Not sure, but those 'peaks' look just like the step response for an abrupt cutoff filter. Or, like the time results using a truncated Fourier series to represent a square wave.
Have you tried convolving the sinx/x function with your data stream? [easy to do in octave] That makes for a remarkably well-behaved waveform. Even satisfies most of the 'violations' done using sampling.
isn't the tanh function like (e^+x + e^-x), or such? Ron Bracewell used that a lot in proving his Fourier Series' He 'bracketed' all the functions with that kind of term and by taking the limit of something to zero proved the existance of the results. Reason? Every derivative existed after adding that function, yet adding the function could be taken to the limit of where it had NO effect on the original, within Engineering limits.
Maybe it was atanh, but it's not the function of a real number, it's the derivative that causes problems. Certain people here swear by this for continuous limiter functions in SPICE, but I've found they can go batty rather often.
Incidentally, The Internet doesn't know that degenerate circuits aren't the only condition that can create singular matrices.