Ve216LectureNotesChapter3Part3

Ve216LectureNotesChapter3Part3 - Ve216 Lecture Notes...

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Unformatted text preview: Ve216 Lecture Notes Dianguang Ma Spring 2009 Chapter 3 (Part III) Fourier Representations of Signals and LTI Systems 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 Ω- ∞ Ω- Ω =-∞ Ω = Ω = ↔ ∫ ∑ 3.6 The Discrete-Time Fourier Transform • Consider the analysis equation of the DTFT pair. It can be shown that the X(e jΩ ) 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- =-∞ =-∞ ∞- Ω Ω =-∞ = ⇒ = = = = ∑ ∑ ∑ ∑ • So it can be represented by the Fourier series. 3.6 The Discrete-Time Fourier Transform • Mathematically, the analysis equation is just the Fourier series expansion of X(e jΩ ), with x[n] being the corresponding Fourier series coefficients. Thus x[n] may be recovered from X(e jΩ ) 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 =-∞ =-∞ ∞ ∞ Ω Ω Ω- Ω- =-∞ =-∞ ∞ Ω Ω =-∞ = = ⇒ Ω = Ω = Ω =- = ⇒ = Ω ∑ ∑ ∑ ∑ ∫ ∫ ∫ ∑ ∫ 3.6 The Discrete-Time Fourier Transform • Common DTFT pairs 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- Ω = < ↔- = ⇒ = = = = <- ∑ ∑ ∑ 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 +- Ω- Ω Ω- Ω ≤ Ω + = ↔ Ω = =-- = =-- Ω + = Ω ∑ ∑ 3.6 The Discrete-Time Fourier Transform3....
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This note was uploaded on 10/26/2009 for the course EECS EECS 216 taught by Professor Dianguangma during the Spring '09 term at University of Michigan-Dearborn.

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Ve216LectureNotesChapter3Part3 - Ve216 Lecture Notes...

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