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Symmetry

The integral of (6.12) yields the average (DC ) value of f t() .

6.3 Symmetry

With a few exceptions such as the waveform of Example 6.6, the most common waveforms used
in science and engineering, do not have the average, cosine, and sine terms all present. Some
waveforms have cosine terms only, while others have sine terms only. Still other waveforms have
or have not DC components. Fortunately, it is possible to predict which terms will be present in
the trigonometric Fourier series, by observing whether or not the given waveform possesses some
kind of symmetry.

We will discuss three types of symmetry that can be used to facilitate the computation of the trig-
onometric Fourier series form. These are:

1. Odd symmetry − If a waveform has odd symmetry, that is, if it is an odd function, the series will
consist of sine terms only. In other words, if ft() is an odd function, all the coefficients
a
i
including , will be zero.
a
0
2. Even symmetry − If a waveform has even symmetry, that is, if it is an even function, the series
will consist of cosine terms only, and a 0 may or may not be zero. In other words, if ft() is an
even function, all the coefficients will be zero.
b
i
3. Half−wave symmetry − If a waveform has half−wave symmetry (to be defined shortly), only odd
(odd cosine and odd sine) harmonics will be present. In other words, all even (even cosine and
even sine) harmonics will be zero.

We will now define even and odd functions and we should remember that even functions have
nothing to do with even harmonics, and odd functions have nothing to do with odd harmonics.

A function ft() is an even function of time if the following relation holds.

t
f –() = ft() (6.15)
that is, if in an even function we replace with t– , the function ft() does not change. Thus,
t
polynomials with even exponents only, and with or without constants, are even functions. For
instance, the cosine function is an even function because it can be written as the power series *

2
6
4
t
t
t
cos t = 1 – ---- + ----- – ----- + …
-
2! 4! 6!
Other examples of even functions are shown in Figure 6.7.
* We will discuss power series later in this chapter.

Numerical Analysis Using MATLAB® and Excel®, Third Edition 6−7
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