Матричное представление дискретного преобразования Фурье

Discrete fourier transform expressed as a matrix
In applied mathematics, a DFT matrix is an expression of a discrete Fourier transform (DFT) as a transformation matrix, which can be applied to a signal through matrix multiplication.

The two point DFT is a simple case, in which the first entry is the DC (sum) and the second entry is the AC (difference). The first row performs the sum, and the second row performs the difference. The factor of is to make the transform unitary (see below).

The DFT is (or can be, through appropriate selection of scaling) a unitary transform, i. e., one that preserves energy. The appropriate choice of scaling to achieve unitarity is , so that the energy in the physical domain will be the same as the energy in the Fourier domain, i. e., to satisfy Parseval's theorem. (Other, non unitary, scalings, are also commonly used for computational convenience; e. g., the convolution theorem takes on a slightly simpler form with the scaling shown in the discrete Fourier transform article.)

For other properties of the DFT matrix, including its eigenvalues, connection to convolutions, applications, and so on, see the discrete Fourier transform article.

The notion of a Fourier transform is readily generalized. One such formal generalization of the N point DFT can be imagined by taking N arbitrarily large. In the limit, the rigorous mathematical machinery treats such linear operators as so called integral transforms. In this case, if we make a very large matrix with complex exponentials in the rows (i. e., cosine real parts and sine imaginary parts), and increase the resolution without bound, we approach the kernel of the Fredholm integral equation of the 2nd kind, namely the Fourier operator that defines the continuous Fourier transform. A rectangular portion of this continuous Fourier operator can be displayed as an image, analogous to the DFT matrix, as shown at right, where greyscale pixel value denotes numerical quantity.