Linear and Time Invariant Systems

From LNTwww

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:

$(1)$    $\text{Exercises}$

$(2)$    $\text{Learning videos}$

$(3)$    $\text{Applets}$

Further links