Digital Signal Processing Reference
In-Depth Information
TABLE 2.1
Z-Transform Theorems
Property
f(n)
F(z)
Linearity
A x
1
(
n
) +
B x
2
(
n
), for constants
A
,
BA
1
(
z
) +
B X
2
(
z
)
Convolution
x
(
n
)
∗
h
(
n
)
X
(
z
)
H
(
z
)
Time Shift
x
(
n
-
n
0
)
X
(
z
)
z
-
n
.
Frequency Scaling
z
0
n
x
(
n
)
X
(
z
/
z
0
)
z
dX z
dz
()
Differentiation
n x
(
n
)
−
Similarly, using property (3) from Table 2.1 in Equation 2.3, we get the
transformed equation:
N
M
∑∑
−
k
−
k
(2.6)
Yz
()
az
=
Xz
()
bz
k
k
k
=
0
k
=
0
Comparing Equation 2.5 and Equation 2.6, we get the following
system
function
H
(
z
):
M
∑
−
k
bz
k
Yz
Xz
()
()
k
=
0
(2.7)
=
Hz
()
=
N
∑
−
k
az
k
k
=
0
Properties of the System Function
H
(
z
)
The system function or transfer function
H
(
z
) can be expressed in the fol-
lowing forms
• Polynomial form
This form can be obtained by expanding Equation 2.7 to yield:
−
1
−
2
−
M
bbz bz
+
+
+ …
bz
0
1
2
M
(2.8)
Hz
()
=
−
1
−
2
z
−
N
aaz az
+
+
+ …
a
0
1
2
N
• Pole-zero form and stability of the system
This form can be obtained by factorizing the numerator and denominator
polynomials in Equation 2.8 to yield:
(
)
(
)
…−
(
)
−
1
−
1
−
1
b
1
−
c z
1
−
c z
1
c
z
0
1
2
M
(2.9)
Hz
()
=
(
)
(
)
…−
(
)
−
1
−
1
−
1
a
1
−
d z
1
−
dz
1
d z
N
0
1
2
