a block ones up to the cut-olf frequency, and zeros above. The back transform of a block is a sine function, i.e. the function sin(x)/x. The wider the block, the narrower the sine. The equivalent operation in the time domain would thus be a convolution of the signal with this sine, as illustrated by Fig. 16. 

The consequences of a filter shape can be visualised most easily by picturing what happens if there is a spike in the signal. The output of the filter than contains a copy of the filter shape on the position of the spike. In other words, we get to see the impulse response of the filter. For a filter shape that is wide and oscillating, phenomena that are purely local in the time domain get spread out and deformed. If we want to have a more reasonable filter shape in the time domain, we have to use a smoother cut-off in the frequency domain. It is the sharpness of the cut that introduces the oscillations, as it disturbs the delicate balance of frequencies required to localise something in 

---