Projects - Cengage Learning

Appendix C C-55
PROJECT
Fourier Series
In the project at the end of Section 8.2,* we discussed square waves, sawtooth
waves, and triangular waves. In this project we use the idea of superposition of
waves to find a “trigonometric” way to describe a square wave.
We start with the square wave given by y f(x) where
f(x) b 1, 0 x 1
1, 1 x 0
f(x) 0 for all integers x and f(x 2) f(x)
This square wave is shown in Figure A.
y
1
-4
-3
-2
-1
1 2 3 4
x
-1
Figure A
A graph of f(x) b 1, 0 x 1 ,
1, 1 x 0
f(x) 0 for integer x, and f(x 2) f(x).
Now, f is an odd function, periodic with period 2, and has domain all real
numbers. That f is periodic suggests it might be related to more familiar periodic
waves, for example, sine waves and cosine waves. Since f is an odd function
we might be able to describe f by using only sine waves. And since the
domain of f is all real numbers, our description of f using sine waves should
coincide with f for most, if not all, real numbers.
It turns out that f can be expressed by adding together infinitely many sine
waves. In fact, using methods from calculus, it can be shown that
f(x) 4 4
4
4
sin px sin 3px sin 5px sin 7px p (1)
p 3p 5p 7p
for all real numbers x. The symbol “ p ” means there are infinitely many more
terms that in this case follow the given pattern. So we are expressing f(x) as an
infinite series, a sum of infinitely many terms. (While we will study some
infinite series of numbers in Chapter 14,* it should be noted that the right-hand
side of equation (1) is an infinite series of functions.) The particular infinite
series on the right-hand side of equation (1) is called the Fourier series for the
function f. Notice that the Fourier series for f consists of sine waves of different
periods, but that 2, the period of the given square wave, is an integer multiple
of each period. When we superimpose or add up the waves, we obtain the
function f of period 2.
Figure B shows the graphs of the first three partial sums of the Fourier series
for f on the closed interval [3, 3] along with the graph of the square wave.
*Precalculus: A Problems-Oriented Approach, 7th edition