M8120 Spectral Analysis II

Faculty of Science
Spring 2011 - only for the accreditation
Extent and Intensity
2/1/0. 3 credit(s) (příf plus uk k 1 zk 2 plus 1 > 4). Type of Completion: zk (examination).
Teacher(s)
doc. RNDr. Martin Kolář, Ph.D. (lecturer)
Guaranteed by
prof. RNDr. Ivanka Horová, CSc.
Department of Mathematics and Statistics – Departments – Faculty of Science
Prerequisites
M7120 Fourier Analysis I
Calculus of complex numbers, Vector and matrix calculus, Linear functional analysis, Basics of Fourier analysis of periodic and nonperiodic functions including convolution operators.
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
At the end of the course students should be able to: understand and explain discrete analogs to the relevant concepts and operations from Spectral Analysis I, in particular the discrete Fourier transform (DFT), the discrete linear (DLC) and cyclic convolution (DCC). Stress is laid on the description of sampling error effects which are due to the finite discretization and on the construction of effective computational algorithms, in particular Fast Fourier Transform (FFT) and fast convoulution operators which play an important role in digital filtration. One chapter gives basic introduction to the theory of generalized functions (distributions) which gives a unifying framework for the entire Fourier analysis, namely for the continuous as well as discrete versions of all mentioned basic operators, both in periodic and nonperiodic case.
Syllabus
  • 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, overlap-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 (Cooley-Tukey FFT) and convolution algorithms. Some more transformations of Fourier type: Hartley, cosine, etc., and their applications.
Literature
  • BRIGHAM, E. Oran. Fast Fourier transform. Englewood Cliffs: Prentice Hall, 1974, 252 s. ISBN 0-13-307496-X. info
  • ČÍŽEK, Václav. Diskretní Fourierova transformace a její použití. 1. vyd. Praha: SNTL - Nakladatelství technické literatury, 1981, 160 s. URL info
  • HOWELL, Kenneth B. Principles of Fourier Analysis. Boca Raton-London-New York-Washington: Chapman & Hall, 2001, 776 pp. Studies in Advanced Mathematics. ISBN 0-8493-8275-0. info
  • VAN LOAN, Charles. Computational frameworks for the fast fourier transform. Philadelphia: Society for Industrial and Applied Mathematics, 1992, 273 s. ISBN 0-89871-285-8. info
  • SCHWARTZ, Laurent. Matematické metody ve fyzice. 1. vyd. Praha, 1972, 357 s. info
Teaching methods
Lectures
Assessment methods
Oral exam
Language of instruction
Czech
Further Comments
The course is taught once in two years.
The course is taught: every week.
The course is also listed under the following terms Spring 2001, Spring 2003, Spring 2005, Spring 2007, Spring 2009, Spring 2011, Spring 2013, Spring 2015, Spring 2017, Spring 2019, Spring 2021, Spring 2023, Spring 2025.