Digital Signal Processing Reference
In-Depth Information
d
(
t
)
1/
x
(
t
)
t
t
−5 3−
3
5
7
0
0
(a)
(b)
Fig. 3.3. Approximation of a CT signal
x
(
t
) by a linear combination of time-shifted unit impulse functions.
(a) Rectangular function δ
(
t
) used to approximate
x
(
t
). (b) CT signal
x
(
t
) and its approximation
x
(
t
)
shown with the staircase function.
3.2 Representation of signals using Dirac delta functions
In this section we will show that any arbitrary signal
x
(
t
) can be represented as
a linear combination of time-shifted impulse functions. To illustrate our result,
we define a new function
δ
(
t
) as follows:
1
/
0
<
t
<
δ
(
t
)
=
(3.20)
0
otherwise.
The waveform for
δ
(
t
) is shown in Fig. 3.3(a); it resembles that of a rectangular
pulse with width
and height 1
/
. To approximate
x
(
t
) as a linear combination
of
δ
(
t
), the time axis is divided into uniform intervals of duration
. Within a
time interval of duration
, say
k
<
t
<
(
k
+
1)
,
x
(
t
) is approximated by
a constant value
x
(
k
)
δ
(
t
−
k
)
. Following the aforementioned procedure
for the entire time axis,
x
(
t
) can be approximated as follows:
x
(
t
)
=+
x
(
−
k
)
δ
(
t
+
k
)
++
x
(
−
)
δ
(
t
+
)
+
x
(0)
δ
(
t
)
+
x
(
)
δ
(
t
−
)
+
+
x
(
k
)
δ
(
t
−
k
)
+ ,
(3.21)
which is shown as the staircase waveform in Fig. 3.3(b). For a given value of
t
,
say
t
=
m
, only one term (
k
=
m
) on the right-hand side of Eq. (3.21) is non-
zero. This is because only one of the shifted functions
δ
(
t
−
k
) corresponding
to
k
=
m
is non-zero. Therefore, a more compact representation for Eq. (3.21)
is obtained by using the following summation:
k
=−∞
x
(
k
)
δ
(
t
−
k
)
.
∞
x
(
t
)
=
(3.22)
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