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Ve216LectureNotesChapter4

# Ve216LectureNotesChapter4 - Ve216 Lecture Notes Dianguang...

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Unformatted text preview: Ve216 Lecture Notes Dianguang Ma Spring 2009 Chapter 4 Applications of Fourier Representations to Mixed Signal Classes 4.1 Introduction • When we use Fourier methods to (1) analyze the interaction between signals and systems or (2) numerically evaluate properties of signals or the behavior of a system, a mixing of the following classes of signals often occur: – Periodic and nonperiodic signals – Continuous- and discrete-time signals • In this chapter, we build bridges between the Fourier representations of different classes of signals. 4.2 Fourier Transform Representations of Periodic Signals • Relating the FT to the FS { } ; ( ) periodic: ( ) [ ] ( ) [ ] ( ) ( ) ( ) FT [ ] [ ]FT 2 [ ] ( ) FS jk t k FT jk t jk t k k k x t x t X k x t X k e x t X j X j X k e X k e X k k ϖ ϖ ϖ ϖ ϖ ϖ π δ ϖ ϖ ∞ =-∞ ∞ ∞ =-∞ =-∞ ∞ =-∞ ↔ = ↔ = = =- ∑ ∑ ∑ ∑ Figure 4.1 (p. 343) FS and FT representation of a periodic continuous-time signal. 4.2 Fourier Transform Representations of Periodic Signals • Relating the DTFT to the DTFS 2 1 1 [ ] periodic: [ ] [ ] [ ] 2 ( ), , or 2 ( 2 ) N N jk n jk n N k k DTFT jk n DTFT jk n m x n x n X k e X k e e k k e k m π πδ π π π π π δ π -- Ω = = Ω ∞ Ω =-∞ = = ↔ Ω - Ω- < Ω <- < Ω < ↔ Ω - Ω - ∑ ∑ ∑ Figure 4.5 (p. 346) Infinite series of frequency-shifted impulses that is 2 π periodic in frequency Ω . ( 2 ) m k m δ π ∞ =-∞ Ω - Ω - ∑ 4.2 Fourier Transform Representations of Periodic Signals { } 1 1 1 1 1 1 [ ] ( ) DTFT [ ] [ ]DTFT [ ] 2 ( 2 ) 2 [ ] ( 2 ) 2 [ ] ( 2 ) 2 [ ] ( ( ) ) N N DTFT jr n jr n j r r N r m N r m N m r N m r x n X e X r e X r e X r r m X r r m X r r m X r mN r mN π δ π π δ π π δ π π δ-- Ω Ω Ω = =- ∞ = =-∞- ∞ = =-∞ ∞- =-∞ =- =- = ↔ = = = Ω - Ω - = Ω - Ω - = Ω - Ω - = + Ω - + Ω ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ ∑ 2 [ ] ( ) k X k k π δ ∞ ∞ ∞ =-∞ = Ω - Ω ∑ ∑ Figure 4.6 (p. 346) DTFS and DTFT representations of a periodic discrete-time signal. 4.3 Convolution and Multiplication with Mixtures of Periodic and Nonperiodic Signals • Convolution of periodic and nonperiodic signals ( ) ( )* ( ) ( ) periodic: ( ) ( ) 2 [ ] ( ) ( ) ( ) ( ) 2 [ ] ( ) ( ) 2 [ ] ( ) ( ) 2 ( ) [ ] ( ) k k k k y t x t h t x t x t X j X k k Y j X j H j X k k H j X k H j k H j X k k ϖ π δ ϖ ϖ ϖ ϖ ϖ π δ ϖ ϖ ϖ π ϖ δ ϖ ϖ π ϖ δ ϖ ϖ ∞ =-∞ ∞ =-∞ ∞ =-∞ ∞ =-∞ = ↔ =- = =- =- =- ∑ ∑ ∑ ∑ Figure 4.8 (p. 449) Convolution property for mixture of periodic and nonperiodic signals. Example 4.4 Periodic Input to an LTI System • Given • Use the convolution property to find the output of this system....
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Ve216LectureNotesChapter4 - Ve216 Lecture Notes Dianguang...

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