Seeing, hearing, doing multivariate statistics

Signal processing is an important task in our days that arise in various areas such as engineering and applied mathematics. A signal represents time-varying or spatially varying physical quantities. Signals of importance can include sound, electromagnetic radiation, images, telecommunication transmission signals, and many others. A signal carries information, and the objective of signal processing is to extract useful information carried by the signal. The received signal is usually disturbed by electrical, atmospheric or deliberate interferences. Due to the random nature of the signal, statistical techniques play an important role in analyzing the signal.

There are many techniques used to analyze these types of data, depending on the focus or research question of the study. Some of these techniques are Principal Component Analysis (PCA) and Fourier transform, in particular discrete Fourier transform (DFT). The main goal in this work is to explore the relations between PCA and others mathematical transforms, based on Toeplitz and circulant matrices. In this sense, the proposed method relates the theory behind the Fourier transform through the Toeplitz and circulant matrices and the PCA. To illustrate the methodology we will consider sounds and images.

Keywords: Circulant Matrix, Fourier Transform, Principal Component Analysis, Signal Processing, Toeplitz Matrix.