Unit8_ZT - EE 3210 Signals and Systems Unit 8 z Transform...

Info iconThis preview shows pages 1–13. Sign up to view the full content.

View Full Document Right Arrow Icon
EE 3210 Signals and Systems Unit 8 z Transform
Background image of page 1

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

View Full DocumentRight Arrow Icon
Unit 8.1 What is z-Transform? After studying this unit, you will be able to 1. describe the definition of z-transform and its relationship with Fourier transform 2. find the z-transform of some simple signals 3. describe the properties of z-transform
Background image of page 2
EE3210 Signals and Systems Unit 8 3 z-Transform Definition of z-transform : Comparison with DTFT: What have you found? −∞ = ⎯→ n n z z n x z Χ n x ] [ ) ( ] [ −∞ = n n j j F e n x e Χ n x ω ] [ ) ( ] [
Background image of page 3

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

View Full DocumentRight Arrow Icon
EE3210 Signals and Systems Unit 8 4 Generalization of DTFT z-Transform reduces to DTFT when In general, z is a complex number: ω j e z = . 0 where form), (polar = r re z j z-transform DTFT when |z| =1 j e z = Im(z) Re(z) unit circle 1 -1
Background image of page 4
EE3210 Signals and Systems Unit 8 5 Region of Convergence The summation may or may not converge. The region of z in which the summation converges is called the region of convergence (ROC) of X(z). The ROC depends on r = |z| only. −∞ = = Χ n n z n x z ] [ ) ( () ( ) [] {[] } j nj n n n zr e x n r e zF x n r ω +∞ −− =−∞ Χ= Χ =
Background image of page 5

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

View Full DocumentRight Arrow Icon
EE3210 Signals and Systems Unit 8 6 Find the z-transform of ] [ ] [ n u a n x n = n 0 [] n aun (a < 1) Example 1 (Example 10.1 in textbook) Note: The DTFT of a n u[n] does not exist if |a|>1.
Background image of page 6
EE3210 Signals and Systems Unit 8 7 Example 1 Solution: This is a geometric series. It converges if | az -1 | < 1 , i.e., | z | > | a |. = = −∞ = = = = 0 1 0 ) ( ] [ ) ( n n n n n n n n az z a z n u a z Χ 1 1 () , | || | 1 z Xz z a az z a = => −− 1 0 1 1 N N n n a a a = =
Background image of page 7

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

View Full DocumentRight Arrow Icon
EE3210 Signals and Systems Unit 8 8 Example 2 (Example 10.2 in textbook) ] 1 [ ] [ = n u a n x n Find the z-transform of Find the answer yourselves! What is its ROC?
Background image of page 8
EE3210 Signals and Systems Unit 8 9 Complete Specification From the previous two examples, the z-transforms of the two signals have the same algebraic expression. But they have different ROC. To completely specify the z-Transform of a signal, we need to provide Algebraic expression for X(z) ROC
Background image of page 9

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

View Full DocumentRight Arrow Icon
EE3210 Signals and Systems Unit 8 10 Relationship between Transforms ω j e z = Im(z) Re(z) unit circle 1 -1 Im(s) Re(s) -a j s = z transform Laplace transform DTFT can be obtained from z-Transform if ROC contains the unit circle. CTFT can be obtained from Laplace Transform if ROC contains the imaginary axis.
Background image of page 10
EE3210 Signals and Systems Unit 8 11 Find the z-transform of Solution: When will the two summations converge? ] [ ) 2 1 ( 6 ] [ ) 3 1 ( 7 ] [ n u n u n x n n = = = −∞ = −∞ = −∞ = = = = 0 1 0 1 2 1 6 3 1 7 ] [ ) 2 1 ( 6 ] [ ) 3 1 ( 7 ] [ ) ( n n n n n n n n n n n n z z z n u z n u z n x z X Example 3 (Example 10.3 in textbook)
Background image of page 11

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

View Full DocumentRight Arrow Icon
EE3210 Signals and Systems Unit 8 12 Example 3 Both summation converges when |z| > ½.
Background image of page 12
Image of page 13
This is the end of the preview. Sign up to access the rest of the document.

This note was uploaded on 11/22/2009 for the course ELECTRONIC EE3210 taught by Professor Sungchiwan during the Spring '09 term at École Normale Supérieure.

Page1 / 54

Unit8_ZT - EE 3210 Signals and Systems Unit 8 z Transform...

This preview shows document pages 1 - 13. Sign up to view the full document.

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