Aufgaben:Exercise 4.4Z: Signal-to-Noise Ratio with PCM - Revision history

Hwang at 15:18, 23 January 2023

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Guenter at 13:53, 16 March 2022

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Guenter: Guenter moved page Aufgaben:Aufgabe 4.4Z: Störabstand bei PCM to Aufgaben:Exercise 4.4Z: Signal-to-Noise Ratio with PCM

2022-03-16T13:46:03Z

Guenter moved page Aufgabe 4.4Z: Störabstand bei PCM to Exercise 4.4Z: Signal-to-Noise Ratio with PCM

@@ Line 10: / Line 10: @@
 summarizes the following system parameters:
 :* the frequency-independent attenuation factor &nbsp;$α$&nbsp; of the transmission channel,
-:* the power &nbsp;$P_{\rm S}$&nbsp; of the transmit signal &nbsp;$s(t)$,&nbsp, also called&nbsp; "transmit power"&nbsp; for short,
+:* the power &nbsp;$P_{\rm S}$&nbsp; of the transmitted signal &nbsp;$s(t)$,&nbsp, also called&nbsp; "transmit power"&nbsp; for short,
 :* the message frequency &nbsp;$f_{\rm N}$&nbsp; (bandwidth)&nbsp; of the cosine source signal &nbsp;$q(t)$,
 :* the noise power density &nbsp;$N_0$&nbsp; of the AWGN noise.

@@ Line 15: / Line 15: @@
-For the PCM system,&nbsp; the following approximation for the sink SNR was given on the page&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Estimation_of_SNR_degradation_due_to_transmission_errors|"Estimation of SNR degradation due to bit errors"]],&nbsp; which also takes into account transmission errors due to AWGN noise:
+For the PCM system,&nbsp; the following approximation for the sink SNR was given in the section&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Estimation_of_SNR_degradation_due_to_transmission_errors|"Estimation of SNR degradation due to bit errors"]],&nbsp; which also takes into account transmission errors due to AWGN noise:
 :$$ \rho_{\upsilon}= \frac{1}{ 2^{-2N } + 4 \cdot p_{\rm B}} \hspace{0.05cm}.$$
 *Here &nbsp;$N$&nbsp; denotes the number of bits per sample and &nbsp;$p_{\rm B}$&nbsp; the bit error probability.
@@ Line 27: / Line 27: @@
 Hints:
 *The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Pulse_Code_Modulation|"Pulse Code Modulation"]].
-*Reference is made in particular to the pages&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Influence_of_transmission_errors|"Influence of transmission errors"]] &nbsp; and &nbsp; [[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&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Influence_of_transmission_errors|"Influence of transmission errors"]] &nbsp; and &nbsp; [[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&nbsp; '''PCM 30/32''',&nbsp; whose system parameters are given,&nbsp; e.g. in&nbsp; [[Aufgaben:Exercise_4.1:_PCM_System_30/32 |Exercise 4.1]]&nbsp;.
@@ Line 77: / Line 77: @@
 '''(3)'''&nbsp; For double-sideband amplitude modulation&nbsp; $\rm (DSB-AM)$,&nbsp; this would require&nbsp; $10 ⋅ \lg \ ξ = 40\ \rm dB$&nbsp;.
-*As can be seen from the graph on the data page,&nbsp; this abscissa value for the given PCM is lower by&nbsp; $30 \ \rm dB$&nbsp;⇒&nbsp; $10 ⋅ \lg \ ξ_{40\ \rm dB}\hspace{0.15cm}\underline { = 10 \ \rm dB}$.
+*As can be seen from the graph in the data section,&nbsp; this abscissa value for the given PCM is lower by&nbsp; $30 \ \rm dB$&nbsp;⇒&nbsp; $10 ⋅ \lg \ ξ_{40\ \rm dB}\hspace{0.15cm}\underline { = 10 \ \rm dB}$.
@@ Line 85: / Line 85: @@
-'''(5)'''&nbsp; From the graph on the information page,&nbsp; it can be seen that the abscissa value&nbsp; $10 ⋅ \lg \ ξ= 6 \ \rm dB$&nbsp; results in the signal-to-noise ratio&nbsp; $20 \ \rm dB$.
+'''(5)'''&nbsp; From the graph in the information section,&nbsp; it can be seen that the abscissa value&nbsp; $10 ⋅ \lg \ ξ= 6 \ \rm dB$&nbsp; results in the signal-to-noise ratio&nbsp; $20 \ \rm dB$.
 *From&nbsp; $10 ⋅ \lg \ ρ_v = 20 \ \rm dB$&nbsp; follows&nbsp; $ρ_v = 100$&nbsp; and thus further&nbsp; $($with&nbsp; $N = N_1 = 8)$:
 :$$\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}

@@ Line 64: / Line 64: @@
 {{ML-Kopf}}
 '''(1)'''&nbsp; The horizontal section of the PCM curve is determined by the quantization noise alone.&nbsp;
-*Here, with the quantization step number&nbsp; $M = 2^N$:
+*Here,&nbsp; with the quantization step number&nbsp; $M = 2^N$:
 :$$ \rho_{v} (\xi \rightarrow \infty) = \rho_{\rm Q} = M^2 = 2^{2N} \hspace{0.3cm}\rightarrow \hspace{0.3cm} 10 \cdot {\rm lg}\hspace{0.1cm}\rho_{v} \approx 6\,{\rm dB} \cdot N\hspace{0.05cm}.$$
-*From the readable signal-to-noise ratio&nbsp; $10 ⋅ \lg \ ρ_v ≈ 48 \ \rm dB$&nbsp; it follows&nbsp; $N_1\hspace{0.15cm}\underline { = 8}$&nbsp; bits per sample and for the quantization level number&nbsp; $M = 256$.
+*From the readable signal-to-noise ratio&nbsp; $10 ⋅ \lg \ ρ_v ≈ 48 \ \rm dB$&nbsp; it follows:<br>&nbsp; $N_1\hspace{0.15cm}\underline { = 8}$&nbsp; bits per sample and for the quantization level number&nbsp; $M = 256$.
-'''(2)'''&nbsp; From the above approximation, we obtain for&nbsp; $N_2\hspace{0.15cm}\underline { = 11}$&nbsp; bits per sample &nbsp; ⇒ &nbsp; $M = 2048$&nbsp; the signal-to-noise ratio&nbsp; $66 \rm dB$.
+'''(2)'''&nbsp; From the above approximation,&nbsp; we obtain for&nbsp; $N_2\hspace{0.15cm}\underline { = 11}$&nbsp; bits per sample &nbsp; ⇒ &nbsp; $M = 2048$&nbsp; the signal-to-noise ratio&nbsp; $66 \ \rm dB$.
 *With&nbsp; $N = 10$ &nbsp; ⇒ &nbsp; $M = 1024$&nbsp; one reaches only approx.&nbsp; $60 \ \rm dB$.
-*For the compact disc (CD), the PCM parameters&nbsp; $N = 16$ &nbsp; ⇒ &nbsp; $M = 65536$ &nbsp; ⇒ &nbsp; $10 ⋅ \lg \ ρ_v > 96 \ \rm dB$&nbsp; are used.
+*For the compact disc&nbsp; $\rm (CD)$,&nbsp; the PCM parameters&nbsp; $N = 16$ &nbsp; ⇒ &nbsp; $M = 65536$ &nbsp; ⇒ &nbsp; $10 ⋅ \lg \ ρ_v > 96 \ \rm dB$&nbsp; are used.
-'''(3)'''&nbsp; For double sideband amplitude modulation, this would require&nbsp; $10 ⋅ \lg \ ξ = 40\ \rm dB$&nbsp;.
+'''(3)'''&nbsp; For double-sideband amplitude modulation&nbsp; $\rm (DSB-AM)$,&nbsp; this would require&nbsp; $10 ⋅ \lg \ ξ = 40\ \rm dB$&nbsp;.
-*As can be seen from the graph on the data page, this abscissa value for the given PCM is lower by&nbsp; $30 \ \rm dB$&nbsp;⇒&nbsp; $10 ⋅ \lg \ ξ_{40\ \rm dB}\hspace{0.15cm}\underline { = 10 \ \rm dB}$.
+*As can be seen from the graph on the data page,&nbsp; this abscissa value for the given PCM is lower by&nbsp; $30 \ \rm dB$&nbsp;⇒&nbsp; $10 ⋅ \lg \ ξ_{40\ \rm dB}\hspace{0.15cm}\underline { = 10 \ \rm dB}$.
@@ Line 85: / Line 85: @@
-'''(5)'''&nbsp; From the graph on the information page, it can be seen that the abscissa value&nbsp; $10 ⋅ \lg \ ξ= 6 \ \rm dB$&nbsp; results in the signal-to-noise ratio&nbsp; $20 \ \rm dB$&nbsp; .
+'''(5)'''&nbsp; From the graph on the information page,&nbsp; it can be seen that the abscissa value&nbsp; $10 ⋅ \lg \ ξ= 6 \ \rm dB$&nbsp; results in the signal-to-noise ratio&nbsp; $20 \ \rm dB$.
 *From&nbsp; $10 ⋅ \lg \ ρ_v = 20 \ \rm dB$&nbsp; follows&nbsp; $ρ_v = 100$&nbsp; and thus further&nbsp; $($with&nbsp; $N = N_1 = 8)$:
 :$$\rho_{\upsilon}= \frac{1}{ 2^{-2N } + 4 \cdot p_{\rm B}} \approx \frac{1}{ 1.5 \cdot 10^{-5} + 4 \cdot p_{\rm B}} = 100 \hspace{0.3cm}
@@ Line 91: / Line 91: @@
-'''(6)'''&nbsp; With the same&nbsp; $ξ$&nbsp; the bit error probability is still&nbsp; $p_{\rm B} = 0.025$.&nbsp; Thus, with&nbsp; $N = 3$&nbsp; (bits per sample):
+'''(6)'''&nbsp; With the same performance parameter&nbsp; $ξ$,&nbsp; the bit error probability is still&nbsp; $p_{\rm B} = 0.025$.&nbsp; Thus, with&nbsp; $N = 3$&nbsp; (bits per sample):
 :$$\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}.$$
-Further, it should be noted:
+Further,&nbsp; it should be noted:
-*With only three bits per sample, the quantization noise power&nbsp; $(P_{\rm Q} = 0.015625)$&nbsp; is already larger than the error noise power&nbsp; $(P_{\rm F} = 0.01)$.
+*With only three bits per sample,&nbsp; the quantization noise power&nbsp; $(P_{\rm Q} = 0.015625)$&nbsp; is already larger than the error noise power&nbsp; $(P_{\rm E} = 0.01)$.
-*By increasing the transmit power, the signal-to-noise ratio could be maximum&nbsp; $10 ⋅ \lg \ ρ_v =18 \ \rm dB$&nbsp; because of quantization, if no bit errors occur&nbsp; $(P_{\rm F} = 0)$.
+*By increasing the transmit power,&nbsp; the signal-to-noise ratio could be maximum&nbsp; $10 ⋅ \lg \ ρ_v =18 \ \rm dB$&nbsp; because of quantization, if no bit errors occur&nbsp; $(P_{\rm E} = 0)$.
 {{ML-Fuß}}

@@ Line 27: / Line 27: @@
 Hints:
 *The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Pulse_Code_Modulation|"Pulse Code Modulation"]].
-*Reference is made in particular to the pages&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Influence_of_transmission_errors|"Influence of transmission errors"]] &nbsp; and &nbsp; [[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&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Influence_of_transmission_errors|"Influence of transmission errors"]] &nbsp; and &nbsp; [[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&nbsp; '''PCM 30/32,'''&nbsp; whose system parameters are given,&nbsp; e.g. in&nbsp; [[Aufgaben:Exercise_4.1:_PCM_System_30/32 |Exercise 4.1]]&nbsp;.
+*The PCM considered here is the&nbsp; '''PCM 30/32''',&nbsp; whose system parameters are given,&nbsp; e.g. in&nbsp; [[Aufgaben:Exercise_4.1:_PCM_System_30/32 |Exercise 4.1]]&nbsp;.

@@ Line 4: / Line 4: @@
 [[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 &nbsp;$10 ⋅ \lg \ ρ_v$&nbsp; for pulse code modulation&nbsp; $\rm (PCM)$&nbsp; compared to analog double-sideband amplitude modulation, abbreviated as&nbsp; $\rm ZSB-AM$.&nbsp;
+The graph shows the signal-to-noise ratio &nbsp;$10 ⋅ \lg \ ρ_v$&nbsp; for pulse code modulation&nbsp; $\rm (PCM)$&nbsp; compared to analog double-sideband amplitude modulation, abbreviated as&nbsp; "$\rm DSB-AM$".&nbsp;
-For the latter, &nbsp;$ρ_v = ξ$, where the power parameter.
+For the latter,&nbsp; &nbsp;$ρ_v = ξ$,&nbsp; where the persormanc parameter
 :$$\xi = \frac{\alpha^2 \cdot P_{\rm S}}{N_0 \cdot f_{\rm N}} $$
-Summarizes the following system parameters:
+summarizes the following system parameters:
-:* the frequency-independent transmission factor &nbsp;$α$&nbsp; of the transmission channel,
+:* the frequency-independent attenuation factor &nbsp;$α$&nbsp; of the transmission channel,
-:* the power &nbsp;$P_{\rm S}$&nbsp; of the transmit signal &nbsp;$s(t)$, also called transmit power for short,
+:* the power &nbsp;$P_{\rm S}$&nbsp; of the transmit signal &nbsp;$s(t)$,&nbsp, also called&nbsp; "transmit power"&nbsp; for short,
-:* the message frequency &nbsp;$f_{\rm N}$&nbsp; (bandwidth) of the cosine source signal &nbsp;$q(t)$,
+:* the message frequency &nbsp;$f_{\rm N}$&nbsp; (bandwidth)&nbsp; of the cosine source signal &nbsp;$q(t)$,
 :* the noise power density &nbsp;$N_0$&nbsp; of the AWGN noise.
-For the PCM system, the following approximation for the sink SNR was given on the page&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Estimation_of_SNR_degradation_due_to_transmission_errors|Estimation of SNR degradation due to bit errors]]&nbsp; which also takes into account transmission errors due to AWGN noise:
+For the PCM system,&nbsp; the following approximation for the sink SNR was given on the page&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Estimation_of_SNR_degradation_due_to_transmission_errors|"Estimation of SNR degradation due to bit errors"]],&nbsp; which also takes into account transmission errors due to AWGN noise:
 :$$ \rho_{\upsilon}= \frac{1}{ 2^{-2N } + 4 \cdot p_{\rm B}} \hspace{0.05cm}.$$
 *Here &nbsp;$N$&nbsp; denotes the number of bits per sample and &nbsp;$p_{\rm B}$&nbsp; the bit error probability.
-* Since &nbsp;$ξ$&nbsp; in digital modulation can also be interpreted as the ''signal energy per bit''&nbsp; related to the ''noise power density'' $(E_{\rm B}/N_0)$, with the complementary Gaussian error signal &nbsp;${\rm Q}(x)$&nbsp; approximately:
+* Since &nbsp;$ξ$&nbsp; can in digital modulation also be interpreted as the&nbsp; "signal energy per bit"&nbsp; related to the&nbsp; "noise power density"&nbsp; $(E_{\rm B}/N_0)$,&nbsp; with the complementary Gaussian error signal &nbsp;${\rm Q}(x)$&nbsp; approximately:
 :$$ p_{\rm B}= {\rm Q} \left ( \sqrt{2 \xi }\right ) \hspace{0.05cm}.$$
@@ Line 29: / 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 pages&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Influence_of_transmission_errors|Influence of transmission errors]]&nbsp; and&nbsp; [[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&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Influence_of_transmission_errors|"Influence of transmission errors"]] &nbsp; and &nbsp; [[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&nbsp; '''PCM 30/32''' whose system parameters are given, for example, in the&nbsp; [[Aufgaben:Exercise_4.1:_PCM_System_30/32 |Exercise 4.1]]&nbsp;.
+*The PCM considered here is the&nbsp; '''PCM 30/32,'''&nbsp; whose system parameters are given,&nbsp; e.g. in&nbsp; [[Aufgaben:Exercise_4.1:_PCM_System_30/32 |Exercise 4.1]]&nbsp;.
@@ Line 42: / Line 39: @@
 $N_1 \ = \ $ { 8 3% }
-{How many bits per sample &nbsp; &rArr; &nbsp; $N = N_2$&nbsp; would have to be used to achieve &nbsp;$10 ⋅ \lg \ ρ_v > 64 \ \rm dB$&nbsp; (music quality)?
+{How many bits per sample &nbsp; &rArr; &nbsp; $N = N_2$&nbsp; would have to be used to achieve &nbsp;$10 ⋅ \lg \ ρ_v > 64 \ \rm dB$&nbsp; ("music quality")?
 |type="{}"}
 $N_2 \ = \ $ { 11 3% }
-{What (logarithmized) power parameter &nbsp;$ξ_{40\ \rm dB}$&nbsp; is required for 8-bit PCM to have a signal-to-noise ratio equal to &nbsp;$40\ \rm dB$&nbsp;?
+{What&nbsp; (logarithmized)&nbsp; performance parameter &nbsp;$ξ_{40\ \rm dB}$&nbsp; is required for 8-bit PCM to have a signal-to-noise ratio of &nbsp;$40\ \rm dB$&nbsp;?
 |type="{}"}
 $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 &nbsp;$10 ⋅ \lg \ ρ_v = 40\ \rm dB$&nbsp;?
+{By what factor could the transmit power be reduced for PCM compared to&nbsp; "DSB-AM"&nbsp; to still achieve &nbsp;$10 ⋅ \lg \ ρ_v = 40\ \rm dB$&nbsp;?
 |type="{}"}
 $K_\text{AM → PCM} \ = \ $ { 1000 3% }

@@ Line 3: / Line 3: @@
 }}
-[[File:EN_Mod_Z_4_4.png|right|frame|Sink-to-drop ratio of PCM 30/32 compared to ZSB amplitude modulation]]
+[[File:EN_Mod_Z_4_4.png|right|frame|Sink-to-noise ratio of PCM 30/32 compared to ZSB amplitude modulation]]
-The graph shows the sink-to-noise ratio &nbsp;$10 - \lg \ ρ_v$&nbsp; for pulse code modulation&nbsp; $\rm (PCM)$&nbsp; compared to analog double-sideband amplitude modulation, abbreviated as&nbsp; $\rm ZSB-AM$.&nbsp;
+The graph shows the sink-to-noise ratio &nbsp;$10 ⋅ \lg \ ρ_v$&nbsp; for pulse code modulation&nbsp; $\rm (PCM)$&nbsp; compared to analog double-sideband amplitude modulation, abbreviated as&nbsp; $\rm ZSB-AM$.&nbsp;
 For the latter, &nbsp;$ρ_v = ξ$, where the power parameter.
@@ Line 38: / Line 38: @@
 <quiz display=simple>
-{Wieviele Bit pro Abtastwert &nbsp; &rArr; &nbsp; $N = N_1$&nbsp; verwendet das betrachtete PCM–System?
+{How many bits per sample &nbsp; &rArr; &nbsp; $N = N_1$&nbsp; does the PCM system under consideration use?
 |type="{}"}
 $N_1 \ = \ $ { 8 3% }
-{Wieviele Bit pro Abtastwert  &nbsp; &rArr; &nbsp; $N = N_2$&nbsp; müsste man verwenden, damit &nbsp;$10 · \lg \ ρ_v > 64 \ \rm dB$&nbsp; (Musikqualität) erreicht wird?
+{How many bits per sample &nbsp; &rArr; &nbsp; $N = N_2$&nbsp; would have to be used to achieve &nbsp;$10 ⋅ \lg \ ρ_v > 64 \ \rm dB$&nbsp; (music quality)?
 |type="{}"}
-$N_2 \ = \ $ { 11 3%  }
+$N_2 \ = \ $ { 11 3% }
-{Welche (logarithmierte) Leistungskenngröße &nbsp;$ξ_{40\ \rm dB}$&nbsp; ist erforderlich, damit bei 8–Bit–PCM der Sinkenstörabstand gleich &nbsp;$40\ \rm  dB$&nbsp; ist?
+{What (logarithmized) power parameter &nbsp;$ξ_{40\ \rm dB}$&nbsp; is required for 8-bit PCM to have a sink-to-noise ratio equal to &nbsp;$40\ \rm dB$&nbsp;?
 |type="{}"}
-$10 · \lg \ ξ_{40\ \rm dB} \ = \ $ { 10 3% } $\ \rm dB$
+$10 ⋅ \lg \ ξ_{40\ \rm dB} \ = \ $ { 10 3% } $\ \rm dB$
-{Um welchen Faktor könnte man bei PCM die Sendeleistung gegenüber der ZSB–AM reduzieren, um trotzdem &nbsp;$10 · \lg  \ ρ_v = 40\ \rm   dB$&nbsp; zu erreichen?
+{By what factor could the transmit power be reduced for PCM compared to ZSB-AM to still achieve &nbsp;$10 ⋅ \lg \ ρ_v = 40\ \rm dB$&nbsp;?
 |type="{}"}
 $K_\text{AM → PCM} \ = \ $ { 1000 3% }
-{Welche Bitfehlerwahrscheinlichkeit &nbsp;$p_{\rm B}$&nbsp; ergibt sich für &nbsp;$10 · \lg \ ξ = 6\ \rm  dB$&nbsp; und &nbsp;$N = N_1$ &nbsp; &rArr; &nbsp; Ergebnis der Teilaufgabe&nbsp; '''(1)'''?
+{What is the bit error probability &nbsp;$p_{\rm B}$&nbsp; for &nbsp;$10 ⋅ \lg \ ξ = 6\ \rm dB$&nbsp; and &nbsp;$N = N_1$ &nbsp; &rArr; &nbsp; result of the subtask&nbsp; '''(1)'''?
 |type="{}"}
 $p_{\rm B} \ = \ $ { 2.5 3% } $\ \%$
-{Welches SNR würde sich bei gleichem &nbsp;$ξ$&nbsp; mit einer 3–Bit–PCM &nbsp; &rArr; &nbsp; $N = 3$&nbsp; ergeben?
+{What would be the SNR for the same &nbsp;$ξ$&nbsp; with a 3-bit PCM &nbsp; &rArr; &nbsp; $N = 3$&nbsp;?
 |type="{}"}
-$10 · \lg  \ ρ_v \ = \ $ { 15.9 3% } $\ \rm dB$
+$10 ⋅ \lg \ ρ_v \ = \ $ { 15.9 3% } $\ \rm dB$
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp;  Der horizontale Abschnitt der PCM–Kurve wird allein durch das Quantisierungsrauschen bestimmt.&nbsp;
+'''(1)'''&nbsp; The horizontal section of the PCM curve is determined by the quantization noise alone.&nbsp;
-*Hier gilt mit der Quantisierungsstufenzahl&nbsp; $M = 2^N$:
+*Here, with the quantization step number&nbsp; $M = 2^N$:
-:$$ \rho_{v} (\xi \rightarrow \infty) = \rho_{\rm Q} = M^2 = 2^{2N} \hspace{0.3cm}\Rightarrow \hspace{0.3cm} 10 \cdot {\rm lg}\hspace{0.1cm}\rho_{v} \approx 6\,{\rm dB} \cdot N\hspace{0.05cm}.$$
+:$$ \rho_{v} (\xi \rightarrow \infty) = \rho_{\rm Q} = M^2 = 2^{2N} \hspace{0.3cm}\rightarrow \hspace{0.3cm} 10 \cdot {\rm lg}\hspace{0.1cm}\rho_{v} \approx 6\,{\rm dB} \cdot N\hspace{0.05cm}.$$
-*Aus dem ablesbaren Störabstand&nbsp; $10 · \lg \ ρ_v ≈ 48 \ \rm dB$&nbsp; folgt daraus&nbsp; $N_1\hspace{0.15cm}\underline { = 8}$&nbsp; Bit pro Abtastwert und für die Quantisierungsstufenzahl&nbsp; $M = 256$.
+*From the readable signal-to-noise ratio&nbsp; $10 ⋅ \lg \ ρ_v ≈ 48 \ \rm dB$&nbsp; it follows&nbsp; $N_1\hspace{0.15cm}\underline { = 8}$&nbsp; bits per sample and for the quantization level number&nbsp; $M = 256$.
-'''(2)'''&nbsp;  Aus der obigen Näherung erhält man für&nbsp; $N_2\hspace{0.15cm}\underline { = 11}$&nbsp; Bit pro Abtastwert   &nbsp; ⇒  &nbsp; $M = 2048$&nbsp; den Störabstand&nbsp; $66  \ \rm dB$.
+'''(2)'''&nbsp; From the above approximation, we obtain for&nbsp; $N_2\hspace{0.15cm}\underline { = 11}$&nbsp; bits per sample &nbsp; ⇒ &nbsp; $M = 2048$&nbsp; the signal-to-noise ratio&nbsp; $66 \rm dB$.
-*Mit&nbsp; $N = 10$ &nbsp; ⇒ &nbsp;  $M = 1024$&nbsp; erreicht man nur ca.&nbsp; $60  \ \rm dB$.
+*With&nbsp; $N = 10$ &nbsp; ⇒ &nbsp; $M = 1024$&nbsp; one reaches only approx.&nbsp; $60 \ \rm dB$.
-*Bei der Compact Disc (CD) werden die PCM–Parameter&nbsp; $N = 16$  &nbsp; ⇒  &nbsp;   $M = 65536$  &nbsp; ⇒  &nbsp;   $10 · \lg  \ ρ_v > 96 \ \rm dB$&nbsp; verwendet.
+*For the compact disc (CD), the PCM parameters&nbsp; $N = 16$ &nbsp; ⇒ &nbsp; $M = 65536$ &nbsp; ⇒ &nbsp; $10 ⋅ \lg \ ρ_v > 96 \ \rm dB$&nbsp; are used.
-'''(3)'''&nbsp;  Bei Zweiseitenband–Amplitudenmodulation wären hierfür&nbsp; $10 · \lg  \ ξ = 40\ \rm  dB$&nbsp; erforderlich.
+'''(3)'''&nbsp; For double sideband amplitude modulation, this would require&nbsp; $10 ⋅ \lg \ ξ = 40\ \rm dB$&nbsp;.
-*Wie aus der Grafik auf der Angabenseite hervorgeht, ist dieser Abszissenwert für die vorgegebene PCM um&nbsp; $30 \ \rm dB$ &nbsp;geringer &nbsp; ⇒ &nbsp; $10 · \lg \ ξ_{40\ \rm dB}\hspace{0.15cm}\underline { = 10 \ \rm dB}$.
+*As can be seen from the graph on the data page, this abscissa value for the given PCM is lower by&nbsp; $30 \ \rm dB$&nbsp;⇒&nbsp; $10 ⋅ \lg \ ξ_{40\ \rm dB}\hspace{0.15cm}\underline { = 10 \ \rm dB}$.
-'''(4)'''&nbsp;  Der logarithmische Wert&nbsp; $30  \ \rm  dB$&nbsp; entspricht einer um den Faktor&nbsp; $10^3\hspace{0.15cm}\underline {  = 1000}$&nbsp; reduzierten Leistung.
+'''(4)'''&nbsp; The logarithmic value&nbsp; $30 \ \rm dB$&nbsp; corresponds to a power reduced by a factor&nbsp; $10^3\hspace{0.15cm}\underline { = 1000}$&nbsp; .
-'''(5)'''&nbsp;  Aus der Grafik auf der Angabenseite erkennt man, dass der Abszissenwert&nbsp; $10 · \lg \ ξ= 6 \ \rm  dB$&nbsp; den Störabstand&nbsp; $20 \ \rm  dB$&nbsp; zur Folge hat.
+'''(5)'''&nbsp; From the graph on the information page, it can be seen that the abscissa value&nbsp; $10 ⋅ \lg \ ξ= 6 \ \rm dB$&nbsp; results in the signal-to-noise ratio&nbsp; $20 \ \rm dB$&nbsp; .
-*Aus&nbsp; $10 · \lg \ ρ_v = 20  \ \rm   dB$&nbsp; folgt&nbsp; $ρ_v = 100$&nbsp; und damit weiter&nbsp; $($mit&nbsp; $N = N_1 = 8)$:
+*From&nbsp; $10 ⋅ \lg \ ρ_v = 20 \ \rm dB$&nbsp; follows&nbsp; $ρ_v = 100$&nbsp; and thus further&nbsp; $($with&nbsp; $N = N_1 = 8)$:
 :$$\rho_{\upsilon}= \frac{1}{ 2^{-2N } + 4 \cdot p_{\rm B}} \approx \frac{1}{ 1.5 \cdot 10^{-5} + 4 \cdot p_{\rm B}} = 100 \hspace{0.3cm}
 \Rightarrow \hspace{0.3cm} p_{\rm B} = \frac{0.01 - 1.5 \cdot 10^{-5}}{ 4} \hspace{0.15cm}\underline {\approx 2.5\%} \hspace{0.05cm}.$$
-'''(6)'''&nbsp;  Bei gleichem&nbsp; $ξ$&nbsp; ist die Bitfehlerwahrscheinlichkeit weiterhin&nbsp; $p_{\rm B} = 0.025$.&nbsp; Damit erhält man mit&nbsp; $N = 3$&nbsp; (Bit pro Abtastwert):
+'''(6)'''&nbsp; With the same&nbsp; $ξ$&nbsp; the bit error probability is still&nbsp; $p_{\rm B} = 0.025$.&nbsp; Thus, with&nbsp; $N = 3$&nbsp; (bits per sample):
 :$$\rho_{\upsilon}= \frac{1}{ 2^{-6 } + 4 \cdot p_{\rm B}} = \frac{1}{ 0.015625 + 0.01} \approx 39 \hspace{0.3cm} \Rightarrow \hspace{0.3cm}10 \cdot {\rm lg} \hspace{0.15cm}\rho_{\upsilon}\hspace{0.15cm}\underline {\approx 15.9\,{\rm dB}} \hspace{0.05cm}.$$
-Weiter ist anzumerken:
+Further, it should be noted:
-*Bei nur drei Bit pro Abtastwert ist die Quantisierungsrauschleistung&nbsp; $(P_{\rm Q} = 0.015625)$&nbsp; schon größer als die Fehlerrauschleistung&nbsp; $(P_{\rm F} = 0.01)$.
+*With only three bits per sample, the quantization noise power&nbsp; $(P_{\rm Q} = 0.015625)$&nbsp; is already larger than the error noise power&nbsp; $(P_{\rm F} = 0.01)$.
-*Durch Erhöhung der Sendeleistung könnte wegen der Quantisierung der Sinkenstörabstand maximal&nbsp; $10 · \lg \ ρ_v =18 \ \rm dB$&nbsp; betragen, wenn keine Bitfehler vorkommen&nbsp; $(P_{\rm F} = 0)$.
+*By increasing the transmit power, the signal-to-noise ratio could be maximum&nbsp; $10 ⋅ \lg \ ρ_v =18 \ \rm dB$&nbsp; because of quantization, if no bit errors occur&nbsp; $(P_{\rm F} = 0)$.
 {{ML-Fuß}}

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Modulationsverfahren/Pulscodemodulation
+{{quiz-Header|Buchseite=Modulation_Methods/Pulse_Code_Modulation
 }}
-[[File:EN_Mod_Z_4_4.png|right|frame|Störabstand von PCM 30/32 im Vergleich zur ZSB–Amplitudenmodulation]]
+[[File:EN_Mod_Z_4_4.png|right|frame|Sink-to-drop ratio of PCM 30/32 compared to ZSB amplitude modulation]]
-Die Grafik zeigt den Sinken–Störabstand &nbsp;$10 · \lg \ ρ_v$&nbsp; für die Pulscodemodulation&nbsp; $\rm (PCM)$&nbsp; im Vergleich zur analogen Zweiseitenband–Amplitudenmodulation, abgekürzt mit&nbsp; $\rm  ZSB–AM$.&nbsp;
+The graph shows the sink-to-noise ratio &nbsp;$10 - \lg \ ρ_v$&nbsp; for pulse code modulation&nbsp; $\rm (PCM)$&nbsp; compared to analog double-sideband amplitude modulation, abbreviated as&nbsp; $\rm ZSB-AM$.&nbsp;
-Für letztere gilt &nbsp;$ρ_v = ξ$, wobei die Leistungskenngröße
+For the latter, &nbsp;$ρ_v = ξ$, where the power parameter.
 :$$\xi = \frac{\alpha^2 \cdot P_{\rm S}}{N_0 \cdot f_{\rm N}} $$
-folgende Systemparameter zusammenfasst:
+Summarizes the following system parameters:
-:* den frequenzunabhängigen Übertragungsfaktor &nbsp;$α$&nbsp; des Übertragungskanals,
+:* the frequency-independent transmission factor &nbsp;$α$&nbsp; of the transmission channel,
-:* die Leistung &nbsp;$P_{\rm S}$&nbsp; des Sendsignals &nbsp;$s(t)$, auch kurz Sendeleistung genannt,
+:* the power &nbsp;$P_{\rm S}$&nbsp; of the transmit signal &nbsp;$s(t)$, also called transmit power for short,
-:* die Nachrichtenfrequenz &nbsp;$f_{\rm N}$&nbsp; (Bandbreite) des cosinusförmigen Quellensignals &nbsp;$q(t)$,
+:* the message frequency &nbsp;$f_{\rm N}$&nbsp; (bandwidth) of the cosine source signal &nbsp;$q(t)$,
-:* die Rauschleistungsdichte &nbsp;$N_0$&nbsp; des AWGN–Rauschens.
+:* the noise power density &nbsp;$N_0$&nbsp; of the AWGN noise.
-Für das PCM–System wurde auf der Seite&nbsp; [[Modulation_Methods/Pulscodemodulation#Absch.C3.A4tzung_der_SNR-Degradation_durch_.C3.9Cbertragungsfehler|Abschätzung der SNR-Degradation durch Bitfehler]]&nbsp; folgende Näherung für das Sinken–SNR angegeben, die auch Übertragungsfehler aufgrund des AWGN–Rauschens berücksichtigt:
+For the PCM system, the following approximation for the sink SNR was given on the page&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Estimation_of_SNR_degradation_due_to_transmission_errors|Estimation of SNR degradation due to bit errors]]&nbsp; which also takes into account transmission errors due to AWGN noise:
 :$$ \rho_{\upsilon}= \frac{1}{ 2^{-2N } + 4 \cdot p_{\rm B}} \hspace{0.05cm}.$$
-*Hierbei bezeichnet &nbsp;$N$&nbsp; die Anzahl der Bit pro Abtastwert und &nbsp;$p_{\rm B}$&nbsp; die Bitfehlerwahrscheinlichkeit.
+*Here &nbsp;$N$&nbsp; denotes the number of bits per sample and &nbsp;$p_{\rm B}$&nbsp; the bit error probability.
-* Da &nbsp;$ξ$&nbsp; bei digitaler Modulation auch als die ''Signalenergie pro Bit''&nbsp; bezogen auf die ''Rauschleistungsdichte'' $(E_{\rm B}/N_0)$ interpretiert werden kann, gilt mit dem komplementären Gaußschen Fehlersignal &nbsp;${\rm Q}(x)$&nbsp; näherungsweise:
+* Since &nbsp;$ξ$&nbsp; in digital modulation can also be interpreted as the ''signal energy per bit''&nbsp; related to the ''noise power density'' $(E_{\rm B}/N_0)$, with the complementary Gaussian error signal &nbsp;${\rm Q}(x)$&nbsp; approximately:
 :$$ p_{\rm B}= {\rm Q} \left ( \sqrt{2 \xi }\right ) \hspace{0.05cm}.$$
@@ Line 28: / Line 28: @@
-''Hinweise:''
+Hints:
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Modulation_Methods/Pulscodemodulation|Pulscodemodulation]].
+*The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Pulse_Code_Modulation|Pulse Code Modulation]].
-*Bezug genommen wird insbesondere auf die Seiten&nbsp;  [[Modulation_Methods/Pulscodemodulation#Einfluss_von_.C3.9Cbertragungsfehlern|Einfluss von Übertragungsfehlern]]&nbsp; und&nbsp; [[Modulation_Methods/Pulscodemodulation#Absch.C3.A4tzung_der_SNR-Degradation_durch_.C3.9Cbertragungsfehler|Abschätzung der SNR-Degradation durch Bitfehler]].
+*Reference is made in particular to the pages&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Influence_of_transmission_errors|Influence of transmission errors]]&nbsp; and&nbsp; [[Modulation_Methods/Pulse_Code_Modulation#Estimation_of_SNR_degradation_due_to_transmission_errors|Estimation of SNR degradation by bit errors]].
-*Bei der hier betrachteten PCM handelt es sich um die&nbsp; '''PCM 30/32''', deren Systemparameter zum Beispiel in der&nbsp; [[Aufgaben:4.1_PCM–System_30/32 |Aufgabe 4.1]]&nbsp; angegeben sind.
+*The PCM considered here is the&nbsp; '''PCM 30/32''' whose system parameters are given, for example, in the&nbsp; [[Aufgaben:Exercise_4.1:_PCM_System_30/32 |Exercise 4.1]]&nbsp;.
-===Fragebogen===
+===Questions===
 <quiz display=simple>

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−	[[Category:Modulation Methods: Exercises\|^4.1 ~~Pulscodemodulation~~^]]	+	[[Category:Modulation Methods: Exercises\|^4.1 Pulse Code Modulation^]]