1. Minimize sum of squares of differences, not sum of absolute values of differences.

2. It is important, that constant AMP (amplitude) is the same for all bits.

3. There is no test with negative amplitude.

Hint should be:
notion of appropriate statistical processing:
extracting a and b in a+b*k from noise
Or
MNK
Or
sequental analysis
Or
maching learning
Or
optimal decision theory
What artical in Wikpedia should read?

Start with this one:
http://en.wikipedia.org/wiki/Frequency-shift_keying

Would you be so kind to give the slightly more detailed description of your solution?
I tried to used DFT but it seems to be too sensitive to noise.

I heard of Arctan-Differentiated Demodulator, but I can't find the literature on this theme. Does anybody know, is it applicable here?

Use OLS (ordinary least squares) estimator for this problem

Thanks a lot. I've never used this method before. But it's worth keeping in mind :)