Tale

1. EEE328
Digital Signal Processing
Ankara University
Faculty of Engineering
Electrical and Electronics Engineering Department
Ankara University Electrical and Electronics Eng. Dept. EEE328

2. The z-Transform
EEE328 Digital Signal Processing
Lecture 7
Ankara University Electrical and Electronics Eng. Dept. EEE328

3. Agenda
• The z-Transform
• Bilateral z-Transform
• Unilateral z-Transform
Ankara University Electrical and Electronics Eng. Dept. EEE328

4. • The z-Transform
𝑋 𝑒𝑗𝜔
= ෍
𝑛=−∞
∞
𝑥[𝑛]𝑒−𝑗𝜔𝑛
𝑧 = 𝑒𝑗𝜔
𝑋 𝑧 = ෍
𝑛=−∞
∞
𝑥[𝑛]𝑧−𝑛
Ankara University Electrical and Electronics Eng. Dept. EEE328
(Bilateral) z-Transform

5. • The z-Transform
𝑥[𝑛] ՞
𝒵
𝑋 𝑧
𝑍 𝑥 𝑛 = ෍
𝑛=−∞
∞
𝑥 𝑛 𝑧−𝑛 = 𝑋(𝑧)
Ankara University Electrical and Electronics Eng. Dept. EEE328

6. • The z-Transform
𝒳 𝑧 = ෍
𝑛=0
∞
𝑥[𝑛]𝑧−𝑛
Ankara University Electrical and Electronics Eng. Dept. EEE328
Unilateral z-Transform

7. • The z-Transform
𝑧 = 𝑟𝑒𝑗𝜔
𝑋 𝑟𝑒𝑗𝜔
= ෍
𝑛=−∞
∞
𝑥 𝑛 𝑟𝑒𝑗𝜔 −𝑛
𝑋 𝑟𝑒𝑗𝜔 = ෍
𝑛=−∞
∞
𝑥 𝑛 𝑟−𝑛 𝑒−𝑗𝜔𝑛
Ankara University Electrical and Electronics Eng. Dept. EEE328

8. • The z-Transform
Ankara University Electrical and Electronics Eng. Dept. EEE328
Unit Circle
Im
Re
ω 1
𝑧 = 𝑒𝑗𝜔
z-plane
The unit circle in the complex z-plane

9. • The z-Transform
෍
𝑛=−∞
∞
|𝑥 𝑛 𝑟−𝑛
| < ∞
෍
𝑛=−∞
∞
|𝑥 𝑛 |𝑟|−𝑛
< ∞
Ankara University Electrical and Electronics Eng. Dept. EEE328
Convergenge

10. • The z-Transform
Example:
𝑥 𝑛 = 𝑎𝑛
𝑢[𝑛]
𝑋 𝑧 = ෍
𝑛=−∞
∞
𝑎𝑛𝑢 𝑛 𝑧−𝑛 = ෍
𝑛=0
∞
(𝑎𝑧−1)𝑛
෍
𝑛=0
∞
|𝑎𝑧−1
|𝑛
< ∞
𝑋 𝑧 = ෍
𝑛=0
∞
(𝑎𝑧−1)𝑛=
1
1 − 𝑎𝑧−1
=
𝑧
𝑧 − 𝑎
, 𝑧 < |𝑎|
Ankara University Electrical and Electronics Eng. Dept. EEE328
Right-sided sequence
For convergence of X(z) 𝑎𝑧−1 < 1 𝑧 < |𝑎|

11. • Example (Cont.)
Ankara University Electrical and Electronics Eng. Dept. EEE328
Im
Re
z-plane
o x 1
a
for |a|<1
Region of Convergence (ROC): |z|>|a|
o Zero
x Pole

12. References
• Signals & Systems, Second Edition, A. V. Oppenheim, A. S. Willsky with S. H.
Nawab, Prentice Hall, 1997
• Discrete-Time Signal Processing, Second Edition, A. V. Oppenheim, R. W. Schafer
with J. R. Buck, Prentice Hall, 1999
Ankara University Electrical and Electronics Eng. Dept. EEE328