Removing Gibbs Phenomenon

Question

I am working with a sample of 20 points given from an unknown 1-periodic function that are plotted like this: Original sample

I am using Inverse Fast Fourier Transform (ifft) to recover the signal resampled in 1000 points at [0,1) that is plotted like this: Resampled

It is showing a Gibbs Phenomenon at the end of the signal. What can be causing this fact? As far as I know Gibbs Phenomenon occurs near a jump discontinuity...

Any idea about why is this happening and how could I solve it?

Answer 1

You could use the Fourier cosine series. This amounts to extending the function to be even on $[-1,1]$ and then making it periodic with period $2$. That function will be continuous.

Answer 2

If you makes the function periodic by pasting copies of it side by side, contiguously, then you will have a jump continuity at each integer.

There's nothing to do about it unless you change from Fourier transform to something else or modify the function to have a smooth step.