470-4118/01 – Integral and Discrete Transforms (ITDT)

Gurantor departmentDepartment of Applied MathematicsCredits8
Subject guarantordoc. Ing. David Horák, Ph.D.Subject version guarantordoc. Ing. David Horák, Ph.D.
Study levelundergraduate or graduateRequirementCompulsory
Year1Semestersummer
Study languageCzech
Year of introduction2012/2013Year of cancellation
Intended for the facultiesFEI, HGFIntended for study typesFollow-up Master
Instruction secured by
LoginNameTuitorTeacher giving lectures
HOR33 doc. Ing. David Horák, Ph.D.
MRO0010 Ing. Martin Mrovec
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 3+3
Combined Credit and Examination 10+10

Subject aims expressed by acquired skills and competences

Stundent ought to manage the theory and practise of integral and discrete transforms, to get familiar with suitable approaches for the solution of concrete problems, to design an algorithm, to implement it and to make the conclusion of this solution.

Teaching methods

Lectures
Tutorials
Project work

Summary

This subject belongs to the set of basic mathematical subjects of technical university studies. Student gets know the theory and use of the Laplace transform, Z-transform, Fourier series, Fourier and Window-Fourier transform, Wavelet transform in the continuous form and discrete form as well including algorithms, efficient implementations and applications for signal processing, e.g. time-frequency analysis, compression, filtering etc.

Compulsory literature:

• Bachman G., Narici L., Becktenstein E.: Fourier and wavelet analysis, Springer, 2000. • William L. Briggs, Van Emden Henson: THE DFT, An Owner´s Manual for the Discrete Fourier Transform, SIAM, 1995,ISBN 0-89871-342-0.

Recommended literature:

• William L. Briggs, Van Emden Henson: THE DFT, An Owner´s Manual for the Discrete Fourier Transform, SIAM, 1995,ISBN 0-89871-342-0.

Way of continuous check of knowledge in the course of semester

Průběžná kontrola studia: • Test - max. 10 bodů. • Individuální úlohy na téma Fourierova, Laplaceova a Z-transformace - max. 20 bodů. • Individuální projekt na implementaci a aplikaci diskrétních transformací nebo vypracování pěti implementačních úloh - max. 10 bodů. Podmínky udělení zápočtu: • Napsání testu - max. 10 bodů. • Odevzdání individuálních úloh - max. 20 bodů. • Odevzdání a obhajoba aplikačního projektu nebo pěti implementačních úloh – max. 10 bodů. Maximální počet bodů, které lze získat ve cvičení je 40 bodů. Minimální počet bodů pro udělení zápočtu je 15 bodů.

E-learning

Další požadavky na studenta

There are not defined other requirements for student.

Prerequisities

Subject has no prerequisities.

Co-requisities

Subject has no co-requisities.

Subject syllabus:

Lectures: • Introduction, keywords, general insight to integral and discrete transforms • Convolution as the basic IT (convolution of functions, sequencies, vectors, n-dimensional convolution) • Orthonormal systems and discrete orthonormal systems (Rademacher, Walsh, modified Walsh, Haar systems) • Generalised Fourier serie and generalised discrete Fourier transform (Discrete generalised Fourier serie vs. Generalised discrete Fourier transform, harmonic analysis, Fourier serie in real and complex form, spectrum, Dirichlet's conditions, use of Fourier series for the PDE solution) • Fourier transform (FT) (Definition of continuous and discrete FT (DFT), properties, inverse FT, matrix MF properties, two-sides DFT, two-dimensional DFT, Fast FT (FFT) • Window FT (WFT) (Definition of window function, continuous and discrete WFT (DWFT), applications) • Wavelet transform (WT) (Multiresolution analysis, definition of the continuous WT, properties, construction of orthonormal wavelets, discrete WT (DWT), Mallat's algorithm, fast DWT (FWT), packet decomposition, two-dimensional WT, applications) • Laplace transform (LT) (Definition, properties, inverse LT, existence and convergence questions, use of LT for PDE solution) • Z-transform (ZT) (Definition, inverse ZT, properties, relation to DLT, two-sides ZT, use for the solution of difference equations) Exercises: • Laplace transform and inverse LT • Solution of PDE using LT • Orthogonal and orthonormal systems of functions, Fourier serie, amplitude and phase spectrum • Solution of PDE using Fourier series • Fourier transform, inverse FT, convolution • Z-transform, solution of difference equations Computer Labs: • Introduction of software Matlab and its toolboxes • Discrete orthogonal systems, implementation, methods of numerical convolution • Analysis of one-dimensional signals using DFT • FFT algorithm and its implementation • Discrete Window Fourier transform implementation • Discrete Wavelet transform implementation • Algoritms usage for analysis of signals and their filtering Projects: • Fourier series, Fourier transform • Laplace transform, Z-transform • Application project according to student's choice

Conditions for subject completion

Full-time form (validity from: 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 (100) 51
        Exercises evaluation Credit 40  15
        Examination Examination 60  11
Mandatory attendence parzicipation:

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Occurrence in study plans

Academic yearProgrammeField of studySpec.FormStudy language Tut. centreYearWSType of duty
2019/2020 (N2647) Information and Communication Technology (1103T031) Computational Mathematics P Czech Ostrava 1 Compulsory study plan
2019/2020 (N2647) Information and Communication Technology (1103T031) Computational Mathematics K Czech Ostrava 1 Compulsory study plan
2019/2020 (N0541A170007) Computational and Applied Mathematics (S01) Applied Mathematics P Czech Ostrava 2 Compulsory study plan
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2019/2020 (N0541A170007) Computational and Applied Mathematics (S01) Applied Mathematics K Czech Ostrava 2 Compulsory study plan
2019/2020 (N0541A170007) Computational and Applied Mathematics (S02) Computational Methods and HPC K Czech Ostrava 2 Compulsory study plan
2018/2019 (N2647) Information and Communication Technology (1103T031) Computational Mathematics P Czech Ostrava 1 Compulsory study plan
2018/2019 (N2647) Information and Communication Technology (1103T031) Computational Mathematics K Czech Ostrava 1 Compulsory study plan
2017/2018 (N2647) Information and Communication Technology (1103T031) Computational Mathematics P Czech Ostrava 1 Compulsory study plan
2017/2018 (N2647) Information and Communication Technology (1103T031) Computational Mathematics K Czech Ostrava 1 Compulsory study plan
2016/2017 (N2647) Information and Communication Technology (1103T031) Computational Mathematics P Czech Ostrava 1 Compulsory study plan
2016/2017 (N2647) Information and Communication Technology (1103T031) Computational Mathematics K Czech Ostrava 1 Compulsory study plan
2015/2016 (N2647) Information and Communication Technology (1103T031) Computational Mathematics P Czech Ostrava 1 Compulsory study plan
2015/2016 (N2647) Information and Communication Technology (1103T031) Computational Mathematics K Czech Ostrava 1 Compulsory study plan
2014/2015 (N2647) Information and Communication Technology (1103T031) Computational Mathematics P Czech Ostrava 1 Choice-compulsory study plan
2014/2015 (N2647) Information and Communication Technology (1103T031) Computational Mathematics K Czech Ostrava 1 Choice-compulsory study plan
2013/2014 (N2647) Information and Communication Technology (1103T031) Computational Mathematics P Czech Ostrava 1 Choice-compulsory study plan
2013/2014 (N2647) Information and Communication Technology (1103T031) Computational Mathematics K Czech Ostrava 1 Choice-compulsory study plan
2012/2013 (N2647) Information and Communication Technology (1103T031) Computational Mathematics P Czech Ostrava 1 Choice-compulsory study plan
2012/2013 (N2647) Information and Communication Technology (1103T031) Computational Mathematics K Czech Ostrava 1 Choice-compulsory study plan

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