Independent component analysis for one observation of a Signal So I am a complete novice to ICA, so excuse my question if it is bad one, but I have the signal: 

$$\sin(2\pi x) + \sin(4\pi x) + \textrm{Additive White Gaussian Noise}$$

I want to try to separate the two signals and apply a Fourier Transform on each one individually, to see if I can get a better spectral estimate than if they were together. I was thinking of using scikit's FastICA Algorithm. So my plan was to take the signal of length $3200$ samples, split it into $16$ windows each with length $200$ samples, apply a Hanning Function on each window, and do an ICA on 4 subsequent windows at a time. 


Would this theoretically work, or does the ICA have to act on two different observations of a signal? 
Is the ICA dependent on each observation of the signal having a different mixing matrix, or can it be the same?
You have single channel / observation along x. Splitting it along x would not make it multiple channels. I think you should not split it. You should rather approach it as underdetermined ICA problem, i.e., the number of sources are more than the number of observations. 
For the point, theoretically, all the sources should be transformed via a system. So the observations, generated using different mixing matrices cannot be utilized jointly in a single ICA process.