Background

My initial reason for getting into this debacle was the search for a universal window size for a Fourier transform of an acoustic time series. It always disturbed that when you perform a DFT on a time series, sooner or later the correlation waves will not be pinned to zero at each end.
For example, if you take a 12 sample time series, you get one wavelength with 12 samples, 2 wavelengths with 6 samples, 3 wavelengths with 4 samples each, 4 wavelengths with 3 samples each, but 5 wavelengths will have 2.4 samples each... Clearly it is impossible to have a fraction of a sample, and this is the problem, 12 does not divide equally by 5.
Furthermore, using Filters, or convolutions, also left me slightly wary because normal band, high or low pass convolution filters distort the signal. Ok, it may not be much of a distortion, but it’s a departure from the original signal none the less. During my first trials with the Fourier Transform I realised I could use the DFT and reverse DFT as a filter in the frequency domain, a rather expensive filter in terms of processing power, but a filter none the less that promised to keep true to the original signal.