Frequency-Domain Methods

Spectral methods based on the singular value decomposition (SVD) of the data matrix, as studied in the previous chapter, Chap. 3, apply to the physical (or spatial) domain of the data set. In this chapter, the concepts of frequency and frequency representation are introduced, and various frequency-domain methods along with efficient computational algorithms are derived. The root of these methods is the Fourier series representation of functions on a bounded interval, which is one of the most important topics in Applied Mathematics and will be investigated in some depth in Chap. 6. While the theory and methods of Fourier series require knowledge of mathematical analysis, the discrete version of the Fourier coefficients (of the Fourier series) is simply matrix multiplication of the data vector by some \(n \times n\) square matrix \(F_n\). This matrix is called the discrete Fourier transformation (DFT), which has the important property that with the multiplicative factor of \(1\over {\sqrt{n}}\), it becomes a unitary matrix, so that the inverse discrete Fourier transformation matrix (IDFT) is given by the \(1\over {n}\)- multiple of the adjoint (that is, transpose of the complex conjugate) of \(F_n\). This topic will be studied in Sect. 4.1.