Digital Signal Processing Reference
In-Depth Information
The term in brackets is similar to the defining equation for the Fourier transform
(5.1); the difference is the sign of the exponent. Substitution of
-v
for
v
in the
Fourier transform equation yields
q
f(t)e
jvt
dt.
F(-v) =
L
-
q
Substituting this result into the energy equation (5.5) yields
q
1
2p
L
E =
F(v)F(-v)dv.
-
q
For signals that are real valued (this includes all voltage and current waveforms
that can be produced by a physical circuit),
f(t)
F(-v) = F
*
(v),
F
*
(v)
where
is the complex conjugate of the function
F(v).
Hence,
q
q
1
2p
L
1
2p
L
F(v)F
*
(v)dv =
2
dv.
E =
ƒF(v) ƒ
-
q
-
q
The final, important result that we want to recognize is that
q
-
q
ƒ f(t) ƒ
q
-
q
ƒF(v) ƒ
1
2p
L
2
dt =
2
dv.
E =
L
(5.48)
The relationship described by equation (5.48) is known as
Parseval's theorem
.
It can be shown that (5.48) is valid for both real- and complex-valued signals.
Because the function is a real and even function of frequency, we can
rewrite the energy equation in the frequency spectrum as
2
ƒF(v) ƒ
q
-
q
ƒF(v) ƒ
q
1
2p
L
1
p
L
2
dv =
2
dv.
E =
ƒF(v) ƒ
0
The
energy spectral density
function of the signal
f(t)
is defined as
1
p
ƒF(v) ƒ
1
p
F(v)F(v)
*
2
f
(v) K
=
(5.49)
and describes the distribution of signal energy over the frequency spectrum. With the
energy density function thus defined, the energy equation (5.48) can be rewritten as
q
0
f
(v)dv.
E =
L
(5.50)
