If I have a convolution

$$z(t) = x(t) * y(t)$$

where I know $x(t)$ and $z(t)$, is there a way to determine $y(t)$? Is there a "reverse" convolution theorem for this? I know there are numerical methods used in data processing, but I'm looking for an analytical method.

  • 1
    $\begingroup$ My guess is there wouldn't be a numerical method if analytical method existed. $\endgroup$ – Kaster Jun 1 '16 at 4:53

There is not (generally). What you are looking for is deconvolution.

Consider the simple case where the Fourier transform of $x$ is zero somewhere. Then the other part can be arbitrary, since then:

$$ Y(\omega) = \frac{Z(\omega)}{X(\omega)} $$

  • $\begingroup$ That sounds reasonable. But when you say there is generally no way to reverse a convolution, are you implying that there are special cases where it's possible? $\endgroup$ – Medulla Oblongata Jun 1 '16 at 7:20
  • $\begingroup$ Yes, as said, when $Z$ and $X$ have infinite support, for example. Then the analytical solution is trivially $\mathcal{F}^{-1}\left\{ \frac{\mathcal{F}\left\{ z \right\}}{\mathcal{F}\left\{ x \right\}} \right\}$ $\endgroup$ – divB Jun 1 '16 at 8:02

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