457-0905/01 – Integral Transforms and Spectral Analysis (ITSA)

Gurantor departmentDepartment of Applied MathematicsCredits10
Subject guarantorprof. Ing. Tomáš Kozubek, Ph.D.Subject version guarantorprof. Ing. Tomáš Kozubek, Ph.D.
Study levelpostgraduateRequirementChoice-compulsory
YearSemesterwinter + summer
Study languageCzech
Year of introduction1998/1999Year of cancellation2010/2011
Intended for the facultiesIntended for study typesDoctoral
Instruction secured by
LoginNameTuitorTeacher giving lectures
KOZ75 prof. Ing. Tomáš Kozubek, Ph.D.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 2+0
Combined Credit and Examination 2+0

Subject aims expressed by acquired skills and competences

Student must manage basic terms and must be able to apply them to the practical problems.

Teaching methods

Lectures
Project work

Summary

The lectures give us the general insight into the theory of integral transforms in terms of functional analysis. They contain orthogonal systems of functions, Fourier series, Fourier, Window Fourier and Wavelet transforms in continuous and discrete form, and Z-transform as the special case of discrete Laplace transform. Importance is focused to the applications of their usage for the signal processing, especially time-frequency analysis, compression and their denoising. The cources are extended by exercises and Matlab's examples of efficient algorithms implementations.

Compulsory literature:

Michael W. Frazier: An introduction to wavelets through Linear Algebra, Springer,1999, ISBN 0-387-98639-1. G. Bachman, L. Narici, E. Beckenstein: Fourier and Wavelet Analysis, Springer, ISBN 0-387-98899-8.

Recommended literature:

William L. Briggs, Van Emden Henson: An Owner's Manual for the Discrete Fourier Transform, SIAM, 1995, ISBN 0-89871-342-0. Bruel & Kjaer: Frequency analysis, Denmark, ISBN 87-87355-07-8. G. Strang, T. Nguyen: Wavelets and Filter Banks, Wellesley-Cambridge Press, ISBN 0-9614088-7-1.

Way of continuous check of knowledge in the course of semester

E-learning

Další požadavky na studenta

Prerequisities

Subject has no prerequisities.

Co-requisities

Subject has no co-requisities.

Subject syllabus:

Přednášky: Úvod do problematiky integrálních transformací. Lineární prostory, norma, metrika. Prostory, Hilbertův prostor. Konvergence řad a integrálů. Ortogonální systémy spojitých a diskétních funkcí. Konvoluce dvou posloupností. Konvoluce dvou funkcí. Konvoluce jako integrální transformace. Inverzní konvoluce. Fourierova transformace. Fourierova transformace. Zpětná Fourierova transformace. Fourierova-Laplaceova transformace. Diskrétní Fourierova transformace. Chyby diskretizace. Algoritmy FFT. Regularizace zpětné transformace. Distribuce. Distribuce, Diracův impuls, F-obraz distribuce. Reprezentace distribuce. Konstrukce okenních funkcí. Matematické filtry. Základy harmonické analýzy. Harmonická analýza, nerozlišitelné kmity. Shanonn - Kotelnikovova věta na distribucích, okenní Fourierova transformace. Zobecněná Fourierová řada a wavelety (časově-frekvenční analýza). Zobecněná Fourierová řada. Waveletová báze, konstrukce. Spojitá a diskrétní waveletová transformace. Multirozklad (MRA). Komprese dat, rekonstrukce signálu, další užití. Software. Náhodné procesy a spektrální analýza v ortogonální bázi. Náhodné funkce. Rozklad v ortogonální bázi. Autokorelační funkce ergodických procesů. Funkce vzájemné korelace. Časové řady. Filtrace a predikce. Spektrální analýza ergodického procesu. Rozklad pomocí dyadických ortogonálních waveletů reálných, komplexních). Waveletovské spektrum. Časově-frekvenční analýza.

Conditions for subject completion

Full-time form (validity from: 1960/1961 Summer semester, validity until: 2012/2013 Summer semester)
Task nameType of taskMax. number of points
(act. for subtasks)
Min. number of points
Exercises evaluation and Examination Credit and Examination 100 (145) 51
        Examination Examination 100  0
        Exercises evaluation Credit 45  0
Mandatory attendence parzicipation:

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