Unstable deconvolution in frequency domain (spurious high frequency component?)

Question

I have a black box-like deterministic model $h(t)$, which provided with a real-valued input $x(t)$ returns a real-valued response $y(t)$. The model should produce little to no noise. My goal is to determine an input signal $x^*(t)$ that produces a specific output $y^*(t)$.

I'm trying to use deconvolution in the frequency domain to do this, as explained in this answer. So basically I'm estimating $X^*(\omega)$ by multiplication and division of the signals' FFTs:

$$ X^*(\omega) = Y^*(\omega) \,\, \frac{X(\omega)}{Y(\omega)}, $$

and then performing an IFFT to recover the time-domain signal $x^*(t)$. I'm using MATLAB to do this.

The whole procedure is done iteratively, each time running the model with the newly determined input. But I'm facing a problem with the first few seconds of my input, which seems to be due to high-frequency components that should not be there.

To be clearer, this is a representation of what the output and the target look like after the first run, after four iterations and after seven iterations. Please take a look at the first few seconds of the signal.

What could be causing this? I apologise, should the question be trivial or old, but I'm pretty inexperienced in the field, so I searched but couldn't find a specific question. My knowledge of the lingo is also pretty limited, so that may be part of it.

Answer 1

Without seeing your actual data and code it's not easy to debug this.

A few things to consider.

1. Dividing spectra requires you to have good signal to noise ratio over the entire frequency range. At frequencies where $Y(\omega)$ is small $1/Y(\omega)$ is very large and hence any noise would be greatly amplified there.
2. Division in the frequency domain is circular de-convolution on the time domain. You probably need to manage this properly with zero-padding.
3. Not all systems are invertible. To be precise: only a minimum-phase system is invertible (and the inverse is minimum phase too). The inverse of a non-minimum phase systems tends to be non-causal.