Transformações rápidas, convoluções e aplicações - análise dos algoritmos e implementação.

Abstract

Fourier transforms are often used in Numerical analysis and digital signal processing is a indispensable tool for analyzing and synthesizing devices such as digital filters. The evaluation of discrete Fourier using its definition directly is a process computationally expensive; fortunately there are algorithms that make the process more efficient, known as Fourier transform algorithms, which we refer to by its initials in English FFT, of Fast Fourier Transform. Motivated by the facts above and also by the use of FFT in efficient convolution computation, in this work a) we did a study of the mathematical foundations of rapid transformations; b) we analyzed the various formulations of processing, in particular FFT, Mersenne and the Chinese remainder theorem; we study and implement the implementation of some fast transformation algorithms to evaluate the differences between mathematically defined algorithms and its implementations as robust software, efficient and portable; and we study and implement the implementation of some algorithms for fast computation of convolutions.