e178-L8.ppt

e178-L8.ppt - 2D Fourier Transform 2-D DFT & Properties 2D...

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2D Fourier Transform 1 2D Fourier Transform 2-D DFT & Properties 2D Fourier Transform 2 Fourier Transform - review 1-D: 2-D: F u ( ) ≡ ℑ f ( x ) { } = f ( x ) e j 2 π u x −∞ dx f ( x ) ≡ ℑ 1 F ( u ) { } = F ( u ) e j 2 u x −∞ du F ( u , v ) = f ( x , y ) ∫∫ e j 2 ux + vy ( ) dx dy f ( x , y ) = F ( u , v ) e j 2 ux + vy ( ) du dv ∫∫
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2D Fourier Transform 3 2D FT: Properties Convolution: f(x,y) g(x,y) = F(u,v) G(u,v) Multiplication: f(x,y) g(x,y) = F(u,v) G(u,v) Separable functions: Suppose f(x,y) = g(x) h(y), Then F(u,v)=G(u)H(v) Shifting: f(x+ x 0 , y+ y 0 ) exp[2 π j (+ x 0 u + y 0 v)] F(u,v) Linearity: a f(x,y) + b g(x,y) a F(u,v) + b G(u,v) 2D Fourier Transform 4 Separability of the FT F ( u , v ) = f ( x , y ) e j 2 π ux dx −∞ −∞ e j 2 vy dy = F ( u , y ) −∞ e j 2 vy dy
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2D Fourier Transform 5 Separability (contd.) f(x,y) F(u,y) F(u,v) Fourier Transform along X. Fourier Transform along Y. We can implement the 2D Fourier transform as a sequence of 1-D Fourier transform operations. 2D Fourier Transform 6 Eigenfunctions of LSI Systems A function f(x,y) is an Eigenfunction of a system T if T[ f(x,y) ] = α f(x,y) for some constant ( Possibly complex) α . For LSI systems, complex exponentials of the form exp{ j2 π (ux+vy) }, for any (u,v), are the Eigenfunctions.
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2D Fourier Transform 7 Impulse Response and Eigenfunctions g ( x , y ) = h ( x s , y t ) e j 2 π ( us + vt ) ds −∞ dt = h ( x , y ) ∫∫ e j 2 ( ux + vy ) e j 2 ( u x + v y ) d x d y = H ( u , v ) e j 2 ( ux + vy ) Consider a LSI system with impulse response h(x,y). Its output to the complex exponential is 2D Fourier Transform 8 2-D FT: Example f(x,y) x y Y X A F ( u , v ) = f ( x , y ) e j 2 ( ux + vy ) dxdy −∞ = A e j 2 ux 0 X dx e j 2 vy 0 Y dy = AXY sin uX uX sin vY vY e j ( uX + vY )
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2D Fourier Transform 9 Example (contd.) 2D Fourier Transform 10 Example2
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2D Fourier Transform 11 Discrete Fourier Transform Consider a sequence {u(n), n=0,1,2,. ...., N-1}. The DFT of u(n) is v k u n W k N W e u n N v k W n N n N N kn N j N k N N kn ( ) ( ) , , ,.
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This note was uploaded on 12/28/2011 for the course ECE 178 taught by Professor Manjunath during the Fall '08 term at UCSB.

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e178-L8.ppt - 2D Fourier Transform 2-D DFT & Properties 2D...

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