This preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full DocumentThis preview has intentionally blurred sections. Sign up to view the full version.
View Full Document
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 discretetime 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 continuoustime 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 frequencyshifted 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 discretetime 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....
View
Full Document
 Spring '09
 DianguangMa
 Digital Signal Processing, Signal Processing, jω

Click to edit the document details