Fourier Series Symmetries.
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 pink 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
coincides with the
green ie fn(t) =
fn(t + T/2). n is changed by scrollbar 1.
Different aspects of symmetry can be seen by changing Fn which is
set by scrollbar 0. The value of Fn is shown in the first line
Fn = 1 shows the Fourier Series of an 8 segment function. The
value of the segments can be changed by scrollbar 10 to 17.
Fn = 2 to 9 show special cases of 1. Fn = 2 shows an even function f(t) with f(t) = -f(t + T/2). There are no even terms in the series.
Fn = 3 shows a function with f(t) = f(t + T/2). There are no
odd terms in the Fourier series.
Fn = 4 shows an odd function with f(t) = f(t + T/2). Here the period is effectively T/2 so there are even terms. Because of the
symmetry, terms as 4, 8, 12 etc are missing.
Fn = 5 shows another even function with f(t) = -f(t + T/2).
There are no evem terms.
Fn = 6, 7, 8 show functions without any symmetry.
Fn = 9 is a function with fundamental period T/4. Only 4n terms
are present with n even (8, 16, etc) terms missing.
Fn = 10 shows the function being represented as the sum of 2 functions
(Yellow) with f(t) = -f(t + T/2) and (Blue) with f(t) = f(t + T/2).
Fn = 11 to 14 show special cases of 10. Fn = 11 is a
function with f(t) = -f(t + T/2). Fn = 12 is a function
with f(t) = f(t + T/2). Fn = 13 and 14 has both symmetries present.
Fn = 15 shows the spectrum as a complex spectrum with +ve and -ve
Fn = 16 to 18 show special cases of 15. Fn = 16 is an even
function with a real and even spectrum shown in cyan. Fn = 17
is an odd function with an imaginary and odd spectrum shown in pink. Fn = 18 has both even and odd components present, and this is
reflected in the spectrum.
Fn = 19 shows a sine and a cosine superimposed on the function. This
can be used to see the relation between the absent lines in the spectrum
and the function.
Fn = 20 to 22 shows special cases of 19. Fn = 20 shows the
first 2 cosines which are zero in the spectrum. Because there
are an integral number of periods 'within' the function in each
case, then the f(t) cos(nwt) averages to zero. Fn = 21 and 22 illustrate the first cosine and sine terms
which are zero. It is the even and odd parts of the function
which are relevant here.
Fn = 23 to 29 shows a 6 segment function.
Fn = 30 to 32 shows a 10 segment function.
Both of these can be used to compare their symmetry behaviour
and series zeros with the 8 segment function.
Fn = 33 shows the expressions used for the Fourier Series. The
expressions follow a pattern which can be used to
find the series for any even number of flat segments.
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COPYRIGHT © 1996,2010 Cuthbert Nyack.