# part5 - Part 5: Fourier Transform Fourier Transform Fourier...

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Part 5: Fourier Transform Fourier Transform z Fourier series of signal with periodic and z How about aperiodic signal? z Rewrite ( ) as and z Extending the periodic to infinity, , then z We have and z Fourier Transform: () 0 0 0 0 2 2 1 ω π n e c t x n t jn n −∞ = = −∞ = = n t jn n e c t x 0 = 2 0 2 0 0 0 1 T T dt e t x T c t jn n = = 0 0 0 0 0 2 dt e t x c n X t jn n ( ) 0 0 0 T 0 n () ( ) = dt e t x X t j = d e X t x t j 2 1 ( ) δω 0 n () ( ) X t x F

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Properties of Fourier Transform z Symmetrical Properties ¾ Even Function x(t) = x(-t): is a real, even function of ¾ Odd Function x(t) = -x(-t): is an imaginary, odd function of ¾ Real Function x(t): the magnitude of is an even function of and the phase is an odd function of ω ( ) X ( ) X ( ) X Properties of Fourier Transform z Linearity z Duality z Scaling z Convolution z Multiplication ( ) ( ) [] ( ) ( ) 2 2 1 1 2 2 1 1 X a X a t x a t x a + = + F () ( ) π = x t X 2 F  ω = a X a at x 1 F ()( ) () () τ H X d t h x = F () () ()( ) = du u Y u X t y t x 2 1 F
Properties of Fourier Transform z Time Shifting z Frequency Shifting z Time Domain Differentiation z Time Domain Integration z Frequency Differentiation z Frequency Integration ( ) [ ] ( ) ( ) ω X t j t t x 0 0 exp = F ( ) ( ) [ ] ( ) 0 0 exp = X t x t j F ( ) ()( ) X j dt t x d n n = F () ( ) () ( ) δ π τ 0 X j X d x t + = F [] ( ) d dX j t x t = F = du u X j t x t 1 F Remarks z Fourier Series is used to represent the periodic signal z Fourier Transform is usually applied for aperiodic signal z Any periodic signal that can be represented by a

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## This note was uploaded on 01/11/2011 for the course EE 3118 taught by Professor Kitsangtsang during the Spring '08 term at City University of Hong Kong.

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part5 - Part 5: Fourier Transform Fourier Transform Fourier...

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