Difference between revisions of "Aufgaben:Exercise 1.1: Music Signals"

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{{quiz-Header|Buchseite=Signaldarstellung/Prinzip der Nachrichtenübertragung}}
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{{quiz-Header|Buchseite=Signal_Representation/Principles_of_communication}}
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===Aufgabe zu [[Signaldarstellung/Prinzip_der_Nachrichtenübertragung|Prinzip der Nachrichtenübertragung]]===
 
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[[File:P_ID339__Sig_A_1_1.png|right|Musiksignale, verrauscht und verzerrt]]
 
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Nebenstehend sehen Sie einen ca. 30 ms langen Ausschnitt eines Musiksignals <math>q(t)</math>. Es handelt sich um das Stück &bdquo;Für Elise&rdquo; von Ludwig van Beethoven.
 
  
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Darunter gezeichnet sind zwei Sinkensignale <math>v_1(t)</math> und <math>v_2(t)</math>, die nach der Übertragung des Musiksignals <math>q(t)</math> über zwei unterschiedliche Kanäle aufgezeichnet wurden. Mit Hilfe der nachfolgenden Buttons können Sie sich die jeweils ersten dreizehn Sekunden der drei Audiosignale <math>q(t)</math>, <math>v_1(t)</math> und <math>v_2(t)</math> anhören.
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[[File:P_ID339__Sig_A_1_1.png|right|frame|Music signals, <br>original, noisy and/or distorted?]]
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On the right you see a&nbsp; $\text{30 ms}$&nbsp; long section of a music signal&nbsp; <math>q(t)</math>.&nbsp; It is the piece&nbsp; &raquo;For Elise&laquo;&nbsp; by Ludwig van Beethoven.
  
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*Underneath are drawn two sink signals&nbsp; <math>v_1(t)</math>&nbsp; and&nbsp; <math>v_2(t)</math>, which were recorded after the transmission of the music signal&nbsp; <math>q(t)</math>&nbsp; over two different channels.
  
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Originalsignal <math>q(t)</math>
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*The following operating elements allow you to listen to the first fourteen seconds of each of the three audio signals&nbsp; <math>q(t)</math>,&nbsp; <math>v_1(t)</math>&nbsp; and&nbsp; <math>v_2(t)</math>.
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Original signal&nbsp; <math>q(t)</math>:
  
 
<lntmedia>file:A_ID9__Sig_A1_1Elise10sek22kb.mp3</lntmedia>
 
<lntmedia>file:A_ID9__Sig_A1_1Elise10sek22kb.mp3</lntmedia>
  
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Sinkensignal <math>v_1(t)</math>
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Sink signal&nbsp; <math>v_1(t)</math>:
  
 
<lntmedia>file:A_ID10__Sig_A1_1Elise10sek30Prozent22kb.mp3</lntmedia>
 
<lntmedia>file:A_ID10__Sig_A1_1Elise10sek30Prozent22kb.mp3</lntmedia>
  
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Sinkensignal <math>v_2(t)</math>
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Sink signal&nbsp; <math>v_2(t)</math>:
  
 
<lntmedia>file:A_ID12__Sig_A1_1elise10sek30dB22kb.mp3</lntmedia>
 
<lntmedia>file:A_ID12__Sig_A1_1elise10sek30dB22kb.mp3</lntmedia>
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===Fragebogen zu &bdquo;Aufgabe 1.1: Musiksignale&rdquo;===
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<u>Notes:</u>&nbsp; The exercise belongs to the chapter&nbsp;[[Signal_Representation/Principles_of_Communication|&raquo;Principles of Communication&laquo;]].
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===Questions===
  
 
<quiz display=simple>
 
<quiz display=simple>
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{Schätzen Sie die Signalfrequenz von <math>q(t)</math> im dargestellen Ausschnitt ab.
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{Estimate the signal frequency of&nbsp; <math>q(t)</math>&nbsp; in the displayed section.
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|type="[]"}
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|type="()"}
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- Die Signalfrequenz beträgt etwa <math>f = 250\,\text{Hz}</math>.
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- The signal frequency is approximately&nbsp; <math>f = 250\,\text{Hz}</math>.
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+ Die Signalfrequenz beträgt etwa <math>f = 500\,\text{Hz}</math>.
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+ The signal frequency is approximately&nbsp; <math>f = 500\,\text{Hz}</math>.
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- Die Signalfrequenz beträgt etwa <math>f = 1\,\text{kHz}</math>.
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- The signal frequency is approximately&nbsp; <math>f = 1\,\text{kHz}</math>.
  
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{Welche Aussagen sind für das Signal <math>v_1(t)</math> zutreffend?
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{Which statements are true for the signal&nbsp; <math>v_1(t)</math>&nbsp;?
 
|type="[]"}
 
|type="[]"}
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+  Das Signal <math>v_1(t)</math> ist gegenüber <math>q(t)</math> unverzerrt.
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+  The signal&nbsp; <math>v_1(t)</math>&nbsp; is undistorted compared to&nbsp; <math>q(t)</math>.
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-  Das Signal <math>v_1(t)</math> weist gegenüber <math>q(t)</math> Verzerrungen auf.
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-  The signal&nbsp; <math>v_1(t)</math>&nbsp; shows distortions compared to&nbsp; <math>q(t)</math>&nbsp;.
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-  Das Signal <math>v_1(t)</math> ist gegenüber <math>q(t)</math> verrauscht.
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-  The signal&nbsp; <math>v_1(t)</math>&nbsp; is noisy compared to&nbsp; <math>q(t)</math>&nbsp;.
 
 
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{Welche Aussagen sind für das Signal <math>v_2(t)</math> zutreffend?
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{Which statements are true for the signal&nbsp; <math>v_2(t)</math>&nbsp;?
 
|type="[]"}
 
|type="[]"}
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+ Das Signal <math>v_2(t)</math> ist gegenüber <math>q(t)</math> unverzerrt.
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+ The signal&nbsp; <math>v_2(t)</math>&nbsp; is undistorted compared to&nbsp; <math>q(t)</math>&nbsp;.
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- Das Signal <math>v_2(t)</math> weist gegenüber <math>q(t)</math> Verzerrungen auf.
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- The signal&nbsp; <math>v_2(t)</math>&nbsp; shows distortions compared to&nbsp; <math>q(t)</math>&nbsp;.
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+ Das Signal <math>v_2(t)</math> ist gegenüber <math>q(t)</math> verrauscht.
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+ The signal&nbsp; <math>v_2(t)</math>&nbsp; is noisy compared to&nbsp; <math>q(t)</math>&nbsp;.
  
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{Eines der Signale ist gegenüber dem Orginal <math>q(t)</math> unverzerrt und nicht verrauscht. Schätzen Sie hierfür den Dämpfungsfaktor und die Laufzeit ab.
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{One of the signals is undistorted and not noisy compared to the original &nbsp; <math>q(t)</math>&nbsp;. <br> Estimate the attenuation factor and the delay time for this.
 
|type="{}"}
 
|type="{}"}
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<math> \alpha = </math> { 0.2-0.4 }
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<math> \alpha \ = \ </math> { 0.2-0.4 }
  
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<math> \tau = </math>  { 5-15 } $\text{ms}$
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<math> \tau \ = \  </math>  { 5-15 } $\ \text{ms}$
  
 
</quiz>
 
</quiz>
  
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===Musterlösung zu &bdquo;Aufgabe 1.1: Musiksignale&rdquo;===
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===Solution===
 
{{ML-Kopf}}
 
{{ML-Kopf}}
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'''1.'''  Im markierten Bereich (20 Millisekunden) sind ca 10 Schwingungen zu erkennen. Daraus folgt für die Signalfrequenz näherungsweise
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'''(1)'''&nbsp;  Correct is <u>solution 2</u>:
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*In the marked range of&nbsp; $20$&nbsp; milliseconds &nbsp; &rArr; &nbsp; approx.&nbsp; $10$&nbsp; oscillations can be detected.
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*From this the result&nbsp; follows approximately for the signal frequency:&nbsp; $f = {10}/(20 \,\text{ms}) =  500 \,\text{Hz}$.
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'''(2)'''&nbsp; Correct is <u>solution 1</u>:
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*The signal&nbsp; <math>v_1(t)</math>&nbsp; is undistorted compared to the original signal <math>q(t)</math>.&nbsp; The following applies: &nbsp; $v_1(t)=\alpha \cdot q(t-\tau)$.
  
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<math>f = \frac{10}{20ms} = </math> '''500 Hz''' ⇒  '''Lösungsvorschlag 2'''.
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*An attenuation&nbsp; <math>\alpha</math>&nbsp; and a delay time&nbsp; <math>\tau</math>&nbsp; do not cause distortion, but the signal is then only quieter and delayed in time, compared to the original.
  
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'''2.''' Das Signal <math>v_1(t)</math> ist gegenüber dem Orginalsignal <math>q(t)</math> unverzerrt. Es gilt:
 
  
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<math>v_1(t)=\alpha \cdot q(t-\tau) </math>
 
  
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Eine '''Dämpfung <math>\alpha</math>''' und eine '''Laufzeit <math>\tau</math>''' führen nicht zu Verzerrungen, sondern das Signal ist leiser und es kommt später als das Original  ⇒  '''Lösungsvorschlag 1'''.
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'''(3)'''&nbsp; Correct are the <u>solutions 1 and 3</u>:
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*One can recognize additive noise both in the displayed signal&nbsp; <math>v_2(t)</math>&nbsp; and in the audio signal&nbsp; &nbsp; ⇒ &nbsp; <u>solution 3</u>.
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*The signal-to-noise ratio is approx.&nbsp; $\text{30 dB}$&nbsp; $($but this cannot be seen from the mentioned data$)$.
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*Correct is also <u>solution 1</u>: &nbsp; Without this noise component&nbsp; <math>v_2(t)</math>&nbsp; would be identical with&nbsp; <math>q(t)</math>.
  
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'''3.''' Man erkennt sowohl im dargestellten Signalverlauf als auch im Audiosignal das '''additive Rauschen'''  ⇒  '''Lösungsvorschläge 1 und 3'''. Der Signalrauschabstand beträgt dabei ca. 30 dB; dies ist aber aus dieser Darstellung nicht erkennbar.
 
  
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'''4.'''  Das Signal <math>v_1(t)</math> ist formgleich mit dem Originalsignal <math>q(t)</math> und unterscheidet sich von diesem lediglich durch den Amplitudenfaktor <math>\alpha</math> = '''0.3''' (entspricht etwa –10 dB) und die Laufzeit <math>\tau</math> = '''10 ms'''.
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'''(4)'''&nbsp; The signal&nbsp; <math>v_1(t)</math>&nbsp; is identical in shape to the original signal&nbsp; <math>q(t)</math>&nbsp; and differs from it only
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*by the attenuation factor&nbsp; $\alpha = \underline{\text{0.3}}$ &nbsp;  $($this corresponds to about&nbsp; $\text{–10 dB)}$,
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*and the delay time&nbsp;  $\tau = \underline{10\,\text{ms}}$.
 
{{ML-Fuß}}
 
{{ML-Fuß}}
  
 
__NOEDITSECTION__
 
__NOEDITSECTION__
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[[Category:Aufgaben zu Signaldarstellung|^1. Grundbegriffe der Nachrichtentechnik^]]
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[[Category:Signal Representation: Exercises|^1.1 Principles of Communication^]]

Latest revision as of 15:29, 12 January 2024

Music signals,
original, noisy and/or distorted?

On the right you see a  $\text{30 ms}$  long section of a music signal  \(q(t)\).  It is the piece  »For Elise«  by Ludwig van Beethoven.

  • Underneath are drawn two sink signals  \(v_1(t)\)  and  \(v_2(t)\), which were recorded after the transmission of the music signal  \(q(t)\)  over two different channels.
  • The following operating elements allow you to listen to the first fourteen seconds of each of the three audio signals  \(q(t)\),  \(v_1(t)\)  and  \(v_2(t)\).


Original signal  \(q(t)\):

Sink signal  \(v_1(t)\):

Sink signal  \(v_2(t)\):



Notes:  The exercise belongs to the chapter »Principles of Communication«.



Questions

1

Estimate the signal frequency of  \(q(t)\)  in the displayed section.

The signal frequency is approximately  \(f = 250\,\text{Hz}\).
The signal frequency is approximately  \(f = 500\,\text{Hz}\).
The signal frequency is approximately  \(f = 1\,\text{kHz}\).

2

Which statements are true for the signal  \(v_1(t)\) ?

The signal  \(v_1(t)\)  is undistorted compared to  \(q(t)\).
The signal  \(v_1(t)\)  shows distortions compared to  \(q(t)\) .
The signal  \(v_1(t)\)  is noisy compared to  \(q(t)\) .

3

Which statements are true for the signal  \(v_2(t)\) ?

The signal  \(v_2(t)\)  is undistorted compared to  \(q(t)\) .
The signal  \(v_2(t)\)  shows distortions compared to  \(q(t)\) .
The signal  \(v_2(t)\)  is noisy compared to  \(q(t)\) .

4

One of the signals is undistorted and not noisy compared to the original   \(q(t)\) .
Estimate the attenuation factor and the delay time for this.

\( \alpha \ = \ \)

\( \tau \ = \ \)

$\ \text{ms}$


Solution

Solution

(1)  Correct is solution 2:

  • In the marked range of  $20$  milliseconds   ⇒   approx.  $10$  oscillations can be detected.
  • From this the result  follows approximately for the signal frequency:  $f = {10}/(20 \,\text{ms}) = 500 \,\text{Hz}$.


(2)  Correct is solution 1:

  • The signal  \(v_1(t)\)  is undistorted compared to the original signal \(q(t)\).  The following applies:   $v_1(t)=\alpha \cdot q(t-\tau)$.
  • An attenuation  \(\alpha\)  and a delay time  \(\tau\)  do not cause distortion, but the signal is then only quieter and delayed in time, compared to the original.


(3)  Correct are the solutions 1 and 3:

  • One can recognize additive noise both in the displayed signal  \(v_2(t)\)  and in the audio signal    ⇒   solution 3.
  • The signal-to-noise ratio is approx.  $\text{30 dB}$  $($but this cannot be seen from the mentioned data$)$.
  • Correct is also solution 1:   Without this noise component  \(v_2(t)\)  would be identical with  \(q(t)\).


(4)  The signal  \(v_1(t)\)  is identical in shape to the original signal  \(q(t)\)  and differs from it only

  • by the attenuation factor  $\alpha = \underline{\text{0.3}}$   $($this corresponds to about  $\text{–10 dB)}$,
  • and the delay time  $\tau = \underline{10\,\text{ms}}$.

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