# Difference between revisions of "Convolution Theorem"

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− | \section{Fourier Transform} | + | \section*{Fourier Transform} |

Here are the definitions we will use for the forward ($\mathcal{F}$) and inverse ($\mathcal{F}^{-1}$) | Here are the definitions we will use for the forward ($\mathcal{F}$) and inverse ($\mathcal{F}^{-1}$) | ||

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and a top-hat is generally used to denote Fourier-domain quantities. | and a top-hat is generally used to denote Fourier-domain quantities. | ||

− | \section{Convolution Theorem} | + | \section*{Convolution Theorem} |

The {\it convolution} is a useful operation with applications ranging from photo editing (blurring) to | The {\it convolution} is a useful operation with applications ranging from photo editing (blurring) to | ||

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\end{equation} | \end{equation} | ||

− | \subsection{Convolution vs. Correlation} | + | \subsection*{Convolution vs. Correlation} |

{\it Correlation} is very similar to convolution, and it is best defined through its | {\it Correlation} is very similar to convolution, and it is best defined through its |

## Revision as of 17:19, 30 August 2011

# 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: