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| + | ===Prerequisites=== |
| + | * Fourier Transforms (link?) |
| + | * Integral Calculus (link?) |
| + | |
| + | ===Short Topical Videos=== |
| + | * [http://www.khanacademy.org/video/introduction-to-the-convolution?playlist=Differential%20Equations Introduction to the Convolution] by Khan Academy |
| + | |
| + | ===Reference Material=== |
| + | |
| <latex> | | <latex> |
| \documentclass[]{article} | | \documentclass[]{article} |
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| \end{document} | | \end{document} |
| </latex> | | </latex> |
| + | |
| + | ===Related Subjects=== |
| + | * [[Interferometric Imaging]] |
Revision as of 14:00, 31 August 2011
Prerequisites
- Fourier Transforms (link?)
- Integral Calculus (link?)
Short Topical Videos
Reference Material
Convolution Theorem
Fourier Transform
Here are the definitions we will use for the forward (
) and inverse (
) Fourier transforms:
where
is the angular frequency coordinate that is the Fourier complement of time
, and a top-hat is generally used to denote Fourier-domain quantities.
Convolution Theorem
The convolution is a useful operation with applications ranging from photo editing (blurring) to crystallography to astronomy.
Renaming
to be
(which we are totally free to do), we get a statement of the convolution theorem:
Convolution vs. Correlation
Correlation is very similar to convolution, and it is best defined through its equivalent “correlation theorem”:
The difference between correlation and convolution is that that when correlating two signals, the Fourier transform of the second function (
in equation None) is conjugated before multiplying and integrating. Using that
we can show that correlating
and
is equivalent to convolving
with a conjugated, time-reversed version of
:
Although this relation between convolution and correlation is often mentioned in the literature, I don’t personally find it very intuitively illuminating. I much prefer the “correlation theorem” in equation (None), because when it is combined with the expression of a time-shifted signal in Fourier domain:
it shows that correlating a flat-spectrum signal with a time-shifted version of itself yields a measure of the power of the signal at the delay corresponding to the time shift:
Related Subjects