Biomedical Engineering Reference
In-Depth Information
CHAPTER
12
Fourier Series:
Trigonometric
The objective of this chapter is for the reader to gain a better understanding of the
Basic Fourier Trigonometric Series Transformation and waveform synthesis. If we are
interested in studying time-domain responses in networks subjected to periodic inputs
in terms of their frequency content, the Fourier Series can be used to represent arbitrary
periodic functions as an infinite series of sinusoids of harmonically (not harmoniously)
related frequencies as shown in (12.1).
A signal
f
(
t
) is periodic with period,
T
,if
f
(
t
)
T
) for all
t
, then the
Fourier Trigonometric Series representation of the periodic function may be expressed as
(12.1). It should be noted that both “cosine” and “sine” trigonometric terms are necessary
and that an infinite number of terms may be necessary.
=
f
(
t
+
f
(
t
)
=
a
0
+
a
1
cos
ω
0
t
+
a
2
cos 2
ω
0
t
+···
a
n
cos
n
ω
0
t
+
b
1
sin
ω
0
t
(12.1)
+
b
2
sin 2
ω
0
t
+···
b
n
sin
n
ω
0
t
+···
n
=
1
,
2
,
3
,...
∞
2
n
T
1
where
ω
0
=
;
T
=
f
0
; and
n
is the
n
th harmonic of the fundamental frequency,
ω
0
.
12.1 FOURIER ANALYSIS
Recall the chapter on basis functions. In taking the integral transform of a signal, the
integral of the product of the signal with any basis function transforms the work from
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