03Fourier - Fourier Series vs Fourier Transform FOURIER...

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Fourier Series vs Fourier Transform FOURIER SERIES FOURIER TRANSFORM periodic function non-periodic, random T (finite) period infinite f T = 1 fundamental (loop) frequency, Hz fd f f k T kf k k == ± , , , ,..... 012 frequency f (, ) −∞ ∞ () k f =−∞ operator −∞ df A T xte d t k jf t T k = 1 2 0 π k =±± , , , Fourier series coefficient Fourier Transform Xf x te d t t ( ) = 2 π xt Ae k k t k = 2 Fourier series expansion Inverse Fourier Transform X f e d f t ( ) = 2 π −∞< <∞ t discrete spectrum continuous Finite Fourier Transform : X fT x t e d T t T (,) 2 0 t Relation between Finite F.T. and Fourier Series : ,....... 2 , 1 , 0 , ) , ( ± ± = = k TA T f X k k Remark : ± ,....... 2 , 1 , 0 , ) , ( 2 2 2 ± ± = = k A T T f X k k ,....... 2 , 1 , 0 , 2 ± ± = k A k : Power Spectrum ,....... 2 , 1 , 0 ), , ( ) , ( 2 2 2 ± ± = = = = k T f S T T f X A T f A k xx k k k : Power Spectral Density T T f X T T f X E f G T xx
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This note was uploaded on 11/21/2009 for the course ME . taught by Professor . during the Spring '09 term at Korea University.

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03Fourier - Fourier Series vs Fourier Transform FOURIER...

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