# Fourier Series Pulse Train Applet

Cuthbert Nyack
Here the Fourier series of the pulse train shown below is examined. the period of the pulses is T and the width of each pulse is t. Using the complex form of thr Fourier Series, the coefficients d0, d-n and dn are given by the expression below. Spectrum consists of a set of lines under a sinc envelope. The applet below hows how the spectrum depends on T and t. Two periods of the pulse train are shown in magenta.
Fn = 1 shows the Fourier Series of a rectangular pulse train as a function of T and t.
Fn = 2 to 6 show special cases of Fn = 1.
Fn = 6 shows that as T/t increases the lines get closer together and the spectrum begins to look like that of the Fourier Transform.

Fn = 7 shows the Fourier Series of a triangular pulse train.
Fn = 8 to 12 show special cases of Fn = 7.
Fn = 13 shows the Fourier Series of a raised cosine pulse train.
Fn = 14 to 19 show special cases of Fn = 13.

If the channel does not pass zero frequency, then odd pulses must be used.
Fn = 20 shows the Fourier Series of an odd rectangular pulse train.
Fn = 21 to 27 show special cases of Fn = 20.
Fn = 28 shows the Fourier Series of an odd triangular pulse train.
Fn = 29 to 34 show special cases of Fn = 28.
Fn = 35 shows the Fourier Series of an odd ramp pulse train.
Fn = 36 to 41 show special cases of Fn = 35.

Fn = 42 shows the Fourier Series of a ramp pulse train with both even and odd parts.
Fn = 43 to 48 show special cases of Fn = 42.
Fn = 49 shows the Fourier Series of a parabolic pulse train with both even and odd parts.
Fn = 50 to 54 show special cases of Fn = 49.
Fn = 55 shows the Fourier Series of a 3 variable pulse train which can have both even and odd parts. The shape of the pulse is set by 'a', 'b' and 'c'.
Fn = 56 to 60 show special cases of Fn = 55.

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COPYRIGHT © 1996, 2012 Cuthbert Nyack.