Aufgaben:Exercise 4.4: About the Quantization Noise - Revision history

Guenter at 13:57, 11 April 2022

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Guenter at 11:55, 9 April 2022

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Guenter at 10:26, 9 April 2022

2022-04-09T10:26:13Z

Guenter at 15:24, 8 April 2022

2022-04-08T15:24:23Z

Noah at 16:44, 6 April 2022

2022-04-06T16:44:30Z

Noah at 19:11, 19 March 2022

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Guenter at 13:52, 16 March 2022

2022-03-16T13:52:23Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.4: Zum Quantisierungsrauschen to Aufgaben:Exercise 4.4: About the Quantization Noise

2022-03-16T13:44:35Z

Guenter moved page Aufgabe 4.4: Zum Quantisierungsrauschen to Exercise 4.4: About the Quantization Noise

Javier: Text replacement - "Category:Aufgaben zu Modulationsverfahren" to "Category:Modulation Methods: Exercises"

2021-03-23T12:52:40Z

Text replacement - "Category:Aufgaben zu Modulationsverfahren" to "Category:Modulation Methods: Exercises"

Javier: Text replacement - "[[Modulationsverfahren" to "[[Modulation_Methods"

2020-07-09T12:42:16Z

Text replacement - "[[Modulationsverfahren" to "[[Modulation_Methods"

@@ Line 77: / Line 77: @@
-[[File:P_ID1616__Mod_A_4_4.png|right|frame|Error signal for&nbsp; $Q_{\rm max} = q_{\rm max}$]]
+[[File:Mod_A_4_4b_neu.png|right|frame|Error signal for&nbsp; $Q_{\rm max} = q_{\rm max}$]]
 '''(2)'''&nbsp; Correct are&nbsp; <u>suggested solutions 1, 3, and 4</u>:
 *We assume here&nbsp; $Q_{\rm max} = q_{\rm max} = 6 \ \rm V$.

@@ Line 39: / Line 39: @@
 $P_{\rm S} \ = \ $ { 12 3% } $\ \rm V^2$
-{Which statements are true for the error signal &nbsp;$ε(t)$&nbsp;?
+{Which statements are true for the error signal &nbsp;$ε(t)= q_{\rm Q}(t)-q(t)$&nbsp;?
 |type="[]"}
 + $ε(t)$&nbsp; has a sawtooth shape.

@@ Line 70: / Line 70: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The signal power&nbsp; $P_{\rm S} $&nbsp; is equal to the root mean square of&nbsp; $q(t)$ if the reference resistance&nbsp; $1 \rm Ω$&nbsp; is used and therefore the unit&nbsp; $\rm V^2$&nbsp; is accepted for the power.
+'''(1)'''&nbsp; The signal power&nbsp; $P_{\rm S} $&nbsp; is equal to the second moment of&nbsp; $q(t)$ if the reference resistance&nbsp; $1 \rm Ω$&nbsp; is used and therefore the unit&nbsp; $\rm V^2$&nbsp; is accepted for the power.
-*Due to periodicity and symmetry, averaging over the time domain&nbsp; $T_0/2$ is sufficient:
+*Due to periodicity and symmetry,&nbsp; averaging over the time domain&nbsp; $T_0/2$&nbsp; is sufficient:
 :$$P_{\rm S} = \frac{1}{T_0/2} \cdot \int_{0}^{T_0/2}q^2(t) \hspace{0.05cm}{\rm d}t = \frac{2 \cdot q_{\rm max}^2}{T_0} \cdot \int_{0}^{T_0/2}\left ( { 2 \cdot t}/{T_0} \right )^2 \hspace{0.05cm}{\rm d}t= \frac{2 \cdot q_{\rm max}^2}{T_0} \cdot \frac{T_0}{2} \cdot \int_{0}^{1}x^2 \hspace{0.05cm}{\rm d}x = \frac{q_{\rm max}^2}{3} \hspace{0.05cm}.$$
 *Here the substitution&nbsp; $x = 2 - t/T_0$&nbsp; was used.&nbsp; With&nbsp; $q_{\rm max} = 6 \ \rm V$&nbsp; one gets&nbsp; $P_\rm S\hspace{0.15cm}\underline { = 12 \ V^2}$.
@@ Line 78: / Line 78: @@
 [[File:P_ID1616__Mod_A_4_4.png|right|frame|Error signal for&nbsp; $Q_{\rm max} = q_{\rm max}$]]
-'''(2)'''&nbsp; Correct are <u>suggested solutions 1, 3, and 4</u>:
+'''(2)'''&nbsp; Correct are&nbsp; <u>suggested solutions 1, 3, and 4</u>:
-*We assume here&nbsp; $Q_{\rm max} = q_{\rm max} = 6 \ \rm V$&nbsp; .
+*We assume here&nbsp; $Q_{\rm max} = q_{\rm max} = 6 \ \rm V$.
 *This gives the sawtooth-shaped error signal&nbsp; $ε(t)$&nbsp; between&nbsp; $±1\ \rm V$.
 *The period duration is&nbsp; $T_0' = T_0/6$.
@@ Line 86: / Line 86: @@
 '''(3)'''&nbsp; The error signal&nbsp; $ε(t)$&nbsp; proceeds in the same way as&nbsp; $q(t)$&nbsp; sawtooth.
-*Thus, the same equation as in subtask&nbsp; '''(1)'''' is suitable for calculating the root mean square.
+*Thus,&nbsp; the same equation as in subtask&nbsp; '''(1)'''&nbsp; is suitable for calculating the power.
-*Note, however, that the amplitude is smaller by a factor&nbsp; $M$&nbsp; while the different period duration does not matter for the averaging:
+*Note,&nbsp; however,&nbsp; that the amplitude is smaller by a factor&nbsp; $M$&nbsp; while the different period duration does not matter for the averaging:
 :$$P_{\rm Q} = \frac{P_{\rm S}}{M^2} = \frac{12\,{\rm V}^2}{36}\hspace{0.15cm}\underline {= 0.333\,{\rm V}^2 }\hspace{0.05cm}.$$
@@ Line 104: / Line 104: @@
+:In both cases,&nbsp; the quantization noise power is significantly larger than calculated in sub-task&nbsp; '''(3)'''.

@@ Line 9: / Line 9: @@
-The signal &nbsp;$q(t)$&nbsp; is quantized according to the graph with &nbsp;$M = 6$&nbsp; stages.&nbsp; The quantized signal is &nbsp;$q_{\rm Q}(t)$, where:
+The signal &nbsp;$q(t)$&nbsp; is quantized according to the graph with &nbsp;$M = 6$&nbsp; steps.&nbsp; The quantized signal is &nbsp;$q_{\rm Q}(t)$,&nbsp; where:
-*The linear quantizer is designed for the amplitude range &nbsp;$±Q_{\rm max}$&nbsp; such that each quantization interval has width &nbsp;${\it Δ} = 2/M \cdot Q_{\rm max}$&nbsp;.
+*The linear quantizer is designed for the amplitude range &nbsp;$±Q_{\rm max}$&nbsp; such that each quantization interval has width &nbsp;${\it Δ} = 2/M \cdot Q_{\rm max}$.
-*The diagram shows this fact for &nbsp;$Q_{\rm max} = q_{\rm max} = 6 \ \rm V$.&nbsp; These numerical values shall be assumed up to and including the subtask&nbsp; '''(5)'''&nbsp;.
+*The diagram shows this fact for &nbsp;$Q_{\rm max} = q_{\rm max} = 6 \ \rm V$.&nbsp; These numerical values shall be assumed up to and including the subtask&nbsp; '''(5)'''.
-The so-called&nbsp; '''quantization noise power'''&nbsp; is defined as the root mean square of the difference signal &nbsp;$ε(t) = q_{\rm Q}(t) - q(t)$&nbsp; . It holds
+The&nbsp; "quantization noise power"&nbsp; is defined as the second moment of the difference signal &nbsp;$ε(t) = q_{\rm Q}(t) - q(t)$.&nbsp;  It holds:
 :$$P_{\rm Q} = \frac{1}{T_0' } \cdot \int_{0}^{T_0'}\varepsilon(t)^2 \hspace{0.05cm}{\rm d}t \hspace{0.05cm},$$
-where the time &nbsp;$T_0'$&nbsp; is to be chosen appropriately.
+where the time &nbsp;$T_0'$&nbsp; is to be chosen appropriately.&nbsp; The&nbsp; "quantization SNR"&nbsp; is the ratio &nbsp; &nbsp;$\rho_{\rm Q} = {P_{\rm S}}/{P_{\rm Q}}\hspace{0.05cm}$,&nbsp; which is usually given logarithmically&nbsp; (in dB).
@@ Line 28: / Line 26: @@
 Hints:
-*The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Pulse_Code_Modulation|Pulse Code Modulation]].
+*The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Pulse_Code_Modulation|"Pulse Code Modulation"]].
-*Reference is made in particular to the page&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Quantization_and_quantization_noise|Quantization and quantization Noise]].
+*Reference is made in particular to the page&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Quantization_and_quantization_noise|"Quantization and quantization Noise"]].
@@ Line 37: / Line 35: @@
 <quiz display=simple>
-{Calculate the signal power &nbsp;$P_{\rm S}$&nbsp; (referred to the resistor $1 \ \rm Ω$&nbsp;).
+{Calculate the signal power &nbsp;$P_{\rm S}$&nbsp; $($referred to the resistor $1 \ \rm Ω)$.
 |type="{}"}
 $P_{\rm S} \ = \ $ { 12 3% } $\ \rm V^2$
@@ Line 45: / Line 43: @@
 + $ε(t)$&nbsp; has a sawtooth shape.
 - $ε(t)$&nbsp; has a step-like progression.
-+ $ε(t)$&nbsp; is restricted to the range &nbsp;$±{\it Δ}/2 = ±1 \ \rm V$&nbsp;.
++ $ε(t)$&nbsp; is restricted to the range &nbsp;$±{\it Δ}/2 = ±1 \ \rm V$.
 + $ε(t)$&nbsp; has period &nbsp;$T_0' = T_0/M$.
@@ Line 54: / Line 52: @@
 {Calculate the quantization noise ratio for &nbsp;$M = 6$.
 |type="{}"}
-$10 - \lg \ ρ_{\rm Q} \ = \ $ { 15.56 3% } $\ \rm dB$
+$10 \cdot \lg \ ρ_{\rm Q} \ = \ $ { 15.56 3% } $\ \rm dB$
 {What values result from quantization with &nbsp;$N = 8$&nbsp; or &nbsp;$N = 16$ bits?
@@ Line 61: / Line 59: @@
 $N = 16\text{:}\hspace{0.15cm}10 ⋅ \lg \ ρ_{\rm Q} \ = \ ${ 96.32 3% } $\ \rm dB$
-{What conditions must be met for the derived equation to apply to &nbsp;$ρ_{\rm Q}$&nbsp;?
+{What conditions must be met for the derived equation to apply to &nbsp;$ρ_{\rm Q}$?
 |type="[]"}
 + All amplitude values are equally probable.
@@ Line 79: / Line 77: @@
-[[File:P_ID1616__Mod_A_4_4.png|right|frame|error signal for&nbsp; $Q_{\rm max} = q_{\rm max}$]]
+[[File:P_ID1616__Mod_A_4_4.png|right|frame|Error signal for&nbsp; $Q_{\rm max} = q_{\rm max}$]]
 '''(2)'''&nbsp; Correct are <u>suggested solutions 1, 3, and 4</u>:
 *We assume here&nbsp; $Q_{\rm max} = q_{\rm max} = 6 \ \rm V$&nbsp; .
@@ Line 113: / Line 111: @@
-[[File:P_ID1618__Mod_A_4_4f.png|right|frame|Quantisierung mit&nbsp; $Q_{\rm max} \ne q_{\rm max}$]]
+[[File:P_ID1618__Mod_A_4_4f.png|right|frame|Quantization with&nbsp; $Q_{\rm max} \ne q_{\rm max}$]]
 <br><br><br><br>
 The graph shows the error signals &nbsp;$ε(t)$&nbsp; for &nbsp;$Q_{\rm max} > q_{\rm max}$&nbsp; (left) and &nbsp;$Q_{\rm max} < q_{\rm max}$&nbsp; (right).

@@ Line 10: / Line 10: @@
 The signal &nbsp;$q(t)$&nbsp; is quantized according to the graph with &nbsp;$M = 6$&nbsp; stages.&nbsp; The quantized signal is &nbsp;$q_{\rm Q}(t)$, where:
-*The linear quantizer is designed for the amplitude range &nbsp;$±Q_{\rm max}$&nbsp; such that each quantization interval has width &nbsp;${\it Δ} = 2/M - Q_{\rm max}$&nbsp;.
+*The linear quantizer is designed for the amplitude range &nbsp;$±Q_{\rm max}$&nbsp; such that each quantization interval has width &nbsp;${\it Δ} = 2/M \cdot Q_{\rm max}$&nbsp;.
 *The diagram shows this fact for &nbsp;$Q_{\rm max} = q_{\rm max} = 6 \ \rm V$.&nbsp; These numerical values shall be assumed up to and including the subtask&nbsp; '''(5)'''&nbsp;.

← Older revision		Revision as of 13:52, 16 March 2022
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−	[[Category:Modulation Methods: Exercises\|^4.1 ~~Pulscodemodulation~~^]]	+	[[Category:Modulation Methods: Exercises\|^4.1 Pulse Code Modulation^]]

← Older revision	Revision as of 13:44, 16 March 2022
(No difference)

← Older revision		Revision as of 12:52, 23 March 2021
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−	[[Category:~~Aufgaben zu Modulationsverfahren~~\|^4.1 Pulscodemodulation^]]	+	[[Category:Modulation Methods: Exercises\|^4.1 Pulscodemodulation^]]

@@ Line 28: / Line 28: @@
 ''Hinweise:''
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Modulationsverfahren/Pulscodemodulation|Pulscodemodulation]].
+*Die Aufgabe gehört zum  Kapitel&nbsp; [[Modulation_Methods/Pulscodemodulation|Pulscodemodulation]].
-*Bezug genommen wird insbesondere auf die Seite&nbsp;  [[Modulationsverfahren/Pulscodemodulation#Quantisierung_und_Quantisierungsrauschen|Quantisierung und Quantisierungsrauschen]].
+*Bezug genommen wird insbesondere auf die Seite&nbsp;  [[Modulation_Methods/Pulscodemodulation#Quantisierung_und_Quantisierungsrauschen|Quantisierung und Quantisierungsrauschen]].