Graphics Reference
In-Depth Information
Whatever function we synthesize, however, will be periodic of period 1, because
each term in the sum is periodic with that period. Thus, if
f
is a sum of sines and
cosines of different integer frequencies, we'll have
f
(
2
)=
f
(
1.5
1
1
2
)
.
−
0.5
We're using the term “frequency” here quite specifically: We say that
x
→
0
cos(
2
cos(
x
)
is a function of frequency 1. In the same way, some topics prefer to define the
Fourier transform on the interval
[
π
x
)
is a function of frequency 1. Some other texts say that
x
→
−
0.5
]
; depending on the
interval, this introduces a multiplicative constant in the definition of the inner
product. We're following the convention of Dym and McKean [DM85] so that
the interested reader may refer there for proofs, but there is no universal stan-
dard. Fortunately for us, we'll mostly be concerned with
qualitative
properties
of the Fourier transform, for which the interval of definition and multiplicative
constants are not important.
−
1, 1
]
,or
[
0, 1
]
,or
[
0, 2
π
−
1
−
1.5
−
0.5
0
0.5
Figure 18.31: A “square wave”
function.
More surprising, perhaps, is that
any
even continuous function
f
on the interval
H
that satisfies
f
(
2
)=
f
(
1
1
2
)
can be written as a sum of cosines of various integer
frequencies. Even
discontinuous
functions can be
almost
written as such a sum.
For instance, the square-wave function (see Figure 18.31) defined by
f
(
x
)=
1
−
0
−
1
1
1
4
−
4
≤
x
≤
(18.34)
−
0.5
0
0.5
−
1
otherwise
1
can be almost expressed by the infinite sum
cos(
2
0
f
(
x
)=
4
π
cos(
6
π
x
)
+
cos(
10
π
x
)
π
x
)
−
−...
(18.35)
3
5
−
1
∞
4
1
)
k
cos(
2
=
(
2
k
+
1
)
(
−
π
(
2
k
+
1
)
x
)
.
(18.36)
−
0.5
0
0.5
π
k
=
0
1
We say “almost expressed” because
f
(
1
±
4
)=
0—the average of the values of
f
to
1
1
the left and right of
4
)=
1.
The sequence in Equation 18.35 can be approximated by taking a finite num-
ber of terms; a few of those approximations are shown in Figure 18.32.
±
4
—rather than being equal to
f
(
±
0
−
1
As a more graphically oriented example, let's take one row of pixels from a
symmetric image like that shown in Figure 18.33. We've actually taken half a row
and flipped it over to get a perfectly symmetric line of 3,144 pixels, shown in
Figure 18.34.
If we write this function as a sum of cosines, the sum will have 3,144 terms,
which is hard to read. It starts out as
−
0.5
0
0.5
Figure 18.32: The square wave
approximated by
2
,
10
, and
100
terms. The slight overshoot in the
approximations is called
ringing.
f
(
x
)=
129.28
cos(
0
x
)+
5.67
cos(
2
π
x
)
−
2.35
cos(
4
π
x
)+
...
.
(18.37)
We can make an abstract
picture
of this summation by plotting the
coefficients
:
At 0 we plot 129.28, at 1 we plot 5.67, at 2 we plot
2.34, etc. The result is
shown in Figure 18.35, except that since the coefficient of
cos(
0
x
)
is actually
141.8, we've adjusted the
y
-axis so that you can see the other details, thus hiding
the large coefficient for the
cos(
0
x
)
term. (That coefficient is just the average of
all the pixel values.)
−
