Page 703 - The Mechatronics Handbook

0066_Frame_C23 Page 11 Wednesday, January 9, 2002 1:53 PM

FIGURE 23.8 Illustration of the differentiation technique for Fourier series coefficient computation.

which can be written as
∞
x″ t() = ∑ β k e j2πkf t
0
k=∞
–
Figure 23.8(c) shows the result of differentiating x(t). It is noted that if a signal is periodic, its derivatives
will also be periodic. This implies that β k is the complex Fourier series coefficient of x″(t) and can be
computed from

β k = 1 ∫ T/2 x″ t()e – j2πkf t t d
0
---
T – T/2
where


A
x″ t() = 2--- δ t + T  2δ t() + δ t – T 
---
--- –
T  2   2 
Thus,

A
-------------- =
β k = – 8------sin 2 πkTf 0 ( j2πf 0 k) c k
2

T 2 2 
That is,
 2A
c k =  ----------, k odd
2 2

π k

 0, otherwise
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