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 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. 
