Ve216LectureNotesChapter3Part3 - Ve216 Lecture Notes...

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Ve216 Lecture Notes Dianguang Ma Spring 2009
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Chapter 3 (Part III) Fourier Representations of Signals and LTI Systems
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3.6 The Discrete-Time Fourier Transform The discrete-time Fourier transform (DTFT) is used to represent a discrete- time nonperiodic signal as a superposition of complex sinusoids. 1 IDTFT: [ ] ( ) (Synthesis equation) 2 DTFT: ( ) [ ] (Analysis equation) [ ] ( ) j j n j j n n DTFT j x n X e e d X e x n e x n X e π π π - - Ω =-∞ = =
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3.6 The Discrete-Time Fourier Transform Consider the analysis equation of the DTFT pair. It can be shown that the X(e ) is a periodic function of Ω with period 2π: ( 2 ) ( 2 ) 2 ( ) [ ] ( ) [ ] [ ] [ ] ( ) j j n n j j n j n j n n n j n j n X e x n e X e x n e x n e e x n e X e π π π - Ω =-∞ Ω+ - Ω+ - Ω - =-∞ =-∞ - Ω =-∞ = = = = = So it can be represented by the Fourier series.
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3.6 The Discrete-Time Fourier Transform Mathematically, the analysis equation is just the Fourier series expansion of X(e ), with x[n] being the corresponding Fourier series coefficients. Thus x[n] may be recovered from X(e ) by the inverse Fourier series. 2 2 ( ) ( ) 2 2 2 2 ( ) [ ] [ ] ( ) [ ] [ ] 1 [ ]2 [ ] 2 [ ] [ ] ( ) 2 jm j j m m m j j n j n m j n m m m j j n m X e x m e x m e X e e d x m e d x m e d x m n m x n x n X e e d π π π π π π πδ π π - - Ω =-∞ =-∞ - - =-∞ =-∞ =-∞ = = Ω = Ω = = - = =
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3.6 The Discrete-Time Fourier Transform Common DTFT pairs 0 0 1 [ ], 1 (Example 3.17) 1 [ ] [ ] ( ) [ ] 1 ( ) , 1 1 n j n j n j n n j n n n j n j n u n e x n u n X e u n e e e e α α α α α α α α α - Ω - Ω - Ω =-∞ = - Ω - Ω = < - = = = = = < -
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3.6 The Discrete-Time Fourier Transform Common DTFT pairs / 2 (2 1)/ 2 (2 1)/ 2 / 2 / 2 / 2 1, sin( (2 1) / 2) [ ] (Example 3.18) 0, sin( / 2) ( ) [ ] ( ) 1 ( ) sin( (2 1) / 2) sin( / 2) M j j n j n n n M j M j M j j j M j M j j j j n M M x n n M X e x n e e e e e e e e e e e e M - Ω - Ω =-∞ =- - Ω - Ω - Ω + - Ω + - Ω - Ω - Ω + = = = - - = = - - + =
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3.6 The Discrete-Time Fourier Transform Common DTFT pairs 2 1, sin( ) [ ] (Example 3.19) 0, 1, ( ) 0, 1 1 [ ] ( ) 2 2 1 1 sin( ) 2 2 j W j j n j n W W j n jWn jWn W W Wn x n W n W X e W x n X e e d e d e e e Wn jn jn n π π π π π π π π π
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