Digital Signal Processing Reference
In-Depth Information
TABLE 2.3 Properties of
z
Transforms
Operation
x(n)u(n)
X(z)
Addition
x
1
(n)
+
x
2
(n)
X
1
(z)
+
X
2
(z)
Scalar multiplication
Kx(n)
KX(z)
z
−
1
X(z)
Delay
x(n
−
1
)u(n
−
1
)
+
x(
−
1
)
z
−
2
X(z)
z
−
1
x(
x(n
−
2
)u(n
−
2
)
+
−
1
)
+
x(
−
2
)
z
−
3
X(z)
z
−
2
x(
x(n
−
3
)u(n
−
3
)
+
−
1
)
+
z
−
1
x(
−
2
)
+
x(
−
3
)
+
m
−
1
z
−
m
X(z)
m)z
−
n
x(n
−
m)u(n
−
m)
n
=
0
x(n
−
X(z
−
1
)
Time reversal
x(
−
n)u(
−
n)
z
dX(z)
dz
Multiplication by
n
nx(n)
−
Multiplication by
r
n
r
n
x(n)
X(r
−
1
z)
Time convolution
x
1
(n)
∗
x
2
(n)
X
1
(z)X
2
(z)
(
1
/
2
πj)
C
X
1
(z)X
2
z
u
u
−
1
du
Modulation
x
1
(n)x
2
(n)
Initial value
x(
0
)
lim
z
→∞
X(z)
Final value
lim
N
→∞
x(N)
lim
z
→
1
(z
−
1
)X(z)
, when poles
of
(z
−
1
)X(z)
are inside the
unit circle
2.8 STABILITY
It is essential that every system designed by an engineer be extremely stable in
practical use. Hence we must always analyze the stability of the system under
various operating conditions and environments. The basic requirement is that
when it is disturbed by a small input, the response of the system will eventually
attain a zero or a constant value or at most be bounded within a finite limit.
There are definitions for various kinds of stability, but the definition used most
often is that the output asymptotically approaches a constant or bounded value
when a bounded input is applied. This is known as the
bounded input-bounded
output
(BIBO) stability condition. It satisfies the condition when the unit impulse
response
h(n)
satisfies the condition
n
=
0
|
h(n)
|
<M<
∞
. To prove this result,
let us assume that
H(z)
=
z/z
−
γ
i
,where
γ
i
is the pole of
H(z)
. The unit
impulse response is
γ
i
for
n
≥
0:
∞
∞
n
0
|
h(n)
| =
0
|
γ
i
|
n
=
n
=
1
1
− |
=
|
γ
i
|
<
1
when
γ
i
|
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