Integral and discrete transforms

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Department of Technical Mathematics

Synopsis:

Complex function of complex variable: basic functions, derivatives, analytic function, Cauchy-Riemann conditions, line integral, Cauchy integral theorem, Cauchy integral formula, Taylor serie of analytic function, Laurent serie, singular points, residual of function in singular point.

Laplace transform: basic properties, inverse Laplace transform, Laplace transform of Dirac a Heaviside function, application of Laplace transform to solution of ODE and PDE.

Discrete Laplace a Z transform: basic properties, inverse transform, application of Z transform to solution of difference equations.

Fourier series: Fourier serie of periodic function, amplitude spectra, application to solutions of ODE with periodical forcing term, solution of PDE by Fourier method, extension to nonperiodic functions, Fourier integral.

Fourier transform: basic properties, amplitude spectra of nonperiodic function, application to solution of PDE, discrete Fourier transform (DFT), fast Fourier transform (FFT).

Todays techniques used for real time transfer of signal: windowed Fourier transform, wavelet transform, Hilbert-Huang transform.

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Study materials:

Schiff J. L.: The Laplace Transform - Theory and Applications, Springer-Verlag New York, 1999.

Gasquet C., Witomski P.: Fourier Analysis and Applications - Filtering, Numerical Computation, Wavelets, Springer-Verlag New York, 1999.

Mallat S.: A Wavelet Tour of Signal Processing, Academic Press, 2008.

Veit J.: Integrální transformace, SNTL, 1979.

Distant learning references: lecture texts (online)

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Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: