Aufgaben:Exercise 4.4Z: Signal-to-Noise Ratio with PCM - 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.~~4Z_Störabstand_bei_PCM~~]]		[[de:Aufgaben:Aufgabe 4.4Z: Störabstand bei PCM]]

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	'''(5)'''  From the graph in the information section,  it can be seen that the abscissa value  $10 ⋅ \lg \ ξ= 6 \ \rm dB$  results in the signal-to-noise ratio  $20 \ \rm dB$.		'''(5)'''  From the graph in the information section,  it can be seen that the abscissa value  $10 ⋅ \lg \ ξ= 6 \ \rm dB$  results in the signal-to-noise ratio  $20 \ \rm dB$.
	*From  $10 ⋅ \lg \ ρ_v = 20 \ \rm dB$  follows  $ρ_v = 100$  and thus further  $($with  $N = N_1 = 8)$:		*From  $10 ⋅ \lg \ ρ_v = 20 \ \rm dB$  follows  $ρ_v = 100$  and thus further  $($with  $N = N_1 = 8)$:
	:$$\rho_{\upsilon}= \frac{1}{ 2^{-2N } + 4 \cdot p_{\rm B}} \approx \frac{1}{ 1.5 \cdot 10^{-5} + 4 \cdot p_{\rm B}} = 100 \hspace{0.3cm}		:$$\rho_{\upsilon}= \frac{1}{ 2^{-2N } + 4 \cdot p_{\rm B}} \approx \frac{1}{ 1.5 \cdot 10^{-5} + 4 \cdot p_{\rm B}} = 100 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} p_{\rm B} = \frac{0.01 - 1.5 \cdot 10^{-5}}{ 4} \hspace{0.15cm}\underline {\approx 2.5\%} \hspace{0.05cm}.$$
	\Rightarrow \hspace{0.3cm} p_{\rm B} = \frac{0.01 - 1.5 \cdot 10^{-5}}{ 4} \hspace{0.15cm}\underline {\approx 2.5\%} \hspace{0.05cm}.$$

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

Hwang at 15:18, 23 January 2023

2023-01-23T15:18:43Z

← Older revision		Revision as of 17:18, 23 January 2023
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	summarizes the following system parameters:		summarizes the following system parameters:
	:* the frequency-independent attenuation factor  $α$  of the transmission channel,		:* the frequency-independent attenuation factor  $α$  of the transmission channel,
	:* the power  $P_{\rm S}$  of the ~~transmit~~ signal  $s(t)$,&nbsp, also called  "transmit power"  for short,		:* the power  $P_{\rm S}$  of the transmitted signal  $s(t)$,&nbsp, also called  "transmit power"  for short,
	:* the message frequency  $f_{\rm N}$  (bandwidth)  of the cosine source signal  $q(t)$,		:* the message frequency  $f_{\rm N}$  (bandwidth)  of the cosine source signal  $q(t)$,
	:* the noise power density  $N_0$  of the AWGN noise.		:* the noise power density  $N_0$  of the AWGN noise.

Hwang at 12:47, 17 November 2022

2022-11-17T12:47:30Z

Guenter at 12:30, 9 April 2022

2022-04-09T12:30:25Z

Guenter at 12:17, 9 April 2022

2022-04-09T12:17:48Z

← Older revision		Revision as of 14:17, 9 April 2022
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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 pages  [[Modulation_Methods/Pulse_Code_Modulation#Influence_of_transmission_errors\|"Influence of transmission errors"]]   and   [[Modulation_Methods/Pulse_Code_Modulation#Estimation_of_SNR_degradation_due_to_transmission_errors\|"Estimation of SNR degradation ~~by bit~~ errors"]].		*Reference is made in particular to the pages  [[Modulation_Methods/Pulse_Code_Modulation#Influence_of_transmission_errors\|"Influence of transmission errors"]]   and   [[Modulation_Methods/Pulse_Code_Modulation#Estimation_of_SNR_degradation_due_to_transmission_errors\|"Estimation of SNR degradation due to transmission errors"]].
	*The PCM considered here is the  '''PCM 30/32,'''  whose system parameters are given,  e.g. in  [[Aufgaben:Exercise_4.1:_PCM_System_30/32 \|Exercise 4.1]] .		*The PCM considered here is the  '''PCM 30/32''',  whose system parameters are given,  e.g. in  [[Aufgaben:Exercise_4.1:_PCM_System_30/32 \|Exercise 4.1]] .

Guenter at 12:15, 9 April 2022

2022-04-09T12:15:48Z

Noah at 20:20, 19 March 2022

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← Older revision		Revision as of 22:20, 19 March 2022
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	}}		}}

	[[File:EN_Mod_Z_4_4.png\|right\|frame\|~~Sink~~-to-noise ratio of PCM 30/32 compared to ZSB amplitude modulation]]		[[File:EN_Mod_Z_4_4.png\|right\|frame\|Signal-to-noise ratio of PCM 30/32 compared to ZSB amplitude modulation]]
	The graph shows the ~~sink~~-to-noise ratio  $10 ⋅ \lg \ ρ_v$  for pulse code modulation  $\rm (PCM)$  compared to analog double-sideband amplitude modulation, abbreviated as  $\rm ZSB-AM$.		The graph shows the signal-to-noise ratio  $10 ⋅ \lg \ ρ_v$  for pulse code modulation  $\rm (PCM)$  compared to analog double-sideband amplitude modulation, abbreviated as  $\rm ZSB-AM$.

	For the latter,  $ρ_v = ξ$, where the power parameter.		For the latter,  $ρ_v = ξ$, where the power parameter.
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	$N_2 \ = \ $ { 11 3% }		$N_2 \ = \ $ { 11 3% }

	{What (logarithmized) power parameter  $ξ_{40\ \rm dB}$  is required for 8-bit PCM to have a ~~sink~~-to-noise ratio equal to  $40\ \rm dB$ ?		{What (logarithmized) power parameter  $ξ_{40\ \rm dB}$  is required for 8-bit PCM to have a signal-to-noise ratio equal to  $40\ \rm dB$ ?
	\|type="{}"}		\|type="{}"}
	$10 ⋅ \lg \ ξ_{40\ \rm dB} \ = \ $ { 10 3% } $\ \rm dB$		$10 ⋅ \lg \ ξ_{40\ \rm dB} \ = \ $ { 10 3% } $\ \rm dB$

Noah at 20:19, 19 March 2022

2022-03-19T20:19:54Z

Show changes

← Older revision		Revision as of 14:47, 17 November 2022
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	For the PCM system,  the following approximation for the sink SNR was given on the ~~page~~  [[Modulation_Methods/Pulse_Code_Modulation#Estimation_of_SNR_degradation_due_to_transmission_errors\|"Estimation of SNR degradation due to bit errors"]],  which also takes into account transmission errors due to AWGN noise:		For the PCM system,  the following approximation for the sink SNR was given in the section  [[Modulation_Methods/Pulse_Code_Modulation#Estimation_of_SNR_degradation_due_to_transmission_errors\|"Estimation of SNR degradation due to bit errors"]],  which also takes into account transmission errors due to AWGN noise:
	:$$ \rho_{\upsilon}= \frac{1}{ 2^{-2N } + 4 \cdot p_{\rm B}} \hspace{0.05cm}.$$		:$$ \rho_{\upsilon}= \frac{1}{ 2^{-2N } + 4 \cdot p_{\rm B}} \hspace{0.05cm}.$$
	*Here  $N$  denotes the number of bits per sample and  $p_{\rm B}$  the bit error probability.		*Here  $N$  denotes the number of bits per sample and  $p_{\rm B}$  the bit error probability.
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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 ~~pages~~  [[Modulation_Methods/Pulse_Code_Modulation#Influence_of_transmission_errors\|"Influence of transmission errors"]]   and   [[Modulation_Methods/Pulse_Code_Modulation#Estimation_of_SNR_degradation_due_to_transmission_errors\|"Estimation of SNR degradation due to transmission errors"]].		*Reference is made in particular to the sections  [[Modulation_Methods/Pulse_Code_Modulation#Influence_of_transmission_errors\|"Influence of transmission errors"]]   and   [[Modulation_Methods/Pulse_Code_Modulation#Estimation_of_SNR_degradation_due_to_transmission_errors\|"Estimation of SNR degradation due to transmission errors"]].
	*The PCM considered here is the  '''PCM 30/32''',  whose system parameters are given,  e.g. in  [[Aufgaben:Exercise_4.1:_PCM_System_30/32 \|Exercise 4.1]] .		*The PCM considered here is the  '''PCM 30/32''',  whose system parameters are given,  e.g. in  [[Aufgaben:Exercise_4.1:_PCM_System_30/32 \|Exercise 4.1]] .

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	'''(3)'''  For double-sideband amplitude modulation  $\rm (DSB-AM)$,  this would require  $10 ⋅ \lg \ ξ = 40\ \rm dB$ .		'''(3)'''  For double-sideband amplitude modulation  $\rm (DSB-AM)$,  this would require  $10 ⋅ \lg \ ξ = 40\ \rm dB$ .
	*As can be seen from the graph on the data ~~page~~,  this abscissa value for the given PCM is lower by  $30 \ \rm dB$ ⇒  $10 ⋅ \lg \ ξ_{40\ \rm dB}\hspace{0.15cm}\underline { = 10 \ \rm dB}$.		*As can be seen from the graph in the data section,  this abscissa value for the given PCM is lower by  $30 \ \rm dB$ ⇒  $10 ⋅ \lg \ ξ_{40\ \rm dB}\hspace{0.15cm}\underline { = 10 \ \rm dB}$.


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	'''(5)'''  From the graph on the information ~~page~~,  it can be seen that the abscissa value  $10 ⋅ \lg \ ξ= 6 \ \rm dB$  results in the signal-to-noise ratio  $20 \ \rm dB$.		'''(5)'''  From the graph in the information section,  it can be seen that the abscissa value  $10 ⋅ \lg \ ξ= 6 \ \rm dB$  results in the signal-to-noise ratio  $20 \ \rm dB$.
	*From  $10 ⋅ \lg \ ρ_v = 20 \ \rm dB$  follows  $ρ_v = 100$  and thus further  $($with  $N = N_1 = 8)$:		*From  $10 ⋅ \lg \ ρ_v = 20 \ \rm dB$  follows  $ρ_v = 100$  and thus further  $($with  $N = N_1 = 8)$:
	:$$\rho_{\upsilon}= \frac{1}{ 2^{-2N } + 4 \cdot p_{\rm B}} \approx \frac{1}{ 1.5 \cdot 10^{-5} + 4 \cdot p_{\rm B}} = 100 \hspace{0.3cm}		:$$\rho_{\upsilon}= \frac{1}{ 2^{-2N } + 4 \cdot p_{\rm B}} \approx \frac{1}{ 1.5 \cdot 10^{-5} + 4 \cdot p_{\rm B}} = 100 \hspace{0.3cm}

← Older revision		Revision as of 14:30, 9 April 2022
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	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  The horizontal section of the PCM curve is determined by the quantization noise alone.		'''(1)'''  The horizontal section of the PCM curve is determined by the quantization noise alone.
	*Here, with the quantization step number  $M = 2^N$:		*Here,  with the quantization step number  $M = 2^N$:
	:$$ \rho_{v} (\xi \rightarrow \infty) = \rho_{\rm Q} = M^2 = 2^{2N} \hspace{0.3cm}\rightarrow \hspace{0.3cm} 10 \cdot {\rm lg}\hspace{0.1cm}\rho_{v} \approx 6\,{\rm dB} \cdot N\hspace{0.05cm}.$$		:$$ \rho_{v} (\xi \rightarrow \infty) = \rho_{\rm Q} = M^2 = 2^{2N} \hspace{0.3cm}\rightarrow \hspace{0.3cm} 10 \cdot {\rm lg}\hspace{0.1cm}\rho_{v} \approx 6\,{\rm dB} \cdot N\hspace{0.05cm}.$$
	*From the readable signal-to-noise ratio  $10 ⋅ \lg \ ρ_v ≈ 48 \ \rm dB$  it follows  $N_1\hspace{0.15cm}\underline { = 8}$  bits per sample and for the quantization level number  $M = 256$.		*From the readable signal-to-noise ratio  $10 ⋅ \lg \ ρ_v ≈ 48 \ \rm dB$  it follows:<br>  $N_1\hspace{0.15cm}\underline { = 8}$  bits per sample and for the quantization level number  $M = 256$.



	'''(2)'''  From the above approximation, we obtain for  $N_2\hspace{0.15cm}\underline { = 11}$  bits per sample   ⇒   $M = 2048$  the signal-to-noise ratio  $66 \rm dB$.		'''(2)'''  From the above approximation,  we obtain for  $N_2\hspace{0.15cm}\underline { = 11}$  bits per sample   ⇒   $M = 2048$  the signal-to-noise ratio  $66 \ \rm dB$.
	*With  $N = 10$   ⇒   $M = 1024$  one reaches only approx.  $60 \ \rm dB$.		*With  $N = 10$   ⇒   $M = 1024$  one reaches only approx.  $60 \ \rm dB$.
	*For the compact disc (CD), the PCM parameters  $N = 16$   ⇒   $M = 65536$   ⇒   $10 ⋅ \lg \ ρ_v > 96 \ \rm dB$  are used.		*For the compact disc  $\rm (CD)$,  the PCM parameters  $N = 16$   ⇒   $M = 65536$   ⇒   $10 ⋅ \lg \ ρ_v > 96 \ \rm dB$  are used.



	'''(3)'''  For double sideband amplitude modulation, this would require  $10 ⋅ \lg \ ξ = 40\ \rm dB$ .		'''(3)'''  For double-sideband amplitude modulation  $\rm (DSB-AM)$,  this would require  $10 ⋅ \lg \ ξ = 40\ \rm dB$ .
	*As can be seen from the graph on the data page, this abscissa value for the given PCM is lower by  $30 \ \rm dB$ ⇒  $10 ⋅ \lg \ ξ_{40\ \rm dB}\hspace{0.15cm}\underline { = 10 \ \rm dB}$.		*As can be seen from the graph on the data page,  this abscissa value for the given PCM is lower by  $30 \ \rm dB$ ⇒  $10 ⋅ \lg \ ξ_{40\ \rm dB}\hspace{0.15cm}\underline { = 10 \ \rm dB}$.


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	'''(5)'''  From the graph on the information page, it can be seen that the abscissa value  $10 ⋅ \lg \ ξ= 6 \ \rm dB$  results in the signal-to-noise ratio  $20 \ \rm dB$~~ ~~ .		'''(5)'''  From the graph on the information page,  it can be seen that the abscissa value  $10 ⋅ \lg \ ξ= 6 \ \rm dB$  results in the signal-to-noise ratio  $20 \ \rm dB$.
	*From  $10 ⋅ \lg \ ρ_v = 20 \ \rm dB$  follows  $ρ_v = 100$  and thus further  $($with  $N = N_1 = 8)$:		*From  $10 ⋅ \lg \ ρ_v = 20 \ \rm dB$  follows  $ρ_v = 100$  and thus further  $($with  $N = N_1 = 8)$:
	:$$\rho_{\upsilon}= \frac{1}{ 2^{-2N } + 4 \cdot p_{\rm B}} \approx \frac{1}{ 1.5 \cdot 10^{-5} + 4 \cdot p_{\rm B}} = 100 \hspace{0.3cm}		:$$\rho_{\upsilon}= \frac{1}{ 2^{-2N } + 4 \cdot p_{\rm B}} \approx \frac{1}{ 1.5 \cdot 10^{-5} + 4 \cdot p_{\rm B}} = 100 \hspace{0.3cm}
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	'''(6)'''  With the same  $ξ$  the bit error probability is still  $p_{\rm B} = 0.025$.  Thus, with  $N = 3$  (bits per sample):		'''(6)'''  With the same performance parameter  $ξ$,  the bit error probability is still  $p_{\rm B} = 0.025$.  Thus, with  $N = 3$  (bits per sample):
	:$$\rho_{\upsilon}= \frac{1}{ 2^{-6 } + 4 \cdot p_{\rm B}} = \frac{1}{ 0.015625 + 0.01} \approx 39 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}10 \cdot {\rm lg} \hspace{0.15cm}\rho_{\upsilon}\hspace{0.15cm}\underline {\approx 15.9\,{\rm dB}} \hspace{0.05cm}.$$		:$$\rho_{\upsilon}= \frac{1}{ 2^{-6 } + 4 \cdot p_{\rm B}} = \frac{1}{ 0.015625 + 0.01} \approx 39 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}10 \cdot {\rm lg} \hspace{0.15cm}\rho_{\upsilon}\hspace{0.15cm}\underline {\approx 15.9\,{\rm dB}} \hspace{0.05cm}.$$
	Further, it should be noted:		Further,  it should be noted:
	*With only three bits per sample, the quantization noise power  $(P_{\rm Q} = 0.015625)$  is already larger than the error noise power  $(P_{\rm F} = 0.01)$.		*With only three bits per sample,  the quantization noise power  $(P_{\rm Q} = 0.015625)$  is already larger than the error noise power  $(P_{\rm E} = 0.01)$.
	*By increasing the transmit power, the signal-to-noise ratio could be maximum  $10 ⋅ \lg \ ρ_v =18 \ \rm dB$  because of quantization, if no bit errors occur  $(P_{\rm F} = 0)$.		*By increasing the transmit power,  the signal-to-noise ratio could be maximum  $10 ⋅ \lg \ ρ_v =18 \ \rm dB$  because of quantization, if no bit errors occur  $(P_{\rm E} = 0)$.

	{{ML-Fuß}}		{{ML-Fuß}}

← Older revision		Revision as of 14:15, 9 April 2022
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	[[File:EN_Mod_Z_4_4.png\|right\|frame\|Signal-to-noise ratio of PCM 30/32 compared to ZSB amplitude modulation]]		[[File:EN_Mod_Z_4_4.png\|right\|frame\|Signal-to-noise ratio of PCM 30/32 compared to ZSB amplitude modulation]]
	The graph shows the signal-to-noise ratio  $10 ⋅ \lg \ ρ_v$  for pulse code modulation  $\rm (PCM)$  compared to analog double-sideband amplitude modulation, abbreviated as  $\rm ~~ZSB~~-AM$.		The graph shows the signal-to-noise ratio  $10 ⋅ \lg \ ρ_v$  for pulse code modulation  $\rm (PCM)$  compared to analog double-sideband amplitude modulation, abbreviated as  "$\rm DSB-AM$".

	For the latter,  $ρ_v = ξ$, where the ~~power~~ parameter.		For the latter,   $ρ_v = ξ$,  where the persormanc parameter
	:$$\xi = \frac{\alpha^2 \cdot P_{\rm S}}{N_0 \cdot f_{\rm N}} $$		:$$\xi = \frac{\alpha^2 \cdot P_{\rm S}}{N_0 \cdot f_{\rm N}} $$
	~~Summarizes~~ the following system parameters:		summarizes the following system parameters:
	:* the frequency-independent ~~transmission~~ factor  $α$  of the transmission channel,		:* the frequency-independent attenuation factor  $α$  of the transmission channel,
	:* the power  $P_{\rm S}$  of the transmit signal  $s(t)$, also called transmit power for short,		:* the power  $P_{\rm S}$  of the transmit signal  $s(t)$,&nbsp, also called  "transmit power"  for short,
	:* the message frequency  $f_{\rm N}$  (bandwidth) of the cosine source signal  $q(t)$,		:* the message frequency  $f_{\rm N}$  (bandwidth)  of the cosine source signal  $q(t)$,
	:* the noise power density  $N_0$  of the AWGN noise.		:* the noise power density  $N_0$  of the AWGN noise.


	For the PCM system, the following approximation for the sink SNR was given on the page  [[Modulation_Methods/Pulse_Code_Modulation#Estimation_of_SNR_degradation_due_to_transmission_errors\|Estimation of SNR degradation due to bit errors]]  which also takes into account transmission errors due to AWGN noise:		For the PCM system,  the following approximation for the sink SNR was given on the page  [[Modulation_Methods/Pulse_Code_Modulation#Estimation_of_SNR_degradation_due_to_transmission_errors\|"Estimation of SNR degradation due to bit errors"]],  which also takes into account transmission errors due to AWGN noise:
	:$$ \rho_{\upsilon}= \frac{1}{ 2^{-2N } + 4 \cdot p_{\rm B}} \hspace{0.05cm}.$$		:$$ \rho_{\upsilon}= \frac{1}{ 2^{-2N } + 4 \cdot p_{\rm B}} \hspace{0.05cm}.$$
	*Here  $N$  denotes the number of bits per sample and  $p_{\rm B}$  the bit error probability.		*Here  $N$  denotes the number of bits per sample and  $p_{\rm B}$  the bit error probability.
	* Since  $ξ$  in digital modulation ~~can~~ also be interpreted as the ''signal energy per bit''  related to the ''noise power density'' $(E_{\rm B}/N_0)$, with the complementary Gaussian error signal  ${\rm Q}(x)$  approximately:		* Since  $ξ$  can in digital modulation also be interpreted as the  "signal energy per bit"  related to the  "noise power density"  $(E_{\rm B}/N_0)$,  with the complementary Gaussian error signal  ${\rm Q}(x)$  approximately:
	:$$ p_{\rm B}= {\rm Q} \left ( \sqrt{2 \xi }\right ) \hspace{0.05cm}.$$		:$$ p_{\rm B}= {\rm Q} \left ( \sqrt{2 \xi }\right ) \hspace{0.05cm}.$$





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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 pages  [[Modulation_Methods/Pulse_Code_Modulation#Influence_of_transmission_errors\|Influence of transmission errors]]  and  [[Modulation_Methods/Pulse_Code_Modulation#Estimation_of_SNR_degradation_due_to_transmission_errors\|Estimation of SNR degradation by bit errors]].		*Reference is made in particular to the pages  [[Modulation_Methods/Pulse_Code_Modulation#Influence_of_transmission_errors\|"Influence of transmission errors"]]   and   [[Modulation_Methods/Pulse_Code_Modulation#Estimation_of_SNR_degradation_due_to_transmission_errors\|"Estimation of SNR degradation by bit errors"]].
	*The PCM considered here is the  '''PCM 30/32''' whose system parameters are given, ~~for example,~~ in ~~the~~  [[Aufgaben:Exercise_4.1:_PCM_System_30/32 \|Exercise 4.1]] .		*The PCM considered here is the  '''PCM 30/32,'''  whose system parameters are given,  e.g. in  [[Aufgaben:Exercise_4.1:_PCM_System_30/32 \|Exercise 4.1]] .


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	$N_1 \ = \ $ { 8 3% }		$N_1 \ = \ $ { 8 3% }

	{How many bits per sample   ⇒   $N = N_2$  would have to be used to achieve  $10 ⋅ \lg \ ρ_v > 64 \ \rm dB$  (music quality)?		{How many bits per sample   ⇒   $N = N_2$  would have to be used to achieve  $10 ⋅ \lg \ ρ_v > 64 \ \rm dB$  ("music quality")?
	\|type="{}"}		\|type="{}"}
	$N_2 \ = \ $ { 11 3% }		$N_2 \ = \ $ { 11 3% }

	{What (logarithmized) ~~power~~ parameter  $ξ_{40\ \rm dB}$  is required for 8-bit PCM to have a signal-to-noise ratio ~~equal to~~  $40\ \rm dB$ ?		{What  (logarithmized)  performance parameter  $ξ_{40\ \rm dB}$  is required for 8-bit PCM to have a signal-to-noise ratio of  $40\ \rm dB$ ?
	\|type="{}"}		\|type="{}"}
	$10 ⋅ \lg \ ξ_{40\ \rm dB} \ = \ $ { 10 3% } $\ \rm dB$		$10 ⋅ \lg \ ξ_{40\ \rm dB} \ = \ $ { 10 3% } $\ \rm dB$

	{By what factor could the transmit power be reduced for PCM compared to ~~ZSB~~-AM to still achieve  $10 ⋅ \lg \ ρ_v = 40\ \rm dB$ ?		{By what factor could the transmit power be reduced for PCM compared to  "DSB-AM"  to still achieve  $10 ⋅ \lg \ ρ_v = 40\ \rm dB$ ?
	\|type="{}"}		\|type="{}"}
	$K_\text{AM → PCM} \ = \ $ { 1000 3% }		$K_\text{AM → PCM} \ = \ $ { 1000 3% }