470-4118/03 – Integral and Discrete Transforms (ITDT)

Gurantor departmentDepartment of Applied MathematicsCredits4
Subject guarantordoc. Ing. David Horák, Ph.D.Subject version guarantordoc. Ing. David Horák, Ph.D.
Study levelundergraduate or graduateRequirementChoice-compulsory
Study languageCzech
Year of introduction2016/2017Year of cancellation2020/2021
Intended for the facultiesUSPIntended for study typesFollow-up Master
Instruction secured by
LoginNameTuitorTeacher giving lectures
HOR33 doc. Ing. David Horák, Ph.D.
Extent of instruction for forms of study
Form of studyWay of compl.Extent
Full-time Credit and Examination 3+3
Part-time 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

Project work


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.


Other requirements

There are not defined other requirements for student.


Subject has no prerequisities.


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: 2016/2017 Summer semester, validity until: 2020/2021 Summer semester)
Task nameType of taskMax. number of points
(act. for subtasks)
Min. number of pointsMax. počet pokusů
Credit and Examination Credit and Examination 100 (100) 51
        Credit Credit 40  15
        Examination Examination 60  11 3
Mandatory attendence participation:

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Conditions for subject completion and attendance at the exercises within ISP:

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Occurrence in study plans

Academic yearProgrammeBranch/spec.Spec.ZaměřeníFormStudy language Tut. centreYearWSType of duty
2018/2019 (N2658) Computational Sciences (2612T078) Computational Sciences P Czech Ostrava 2 Choice-compulsory study plan
2017/2018 (N2658) Computational Sciences (2612T078) Computational Sciences P Czech Ostrava 2 Choice-compulsory study plan
2016/2017 (N2658) Computational Sciences (2612T078) Computational Sciences P Czech Ostrava 2 Choice-compulsory study plan

Occurrence in special blocks

Block nameAcademic yearForm of studyStudy language YearWSType of blockBlock owner

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