# Fourier Transform Rectangular Pulse Reconstruction

Cuthbert Nyack
Here a rectangular pulse is reconstructed from its spectrum and the effect of using a truncated part of the spectrum to reconstruct the pulse is illustrated. The pulse is specified by the equation below and is zero outside of the region stated.
The Fourier Transform of f(t) is readily found to be the following function.
Zeros occur in the transform at values of n in the equation below.
When integrated over a finite bandwidth, the reconstructed pulse is given by the following expression. This is also the same pulse which would be obtained if the rectangular pulse was passed through an ideal low pass filter.

If the pulse is reconstructed from a finite bandwidth, then the result is illustrated by the applet below.

Fn = 1 shows a pulse in cyan and its spectrum in red. The area of the pulse is 1 and its width is t. The reconstructed pulse is reconstructed from the spectrum between ± wm shown in green and the resulting pulse is shown in yellow.
Fn = 2 to 7 show special cases of Fn = 1.
Fn = 8 shows a double pulse in cyan and its spectrum in red. The width of each is t and their inner edges is separated by T.
The reconstructed pulse is reconstructed from the spectrum between ± wm shown in green and the resulting pulse is shown in yellow.
Fn = 9 to 12 show special cases of Fn = 8.
Fn = 13 shows a 2 level pulse in cyan and its spectrum in red. The width of the lower level is 2T and of the upper level is 2t .
Fn = 14 to 15 show special cases of Fn = 13.
Fn = 16 shows a pulse in cyan and its spectrum in red. The width of the pulse is 2t and the width of the linear edge is T.
Fn = 17 to 20 show special cases of Fn = 16.
Fn = 21 shows a odd rectangular pulse in pink and its spectrum in orange. The width of the pulse is 2t.
Fn = 22 to 24 show special cases of Fn = 21.
Fn = 25 shows a odd triangular pulse in pink and its spectrum in orange. The width of the pulse is 2t.
Fn = 26 to 27 show special cases of Fn = 25.
Fn = 28 shows another odd triangular pulse in pink and its spectrum in orange. The width of the pulse is 2t.
Fn = 29 to 30 show special cases of Fn = 28.
Fn = 31 shows a pulse with both even and odd parts with a spectrum including real and imaginary parts. The reconstruction from the real spectrum is in light green and the reconstruction from the imaginary spectrum is in light cyan. The width of the pulse is 2t.
Fn = 32 to 33 show special cases of Fn = 31.

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COPYRIGHT © 1996, 2012 Cuthbert Nyack.