After a lecture on harmonic analysis and time/frequency methods, I reconsidered the Gaussian kernel, defined in continuous time.
It is unimodal and symmetric, and its continuous Fourier transform is another Gaussian, thus unimodal and symmetric (from Fourier transform of a Gaussian is not a Gaussian, but that's wrong!):
As a filter or a window, the Gaussian ensure a monotonic weight decay around its center. The interpretation in the frequency domain is similar: frequencies are monotonically attenuated away from a center frequency.
Such a property allows a straightforward interpretation in both the time and the frequency domain.
- For continuous kernels, are there generic characterizations (necessary or sufficient conditions) under which unimodal and symmetric windows also have a unimodal and symmetric amplitude spectrum?