I have come across a "weird" behavior of the Discrete Fourier Transform (which I will refer to from now as DFT).
suppose $$x=(1,-1,1,-1)$$ The real part of the DFT of $x$ is $$Real(DFT(x))=(0,0,4,0)$$ From a signal processing perspective, since $x$ oscillates between the maximum and minimum of it's entries in every sequential coordinate, it is reasonable that $DFT(x)$ gets it's maximum at the coordinate which refers to the highest non overlapping frequency. Also $x$ is some what of a "pure" signal (essentially a sinusoidal signal) and does not consist of any other frequency.
Now lets take a look at a different signal $$y=(1,-1,1,-1,1)$$
The real part of the DFT of $y$ is $$Real(DFT(y))=(1,1,1,1,1)$$
Again from a signal processing perspective $y$ resembles $x$ in the time domain, but not in the frequency domain.
This behavior feels unstable: small change in the signal leads to big change in the frequencies.
my questions are:
1. What are other weird behaviors of DFT?
2. Are there any stable alternatives to DFT? (in a sense of capturing the frequencies/repetitiveness of a signal)