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 b_n are zero.

An odd function is defined by:-

For an odd function only the coefficients b_n 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 f_n(t) is in green, f_n(t + T/2) is in pink and - f_n(t + T/2) in orange. For odd n the green curve coincides with the orange ( ie f_n(t) = - f_n(t + T/2) ) while for even n the pink curve coincides with the green ie f_n(t) = f_n(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 of text.
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 frequencies.
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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