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. The sinc envelope is shown in gray. Separation
of spectral lines(in red) is determined by the pulse period T and is (1/T Hz). First
zero of sinc envelope and separation of subsequent zeros are
determined by the width of the pulse
and is 1/t Hz. n = 0 is at the center,
-ve n is to the left and +ve n to the right. T affects the
separation and height of the lines. t
affects the location of the zeros of the envelope and the height of the lines.
eg. For parameters (0.2, 0.1, 0.2, 1.0), the Period is twice the
pulse width. The fundamental frequency is 5kHz and the first zero
of the envelope is at 10kHz.
At the other end, eg parameters (1.0, 0.01, 10.0, 2.0) show there are
200 lines in the main lobe of the spectrum. The lines are at 1kHz
apart and the first envelope zero occurs at 100kHz. As T increases the lines
get closer together and the spectrum begins to look like that of
the Fourier Transform.
When activated the following gif image show how the applet should appear.
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COPYRIGHT © 1996 Cuthbert Nyack.