Fourier Transforms and Convolutions

We will be using the built in discrete Fourier transform routine in Mathematica. This is what is known as a fast Fourier transform algorithm, which is written so as to minimize the number of calculations that are involved. If is a one dimensional array, Mathematica use the following formula to produce a new array that represents the Fourier transform:

where . There are three awkward facts about this formula.

• The first is inevitable, even though the original array had only real entries, the new array has complex entries. So in order to plot this we can plot the real part, the imaginary part, or the absolute value. In what follows, we will plot the absolute value.
• The second problem is that resulting array has its indices shifted by n/2. (You will see this below.) To correct for this, we will use a command to rotate the array circularly by n/2.
• The third is that the computed values are often so small that we may as well replace them by 0. We can do this with the Chop command.

So, if mylist is an array of n real numbers, we can construct the absolute value of its transform as follows

	mytransform = RotateRight[Chop[ Abs[Fourier[mylist]].001], n/2]

We can then plot this list using the ListPlot command above. There is also an inverse Fourier transform command, InverseFourier. It too has these awkward features, and should be invoked with

	mytransform = RotateRight[Chop[ Abs[InverseFourier[mylist]],001], n/2]

We can compute the convolution of two function by using the fact that the convolution is the inverse Fourier transform of the product of the Fourier transforms of the functions. In symbols,

We can define the convolution of two lists with the command

	Convolution[a_,b_]:=Chop[Abs[ InverseFourier[Fourier[a]*Fourier[b]]],.001]

Then we can construct the convolution of two arrays with

	myconvolution = RotateRight[Convolution[mylist,yourlist],n/2];

These same commands will construct the discrete Fourier transform and convolution in two dimensions with a slight modification. The difficulty is that the we must rotate the resulting arrays in the first coordinate and in the second coordinate.

Let us suppose that we have constructed the two-dimensional Fourier transform as above and called it temp,

	temp  = Chop[ Abs[Fourier[mylist]],.001]

Then the following two commands will center the transform:

	dummy=RotateRight[temp,Length[actiontemp]/2]; 
	mytransform=Table[RotateRight[dummy[[i]],Length[dummy]/2],{i,1,Length[dummy]};

We can then plot the resulting array using the above plotting commands.