Difference between revisions of "Linear and Time Invariant Systems"
From LNTwww
| Line 1: | Line 1: | ||
| + | ===Brief summary=== | ||
| − | Based on the book [[Signaldarstellung|"Signal Representation"]], here it is described how to mathematically capture the influence of a filter on deterministic signals. | + | {{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. |
| − | + | # System theory analyses a quadripole $($"system"$)$ using "cause" ⇒ $[$input $ X(f)\bullet\! \!-\!-\!\!\circ\, x( t )]$ and "effect" ⇒ $[$output $ Y(f)\bullet\!-\!-\!\circ\, y( t )]$. | |
| − | + | # 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)$. | |
| − | + | # System distortions ⇒ $ y(t)\ne K \cdot x(t - \tau)$; distortion-free system: output and input differ by attenuation/gain and delay time. | |
| + | # Linear distortions ⇒ $ Y(f)=X(f)\cdot H(f)$ $($possibly reversible$)$; non-linear distortions ⇒ emergence of new frequencies $($irreversible distortuions$)$. | ||
| + | # Peculiarities of causal systems ⇒ $ h(t<0)\equiv 0$; Hilbert transform, Laplace transform; inverse Laplace transform ⇒ residue theorem. | ||
| + | #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|"Theory of Stochastic Signals"]]. | ||
| + | |||
| + | ⇒ First a »'''content overview'''« on the basis of the »'''four main chapters'''« with a total of »'''twelve individual chapters'''«.}} | ||
| − | |||
===Contents=== | ===Contents=== | ||
| Line 35: | Line 43: | ||
}} | }} | ||
{{Collapsible-Fuß}} | {{Collapsible-Fuß}} | ||
| + | |||
| + | ===Exercises and multimedia=== | ||
| + | |||
| + | {{BlaueBox|TEXT= | ||
| + | In addition to these theory pages, we also offer exercises and multimedia modules on this topic, which could help to clarify the teaching material: | ||
| + | |||
| + | $(1)$ [https://en.lntwww.de/Category:Linear_and_Time-Invariant_Systems:_Exercises $\text{Exercises}$] | ||
| + | |||
| + | $(2)$ [[LNTwww:Learning_Videos_to_Linear_and_Time_Invariant_Systems|$\text{Learning videos}$]] | ||
| + | |||
| + | $(3)$ [[LNTwww:LNTwww:Applets_to_"Linear_and_Time_Invariant_Systems"|$\text{Applets}$]]}} | ||
| + | |||
| + | |||
| + | ===Further links=== | ||
| + | |||
| + | {{BlaueBox|TEXT= | ||
| + | $(4)$ [[LNTwww:Bibliography_to_Signal_Representation|$\text{Bibliography for the book}$]] | ||
| + | |||
| + | $(5)$ [[LNTwww:General_notes_about_Signal_Representation|$\text{General notes about the book}$]] $($authors, other participants, materials as a starting point for the book, references$)$}} | ||
| + | <br><br> | ||
In addition to these theory pages, we also offer exercisess and multimedia modules on this topic, which could help to clarify the teaching material: | In addition to these theory pages, we also offer exercisess and multimedia modules on this topic, which could help to clarify the teaching material: | ||
Revision as of 19:40, 17 February 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.
- System theory analyses a quadripole $($"system"$)$ using "cause" ⇒ $[$input $ X(f)\bullet\! \!-\!-\!\!\circ\, x( t )]$ and "effect" ⇒ $[$output $ Y(f)\bullet\!-\!-\!\circ\, y( t )]$.
- 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)$.
- System distortions ⇒ $ y(t)\ne K \cdot x(t - \tau)$; distortion-free system: output and input differ by attenuation/gain and delay time.
- Linear distortions ⇒ $ Y(f)=X(f)\cdot H(f)$ $($possibly reversible$)$; non-linear distortions ⇒ emergence of new frequencies $($irreversible distortuions$)$.
- Peculiarities of causal systems ⇒ $ h(t<0)\equiv 0$; Hilbert transform, Laplace transform; inverse Laplace transform ⇒ residue theorem.
- 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«.
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
$(4)$ $\text{Bibliography for the book}$
$(5)$ $\text{General notes about the book}$ $($authors, other participants, materials as a starting point for the book, references$)$
In addition to these theory pages, we also offer exercisess and multimedia modules on this topic, which could help to clarify the teaching material:
$\text{Further links:}$
$(1)$ $\text{Bibliography for the book}$
$(2)$ $\text{General notes about the book}$ (authors, other participants, materials as a starting point for the book, references)
