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

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	[[Category:Theory of Stochastic Signals: Exercises\|^2.5 Generation of Discrete Random Variables^]]		[[Category:Theory of Stochastic Signals: Exercises\|^2.5 Generation of Discrete Random Variables^]]
	[[de:Aufgaben:2.~~6_PN~~-~~Generator_der_Länge_5~~]]		[[de:Aufgaben:Aufgabe 2.6: PN-Generator der Länge 5]]

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	[[Category:Theory of Stochastic Signals: Exercises\|^2.5 Generation of Discrete Random Variables^]]		[[Category:Theory of Stochastic Signals: Exercises\|^2.5 Generation of Discrete Random Variables^]]
			[[de:Aufgaben:2.6_PN-Generator_der_Länge_5]]

Guenter at 15:04, 28 December 2021

2021-12-28T15:04:39Z

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	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	'''(1)'''  Correct is the <u>proposed solution 2</u>   ⇒   $G(D) = D^5 + D^3 +1$.		'''(1)'''  Correct is the  <u>proposed solution 2</u>   ⇒   $G(D) = D^5 + D^3 +1$.
	*The generator polynomial  $G(D)$  denotes the ~~returns~~ used for modulo-2 addition.		*The generator polynomial  $G(D)$  denotes the feedback coefficients used for modulo-2 addition.
	*$D$  is a formal parameter indicating a delay by one clock.		*$D$  is a formal parameter indicating a delay by one clock.
	*$D^3$  then indicates a delay of three measures.		*$D^3$  then indicates a delay of three measures.
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	'''(2)'''  It is  $g_0 = g_3 = g_5 = 1$.		'''(2)'''  It is  $g_0 = g_3 = g_5 = 1$.
	*All other ~~Rückf~~  ~~coefficients are~~ $0$. It follows that:		*All other feedback coefficients are  $0$.  It follows that:
	:$$(g_{\rm 5}\hspace{0.1cm}g_{\rm 4}\hspace{0.1cm}g_{\rm 3}\hspace{0.1cm}g_{\rm 2}\hspace{0.1cm}g_{\rm 1}\hspace{0.1cm}g_{\rm 0})=\rm (101001)_{bin}\hspace{0.15cm} \underline{=(51)_{oct}}.$$		:$$(g_{\rm 5}\hspace{0.1cm}g_{\rm 4}\hspace{0.1cm}g_{\rm 3}\hspace{0.1cm}g_{\rm 2}\hspace{0.1cm}g_{\rm 1}\hspace{0.1cm}g_{\rm 0})=\rm (101001)_{bin}\hspace{0.15cm} \underline{=(51)_{oct}}.$$



	'''(3)'''  Since the generator polynomial  $G(D)$  is primitive, one obtains an M-sequence.		'''(3)'''  Since the generator polynomial  $G(D)$  is primitive,  one obtains an  "M-sequence".
	*Accordingly, the period is maximal:		*Accordingly,  the period is maximal:
	:$$P_{\rm max} = 2^{L}-1 \hspace{0.15cm}\underline {= 31}.$$		:$$P_{\rm max} = 2^{L}-1 \hspace{0.15cm}\underline {= 31}.$$
	*In the theory part, the table with PN generators of maximum length (M sequences) for degree  $5$  lists the configuration  $(51)_{\rm oct}$.		*In the theory part,  the table with PN generators of maximum length  ("M-sequences")  for degree  $L=5$  lists the configuration  $(51)_{\rm oct}$.



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	:$$G_{\rm R}(D)=D^{\rm 5}\cdot(D^{\rm -5}+\D^{\rm -3}+ 1)= D^{\rm 5}+D^{\rm 2}+1.$$		:$$G_{\rm R}(D)=D^{\rm 5}\cdot(D^{\rm -5}+\D^{\rm -3}+ 1)= D^{\rm 5}+D^{\rm 2}+1.$$

	*Thus, the octal identifier für this configuration  $\rm (100101)_{bin}\hspace{0.15cm} \underline{=(45)_{oct}}.$		*Thus,  the octal identifier für this configuration:   $\rm (100101)_{bin}\hspace{0.15cm} \underline{=(45)_{oct}}.$



	'''(5)'''  The ~~correct solutions are~~ <u>solutions 1, 3, and 4</u>:		'''(5)'''  The  <u>solutions 1,  3,  and  4</u>  are correct:
	*The ~~initial~~ sequence of the reciprocal realization  $G_{\rm R}(D)$  of a primitive polynomial  $G(D)$  is ~~always~~ also an M-sequence.		*The output sequence of the reciprocal realization  $G_{\rm R}(D)$  of a primitive polynomial  $G(D)$  is also an  "M-sequence".
	*Both sequences are inverses of each other. This means:		*Both sequences are inverses of each other.  This means:
	*The ~~initial~~ sequence of  $(45)_{\rm oct}$  is equal to the sequence of  $(51)_{\rm oct}$ when read from right to left ~~and~~ additionally taking into account a phase (cyclic shift).		*The output sequence of  $(45)_{\rm oct}$  is equal to the output sequence of  $(51)_{\rm oct}$  when read from right to left,  additionally taking into account a phase  ("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,  both sequences actually have the same statistical properties.

Guenter at 14:37, 28 December 2021

2021-12-28T14:37:25Z

← Older revision		Revision as of 16:37, 28 December 2021
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	[[File:EN_Sto_A_2_6.png\|right\|frame\|PN generator of length  $L = 5$]]		[[File:EN_Sto_A_2_6.png\|right\|frame\|PN generator of length  $L = 5$]]
	In the ~~diagram~~ you can see a ~~pseudorandom~~ generator of length  $L = 5$, which can be used to generate a ~~bin~~  $\langle z_{\nu} \rangle$~~  to be used.~~		In the graphic you can see a pseudo-random generator of length  $L = 5$,  which can be used to generate a binary random sequence  $\langle z_{\nu} \rangle$.

	*At the start time, let all memory cells be preallocated with ones.
	*At each clock time, the content of the shift register is shifted one place to the right and the currently generated binary value  $z_{\nu}$  $(0$  or  $1)$  is entered into the first memory cell.

			*At the start time,  let all memory cells be preallocated with  "ones".
			*At each clock time,  the content of the shift register is shifted one place to the right.
			* And the currently generated binary value  $z_{\nu}$  $(0$  or  $1)$  is entered into the first memory cell.
	*Hereby  $z_{\nu}$  results from the modulo-2 addition between  $z_{\nu-3}$  and  $z_{\nu-5}$.		*Hereby  $z_{\nu}$  results from the modulo-2 addition between  $z_{\nu-3}$  and  $z_{\nu-5}$.

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	*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Generation_of_Discrete_Random_Variables\|Generation of Discrete Random Variables]].		*The exercise belongs to the chapter  [[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   [[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  (German language)  learning video: <br>     [[Erläuterung_der_PN–Generatoren_an_einem_Beispiel_(Lernvideo)\|"Erläuterung der PN-Generatoren an einem Beispiel"]]   $\Rightarrow$   "Explanation of PN generators using an example".


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	{Assume that the generator polynomial  $G(D)$  is primitive. <br>Is the initial sequence  $〈z_ν \rangle$  an M-sequence? How large is ~~its~~ period  $P$?		{Assume that the generator polynomial  $G(D)$  is primitive. <br>Is the initial sequence  $〈z_ν \rangle$  an M-sequence?  How large is the period  $P$?
	\|type="{}"}		\|type="{}"}
	$P\ = \ $ { 31 }		$P\ = \ $ { 31 }


	{What octal identifier  $O_{\rm R}$  describes the polynomial ~~reciprocal to  $G(D)$~~  $G_{\rm R}(D)$ ?		{What octal identifier  $O_{\rm R}$  describes the polynomial  $G_{\rm R}(D)$  reciprocal to  $G(D)$?
	\|type="{}"}		\|type="{}"}
	$O_{\rm R} \ = \ $ { 45 } $\ \rm (octal)$		$O_{\rm R} \ = \ $ { 45 } $\ \rm (octal)$

Noah at 08:35, 21 December 2021

2021-12-21T08:35:52Z

← Older revision		Revision as of 10:35, 21 December 2021
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	Hints:		Hints:
	*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Generation_of_Discrete_Random_Variables\|Generation of Discrete Random ~~Variable~~]].		*The exercise belongs to the chapter  [[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   [[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 (German language) learning video   [[Erläuterung_der_PN–Generatoren_an_einem_Beispiel_(Lernvideo)\|Erläuterung der PN-Generatoren an einem Beispiel]] $\Rightarrow$ Explanation of PN generators by example.

Noah at 22:50, 20 December 2021

2021-12-20T22:50:15Z

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Guenter at 10:15, 1 December 2021

2021-12-01T10:15:24Z

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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^]]

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

← Older revision	Revision as of 12:10, 1 December 2021
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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"

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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^]]

Guenter at 12:59, 18 August 2020

2020-08-18T12:59:40Z

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	}}		}}

	[[File:~~P_ID105__Sto_A_2_6~~.png\|right\|frame\|PN-Generator der Länge  $L = 5$]]		[[File:EN_Sto_A_2_6.png\|right\|frame\|PN-Generator der Länge  $L = 5$]]
	In der Grafik sehen Sie einen Pseudozufallsgenerator der Länge  $L = 5$, der zur Erzeugung einer Binärfolge  $\langle z_{\nu} \rangle$  eingesetzt werden soll.		In der Grafik sehen Sie einen Pseudozufallsgenerator der Länge  $L = 5$, der zur Erzeugung einer Binärfolge  $\langle z_{\nu} \rangle$  eingesetzt werden soll.