Digital Signal Processing Reference
In-Depth Information
EXAMPLE 5.6
The time-transformation property of the Fourier transform
Consider the rectangular pulse shown in Figure 5.7(a). We will find the Fourier transform of
this function by using a known Fourier transform and the time-transformation property. Given
the rectangular pulse of Figure 5.7(b), we easily determine the Fourier transform from (5.4) to be
F(v) = sinc(v/2).
The magnitude and phase
F(v)
are plotted separately in Figure 5.7(c). From Figure 5.7(a)
and (b), we write
g(t) = 3 rect [(t - 4)/2] = 3f(0.5t - 2).
Then, using the time-transformation property (5.14) with
a = 0.5
and
t
0
= 2
and the linearity
property to account for the magnitude scaling, we can write
G(v) = 6
sinc(v)e
-j4v
.
The magnitude and phase plots of
G(v)
are shown in Figure 5.7(d) for comparison with the
plots of
F(v).
Note the effect of the time shift on the phase of
G(v).
The step changes of
p
ra-
dians in the phase occur because of the changes in the algebraic sign of the sinc function.
■
For signals such as in Example 5.6 that have a phase angle that changes
continuously with frequency, it is usually desirable to plot the magnitude and phase
of the Fourier transform separately. These plots simplify the sketch and display the
information in a way that is easier to interpret. These separate plots are called the
magnitude spectrum
and
phase spectrum
, respectively, of the signal.
G(v)
The symmetry of the Fourier transformation and its inverse in the variables
t
and
can be seen by comparing equations (5.1) and (5.2):
v
q
f(t)e
-jvt
dt;
[eq.(5.1)]
f
5
f(t)
6
= F(v) =
L
-
q
q
1
2p
L
f
-1
F(v)e
jvt
dv.
[eq.(5.2)]
5
F(v)
6
= f(t) =
-
q
The duality property, which is sometimes known as the
symmetry
property, is stated as
F(t)
Î
f
"
2pf(-v)
f(t)
Î
f
"
F(v).
when
(5.15)
This property states that if the mathematical function
f(t)
has the Fourier transform
F(v)
and a function of time exists such that
`
F(t) = F(v)
,
v= t
`
then
f
5
F(t)
6
= 2pf(-v),
where f(-v) = f(t)
.
t =-v
