# Meaning and application of convolution or deconvolution in physical sciences

In which real case scenarios a convolution or deconvolution operation is useful ?

• I think this question is too broad to be answered correctly. We will just have an endless list ... Why are you asking ? – Cedric H. Nov 15 '10 at 22:20
• Anytime there are waves, but I don't think this the answer you are looking for... :-) – Sklivvz Nov 15 '10 at 22:26
• It's useful if it casts your problem in a form that you can solve easily? – John Nov 15 '10 at 22:32
• @Cedric I'm trying to find a real case, simple scenario where a convolution is used. I don't think we are going to have an endless list. I am looking for a very specific answer, the one that explains clearly how a convolution/deconvolution is practically used in real-world science. – Stefano Borini Nov 15 '10 at 22:55
• @Stefano: I advise you to look at the similar question asked at mathoverflow. It's obviously more mathematical than physical but you might gain a nice intuition there. – Marek Nov 15 '10 at 23:08

Any real instrument will have some impulse response. The measured signal is the convolution of the source signal with this impulse response. For example, if you aim a telescope at a point source, you will see not a point source but the point source convolved with the point spread function (2D impulse response) of the telescope. Some kind of (usually approximate) deconvolution is applied to correct this and better estimate the source signal.

• oh thanks. next stop lmgtfy ? :P – Stefano Borini Nov 16 '10 at 7:52
• Well, the next time before asking spend at least a few minutes checking Google and Wikipedia. Sometimes you can find easily good articles. – Piotr Migdal Nov 16 '10 at 10:18
• the point of googling stuff has been discussed at length on all the meta sites since the very inception of SE, and the conclusion forms the very philosophy behind SE sites. We want to localize knowledge, not point to links that can become outdated. The goal is to have this answer as the first google entry (see: "google as an interface" SE concept) so that people can learn immediately a well-written answer to a very specific question. I encourage you to browse the meta sites (not only the physics meta, but stackoverflow meta as well), to see the rationale behind my point. – Stefano Borini Nov 16 '10 at 16:43

This is a very general topic to discuss, (I'm not sure there's a right answer here), but I think it's very much worth pointing out the convolution theorem. It's not a "practical" application per se, but it's a very useful method, since Fourier analysis enters physics in so many areas. Anywhere where you encounter spectrum analysis, convolutions will quite possibly come into play somewhere.

The convolution theorem states:

$$\mathcal{F}\{f*g\} = \mathcal{F}\{f\} \cdot \mathcal{F}\{g\}$$

$$\mathcal{F}\{f \cdot g\}= \mathcal{F}\{f\}*\mathcal{F}\{g\}$$

where $\mathcal{F}$ represents the Fourier transform operator.

These are remarkably useful identities when performing Fourier transforms. Since Fourier transforms can in fact be used for solving some differential equations, they have a notable application there too.

Solving the Fresnel diffraction integral, to find the amplitude and phase of waves near a source: http://en.wikipedia.org/wiki/Fresnel_diffraction#Convolution

Convolution is applied whenever the amount of a quantity is simultaneously being produced and undergoing another change and then these effects added up over time.

A practical example I've heard is when you produce radioactive material at a certain rate $f(t)$ (which decays from the moment it appears). At any moment $t$, we have total amount equal to $(f\star g)(t)$, where $g(t) = e^{-kt}$.