# 5. Harmonic Analysis

Recall the Fourier series (that we met in Full Range Fourier Series):

f(t)=(a_0)/2sum_(n=1)^ooa_n\ cos nt+sum_(n=1)^oob_n\ sin nt

=(a_0)/2+a_1\ cos t  +\ a_2\ cos 2t  +\ a_3\ cos 3t+...   +\ b_1\ sin t +\ b_2\ sin 2t  +\ b_3\ sin 3t+...

We can re-arrange this series and write it as:

f(t)=(a_0)/2+(a_1\ cos t + b_1\ sin t)  + (a_2\ cos 2t + b_2\ sin 2t)  + (a_3\ cos 3t+ b_3\ sin 3t)+...

The term (a1 cos t + b1 sin t) is known as the fundamental.

The term (a2 cos 2t + b2 sin 2t) is called the second harmonic.

The term (a3 cos 3t + b3 sin 3t) is called the third harmonic, etc.

## Odd Harmonics

The Fourier series will contain odd harmonics if f(t + π) = - f(t).

### Example 1

In this case, the Fourier expansion will be of the form:

f(t)=(a_0)/2+(a_1\ cos t + b_1\ sin t)  +\ (a_3\ cos 3t + b_3\ sin 3t)  +\ (a_5\ cos 5t+ b_5\ sin 5t)+...

All of the harmonics are odd.

## Even Harmonics

The Fourier series will contain even harmonics if f(t + π) = f(t). That is, it has period pi.

In this case, the Fourier expansion will be of the form:

f(t)=(a_0)/2+(a_2\ cos 2t + b_2\ sin 2t)  +\ (a_4\ cos 4t + b_4\ sin 4t)  +\ (a_6\ cos 6t+ b_6\ sin 6t)+...

All of the harmonics are even.

### Example 2

Determine the existence of odd or even harmonics for the following functions.

(a) f(t)= { {: (-t-pi/2",", -pi <=t <0),(t-pi/2",", 0 <= t < pi) :}

f(t) = f(t + 2π).

### Aside: Music Harmonics

Playing harmonics on a guitar. If you just lightly touch a string with the left hand and then pluck it, you hear a high pitched sound called the harmonic.

Music sounds "in tune" because the harmonics contained in each note sound "right" with certain other notes.

(b) f(t)={ {:(e^(-t),if , 0 {:<=:}t < pi), (e^(-t+pi),if, pi {:<=:} t < 2pi) :}

### Useful Background

Check out Graphs of Exponential Functions.

f(t) = f(t + π).

top

### Online Algebra Solver

This algebra solver can solve a wide range of math problems.

### Calculus Lessons on DVD

Easy to understand calculus lessons on DVD. See samples before you commit.