Syllabus
1. Hilbert space - basic properties. Orthogonal/orthonormal series. Complete systems. General Fourier series, trigonometric and exponential form of F. series.
2. Convergention of Fourier series, Gibs phenomenon. Basic properties of Fourier series.
3. Generalized Fourier series of eigenfuncions and orthogonal polynomials. Multidimensional Fourier series.
4. Fourier theorem. Fourier transform. Sine and cosine transform.
5. Properties of Fourier transform. Multidimensional Fourier transform.
6. Fourier transform of special functions (periodic, Dirac distribution, Heaviside function, signum). Shah function and its properties.
7. Linear filters. Transfer function and impulse response.
8. Hilbert transform - definition, basic properties. Fourier spectra of causal functions. Analytic signals. Instant frequency.
9. Fourier transform of discrete signals. Definition, basic properties. Alias in frequency domain.
10. Fourier series of discrete signals. Definition, basic properties. Alias in time domain.
11. Discrete Fourier transform (DFT). Fast Fourier transform (FFT) algorithm. Fourier interpolation.
12. Fundamentals of time-frequency analysis. * Bibliography - Červený, V.: Spectral analysis in geophysics I, SPN, Prague 1979, lecture notes in Czech. - Červený, V.: Fourier spectral analysis, MFF UK, Prague 1987, lecture notes in Czech - Bezvoda V., Ježek J., Saic S., Segeth K.: Twodimwnsional discrete Fourier transform and its applications. I Theory. SPN, Prague 1987, lecture notes in Czech - Bracewell R.N: The Fourier transform and its applications, McGraw-Hill, 1978
Annotation
Fourier series. Fourier transform.
Filters. Hilbert transform.
Analytic signals. Fourier transform and Fourier series of discrete signals.
Dicrete Fourier transform. Aliasing.
Fast Fourier transform. Time-frequency analysis.