# Linear and Time Invariant Systems

## Contents

### Brief summary

Based on the book  »Signal Representation«,  here it is described how to mathematically capture the influence of a filter on deterministic signals.

1. System theory analyses a quadripole  $($»system«$)$  using  »cause«   ⇒   $[$input   $X(f)\ \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\,\ x(t)]$  and  »effect«   ⇒   $[$output  $Y(f)\ \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\,\, y( t )]$.
2. Indicator in frequency domain is the  »frequency response«  $H(f)=Y(f)/X(f)$,  in time domain the  »impulse response»  $h(t)$,  where  $y(t)=x(t)\star h(t)$.
3. System distortions   ⇒   $y(t)\ne K \cdot x(t - \tau)$;  distortion-free system:  output and input differ only by attenuation/gain and/or delay time.
4. Linear distortions $($possibly reversible$)$   ⇒   $Y(f)=X(f)\cdot H(f)$;   non-linear distortions   ⇒   emergence of new frequencies  $($irreversible distortions$)$.
5. Peculiarities of causal systems   ⇒   $h(t<0)\equiv 0$;  Hilbert transform,  Laplace transform; inverse Laplace transform   ⇒   residue theorem.
6. Some results of line transmission theory;  coaxial cable systems   ⇒   "white noise";  copper twisted pairs   ⇒   dominant is  "near-end crosstalk".

The filter influence on a random signal is only dealt with in the last chapter of the book  »Theory of Stochastic Signals«.

⇒   First the  »content overview«  on the basis of the  »four main chapters«  with a total of  »twelve individual chapters«  and  »93 sections«.

### Exercises and multimedia

In addition to these theory pages,  we also offer exercises and multimedia modules on this topic,  which could help to clarify the teaching material:

$(1)$    $\text{Exercises}$

$(3)$    $\text{Applets}$

$(5)$    $\text{Impressum}$