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

Square wave - odd harmonics

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`.

Even harmonics

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

guitar 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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