5. Harmonic Analysis

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

math expression

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

f(t) = a0/2 + (a1 cos t + b1 sin t) + (a2 cos 2t + b2 sin 2t) + (a3 cos 3t + b3 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:

math expression

Even Harmonics

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

math expression

EXAMPLES

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

(a)

math expression

f(t) = f(t + 2π).


Answer


Aside: Music Harmonics

guitar harmonics
Playing harmonics on a guitar. [Image source.]

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

(b)

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Useful Background

Check out Graphs of Exponential Functions.

f(t) = f(t + π).



Answer




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