# Solving nonlinear Fourier relation

I'm trying to solve the following nonlinear cross-correlation problem for the time-domain signal $$f(t)$$:

$$S(\omega) = \overline{\mathcal{F}\left[f(t)\right]} \mathcal{F}\left[f^n(t)\right]$$

with $$n>1$$, the bar denoting the complex conjugate, and $$S(\omega)$$ is a given (complex-valued) spectrum. $$f(t)$$ is the unknown in this implicit equation and I want to solve for it.

I know the problem is undetermined for $$n=1$$ because the phase cancels out. However, for $$n>1$$, the phase is nontrivial and the solution for $$f(t)$$ is somewhat constrained. With a good enough initial guess I can use an iterative algorithm, which converges to the correct solution. However, I'm wondering if there's a smarter way to "decorrelate" $$f(t)$$ by exploiting the relationship between the two parts of the cross-correlation. Certainly, there is no unique solution (e.g. it's invariant against delay or phase shift), but maybe there's an analytical way to retrieve one of the solutions or a partial solution.

Thanks for any ideas!

• Why do you say it is undetermined when n =1? Isn't the result then real and is the power spectral density of the signal? As far as a "Cross-Correlation", this would be the (circular) cross-correlation if you subsequently computed the inverse-FFT of $S(\omega)$. I don't yet see why the solution isn't unique either if you simply do the math you get one result for any f(t). (right?) – Dan Boschen Mar 18 '20 at 16:11
• Yes, for $n=1$ the result is the PSD with no imaginary part or zero phase. That's why a "decorrelation" is impossible in that case: If you simply square-root and inverse-FFT $S(\omega)$, you get a "transform-limited" version of $f(t)$, i.e. where the spectral phase is zero. Any signal with a PSD of $S(\omega)$ but with arbitrary spectral phase would fulfill the equation for $n=1$ but have distinctly different shapes in the time domain (think about chirped signals for example). I'm looking to solve the equation for $f(t)$ for a given (complex) $S(\omega)$. – Novgorod Mar 18 '20 at 17:10
• Interesting problem! I think your question would be clearer and you may invite a better answer if you stated what exactly you are trying to solve.I see later in your question you mention "decorrelate" but what do you mean mathematically? The opening two sentences suggest that you are saying S(ω) is undetermined. – Dan Boschen Mar 18 '20 at 17:20
• I thought the first sentence made it clear but I see it might be misunderstood. I added the purpose now explicitly right after the equation. I'm used to the term "decorrelate f" as "retrieve signal f from some correlation S". The context would be the measurement of a signal $f(t)$, which is not directly accessible by sampling, but whose auto- or cross-correlation with some other signal (in my case a power of itself) is available. – Novgorod Mar 18 '20 at 17:30