Inverse Fourier Transform of Pulse and Double Pulse Spectrum

Cuthbert Nyack

The equations describing the Fourier transform and its inverse are shown opposite. The Fourier transform G(w) is a continuous function of frequency with real and imaginary parts. The inverse Fourier Transform f(t) can be obtained by substituting the known function G(w) into the second equation opposite and integrating. On this page the inverse Fourier Transform f(t) of some frequency spectra (or Fourier transform G(w) ) are illustrated.

For illustrative purposes the inverse of spectra which can be described by piecewise continuous functions as shown opposite will be considered. With this assumption, the integration to find the inverse can easily be carried out. It also illustrates the duality property of the Fourier Transform. As shown in the diagram opposite the real and imaginary parts of G(w) can be specified.

The real part of G(w) is determined by a, b and f in the diagram and is shown as g_r(w).

The imaginary part is determined by c, d and f and is shown as g_i(w).
The real part of the transform is even and its inverse produces the even part of f(t) while the imaginary(odd) part of the transform produces the odd part of f(t).

The above applet shows the even and odd parts of f(t) the Inverse Fourier transform as the Frequency Spectrum is changed. g(w) is the purple line( applet vertical is ą 1, horizontal is ą 2). The frequency spectrum is usually complex with real and imaginary parts. a and b determines the real part of g(w) and c and d determine the imaginary part of g(w). Purple line shows a simple sum of the real and imaginary parts of g(w). The even part of f(t) is shown by the red line (vertical is ą p, horizontal is ą 4 p when the horizontal gain is 1). The odd part of f(t) is shown by the orange line (same scale as even part). The sum of the even and odd parts of f(t) is shown by the green line. More of f(t) can be seen by changing the horizontal gain hgain and f(t) can be amplified by the vertical gain vgain.

The diagram opposite shows a spectrum consisting of 2 separate pieces symmetrically placed about the origin.

The real(even) part of the spectrum is determined by a, b, e and f and is shown as g_r(w) in the diagram opposite.

The imaginary(odd) part of the spectrum is determined by c, d, e and f and is shown as g_i(w). opposite.

The above applet shows the even (a and b) and odd (c and d) parts of f(t) as g(w) is changed. g(w) is the purple line( applet vertical is ą 1, horizontal is ą 2). Purple line shows a simple sum of real and imaginary parts. The function f(t) is shown by the red line. More of f(t) can be seen by changing the horizontal gain.
Continued on next page

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