# Fourier Transform Triangular Pulse Reconstruction

Cuthbert Nyack
The following 2 equations specify the value of f(t) over the range of t for which it is nonzero. Here the effect on the pulse of reconstructing it from a limited part of its spectrum is illustrated. The Fourier Transform of this function is a sinc2 function shown below. The transform has zeros at integer values of n in the expression below. The reconstructed pulse is given by the following equation. Fn = 1 in the applet below shows the resulting pulse when it is reconstructed from a limited part of the spectrum. Original pulse is shown in cyan and the reconstructed one in yellow. The spectrum is shown in red and the bandwidth used to reconstitute the pulse is shown by the 2 vertical green lines. Pulse has width 2 t. The sidelobes of the spectrum are very small but can be made visible by increasing the 'Vg' parameter.
Fn = 2 to 5 show special cases of Fn = 1.
Fn = 6 shows a pulse whose shape can be modified by the parameters 'a' and 'b'.
Fn = 7 to 10 show special cases of Fn = 6.
Fn = 11 shows a cosine pulse of width 2t.
Fn = 12 to 19 show special cases of Fn = 11.
Fn = 20 shows a raised cosine pulse of width 2t.
Fn = 21 to 23 show special cases of Fn = 20.
Fn = 24 shows a pulse whose shape can be modified by the parameters 'a', 'b', 'c' and 'd'.
Fn = 25 to 30 show special cases of Fn = 24.
Fn = 31 shows a pulse whose shape can be modified by the parameter 'a'.
Fn = 32 to 35 show special cases of Fn = 31.
Fn = 36 shows a pulse whose shape can be modified by the parameter 'a'.
Fn = 37 to 43 show special cases of Fn = 36.
Fn = 44 shows a pulse whose shape can be modified by the parameter 'a'.
Fn = 45 to 50 show special cases of Fn = 44.