Integral and Discrete Transform

Course guarantor:

Lecturer:

Tutor:

Supervisor:

Department of Technical Mathematics

Synopsis:

Introduction to complex analysis. Laplace transform - basic properties, application to solution of problems for ordinary and partial differential equations. Discrete Laplace transform and Z-transform - basic properties, aplication to solution of diference equations. Fourier series, Fourier integral, Fourier integral transform, Fourier spectra of nonperiodic signal. Solution of exercises is demonstrated using MAPLE software.

Requirements:

Syllabus of lectures:

1. - 3. weekComplex function of complex variable: basic functions exp(z), sin(z), cos(z), ... , function derivative, 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.

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

6. - 8. week1Discrete Laplace a Z transform: basic properties, inverse transform, application of Z transform to solution of difference equations.

8. - 10. weekFourier 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.

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

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

Syllabus of tutorials:

1. - 3. weekComplex function of complex variable: basic functions exp(z), sin(z), cos(z), ... , function derivative, 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.

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

6. - 8. week1Discrete Laplace a Z transform: basic properties, inverse transform, application of Z transform to solution of difference equations.

8. - 10. weekFourier 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.

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

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

Study Objective:

Study materials:

E.Kreyszig: Advanced Engineering Mathematics, John Wiley & Sons, 1993

Note:

Further information:

No time-table has been prepared for this course

The course is a part of the following study plans: