Digital Signal Processing Reference
In-Depth Information
•
Â
[
]
=
(
)
(
)
ZT x n
-
1
x n
-
1
z
-
n
n
=
0
=
()
+
()
()
()
x
1
x
0
z
-
1
+
x
1
z
-
2
+
x
2
z
-
3
+◊◊◊
[
]
=
()
+
()
+
()
()
x
1
z
-
1
x
0
x
1
z
-
1
+
x
2
z
-
2
+◊◊◊
=
()
+
()
(4.16)
x
1
z
-
1
Xz
1) represents the initial condition associated with a
first-order difference equation. Similarly, the
ZT
of
x
(
n
where we used (4.15), and
x
(
-
-
2), equivalent to a second
derivative
d
2
x
(
t
)/
dt
2
is
•
Â
[
]
=
(
)
(
)
ZTxn
-
2
xn
-
2
z
-
n
n
=
0
=
()
+
()
+
()
()
x
2
x
1
z
-
1
x
0
z
-
2
+
x
1
z
-
3
+◊◊◊
[
]
=
()
+
()
+
()
+
()
-
1
-
2
-
1
x
2
x
1
z
z
x
0
x
1
z
+◊◊◊
(4.17)
=
()
+
()
+
(()
x
2
x
1
z
-
1
z
-
2
Xz
1) represent the two initial conditions required to solve a
second-order difference equation. In general,
where
x
(
-
2) and
x
(
-
k
Â
1
[
(
)
]
=
(
)
()
-
k
m
k
(4.18)
ZT x n
-
k
z
x
-
m z
+
z X z
m
If the initial conditions are all zero, then
x
(
-
m
)
=
0 for
m
=
1,2,...,
k
, and (4.18)
reduces to
[
(
)
]
=
()
(4.19)
ZT x n
-
k
z
-
k
X z
4.2 DISCRETE SIGNALS
A discrete signal
x
(
n
) can be expressed as
•
Â
()
=
()
(
)
xn
xm
d
n m
-
(4.20)
=-
•
m
where
d
(
n
-
m
) is the impulse sequence
d
(
n
) delayed by
m
, which is equal to 1 for
n
m
and is 0 otherwise. It consists of a sequence of values
x
(1),
x
(2),...,where
n
is the time, and each sample value of the sequence is taken one sample time apart,
determined by the sampling interval or sampling period
T
=
1/
F
s
.
The signals and systems that we deal with in this topic are linear and time-
invariant, where both superposition and shift invariance apply. Let an input signal
x
(
n
) yield an output response
y
(
n
), or
x
(
n
)
=
Æ
y
(
n
). If
a
1
x
1
(
n
)
Æ
a
1
y
1
(
n
) and
a
2
x
2
(
n
)
Æ
a
2
y
2
(
n
), then
a
1
x
1
(
n
)
+
a
2
x
2
(
n
)
Æ
a
1
y
1
(
n
)
+
a
2
y
2
(
n
), where
a
1
and
a
2
are constants.
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