Aufgaben:Exercise 1.1Z: Sum of Two Ternary Signals - Revision history

Guenter at 15:42, 26 November 2021

2021-11-26T15:42:30Z

Noah at 15:12, 23 November 2021

2021-11-23T15:12:22Z

Guenter at 16:58, 18 November 2021

2021-11-18T16:58:08Z

Guenter at 16:53, 18 November 2021

2021-11-18T16:53:46Z

Noah at 10:37, 15 November 2021

2021-11-15T10:37:31Z

Guenter at 15:34, 12 October 2021

2021-10-12T15:34:19Z

Guenter: Guenter moved page Aufgaben:Aufgabe 1.1Z: Summe zweier Ternärsignale to Aufgaben:Exercise 1.1Z: Sum of Two Ternary Signals

2021-10-12T15:32:06Z

Guenter moved page Aufgabe 1.1Z: Summe zweier Ternärsignale to Exercise 1.1Z: Sum of Two Ternary Signals

Javier: Text replacement - "”" to """

2021-05-28T14:22:08Z

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Javier: Text replacement - "„" to """

2021-05-28T14:21:47Z

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

2021-03-23T12:44:24Z

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

← Older revision		Revision as of 15:42, 26 November 2021
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−	[[File:~~EN_Sto_Z1_1_c~~.png\|right\|frame\|400px\|Sum and difference of ternary random variables]]	+	[[File:EN_Sto_Z1_1_c_neu.png\|right\|frame\|400px\|Sum and difference of ternary random variables]]
	'''(2)'''  $S$  can take a total of  $\underline {I =5}$  values, namely  $0$,  $\pm 1$  and  $\pm 2$.		'''(2)'''  $S$  can take a total of  $\underline {I =5}$  values, namely  $0$,  $\pm 1$  and  $\pm 2$.

← Older revision		Revision as of 15:12, 23 November 2021
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−	{{quiz-Header\|Buchseite=~~Stochastische Signaltheorie~~/~~Einige grundlegende Definitionen~~}}	+	{{quiz-Header\|Buchseite=Theory_of_Stochastic_Signals/Some_Basic_Definitions}}

	[[File:P_ID146__Sto_Z1_1.png\|right\|framed\|Sum $S$ of two <br>ternary signals  $X$  and  $Y$]]		[[File:P_ID146__Sto_Z1_1.png\|right\|framed\|Sum $S$ of two <br>ternary signals  $X$  and  $Y$]]

@@ Line 8: / Line 8: @@
 *At the signal source&nbsp; $X$,&nbsp; the values&nbsp; $-1$,&nbsp; $0$&nbsp; and&nbsp; $+1$&nbsp; occur with equal probability.
 *For source&nbsp; $Y$,&nbsp; the signal value&nbsp; $0$&nbsp; is twice as likely as the other two values&nbsp; $-1$&nbsp; and&nbsp; $+1$, respectively.
@@ Line 59: / Line 57: @@
 '''(2)'''&nbsp; $S$&nbsp; can take a total of&nbsp; $\underline {I =5}$&nbsp; values, namely&nbsp; $0$,&nbsp; $\pm 1$&nbsp; and&nbsp; $\pm 2$.
 '''(3)'''&nbsp; Since&nbsp; $Y$&nbsp; is not equally distributed, one cannot (actually) apply the "Classical Definition of Probability" here.
-*However, if we divide&nbsp; $Y$&nbsp; into four ranges according to the graph, assigning two of the ranges to the event&nbsp; $Y = 0$&nbsp;, we can still proceed according to the classical definition.
+*However,&nbsp; if we divide&nbsp; $Y$&nbsp; into four ranges according to the graph,&nbsp; assigning two of the ranges to the event&nbsp; $Y = 0$,&nbsp; we can still proceed according to the classical definition.
 *One then obtains:
@@ Line 76: / Line 74: @@
 '''(4)'''&nbsp; It is also evident from the graph that the difference signal&nbsp; $D$&nbsp; and the sum signal&nbsp; $S$&nbsp; take the same values with equal probabilities.
-*This was to be expected, since&nbsp; ${\rm Pr}(Y = +1) ={\rm Pr}(Y = -1)$&nbsp; is given &nbsp;  ⇒ &nbsp; <u>Proposed solution 1</u>.
+*This was to be expected,&nbsp; since&nbsp; ${\rm Pr}(Y = +1) ={\rm Pr}(Y = -1)$&nbsp; is given &nbsp;  ⇒ &nbsp; <u>Proposed solution 1</u>.
 {{ML-Fuß}}

@@ Line 3: / Line 3: @@
 [[File:P_ID146__Sto_Z1_1.png|right|framed|Sum $S$ of two <br>ternary signals&nbsp; $X$&nbsp; and&nbsp; $Y$]]
-Let two three-stage message sources&nbsp; $X$&nbsp; and&nbsp; $Y$,  be given, whose output signals can only assume the values&nbsp; $-1$,&nbsp; $0$&nbsp; and&nbsp; $+1$&nbsp; respectively. The signal sources are statistically independent of each other.
+Let two three-stage message sources&nbsp; $X$&nbsp; and&nbsp; $Y$&nbsp;  be given,&nbsp; whose output signals can only assume the values&nbsp; $-1$,&nbsp; $0$&nbsp; and&nbsp; $+1$&nbsp; respectively.&nbsp; The signal sources are statistically independent of each other.
 *A simple circuit now forms the sum signal&nbsp; $S = X + Y$.
-*At the signal source&nbsp; $X$&nbsp; the values&nbsp; $-1$,&nbsp; $0$&nbsp; and&nbsp; $+1$&nbsp; occur with equal probability.
+*At the signal source&nbsp; $X$,&nbsp; the values&nbsp; $-1$,&nbsp; $0$&nbsp; and&nbsp; $+1$&nbsp; occur with equal probability.
-*For source&nbsp; $Y$&nbsp;, the signal value&nbsp; $0$&nbsp; is twice as likely as the other two values&nbsp; $-1$&nbsp; and&nbsp; $+1$, respectively.
+*For source&nbsp; $Y$,&nbsp; the signal value&nbsp; $0$&nbsp; is twice as likely as the other two values&nbsp; $-1$&nbsp; and&nbsp; $+1$, respectively.
@@ Line 18: / Line 18: @@
 *Solve the subtasks&nbsp; '''(3)'''&nbsp; and&nbsp; '''(4)'''&nbsp; according to the classical definition.
-*Nevertheless, consider the different occurrence frequencies of the signal&nbsp; $Y$.
+*Nevertheless,&nbsp; consider the different occurrence frequencies of the signal&nbsp; $Y$.
-*The topic of this section is illustrated with examples in the (German language) learning video &nbsp; [[Klassische_Definition_der_Wahrscheinlickeit_(Lernvideo)|Klassische Definition der Wahrscheinlichkeit]]&nbsp; $\Rightarrow$ "Classical definition of probability".
+*The topic of this section is illustrated with examples in the (German language) learning video &nbsp; <br>[[Klassische_Definition_der_Wahrscheinlickeit_(Lernvideo)|Klassische Definition der Wahrscheinlichkeit]]&nbsp; $\Rightarrow$ "Classical definition of probability".
@@ Line 28: / Line 28: @@
 <quiz display=simple>
-{What are the probabilities of occurrence of the signal values of&nbsp; $Y$? What is the probability that&nbsp; $Y = 0$&nbsp;?
+{What are the probabilities of occurrence of the signal values of&nbsp; $Y$?&nbsp; What is the probability that&nbsp; $Y = 0$&nbsp;?
 |type="{}"}
 ${\rm Pr}(Y=0) \ = \ $ { 0.5 3% }
-{How many different signal values&nbsp; $(I)$&nbsp; can the sum signal&nbsp; $S$&nbsp; assume? Which are these?
+{How many different signal values&nbsp; $(I)$&nbsp; can the sum signal&nbsp; $S$&nbsp; assume?&nbsp; Which are these?
 |type="{}"}
 $ I \ = \ $ { 5 3% }
-{What are the probabilities of the values determined in subtask&nbsp; '''(2)'''&nbsp;? How probable is the maximum value&nbsp; $S_{\rm max}$?
+{What are the probabilities of the values determined in subtask&nbsp; '''(2)'''?&nbsp; How probable is the maximum value&nbsp; $S_{\rm max}$?
 |type="{}"}
 $ {\rm Pr}(S = S_{\rm max} ) \ = \ $ { 0.0833 3% }
-{How do the probabilities change, if now instead of the sum the difference&nbsp; $D = X - Y$&nbsp; is considered?&nbsp; Give reasons for your answer.
+{How do the probabilities change,&nbsp; if now instead of the sum the difference&nbsp; $D = X - Y$&nbsp; is considered?&nbsp; Give reasons for your answer.
 |type="[]"}
 + The probabilities remain the same.

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Stochastische Signaltheorie/Einige grundlegende Definitionen}}
-[[File:P_ID146__Sto_Z1_1.png|right|framed|Summe $S$ zweier <br>Ternärsignale&nbsp; $X$&nbsp; und&nbsp; $Y$]]
+[[File:P_ID146__Sto_Z1_1.png|right|framed|Sum $S$ of two <br>ternary signals&nbsp; $X$&nbsp; and&nbsp; $Y$]]
-Gegeben seien zwei dreistufige Nachrichtenquellen&nbsp; $X$&nbsp; und&nbsp; $Y$, deren Ausgangssignale jeweils nur die Werte&nbsp; $-1$,&nbsp; $0$&nbsp; und&nbsp; $+1$&nbsp; annehmen können. Die Signalquellen sind statistisch voneinander unabhängig.
+Let two three-stage message sources&nbsp; $X$&nbsp; and&nbsp; $Y$,  be given, whose output signals can only assume the values&nbsp; $-1$,&nbsp; $0$&nbsp; and&nbsp; $+1$&nbsp; respectively. The signal sources are statistically independent of each other.
-*Eine einfache Schaltung bildet nun das Summensignal&nbsp; $S = X + Y$.
+*A simple circuit now forms the sum signal&nbsp; $S = X + Y$.
-*Bei der Signalquelle&nbsp; $X$&nbsp; treten die Werte&nbsp; $-1$,&nbsp; $0$&nbsp; und&nbsp; $+1$&nbsp; mit gleicher Wahrscheinlichkeit auf.
+*At the signal source&nbsp; $X$&nbsp; the values&nbsp; $-1$,&nbsp; $0$&nbsp; and&nbsp; $+1$&nbsp; occur with equal probability.
-*Bei der Quelle&nbsp; $Y$&nbsp; ist der Signalwert&nbsp; $0$&nbsp; doppelt so wahrscheinlich wie die beiden anderen Werte&nbsp; $-1$&nbsp; bzw.&nbsp; $+1$.
+*For source&nbsp; $Y$&nbsp;, the signal value&nbsp; $0$&nbsp; is twice as likely as the other two values&nbsp; $-1$&nbsp; and&nbsp; $+1$, respectively.
@@ Line 14: / Line 14: @@
+Hints:
-''Hinweise:''
+*The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Einige_grundlegende_Definitionen | Some basic definitions of probability theory]].
-*Lösen Sie die Teilaufgaben&nbsp; '''(3)'''&nbsp; und&nbsp; '''(4)'''&nbsp; nach der klassischen Definition.
+*Solve the subtasks&nbsp; '''(3)'''&nbsp; and&nbsp; '''(4)'''&nbsp; according to the classical definition.
-*Berücksichtigen Sie trotzdem die unterschiedlichen Auftrittshäufigkeiten des Signals&nbsp; $Y$.
+*Nevertheless, consider the different occurrence frequencies of the signal&nbsp; $Y$.
-*Der Inhalt dieses Abschnitts ist im Lernvideo&nbsp; [[Klassische_Definition_der_Wahrscheinlickeit_(Lernvideo)|Klassische Definition der Wahrscheinlichkeit]]&nbsp; zusammengefasst.
+*The topic of this section is illustrated with examples in the (German language) learning video &nbsp; [[Klassische_Definition_der_Wahrscheinlickeit_(Lernvideo)|Klassische Definition der Wahrscheinlichkeit]]&nbsp; $\Rightarrow$ "Classical definition of probability".
@@ Line 26: / Line 25: @@
-===Fragebogen===
+===Questions===
 <quiz display=simple>
-{Wie groß sind die Auftrittswahrscheinlichkeiten der Signalwerte von&nbsp; $Y$? Wie groß ist die Wahrscheinlichkeit, dass&nbsp; $Y = 0$&nbsp; ist ?
+{What are the probabilities of occurrence of the signal values of&nbsp; $Y$? What is the probability that&nbsp; $Y = 0$&nbsp;?
 |type="{}"}
 ${\rm Pr}(Y=0) \ = \ $ { 0.5 3% }
-{Wieviele unterschiedliche Signalwerte&nbsp; $(I)$&nbsp; kann das Summensignal&nbsp; $S$&nbsp; annehmen? Welche sind dies?
+{How many different signal values&nbsp; $(I)$&nbsp; can the sum signal&nbsp; $S$&nbsp; assume? Which are these?
 |type="{}"}
 $ I \ = \ $ { 5 3% }
-{Mit welchen Wahrscheinlichkeiten treten die in der Teilaufgabe&nbsp; '''(2)'''&nbsp; ermittelten Werte auf? Wie wahrscheinlich ist der Maximalwert&nbsp; $S_{\rm max}$?
+{What are the probabilities of the values determined in subtask&nbsp; '''(2)'''&nbsp;? How probable is the maximum value&nbsp; $S_{\rm max}$?
 |type="{}"}
 $ {\rm Pr}(S = S_{\rm max} ) \ = \ $ { 0.0833 3% }
-{Wie ändern sich die Wahrscheinlichkeiten, wenn nun anstelle der Summe die Differenz&nbsp; $D = X - Y$&nbsp; betrachtet wird?&nbsp; Begründen Sie Ihre Antwort.
+{How do the probabilities change, if now instead of the sum the difference&nbsp; $D = X - Y$&nbsp; is considered?&nbsp; Give reasons for your answer.
 |type="[]"}
-+ Die Wahrscheinlichkeiten bleiben gleich.
++ The probabilities remain the same.
-- Die Wahrscheinlichkeiten ändern sich.&nbsp; Wie ändern sie sich?
+- The probabilities change.&nbsp; How do they change?
@@ Line 53: / Line 52: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Da die Wahrscheinlichkeiten von&nbsp; $ \pm 1$&nbsp; gleich sind und&nbsp; ${\rm Pr}(Y = 0) = 2 \cdot {\rm Pr}(Y = 1)$&nbsp; gilt, erhält man:
+'''(1)'''&nbsp; Since the probabilities of&nbsp; $ \pm 1$&nbsp; are the same and&nbsp; ${\rm Pr}(Y = 0) = 2 \cdot {\rm Pr}(Y = 1)$&nbsp; holds, we get:
 :$${\rm Pr}(Y = 1) + {\rm Pr}(Y = 0) + {\rm Pr}(Y = -1) = 1/2 \cdot {\rm Pr}(Y = 0) + {\rm Pr}(Y = 0) + 1/2\cdot {\rm Pr}(Y = 0) = 1\hspace{0.3cm} \Rightarrow \hspace{0.3cm}{\rm Pr}(Y = 0)\;\underline { = 0.5}. $$
-'''(2)'''&nbsp; $S$&nbsp; kann insgesamt&nbsp; $\underline {I =5}$&nbsp; Werte annehmen, nämlich&nbsp; $0$,&nbsp; $\pm 1$&nbsp; und&nbsp; $\pm 2$.
+'''(2)'''&nbsp; $S$&nbsp; can take a total of&nbsp; $\underline {I =5}$&nbsp; values, namely&nbsp; $0$,&nbsp; $\pm 1$&nbsp; and&nbsp; $\pm 2$.
-[[File:EN_Sto_Z1_1_c.png|right|frame|400px|Summe und Differenz ternärer Zufallsgrößen]]
+[[File:EN_Sto_Z1_1_c.png|right|frame|400px|Sum and difference of ternary random variables]]
-'''(3)'''&nbsp; Da&nbsp; $Y$&nbsp; nicht gleichverteilt ist, kann man hier  (eigentlich) die "Klassische Definition der Wahrscheinlichkeit" nicht anwenden.
+'''(3)'''&nbsp; Since&nbsp; $Y$&nbsp; is not equally distributed, one cannot (actually) apply the "Classical Definition of Probability" here.
-*Teilt man&nbsp; $Y$&nbsp; jedoch gemäß der Grafik in vier Bereiche auf, wobei man zwei der Bereiche dem Ereignis&nbsp; $Y = 0$&nbsp; zuordnet, so kann man trotzdem gemäß der klassischen Definition vorgehen.
+*However, if we divide&nbsp; $Y$&nbsp; into four ranges according to the graph, assigning two of the ranges to the event&nbsp; $Y = 0$&nbsp;, we can still proceed according to the classical definition.
-*Man erhält dann:
+*One then obtains:
 :$${\rm Pr}(S = 0) = {4}/{12} = {1}/{3},$$
@@ Line 75: / Line 74: @@
-'''(4)'''&nbsp; Aus der Grafik ist auch ersichtlich, dass das Differenzsignal&nbsp; $D$&nbsp; und das Summensignal&nbsp; $S$&nbsp; die gleichen Werte mit gleichen Wahrscheinlichkeiten annehmen.
+'''(4)'''&nbsp; It is also evident from the graph that the difference signal&nbsp; $D$&nbsp; and the sum signal&nbsp; $S$&nbsp; take the same values with equal probabilities.
-*Dies war zu erwarten, da&nbsp; ${\rm Pr}(Y = +1) ={\rm Pr}(Y = -1)$&nbsp; vorgegeben ist &nbsp;  ⇒ &nbsp; <u>Lösungsvorschlag 1</u>.
+*This was to be expected, since&nbsp; ${\rm Pr}(Y = +1) ={\rm Pr}(Y = -1)$&nbsp; is given &nbsp;  ⇒ &nbsp; <u>Proposed solution 1</u>.
 {{ML-Fuß}}

← Older revision		Revision as of 15:34, 12 October 2021
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−	[[Category:Theory of Stochastic Signals: Exercises\|^1.1 ~~Einige grundlegende Definitionen~~	+	[[Category:Theory of Stochastic Signals: Exercises\|^1.1 Some Basic Definitions
	^]]		^]]

← Older revision	Revision as of 15:32, 12 October 2021
(No difference)

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 [[File:EN_Sto_Z1_1_c.png|right|frame|400px|Summe und Differenz ternärer Zufallsgrößen]]
-'''(3)'''&nbsp; Da&nbsp; $Y$&nbsp; nicht gleichverteilt ist, kann man hier  (eigentlich) die &bdquo;Klassische Definition der Wahrscheinlichkeit&rdquo; nicht anwenden.
+'''(3)'''&nbsp; Da&nbsp; $Y$&nbsp; nicht gleichverteilt ist, kann man hier  (eigentlich) die "Klassische Definition der Wahrscheinlichkeit&rdquo; nicht anwenden.
 *Teilt man&nbsp; $Y$&nbsp; jedoch gemäß der Grafik in vier Bereiche auf, wobei man zwei der Bereiche dem Ereignis&nbsp; $Y = 0$&nbsp; zuordnet, so kann man trotzdem gemäß der klassischen Definition vorgehen.

← Older revision		Revision as of 12:44, 23 March 2021
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−	[[Category:~~Aufgaben zu Stochastische Signaltheorie~~\|^1.1 Einige grundlegende Definitionen	+	[[Category:Theory of Stochastic Signals: Exercises\|^1.1 Einige grundlegende Definitionen
	^]]		^]]