Reverse of convolution theorem

Question

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.

My guess is there wouldn't be a numerical method if analytical method existed. — Kaster, Jun 01 '16 at 04:53

divB · Accepted Answer · 2016-06-01T07:01:33.190

3

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:

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

edited Jun 01 '16 at 07:01

answered Jun 01 '16 at 05:00

divB

490

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? – Medulla Oblongata Jun 01 '16 at 07:20
Yes, as said, when $Z$ and $X$ have infinite support, for example. Then the analytical solution is trivially $\mathcal{F}^{-1}\left{ \frac{\mathcal{F}\left{ z \right}}{\mathcal{F}\left{ x \right}} \right}$ – divB Jun 01 '16 at 08:02

Reverse of convolution theorem

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