Digital Signal Processing Reference
In-Depth Information
EXAMPLE 5.5
Fourier transform of a time-delayed sinusoidal signal
Consider the time-shifted cosine wave of frequency
v = 200p
and a delay of 1.25 ms in its
propagation:
x(t) = 10
cos
[200p(t - 1.25 * 10
-3
)].
This signal can be viewed as a phase-shifted cosine wave where the amount of phase shift is
radians:
p/4
x(t) = 10
cos
(200pt - p/4).
Using the linearity and time-shifting property, we find the Fourier transform of this delayed
cosine wave:
e
-j.00125v
5
6
5
6
f
x(t)
= X(v) = 10f
cos
(200pt)
= 10p[d(v - 200p) + d(v + 200p)]e
-j.00125v
= 10p[d(v - 200p)e
-jp/4
+ d(v + 200p)e
jp/4
].
e
-j.00125v
,
The rotating phasor, is reduced to the two fixed phasors shown in the final equation,
because the frequency spectrum has zero magnitude except at
Recall, from Table 2.3, that
v = 200p
and
v =-200p.
F(v)d(v - v
0
) = F(v
0
)d(v - v
0
).
Notice that the phase shift of radians, which is the result of the 1.25-ms delay in the
propagation of the cosine wave, is shown explicitly in the frequency spectrum.
-p/4
■
The properties of time scaling and time shifting can be combined into a more general
property of time transformation. The concept of the time-transformation property
was introduced in Section 2.2, and for the Fourier series in Section 4.6.
Let
t = at - t
0
,
where
a
is a scaling factor and
t
0
is a time shift. Application of the time-scaling
property (5.12) gives
1
ƒ a ƒ
v
a
f(at)
Î
f
"
¢
≤
F
.
Application of the time-shift property (5.13) to this time-scaled function gives
us the time-transformation property:
1
ƒ a ƒ
v
a
f(at - t
0
)
Î
f
"
¢
≤
e
-jt
0
(v/a)
.
F
(5.14)
