Graphics Reference
In-Depth Information
synthetic image, which has sharp edges (e.g., a checkerboard), you might expect
a1
falloff, as we saw in the case of a square wave. In general, we'll plot signals
in blue and their transforms in magenta, although on a few occasions we'll plot
several signals on one axis and their transforms on another axis, using the same
color for each signal and its transform. We'll plot discrete signals—ones whose
domain is
Z
—using stemplots.
/ω
b
(
x
)=
1
−
≤
≤
0. 5
x
0. 5
(18.55)
0
otherwise,
be a box function defined on the real line. Because it's an even function and it's
real-valued, its Fourier transform will be even and real-valued. We can evaluate
F
(
b
)(
ω
)
directly from the definition.
)=
∞
−∞
F
(
b
)(
ω
b
(
x
)
e
ω
(
x
)
dx
(18.56)
=
1
2
1
2
e
ω
(
x
)
dx
because
b
(
x
)=
0for
|
x
| >
(18.57)
1
2
−
=
i
1
2
1
2
cos(
2
πω
x
)
dx
−
sin(
2
πω
x
)
dx
(18.58)
1
2
1
2
−
−
=
1
2
cos(
2
πω
x
)
dx
because
sin
is odd
(18.59)
1
2
−
1
2
=
sin(
2
πω
x
)
(18.60)
2
πω
−
1
2
=
sin(
πω
)
.
(18.61)
πω
The calculation above works for all
ω
=
0; for
ω
=
0, we have
F
(
b
)(
0
)=
1,
which you should verify by writing out the integral.
This computation is shown pictorially in Figure 18.39, which is adapted from
Bracewell [Bra99], an excellent reference for those interested in practical signal
processing.
Inline Exercise 18.6:
Repeat the preceding computation to compute the
Fourier transform of a box of width
a
, that is, a function that's one on the inter-
val
[
−
a
/
2,
a
/
2
]
and zero elsewhere. Hint: Substitute
u
=
x
/
a
in the integral to
avoid doing any further work at all.
This is the only Fourier transform of a function on the real line that we'll
actually compute directly like this. The resultant function is so important that it
gets its own name:
sinc
(
x
)=
sin(
π
x
)
x
=
0
π
x
x
=
0
.
(18.62)
1
