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Alternate Forms of the Trigonometric Fourier Series

2A 4A ∞ 1
ft() = ------- – ------- ∑ ------------------- cos nωt (6.65)
π π ( n – 1 )
2
,,
,
n = 246 …
This series of (6.65) shows that there is no component of the fundamental frequency. This is
because we chose the period to be from π and +π . Generally, the period is defined as the short-
–
est period of repetition. In any waveform where the period is chosen appropriately, it is very
unlikely that a Fourier series will consist of even harmonic terms only.

6.5 Alternate Forms of the Trigonometric Fourier Series

We recall that the trigonometric Fourier series is expressed as

1
-
ft() = --a + a cos ωt + a cos 2ωt + a cos 3ωt + a cos 4ωt + …
2 0 1 2 3 4 (6.66)
+ b sin ωt + b sin 2ωt + b sin 3ωt + b sin 4ωt + …
2
4
1
3
If a given waveform does not have any kind of symmetry, it may be advantageous of using the
alternate form of the trigonometric Fourier series where the cosine and sine terms of the same fre-
quency are grouped together, and the sum is combined to a single term, either cosine or sine.
However, we still need to compute the a n and b n coefficients separately.

We use the triangle shown in Figure 6.20 for the derivation of the alternate forms.

c = a + b
ϕ n n n
c n n b n cos θ = ---------------------- = a n sin θ = ---------------------- = b n
b
a
n
n
-----
------
θ n n a + b n c n n a + b n c n
n
n
a n b a
n
n
cos θ = sin ϕ θ = atan ------ ϕ = atan ------
n n n a n n b n
Figure 6.20. Derivation of the alternate form of the trigonometric Fourier series
,,,
We assume ω = 1 , and for n = 1 2 3 … , we rewrite (6.66) as

Numerical Analysis Using MATLAB® and Excel®, Third Edition 6−25
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