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28/05/2014, 11:30 — 12:30 — Room P3.10, Mathematics Building

Eunice Carrasquinha, *CEMAT, IST*

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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.