# 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 gr(w). The imaginary part is determined by c, d and f and is shown as gi(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 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 gr(w) in the diagram opposite. The imaginary(odd) part of the spectrum is determined by c, d, e and f and is shown as gi(w). opposite. The applet below shows the inverse Fourier transform of 1 and 2 pulses and related functions.
Fn = 2 shows the inverse Fourier Transform of a spectrum consisting of real(cyan) and imaginary(pink) parts. The inverse of the real part of the spectrum is the even function in red and the inverse of the imaginary part of the spectrum is the odd function in green.
The real part of the spectrum is changes by 'a', 'b' and 'f' while the imaginary part is changed by 'c', 'd' and 'f'.
Fn = 3 to 10 show special cases of Fn = 2.
Fn = 11 shows the inverse Fourier Transform of a double pulse spectrum consisting of real(cyan) and imaginary(pink) parts.
Fn = 12 to 21 show special cases of Fn = 11.
Fn = 22 shows the inverse Fourier Transform of a 6 segment spectrum which can be changed by 'a' to 'g'.
Fn = 23 to 28 show special cases of Fn = 22.
Fn = 29 shows the inverse Fourier Transform of another 6 segment spectrum which can be changed by 'a' to 'k'.
Fn = 30 to 43 show special cases of Fn = 29.
Fn = 33 to 39 show cases where the spectrum and its inverse are similar in shape.
Fn = 44 shows the inverse Fourier Transform of a 4 pulse spectrum which can be changed by 'a' to 'g'.
Fn = 45 to 48 show special cases of Fn = 44.

The image below shows how the equations should appear. Unfortunately the appearance depends on the version of the java virtual machine used. 