z- transform and its properties dr. wajiha shah. the z-transform given the causal sequence {x[1],...

Z- Transform and Its Properties

Dr. Wajiha Shah

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

0)()(

k

kzkXzX

Example

• Find the z-transform of the sequence

x = {1, 1, 3, 2, 0, 4}

Solution:

5432105

0)5()4()3()2()1()0()()(

zXzXzXzXzXzXzkXzX

k

k

5321 4231)( zzzzzX

Exercise

• Find the z-transform of the following sequences:

(a) {0, 1, 2, 4, 0, 0, . . .}

(b) {0, 0, 0, 1, 1, 1, 0, 0, 0, . . .}

(c) {0, 2−0.5, 1, 2−0.5, 0, 0, 0, . . .}

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

0 k 0

0 k 1)(

kkx

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

lsewhere 0

... 2, 0,1, k 1

e

ku

0)()(

k

kzkuzU

.....1 54321 zzzzz

11

11

z

z

z

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

lsewhere 0

... 2, 0,1, k k

e

akx

0)()(

k

kzkxzX

.....1 554433221 zazazazaaz

az

z

az

11

1

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:

1

464

12)(

z

z

z

zzX

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

0)()(

k

kzDkxDkxz

)(

)(

)(

)()(

0

0

0

)(

zXz

zmx z

zzmx

zmxmxz

-D

k

m-D

D

k

m

k

Dm

Contd…

• Example

Find the z-transform of the causal sequence

Solution:

lsewhere 0

... 3, 2, k 4

e

kx

1

4

1

4

14)(

22

zzzz

z

z

zzzX

Contd…

Inverse of the z-Transform

Contd…

Convolution Sum

Contd…

Transfer Function

Contd…

z- transform and its properties dr. wajiha shah. the z-transform given the causal sequence {x[1],...

Documents

variable z

wajiha shah slide

causal sequence

sequence x

time delay operator

following sequences