Aufgaben:Exercise 1.2: Bit Error Rate - Revision history

Hwang at 14:51, 9 May 2022

2022-05-09T14:51:01Z

Guenter at 12:24, 29 April 2022

2022-04-29T12:24:32Z

Guenter at 12:08, 29 April 2022

2022-04-29T12:08:16Z

Hwang at 14:45, 2 March 2022

2022-03-02T14:45:56Z

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Guenter at 15:23, 2 February 2022

2022-02-02T15:23:23Z

Guenter: Guenter moved page Aufgaben:Aufgabe 1.2: Bitfehlerquote (BER) to Aufgaben:Exercise 1.2: Bit Error Rate

2022-02-02T15:18:31Z

Guenter moved page Aufgabe 1.2: Bitfehlerquote (BER) to Exercise 1.2: Bit Error Rate

Javier: Text replacement - "„" to """

2021-05-28T14:28:47Z

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Javier: Text replacement - "”" to """

2021-05-28T14:20:10Z

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Javier: Text replacement - "Category:Aufgaben zu Digitalsignalübertragung" to "Category:Digital Signal Transmission: Exercises"

2021-03-23T12:49:37Z

Text replacement - "Category:Aufgaben zu Digitalsignalübertragung" to "Category:Digital Signal Transmission: Exercises"

Javier: Text replacement - "[[Digitalsignalübertragung" to "[[Digital_Signal_Transmission"

2020-07-09T12:49:10Z

Text replacement - "[[Digitalsignalübertragung" to "[[Digital_Signal_Transmission"

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 The larger the simulation parameter &nbsp;$N$&nbsp; is chosen,&nbsp; the more accurate is this approximation.
-From &nbsp;[[Aufgaben:Exercise_3.7:_Bit_Error_Rate_(BER)|Exercise 3.7]]&nbsp; in the book&nbsp; "Stochastic Signal Theory"&nbsp; it is known that the random variable&nbsp; "BER"&nbsp; is actually binomially distributed,&nbsp; but can be approximated with good approximation by a&nbsp; (discrete)&nbsp; Gaussian distribution with mean &nbsp;$p$&nbsp; and standard deviation &nbsp;$\sigma$:&nbsp;
+From &nbsp;[[Aufgaben:Exercise_3.7:_Bit_Error_Rate_(BER)|"Exercise 3.7"]]&nbsp; in the book&nbsp; "Stochastic Signal Theory"&nbsp; it is known that the random variable&nbsp; "BER"&nbsp; is actually binomially distributed,&nbsp; but can be approximated with good approximation by a&nbsp; (discrete)&nbsp; Gaussian distribution with mean &nbsp;$p$&nbsp; and standard deviation &nbsp;$\sigma$:&nbsp;
 :$$\sigma =   \sqrt{\frac{ p\cdot (\rm 1- \it p)}{N}}.$$

@@ Line 67: / Line 67: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; <u>Only the second statement</u> is correct:
+'''(1)'''&nbsp; <u>Only the second statement</u>&nbsp; is correct:
-*$\rm BER$ is the quotient of the number $n_{\rm B}$ of detected symbol errors and the number $N$ of all simulated symbols and thus actually a relative frequency.
+*$\rm BER$&nbsp; is the quotient of the number&nbsp; $n_{\rm B}$&nbsp; of detected symbol errors and the number&nbsp; $N$&nbsp; of all simulated symbols and thus actually a relative frequency.
-*The probability that ${\rm BER} = p$ is always exactly zero, since $\rm BER$ is a continuous random variable.
+*The probability that&nbsp; ${\rm BER} = p$&nbsp; is always exactly zero,&nbsp; since&nbsp; $\rm BER$&nbsp; is a continuous random variable.
-*However, the probability that $\rm BER$ lies in a narrow interval around $p$ increases as $N$ increases.
+*However,&nbsp; the probability that&nbsp; $\rm BER$&nbsp; lies in a narrow interval around&nbsp; $p$&nbsp; increases as&nbsp; $N$&nbsp; increases.
-'''(2)'''&nbsp; The standard deviation of the Gaussian random variable $\rm BER$ with $N = 10^6$ and $p = 10^{-2}$ is given by
+'''(2)'''&nbsp; The standard deviation of the Gaussian random variable&nbsp; $\rm BER$&nbsp; with&nbsp; $N = 10^6$&nbsp; and&nbsp; $p = 10^{-2}$&nbsp; is given by
 :$$\sigma =   \sqrt{{ p\cdot (\rm 1- \it p)}/{N}}\approx \sqrt{{ p}/{N}}\hspace{0.1cm}\underline {= 10^{-4}}\hspace{0.05cm}.$$
-'''(3)'''&nbsp; The probability that the $\rm BER$ takes a value outside the range
+'''(3)'''&nbsp; The probability that the&nbsp; $\rm BER$&nbsp; takes a value outside the range&nbsp;
-$0.95 \cdot p$ ... $1.05 \cdot p$ is obtained with $\varepsilon = 5 \cdot 0^{-4}$ ($p = 10^{-2}$) as follows.
+$0.95 \cdot p$ ... $1.05 \cdot p$&nbsp; is obtained with&nbsp; $\varepsilon = 5 \cdot 10^{-4}$&nbsp; $(p = 10^{-2})$&nbsp; as follows.
 :$${\rm Pr} \left( {\rm BER} < 0.95 \cdot 10^{-2} \right)
    = {\rm Pr} \left( {\rm BER} > 1.05 \cdot 10^{-2} \right)
@@ Line 87: / Line 87: @@
-'''(4)'''&nbsp; With $p = 10^{-4}$, the comparable probability is:
+'''(4)'''&nbsp; With&nbsp; $p = 10^{-4}$,&nbsp; the comparable probability is:
 :$${\rm Pr} \left( |{\rm BER} - 10^{-4}| > 0.05 \cdot 10^{-4} \right)
    = 2 \cdot {\rm Q} \left( {\varepsilon}/{\sigma}
@@ Line 98: / Line 98: @@
-'''(5)'''&nbsp; This condition can be formulated with $\varepsilon = 5 \cdot 10^{-6}$ as follows:
+'''(5)'''&nbsp; This condition can be formulated with&nbsp; $\varepsilon = 5 \cdot 10^{-6}$&nbsp; as follows:
 :$${\rm Q} \left( {\varepsilon}/{\sigma} \right) < 0.1  \hspace{0.3cm}\Rightarrow \hspace{0.3cm}
 {\varepsilon}/{\sigma} > {\rm Q}^{-1}(0.05) \approx 1.64

@@ Line 4: / Line 4: @@
-[[File:P_ID1264__Dig_A_1_2.png|right|frame|Table of Gaussian error functions]]
+[[File:P_ID1264__Dig_A_1_2.png|right|frame|Table of two Gaussian error functions]]
 It is known from a digital transmission system that it can be approximated by a
-BSC model (<i>Binary Symmetrical Channel</i>&nbsp;) with error probability &nbsp;$p$.&nbsp;
+BSC model&nbsp; ("Binary Symmetrical Channel") with error probability &nbsp;$p$.&nbsp;
-For verification, the bit error rate is to be determined by comparing the sink symbol sequence &nbsp;$ \langle v_\nu \rangle $&nbsp;  with the source symbol sequence
+For verification,&nbsp; the bit error rate is to be determined by comparing the sink symbol sequence &nbsp;$ \langle v_\nu \rangle $&nbsp;  with the source symbol sequence
 &nbsp;$ \langle q_\nu \rangle $&nbsp; and finding the error sequence &nbsp;$ \langle e_\nu \rangle $.&nbsp; Thereby holds:
-$$e_\nu  =   \left\{ \begin{array}{c} 0  \\
+:$$e_\nu  =   \left\{ \begin{array}{c} 0  \\
 \\  \end{array} \right.\quad
 \begin{array}{*{1}c} {\rm{for}}
@@ Line 18: / Line 18: @@
 v_\nu \ne q_\nu .  \\
 \end{array}$$
-The bit error rate is an approximation for the bit error probability &nbsp;$p$&nbsp;:
+The bit error rate is an approximation for the bit error probability &nbsp;$p$:
 :$${\rm BER} = \frac{1}{N}\cdot\sum_{\nu=1}^N e_\nu.$$
-The larger the simulation parameter &nbsp;$N$&nbsp; is chosen, the more accurate this approximation is.
+The larger the simulation parameter &nbsp;$N$&nbsp; is chosen,&nbsp; the more accurate is this approximation.
-From &nbsp;[[Aufgaben:Exercise_3.7:_Bit_Error_Rate_(BER)|Exercise 3.7]]&nbsp; in the book "Stochastic Signal Theory" it is known that the random variable "BER" is actually binomially distributed, but can be approximated with good approximation by a (discrete) Gaussian distribution with mean &nbsp;$p$&nbsp; and standard deviation &nbsp;$\sigma$:&nbsp;
+From &nbsp;[[Aufgaben:Exercise_3.7:_Bit_Error_Rate_(BER)|Exercise 3.7]]&nbsp; in the book&nbsp; "Stochastic Signal Theory"&nbsp; it is known that the random variable&nbsp; "BER"&nbsp; is actually binomially distributed,&nbsp; but can be approximated with good approximation by a&nbsp; (discrete)&nbsp; Gaussian distribution with mean &nbsp;$p$&nbsp; and standard deviation &nbsp;$\sigma$:&nbsp;
 :$$\sigma =   \sqrt{\frac{ p\cdot (\rm 1- \it p)}{N}}.$$
@@ Line 29: / Line 29: @@
-''Notes:''
+Notes:
-*The exercise belongs to the chapter&nbsp; [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission|Error Probability for Baseband Transmission]].
+*The exercise belongs to the chapter&nbsp; [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission|"Error Probability for Baseband Transmission"]].
-*Reference is also made to the chapter &nbsp;[[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|Gaussian Distributed Random Variables]]&nbsp; in the book "Stochastic Signal Theory".
+*Reference is also made to the chapter &nbsp;[[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables|"Gaussian Distributed Random Variables"]]&nbsp; in the book&nbsp; "Stochastic Signal Theory".
-*In the table some values of the Gaussian error functions &nbsp;$\rm \phi(x)$&nbsp; and &nbsp;$\rm Q(x)$&nbsp; are given.
+*In the table some values of the Gaussian error functions &nbsp;${\rm \phi}(x)$&nbsp; and &nbsp;${\rm Q}(x)$&nbsp; are given.
@@ Line 43: / Line 43: @@
 - $\rm BER$&nbsp; is a probability.
 + $\rm BER$&nbsp; is a relative frequency.
-- If &nbsp;$N$&nbsp; is sufficiently large, &nbsp;$\rm BER$&nbsp; coincides <b>exactly</b> with &nbsp;$p$&nbsp;
+- If &nbsp;$N$&nbsp; is sufficiently large, &nbsp;$\rm BER$&nbsp; coincides <b>exactly</b> with &nbsp;$p$.

← Older revision		Revision as of 15:23, 2 February 2022
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−	[[Category:Digital Signal Transmission: Exercises\|^1.2 BER ~~bei Basisbandsystemen~~^]]	+	[[Category:Digital Signal Transmission: Exercises\|^1.2 BER for Baseband Systems^]]

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 Je größer der Simulationsparameter &nbsp;$N$&nbsp; gewählt wird, um so genauer ist diese Näherung.
-Aus der &nbsp;[[Aufgaben:3.7_Bitfehlerquote_(BER)|Aufgabe 3.7]]&nbsp; im Buch „Stochastische Signaltheorie” ist bekannt, dass die Zufallsgröße &bdquo;BER" eigentlich binominalverteilt ist, aber mit guter Näherung durch eine (diskrete) Gaußverteilung mit Mittelwert &nbsp;$p$&nbsp; und Streuung &nbsp;$\sigma$&nbsp; angenähert werden kann:
+Aus der &nbsp;[[Aufgaben:3.7_Bitfehlerquote_(BER)|Aufgabe 3.7]]&nbsp; im Buch „Stochastische Signaltheorie” ist bekannt, dass die Zufallsgröße "BER" eigentlich binominalverteilt ist, aber mit guter Näherung durch eine (diskrete) Gaußverteilung mit Mittelwert &nbsp;$p$&nbsp; und Streuung &nbsp;$\sigma$&nbsp; angenähert werden kann:
 :$$\sigma =   \sqrt{\frac{ p\cdot (\rm 1- \it p)}{N}}.$$

← Older revision	Revision as of 15:18, 2 February 2022
(No difference)

← Older revision		Revision as of 12:49, 23 March 2021
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−	[[Category:~~Aufgaben zu Digitalsignalübertragung~~\|^1.2 BER bei Basisbandsystemen^]]	+	[[Category:Digital Signal Transmission: Exercises\|^1.2 BER bei Basisbandsystemen^]]

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 ''Hinweise:''
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Digitalsignalübertragung/Fehlerwahrscheinlichkeit_bei_Basisbandübertragung|Fehlerwahrscheinlichkeit bei Basisbandübertragung]].
+*Die Aufgabe gehört zum  Kapitel&nbsp; [[Digital_Signal_Transmission/Fehlerwahrscheinlichkeit_bei_Basisbandübertragung|Fehlerwahrscheinlichkeit bei Basisbandübertragung]].
 *Bezug genommen wird auch auf das Kapitel &nbsp;[[Theory_of_Stochastic_Signals/Gaußverteilte_Zufallsgrößen|Gaußverteilte Zufallsgrößen]]&nbsp; im Buch „Stochastische Signaltheorie”.
 *In der Tabelle sind einige Werte der Gaußschen Fehlerfunktionen &nbsp;$\rm \phi(x)$&nbsp; und &nbsp;$\rm Q(x)$&nbsp; angegeben.