Page 240 - Numerical Analysis Using MATLAB and Excel

Waveforms in Trigonometric Form of Fourier Series

By inspection, the average is a non−zero value. We choose the period of the input sinusoid so that
the output will be expressed in terms of the fundamental frequency. We also choose the limits of
integration as π and +π , we observe that the waveform has even symmetry.
–
Therefore, we expect only cosine terms to be present. The a n coefficients are found from

1
d
a = --- ∫ 2π ft()cos nt t
n
π
where for this example, 0
1 π 2A π
d
a = --- ∫ – π Asin tcos nt t = ------- ∫ 0 sin tcos nt t (6.55)
d
π
n
π
and from tables of integrals,
)
(
)
cos
m +
( cos
–
mn x
∫ ( sin mx cos nx x = ------------------------------ – ------------------------------- m ≠( n n x 2 n ) 2
)
(
)
d
(
)
(
2m +
)
m
2n –
Since
)
)
–
y
( cos xy = ( cos y – x = cos xcos + sin xsiny
we express (6.55) as
2A 1⎧ ( cos n – 1 t ( cos n + 1 t π ⎫
)
)
a = ------- -- - ⎩ --------------------------- – ---------------------------- 0 ⎬ ⎭
⋅ ⎨
n
π
2
n +
1
1
n –
A⎧ ( cos n1 π – ) ( cos n + 1 π ) 1 1 ⎫
= ---- ⎨ ----------------------------- – ----------------------------- – ------------ – ------------ ⎬ (6.56)
π ⎩ n1 n + 1 n – 1 n + 1 ⎭
–
)
A 1 – cos n + ) π( π cos ( nππ – 1
–
= ---- --------------------------------------- + --------------------------------------
π n + 1 n – 1
To simplify the last expression in (6.56), we make use of the trigonometric identities
π
cos n + ) π( π = n cos – sin nπsinπ = – cos nπ
π cos
and
π
cos n – ) π( π = n cos + sin nπsinπ = – cos nπ
π cos
Then, (6.56) simplifies to
)
)
–
2
A 1 + cos nπ 1 + cos nπ A – + ( n1 cos nπ – ( n + 1 cos nπ
a = ---- ------------------------- – ------------------------- = ---- -------------------------------------------------------------------------------------
n
π
π
1
n +
2
n –
1
n –
1
(
– 2A cos nπ + 1 ) (6.57)
= ----------------------------------------- n ≠ 1
π n –( 2 1 )
Now, we can evaluate all the a n coefficients, except , from (6.57). First, we will evaluate to
a
a
1
0
Numerical Analysis Using MATLAB® and Excel®, Third Edition 6−23
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