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The Exponential Form of the Fourier Series

3 1
)
ft() = -- + --- [22cos ( ωt – 45° + cos ( 2ωt – 90° )
-
2 π
(6.84)
22 1
)
)
+ ---------- cos ( 3ωt – 135° + -- cos ( 4ωt – 90° + … ]
-
3 2
6.6 The Exponential Form of the Fourier Series
The Fourier series are often expressed in exponential form. The advantage of the exponential

form is that we only need to perform one integration rather than two, one for the a n , and
another for the b n coefficients in the trigonometric form of the series. Moreover, in most cases
the integration is simpler.

The exponential form is derived from the trigonometric form by substitution of

cos ωt = e jωt + e – jωt (6.85)
----------------------------
2
and
e jωt – e – jωt
sin ωt = --------------------------- (6.86)
j2
into ft() . Thus,

1 ⎛ e jωt + e – jωt ⎞ ⎛ e j2ωt + e – j2ωt ⎞
-
ft() = --a + a ---------------------------- + a --------------------------------- + (6.87)
2 0 1 ⎝ 2 ⎠ 2 ⎝ 2 ⎠
j2ωt
j2ωt
–
jωt
–
jωt
e
–
–
e
e
e
⎛
⎞
⎞
⎛
… + b --------------------------- + b -------------------------------- + …
⎝
⎠
⎝
⎠
1
2
j2
j2
and grouping terms with same exponents, we get
2
--a +
----- e
----- e
---- +
----- e
ft() = … + ⎛ ⎝ a 2 2 b 2 ⎞ ⎠ – j2ωt + ⎛ ⎝ a 2 1 b 1 ⎞ ⎠ – jωt + 1 - 0 ⎛ ⎝ a 2 1 b 1⎞ ⎠ jωt + ⎛ ⎝ a ---- + b 2⎞ ⎠ j2ωt (6.88)
---- –
----- e
---- –
2
j2
2
j2
j2
j2
The terms of (6.88) in parentheses are usually denoted as
1 ⎛ b n⎞ 1
-
C – n = -- a – ----- = -- a +( 2 - n jb ) n (6.89)
⎠
2 ⎝
n
j
1 ⎛ b n⎞ 1
-
C = -- a + ----- = --- a j– b( 2 n n ) (6.90)
⎠
2 ⎝
n
n
j
1
-
C = --a 0 (6.91)
0
2
Numerical Analysis Using MATLAB® and Excel®, Third Edition 6−29
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