Aufgaben:Exercise 2.6: PN Generator of Length 5 - Revision history

Guenter at 15:04, 28 December 2021

2021-12-28T15:04:39Z

Guenter at 14:37, 28 December 2021

2021-12-28T14:37:25Z

Noah at 08:35, 21 December 2021

2021-12-21T08:35:52Z

Noah at 22:50, 20 December 2021

2021-12-20T22:50:15Z

Guenter at 10:15, 1 December 2021

2021-12-01T10:15:24Z

Guenter: Guenter moved page Aufgaben:Aufgabe 2.6: PN-Generator der Länge 5 to Aufgaben:Exercise 2.6: PN Generator of Length 5

2021-12-01T10:10:11Z

Guenter moved page Aufgabe 2.6: PN-Generator der Länge 5 to Exercise 2.6: PN Generator of Length 5

Javier: Text replacement - "Category:Aufgaben zu Stochastische Signaltheorie" to "Category:Theory of Stochastic Signals: Exercises"

2021-03-23T12:45:08Z

Text replacement - "Category:Aufgaben zu Stochastische Signaltheorie" to "Category:Theory of Stochastic Signals: Exercises"

Guenter at 12:59, 18 August 2020

2020-08-18T12:59:40Z

Javier: Text replacement - "[[Stochastische_Signaltheorie/" to "[[Theory_of_Stochastic_Signals/"

2020-07-09T09:51:57Z

Text replacement - "[[Stochastische_Signaltheorie/" to "[[Theory_of_Stochastic_Signals/"

Guenter at 17:57, 13 November 2019

2019-11-13T17:57:38Z

@@ Line 62: / Line 62: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Correct is the <u>proposed solution 2</u> &nbsp; &#8658; &nbsp; $G(D) = D^5 + D^3 +1$.
+'''(1)'''&nbsp; Correct is the&nbsp; <u>proposed solution 2</u> &nbsp; &#8658; &nbsp; $G(D) = D^5 + D^3 +1$.
-*The generator polynomial&nbsp; $G(D)$&nbsp; denotes the returns used for modulo-2 addition.
+*The generator polynomial&nbsp; $G(D)$&nbsp; denotes the feedback coefficients used for modulo-2 addition.
 *$D$&nbsp; is a formal parameter indicating a delay by one clock.
 *$D^3$&nbsp; then indicates a delay of three measures.
@@ Line 70: / Line 70: @@
 '''(2)'''&nbsp; It is&nbsp; $g_0 = g_3 = g_5 = 1$.&nbsp;
-*All other R&uuml;ckf&nbsp; coefficients are $0$. It follows that:
+*All other feedback coefficients are&nbsp; $0$.&nbsp; It follows that:
 :$$(g_{\rm 5}\hspace{0.1cm}g_{\rm 4}\hspace{0.1cm}g_{\rm 3}\hspace{0.1cm}g_{\rm 2}\hspace{0.1cm}g_{\rm 1}\hspace{0.1cm}g_{\rm 0})=\rm (101001)_{bin}\hspace{0.15cm} \underline{=(51)_{oct}}.$$
-'''(3)'''&nbsp; Since the generator polynomial&nbsp; $G(D)$&nbsp; is primitive, one obtains an M-sequence.
+'''(3)'''&nbsp; Since the generator polynomial&nbsp; $G(D)$&nbsp; is primitive,&nbsp; one obtains an&nbsp; "M-sequence".
-*Accordingly, the period is maximal:
+*Accordingly,&nbsp; the period is maximal:
 :$$P_{\rm max} = 2^{L}-1 \hspace{0.15cm}\underline {= 31}.$$
-*In the theory part, the table with PN generators of maximum length (M sequences) for degree&nbsp; $5$&nbsp; lists the configuration&nbsp; $(51)_{\rm oct}$.
+*In the theory part,&nbsp; the table with PN generators of maximum length&nbsp; ("M-sequences")&nbsp; for degree&nbsp; $L=5$&nbsp; lists the configuration&nbsp; $(51)_{\rm oct}$.
@@ Line 84: / Line 85: @@
 :$$G_{\rm R}(D)=D^{\rm 5}\cdot(D^{\rm -5}+\D^{\rm -3}+ 1)= D^{\rm 5}+D^{\rm 2}+1.$$
-*Thus, the octal identifier für this configuration&nbsp; $\rm (100101)_{bin}\hspace{0.15cm} \underline{=(45)_{oct}}.$
+*Thus,&nbsp; the octal identifier für this configuration: &nbsp; $\rm (100101)_{bin}\hspace{0.15cm} \underline{=(45)_{oct}}.$
-'''(5)'''&nbsp; The correct solutions are <u>solutions 1, 3, and 4</u>:
+'''(5)'''&nbsp; The&nbsp; <u>solutions 1,&nbsp; 3,&nbsp; and&nbsp; 4</u>&nbsp;  are correct:
-*The initial sequence of the reciprocal realization&nbsp; $G_{\rm R}(D)$&nbsp; of a primitive polynomial&nbsp; $G(D)$&nbsp; is always also an M-sequence.
+*The output sequence of the reciprocal realization&nbsp; $G_{\rm R}(D)$&nbsp; of a primitive polynomial&nbsp; $G(D)$&nbsp; is also an&nbsp; "M-sequence".
-*Both sequences are inverses of each other. This means:
+*Both sequences are inverses of each other.&nbsp; This means:
-*The initial sequence of&nbsp; $(45)_{\rm oct}$&nbsp; is equal to the sequence of&nbsp; $(51)_{\rm oct}$ when read from right to left and additionally taking into account a phase (cyclic shift).
+*The output sequence of&nbsp; $(45)_{\rm oct}$&nbsp; is equal to the output sequence of&nbsp; $(51)_{\rm oct}$&nbsp; when read from right to left,&nbsp; additionally taking into account a phase&nbsp; ("cyclic shift").
-*A precondition is again that not all memory cells are preallocated with zeros.
+*The prerequisite is again that not all memory cells are preallocated with zeros.
-*Under this condition, both sequences actually also have the same statistical properties.
+*Under this condition,&nbsp; both sequences actually have the same statistical properties.

@@ Line 4: / Line 4: @@
 [[File:EN_Sto_A_2_6.png|right|frame|PN generator of length&nbsp; $L = 5$]]
-In the diagram you can see a pseudorandom generator of length&nbsp; $L = 5$, which can be used to generate a bin&nbsp; $\langle z_{\nu} \rangle$&nbsp; to be used.
+In the graphic you can see a pseudo-random generator of length&nbsp; $L = 5$,&nbsp; which can be used to generate a binary random sequence&nbsp; $\langle z_{\nu} \rangle$.
 *Hereby&nbsp; $z_{\nu}$&nbsp; results from the modulo-2 addition between&nbsp; $z_{\nu-3}$&nbsp; and&nbsp; $z_{\nu-5}$.
@@ Line 20: / Line 20: @@
 *The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Generation_of_Discrete_Random_Variables|Generation of Discrete Random Variables]].
-*The topic of this chapter is illustrated with examples in the  (German language)  learning video &nbsp; [[Erläuterung_der_PN–Generatoren_an_einem_Beispiel_(Lernvideo)|Erläuterung  der PN-Generatoren an einem Beispiel]] $\Rightarrow$ Explanation of PN generators by example.
+*The topic of this chapter is illustrated with examples in the&nbsp; (German language)&nbsp;  learning video: <br> &nbsp; &nbsp; [[Erläuterung_der_PN–Generatoren_an_einem_Beispiel_(Lernvideo)|"Erläuterung der PN-Generatoren an einem Beispiel"]] &nbsp; $\Rightarrow$ &nbsp; "Explanation of PN generators using an example".
@@ Line 38: / Line 38: @@
-{Assume that the generator polynomial&nbsp; $G(D)$&nbsp; is primitive. <br>Is the initial sequence&nbsp; $〈z_ν \rangle$&nbsp; an M-sequence? How large is its period&nbsp; $P$?
+{Assume that the generator polynomial&nbsp; $G(D)$&nbsp; is primitive. <br>Is the initial sequence&nbsp; $〈z_ν \rangle$&nbsp; an M-sequence?&nbsp; How large is the period&nbsp; $P$?
 |type="{}"}
 $P\ = \ $ { 31 }
-{What octal identifier&nbsp; $O_{\rm R}$&nbsp; describes the polynomial reciprocal to&nbsp; $G(D)$&nbsp; $G_{\rm R}(D)$&nbsp;?
+{What octal identifier&nbsp; $O_{\rm R}$&nbsp; describes the polynomial&nbsp; $G_{\rm R}(D)$&nbsp; reciprocal to&nbsp; $G(D)$?
 |type="{}"}
 $O_{\rm R} \ = \ $ { 45 } $\ \rm (octal)$

@@ Line 18: / Line 18: @@
 Hints:
-*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Generation_of_Discrete_Random_Variables|Generation of Discrete Random Variable]].
+*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Generation_of_Discrete_Random_Variables|Generation of Discrete Random Variables]].
 *The topic of this chapter is illustrated with examples in the  (German language)  learning video &nbsp; [[Erläuterung_der_PN–Generatoren_an_einem_Beispiel_(Lernvideo)|Erläuterung  der PN-Generatoren an einem Beispiel]] $\Rightarrow$ Explanation of PN generators by example.

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Stochastische Signaltheorie/Erzeugung von diskreten Zufallsgrößen
+{{quiz-Header|Buchseite=Theory_of_Stochastic_Signals/Generation_of_Discrete_Random_Variables
 }}
-[[File:EN_Sto_A_2_6.png|right|frame|PN-Generator der Länge&nbsp; $L = 5$]]
+[[File:EN_Sto_A_2_6.png|right|frame|PN generator of length&nbsp; $L = 5$]]
-In der Grafik sehen Sie einen Pseudozufallsgenerator der L&auml;nge&nbsp; $L = 5$, der zur Erzeugung einer Bin&auml;rfolge&nbsp; $\langle z_{\nu} \rangle$&nbsp; eingesetzt werden soll.
+In the diagram you can see a pseudorandom generator of length&nbsp; $L = 5$, which can be used to generate a bin&nbsp; $\langle z_{\nu} \rangle$&nbsp; to be used.
-*Zum Startzeitpunkt seien alle Speicherzellen mit Einsen vorbelegt.
+*At the start time, let all memory cells be preallocated with ones.
-*Zu jedem Taktzeitpunkt wird der Inhalt des Schieberegisters um eine Stelle nach rechts verschoben und der aktuell erzeugte Bin&auml;rwert&nbsp; $z_{\nu}$&nbsp; $(0$&nbsp; oder&nbsp; $1)$&nbsp; in die erste Speicherzelle eingetragen.
+*At each clock time, the content of the shift register is shifted one place to the right and the currently generated binary value&nbsp; $z_{\nu}$&nbsp; $(0$&nbsp; or&nbsp; $1)$&nbsp; is entered into the first memory cell.
-*Hierbei ergibt sich&nbsp; $z_{\nu}$&nbsp; aus der Modulo-2-Addition zwischen&nbsp; $z_{\nu-3}$&nbsp; und&nbsp; $z_{\nu-5}$.
+*Hereby&nbsp; $z_{\nu}$&nbsp; results from the modulo-2 addition between&nbsp; $z_{\nu-3}$&nbsp; and&nbsp; $z_{\nu-5}$.
@@ Line 17: / Line 17: @@
-''Hinweise:''
+Hints:
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Theory_of_Stochastic_Signals/Erzeugung_von_diskreten_Zufallsgrößen|Erzeugung von diskreten Zufallsgrößen]].
+*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Generation_of_Discrete_Random_Variables|Generation of Discrete Random Variable]].
-*Wir verweisen hier auch auf das  Lernvideo &nbsp; [[Erläuterung_der_PN–Generatoren_an_einem_Beispiel_(Lernvideo)|Erläuterung  der PN-Generatoren an einem Beispiel]].
+*The topic of this chapter is illustrated with examples in the  (German language)  learning video &nbsp; [[Erläuterung_der_PN–Generatoren_an_einem_Beispiel_(Lernvideo)|Erläuterung  der PN-Generatoren an einem Beispiel]] $\Rightarrow$ Explanation of PN generators by example.
-===Fragebogen===
+===Question===
 <quiz display=simple>
-{Wie lautet das Generatorpolynom&nbsp; $G(D)$&nbsp; des dargestellten PN-Generators?
+{What is the generator polynomial&nbsp; $G(D)$&nbsp; of the PN generator shown?
 |type="()"}
 - $G(D) = D^5 + D^2 +1$.
@@ Line 33: / Line 33: @@
-{Welche Oktalkennung&nbsp; $O_{\rm G}$&nbsp; hat dieser PN-Generator?
+{What octal identifier&nbsp; $O_{\rm G}$&nbsp; does this PN generator have?
 |type="{}"}
-$O_{\rm G} \ = \ $  { 51 } $\ \rm (oktal)$
+$O_{\rm G} \ = \ $ { 51 } $\ \rm (octal)$
-{Gehen Sie davon aus, dass das Generatorpolynom&nbsp; $G(D)$&nbsp; primitiv ist. <br>Ist die Ausgangsfolge&nbsp; $〈z_ν \rangle$&nbsp; eine M-Sequenz? Wie gro&szlig; ist deren Periodendauer&nbsp; $P$?
+{Assume that the generator polynomial&nbsp; $G(D)$&nbsp; is primitive. <br>Is the initial sequence&nbsp; $〈z_ν \rangle$&nbsp; an M-sequence? How large is its period&nbsp; $P$?
 |type="{}"}
-$P\ =  \ $ { 31 }
+$P\ = \ $ { 31 }
-{Welche Oktalkennung&nbsp; $O_{\rm R}$&nbsp; beschreibt das zu&nbsp; $G(D)$&nbsp; reziproke Polynom&nbsp; $G_{\rm R}(D)$&nbsp;?
+{What octal identifier&nbsp; $O_{\rm R}$&nbsp; describes the polynomial reciprocal to&nbsp; $G(D)$&nbsp; $G_{\rm R}(D)$&nbsp;?
 |type="{}"}
-$O_{\rm R} \ = \ $ { 45 } $\ \rm (oktal)$
+$O_{\rm R} \ = \ $ { 45 } $\ \rm (octal)$
-{Welche Aussagen gelten für die Konfiguration mit dem Polynom&nbsp; $G_{\rm R}(D)$?
+{What statements hold for the configuration with the polynomial&nbsp; $G_{\rm R}(D)$?
 |type="[]"}
-+ Es handelt sich ebenfalls um eine Folge maximaler L&auml;nge.
++ It is also a sequence of maximum length.
-- Die Ausgangsfolge von&nbsp;  $G_{\rm R}(D)$&nbsp; ist die gleiche wie die des Generatorpolynoms&nbsp; $G(D)$.
+- The output sequence of&nbsp; $G_{\rm R}(D)$&nbsp; is the same as that of the generator polynomial&nbsp; $G(D)$.
-+ Die Ausgangsfolgen von&nbsp; $G_{\rm R}(D)$&nbsp; und&nbsp; $G(D)$&nbsp; sind zueinander invers.
++ The output sequences of&nbsp; $G_{\rm R}(D)$&nbsp; and&nbsp; $G(D)$&nbsp; are inverses of each other.
-+ Beide Folgen zeigen gleiche statistische Eigenschaften.
++ Both sequences show the same statistical properties.
-- Bei&nbsp; $G_{\rm R}(D)$&nbsp; k&ouml;nnen alle Speicherelemente mit Nullen vorbelegt sein.
+- In&nbsp; $G_{\rm R}(D)$&nbsp; all memory elements can be preallocated with zeros.
@@ Line 60: / Line 60: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Richtig ist der <u>Lösungsvorschlag 2</u> &nbsp; &#8658; &nbsp; $G(D) = D^5 + D^3 +1$.
+'''(1)'''&nbsp; Correct is the <u>proposed solution 2</u> &nbsp; &#8658; &nbsp; $G(D) = D^5 + D^3 +1$.
-*Das Generatorpolynom&nbsp; $G(D)$&nbsp; kennzeichnet die R&uuml;ckf&uuml;hrungen, die zur Modulo-2-Addition herangezogen werden.
+*The generator polynomial&nbsp; $G(D)$&nbsp; denotes the returns used for modulo-2 addition.
-*$D$&nbsp; ist ein formaler Parameter, der eine Verz&ouml;gerung um einen Takt angibt.
+*$D$&nbsp; is a formal parameter indicating a delay by one clock.
-*$D^3$&nbsp; kennzeichnet dann eine Verz&ouml;gerung um drei Takte.
+*$D^3$&nbsp; then indicates a delay of three measures.
-'''(2)'''&nbsp; Es ist&nbsp; $g_0 = g_3 = g_5 = 1$.&nbsp;
+'''(2)'''&nbsp; It is&nbsp; $g_0 = g_3 = g_5 = 1$.&nbsp;
-*Alle anderen R&uuml;ckf&uuml;hrungskoeffizienten sind $0$. Daraus folgt:
+*All other R&uuml;ckf&nbsp; coefficients are $0$. It follows that:
 :$$(g_{\rm 5}\hspace{0.1cm}g_{\rm 4}\hspace{0.1cm}g_{\rm 3}\hspace{0.1cm}g_{\rm 2}\hspace{0.1cm}g_{\rm 1}\hspace{0.1cm}g_{\rm 0})=\rm (101001)_{bin}\hspace{0.15cm} \underline{=(51)_{oct}}.$$
-'''(3)'''&nbsp; Da das Generatorpolynom&nbsp; $G(D)$&nbsp; primitiv ist, erh&auml;lt man eine M-Sequenz.
+'''(3)'''&nbsp; Since the generator polynomial&nbsp; $G(D)$&nbsp; is primitive, one obtains an M-sequence.
-*Dementsprechend ist die Periodendauer maximal:
+*Accordingly, the period is maximal:
 :$$P_{\rm max} = 2^{L}-1 \hspace{0.15cm}\underline {= 31}.$$
-*Im Theorieteil ist in der Tabelle mit den PN-Generatoren maximaler L&auml;nge (M-Sequenzen) für den Grad&nbsp; $5$&nbsp; die Konfiguration&nbsp; $(51)_{\rm oct}$ aufgef&uuml;hrt.
+*In the theory part, the table with PN generators of maximum length (M sequences) for degree&nbsp; $5$&nbsp; lists the configuration&nbsp; $(51)_{\rm oct}$.
-'''(4)'''&nbsp; Das reziproke Polynom lautet:
+'''(4)'''&nbsp; The reciprocal polynomial is:
 :$$G_{\rm R}(D)=D^{\rm 5}\cdot(D^{\rm -5}+\D^{\rm -3}+ 1)= D^{\rm 5}+D^{\rm 2}+1.$$
-*Somit ist die Oktalkennung f&uuml;r diese Konfiguration&nbsp; $\rm (100101)_{bin}\hspace{0.15cm} \underline{=(45)_{oct}}.$
+*Thus, the octal identifier für this configuration&nbsp; $\rm (100101)_{bin}\hspace{0.15cm} \underline{=(45)_{oct}}.$
-'''(5)'''&nbsp; Richtig sind die <u>Lösungsvorschläge 1, 3 und 4</u>:
+'''(5)'''&nbsp; The correct solutions are <u>solutions 1, 3, and 4</u>:
-*Die Ausgangsfolge der reziproken Realisierung&nbsp; $G_{\rm R}(D)$&nbsp; eines primitiven Polynoms&nbsp; $G(D)$&nbsp; ist immer ebenfalls eine M-Sequenz.
+*The initial sequence of the reciprocal realization&nbsp; $G_{\rm R}(D)$&nbsp; of a primitive polynomial&nbsp; $G(D)$&nbsp; is always also an M-sequence.
-*Beide Folgen sind zueinander invers. Das bedeutet:
+*Both sequences are inverses of each other. This means:
-*Die Ausgangsfolge von&nbsp; $(45)_{\rm oct}$&nbsp; ist gleich der Folge von&nbsp; $(51)_{\rm oct}$, wenn man diese von rechts nach links liest und zusätzlich eine Phase (zyklische Verschiebung) ber&uuml;cksichtigt.
+*The initial sequence of&nbsp; $(45)_{\rm oct}$&nbsp; is equal to the sequence of&nbsp; $(51)_{\rm oct}$ when read from right to left and additionally taking into account a phase (cyclic shift).
-*Voraussetzung ist wieder, dass nicht alle Speicherzellen mit Nullen vorbelegt sind.
+*A precondition is again that not all memory cells are preallocated with zeros.
-*Unter dieser Bedingung weisen beide Folgen tatsächlich auch gleiche statistische Eigenschaften auf.
+*Under this condition, both sequences actually also have the same statistical properties.

← Older revision		Revision as of 10:15, 1 December 2021
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−	[[Category:Theory of Stochastic Signals: Exercises\|^2.5 ~~Erzeugung diskreter Zufallsgrößen~~^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^2.5 Generation of Discrete Random Variables^]]

← Older revision	Revision as of 10:10, 1 December 2021
(No difference)

← Older revision		Revision as of 12:45, 23 March 2021
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−	[[Category:~~Aufgaben zu Stochastische Signaltheorie~~\|^2.5 Erzeugung diskreter Zufallsgrößen^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^2.5 Erzeugung diskreter Zufallsgrößen^]]