Difference between revisions of "Linear and Time Invariant Systems"

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Based on the book [[Signaldarstellung|$\text{Signal Representation}$]], it is described how to mathematically capture the influence of a filter on deterministic signals.
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===Brief summary===
*The book defines distortions and describes the Laplace transform for causal systems as well as the properties of electric leads.
 
*The filter influence on a random signal is covered later in Chapter 5 of the book [[Stochastische Signaltheorie|$\text{Theory of Stochastic Signals}$]] .
 
  
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{{BlaueBox|TEXT=Based on the book  [[Signaldarstellung|»Signal Representation«]],  here it is described how to mathematically capture the influence of a filter on deterministic signals.
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# System theory analyses a quadripole  $($»system«$)$  using  »cause«   ⇒   $[$input   $ X(f)\ \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\,\ x(t)]$  and  »effect«   ⇒   $[$output  $ Y(f)\ \bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\,\, y( t )]$. 
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# 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)$.
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# 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.
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# Linear distortions $($possibly reversible$)$   ⇒   $ Y(f)=X(f)\cdot H(f)$;    non-linear distortions   ⇒   emergence of new frequencies  $($irreversible distortions$)$.
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# Peculiarities of causal systems &nbsp; &rArr; &nbsp; $ h(t<0)\equiv 0$;&nbsp; Hilbert transform,&nbsp; Laplace transform; inverse Laplace  transform &nbsp; &rArr; &nbsp; residue theorem.
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#Some results of line transmission theory;&nbsp; coaxial cable systems &nbsp; &rArr; &nbsp; "white noise";&nbsp; copper twisted pairs &nbsp; &rArr; &nbsp; dominant is&nbsp; "near-end crosstalk".
  
The subject matter corresponds to a&nbsp; $\text{lecture with two semester hours per week (SWS) and another SWS of exercises}$.
 
  
First of all, here is an overview of the contents based on the&nbsp; $\text{four main chapters}$&nbsp; with a total of&nbsp; $\text{twelve individual chapters}$.
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The filter influence on a random signal is only dealt with in the last chapter of the book &nbsp;[[Theory_of_Stochastic_Signals|&raquo;Theory of Stochastic Signals&laquo;]].
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&rArr; &nbsp; First the&nbsp; &raquo;'''content overview'''&laquo;&nbsp; on the basis of the&nbsp; &raquo;'''four main chapters'''&laquo;&nbsp; with a total of&nbsp; &raquo;'''twelve individual chapters'''&laquo;&nbsp; and&nbsp; &raquo;'''93 sections'''&laquo;.}}
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|submenu=
 
|submenu=
 
*[[/Classification of the Distortions/]]
 
*[[/Classification of the Distortions/]]
*[[/Nonlinear Distortion/]]
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*[[/Nonlinear Distortions/]]
 
*[[/Linear Distortions/]]
 
*[[/Linear Distortions/]]
 
}}
 
}}
{{Collapse3 | header=Description of Causal  Realisable Systems
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{{Collapse3 | header=Description of Causal  Realizable Systems
 
|submenu=
 
|submenu=
 
*[[/Conclusions from the Allocation Theorem/]]
 
*[[/Conclusions from the Allocation Theorem/]]
*[[/Laplace Transform and P-Transfer Function/]]
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*[[/Laplace Transform and p-Transfer Function/]]
 
*[[/Inverse Laplace Transform/]]
 
*[[/Inverse Laplace Transform/]]
 
}}
 
}}
 
{{Collapse4 | header=Properties of Electrical Cables
 
{{Collapse4 | header=Properties of Electrical Cables
 
|submenu=
 
|submenu=
*[[/Some Results from Transmission Line Theory/]]
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*[[/Some Results from Line Transmission Theory/]]  
 
*[[/Properties of Coaxial Cables/]]
 
*[[/Properties of Coaxial Cables/]]
*[[/Properties of Balanced Copper Pairs/]]
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*[[/Properties of Balanced Copper Pairs/]]  
 
}}
 
}}
 
{{Collapsible-Fuß}}
 
{{Collapsible-Fuß}}
  
In addition to these theory pages, we also offer assignments and multimedia modules on this topic that could help clarify the subject matter:
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===Exercises and multimedia===
*[https://en.lntwww.de/Category:Linear_and_Time-Invariant_Systems:_Exercises  $\text{Exercises}$]
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*[[LNTwww:Lernvideos_zu_Lineare_zeitinvariante_Systeme|$\text{Lernvideos}$]]
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{{BlaueBox|TEXT=
*[[LNTwww:HTML5-Applets_zu_Lineare_zeitinvariante_Systeme|$\text{neu gestaltete Applets}$]]&nbsp; (basierend auf HTML5 und JavaScript, auch auf Smartphones lauffähig)
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In addition to these theory pages,&nbsp; we also offer exercises and multimedia modules on this topic,&nbsp; which could help to clarify the teaching material:
*[[LNTwww:SWF-Applets_zu_Lineare_zeitinvariante_Systeme|$\text{frühere Applets}$]]&nbsp; (basierend auf SWF, lauffähig nur unter WINDOWS mit "Adobe Flash Player")
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$(1)$&nbsp; &nbsp; [https://en.lntwww.de/Category:Linear_and_Time-Invariant_Systems:_Exercises  $\text{Exercises}$]
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$(2)$&nbsp; &nbsp; [[LNTwww:Learning_Videos_to_Linear_and_Time_Invariant_Systems|$\text{Learning videos}$]]
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$(3)$&nbsp; &nbsp; [[LNTwww:LNTwww:Applets_to_"Linear_and_Time_Invariant_Systems"|$\text{Applets}$]]}}
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===Further links===
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{{BlaueBox|TEXT=
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$(4)$&nbsp; &nbsp; [[LNTwww:Bibliography_to_"Linear_and_Time_Invariant_Systems"|$\text{Bibliography}$]]
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$(5)$&nbsp; &nbsp; [[LNTwww:Imprint_for_the_book_"Linear_and_Time_Invariant_Systems"|$\text{Impressum}$]] }}
 
<br><br>
 
<br><br>
$\text{Weitere Links:}$
 
<br><br>
 
$(1)$&nbsp; &nbsp; [[LNTwww:Literaturempfehlung_zu_Lineare_zeitinvariante_Systeme|$\text{Literaturempfehlungen zum Buch}$]]
 
  
$(2)$&nbsp; &nbsp; [[LNTwww:Weitere_Hinweise_zum_Buch_Lineare_zeitinvariante_Systeme|$\text{Allgemeine Hinweise zum Buch}$]] &nbsp; (Autoren,&nbsp; Weitere Beteiligte,&nbsp; Materialien als Ausgangspunkt des Buches,&nbsp; Quellenverzeichnis)
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{{Display}}
 
{{Display}}

Latest revision as of 17:38, 1 November 2023

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«.


Contents

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