Difference between revisions of "Linear and Time Invariant Systems"

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$\Rightarrow \hspace{0.5cm}\text{The English translation is completed with the exception of the last Chapter.}$
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===Brief summary===
  
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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".
  
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Based on the book&nbsp;[[Signaldarstellung|$\text{Signal Representation}$]], it is described how to mathematically capture the influence of a filter on deterministic signals.
 
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*The book defines distortions and describes the Laplace transform for causal systems as well as the properties of electric leads.
 
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*The filter influence on a random signal is covered later in Chapter 5 of the book&nbsp;[[Stochastische Signaltheorie|$\text{Theory of Stochastic Signals}$]]&nbsp;.
 
  
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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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The subject matter corresponds to a&nbsp; $\text{lecture with two semester hours per week (sh/w) and another sh/w of exercises}$.
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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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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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In addition to these theory pages, we also offer tasks and multimedia modules on this topic, which could help to clarify the teaching material:
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===Exercises and multimedia===
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*[https://en.lntwww.de/Category:Linear_and_Time-Invariant_Systems:_Exercises  $\text{Exercises}$]
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*[[LNTwww:Learning_Videos_to_Linear_and_Time_Invariant_Systems|$\text{Learning videos}$]]
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{{BlaueBox|TEXT=
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*[[LNTwww:LNTwww:Applets_to_"Linear_and_Time_Invariant_Systems"|$\text{Applets}$]]&nbsp;  
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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:
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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>
 
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$\text{Further links:}$
 
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<br><br>
 
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$(1)$&nbsp; &nbsp; [[LNTwww:Bibliography_to_Linear_and_Time_Invariant_Systems|$\text{Bibliography for the book}$]]
 
  
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$(2)$&nbsp; &nbsp; [[LNTwww:General_notes_about_"Linear_and_Time_Invariant_Systems"|$\text{General notes about the book}$]] &nbsp; (authors,&nbsp; other participants,&nbsp; materials as a starting point for the book,&nbsp; list of sources)
 
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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

$(4)$    $\text{Bibliography}$

$(5)$    $\text{Impressum}$






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