# Problems (Linear Systems)

#### 1.)

Let be a Fourier transformation of . Determine the inverse Fourier transformation i.e. function under the following condition:

#### 2.)

Determine the Laplace transformation of the following functions:

, and

#### 3.)

Determine the inverse Laplace transformation of the following function:

#### 4.)

Find the solution of the following differential equation!

where is the initial condition.

#### 5.)

Find the solution of the following linear second ordered differential equation!

, with the initial conditions

#### 6.)

Find the solution of the following linear third ordered differential equation!

, with the initial conditions

#### 7.)

Let be the step response function of a linear shift invariant system. Determine the system response function if the input function is !

#### 8.)

Two diagrams of a characteristic transfer function (amplitude and phase) are presented below. Shape-preserving transfer is executing within the frequency band. Please, determine the weighting function of the linear invariant system.

Amplitude diagram

Phase diagram

#### 9.)

Step response function of a system is : . Please determine the weighting function of the system!

#### 10.)

Transfer characteristic of a system is as follow:

Please determine the step response function of the system by the transfer characteristic!

#### 11.)

Solve the following partial differential equation with the following initial conditions

And