1.Duality of the Fourier Transform

Notice a certain symmetry in these two system transformations.

Say y(t) has a Fourier Transform Y(f), then :

What is the transform of Y(t) ? Or, which signal on Inverse Fourier transformation would yield Y(t) ?

Recall the Inverse Transformation equation above, and put in the equation for Y:

Therefore, y(-f ) is the Fourier transform of Y(t) (where Y(f) is the Fourier transform of y(t) ) !

This remarkable relationship between a signal and its Fourier transform is called the Duality of the Fourier Transform.

i.e:

Duality implies a very remarkable relationship between the Fourier transform and its inverse. Notice the relationship between the Fourier Transform and the Fourier Inverse of X above:

This gives us a very important insight into the nature of the Fourier transform. We will use it to prove many “dual” relationships: if some result holds for the Fourier Transform, a dual result will hold for the Inverse transform as well. We will encounter some examples soon.