| Course Unit Code | 470-4118/01 |
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| Number of ECTS Credits Allocated | 8 ECTS credits |
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| Type of Course Unit * | Compulsory |
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| Level of Course Unit * | Second Cycle |
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| Year of Study * | Second Year |
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| Semester when the Course Unit is delivered | Winter Semester |
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| Mode of Delivery | Face-to-face |
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| Language of Instruction | Czech |
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| Prerequisites and Co-Requisites | Course succeeds to compulsory courses of previous semester |
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| Name of Lecturer(s) | Personal ID | Name |
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| HOR33 | doc. Ing. David Horák, Ph.D. |
| Summary |
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| 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. |
| Learning Outcomes of the Course Unit |
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| 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. |
| Course Contents |
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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 |
| Recommended or Required Reading |
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| Required Reading: |
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• 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.
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• Častová, N.,Kozubek,T: Integrální transformace, elektr. verze. www.am.vsb.cz
• Horák D., Diskrétní transformace, elektronická verze http://mi21.vsb.cz/modul/diskretni-transformace
• Galajda P., Schrötter Š.: Funkce komplexní proměnné a operátorový počet, Alfa-Bratislava, 1991.
• G.James and D.Burley, P.Dyke, J.Searl, N.Steele, J.Wright: Moderní inženýrská matematika,Addison-Wesley Publishing Company, 1994.
• Čížek, V: Diskrétní Fourierova transformace a její použití, SNTL, Praha, 1981.
• Častová N.: Sylaby k předmětu Diskrétní transformace.
• 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.
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| Recommended Reading: |
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| • William L. Briggs, Van Emden Henson: THE DFT, An Owner´s Manual for the Discrete Fourier Transform, SIAM, 1995,ISBN 0-89871-342-0. |
| • Škrášek J., Tichý Z.: Základy aplikované matematiky II, SNTL, Praha, 1986. |
| Planned learning activities and teaching methods |
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| Lectures, Tutorials, Project work |
| Assesment methods and criteria |
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| Task Title | Task Type | Maximum Number of Points
(Act. for Subtasks) | Minimum Number of Points for Task Passing |
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| Exercises evaluation and Examination | Credit and Examination | 100 (100) | 51 |
| Exercises evaluation | Credit | 40 | 15 |
| Examination | Examination | 60 | 11 |