Fourier transform filtering

A harmonic filter is a device that selectively modifies the amplitude and phase of the spectral components of the input signal. If you have a Fourier transform algorithm, it's only natural to try to use it to perform this operation.

You might think that all you have to do is perform the Fourier transform of the input signal, multiply the resulting spectrum by the filter transfer function and perform an inverse transform. Unfortunately, the Fourier transform is a mathematical operation that operates on the signal as a whole, involving integration from $-\infty$ to $+\infty$ . However, the discrete Fourier transform is defined on the basis...