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lecture4_2DFT

# lecture4_2DFT - 2-D Fourier Transforms Yao Wang Polytechnic...

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2-D Fourier Transforms Yao Wang olytechnic University Brooklyn NY 11201 Polytechnic University, Brooklyn, NY 11201 With contribution from Zhu Liu, Onur Guleryuz, and Gonzalez/Woods, Digital Image Processing, 2ed

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Lecture Outline Continuous Fourier Transform (FT) 1D FT (review) 2D FT Fourier Transform for Discrete Time Sequence TFT) (DTFT) 1D DTFT (review) 2D DTFT Linear Convolution 1D, Continuous vs. discrete signals (review) 2D Filter Design Computer Implementation Yao Wang, NYU-Poly EL5123: Fourier Transform 2
What is a transform? Transforms are decompositions of a function f(x) to some asis functions (x, u). u is typically into some basis functions Ø(x, u). u is typically the freq. index. Yao Wang, NYU-Poly EL5123: Fourier Transform 3

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Illustration of Decomposition Φ 3 f α 3 f = α 1 Φ 1 + α 2 Φ 2 + α 3 Φ 3 Φ 2 o α 1 α 2 Yao Wang, NYU-Poly EL5123: Fourier Transform 4 Φ 1
Decomposition Ortho-normal basis function 2 1 2 1 2 1 , 0 , 1 ) , ( * ) , ( u u u u dx u x u x Forward jection of verse   dx u x x f u x x f u F ) , ( ) ( ) , ( ), ( ) ( * Projection of f(x) onto (x,u) Inverse du u x u F x f ) , ( ) ( ) ( Representing f(x) as sum of (x,u) for all u , with weight Yao Wang, NYU-Poly EL5123: Fourier Transform 5 F(u)

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Fourier Transform Basis function   . , , ) , ( 2  u e u x ux j Forward Transform x verse Transform dx e x f x f F u F ux j 2 ) ( )} ( { ) ( Inverse Transform du e u F u F F x f ux j 2 1 ) ( )} ( { ) ( Yao Wang, NYU-Poly EL5123: Fourier Transform 6
Important Transform Pairs u u F x f ) ( ) ( 1 ) (  f u f u u F x f x f f u u F e x f x f j ) ( ) ( 2 1 ) ( ) 2 cos( ) ( ) ( ) ( ) ( 0 0 0 0 2 0 f u f u j u F x f x f ) ( ) ( 2 1 ) ( ) 2 sin( ) ( 0 0 0 u x x u u x u F otherwise x x x f n( ) 2 sinc( 2 ) 2 sin( ) ( , 0 , 1 ) ( 0 0 0 0 t t t where ) sin( ) sinc( , Yao Wang, NYU-Poly EL5123: Fourier Transform 7 Derive the last transform pair in class

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FT of the Rectangle Function t t t where u x x u u x u F ) sin( ) sinc( , ) 2 sinc( 2 ) 2 sin( ) ( 0 0 0 f(x) x 0 =1 f(x) x 0 =2 x 1 -1 x 2 -2 Yao Wang, NYU-Poly EL5123: Fourier Transform 8 Note first zero occurs at u 0 =1/(2 x 0 )=1/pulse-width, other zeros are multiples of this.
IFT of Ideal Low Pass Signal What is f(x)? F(u) u u 0 -u 0 Yao Wang, NYU-Poly EL5123: Fourier Transform 9

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Representation of FT Generally, both f(x) and F(u) are complex Two representations Real and Imaginary Magnitude and Phase ) ( ) ( ) ( u jI u R u F ) ( tan ) ( , ) ( ) ( ) ( , ) ( ) ( 1 2 2 ) ( u I u u I u R u A where e u A u F u j F(u) I I(u) Relationship ) ( u R R R(u) Φ (u) Power spectrum ) ( sin ) ( ) ( ), ( cos ) ( ) ( u u A u I u u A u R 2 * 2 Yao Wang, NYU-Poly EL5123: Fourier Transform 10 ) ( ) ( ) ( ) ( ) ( u F u F u F u A u P
What if f(x) is real?

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lecture4_2DFT - 2-D Fourier Transforms Yao Wang Polytechnic...

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