Biomedical Engineering Reference
In-Depth Information
1.
The first operation is to determine the values of the coefficients
a
0
,
a
1
,...
a
n
,
and
b
1
,...
b
n
.
2.
The second operation is to decide how many terms to include in a truncated series
such that the partial sum will represent the function within allowable error. It is
not possible to calculate an infinite number of coefficients.
12.2 EVALUATION OF THE FOURIER COEFFICIENTS
In evaluating the coefficients, the limits of integration must be set. Since the Fourier
Basis functions are orthogonal over the interval
t
0
to (
t
0
+
T
) for any
t
0
, we often use
0or
t
0
=
−
T
the value
t
0
=
/
2
with the understanding that any period may be used as the
period of integration. Thus, we replace the interval of the integral from,
t
0
to (
t
0
+
T
),
with the limits of integration from, 0 to
T
.
The next step would be to calculate the
a
0
term, which is simply the average
value of
f
(
t
) over a period;
a
0
is often referred to as the DC value of a sinusoid over
N
complete cycles in the period. Next the complex coefficients,
a
n
and
b
n
are evaluated
with the expressions in (12.2), (12.3), and (12.4). To obtain the
a
n
terms, which
constitute the “real-part” of the complex Fourier coefficient, the signal or function is
multiplied by the corresponding cosine term, integrated, and then normalized by the
fundamental period,
T
, as shown in (12.3). In a similar manner, to obtain the
b
n
terms,
which constitute the “imaginary-part” of the complex Fourier coefficient, the signal or
function is multiplied by the corresponding sine term, integrated, and then normalized
by the fundamental period,
T
, as shown in (12.4).
,
T
/
+
t
0
T
T
1
T
1
T
1
T
2
DC Value of
f
(
t
):
a
0
=
c
0
=
f
(
t
)
dt
=
f
(
t
)
dt
=
f
(
t
)
dt
−
T
/
2
t
0
0
(12.2)
T
/
T
2
T
2
Real Term:
a
n
=
f
(
t
) cos
n
ω
0
tdt
or
(12.3)
0
−
T
/
2
T
2
T
Imaginary term:
b
n
=
f
(
t
) sin
n
ω
0
tdt
(12.4)
0
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