Digital Signal Processing Reference
In-Depth Information
+
0
.
04
z
−
3
X(z)
2
z
−
2
=
1
+
0
.
3
z
−
1
+
0
.
42
z
−
2
X(z)
2
z
=
z
3
+
0
.
3
z
2
+
0
.
42
z
+
0
.
04
When we substitute the
z
transform of the given input above and find the inverse
z
transform, we get the output
y
2
(n)
.
In this example, the natural frequencies of the system are computed as the
zeros of the system determinant
0
.
2
z
−
1
0
.
4
z
−
2
)
(
1
+
+
0
−
2
z
−
1
(
1
+
0
.
1
z
−
1
)
0
.
21
z
−
1
0
.
42
z
−
2
0
.
04
z
−
3
z
−
3
[
z
3
0
.
3
z
2
which is 1
0
.
04]. It
is obvious that the zeros of this determinant are the same as the poles of the
transfer function
+
+
+
=
+
+
0
.
42
z
+
Y
2
(z)
X(z)
2
z
H(z)
=
=
z
3
+
0
.
3
z
2
+
0
.
42
z
+
0
.
04
As long as these poles of
H(z)
are not canceled by its zeros, that is, if there
are no common factors between its numerator and the denominator, its inverse
z
transform will display all three natural frequencies. If some poles of the transfer
function are canceled by its zeros, and it is therefore given in its reduced form, we
may not be able to identify all the natural frequencies of the system. Therefore
the only way to find all the natural frequencies of the system is to look for
the zeros of the system determinant or the characteristic polynomial. We see
that in this example, the system response does contain three terms in its natural
response, corresponding to the three natural frequencies of the system. But if
and when there is a cancellation of its poles by some zeros, the natural response
components corresponding to the canceled poles will not be present in the zero
state response
h(n)
. So we repeat that in some cases, the poles of the transfer
function may not display all the natural frequencies of the system.
Note that the inverse
z
transform of
Y
2
(z)
is computed from
Y
2
(z)
H (z)X(z)
when the initial states are zero. Therefore the response
y
2
(n)
is just the zero state
response of the system, for the given input
x(n)
.
=
2.4.1 Transient Response and Steady-State Response
The total response can also be expressed as the sum of its transient response and
steady-state response. But there is again a misconception that the natural response
of a system is the same as the transient response, and hence an explanation is
given below to clarify this misconception.
The transient response is the component of the total response, which approaches
zero as
n
→∞
, whereas the steady-state response is the part that is left as the
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