LECTURE 2
Z - TRANSFORM
Z– Transform plays an important role in discrete analysis and may be seen as discrete
analogue of Laplace transform
What is Z-Transform?
The Z-transform (ZT) is a mathematical tool which is used to convert the difference equations in
time domain into the algebraic equations in z-domain. (Just like Laplace transform is used to convert
differential equations in time domain into the algebraic equations in s-domain).
The Z-transform is a very useful tool in the analysis of a linear shift invariant (LSI) system. An LSI
discrete time system is represented by difference equations. To solve these difference equations
which are in time domain, they are converted first into algebraic equations in z-domain using the Z-
transform, then the algebraic equations are manipulated in z-domain and the result obtained is
converted back into time domain using the inverse Z-transform.
Consider a discrete-time signal x(t) below sampled every T sec.
The Laplace transform of x(t) is therefore
Note
from Laplace transform
1
2
Example
Find the z-transforms of
1)δ[n]
2) u[n]
3)cosβn u[n]
Solution
1) Remember that by definition:
2)
3
2) cosβn u[n]?
We already know that
Then,
Which leads to
Ex. Find the Z transform of
By definition the z transform is
4
Z - TRANSFORMTABLE
5
PROPERTIES OF Z TRANSFORM
6
7
8
INVERSE Z TRANSFORM
The inverse of the z-transform may be obtained by at least three methods:
1. Partial fraction expansion and look-up table (z-transform function must be rational).
2. Power series expansion.
3. Inversion Formula Method.
9
Find inverse z-transform – real unique poles
10
Find inverse z-transform – complex poles
11
Find inverse z-transform – repeat real poles
Find the inverse z-transform of:
Divide both sides by z and expand:
Use covering method to find k and a0:
12
To find a2, multiply both sides by z and let z→∞:
To find a1, let z = 0:
Therefore, we find
From the table:
13
14
SOLUTION OF DIFFERENCE EQUATIONS USING THE Z-TRANSFORM
But
which is the same as the shift theorem
In solving a difference equation the following steps are followed
1. Apply z-transform to the difference equation.
2. Substitute the initial conditions.
3. Solve the difference equation in z-transform domain.
4. Find the solution in time domain by applying the inverse z-transform.
Example
A digital signal processing (DSP) system is described by a difference equation
Applying the z-transform on both sides of the difference equation we have
15
ASSIGNMENT
1. A system is described by the difference equation
2. Using the long division method, solve to the third term
3.Using the partial fraction expansion method, find the inverse of the following z-transforms:
16
3. Given X[z], find x(n)
17
