Aufgaben:Exercise 4.09Z: Periodic ACF - Revision history

Guenter at 16:33, 19 March 2022

2022-03-19T16:33:03Z

Guenter at 16:20, 19 March 2022

2022-03-19T16:20:51Z

Noah at 20:27, 28 February 2022

2022-02-28T20:27:57Z

Noah at 17:32, 28 February 2022

2022-02-28T17:32:30Z

Noah at 17:31, 28 February 2022

2022-02-28T17:31:03Z

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Guenter at 08:56, 28 January 2022

2022-01-28T08:56:23Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.09Z: Periodische AKF to Aufgaben:Exercise 4.09Z: Periodic ACF

2022-01-28T08:40:52Z

Guenter moved page Aufgabe 4.09Z: Periodische AKF to Exercise 4.09Z: Periodic ACF

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

2021-03-23T12:44:46Z

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

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

2020-07-09T09:48:55Z

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

Guenter at 15:27, 29 November 2019

2019-11-29T15:27:44Z

@@ Line 52: / Line 52: @@
 ===Solution===
 {{ML-Kopf}}
-[[File:P_ID382__Sto_Z_4_9_d.png|right|frame|For ACF&ndash;calculation]]
+[[File:P_ID382__Sto_Z_4_9_d.png|right|frame|For the ACF calculation]]
-'''(1)'''&nbsp; The (normalized) period duration is&nbsp; $T_0/T \hspace{0.15cm}\underline{= 5}.$
+'''(1)'''&nbsp; The&nbsp; (normalized)&nbsp; period duration is&nbsp; $T_0/T \hspace{0.15cm}\underline{= 5}.$
-'''(2)'''&nbsp; Due to periodicity, the averaging &uuml;g over a periodic time&nbsp; $T_0$:
+'''(2)'''&nbsp; Due to periodicity,&nbsp; the averaging over a periodic time&nbsp; $T_0$:
 :$$m_x = \frac{1}{T_0} \cdot \int_0^{T_0} x(t) \hspace{0.1cm}{\rm d} t = \frac{1}{5 T} \cdot (2\hspace{0.05cm}{\rm V} \cdot 2 T - 2\hspace{0.05cm}{\rm V} \cdot 2 T) \hspace{0.15cm}\underline{= \rm 0.4 \,V}.$$
-'''(3)'''&nbsp; In analogy to the last subtask, we obtain for the mean power:
+'''(3)'''&nbsp; In analogy to the last subtask,&nbsp; we obtain for the mean power:
 :$$P_x = \frac{2 T}{5 T} \cdot \big[(\rm 2V)^2 +(- \rm 1V)^2 \big]\hspace{0.15cm}\underline{ = \rm 2 \,V^2}.$$
 '''(4)'''&nbsp; The accompanying graph shows in each case in the range from&nbsp; $0$&nbsp; to&nbsp; $T_0 = 5T$.
-*above the product&nbsp; $x(t) \cdot x(t+T)$,
+:*above the product&nbsp; $x(t) \cdot x(t+T)$,
-*down the product&nbsp; $x(t) \cdot x(t+2T)$.
+:*down the product&nbsp; $x(t) \cdot x(t+2T)$.
-Note that&nbsp; $x(t+T)$&nbsp; means a shift of the signal&nbsp; $x(t)$&nbsp; by&nbsp; $T$&nbsp; to the left.
+*Note that&nbsp; $x(t+T)$&nbsp; means a shift of the signal&nbsp; $x(t)$&nbsp; by&nbsp; $T$&nbsp; to the left.
-From these sketches follow the relations:
+*From these sketches follow the relations:
 :$$\varphi_x (T)= \rm {1}/{5 } \cdot (\rm 4V^2 + \rm 1V^2 - \rm 2V^2) \hspace{0.15cm}\underline{= \rm 0.6\, V^2},$$
 :$$\varphi_x ( 2 T)= \rm {1}/{5 } \cdot(-\rm 2V^2 \cdot 3) \hspace{0.15cm}\underline{= - \rm 1.2 \,V^2}.$$
 [[File:P_ID383__Sto_Z_4_9_e.png|right|frame|Wanted auto-correlation function]]
 :$$\varphi_x ( 0) = \varphi_x (5 T) = \varphi_x (10 T) = \ \text{...} \ = \it P_x = \rm 2 \,V^2,$$
@@ Line 84: / Line 85: @@
 :$$\varphi_x (4 T) = \varphi_x (-4 T) =\varphi_x ( T) = \ \text{...} \ \hspace{0.15cm}\underline{= \rm 0.6 \,V^2}.$$
-*The calculated ACF values can be connected by straight line sections, since integration &uuml;over rectangular functions always yields linear subsections.
+*The calculated ACF values can be connected by straight line sections,&nbsp; since integration over rectangular functions always yields linear subsections.
@@ Line 90: / Line 91: @@
 '''(6)'''&nbsp; The five intervals&nbsp; $(0$ to $T)$,&nbsp; $(T$ to $2T)$, ... ,&nbsp; $(4T$ to $5T)$&nbsp; provide the contributions.
 :$$(+1.3; -0.3; -1.2; -0.3; +1.3) \cdot \rm V^2.$$
-*This gives the expected value (linear mean):
+*This gives the expected value&nbsp; (linear mean):
 :$${\rm E}\big[\varphi_x(\tau)\big] = 1/5 \cdot (1.3-0.3 -1.2 -0.3 +1.3]\hspace{0.15cm}\underline{= \rm 0.16 \,V^2}.$$

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID380__Sto_Z_4_9.png|right|frame|Periodic multistage rectangular signal]]
+[[File:P_ID380__Sto_Z_4_9.png|right|frame|Periodic multilevel rectangular signal]]
-We consider in this exercise a periodic and simultaneously ergodic stochastic process&nbsp; $\{x_i(t)\}$, which is fully characterized by the presented pattern function&nbsp; $x(t)$&nbsp; .
+We consider in this exercise a periodic and simultaneously ergodic stochastic process&nbsp; $\{x_i(t)\}$,&nbsp; which is fully characterized by the presented pattern function&nbsp; $x(t)$.
-Further pattern signals of the random process&nbsp; $\{x_i(t)\}$&nbsp; are obtained by shifting by delays of different sizes&nbsp; $\tau_i$, where&nbsp; $\tau_i$&nbsp; is assumed to be uniformly distributed between&nbsp; $0$&nbsp; and the period&nbsp; $T_0$&nbsp; .
+Further pattern signals of the random process&nbsp; $\{x_i(t)\}$&nbsp; are obtained by shifting by  different delays&nbsp; $\tau_i$,&nbsp; where&nbsp; $\tau_i$&nbsp; is assumed to be uniformly distributed between&nbsp; $0$&nbsp; and the period&nbsp; $T_0$.
+'''Hint:'''&nbsp; The exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Auto-Correlation_Function|Auto-Correlation Function]].
@@ Line 22: / Line 16: @@
 <quiz display=simple>
-{Determine the period duration&nbsp; $T_0$ normalized to the period duration&nbsp; $T$ defined in the sketch.
+{Determine the period duration&nbsp; $T_0$&nbsp; normalized to the period duration&nbsp; $T$&nbsp; defined in the sketch.
 |type="{}"}
 $T_0/T \ = \ $ { 5 3% }
-{What is the size of the DC signal component (linear mean)&nbsp; $m_x$&nbsp; of the described process&nbsp; $\{x_i(t)\}$?
+{What is the size of the DC signal component &nbsp; &rArr; &nbsp; linear mean&nbsp; $m_x$&nbsp; of the described process&nbsp; $\{x_i(t)\}$?
 |type="{}"}
 $m_x \ = \ $ { 0.4 3% } $\ \rm V$
-{What is the process power (related to the resistor&nbsp; $1 \hspace{0.05cm} \rm \Omega$&nbsp;)?
+{What is the process power&nbsp; (related to the resistor&nbsp; $1 \hspace{0.05cm} \rm \Omega$&nbsp;)?
 |type="{}"}
 $P_x \ = \ $ { 2 3% } $\ \rm V^2$
-{Calculate the ASF values f&uuml;r&nbsp; $\tau = T$&nbsp; and&nbsp; $\tau = 2T$.
+{Calculate the ACF values for&nbsp; $\tau = T$&nbsp; and&nbsp; $\tau = 2T$.
 |type="{}"}
 $\varphi_x(\tau = T) \ = \ $ { 0.6 3% } $\ \rm V^2$

← Older revision		Revision as of 20:27, 28 February 2022
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−	'''(3)'''  In analogy to the last ~~sub-task~~, we obtain for the mean power:	+	'''(3)'''  In analogy to the last subtask, we obtain for the mean power:
	:$$P_x = \frac{2 T}{5 T} \cdot \big[(\rm 2V)^2 +(- \rm 1V)^2 \big]\hspace{0.15cm}\underline{ = \rm 2 \,V^2}.$$		:$$P_x = \frac{2 T}{5 T} \cdot \big[(\rm 2V)^2 +(- \rm 1V)^2 \big]\hspace{0.15cm}\underline{ = \rm 2 \,V^2}.$$

@@ Line 84: / Line 84: @@
 '''(5)'''&nbsp; An auto-correlation function is always even: &nbsp; $\varphi_x (-\tau)= \varphi_x (\tau)$.&nbsp;
 *In addition, for periodic processes, the ACF is also periodic with the same periodic time&nbsp; $T_0$&nbsp; as the individual pattern functions. It follows that:
-[[File:P_ID383__Sto_Z_4_9_e.png|right|frame|Sought auto-correlation function]]
+[[File:P_ID383__Sto_Z_4_9_e.png|right|frame|Wanted auto-correlation function]]
 :$$\varphi_x ( 0) = \varphi_x (5 T) = \varphi_x (10 T) = \ \text{...} \ = \it P_x = \rm 2 \,V^2,$$

← Older revision		Revision as of 08:56, 28 January 2022
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−	[[Category:Theory of Stochastic Signals: Exercises\|^4.4 ~~Autokorrelationsfunktion (AKF)~~^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^4.4 Auto-Correlation Function^]]

← Older revision	Revision as of 08:40, 28 January 2022
(No difference)

← Older revision		Revision as of 12:44, 23 March 2021
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−	[[Category:~~Aufgaben zu Stochastische Signaltheorie~~\|^4.4 Autokorrelationsfunktion (AKF)^]]	+	[[Category:Theory of Stochastic Signals: Exercises\|^4.4 Autokorrelationsfunktion (AKF)^]]

← Older revision		Revision as of 09:48, 9 July 2020
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	''Hinweis:''		''Hinweis:''
−	*Die Aufgabe gehört zum Kapitel  [[~~Stochastische_Signaltheorie~~/Autokorrelationsfunktion_(AKF)\|Autokorrelationsfunktion]].	+	*Die Aufgabe gehört zum Kapitel  [[Theory_of_Stochastic_Signals/Autokorrelationsfunktion_(AKF)\|Autokorrelationsfunktion]].

@@ Line 59: / Line 59: @@
 {{ML-Kopf}}
 [[File:P_ID382__Sto_Z_4_9_d.png|right|frame|Zur AKF&ndash;Berechnung]]
-'''(1)'''&nbsp; Die (normierte) Periodendauer betr&auml;gt $T_0/T \hspace{0.15cm}\underline{= 5}.$
+'''(1)'''&nbsp; Die (normierte) Periodendauer betr&auml;gt&nbsp; $T_0/T \hspace{0.15cm}\underline{= 5}.$
-'''(2)'''&nbsp; Aufgrund der Periodizit&auml;t gen&uuml;gt die Mittelung &uuml;ber eine Periodendauer $T_0$:
+'''(2)'''&nbsp; Aufgrund der Periodizit&auml;t gen&uuml;gt die Mittelung &uuml;ber eine Periodendauer&nbsp; $T_0$:
 :$$m_x = \frac{1}{T_0} \cdot \int_0^{T_0} x(t) \hspace{0.1cm}{\rm d}  t  =  \frac{1}{5  T} \cdot  (2\hspace{0.05cm}{\rm V} \cdot 2  T - 2\hspace{0.05cm}{\rm V} \cdot 2 T) \hspace{0.15cm}\underline{= \rm 0.4 \,V}.$$
@@ Line 70: / Line 70: @@
-'''(4)'''&nbsp; Die nebenstehende Grafik zeigt jeweils im Bereich von $0$ bis $T_0  = 5T$
+'''(4)'''&nbsp; Die nebenstehende Grafik zeigt jeweils im Bereich von&nbsp; $0$&nbsp; bis&nbsp; $T_0  = 5T$
-*oben das Produkt $x(t) \cdot x(t+T)$,
+*oben das Produkt&nbsp; $x(t) \cdot x(t+T)$,
-*unten das Produkt $x(t) \cdot x(t+2T)$.
+*unten das Produkt&nbsp; $x(t) \cdot x(t+2T)$.
-Zu beachten ist, dass $x(t+T)$ eine Verschiebung des Signals $x(t)$ um $T$ nach links bedeutet.
+Zu beachten ist, dass&nbsp; $x(t+T)$&nbsp; eine Verschiebung des Signals&nbsp; $x(t)$&nbsp; um&nbsp; $T$&nbsp; nach links bedeutet.
 Aus diesen Skizzen folgen die Beziehungen:
@@ Line 82: / Line 82: @@
-'''(5)'''&nbsp; Eine Autokorrelationsfunktion ist stets gerade: &nbsp; $\varphi_x (-\tau)= \varphi_x (\tau)$. Bei periodischen Prozessen ist die AKF zudem ebenfalls periodisch und zwar mit genau der gleichen Periodendauer $T_0$ wie die einzelnen Musterfunktionen. Daraus folgt:
+'''(5)'''&nbsp; Eine Autokorrelationsfunktion ist stets gerade: &nbsp; $\varphi_x (-\tau)= \varphi_x (\tau)$.&nbsp;
-[[File:P_ID383__Sto_Z_4_9_e.png|right|frame|AKF&ndash;Berechnung von Rechtecksignalen]]
+*Bei periodischen Prozessen ist die AKF zudem ebenfalls periodisch und zwar mit der gleichen Periodendauer&nbsp; $T_0$&nbsp; wie die einzelnen Musterfunktionen. Daraus folgt:
 :$$\varphi_x ( 0) =  \varphi_x (5 T) = \varphi_x (10 T) = \ \text{...} \ = \it P_x = \rm 2 \,V^2,$$
@@ Line 89: / Line 90: @@
 :$$\varphi_x (4 T) = \varphi_x (-4 T) =\varphi_x ( T) = \ \text{...} \  \hspace{0.15cm}\underline{=  \rm 0.6 \,V^2}.$$
-Die berechneten AKF-Werte k&ouml;nnen durch Geradenabschnitte miteinander verbunden werden, da die Integration &uuml;ber Rechteckfunktionen stets lineare Teilabschnitte ergibt.
+*Die berechneten AKF-Werte k&ouml;nnen durch Geradenabschnitte miteinander verbunden werden, da die Integration &uuml;ber Rechteckfunktionen stets lineare Teilabschnitte ergibt.
-'''(6)'''&nbsp; Die fünf Intervalle ($0$ bis $T$), ($T$ bis $2T$), ... , ($4$ bis $5T$) liefern die Beiträge
 :$$(+1.3;   -0.3;    -1.2;   -0.3;   +1.3) \cdot \rm V^2.$$
-Daraus ergibt sich der Erwartungswert (lineare Mittelwert):
+*Daraus ergibt sich der Erwartungswert (lineare Mittelwert):
 :$${\rm E}\big[\varphi_x(\tau)\big] = 1/5 \cdot (1.3-0.3 -1.2 -0.3 +1.3]\hspace{0.15cm}\underline{=  \rm 0.16 \,V^2}.$$
-Dies entspricht dem Quadrat des Mittelwertes $m_x$ &nbsp; &rArr; &nbsp; siehe Teilaufgabe '''(2)'''.
+*Dies entspricht dem Quadrat des Mittelwertes&nbsp; $m_x$ &nbsp; &rArr; &nbsp; siehe Teilaufgabe&nbsp; '''(2)'''.
 {{ML-Fuß}}