Ve216LectureNotesChapter7Part1

Ve216LectureNotesChapter7Part1 - Ve216 Lecture Notes...

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Unformatted text preview: Ve216 Lecture Notes Dianguang Ma Spring 2009 Chapter 7 (Part I) Representing Signals by Using Discrete-Time Complex Exponentials: The z-Transform 7.1 Introduction The z-transform is a generalization of the DTFT. It provides a broader characterization of discrete-time LTI systems and their interaction with signals than is possible with the DTFT. The primary roles of the z-transform in engineering practice are the study of system characteristics and the derivation of computational structures for implementing discrete-time systems on computers. The unilateral z-transform is also used to solve difference equations subject to initial conditions. 7.2 The z-Transform Consider applying a complex exponential input of the form x[n]= z n to an LTI system with impulse response h[n]. The system output is given by [ ] [ ]* [ ] [ ] [ ] [ ] [ ] ( ) ( ) [ ] : transfer function k n k n k k k n k k y n h n x n h k x n k h k z z h k z H z z H z h k z =- -- =- =- - =- = =- = = = @ 7.2 The z-Transform Given the simplicity of describing the action of the system on inputs of the form z n , we now seek a representation of arbitrary signals as a weighted superposition of eigenfunction z n . Recall that an eigenfunction is a signal that passes through the system without being modified except for multiplication by a scalar. Hence, we identify z n as an eigenfunction of the LTI system and H(z) as the corresponding eigenvalue. n z z H n y ) ( ] [ = 7.2 The z-Transform The transfer function H(z) is the z-transform of h[n], or the DTFT of h[n]r-n . ( 29 { } n n n j n n n j n n r n h DTFT e r n h re n h z n h z H- - = -- - =- - =- = = = = ] [ ] [ ) ]( [ ] [ ) ( We may recover h[n] by the inverse z-transform of H(z). - - = = = dz z z H j d re re H n h d e re H r n h n n j j n j j n 1 2 2 ) ( 2 1 ) )( ( 2 1 ] [ ) ( 2 1 ] [ 7.2 The z-Transform For an arbitrary signal x[n], the z-transform of x[n] is - =- = n n z n x z X ] [ ) ( And the inverse z-transform of X(z) is We express this relationship with the notation - = dz z z X j n x n 1 ) ( 2 1 ] [ ) ( ] [ z X n x z 7.2 The z-Transform = - - =- = = = j e z j n n n z X e X r n x DTFT z n x z X ) ( ) ( } ] [ { ] [ ) ( The z-transform of x[n] can be interpreted as the DTFT of x[n] after multiplication by a real exponential signal r-n . Hence, a necessary condition for convergence of the z- transform is the absolute summability of x[n]r-n . That is, we must have &lt; = - =- - =- n n n n r n x r n x ] [ ] [ 7.2 The z-Transform The range of r for which the z-transform converges is termed the region of convergence (ROC)....
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Ve216LectureNotesChapter7Part1 - Ve216 Lecture Notes...

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