Linear and Time Invariant Systems

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 by attenuation/gain and delay time.
4. Linear distortions   ⇒   $ Y(f)=X(f)\cdot H(f)$  $($possibly reversible$)$;  non-linear distortions   ⇒   emergence of new frequencies  $($irreversible distortuions$)$.
5. Peculiarities of causal systems   ⇒   $ h(t<0)\equiv 0$;  Hilbert transform,  Laplace transform; inverse Laplace transform   ⇒   residue theorem.
6. Some results of line 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 a  »content overview«  on the basis of the  »four main chapters«  with a total of  »twelve individual chapters«  and  »93 sections«.

Contents

Basics of System Theory

• System Description in Frequency Domain
• System Description in Time Domain
• Some Low-Pass Functions in Systems Theory

Signal Distortion and Equalization

• Classification of the Distortions
• Nonlinear Distortions
• Linear Distortions

Description of Causal Realizable Systems

• Conclusions from the Allocation Theorem
• Laplace Transform and p-Transfer Function
• Inverse Laplace Transform

Properties of Electrical Cables

• Some Results from Line Transmission Theory
• Properties of Coaxial Cables
• Properties of Balanced Copper Pairs

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:

$(1)$    $\text{Exercises}$

$(2)$    $\text{Learning videos}$

$(3)$    $\text{Applets}$

Further links

$(4)$    $\text{Bibliography}$

$(5)$    $\text{Impressum}$

$(1)$    $\text{Bibliography for the book}$

$(2)$    $\text{General notes about the book}$   (authors,  other participants,  materials as a starting point for the book,  references)