03Fourier

# 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 f d f f k T k f k k = = = ± ± , , , , ..... 0 1 2 frequency f ( , ) −∞ ∞ ( ) k f =−∞ operator ( ) −∞ df A T x t e dt k j f t T k = 1 2 0 ( ) π k = ± ± 0 1 2 , , , ..... , Fourier series coefficient Fourier Transform X f x t e dt j ft ( ) ( ) = −∞ 2 π x t A e k k j f t k ( ) = =−∞ 2 π Fourier series expansion Inverse Fourier Transform x t X f e df j ft ( ) ( ) = −∞ 2 π − ∞ < < ∞ t discrete spectrum continuous Finite Fourier Transform : X f X f T x t e d T j ft T ( ) ( , ) ( ) = = 2 0 π 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

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