Exercise 1.1: Music Signals

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Music signals, original,
noisy and/or distorted?

On the right you see a ca.  $\text{30 ms}$  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 controls 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 task 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 about  $f = 1\,\text{kHz}$.

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 running time for this.

$\alpha \ = \ $

$\tau \ = \ $ $\ \text{ms}$

Solution

Solution

(1)  Correct is the 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 the 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  $\tau$  do not cause distortion, but the signal is then only quieter and comes later than the original.

(3)  Correct are the solutions 1 and 3:

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

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

• by the attenuation factor  $\alpha = \underline{\text{0.3}}$  (dies entspricht etwa  $\text{–10 dB)}$
• and the delay  $\tau = \underline{10\,\text{ms}}$.