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. The rectangular pulse is shown in magenta and its reconstruction in yellow. When reconstructed up to frequency 1 (2 pi / tau), then frequencies up to the first zero of the spectrum are included. The spectrum is shown in cyan and the range of frequencies included in the reconstruction is shown by the vertical red lines.

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