Difference between revisions of "Aufgaben:Exercise 2.1: Linear? Or Non-Linear?"

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{{quiz-Header|Buchseite=Lineare zeitinvariante Systeme/Klassifizierung der Verzerrungen
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{{quiz-Header|Buchseite=Linear_and_Time_Invariant_Systems/Classification_of_the_Distortions}}
}}
 
  
[[File:P_ID879__LZI_A_2_1.png|right|frame|Zusammengeschaltetes System]]
+
[[File:P_ID879__LZI_A_2_1.png|right|frame|Interconnected system]]
Wir betrachten die skizzierte Anordnung mit Eingang  $x(t)$  und Ausgang  $z(t)$:
+
We consider the sketched arrangement with input  $x(t)$  and output $z(t)$:
  
*Das System  $S_1$  ist durch folgende Gleichung beschreibbar:
+
*The system $S_1$  is describable by the following equation:
 
:$$y(t) = x(t) + {1 \, \rm V}^{\rm -1} \cdot x^2(t) .$$
 
:$$y(t) = x(t) + {1 \, \rm V}^{\rm -1} \cdot x^2(t) .$$
  
*Über das System $S_2$ mit Eingang $y(t)$  und Ausgang $z(t)$ ist nichts weiter bekannt.
+
*Nothing else is known about the system  $S_2$  with input  $y(t)$  and output  $z(t)$ .
  
*Das System  $S_3$  ist die Zusammenschaltung von $S_1$ und $S_2$.
+
*The system $S_3$  is the interconnection of  $S_1$  and  $S_2$.
  
  
An den Eingang wird eine Schwingung mit der Frequenz $f_0 = 5 \ \rm kHz$ angelegt:
+
An oscillation with frequency  $f_0 = 5 \ \rm kHz$  is applied to the input:
 
:$$x(t) = {2 \, \rm V} \cdot {\rm cos}(2\pi  f_0  t ) .$$
 
:$$x(t) = {2 \, \rm V} \cdot {\rm cos}(2\pi  f_0  t ) .$$
  
Damit erhält man am Ausgang des Gesamtsystems $S_3$:
+
Hence, at the output of the overall system  $S_3$ the following is obtained:
 
:$$z(t) = {1 \, \rm V} \cdot {\rm sin}(2\pi  f_0  t ) .$$
 
:$$z(t) = {1 \, \rm V} \cdot {\rm sin}(2\pi  f_0  t ) .$$
  
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''Hinweise:''  
+
 
*Die Aufgabe gehört zum Kapitel   [[Lineare_zeitinvariante_Systeme/Klassifizierung_der_Verzerrungen|Klassifizierung der Verzerrungen]].  
+
 
 +
 
 +
''Please note:''  
 +
*The exercise belongs to the chapter   [[Linear_and_Time_Invariant_Systems/Classification_of_the_Distortions|Classification of the Distortions]].  
 
   
 
   
*Gegeben ist die folgende trigonometrische Beziehung:
+
*The following trigonometric relation is given:
 
:$$\cos^2(\alpha) = {1}/{2} \cdot \big[ 1 + \cos(2\alpha)\big].$$
 
:$$\cos^2(\alpha) = {1}/{2} \cdot \big[ 1 + \cos(2\alpha)\big].$$
  
  
===Fragebogen===
+
===Questions===
  
 
<quiz display=simple>
 
<quiz display=simple>
{Wie lautet das Signal &nbsp;$y(t)$? Welcher Signalwert ergibt sich zum Zeitnullpunkt?
+
{What is the signal&nbsp;$y(t)$? What is the signal value at time zero?
 
|type="{}"}
 
|type="{}"}
 
$y(t = 0) \ = \ $ { 6 1% } $\ \rm V$
 
$y(t = 0) \ = \ $ { 6 1% } $\ \rm V$
  
  
{Welche richtigen Schlüsse könnte ein Beobachter ziehen, der nur die Signale &nbsp;$x(t)$&nbsp; und &nbsp;$z(t)$&nbsp; kennt, aber nicht den Aufbau von $S_3$?
+
{What correct conclusions could be drawn by an observer who only knows the signals&nbsp;$x(t)$&nbsp; and &nbsp;$z(t)$&nbsp; but not the structure of&nbsp; $S_3$?
 
|type="[]"}
 
|type="[]"}
- $S_3$&nbsp; ist ein ideales System.
+
- $S_3$&nbsp; is an ideal system.
+ $S_3$&nbsp; ist ein verzerrungsfreies System.
+
+ $S_3$&nbsp; is a distortion-free system.
+ $S_3$&nbsp; ist ein linear verzerrendes System.
+
+ $S_3$&nbsp; is a linearly distorting system.
- $S_3$&nbsp; ist ein nichtlinear verzerrendes System.
+
- $S_3$&nbsp; is a non-linearly distorting system.
  
  
{Welche Schlüsse müsste der Beobachter ziehen, wenn ihm alle Informationen von der Angabenseite bekannt sind?
+
{What conclusions would the observer have to draw if all the information from the information section is known to them?
 
|type="[]"}
 
|type="[]"}
- $S_2$&nbsp; ist ein verzerrungsfreies System.
+
- $S_2$&nbsp; is a distortion-free system.
+ $S_2$&nbsp; ist ein linear verzerrendes System.
+
+ $S_2$&nbsp; is a linearly distorting system.
- $S_2$&nbsp; ist ein nichtlinear verzerrendes System.
+
- $S_2$&nbsp; is a non-linearly distorting system.
  
  
{Welches Signal &nbsp;$z(t)$&nbsp; könnte sich mit der Eingangsfrequenz &nbsp;$f_0 = 10 \ \rm kHz$ ergeben?
+
{What signal&nbsp;$z(t)$&nbsp; could arise as a result with the input frequency&nbsp;$f_0 = 10 \ \rm kHz$&nbsp;?
 
|type="[]"}
 
|type="[]"}
+ Das Signal $z(t)$ ist für alle Zeiten Null.
+
+ The signal&nbsp; $z(t)$&nbsp; is zero for all times.
- Ein Signal der Form &nbsp;$z(t) = A \cdot {\rm cos}(2\pi  \cdot  10 \ {\rm kHz}  \cdot  t ) ,$ mit $A \ne 0.$
+
- A signal of the form&nbsp;$z(t) = A \cdot {\rm cos}(2\pi  \cdot  10 \ {\rm kHz}  \cdot  t ) ,$ with $A \ne 0.$
+ Ein Signal der Form &nbsp;$z(t) = A \cdot {\rm cos}(2\pi  \cdot  20 \ {\rm kHz}  \cdot  t ) ,$ mit $A \ne 0.$
+
+ A signal of the form&nbsp;$z(t) = A \cdot {\rm cos}(2\pi  \cdot  20 \ {\rm kHz}  \cdot  t ) ,$ with $A \ne 0.$
  
  
 
</quiz>
 
</quiz>
  
===Musterlösung===
+
===Solution===
 
{{ML-Kopf}}
 
{{ML-Kopf}}
  
'''(1)'''&nbsp; Aufgrund der Kennlinie mit linearem und quadratischem Anteil gilt:
+
'''(1)'''&nbsp; The following holds due to the characteristic curve with linear and quadratic components:
 
:$$y(t) = {2 \, \rm V} \cdot {\rm cos}(2\pi  f_0  t ) + {1 \,
 
:$$y(t) = {2 \, \rm V} \cdot {\rm cos}(2\pi  f_0  t ) + {1 \,
 
\rm V}^{\rm -1} \cdot ({2 \, \rm V})^2 \cdot {\rm cos}^2(2\pi  f_0
 
\rm V}^{\rm -1} \cdot ({2 \, \rm V})^2 \cdot {\rm cos}^2(2\pi  f_0
t )  = {2 \, \rm V} \cdot \left[ 1 + {\rm cos}(2\pi \cdot  f_0 \cdot  t
+
t )  = {2 \, \rm V} \cdot \big[ 1 + {\rm cos}(2\pi \cdot  f_0 \cdot  t
) +{\rm cos}(2\pi  \cdot 2f_0 \cdot  t )  \right].$$
+
) +{\rm cos}(2\pi  \cdot 2f_0 \cdot  t )  \big].$$
 +
 
 +
*Therefore, the <u>signal value 6 V</u> occurs at time&nbsp; $t= 0$&nbsp;.
 +
 
 +
 
  
Zum Zeitpunkt $t= 0$ tritt somit der <u>Signalwert 6 V</u> auf.
+
'''(2)'''&nbsp; <u>Alternatives 2 snd 3</u> are possible:
 +
*An ideal system is out of the question because of&nbsp; $z(t) &ne; x(t)$&nbsp;.
 +
*With only one input frequency&nbsp; $(f_0 = 5 \ \rm kHz)$&nbsp; in the test signal it is not possible to make a statement about whether a second frequency component with&nbsp; $f \ne f_0$&nbsp; would also be attenuated by&nbsp;$\alpha = 0.5$&nbsp; and delayed by&nbsp;$\tau = T_0/4  = 50 \ &micro;\rm  s$&nbsp;.
 +
*If for the second frequency component&nbsp;$\alpha = 0.5$&nbsp; and &nbsp;$\tau = T_0/4  = 50 \ &micro; \rm s$ arose as a result, a ''distortion-free system'' could exist.
 +
*If for the second frequency component&nbsp;$\alpha \ne 0.5$&nbsp; and/or &nbsp;$\tau \ne T_0/4$ arose as a result, then the system would be ''linearly distorting''.
 +
*The last alternative would have to be logically denied &ndash; although partially true &ndash; by the observer.
  
  
'''(2)'''&nbsp;  Möglich sind die <u>Alternativen 2 und 3</u>:
 
*Ein ideales System kommt wegen $z(t) &ne; x(t)$ nicht in Frage.
 
*Bei nur einer Eingangsfrequenz ($f_0 = 5 \ \rm kHz$) im Testsignal ist keine Aussage möglich, ob eine zweite Frequenzkomponente mit $f \ne f_0$ ebenfalls um $\alpha = 0.5$ gedämpft und um $\tau = T_0/4  = 50 \ &micro; s$ verzögert würde.
 
*Ergäbe sich für die zweite Frequenzkomponente $\alpha = 0.5$ und $\tau = T_0/4  = 50 \ &micro; s$, so würde ein ''verzerrungsfreies System'' vorliegen.
 
*Ergäbe sich für die zweite Frequenzkomponente $\alpha \ne 0.5$ und/oder $\tau \ne T_0/4$, so wäre das System ''linear verzerrend''.
 
*Die letzte Alternative müsste der Beobachter &ndash; obwohl teilweise zutreffend &ndash; logischerweise verneinen.
 
  
 +
'''(3)'''&nbsp; <u>Approach 2</u> is correct:
 +
*The observer would realise that&nbsp; $S_2$&nbsp; is a linearly distorting system.
 +
*For a distortion-free system,&nbsp; $z(t)$&nbsp; would additionally have to include a direct (DC) component and a&nbsp;$10 \ \rm kHz$&ndash;component,
 +
*for a non-linearly distorting system, even larger frequency components&nbsp; $($at multiples of &nbsp;$10 \ \rm kHz)$.
  
'''(3)'''&nbsp;  Richtig ist der <u>Lösungsvorschlag 2</u>:
 
*Der Beobachter würde erkennen, dass $S_2$ ein linear verzerrendes System ist.
 
*Bei einem verzerrungsfreien System müsste $z(t)$ zusätzlich noch eine Gleichkomponente und eine $10 \ \rm kHz$-Komponente beinhalten, bei einem nichtlinear verzerrenden System noch größere Frequenzanteile (bei Vielfachen von $10 \ \rm kHz$.
 
  
  
'''(4)'''&nbsp; In diesem Fall würde gelten:
+
'''(4)'''&nbsp; In this case the following would hold:
:$$y(t) = {2 \, \rm V} \cdot \left[ 1 + {\rm cos}(2\pi \cdot 10 \ {\rm kHz} \cdot  t
+
:$$y(t) = {2 \, \rm V} \cdot \big[ 1 + {\rm cos}(2\pi \cdot 10 \ {\rm kHz} \cdot  t
) +{\rm cos}(2\pi  \cdot 20 \ {\rm kHz} \cdot  t )  \right].$$
+
) +{\rm cos}(2\pi  \cdot 20 \ {\rm kHz} \cdot  t )  \big].$$
Das heißt: $Y(f)$ würde Spektrallinien bei $f = 0$, $10 \ \rm kHz$ und $20 \ \rm kHz$ aufweisen.  
+
*That is: &nbsp; $Y(f)$&nbsp; would have spectral lines at&nbsp; $f = 0$,&nbsp; $10 \ \rm kHz$&nbsp; and&nbsp; $20 \ \rm kHz$&nbsp;.  
  
Die auf der Angabenseite beschriebene Messung mit $f_0 = 5 \ \rm kHz$ hat aber gezeigt, dass $H_2(f = 0) = H_2(f = 10 \ {\rm kHz}) = 0$ gelten muss. Die einzig mögliche Signalform ist somit
+
*However, the measurement described in the information section with&nbsp; $f_0 = 5 \ \rm kHz$&nbsp; has shown that&nbsp; $H_2(f = 0) = H_2(f = 10 \ {\rm kHz}) = 0$&nbsp; must hold.  
 +
*The only possible signal form is therefore
 
:$$z(t) = {2 \, \rm V} \cdot H_2 (f = {20 \, \rm kHz})\cdot {\rm
 
:$$z(t) = {2 \, \rm V} \cdot H_2 (f = {20 \, \rm kHz})\cdot {\rm
 
cos}(2\pi \cdot {20 \, \rm kHz} \cdot t ) .$$
 
cos}(2\pi \cdot {20 \, \rm kHz} \cdot t ) .$$
  
Möglich sind also die <u>Lösungsvorschläge 1 und 3</u>, je nachdem, ob das System $S_2$ die Frequenz $20 \ {\rm kHz}$ unterdrückt oder durchlässt.
+
*So, <u>approaches 1 and 3</u> are possible depending on whether the system&nbsp; $S_2$&nbsp; suppresses or passes the frequency&nbsp; $20 \ {\rm kHz}$&nbsp;.
  
  
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[[Category:Aufgaben zu Lineare zeitinvariante Systeme|^2.1 Klassifizierung der Verzerrungen^]]
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[[Category:Linear and Time-Invariant Systems: Exercises|^2.1 Classification of the Distortions^]]

Latest revision as of 13:45, 17 November 2022

Interconnected system

We consider the sketched arrangement with input  $x(t)$  and output $z(t)$:

  • The system $S_1$  is describable by the following equation:
$$y(t) = x(t) + {1 \, \rm V}^{\rm -1} \cdot x^2(t) .$$
  • Nothing else is known about the system  $S_2$  with input  $y(t)$  and output  $z(t)$ .
  • The system $S_3$  is the interconnection of  $S_1$  and  $S_2$.


An oscillation with frequency  $f_0 = 5 \ \rm kHz$  is applied to the input:

$$x(t) = {2 \, \rm V} \cdot {\rm cos}(2\pi f_0 t ) .$$

Hence, at the output of the overall system  $S_3$ the following is obtained:

$$z(t) = {1 \, \rm V} \cdot {\rm sin}(2\pi f_0 t ) .$$





Please note:

  • The following trigonometric relation is given:
$$\cos^2(\alpha) = {1}/{2} \cdot \big[ 1 + \cos(2\alpha)\big].$$


Questions

1

What is the signal $y(t)$? What is the signal value at time zero?

$y(t = 0) \ = \ $

$\ \rm V$

2

What correct conclusions could be drawn by an observer who only knows the signals $x(t)$  and  $z(t)$  but not the structure of  $S_3$?

$S_3$  is an ideal system.
$S_3$  is a distortion-free system.
$S_3$  is a linearly distorting system.
$S_3$  is a non-linearly distorting system.

3

What conclusions would the observer have to draw if all the information from the information section is known to them?

$S_2$  is a distortion-free system.
$S_2$  is a linearly distorting system.
$S_2$  is a non-linearly distorting system.

4

What signal $z(t)$  could arise as a result with the input frequency $f_0 = 10 \ \rm kHz$ ?

The signal  $z(t)$  is zero for all times.
A signal of the form $z(t) = A \cdot {\rm cos}(2\pi \cdot 10 \ {\rm kHz} \cdot t ) ,$ with $A \ne 0.$
A signal of the form $z(t) = A \cdot {\rm cos}(2\pi \cdot 20 \ {\rm kHz} \cdot t ) ,$ with $A \ne 0.$


Solution

(1)  The following holds due to the characteristic curve with linear and quadratic components:

$$y(t) = {2 \, \rm V} \cdot {\rm cos}(2\pi f_0 t ) + {1 \, \rm V}^{\rm -1} \cdot ({2 \, \rm V})^2 \cdot {\rm cos}^2(2\pi f_0 t ) = {2 \, \rm V} \cdot \big[ 1 + {\rm cos}(2\pi \cdot f_0 \cdot t ) +{\rm cos}(2\pi \cdot 2f_0 \cdot t ) \big].$$
  • Therefore, the signal value 6 V occurs at time  $t= 0$ .


(2)  Alternatives 2 snd 3 are possible:

  • An ideal system is out of the question because of  $z(t) ≠ x(t)$ .
  • With only one input frequency  $(f_0 = 5 \ \rm kHz)$  in the test signal it is not possible to make a statement about whether a second frequency component with  $f \ne f_0$  would also be attenuated by $\alpha = 0.5$  and delayed by $\tau = T_0/4 = 50 \ µ\rm s$ .
  • If for the second frequency component $\alpha = 0.5$  and  $\tau = T_0/4 = 50 \ µ \rm s$ arose as a result, a distortion-free system could exist.
  • If for the second frequency component $\alpha \ne 0.5$  and/or  $\tau \ne T_0/4$ arose as a result, then the system would be linearly distorting.
  • The last alternative would have to be logically denied – although partially true – by the observer.


(3)  Approach 2 is correct:

  • The observer would realise that  $S_2$  is a linearly distorting system.
  • For a distortion-free system,  $z(t)$  would additionally have to include a direct (DC) component and a $10 \ \rm kHz$–component,
  • for a non-linearly distorting system, even larger frequency components  $($at multiples of  $10 \ \rm kHz)$.


(4)  In this case the following would hold:

$$y(t) = {2 \, \rm V} \cdot \big[ 1 + {\rm cos}(2\pi \cdot 10 \ {\rm kHz} \cdot t ) +{\rm cos}(2\pi \cdot 20 \ {\rm kHz} \cdot t ) \big].$$
  • That is:   $Y(f)$  would have spectral lines at  $f = 0$,  $10 \ \rm kHz$  and  $20 \ \rm kHz$ .
  • However, the measurement described in the information section with  $f_0 = 5 \ \rm kHz$  has shown that  $H_2(f = 0) = H_2(f = 10 \ {\rm kHz}) = 0$  must hold.
  • The only possible signal form is therefore
$$z(t) = {2 \, \rm V} \cdot H_2 (f = {20 \, \rm kHz})\cdot {\rm cos}(2\pi \cdot {20 \, \rm kHz} \cdot t ) .$$
  • So, approaches 1 and 3 are possible depending on whether the system  $S_2$  suppresses or passes the frequency  $20 \ {\rm kHz}$ .