Digital Signal Processing Reference
In-Depth Information
version of the signal The truncation can be achieved by multiplying the signal
as shown in Figure 5.34(a) by a rectangular pulse having unity amplitude and
duration
T,
as shown in Figure 5.34(b). The truncated signal
f(t).
f(t)
f
T
(t) = f(t) rect(t/T)
has finite energy. This signal meets the other Dirichlet conditions and, therefore,
has a Fourier transform:
f
T
(t)
Î
f
"
F
T
(v).
In working with power signals, it is often desirable to know how the total
power of the signal is distributed in the frequency spectrum. This can be determined
by the development of a
power spectral density
function similar to the energy spec-
tral density function considered earlier. We begin by writing the power equation in
terms of the truncated signal:
q
-
q
ƒ f
T
(t) ƒ
1
2
dt.
P =
lim
T:
q
T
L
Note that the limits of integration have been changed from (5.51). This is justified,
because has zero magnitude for
Because has finite energy, the integral term can be recognized as the
total energy contained in the truncated signal:
f
T
(t)
ƒ t ƒ 7 T/2.
f
T
(t)
q
2
dt.
E =
L
ƒ f
T
(t) ƒ
-
q
By Parseval's theorem (5.48), the energy can be expressed in terms of
to get
f
T
(t)
q
-
q
ƒ
f
T
(t)
ƒ
q
-
q
ƒ
F
T
(v)
ƒ
1
2p
L
2
dt =
2
dv.
E =
L
The frequency-domain expression of the energy in the signal can be substituted into
the power equation to yield
q
-
q
ƒF
T
(v) ƒ
1
2pT
L
2
dv.
P =
lim
T:
q
(5.52)
As the duration of the rectangular pulse increases, it can be seen that the en-
ergy of the signal will also increase. In the limit, as
T
approaches infinity, the energy
will become infinite also. For the average power of the signal to remain finite, the
