Fourier Series Symmetries
Cuthbert Nyack
A periodic function is defined by the following relation:-
The implication of this is that the fourier Spectrum consists
of discrete lines. The separation of the lines is 1/T Hz and
the fundamental frequency is also 1/T Hz.
An even function is defined by the realation:-
This symmetry implies that the coefficients bn are zero.
An odd function is defined by:-
For an odd function only the coefficients bn are nonzero.
A function with the following symmetry does not have any even
harmonics in its spectrum.
The effect is illustrated by the applet below. Fundamental is in
red, nth harmonic
fn(t) is in green,
fn(t + T/2)
is in blue and
- fn(t + T/2) in orange.
For odd n the
green curve coincides with the
orange ( ie fn(t) =
- fn(t + T/2) ) while for even n the
blue curve
coincides with the
green ie fn(t) =
fn(t + T/2).
When activated the following gif image show how the applet should appear.
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COPYRIGHT © 1996 Cuthbert Nyack.