Image Processing Reference
In-Depth Information
A
if -
T
/ 2 / 2
t
T
pt
() =
(2.2)
0
otherwise
To obtain the Fourier transform, we substitute for
p
(
t
) in Equation 2.1.
p
(
t
) =
A
only for a
specified time so we choose the limits on the integral to be the start and end points of our
pulse (it is zero elsewhere) and set
p
(
t
) =
A
, its value in this time interval. The Fourier
transform of this pulse is the result of computing:
T
/2
-
jt
Fp
( =
-/2
)
Ae
dt
(2.3)
T
When we solve this we obtain an expression for
Fp
(ω ):
Ae
-/2
jT
-
Ae
jT/
2
Fp
() = -
(2.4)
j
) = (
e
j
θ
-
e
-
j
θ
)/2
j
, then the Fourier transform of
By simplification, using the relation sin (θ
the pulse is:
2
A
T
sin
if
0
2
(2.5)
Fp
() =
AT
if
0
This is a version of the
sinc
function, sinc(
x
) = sin(
x
)/
x
. The original pulse and its transform
are illustrated in Figure
2.3
. Equation 2.5 (as plotted in Figure
2.3
(a)) suggests that a pulse
is made up of a lot of low frequencies (the main body of the pulse) and a few higher
frequencies (which give us the edges of the pulse). (The range of frequencies is symmetrical
around zero frequency; negative frequency is a necessary mathematical abstraction.) The
plot of the Fourier transform is actually called the
spectrum
of the signal, which can be
considered akin with the spectrum of light.
Fp(
)
p(t)
t
(a) Pulse of amplitude A = 1
(b) Fourier transform
Figure 2.3
A pulse and its Fourier transform
So what actually is this Fourier transform? It tells us what frequencies make up a time
domain signal. The magnitude of the transform at a particular frequency is the amount of
that frequency in the original signal. If we collect together sinusoidal signals in amounts
