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Z- Transform and Its Properties

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1 Z- Transform and Its Properties
Dr. Wajiha Shah

2 The z-Transform Given the causal sequence {x[1], x[2], …., x[k],….}, its z-transform is defined as The variable z may be regarded as a time delay operator

3 Example Find the z-transform of the sequence x = {1, 1, 3, 2, 0, 4}
Solution:

4 Exercise Find the z-transform of the following sequences:
(b) {0, 0, 0, 1, 1, 1, 0, 0, 0, . . .} (c) {0, 2−0.5, 1, 2−0.5, 0, 0, 0, . . .}

5 Z-Transform of Standard Discrete-Time Signals
Unit Impulse Function A discrete time impulse Shown in the Fig. is defined as The z-Transform of δ(k) is X(z) = 1

6 Sampled Step Sampled Step A sampled step function is shown in the Fig.
Mathematically, it is defined as The z-Transform of u(k) is computed as

7 Exponential Exponential:
A sampled exponential function is shown in the Fig. Mathematically, it is defined as The z-Transform of x(k) is computed as

8 Properties of z-Transform
Linearity Let x1(k) ,x2(k) , .... be discrete time sequences and a1, a2, …. be constants, then according to this property Example: Find z-Transform of x(k) = 2u(k) + 4δ(k), where u(k) is a unit step sequence and k = 0, 1, 2, …. Solution:

9 Contd… • Time Delay z[x(k D)]= z X(z) where D denotes time delay.
Proof: By definition Let m = k-D or k = m+D

10 Contd… Example Find the z-transform of the causal sequence Solution:

11 Contd…

12 Contd…

13 Contd…

14 Contd…

15 Inverse of the z-Transform

16 Contd…

17 Contd…

18 Contd…

19 Contd…

20

21 Contd…

22 Contd…

23 Contd…

24 Contd…

25 Contd…

26 Convolution Sum

27 Contd…

28 Contd…

29 Transfer Function

30 Contd…

31 Contd…


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