# 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 (*a*_{1} cos *t* +
*b*_{1} sin *t*) is known as the
**fundamental**.

The term (*a*_{2} cos 2*t* + *b*_{2}
sin 2*t*) is called the **second harmonic.**

The term (*a*_{3} cos 3*t* + *b*_{3}
sin 3*t*) 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,if, -pi <=t <0),(t-pi/2,if, 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**.[Image source.]

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

See also Line Spectrum.

(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)`.

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