Page 226 - Numerical Analysis Using MATLAB and Excel

Symmetry

and

T
∫ f t() t = 0 (6.18)
d
o
– T
A function ft() that is neither even nor odd can be expressed as

1
f t() = -- ft() +[ 2 - f –() ] (6.19)
t
e
or as

1
f t() = -- ft() f– –()[ 2 - t ] (6.20)
o

By addition of (6.16) with (6.17), we get

ft() = f t() + f t() (6.21)
o
e
that is, any function of time can be expressed as the sum of an even and an odd function.

To understand half−wave symmetry, we recall that any periodic function with period , isT
expressed as
(
ft() = ft + T ) (6.22)
that is, the function with value ft() at any time , will have the same value again at a later time
t
t + T .

A periodic waveform with period , has half−wave symmetry if
T
⁄
)
f – ( t + T 2 = ft() (6.23)
that is, the shape of the negative half−cycle of the waveform is the same as that of the positive
half−cycle, but inverted.
We will test the waveforms of Figures 6.9 through 6.13 for any of the three types of symmetry.

1. Square waveform

For the waveform of Figure 6.9, the average value over one period is zero, and therefore,
T
a = 0 . It is also an odd function and has half−wave symmetry since f –()– t = ft() and
0
)
⁄
(
– ft + T 2 = ft() .

Numerical Analysis Using MATLAB® and Excel®, Third Edition 6−9
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