Discrete Fourier Transform

The discrete Fourier transform (DFT) is "the Fourier transform for finite-length sequences" because, unlike the (discrete-space) Fourier transform, the DFT has a discrete argument and can be stored in a finite number of infinite word-length locations. Yet, it turns out that the DFT can be used to exactly implement convolution for finite-size arrays. Our approach to the DFT will be through the discrete Fourier series DFS, which is made possible by the isomorphism between rectangular periodic and finite-length, rectangular-support sequences.

Definition (discrete Fourier transform)

For a finite-support sequence with support , we define its DFT for integers and as follows.

	(4.2.1)

Figure(4.1) and Figure( 4.2)

We note that the DFT maps finite support rectangular sequences into themselves. Pictures of the DFT basis functions of size 8 8 are shown with real parts in Figure (4.1) and imaginary parts in Figure (4.2). The real part of the basis functions represent the components of x that are symmetric with respect to the 8 8 square, while the imaginary parts of these basis functions represent the nonsymmetric parts. In these figures, the color white is maximum positive (+1), mid-gray is 0, and black is minimum negative (-1). Each basis function occupies a small square, all of which are then arranged into a 8 8 mosaic. Note that the highest frequencies are in the middle at =(4,4) and correspond to the Fourier transform at . So the DFT is seen as a projection of the finite-support input sequence x(n₁,n₂) onto these basis functions. The DFT coefficients then are the representation coefficients for this basis. The inverse DFT (IDFT) exists and is given by

	(4.2.2)

The DFT can be seen as a representation for x in terms of the basis functions . and the expansion coefficients .

The correctness of this 2-D IDFT formula can be seen in a number of different ways. Perhaps the easiest, at this point, is to realize that the 2-D DFT is a separable transform, and as such, we can realize it as the concatenation of two 1-D DFTs.

Since we can see that the 2-D IDFT is just the inverse of each of these 1-D DFTs in the given order, say row first and then by column, we have the desired result based on the known validity of the 1-D DFT/IDFT transform pair, applied twice.

A second method of proof is to rely on the DFS. The key concept is that rectangular periodic and rectangular finite-support sequences are isomorphic to one another, i.e., given we can define a finite support x as , and given a finite-support x , we can find the corresponding as where we use the notation , meaning "n mod N'. Still a third method is to simply insert (1) into (2) and perform the 2-D proof directly.