Page 265 - Numerical Analysis Using MATLAB and Excel

Chapter 6 Fourier, Taylor, and Maclaurin Series

6.11 Summary

• Any periodic waveform ft() can be expressed as

1 ∞
-
ft() = --a + ∑ ( a cos nωt + b sin nωt )
2 0 n n
n = 1
where the first term a ⁄ 2 is a constant, and represents the DC (average) component of f t() .
0
a
The terms with the coefficients and b 1 together, represent the fundamental frequency com-
1
ω
ponent . Likewise, the terms with the coefficients a 2 and b 2 together, represent the second
harmonic component 2ω , and so on. The coefficients a 0 , a n , and b n are found from the fol-
lowing relations:

1 1 2π
--a = ------ ∫ ft() t
-
d
2 0 2π 0
1 2π
d
a = --- ∫ 0 ft()cos nt t
π
n
1 2π
d
b = --- ∫ 0 ft()sin nt t
n
π
• If a waveform has odd symmetry, that is, if it is an odd function, the series will consist of sine
terms only. Odd functions are those for which f –()– t = ft() .

• If a waveform has even symmetry, that is, if it is an even function, the series will consist of
cosine terms only, and a 0 may or may not be zero. Even functions are those for which
f –() = ft()
t
• A periodic waveform with period , has half−wave symmetry if
T
f – ( t + T 2 = ft()
)
⁄
that is, the shape of the negative half−cycle of the waveform is the same as that of the positive
half−cycle, but inverted. If a waveform has half−wave symmetry only odd (odd cosine and odd
sine) harmonics will be present. In other words, all even (even cosine and even sine) harmon-
ics will be zero.

• The trigonometric Fourier series for the square waveform with odd symmetry is

4A ⎛ 1 1 ⎞ 4A 1
-
-
ft() = ------- sin ωt + -- sin 3ωt + -- sin 5ωt + … = ------- ∑ -- sin nωt
-
π ⎝ 3 5 ⎠ π n
n = odd

6−48 Numerical Analysis Using MATLAB® and Excel®, Third Edition
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