Problems (Linear Systems)

1.)

Let $$F(j\omega) = e^{-\omega^{2}T^{2}}$ be a Fourier transformation of $f(x) . Determine the inverse Fourier transformation i.e. $f(x) function under the following condition:

$$ \int_{0}^{\infty}e^{-ax_{0}^{2}}\text{cos}(bx_{0})dx_{0} = \frac{1}{2} \sqrt{\frac{\pi}{a}}e^{-(\frac{b}{4a})^{2}}$

Solution

2.)

Determine the Laplace transformation of the following functions:

$$f(x) = e^{-\alpha x}\text{cos}(\omega x)$ , and $$f_{1}(x)=e^{-\alpha x}\text{sin}(\omega x)$

Solution

3.)

Determine the inverse Laplace transformation of the following function:

$$ F(s) = \frac{s^{2}+3s+2}{(s-1)(s-2)^{3}(s-4)} \quad \quad f(x)=\mathcal{L}^{-1}\left\{F(s)\right\} = ?$

Solution

4.)

Find the solution of the following differential equation!

$$ \frac{d^{2}y(x)}{dx^{2}}+4\frac{dy(x)}{dx}+5y(x) = 10,$

where $$y(0)=\frac{dy(x)}{dx}\Big{|}_{x=0}=0$ is the initial condition.

Solution

5.)

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

$$ \frac{d^{2}y(x)}{dx^{2}}+a^{2}y(x)=A\text{cos}(\alpha x),$ , with the initial conditions $y(0)=y'(0)=0.

Solution

6.)

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

$$ \frac{d^{3}y(x)}{dx^{3}}+5\frac{d^{2}y(x)}{dx^{2}}+8\frac{dy(x)}{dx}+4y(x)=1(x),$ , with the initial conditions $y''(0)=y'(0)=y(0)=0

Solution

7.)

Let $$ h(t)=-A(e^{s_1t}- e^{s_2t})$ be the step response function of a linear shift invariant system. Determine the system response function if the input function is $$ f(t)=F_0\text{sin}(\omega t+\varphi)$ !

Solution

8.)

Two diagrams of a characteristic transfer function (amplitude and phase) are presented below. Shape-preserving transfer is executing within the $$ 0<\omega < \omega_0$ frequency band. Please, determine the weighting function of the linear invariant system.

Amplitude diagram

Phase diagram

Solution

9.)

Step response function of a system is : $$ h(t)= \left [ 2-e^{-5t}+2e^{-10t} \right ] 1(t)$ . Please determine the weighting function of the system!

Solution

10.)

Transfer characteristic of a system is as follow:
$$ w(j\omega)= \frac{2(1+j\omega)}{4+3j\omega}$
Please determine the step response function of the system by the transfer characteristic!

Solution

11.)

Solve the following partial differential equation $$ \bigtriangledown^2U(x,y,t)= \frac{1}{c^2}\frac{\partial U(x,y,t) }{\partial t}$ with the following initial conditions
$ U(x,y,0)=f(x,y)