Discrete Fourier Transforms

This chapter deals with the discrete Fourier transform (DFT). In Sect. 3.1, we show that numerical realizations of Fourier methods, such as the computation of Fourier coefficients, Fourier transforms, or trigonometric interpolation, lead to the DFT. We also present barycentric formulas for interpolating trigonometric polynomials. In Sect. 3.2, we study the basic properties of the Fourier matrix and of the DFT. In particular, we consider the eigenvalues of the Fourier matrix with their multiplicities. Further, we present the intimate relations between cyclic convolutions and the DFT. In Sect. 3.3, we show that cyclic convolutions and circulant matrices are closely related and that circulant matrices can be diagonalized by the Fourier matrix. Section 3.4 presents the properties of Kronecker products and stride permutations, which we will need later in Chap. 5 for the factorization of a Fourier matrix. We show that block circulant matrices can be diagonalized by Kronecker products of Fourier matrices. Finally, Sect. 3.5 addresses real versions of the DFT, such as the discrete cosine transform (DCT) and the discrete sine transform (DST). These linear transforms are generated by orthogonal matrices.