Biomedical Engineering Reference
In-Depth Information
15.2 SPECTRAL DENSITY ESTIMATION
A basic question one might ask is whether the Fourier Series or transform is applied
to a voltage signal, how did the unit become power? To answer this question, let us
consider the function
v
(
t
) defined over the interval (
). If
v
(
t
) is periodic or
almost periodic, it can be represented by the Fourier series as in (15.1).
−∞
,
+∞
∞
v
(
t
)
=
+
(
a
n
cos
nwt
+
b
n
sin
nwt
)
(15.1)
a
0
1
where
f
. The complex
Fourier coefficients
a
and
b
are then represented by equations in (15.2).
ω
=
2
π
f
,
f
represents the frequency, and the period is
T
=
1
/
T
1
2
T
a
0
=
v
(
t
)
dt
0
T
1
2
T
(15.2)
a
n
=
v
(
t
) cos(
nw
0
t
)
dt
0
T
1
2
T
b
n
=
v
(
t
) sin(
nw
0
t
)
dt
0
The magnitude of the complex coefficients are obtained from (15.3), which has units of
“Voltage/Hertz.”
(
a
n
+
c
n
=
b
n
)(
a
n
−
b
n
)
(15.3)
The Fourier coefficients are not in units of energy or power. By using Parseval's Theorem,
the power spectrum is obtained from (15.4).
c
n
=
a
n
+
b
n
(15.4)
The squared magnitude of the complex coefficients from (15.4) has units of “Voltage
Squared/Hertz,” “Average energy or power per hertz.”
Parseval's Theorem may be stated as the following equation form (15.5).
+∞
+∞
v
(
t
)
X
(
)
1
2
2
2
dt
=
ω
d
ω
(15.5)
π
−∞
−∞
where
v
(
t
) and
X
(
ω
) are the Fourier Transform pair.
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