Fourier Transform and Convolution Mechanics
Explore the definitions and proofs of Fourier Transform and Convolution, unraveling the mathematics behind these fundamental concepts through detailed step-by-step explanations.
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Presentation Transcript
Fourier Transform Definition: 1 = = ikx ( ) ( ) [ ( )] F k e f x dx f x F 2 1 = = ikx -1 ( ) ( ) [ ( )] f x e F k dk F k F 2
1 = = ikx -1 ( ) ( ) [ ( )] f x e F k dk F k F 2 Proof 1 1 1 = ikx ikx iky ( ) ( ) e F k dk e e f y d ydk 2 2 2 1 = ikx iky ( ) e e dk f y d y 2 = ( ) ( ) f y d y y x = ( ) f x
Convolution in the Fourier Transform Definition: 1 = * ( ) ( ) ( ) f g x f x y g y dy 2 = [ * ] [ ] [ ] f g f g F F F
1 = * ( ) ( ) ( ) f g x f x y g y dy 2 = [ * ] [ ] [ ] f g f g F F F Proof 1 = ikx F-1 [ ] [ ]] [ ] [ ] f g e f g dk [F F F F 2 1 1 1 = ikx ik x ik x ( ) ( ) e e f x d x e g x d x dk 2 2 2 1 1 = ikx ik x ik x ( ) ( ) f x d x g x d x e e e dk 2 2 1 = ( ) ( ) ( ) f x d x g x d x x x x 2 1 = = ( ) ( ) * ( ) f x x g x d x f g x 2
