Aufgaben:Exercise 3.7Z: Error Performance - Revision history

Bene: Text replacement - " rms " to " standard deviation "

2022-02-17T11:34:31Z

Text replacement - " rms " to " standard deviation "

Bene: Text replacement - "rms value" to "standard deviation"

2022-02-17T11:11:50Z

Text replacement - "rms value" to "standard deviation"

Guenter at 13:02, 2 February 2022

2022-02-02T13:02:30Z

Guenter at 11:29, 2 February 2022

2022-02-02T11:29:02Z

Guenter at 11:28, 2 February 2022

2022-02-02T11:28:06Z

Guenter at 11:27, 2 February 2022

2022-02-02T11:27:36Z

Noah at 21:44, 3 January 2022

2022-01-03T21:44:33Z

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Guenter at 10:19, 20 December 2021

2021-12-20T10:19:29Z

Guenter: Guenter moved page Aufgaben:Aufgabe 3.7Z: Error Performance to Aufgaben:Exercise 3.7Z: Error Performance

2021-12-20T10:16:29Z

Guenter moved page Aufgabe 3.7Z: Error Performance to Exercise 3.7Z: Error Performance

Javier: Text replacement - "„" to """

2021-05-28T14:31:38Z

Text replacement - "„" to """

@@ Line 69: / Line 69: @@
-'''(3)'''&nbsp; For the rms one obtains &nbsp;
+'''(3)'''&nbsp; For the standard deviation one obtains &nbsp;
 :$$\it \sigma_f=\rm\sqrt{\rm 64000\cdot 10^{-3}\cdot 0.999}\hspace{0.15cm}\underline{\approx\sqrt{64}=8}.$$
 * The error by applying Poisson distribution instead of binomial distribution here is smaller than&nbsp; $0.05\%$.

@@ Line 18: / Line 18: @@
 *Always assume bit error probability&nbsp; $p = 10^{-3}$&nbsp; for the first three subtasks.
 *In addition,&nbsp; throughout the task,&nbsp; let $N = 64\hspace{0.08cm}000$&nbsp; hold.
-* Under certain conditions &ndash; which are all fulfilled here &ndash; the binomial distribution can be approximated by a Gaussian distribution with equal mean and equal rms value.&nbsp; Use this approximation for the subtask '''(4)'''.
+* Under certain conditions &ndash; which are all fulfilled here &ndash; the binomial distribution can be approximated by a Gaussian distribution with equal mean and equal standard deviation.&nbsp; Use this approximation for the subtask '''(4)'''.
@@ Line 38: / Line 38: @@
-{How large is the rms value?&nbsp; Use appropriate approximations.
+{How large is the standard deviation?&nbsp; Use appropriate approximations.
 |type="{}"}
 $\sigma_f \ = \ $ { 8 3% }

@@ Line 65: / Line 65: @@
-'''(2)'''&nbsp; The mean is obtained as &nbsp;$m_f = N \cdot p \hspace{0.15cm}\underline{= 64}$&nbsp; regardless of whether one assumes the binomial&ndash; or the Poisson distribution.
+'''(2)'''&nbsp; The mean is obtained as &nbsp;$m_f = N \cdot p \hspace{0.15cm}\underline{= 64}$&nbsp; regardless of whether one assumes the binomial distribution or the Poisson distribution.
@@ Line 71: / Line 71: @@
 '''(3)'''&nbsp; For the rms one obtains &nbsp;
 :$$\it \sigma_f=\rm\sqrt{\rm 64000\cdot 10^{-3}\cdot 0.999}\hspace{0.15cm}\underline{\approx\sqrt{64}=8}.$$
-* The error by applying Poissonl distribution instead of binomial distribution here is smaller than&nbsp; $0.05\%$.
+* The error by applying Poisson distribution instead of binomial distribution here is smaller than&nbsp; $0.05\%$.
 '''(4)'''&nbsp; For a Gaussian random variable $f$&nbsp; with mean &nbsp;$m_f {= 64}$&nbsp; the probability&nbsp; ${\rm Pr}(f \le 64) \hspace{0.15cm}\underline{\approx 50\%}$. &nbsp; Note:
-*For a continuous random size, the probability would be exactly $50\%$.
+*For a continuous random size,&nbsp; the probability would be exactly $50\%$.
-*Since $f$&nbsp; can only take integer values, it is slightly larger here.
+*Since $f$&nbsp; can only take integer values,&nbsp; it is slightly larger here.
 '''(5)'''&nbsp; With &nbsp;$\lambda = N \cdot p$&nbsp; the corresponding condition is:
-:$$\rm Q\big (\frac{\rm 64-\it \lambda}{\sqrt{\it \lambda}} \big )\le \rm 0.002\hspace{0.5cm}\rm or.\hspace{0.5cm}\frac{\rm 64-\it \lambda}{\sqrt{\it \lambda}}>\rm 2.9.$$
+:$$\rm Q\big (\frac{\rm 64-\it \lambda}{\sqrt{\it \lambda}} \big )\le \rm 0.002\hspace{0.5cm}\rm or \hspace{0.5cm}\frac{\rm 64-\it \lambda}{\sqrt{\it \lambda}}>\rm 2.9.$$
 *The maximum value of&nbsp; $\lambda$&nbsp; can be determined according to the following equation:

@@ Line 4: / Line 4: @@
 [[File:P_ID132__Sto_Z_3_7.png|right|frame<excerpt from CCITT Recommendation G.821: Error Performance]]
-Every operator of ISDN systems must comply with certain minimum requirements regarding the bit error rate&nbsp; $\rm (BER)$,&nbsp; which are specified for example in the&nbsp; [https://en.wikipedia.org/wiki/G.821 CCITT Recommendation G.821]&nbsp; under the name&nbsp; "Error Performance".
+Every operator of ISDN systems must comply with certain minimum requirements regarding the bit error rate&nbsp; $\rm (BER)$,&nbsp; which are specified for example in the&nbsp; [https://de.wikipedia.org/wiki/G.821 CCITT Recommendation G.821]&nbsp; under the name&nbsp; "Error Performance".
 On the right you can see an excerpt from this recommendation:

@@ Line 4: / Line 4: @@
 [[File:P_ID132__Sto_Z_3_7.png|right|frame<excerpt from CCITT Recommendation G.821: Error Performance]]
-Every operator of ISDN systems must comply with certain minimum requirements regarding the bit error rate&nbsp; $\rm (BER)$,&nbsp; which are specified for example in the&nbsp; [https://de.wikipedia.org/wiki/G.821 CCITT Recommendation G.821]&nbsp; under the name&nbsp; "Error Performance".
+Every operator of ISDN systems must comply with certain minimum requirements regarding the bit error rate&nbsp; $\rm (BER)$,&nbsp; which are specified for example in the&nbsp; [https://en.wikipedia.org/wiki/G.821 CCITT Recommendation G.821]&nbsp; under the name&nbsp; "Error Performance".
 On the right you can see an excerpt from this recommendation:

@@ Line 4: / Line 4: @@
 [[File:P_ID132__Sto_Z_3_7.png|right|frame<excerpt from CCITT Recommendation G.821: Error Performance]]
-Every operator of ISDN systems must comply with certain minimum requirements regarding the bit error rate (BER), which are specified for example in the&nbsp; [https://de.wikipedia.org/wiki/G.821 CCITT Recommendation G.821]&nbsp; under the name "Error Performance".
+Every operator of ISDN systems must comply with certain minimum requirements regarding the bit error rate&nbsp; $\rm (BER)$,&nbsp; which are specified for example in the&nbsp; [https://de.wikipedia.org/wiki/G.821 CCITT Recommendation G.821]&nbsp; under the name&nbsp; "Error Performance".
 On the right you can see an excerpt from this recommendation:
-*This states, among other things,&nbsp; that &ndash; averaged over a sufficiently long time &ndash; at least&nbsp; $99.8\%$&nbsp; of all one-second intervals must have a bit error rate less than&nbsp; $10^{-3}$&nbsp; (one per thousand).
+*This states,&nbsp; among other things,&nbsp; that &ndash; averaged over a sufficiently long time &ndash; at least&nbsp; $99.8\%$&nbsp; of all one-second intervals must have a bit error rate less than&nbsp; $10^{-3}$&nbsp; (one per thousand).
 *For a bit rate of&nbsp; $\text{64 kbit/s}$&nbsp; this corresponds to the condition that in one second&nbsp; $($and thus for&nbsp; $N = 64\hspace{0.08cm}000$&nbsp; transmitted symbols$)$&nbsp; no more than&nbsp; $64$&nbsp; bit errors may occur:
 :$$\rm Pr(\it f \le \rm 64) \ge \rm 0.998.$$
@@ Line 18: / Line 15: @@
 Hints:
 *The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|Gaussian distributed random variables]].
 *Always assume bit error probability&nbsp; $p = 10^{-3}$&nbsp; for the first three subtasks.
-*In addition, throughout the task, let $N = 64\hspace{0.08cm}000$ hold.
+*In addition,&nbsp; throughout the task,&nbsp; let $N = 64\hspace{0.08cm}000$&nbsp; hold.
-* Under certain conditions &ndash; which are all fulfilled here &ndash; the binomial distribution can be approximated by a Gaussian distribution with equal mean and equal rms.
+* Under certain conditions &ndash; which are all fulfilled here &ndash; the binomial distribution can be approximated by a Gaussian distribution with equal mean and equal rms value.&nbsp; Use this approximation for the subtask '''(4)'''.
@@ Line 38: / Line 33: @@
-{What is the mean value of the random variable $f$?
+{What is the mean value of the random variable&nbsp; $f$?
 |type="{}"}
 $m_f \ = \ $ { 64 3% }
-{How large is the rms?&nbsp; Use appropriate approximations.
+{How large is the rms value?&nbsp; Use appropriate approximations.
 |type="{}"}
 $\sigma_f \ = \ $ { 8 3% }
-{Calculate the probability that no more than&nbsp; $64$&nbsp; bit errors occur.&nbsp; Use Gaussian approximation.
+{Calculate the probability that no more than&nbsp; $64$&nbsp; bit errors occur.&nbsp; Use the Gaussian approximation.
 |type="{}"}
 ${\rm Pr}(f ≤ 64) \ = \ $ { 50 3% } $ \ \rm \%$
-{What is the maximum bit error probability&nbsp; $p_\text{B, max}$&nbsp; that the condition "64 (or more) bit errors only in at most 0.2% of the one-second intervals " can be met?&nbsp; It holds&nbsp; ${\rm Q}(2.9) \approx 0.002$.
+{What is the maximum bit error probability&nbsp; $p_\text{B, max}$&nbsp; that the condition&nbsp; "64&nbsp; (or more)&nbsp; bit errors only in at most 0.2% of the one-second intervals"&nbsp; can be met?&nbsp; <br>It holds&nbsp; ${\rm Q}(2.9) \approx 0.002$.
 |type="{}"}
 $p_\text{B, max}\ = \ $ { 0.069 3% } $ \ \rm \%$

← Older revision		Revision as of 10:19, 20 December 2021
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−	[[Category:Theory of Stochastic Signals: Exercises\|^3.5 ~~Gaußverteilte Zufallsgröße~~^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^3.5 Gaussian Random Variable^]]

@@ Line 4: / Line 4: @@
 [[File:P_ID132__Sto_Z_3_7.png|right|frame<Auszug aus der CCITT-Empfehlung G.821: Error Performance]]
-Jeder Betreiber von ISDN-Systemen muss gewisse Mindestanforderungen hinsichtlich der Bitfehlerquote (BER) einhalten, die zum Beispiel in der&nbsp; [https://de.wikipedia.org/wiki/G.821 CCITT-Empfehlung G.821]&nbsp; unter dem Namen &bdquo;Error Performance" spezifiziert sind.
+Jeder Betreiber von ISDN-Systemen muss gewisse Mindestanforderungen hinsichtlich der Bitfehlerquote (BER) einhalten, die zum Beispiel in der&nbsp; [https://de.wikipedia.org/wiki/G.821 CCITT-Empfehlung G.821]&nbsp; unter dem Namen "Error Performance" spezifiziert sind.
 Rechts sehen Sie einen Auszug aus dieser Empfehlung:
@@ Line 52: / Line 52: @@
-{Wie gro&szlig; darf die Bitfehlerwahrscheinlichkeit&nbsp; $p_\text{B, max}$&nbsp; höchstens sein, damit die Bedingung &bdquo;64 (oder mehr) Bitfehler nur in höchstens 0.2% der Einsekunden-Intervalle " eingehalten werden kann?&nbsp; Es gilt&nbsp; ${\rm Q}(2.9) \approx 0.002$.
+{Wie gro&szlig; darf die Bitfehlerwahrscheinlichkeit&nbsp; $p_\text{B, max}$&nbsp; höchstens sein, damit die Bedingung "64 (oder mehr) Bitfehler nur in höchstens 0.2% der Einsekunden-Intervalle " eingehalten werden kann?&nbsp; Es gilt&nbsp; ${\rm Q}(2.9) \approx 0.002$.
 |type="{}"}
 $p_\text{B, max}\ =  \ $ { 0.069 3% } $ \ \rm \%$