Digital Signal Processing Reference
In-Depth Information
In this text, we will use the definition in Eq. (1.36) for the sinc function. The
DT sinc function is defined as follows:
sinc(
Ω
0
k
)
=
sin(
π
Ω
0
k
)
π
Ω
0
k
,
(1.37)
which is plotted in Fig. 1.12(l).
1.2.7 CT exponential function
A CT exponential function, with complex frequency
s
= σ
+
j
ω
0
, is repre-
sented by
x
(
t
)
=
e
st
=
e
(
σ +
j
ω
0
)
t
=
e
σ
t
(cos
ω
0
t
+
j sin
ω
0
t
)
.
(1.38)
The CT exponential function is, therefore, a complex-valued function with the
following real and imaginary components:
Re
e
st
=
e
σ
t
cos
ω
0
t
;
real component
Im
e
st
=
e
σ
t
sin
ω
0
t
.
imaginary component
Depending upon the presence or absence of the real and imaginary components,
there are two special cases of the complex exponential function.
Case 1
Imaginary component is zero (
ω
0
=
0)
Assuming that the imaginary component
ω
of the complex frequency
s
is zero,
the exponential function takes the following form:
x
(
t
)
=
e
σ
t
,
which is referred to as a real-valued exponential function. Figure 1.13 shows the
real-valued exponential functions for different values of
σ
. When the value of
σ
is negative (
σ<
0) then the exponential function decays with increasing time
t
.
x
(
t
)= e
s
t
,
s
< 0
x
(
t
)= e
s
t
,
s
= 0
1
1
t
t
0
0
(a)
(b)
Fig. 1.13. Special cases of
real-valued CT exponential
function
x
(
t
) = exp(σ
t
).
(a) Decaying exponential with
σ<0. (b) Constant with
σ = 0. (c) Rising exponential
with σ>0.
x
(
t
)= e
s
t
,
s
> 0
1
t
0
(c)
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