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The solving of ordinary differential equations by means of the infinite series method.
Dražková, Jana ; Štoudková Růžičková, Viera (referee) ; Čermák, Jan (advisor)
This bachelor's thesis is concerned with the solving of ordinary differential equations by means of the infinite series methods, in particular, the power series and the Fourier series. The aim of this thesis is to find the solution of the initial value problem for ordinary differential equations by use of the power series and compare this approach to traditional analytic methods. Further, the thesis deals with the solving of the second order linear ordinary differential equations with a periodic forcing term via the Fourier series method.
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Semi-Analytical Computations
Herzallah, Ahmad Sudqi Hussein ; Kopřiva, Jan (referee) ; Kunovský, Jiří (advisor)
This thesis discusses the analytical errors emerging from semi-analytical calculations. It also discusses about the modern method of Taylor's series for numerical calculations of ordinary differential equations and chosen methods of charachterized functions. The end solutions indicate favourable results for the semi-analytical calculations in the selected roles and responsible differential equations by direct execution of Taylor's series for solutions of polynomial functions, exponential functions & geometric functions using simulation program "TKSL". It also discusses about the definite & indefinite Intergals and methods for the solution of definite integrals. Followed by a brief introduction of Maple, Matlab & TKSL and further comparison of the result of the three programs & finding the best way of resolving the definite integrals.
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Practical examples of signal processing
Hanzálek, Pavel ; Smékal, Zdeněk (referee) ; Mekyska, Jiří (advisor)
The thesis focuses on the issue of signal processing. Using practical examples, it tries to show the use of individual signal processing operations from a practical point of view. For each of the selected signal processing operations, an application is created in MATLAB, including a graphical interface for easier operation. The division of the thesis is such that each chapter is first analyzed from a theoretical point of view, then it is shown using a practical demonstration of what the operation is used in practice. Individual applications are described here, mainly in terms of how they are handled and their possible results. The results of the practical part are presented in the attachment of the thesis.
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The Fourier Series and Its Properties
Sladká, Pavla ; Žák, Libor (referee) ; Štarha, Pavel (advisor)
The functional series, and especially the Fourier series, are an important mathematical apparatus exploited in the various technical branches. A very essential group of the functional series are the power series, which are applied because of their simplicity for solving of the many problems. An expansion of the function to the power series, i. e. the Taylor expansion, whose sum is the expanded function. These expansions are suitable for evaluation of operations, such as calculation of functional values, limits, derivatives and integrals. Calculations of these expansions are easier than of the functions theirself. The Fourier series are used for studies of events with periodic character. An advantage of the Fourier series is the fact, that the requirements for convergency are weaker than in case of the Taylor expansions. Likewise, calculation of the coefficients can be more simple than in the Taylor expansions. Expansions of functions to the Fourier series are used especially for solving ordinary and partial differential equations. This method of solving is known as the Fourier method or the Fourier method of variable separation.
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Web applications supporting education of signal processing fundamentals
Kuře, Dominik ; Ištvánek, Matěj (referee) ; Rajmic, Pavel (advisor)
The main topic of the thesis is the creation of four web applications which are used as learning material for students who are studying the basics of signal processing. The areas on which the thesis focuses on are root mean square and expected value of signals, basic signal operations (amplification, geometric translation, scale change), the effect these operations have on the Fourier series of the given signal and also resampling of a signal using different methods of interpolation (nearest neighbour method, linear interpolation, cubic interpolation and interpolation using the sinc function). These applications are implemented using the TypeScript programming language which is an extension of the JavaScript language which enhances it with static types. Other libraries that are used are the React library which is used for front-end web applications and a library which allows easy to implement but still very detailed manipulation of charts called Chart.js. The first half of the theoretical part of the thesis focuses on those areas of signal processing which are necessary to understand so the applications can be created. The second half focuses on information technologies used for the implementation of said applications. Besides the already mentioned technologies, the text also briefly mentions the basics of HTML and CSS languages as well as the JSX syntax. The practical part describes how the applications were implemented and also serves as documentation for the source code. This part shows the reader how to create differently shaped signals in code (sine, triangle, sawtooth, square with different duty cycles, noise) and how to obtain the Fourier series of each of these signals, how to implement different signal operations, how to interpolate between multiple points using different interpolation formulas and what are some of the methods which can be used to apply cubic interpolation (finite difference method, cardinal spline, Catmull–Rom spline, natural cubic interpolation), what the applications look like and what is their structure.
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Web applications supporting education of audio signal processing
Malcher, Tomáš ; Mokrý, Ondřej (referee) ; Rajmic, Pavel (advisor)
This bachelors thesis deals with a creation of web applications using JavaScript language. These web applications should be helpful in signal processing classes by showing and explaining different phenomenons to students. Students can see how different parameters affect the outcome of said issue. The theory needed for understanding and implementation is covered by this thesis. Thesis covers key parts of the code of said web applications and the libraries used for their implementaition.
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On convergence of a series
Procházka, Antonín ; Zelený, Miroslav (advisor) ; Kaplický, Petr (referee)
in English This text is devoted to the series, whose n-th term is defined by (−1)n |sin n| /n. The goal of this work is to prove convergence of this series. The solution uses standard convergence tests, the theory of Fourier Series and findings about approximation of number π. 1
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Practical examples of signal processing
Hanzálek, Pavel ; Smékal, Zdeněk (referee) ; Mekyska, Jiří (advisor)
The thesis focuses on the issue of signal processing. Using practical examples, it tries to show the use of individual signal processing operations from a practical point of view. For each of the selected signal processing operations, an application is created in MATLAB, including a graphical interface for easier operation. The division of the thesis is such that each chapter is first analyzed from a theoretical point of view, then it is shown using a practical demonstration of what the operation is used in practice. Individual applications are described here, mainly in terms of how they are handled and their possible results. The results of the practical part are presented in the attachment of the thesis.
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Conjugate series to Fourier's ones
Bathory, Michal ; Opic, Bohumír (advisor) ; Zelený, Miroslav (referee)
This thesis focuses entirely on conjugate series to Fourier's ones. It provides a quick and intuitive introduction to this topic for the reader who is familiar with classical Fourier's series. The thesis contains simple tests for the convergence of conjugate series and for the existence of related conjugate functions. These concepts are illustrated with examples. Extensive and comprehensive works of Antoni Zygmund are dedicated to the conjugate series (among other topics) but the corresponding proofs are far from detailed. Thus, this thesis summarizes the basic assertions systematically, gives proofs in detail and offers author's own solutions to selected examples. Powered by TCPDF (www.tcpdf.org)
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