Fourier Series Made Easy
The Math Lab
2012
Introduction
Hello everybody. This section focuses primarily on fourier series which take the form:
f (x) =
∞
X
a0
mπx
mπx
+
(am cos
+ bm sin
)
2
L
L
m=1
Definition
A Fourier Series is a series of periodic functions (typically sinusoids) that represents another periodic function.
Periodicity
The first thing to notice about a Fourier Series representation of a function is that the
function must be periodic. A function f (x) is said to be Periodic when
f (x + T ) = f (x)
for some value T . The least element of all values T , or, in other words, the smallest value T
can be, is called the Fundamental Period.
Something cool about periodic functions is that if you add or multiply two functions with
the same period, the sum or product will be periodic with the same period. For example,
cos(2πt),
sin(2πt)
both have periods of 1. Let’s look at the sum of the two sinusoids:
cos(2π(t + 1)) + sin(2π(t + 1)) = cos(2πt + 2π) + sin(2πt + 2π) = cos(2πt) + sin(2πt)
You can stew about how the product of two periodic signals with period T is periodic. Let’s
move on, though.
We are going to use the fact that the sum of two periodic functions is periodic to formulate
the Fourier Series.
The Ultimate Goal
Let’s take a look at that big ugly formula that represents a Fourier Series for f (x).
f (x) =
∞
X
mπx
a0
mπx
+
(am cos
+ bm sin
)
2
L
L
m=1
1
Notice first that the sinusoids are periodic with period 2L. This is because
mπ
2π
2πL
2L
=
⇒T =
=
L
T
mπ
m
To obtain the fundamental period, we let m = 1 and T = 2L.
Notice, therefore, that the only information we don’t know is the a0 , am , and bm . If we could
find these easily, then Fourier Series wouldn’t be so daunting. Luckily, our good friend Euler
derived some equations that are now known as the Euler-Fourier Formulas.
Euler-Fourier Formulas
an =
nπx
1ZL
f (x) cos
dx,
L −L
L
bn =
1ZL
nπx
f (x) sin
dx
L −L
L
See pages 587-588 in the Boyce-DiPrima textbook for the derivation.
Fourier Convergence Theorem
This is a theorem that states the following:
If f (x) is a periodic function that is piece-wise continuous over a period 2L
and f 0 (x) is piece-wise continuous over a period 2L, then there exists a Fourier
series that converges to f for all continuous points and [f (x+) + f (x−)]/2 for all
discontinuous points.
Even and Odd Functions - Effects on Fourier Transform
The basic thing to take away from this section is that if f (x) is even or odd, finding the
Fourier series coefficients is much easier.
If f (x) is even, then the Fourier series won’t have any odd components and bn = 0.
If f (x) is odd, then the Fourier series won’t have any even components and an = 0.
2
Examples:
Find the Fourier series of
(
f (x) =
x, −π ≤ x < 0,
0,
0 ≤ x < π;
f (x + 2π) = f (x)
Solution: First off, we notice that f (x) is periodic with period 2π. Therefore L = π. Using
the Euler-Fourier formula for an , we know that
a0 =
1ZL
(0)πx
1Zπ
1Z0
π
f (x) cos
dx =
f (x)dx =
xdx = −
L −L
L
π −π
π −π
2
nπx
1Zπ
nπx
1Z0
1ZL
f (x) cos
dx =
f (x) cos
dx =
x cos nxdx
an =
L −L
L
π −π
π
π −π
1 x sin nx cos nx
=
+
π
n
n2
0
=
−π
1 − (−1)n
n2
And now looking at bn ,
bn =
nπx
nπx
1Zπ
1Z0
1ZL
f (x) sin
f (x) sin
x sin nxdx
dx =
dx =
L −L
L
π −π
π
π −π
1 −x cos nx sin nx
+
π
n
n2
=
0
=
−π
π
(−1)n
n
Therefore:
f (x) =
∞
∞
mπx
1 − (−1)m
a0 X
mπx
π X
π
(am cos
(
+
+bm sin
)=− +
cos
mx+
(−1)m sin mx)
2 m=1
L
L
4 m=1
m2
m
3