# 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.

• So I will give you more insight into the project; I am using a Doppler radar to monitor the respiration and heartbeat signals of humans. So since this is repetitive, on-going data, could another "observation" be taking 2500 samples, waiting 10 seconds and then taking another 2500 samples? – Sachin Konan Jun 27 '16 at 4:23

Basic ICA acts on different mixtures of independent non-gaussian signals.

A requisite for basic ICA is that the mixing constants between recorded signals (x) to be different for every source signal(s). For example:

$$x_1 = a_{11} s_1 + a_{12} s_2$$ $$x_2 = a_{21} s_1 + a_{22} s_2$$

$$a_{11}\neq a_{21} \space \& \space a_{12}\neq a_{22}$$

As you can see, by dividing your signal into windows and trying to apply ICA, you run into two problems. One, your mixing constants must probably are the same, in which case a basic ICA cannot help. And second, even if the mixing constants change over time, your source signals will not be the same, they will have some time offset.