Modul

The assessment of success takes place in the form of a written examination lasting 120 minutes. In addition, the completion of the written work in the workshop is a prerequisite for passing the module.

The students master the basics, properties and calculation rules of the Laplace transformation and can apply these to solve linear differential equations.

• The students are able to use the Laplace transformation to describe time-continuous dynamic systems.

• The students know some basics of complex analysis in the context of integral transformations such as Laurent expansion and theorem of residuals.

• The students know the complex inverse formula of the La-place transformation and can use it for complicated image functions.

• The students know the two-sided Laplace transformation and master the basics, properties and calculation rules of the Fourier transformation.

• Students can use the Fourier transformation to describe time-continuous signals in the frequency domain.

• Students are familiar with the sampling theorem for converting time-continuous into time-discrete signals and can use the discrete Fourier transform to describe time-discrete signals in the frequency domain.

• The students are familiar with the basics, properties and calculation rules of the z-transformation.

• Students can use the z-transformation to describe time-discrete systems.

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• Laplace transform
  • Motivation and Definition
  • Properties and Examples
• Laplace transform of ordinary differential equations
  • Ordinary and generalized differentiation rule
  • Laplace transform of general linear differential equations with constant coefficients
  • Back transformation via the partial fraction decomposition of rational functions
  • Calculation rules of the Laplace transform (1):
    Integration rule and damping rule
  • Back transformation over the convolution rule of the Laplace transformation
  • Calculation rules of the Laplace transform (2):
    Displacement rules and limit theorems

• Characterization of the transfer behavior of dynamic systems with transfer and weight function
• Function theory: Laurent expansion, residual and residual theorem
• Complex inversion formula of the Laplace transformation
  • Derivation of the complex inverse formula
  • Calculation of the complex inverse integral
• Two-sided Laplace Transform and Fourier Transform
  • Two-sided Laplace Transform
  • Definition and properties of the Fourier transform
  • Calculation rules and correspondences of the Fourier transform
• z-Transform
  • Definition, properties and calculation rules of the z-transform
  • Use for the solution of difference equations
• Mathematical basics: Spaces
• Time-continuous signals
  • Fourier series
  • Fourier transform
  • Test signals
  • General signal properties
• Continuous-time systems
  • Properties
  • System description by differential equations
  • Laplace transform
  • System function
  • Frequency selective filters
• Discrete-Time Signals
  • Fourier transform of discrete-time signals
  • Sampling theorem
  • Discrete Fourier Transform
• Discrete-Time Systems
  • Properties
  • System description by difference equations
  • The z-transformation
  • System function
  • Discrete-time representation of continuous systems
  • Frequency selective filters

Knowledge of HM3 is helpful.

Total approx. 240h, of which

Attendance time in lectures and exercises: 75h

Preparation/follow-up of the lectures and exercises: 130h

3. Exam preparation and presence in the same: 40h

Preparation time for the workshop: 5h

Presence time in the workshop: 15h

Preparation of the protocol for the workshop: 5h

Total: 270 LP = 9 LP