Digital Signal Processing Reference
In-Depth Information
The foregoing procedure can also be used for sinusoids with exponentially
varying amplitudes. Now,
e
jbt
+ e
-jbt
2
e
-(a- jb)t
+ e
-(a+ jb)t
2
e
-at
cos
bt = e
-at
B
R
=
;
thus,
1
2
1
s + a - jb
+
1
s + a + jb
l
[e
-at
cos
bt] =
B
R
s + a + jb + s + a - jb
2(s + a - jb)(s + a + jb)
=
s + a
(s + a)
2
=
+ b
2
.
Note the two transform pairs
s
cos
bt
Î
l
"
s
2
+ b
2
and
s + a
(s + a)
2
e
-at
cos
bt
Î
l
"
+ b
2
.
We see that for these two functions, the effect of multiplying a time function by the
exponential function
e
-at
is to replace
s
with
(s + a)
in the Laplace transform. We
now show that this property is general; that is,
q
q
l
[e
-at
f(t)]
=
L
e
-at
f(t)e
-st
dt =
L
f(t)e
-(s +a)t
dt
0
0
`
= F(s)
s;s +a
= F(s + a),
(7.20)
where
F(s) = L[f(t)]
and the notation indicates that
s
is replaced with
Using the transform pair for sin
bt
and this theorem, we see that
s ; (s + a)
(s + a).
b
sin bt
Î
l
"
+ b
2
.
s
2
Therefore,
b
(s + a)
2
e
-at
sin bt
Î
l
"
+ b
2
.
The last transform is that of the product of two time functions. Note that
l
[e
-at
sin bt] Z
l
[e
-at
]
l
[sin bt].
