Digital Signal Processing Reference
In-Depth Information
1
0.8
0.6
0.4
0.2
0
−0.2
−0.4
−0.6
−0.8
−1
0
5
10
15
20
n
0
.
81
z
−
2
), computed using partial fraction
expansion, as well as by filtering a unit impulse sequence using the
b
and
a
filter coefficients. Both results
are plotted as specified by the m-code in the text.
Figure 2.9:
The inverse
z
-transform of the
z
-transform 1/(1
+
is
b
0
H(z)
=
−
a
1
z
−
1
1
and therefore the
z
-transform of the system under excitation by the complex exponential is
b
0
1
Y(z)
=
H (z)X(z)
=
(
a
1
z
−
1
)(
e
jω
z
−
1
)
1
−
1
−
which can be converted into an equivalent time domain expression using a partial fraction expansion
which yields
(C
S
a
1
+
C
E
e
jωn
)u
y
[
n
]=
[
n
]
where
e
jω
)
a
1
e
−
jω
)
C
S
=
b
0
a
1
/(a
1
−
;
C
E
=
b
0
/(
1
−
