Fourier Transform

Fourier Transform - Fourier Transform Fourier Integral f x...

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Fourier Transform [ ] [ ] 0 0 0 Fourier Integral: ( ) ( )cos( ) ( )sin( ) 1 1 where ( ) ( )cos( ) , ( ) ( )sin( ) 1 Therefore, ( ) ( ) cos( )cos( ) sin( )sin( ) 1 ( )cos( ) f x A w wx B w wx dw A w f v wv dv B w f v wv dv f x f v wv wx wv wx dvdw f v wx wv dv d π π π π -∞ -∞ ∞ ∞ -∞ -∞ = + = = = + = - ∫ ∫ ∫ ∫ 0 - - 0 0 1 ( )cos( ) , 2 since [ ] is an even function of w and [ ] [] [] 2 [] w f v wx wv dv dw dw dw dw dw π -∞ -∞ = - = + = ∫ ∫
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0 0 - - 0 Similarly, for an odd function 1 1 ( )sin( ) ( )sin( ) 0 2 since [ ] is an even function of w and [ ] [] [] 0 1 Therefore, ( ) ( ) cos( 2 f v wx wv dv dw f v wx wv dv dw dw dw dw f x f v wx w π π π -∞ -∞ -∞ = - = - = = + = = - ∫ ∫ ∫ ∫ [ ] ( ) ) sin( ) Recall the Euler formula: cos sin , ( 1) 1 ( ) ( ) : Complex Fourier Integral 2 ix iw x v v i wx wv dv dw e x i x i f x f v e dv dw π -∞ -∞ - -∞ -∞ + - = + = - = This term equals zero
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Complex Fourier Transform and the Inverse Transform 1 1 1 ( ) ( ) ( ) 2 2 2 1 ˆ ( ) 2 1 ˆ where ( ) ( ) is called the Fourier Transform of f 2 1 ˆ ( ) ( ) is called 2 iwx iwv iwv iwx iwx iwv iwx f x f v e dv dw f v e dv e
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