Databases Reference
In-Depth Information
1.5
1
0.5
f
(
t
)
0
−
0.5
−1
−1.5
−2
−1.5
−1
−0.5
0
0.5
1
1.5
2
t
F I GU R E 15 . 4
The function
f
(
t
).
where we have used the substitution
x
=
t
/
a
. Thus,
f
t
a
2
2
=
a
f
(
t
)
If we want the sca
le
d function to have the same norm as the original function, we need to
multiply it by 1
/
√
a
. Transformations that change the size of an object but not its shape are
often referred to as
dilation
, which is a term we will use interchangeably with scaling in our
discussion.
Mathematically, we can represent the translation of a function to the right or left by an
amount
b
by replacing
t
by
t
b
. For example, if we want to translate the scaled
function shown in Figure
15.5
by one, we have
f
t
−
b
or
t
+
cos
1
1
2
−
1
(
2
π(
t
−
1
))
−
2
t
−
1
=
0
otherwise
0
.
5
cos
3
2
0 otherwise
The scaled and translated function is shown in Figure
15.6
.
We can use the translation and scaling operations to generate a family of functions from
a single function. Thus, given a
mother
function
1
(
2
π(
t
−
1
))
2
t
=
ψ(
t
)
, we can generate a whole family of
functions
{
ψ
a
,
b
(
t
)
}
with
t
1
√
a
ψ
−
b
ψ
a
,
b
(
t
)
=
(5)
a
and Fourier transforms
)
]
a
,
b
(ω)
=
F
[
ψ
a
,
b
(
(ω)
=
F
[
ψ(
t
t
)
]
(6)
