5. Harmonic Analysis
Recall the Fourier series (that we met in Full Range Fourier Series):
We can re-arrange this series and write it as:
+ (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:
In this case, the Fourier expansion will be of the form:
a0/2 + (a1 cos t + b1 sin t) + (a3 cos 3t + b3 sin 3t) + (a5 cos 5t + b5 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 π.)
In this case, the Fourier expansion will be of the form:
a0/2 + (a2 cos 2t + b2 sin 2t) + (a4 cos 4t + b4 sin 4t) + (a6 cos 6t + b6 sin 6t) + ...
All of the harmonics are even.
EXAMPLES
Determine the existence of odd or even harmonics for the following functions.
(a)
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)
Useful Background
Check out Graphs of Exponential Functions.
f(t) = f(t + π).
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