Ve216LectureNotesChapter7Part1

Ve216LectureNotesChapter7Part1 - Ve216 Lecture Notes...

Info iconThis preview shows pages 1–10. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

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 < = - =- - =- 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)....
View Full Document

Page1 / 50

Ve216LectureNotesChapter7Part1 - Ve216 Lecture Notes...

This preview shows document pages 1 - 10. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online