Digital Signal Processing Reference
In-Depth Information
In Section 10.1, we used a linear, constant-coefficient difference equation to
model an LTID system. A second model is based on the impulse response
h
[
k
]
of a system. This alternative representation leads to a different approach for
analyzing LTID systems. Section 10.2 presents this alternative approach.
10.2 Representatio
n of sequences using Dirac delta functions
In this section, we show that any arbitrary sequence
x
[
k
] may be represented as
a linear combination of time-shifted, DT impulse functions. Recall that a DT
impulse function is defined in Eq. (1.51) as follows:
1
k
=
0
δ
[
k
]
=
(10.3)
0
k
=
0
.
We are interested in representing any DT sequence
x
[
k
] as a linear combina-
tion of shifted impulse functions,
δ
[
k
−
m
], for
−∞ <
m
< ∞
. We illustrate
the procedure using the arbitrary function
x
[
k
] shown in Fig. 10.2(a). Figures
10.2(b)-(f) represent
x
[
k
] as a linear combination of a series of simple functions
x
m
[
k
], for
−∞ <
m
< ∞
. Since
x
m
[
k
] is non-zero only at one location (
k
=
m
),
it represents a scaled and time-shifted impulse function. In other words,
x
m
[
k
]
=
x
[
m
]
δ
[
k
−
m
]
.
(10.4)
Fig. 10.2. Representation of a
DT sequence as a linear
combination of time-shifted
impulse functions. (a) Arbitrary
sequence
x
[
k
]; (b)-(f) its
decomposition using DT impulse
functions.
In terms of
x
m
[
k
], the DT sequence
x
[
k
] is, therefore, represented by
x
[
k
]
=+
x
−
2
[
k
]
+
x
−
1
[
k
]
+
x
0
[
k
]
+
x
1
[
k
]
+
x
2
[
k
]
+
=+
x
[
−
2]
δ
[
k
+
2]
+
x
[
−
1]
δ
[
k
+
1]
+
x
[0]
δ
[
k
]
+
x
[1]
δ
[
k
−
1]
+
x
[2]
δ
[
k
−
2]
+ ,
x
[
k
]
x
−2
[
k
] =
x
[ − 2]
d
[
k
+ 2]
x
−1
[
k
] =
x
[ − 1]
d
[
k
+ 1]
1
k
k
k
−5 −4 −3 −2 −1
2
45
−5 −4 −3 −2 −1
0
2
3
−
−5 −4 −3 −2 −1
0
1
2
3
4
5
(a)
(b)
(c)
x
0
[
k
] =
x
[0]
d
[
k
]
x
1
[
k
] =
x
[1]
d
[
k
− 1]
x
2
[
k
] =
x
[2]
d
[
k
− 2]
1
k
k
k
−5 −4 −3 −2 −1
−5 −4 −3 −2 −1
2
−5 −4 −3 −2 −1
2
(d)
(e)
(f)
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