Home > Convolution, Correlation, & Fourier Transforms

Page 1 |

James R. Graham 11/25/2009

Page 2 |

• 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

Page 3 |

• 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

Page 4 |

• If

– E.g, if

−2��

−�� ��

��

2��

−�� ��

��

Page 5 |

• The Fourier transform is a linear operator

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

12 =

12

(

12

−2��

−�� ��

��

=

1 +

( )

−2��

−�� ��

��

=

1

−2��

−�� ��

��

+

2

−2��

−�� ��

��

=

1 +

Page 6 |

•

– Real, imaginary – Even:

• In the frequency domain these symmetries lead to relations between

Page 7 |

Page 8 |

0

)↔

−2��

0

Time shifting

Page 9 |

• With two functions

– The

−�� ��

��

��)

Page 10 |

•

– 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

Page 11 |

• The

– The correlation lies in the time domain

−�� ��

��

)

Page 12 |

• 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

*

(

Page 13 |

• The correlation of a function with itself is called its

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

*

(

2

Page 14 |

• Mathematically the convolution of

–

Page 15 |

• 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

Page 16 |

• The signal

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

Page 17 |

• 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

Page 18 |

• 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?

Page 19 |

• 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

Page 20 |

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

( )

=

��

( )

↔

Page 21 |

• Convolution of discretely sampled functions

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

Page 22 |

• Java applet demonstrations

– Continuous convolution

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

– Discrete convolution

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

Page 23 |

2��

��

2��

��

-�� ��

��

2��

��

-�� ��

��

−2��

-�� ��

��

⎡ ⎣⎢ ⎤ ⎦⎥

-�� ��

��

2��

[ ]

- Tom's Presentation Tips
- ArcGIS 9.3 manual
- Action Research in High School Physics
- Consumer Behaviour of Luxury Automobiles: A Comparative Study between Thai and UK Customers' Perceptions
- Direct Numerical Simulation of Supersonic Turbulent Boundary Layer over a Compression Ramp
- G-PhoCS – Generalized Phylogenetic Coalescent Sampler
- Design and analysis of a DSP-based Linear Switched Reluctance Motor
- SLOVENSKÁ POĽNOHOSPODÁRSKA UNIVERZITA
- DEPARTMENT OF WOMEN, CHILDREN AND PEOPLE WITH DISABILITIES
- Análise Setorial
- PHASE CHANGE MATERIAL AS A THERMAL ENERGY STORAGE MATERIAL FOR COOLING OF BUILDING
- Public Safety Canada Internal Audit of Talent Management Final Audit Report
- AU_INTERNATIONAL SEMINAR_BROCH.p65
- Lecture 2
- COMPUTER RECYCLING
- Congresso de Métodos Numéricos em Engenharia Congreso de Métodos Numéricos en Ingeniería Congress on Numerical Methods in E
- Title Loan Contract 12-2003 PDF
- Going Green or Gowing Greed:
- V210FP 2-PC Stainless Steel Ball Valve Full Port, 1000 WOG
- CURRICULUM VITAE
- From plain110 at gmail.com Sat Nov 1 00:16:17 2008

- Different specific Position
- Special quality
- The dominant factor
- Culture
- Culture Industry
- the Legislation of the Culture Industry
- Earthquake resisting
- Welded connection
- Experimental research
- Damage analysis
- Calculation model
- Earth's gravity field
- Satellite gravity measurements
- CHAMP
- GPS
- Satellite perturbation motion
- Track Credits
- Geocentric motion
- Precise Orbit
- Bayes least squares estimation block
- Time changes in Earth's gravitational potential coefficients

All Rights Reserved Powered by Free Document Search and Download

Copyright © 2011This site does not host pdf,doc,ppt,xls,rtf,txt files all document are the property of their respective owners. complaint#nuokui.com