Fourier Series Applet, Time Shift.

Cuthbert Nyack

A square wave is shown opposite. It can be seen that the function can be even or odd depending on the location of the origin. There are 2 locations where the function is odd and 2 where it is even. In each case one is the negative of the other.
If the origin is located at any other points than the one above then the function has even and odd parts. An example is shown opposite.

If the signal is advanced by an amount DT then this introduces a phase f_n into the series f_n is given by:-

And the Fourier series is given by:

Depending on the values of a_n, b_n and f_n an even function can become an odd function and vice versa. The magnitude of each Fourier component (a_n² + b_n²)^½ remains constant as the wave is shifted.

The applet below shows the variation of the even coefficients in cyan on the left and the odd coefficients in pink on the right. The magnitude is shown in orange at the bottom. As the wave is time shifted the even and the odd parts change but the magnitude remains constant.

Fn = 1 shows a square wave (shown in magenta) which can be shifted between -0.25T to +0.25T. When the shift is 0, the wave is even, and it is odd when the shift is -0.25T or +0.25T. The origin is at the center of the plot and the plot extends from -0.5T to +0.5T.
Fn = 2 to 6 show special cases of Fn = 1.
Fn = 7 shows how the first 3 components in the spectrum and the corresponding phasors move as the shift is changed.
Fn = 8 to 11 show special cases of Fn = 7.
Fn = 12 show the changes in both the spectrum and the phasors as the wave is shifted. Fn = 13, 14 show special cases of Fn = 12.

Fn = 15 show the changes in the spectrum as the wave is moved for a rectangular wave whose width can be changed by 'a'.
Fn = 16 to 23 show special cases of Fn = 15.
Fn = 24 show the changes in both the spectrum, the harmonics and the phasors of the rectangular wave.
Fn = 25 to 36 show special cases of Fn = 24.
Fn = 37 shows the rectangular wave reconstructed form the shifted values of a_n and b_n. The Series is summed to Ns terms.
Fn = 38 to 43 show special cases of Fn = 37.

Fn = 44 show the changes in the spectrum of a Pulse whose shape can be changed by the parameters 'a', 'b' and 'c'.
Fn = 45 to 56 show special cases of Fn = 44.
Fn = 57 show the changes in both the spectrum and the harmonics of the 3 variable wave.
Fn = 58 to 60 show special cases of Fn = 57.
Fn = 61 show the changes in both the spectrum and the phasors of the 3 variable wave.
Fn = 62 to 67 show special cases of Fn = 61.

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