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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 ω π π ω ω n e c t x n t jn n −∞ = = ( ) −∞ = = n t jn n e c t x 0 ω ( ) = 2 0 2 0 0 0 1 T T dt e t x T c t jn n ω ( ) ( ) = = 0 0 0 0 0 2 ω π ω π ω ω π ω dt e t x c n X t jn n ( ) 0 0 ω 0 T ω ω 0 n ( ) ( ) = dt e t x X t j ω ω ( ) ( ) = ω ω π ω d e X t x t j 2 1 ( ) δω ω 0 n ( ) ( ) ω 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 ( ) ω X ω ( ) ω X ω ω ( ) ω X Properties of Fourier Transform z Linearity z Duality z Scaling z Convolution z Multiplication ( ) ( ) [ ] ( ) ( ) ω ω 2 2 1 1 2 2 1 1 X a X a t x a t x a + = + F ( ) [ ] ( ) ω π = x t X 2 F ( ) [ ]  ω = a X a at x 1 F ( ) ( ) ( ) ( ) ω ω τ τ τ H X d t h x = F ( ) ( ) [ ] ( ) ( ) = du u Y u X t y t x ω π 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 ω X t j t t x 0 0 exp = F ( ) ( ) [ ] ( ) 0 0 exp ω ω ω = X t x t j F ( ) ( ) ( ) ω ω X j dt t x d n n = F ( ) ( ) ( ) ( ) ω δ π ω ω τ τ 0 X j X d x t + = F ( ) [ ] ( ) ω ω d dX j t x t = F ( ) ( ) = ω du u X j t x t 1 F Remarks z

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

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