Lecture-Fourier Transform

# Lecture-Fourier Transform - EE4780 2D Fourier Transform...

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EE 4780 2D Fourier Transform

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Bahadir K. Gunturk 2 Fourier Transform What is ahead? 1D Fourier Transform of continuous signals 2D Fourier Transform of continuous signals 2D Fourier Transform of discrete signals 2D Discrete Fourier Transform (DFT)
Bahadir K. Gunturk 3 Fourier Transform: Concept A signal can be represented as a weighted sum of sinusoids. Fourier Transform is a change of basis, where the basis functions consist of sines and cosines (complex exponentials).

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Bahadir K. Gunturk 4 Fourier Transform Cosine/sine signals are easy to define and interpret. However, it turns out that the analysis and manipulation of sinusoidal signals is greatly simplified by dealing with related signals called complex exponential signals . A complex number has real and imaginary parts: z = x + j*y A complex exponential signal: r*exp(j*a) =r*cos(a) + j*r*sin(a)
Bahadir K. Gunturk 5 Fourier Transform: 1D Cont. Signals Fourier Transform of a 1D continuous signal 2 ( ) ( ) j ux F u f x e dx π - -∞ = Inverse Fourier Transform 2 ( ) ( ) j ux f x F u e du -∞ = ( 29 ( 29 2 cos 2 sin 2 j ux e ux j ux - = - “Euler’s formula”

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Bahadir K. Gunturk 6 Fourier Transform: 2D Cont. Signals Fourier Transform of a 2D continuous signal Inverse Fourier Transform 2 ( ) ( , ) ( , ) j ux vy f x y F u v e dudv π ∞ ∞ + -∞ -∞ = ∫ ∫ 2 ( ) ( , ) ( , ) j ux vy F u v f x y e dxdy ∞ ∞ - + -∞ -∞ = ∫ ∫ f F F and f are two different representations of the same signal.
Bahadir K. Gunturk 7 Fourier Transform: Properties Remember the impulse function ( Dirac delta function ) definition 0 0 ( ) ( ) ( ) x x f x dx f x δ -∞ - = Fourier Transform of the impulse function ( 29 2 ( ) ( , ) ( , ) 1 j ux vy F x y x y e dxdy π ∞ ∞ - + -∞ -∞ = = ∫ ∫ ( 29 0 0 2 ( ) 2 ( ) 0 0 0 0 ( , ) ( , ) j ux vy j ux vy F x x y y x x y y e dxdy e ∞ ∞ - + - + -∞ -∞ - - = - - = ∫ ∫

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Bahadir K. Gunturk 8 Fourier Transform: Properties Fourier Transform of 1 ( 29 2 ( ) 1 ( , ) j ux vy F e dxdy u v π δ ∞ ∞ - + -∞ -∞ = = ∫ ∫ ( 29 1 2 ( ) 2 (0 0) ( , ) ( , ) 1 j ux vy j x v F u v u v e dudv e ∞ ∞ - + + -∞ -∞ = = = ∫ ∫ Take the inverse Fourier Transform of the impulse function

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## This note was uploaded on 11/28/2011 for the course EE 4780 taught by Professor Staff during the Spring '08 term at LSU.

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Lecture-Fourier Transform - EE4780 2D Fourier Transform...

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