Home > Convolution, Correlation, & Fourier Transforms

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James R. Graham 11/25/2009

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• A large class of signal processing techniques fall under the category of

– These methods fall into two broad categories

• Efficient method for accomplishing common data manipulations • Problems related to the Fourier transform or the power spectrum

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• A physical process can be described in two ways

– In the

• In general

• It is useful to think of

– One goes back and forth between these two representations by Fourier transforms

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• If

– E.g, if

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• The Fourier transform is a linear operator

– The transform of the sum of two functions is the sum of the transforms

12 =

12

(

12

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1 +

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•

– Real, imaginary – Even:

• In the frequency domain these symmetries lead to relations between

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0

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Time shifting

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• With two functions

– The

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•

– The convolution is one member of a transform pair

• The Fourier transform of the convolution is the product of the two Fourier transforms!

– This is the

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• The

– The correlation lies in the time domain

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• The correlation is one member of the transform pair

– More generally, the RHS of the pair is

• Multiplying the FT of one function by the complex conjugate of the FT of the other gives the FT of their correlation

– This is the

*

(

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• The correlation of a function with itself is called its

– In this case the correlation theorem becomes the transform pair – This is the

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(

2

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• Mathematically the convolution of

–

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• The effect of convolution is to smear the signal

is

– Smeared into the shape of the response function – Translated from time 0 to time

as

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• The signal

– Since the response function is broader than some features in the original signal, these are smoothed out in the convolution

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• Fourier methods have revolutionized many fields of science & engineering

– Radio astronomy, medical imaging, & seismology

• The wide application of Fourier methods is due to the existence of the

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• The convolution of two functions is defined for the continuous case

– The convolution theorem says that the Fourier transform of the convolution of two functions is equal to the product of their individual Fourier transforms

• We want to deal with the discrete case

– How does this work in the context of convolution?

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• In the discrete case

• The response function is also a discrete set

–

tells what multiple of the input signal in channel

tells what multiple of input signal

tells the multiple of input signal

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• Symbolically the discrete convolution is with a response function of finite duration,

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=

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↔

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• Convolution of discretely sampled functions

– Note the response function for negative times wraps around and is stored at the end of the array

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• Java applet demonstrations

– Continuous convolution

• http://www.jhu.edu/~signals/convolve/

– Discrete convolution

• http://www.jhu.edu/~signals/discreteconv/

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