The z-transform is a useful tool for the
frequency analysis of discrete-time signals
and systems.
)
This type of z-transform
This type of z-transform
is defined as
is defined as
Where z is a complex variable.
The
main disadvantages is its
We can see that X(z) is
convergence conditions . Its the
concerned with the history of x(n)
mathematical conditions for the
prior to n=0
transform to exist and converge
Example : Find the Z-transform of the finite duration
signal
:
The properties of Z-transform
• Linearity
• Shifting Property
• Convolution Theorem
Homework 2
Inverse z-transform
In which x(z) as a
power series of
Example 11-4: Find and sketch the signal
corresponding to the z-transform function
Solution:
0.8
0.7
0.6
x(n)
0.5
0.4
0.3
0.2
0.1
0
0
0.5
1
1.5
2
n
2.5
3
3.5
4
The z-transform used to described a real signal or any LTI
system , it is always a rational function of the frequency
variable z . it can be written as the ratio of numerator
and denominator polynomials in z
• The constants z1 ,z2 ,z3 … are called the zero
of X(z), because they are the values of z for
which X(z) is zero. Conversely p1,p2, … are
known as the poles of X(z), giving values of z
for which X(z) tends to infinity.
• A very useful representation of a z-transform
is obtained by
in
the
. Note
that a zero is shown as an open circular
symbol, and a pole as a cross.
Pole-Zero Map
1
0.8
0.6
Imaginary Axis
0.4
0.2
0
-0.2
-0.4
-0.6
-0.8
-1
-1.5
-1
-0.5
0
Real Axis
0.5
1