# Is there an analogy of the Fourier-decomposition in the Laplace space to decompose a signal to a few components?

I have a signal from which I know, that it is the sum of a few, exponentially decaying components. I want to find these components.

If it would be a sum of some sinusiod waves, it would be easy to Fourier-transform it, and then find the Kronecker-deltas with some heuristics. It is because the Fourier-transform of a sinusiodal signal is a $$\delta$$.

If the signal would be the superposition of some damped oscillators, as it is asked in this question, the problem would be still easily solvable by a 2d fourier transform. If I understand it well, the trick is that we use the fourier space to decompose the decaying signals. I think it would work only if the oscillators would have a different decay time constant, if they belong to different frequencies.

However, my signal is not from dampened oscillators, it is a simple decaying signal without any oscillation, making this question not a dupe.

My first naive try was to Laplace-transform it, and then do like in the Fourier case. It does not work, because $$\mathbb L \{ae^{-bt}\}=\frac{a}{s-b}$$. First, it is singular, making its digital processing problematic, but the major problem is that I can't see any easy way to find the components in the result.

Maybe Laplace is not usable for that, or it should be done with Laplace, but somehow differently?

• – A_A Apr 12 '19 at 8:40
• You might find this and this useful. – A_A Apr 12 '19 at 8:42
• @A_A Thanks! Your first link is not usable in my case (fourier is meaningless or at least impractical for such an aperiodic signal), but your second (Prony's method) looks very interesting. I am currently trying to understand it. – peterh - Reinstate Monica Apr 12 '19 at 10:39
• @A_A My next naive idea: I would create its Taylor-series, transfer into imaginary by substituting $x \rightarrow ix$ in it, then I get a complex signal, and then I try to somehow use some duality between Fourier and Laplace. But this looks to me a quite weird idea yet. Even if it works in theory, it will probably be unusable numerically due to instabilities. – peterh - Reinstate Monica Apr 14 '19 at 14:48
• Yet another idea: Laplace-transforming the signal, and then trying a least-square approximation of it with a $\sum_{k=1}^n\frac{a_k}{s-b_k}$. It would result probably huge formulas, but with luck it will be not much harder than the polynomial regression. Doing the same with exponentials would be non-analytical. – peterh - Reinstate Monica Apr 14 '19 at 19:16