ICA - Statistical Independence & Eigenvalues of Covariance Matrix
Signals 3 and 5 appear to be quite correlated -- they share their first harmonic. If I were given two mixtures of those, I wouldn't be able to separate those, I'd be tempted to put the common harmonic as one signal and the higher harmonics as a second signal. And I would be wrong! This might explain the missing eigenvalue.

Signals 1 and 2 do not look independent too.

A quick and dirty "sanity-check" for the independence of two series is to do a (x, y) plot of one signal against the other:

plot (sig3, sig5)


and then to do the same (x, y) plot with one signal shuffled:

indices = randperm(length(sig3))
plot(sig3(indices), sig5)


If the two plots have a dissimilar look, your signals are not independent. More generally, if the (x, y) plot of the data shows "features", disymmetries etc, it's a bad omen.

Proper tests for independence (and those are the objective functions used in the ICA optimization loop) include, for example, mutual information.

ICA is recovering the most independent signals a linear mixing of which yields your input data. It will work as a signal separation methods and recover the original signals only if those were maximally independent according to the optimization criterion used in your ICA implementation.