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
• 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