Digital Signal Processing Reference
In-Depth Information
5.3
FOURIER TRANSFORMS OF TIME FUNCTIONS
In Sections 5.1 and 5.2, we define the Fourier transform and its inverse. We list and
apply several important properties of the Fourier transform and, in the process, de-
rive the Fourier transforms of several time-domain signals. In this section, we derive
additional Fourier transform pairs for future reference.
Equation (5.9) gives the transform pair
e
jv
0
t
Î
f
"
2pd(v - v
0
).
If we allow
v
0
= 0,
we have
1
Î
f
"
2pd(v),
(5.29)
which, along with the linearity property, allows us to write the Fourier transform of
a dc signal of any magnitude:
K
Î
f
"
2pKd(v).
(5.30)
By comparing this transform pair with that of an impulse function in the time
domain,
d(t)
Î
f
"
1,
[eq(5.8)]
we see another illustration of the duality property (5.15).
The Fourier transform of the unit step function can be derived easily by a consider-
ation of the Fourier transform of the signum function developed in (5.21):
2
jv
.
sgn(t)
Î
f
"
[eq(5.21)]
As illustrated in Figure 5.14, the unit step function can be written in terms of the
signum function:
1
2
[1 + sgn(t)].
u(t) =
Combining the linearity property with (5.21) and (5.30) yields
