Fourier Transform Definition

The Fourier transform is defined in Press[1] to be

(1.1)

The use of  rather than  is my notation as is the use of  rather than .  The use of  for the transform from the time domain to the frequency domain rather than +- k is due to Press.  Most documents on the Fourier transform start with a brief word or two in which they blame Press for all of their failings.  For my part I am very grateful that he has accepted this blame and established a standard to work from. 

A first principles derivation of the Fourier transform along with a reference is detailed in CFOUR.doc .htm.  The more fundamental Discrete Fourier Sum is described in Discrete Fourier Transform.doc.

Usual physics definition of a Fourier Transform

Define

        (1.2)

So that

(1.3)

The delta function

Substituting the second defining equation  into the first yields

                (2.1)

Which establishes the relationship

          (2.2)

For a finite frequency region, the inverse of this becomes

              (2.3)

Integral of the delta function

The integral of the delta function about a short time interval

(2.4)

 

Let

 

(2.5)

 

The limit as  is [2]     (2.6)

 

So that for finite epsilon and infinite F, the delta function relation is satisfied.

A second definition of a delta function

(2.7)

 

This is in the form of a Gabor transform although it is also a standard method of making otherwise difficult integrals tractable.  Evaluate the integral by completing the square

(2.8)

 

Let

 

(2.9)

Note that the value of F in the first definition is on the order of , so that for not so large values of F the two forms can be written as

 

(2.10)

It is of interest to plot 1/F times these two forms for values of Ff between -10 and 10

for\DFUNP.FOR