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

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	[[Category:Modulation Methods: Exercises\|^4.1 Pulse Code Modulation^]]		[[Category:Modulation Methods: Exercises\|^4.1 Pulse Code Modulation^]]
	[[de:Aufgaben:4.~~4_Quantisierungsrauschen~~]]		[[de:Aufgaben:Aufgabe 4.4: Zum Quantisierungsrauschen]]

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	[[Category:Modulation Methods: Exercises\|^4.1 Pulse Code Modulation^]]		[[Category:Modulation Methods: Exercises\|^4.1 Pulse Code Modulation^]]
			[[de:Aufgaben:4.4_Quantisierungsrauschen]]

Guenter at 13:57, 11 April 2022

2022-04-11T13:57:27Z

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

Guenter at 11:55, 9 April 2022

2022-04-09T11:55:41Z

← Older revision		Revision as of 13:55, 9 April 2022
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	$P_{\rm S} \ = \ $ { 12 3% } $\ \rm V^2$		$P_{\rm S} \ = \ $ { 12 3% } $\ \rm V^2$

	{Which statements are true for the error signal  $ε(t)$ ?		{Which statements are true for the error signal  $ε(t)= q_{\rm Q}(t)-q(t)$ ?
	\|type="[]"}		\|type="[]"}
	+ $ε(t)$  has a sawtooth shape.		+ $ε(t)$  has a sawtooth shape.

Guenter at 10:26, 9 April 2022

2022-04-09T10:26:13Z

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

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			[[File:P_ID1618__Mod_A_4_4f.png\|right\|frame\|Quantization with  $Q_{\rm max} \ne q_{\rm max}$]]
			'''(6)'''  <u>All of the above preconditions</u>  must be satisfied:
			*For non-linear quantization,  the simple relation  $ρ_{\rm Q} = M^2$  does not hold.
			*For an PDF other than the uniform distribution  $ρ_{\rm Q} = M^2$  is also only an approximation,  but this is usually accepted.
			*If  $Q_{\rm max} < q_{\rm max}$,  truncation of the peaks occurs,  while with  $Q_{\rm max} > q_{\rm max}$  the quantization intervals are larger than required.

	~~'''(6)'''  <u>All of the above preconditions</u> must be satisfied:~~
	*For nonlinear quantization, the simple relation  $ρ_{\rm Q} = M^2$  does not hold.
	*For an amplitude distribution other than the uniform distribution  $ρ_{\rm Q} = M^2$  is also only an approximation, but this is usually accepted.
	*If  $Q_{\rm max} < q_{\rm max}$, truncation of the peaks occurs, while with  $Q_{\rm max} > q_{\rm max}$  the quantization intervals are larger than required.

			The graph shows the error signals  $ε(t)$
			#for  $Q_{\rm max} > q_{\rm max}$  (left)
			#and  $Q_{\rm max} < q_{\rm max}$  (right):

	~~[[File:P_ID1618__Mod_A_4_4f.png\|right\|frame\|Quantization with  $Q_{\rm max} \ne q_{\rm max}$]]~~
	~~<br><br><br><br>~~
	~~The graph shows the error signals  $ε(t)$  for  $Q_{\rm max} > q_{\rm max}$  (left) and  $Q_{\rm max} < q_{\rm max}$  (right).~~

			:In both cases,  the quantization noise power is significantly larger than calculated in sub-task  '''(3)'''.
	In both cases, the quantization noise power is significantly larger than calculated in ~~point~~  '''(3)'''~~ ~~.

Guenter at 15:24, 8 April 2022

2022-04-08T15:24:23Z

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


	The ~~so-called~~  ~~'''~~quantization noise power~~'''~~  is defined as the ~~root mean square~~ of the difference signal  $ε(t) = q_{\rm Q}(t) - q(t)$  . It holds		The  "quantization noise power"  is defined as the second moment of the difference signal  $ε(t) = q_{\rm Q}(t) - q(t)$.  It holds:
	:$$P_{\rm Q} = \frac{1}{T_0' } \cdot \int_{0}^{T_0'}\varepsilon(t)^2 \hspace{0.05cm}{\rm d}t \hspace{0.05cm},$$		:$$P_{\rm Q} = \frac{1}{T_0' } \cdot \int_{0}^{T_0'}\varepsilon(t)^2 \hspace{0.05cm}{\rm d}t \hspace{0.05cm},$$
	where the time  $T_0'$  is to be chosen appropriately.		where the time  $T_0'$  is to be chosen appropriately.  The  "quantization SNR"  is the ratio    $\rho_{\rm Q} = {P_{\rm S}}/{P_{\rm Q}}\hspace{0.05cm}$,  which is usually given logarithmically  (in dB).

	~~Quantization~~ SNR is the ratio    $\rho_{\rm Q} = {P_{\rm S}}/{P_{\rm Q}}\hspace{0.05cm},$ which is usually given logarithmically (in dB).


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	Hints:		Hints:
	*The exercise belongs to the chapter  [[Modulation_Methods/Pulse_Code_Modulation\|Pulse Code Modulation]].		*The exercise belongs to the chapter  [[Modulation_Methods/Pulse_Code_Modulation\|"Pulse Code Modulation"]].
	*Reference is made in particular to the page  [[Modulation_Methods/Pulse_Code_Modulation#Quantization_and_quantization_noise\|Quantization and quantization Noise]].		*Reference is made in particular to the page  [[Modulation_Methods/Pulse_Code_Modulation#Quantization_and_quantization_noise\|"Quantization and quantization Noise"]].


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	<quiz display=simple>		<quiz display=simple>
	{Calculate the signal power  $P_{\rm S}$  (referred to the resistor $1 \ \rm Ω$~~ )~~.		{Calculate the signal power  $P_{\rm S}$  $($referred to the resistor $1 \ \rm Ω)$.
	\|type="{}"}		\|type="{}"}
	$P_{\rm S} \ = \ $ { 12 3% } $\ \rm V^2$		$P_{\rm S} \ = \ $ { 12 3% } $\ \rm V^2$
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	+ $ε(t)$  has a sawtooth shape.		+ $ε(t)$  has a sawtooth shape.
	- $ε(t)$  has a step-like progression.		- $ε(t)$  has a step-like progression.
	+ $ε(t)$  is restricted to the range  $±{\it Δ}/2 = ±1 \ \rm V$~~ ~~.		+ $ε(t)$  is restricted to the range  $±{\it Δ}/2 = ±1 \ \rm V$.
	+ $ε(t)$  has period  $T_0' = T_0/M$.		+ $ε(t)$  has period  $T_0' = T_0/M$.

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	{Calculate the quantization noise ratio for  $M = 6$.		{Calculate the quantization noise ratio for  $M = 6$.
	\|type="{}"}		\|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  $N = 8$  or  $N = 16$ bits?		{What values result from quantization with  $N = 8$  or  $N = 16$ bits?
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	$N = 16\text{:}\hspace{0.15cm}10 ⋅ \lg \ ρ_{\rm Q} \ = \ ${ 96.32 3% } $\ \rm dB$		$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  $ρ_{\rm Q}$~~ ~~?		{What conditions must be met for the derived equation to apply to  $ρ_{\rm Q}$?
	\|type="[]"}		\|type="[]"}
	+ All amplitude values are equally probable.		+ All amplitude values are equally probable.
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	[[File:P_ID1616__Mod_A_4_4.png\|right\|frame\|~~error~~ signal for  $Q_{\rm max} = q_{\rm max}$]]		[[File:P_ID1616__Mod_A_4_4.png\|right\|frame\|Error signal for  $Q_{\rm max} = q_{\rm max}$]]
	'''(2)'''  Correct are <u>suggested solutions 1, 3, and 4</u>:		'''(2)'''  Correct are <u>suggested solutions 1, 3, and 4</u>:
	*We assume here  $Q_{\rm max} = q_{\rm max} = 6 \ \rm V$  .		*We assume here  $Q_{\rm max} = q_{\rm max} = 6 \ \rm V$  .
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	[[File:P_ID1618__Mod_A_4_4f.png\|right\|frame\|~~Quantisierung mit~~  $Q_{\rm max} \ne q_{\rm max}$]]		[[File:P_ID1618__Mod_A_4_4f.png\|right\|frame\|Quantization with  $Q_{\rm max} \ne q_{\rm max}$]]
	<br><br><br><br>		<br><br><br><br>
	The graph shows the error signals  $ε(t)$  for  $Q_{\rm max} > q_{\rm max}$  (left) and  $Q_{\rm max} < q_{\rm max}$  (right).		The graph shows the error signals  $ε(t)$  for  $Q_{\rm max} > q_{\rm max}$  (left) and  $Q_{\rm max} < q_{\rm max}$  (right).

Noah at 16:44, 6 April 2022

2022-04-06T16:44:30Z

← Older revision		Revision as of 18:44, 6 April 2022
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	The signal  $q(t)$  is quantized according to the graph with  $M = 6$  stages.  The quantized signal is  $q_{\rm Q}(t)$, where:		The signal  $q(t)$  is quantized according to the graph with  $M = 6$  stages.  The quantized signal is  $q_{\rm Q}(t)$, where:
	*The linear quantizer is designed for the amplitude range  $±Q_{\rm max}$  such that each quantization interval has width  ${\it Δ} = 2/M - Q_{\rm max}$ .		*The linear quantizer is designed for the amplitude range  $±Q_{\rm max}$  such that each quantization interval has width  ${\it Δ} = 2/M \cdot Q_{\rm max}$ .
	*The diagram shows this fact for  $Q_{\rm max} = q_{\rm max} = 6 \ \rm V$.  These numerical values shall be assumed up to and including the subtask  '''(5)''' .		*The diagram shows this fact for  $Q_{\rm max} = q_{\rm max} = 6 \ \rm V$.  These numerical values shall be assumed up to and including the subtask  '''(5)''' .

Noah at 19:11, 19 March 2022

2022-03-19T19:11:10Z

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Guenter at 13:52, 16 March 2022

2022-03-16T13:52:23Z

← Older revision		Revision as of 15:52, 16 March 2022
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	[[Category:Modulation Methods: Exercises\|^4.1 ~~Pulscodemodulation~~^]]		[[Category:Modulation Methods: Exercises\|^4.1 Pulse Code Modulation^]]

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

← Older revision		Revision as of 15:44, 16 March 2022
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