Aufgaben:Exercise 1.2: Distortions? Or no Distortion?

• Return to book

Observed Sink signals for the
given input signal   $q(t)$

The communication systems  $S_1$,  $S_2$  and  $S_3$  analyzed in terms of the distortions they cause. For this purpose, the cosine-shaped test signal with signal frequency $f_{\rm N} = 1\text{ kHz}$  is applied to the input of each system:

$$q(t) = 2 \;{\rm V} \cdot \cos(2 \pi f_{\rm N} t )$$

The three signals at the system output are measured, as shown in the graph:

$$v_1(t) = 2 \;{\rm V} \cdot \cos(2 \pi f_{\rm N} t )\hspace{0.05cm},$$

$$v_2(t) = 1 \;{\rm V} \cdot \cos(2 \pi f_{\rm N} t + 1 \;{\rm V} \cdot \sin(2 \pi f_{\rm N} t) \hspace{0.05cm},$$

$$v_3(t)= 1.5 \;{\rm V} \cdot \cos(2 \pi f_{\rm N} t) - 0.3 \;{\rm V} \cdot \cos(6 \pi f_{\rm N} t)\hspace{0.05cm}.$$

The noise components that are always present in practice will be assumed to be negligible here.
Hints:

• This exercise belongs to the chapter  Quality criteria.  Particular reference is made to the page   Signal-to-noise power ratio  and to the chapter   Non-linear distortions  in the book "Linear and Time-Invariant Systems".
• For nonlinear distortion, the sink SNR is  $ρ_v = 1/K^2$, where the distortion factor  $K$  is the ratio of the rms values of all harmonics to the rms value of the fundamental frequency.

Questions

1

What statements can be made about the  $S_1$  system after this measurement?

	$S_1$  could be an ideal system.
	$S_1$  could be a distortionless system.
	$S_1$  could be a linearly distorting system.
	$S_1$  could be a nonlinearly distorting system.

2

Write the second signal in the form  $v_2(t) = α · q(t - τ)$  and determine its paramaters.

$\alpha \ = \ $

$τ \ = \ $ $\ \rm µ s$

3

{What statements can be made about the  $S_2$  system after this measurement?

	$S_2$  could be an ideal system.
	$S_2$  could be a distortionless system.
	$S_2$  could be a linearly distorting system.
	$S_2$  could be a nonlinearly distorting system.

4

What kind of distortions are present in System  $S_3$?

	They are linear distortions.
	They are nonlinear distortions.

5

Calculate the sink SNR  $ρ_{v3}$  of System  $S_3$.

$ρ_{v3} \ = \ $

Solution

Solution

(1)&nbspAnswers 1, 2 and 3 are correct:

• System  $S_1$  could well be an ideal system, namely if for all frequencies $f_{\rm N}$  the condition   $v(t) = q(t)$  were satisfied.
• The second alternative is also possible, since the ideal system is a special case of distortion-free systems.
• However, if at a different message frequency $f_{\rm N} \ne 1$  kHz the condition   $v(t) = q(t)$  were not satisfied, then a linearly distorting system would exist whose frequency response would happen to be equal to $1$  at frequency $f_{\rm N}$ .
• In contrast, a nonlinearly distorting system (Answer 4) can be excluded due to the lack of harmonics.

(2)  Following the explanations in the chapter "Harmonic Oscillation" in the book "Signal Representation" the following equations apply:

$$A \cdot \cos(\omega_{\rm N} t ) + B \cdot \sin(\omega_{\rm N} t ) = C \cdot \cos(\omega_{\rm N} t - \varphi)\hspace{0.3cm} \Rightarrow \hspace{0.3cm} C = \sqrt{A^2 + B^2},\hspace{0.5cm}\varphi ={\rm arctan}\hspace{0.1cm} ({A}/{B})\hspace{0.05cm}$$

• Applied to the present example, one obtains

$$C = \sqrt{(1 \,{\rm V})^2 + (1 \,{\rm V})^2}= 1.414\,{\rm V}\hspace{0.05cm}.$$

• The damping ratio of the system thus takes the value   $α = 1.414/2 \hspace{0.15cm}\underline{= 0.707}$, and the following applies to the phase:

$$ \varphi ={\rm arctan}\hspace{0.1cm}\frac {1 \,{\rm V}}{1 \,{\rm V}} = 45^{\circ} = {\pi}/{4}\hspace{0.05cm}.$$

• The transformation   $\cos(\omega_{\rm N} t - \varphi)= \cos[\omega_{\rm N} (t - \tau)]$  enables claims about the running time:

$$\tau =\frac {\varphi}{2\pi f_{\rm N}} = \frac {\pi /4}{2\pi f_{\rm N}} = \frac {1}{8 \cdot 1 \,{\rm kHz}} \hspace{0.15cm}\underline {= 125\,{\rm µ s}}\hspace{0.05cm}.$$

(3)  Answers 2 and 3 are correct:

• Applying the logic from subtask (1), the system   $S_2$  is neither ideal nor nonlinearly distorting.
• In contrast, options 2 and 3 are possible, depending on whether the calculated values of $α$  and $τ$   are preserved for all frequencies or not.
• However, with just a single measurement at only one frequency, this cannot be clarified.

(4)  Das Signal  $v_3(t)$  beinhaltet eine Oberwelle dritter Ordnung.  Deshalb ist die Verzerrung nichtlinear   ⇒  Lösungsvorschlag 2.

(5)  Mit den Amplituden  $A_1 = 1.5 \ \rm V$  und  $A_3 = -0.3\ \rm V$  erhält man für den Klirrfaktor:

$$ K_3 =\frac {|A_3|}{|A_1|} = 0.2\hspace{0.05cm}.$$

• Deshalb beträgt das Sinken–SNR entsprechend der angegebenen Gleichung  $ρ_{v3} = 1/K_3^{ 2 } = 25$.

Zum gleichen Ergebnis kommt man nach der allgemeinen Berechnung.

• Aus den Amplituden von Quellensignal und Grundwelle des Sinkensignals erhält man für den frequenzunabhängigen Dämpfungsfaktor:

$$ \alpha =\frac {1.5 \,{\rm V}}{2 \,{\rm V}} = 0.75\hspace{0.05cm}.$$

• Das von den nichtlinearen Verzerrungen herrührende Fehlersignal lautet deshalb:  

$$\varepsilon_3(t) = v_3(t) - \alpha \cdot q(t) = - 0.3 \,{\rm V} \cdot \cos(6 \pi f_{\rm N} t)\hspace{0.05cm}.$$

• Damit ergibt sich die Verzerrungsleistung:

$$P_{\varepsilon 3}= {1}/{2} \cdot (0.3 \,{\rm V})^2 = 0.045 \,{\rm V}^2\hspace{0.05cm}.$$

• Mit der Leistung des Quellensignals,

$$P_{q}= {1}/{2} \cdot (2\,{\rm V})^2 = 2 \,{\rm V}^2\hspace{0.05cm},$$

erhält man unter Berücksichtigung des gerade berechneten Dämpfungsfaktors  $ \alpha = 0.75 $:

$$\rho_{v3} = \frac{\alpha^2 \cdot P_{q}}{P_{\varepsilon 3}} = \frac{0.75^2 \cdot 2 {\rm V}^2}{0.045 } \hspace{0.15cm}\underline {= 25}\hspace{0.05cm}.$$