M8800 Fourier Analysis II

Faculty of Science
Spring 2001
Extent and Intensity
2/0/0. 4 credit(s). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Vítězslav Veselý, CSc. (lecturer)
Guaranteed by
prof. RNDr. Ivanka Horová, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Contact Person: doc. RNDr. Vítězslav Veselý, CSc.
Prerequisites (in Czech)
M7120 Spectral Analysis I
Course Enrolment Limitations
The course is also offered to the students of the fields other than those the course is directly associated with.
fields of study / plans the course is directly associated with
Course objectives
Discrete Fourier transform (DFT): DFT as sampled FT in one and more dimensions, properties, sampling and truncation errors, interpolation theorem.
Discrete convolution and correlation (DC): linear and cyclic DC obtained by sampling, properties, relation to polynomial multiplication, the discrete versions of the convolution and correlation theorems, discrete Parseval identities, periodogram, digital filtration, overlapp-add and overlap-save algorithms for long input sequence.
Fourier analysis of generalized functions: a brief survey of the theory of generalized functions (distributions), generalized functions as functionals, Dirac ``function'', carrying over the classical notions and operations to distributions, basic distribution spaces and their properties, unifying approach to FS, FT and the discrete FT in the scope of the theory of distributions.
Algorithms for DFT computation: parallel compuation of two real DFTs of equal length, computing real DFT of length $2N$ using one complex DFT of length $N$, fast Fourier transform and convolution algorithms (Cooley-Tukey FFT, Good's PFA algorithm, Rader's algorithm for prime length, Winograd's method), number-theoretic and polynomial transforms.
Some more transformations of Fourier type: Hartley, cosine, wavelet transforms etc., and their applications.
See http://www.math.muni.cz/~vesely/educ/fa2sylle.ps for more details.
Literature
  • ČÍŽEK, Václav. Diskretní Fourierova transformace a její použití. 1. vyd. Praha: SNTL - Nakladatelství technické literatury, 1981, 160 s. URL info
  • BRIGHAM, E. Oran. Fast Fourier transform. Englewood Cliffs: Prentice Hall, 1974, 252 s. ISBN 0-13-307496-X. info
  • VLADIMIROV, Vasilij Sergejevič. Obobščennyje funkcii v matematičeskoj fizike. Moskva: Nauka, 1976, 280 s. info
Language of instruction
Czech
Further comments (probably available only in Czech)
The course is taught once in two years.
The course is taught: every week.
Teacher's information
http://www.math.muni.cz/~vesely/educ_cz.html#fa2
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