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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 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 = = = = = = = = 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  = = = = =   < < < <   Figure 4.5 (p. 346) Infinite series of frequencyshifted impulses that is 2 periodic in frequency . ( 2 ) m k m =   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  = = = = = =  = = = = = = =   =   =   = +  + 2 [ ] ( ) k X k k = =  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 = = = = = = = = = = 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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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 MichiganDearborn.
 Spring '09
 DianguangMa

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