Difference between revisions of "Convolution Theorem"

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===Prerequisites===
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* Fourier Transforms (link?)
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* Integral Calculus (link?)
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===Short Topical Videos===
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* [http://www.khanacademy.org/video/introduction-to-the-convolution?playlist=Differential%20Equations Introduction to the Convolution] by Khan Academy
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===Reference Material===
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===Related Subjects===
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* [[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