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}}
  
[[File:P_ID339__Sig_A_1_1.png|right|frame|Music signals, original, <br> noisy and/or distorted?]]
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[[File:P_ID339__Sig_A_1_1.png|right|frame|Music signals, <br>original, noisy and/or distorted?]]
On the right you see a ca.&nbsp; $\text{30 ms}$&nbsp; long section of a music signal&nbsp; <math>q(t)</math>. It is the piece &bdquo;For Elise&rdquo; by Ludwig van Beethoven.
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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.
  
 
*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.  
 
*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.  
  
*The following controls 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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*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>.
  
  
Originalsignal&nbsp; <math>q(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>
  
Sinkensignal&nbsp; <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>
  
Sinkensignal&nbsp; <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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''Notes:''
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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;]].
*The task belongs to chapter&nbsp; [[Signal_Representation/Principles_of_Communication|Prinzip der Nachrichtenübertragung]].
 
 
   
 
   
  
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- The signal frequency is approximately&nbsp; <math>f = 250\,\text{Hz}</math>.
 
- The signal frequency is approximately&nbsp; <math>f = 250\,\text{Hz}</math>.
 
+ The signal frequency is approximately&nbsp; <math>f = 500\,\text{Hz}</math>.
 
+ The signal frequency is approximately&nbsp; <math>f = 500\,\text{Hz}</math>.
- The signal frequency is about&nbsp; <math>f = 1\,\text{kHz}</math>.
+
- The signal frequency is approximately&nbsp; <math>f = 1\,\text{kHz}</math>.
  
 
{Which statements are true for the signal&nbsp; <math>v_1(t)</math>&nbsp;?
 
{Which statements are true for the signal&nbsp; <math>v_1(t)</math>&nbsp;?
 
|type="[]"}
 
|type="[]"}
+  The signal&nbsp; <math>v_1(t)</math>&nbsp; is undistorted compared to <math>q(t)</math>.
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+  The signal&nbsp; <math>v_1(t)</math>&nbsp; is undistorted compared to&nbsp; <math>q(t)</math>.
 
-  The signal&nbsp; <math>v_1(t)</math>&nbsp; shows distortions compared to&nbsp; <math>q(t)</math>&nbsp;.
 
-  The signal&nbsp; <math>v_1(t)</math>&nbsp; shows distortions compared to&nbsp; <math>q(t)</math>&nbsp;.
 
-  The signal&nbsp; <math>v_1(t)</math>&nbsp; is noisy compared to&nbsp; <math>q(t)</math>&nbsp;.
 
-  The signal&nbsp; <math>v_1(t)</math>&nbsp; is noisy compared to&nbsp; <math>q(t)</math>&nbsp;.
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+ The signal&nbsp; <math>v_2(t)</math>&nbsp; is noisy compared to&nbsp; <math>q(t)</math>&nbsp;.
 
+ The signal&nbsp; <math>v_2(t)</math>&nbsp; is noisy compared to&nbsp; <math>q(t)</math>&nbsp;.
  
{One of the signals is opposite the original&nbsp; <math>q(t)</math>&nbsp; undistorted and not noisy. <br> Estimate the attenuation factor and the running time for this.
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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="{}"}
 
<math> \alpha \ = \ </math> { 0.2-0.4 }
 
<math> \alpha \ = \ </math> { 0.2-0.4 }
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</quiz>
 
</quiz>
  
===Solutions===
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===Solution===
 
{{ML-Kopf}}
 
{{ML-Kopf}}
'''(1)'''&nbsp;  Correct is the <u>solution 2</u>:
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'''(1)'''&nbsp;  Correct is <u>solution 2</u>:
*In the marked range of $20$ milliseconds approx. &nbsp; $10$&nbsp; oscillations can be detected.  
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*In the marked range of&nbsp; $20$&nbsp; milliseconds &nbsp; &rArr; &nbsp; approx.&nbsp; $10$&nbsp; oscillations can be detected.
*From this the result&nbsp; follows approximately for the signal frequency; $f = {10}/(20 \,\text{ms}) =  500 \,\text{Hz}$.
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 +
*From this the result&nbsp; follows approximately for the signal frequency:&nbsp; $f = {10}/(20 \,\text{ms}) =  500 \,\text{Hz}$.
  
  
  
'''(2)'''&nbsp; Correct is the <u>solution 1</u>:
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'''(2)'''&nbsp; Correct is <u>solution 1</u>:
*The signal&nbsp; <math>v_1(t)</math>&nbsp; is undistorted compared to the original signal <math>q(t)</math>. The following applies: &nbsp; $v_1(t)=\alpha \cdot q(t-\tau) .$
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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; $v_1(t)=\alpha \cdot q(t-\tau)$.
  
*An attenuation&nbsp; <math>\alpha</math>&nbsp; and a delay&nbsp; <math>\tau</math>&nbsp; do not cause distortion, but the signal is then only quieter and comes later than the original.
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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.
  
  
  
 
'''(3)'''&nbsp; Correct are the <u>solutions 1 and 3</u>:
 
'''(3)'''&nbsp; Correct are the <u>solutions 1 and 3</u>:
*One can recognize both in the displayed signal&nbsp; <math>v_2(t)</math>&nbsp; and in the audio signal&nbsp; ''additive noise'' &nbsp; ⇒ &nbsp; <u>solution 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>.
*The signal-to-noise ratio is approx. &nbsp; $\text{30 dB}$; but this cannot be seen from this representation.  
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*Correct is also the <u>solution 1</u>: &nbsp; Without this noise component&nbsp; <math>v_2(t)</math>&nbsp; identical with&nbsp; <math>q(t)</math>.
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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$)$.
 +
 +
*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>.
  
  
'''(4)'''&nbsp;  The signal&nbsp; <math>v_1(t)</math>&nbsp; is identical in form to the original signal&nbsp; <math>q(t)</math>&nbsp; and differs from it only  
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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  
*by the attenuation factor&nbsp; $\alpha = \underline{\text{0.3}}$&nbsp;  (dies entspricht etwa&nbsp; $\text{–10 dB)}$  
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*by the attenuation factor&nbsp; $\alpha = \underline{\text{0.3}}$ &nbsp;  $($this corresponds to about&nbsp; $\text{–10 dB)}$,
*and the delay&nbsp;  $\tau = \underline{10\,\text{ms}}$.
+
 +
*and the delay time&nbsp;  $\tau = \underline{10\,\text{ms}}$.
 
{{ML-Fuß}}
 
{{ML-Fuß}}
  
 
__NOEDITSECTION__
 
__NOEDITSECTION__
[[Category:Exercises for Signal Representation|^1.1 Principles of Communication^]]
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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

(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  $\tau = \underline{10\,\text{ms}}$.