Aufgaben:Exercise 4.09Z: Periodic ACF - Revision history

Maintenance script: Fix interlanguage link: resolve redirect chain

2026-03-16T15:53:50Z

Fix interlanguage link: resolve redirect chain

← Older revision		Revision as of 17:53, 16 March 2026
Line 101:		Line 101:

	[[Category:Theory of Stochastic Signals: Exercises\|^4.4 Auto-Correlation Function^]]		[[Category:Theory of Stochastic Signals: Exercises\|^4.4 Auto-Correlation Function^]]
	[[de:~~Zusatzaufgaben~~:4.~~09_Periodische_AKF~~]]		[[de:Aufgaben:Aufgabe 4.09Z: Periodische AKF]]

Maintenance script: Add German interlanguage link

2026-03-16T12:29:17Z

Add German interlanguage link

← Older revision		Revision as of 14:29, 16 March 2026
Line 101:		Line 101:

	[[Category:Theory of Stochastic Signals: Exercises\|^4.4 Auto-Correlation Function^]]		[[Category:Theory of Stochastic Signals: Exercises\|^4.4 Auto-Correlation Function^]]
			[[de:Zusatzaufgaben:4.09_Periodische_AKF]]

Guenter at 16:33, 19 March 2022

2022-03-19T16:33:03Z

← Older revision		Revision as of 18:33, 19 March 2022
Line 52:		Line 52:
	===Solution===		===Solution===
	{{ML-Kopf}}		{{ML-Kopf}}
	[[File:P_ID382__Sto_Z_4_9_d.png\|right\|frame\|For ACF~~–~~calculation]]		[[File:P_ID382__Sto_Z_4_9_d.png\|right\|frame\|For the ACF calculation]]
	'''(1)'''  The (normalized) period duration is  $T_0/T \hspace{0.15cm}\underline{= 5}.$		'''(1)'''  The  (normalized)  period duration is  $T_0/T \hspace{0.15cm}\underline{= 5}.$


	'''(2)'''  Due to periodicity, the averaging ~~üg~~ over a periodic time  $T_0$:		'''(2)'''  Due to periodicity,  the averaging over a periodic time  $T_0$:
	:$$m_x = \frac{1}{T_0} \cdot \int_0^{T_0} x(t) \hspace{0.1cm}{\rm d} t = \frac{1}{5 T} \cdot (2\hspace{0.05cm}{\rm V} \cdot 2 T - 2\hspace{0.05cm}{\rm V} \cdot 2 T) \hspace{0.15cm}\underline{= \rm 0.4 \,V}.$$		:$$m_x = \frac{1}{T_0} \cdot \int_0^{T_0} x(t) \hspace{0.1cm}{\rm d} t = \frac{1}{5 T} \cdot (2\hspace{0.05cm}{\rm V} \cdot 2 T - 2\hspace{0.05cm}{\rm V} \cdot 2 T) \hspace{0.15cm}\underline{= \rm 0.4 \,V}.$$


	'''(3)'''  In analogy to the last subtask, we obtain for the mean power:		'''(3)'''  In analogy to the last subtask,  we obtain for the mean power:
	:$$P_x = \frac{2 T}{5 T} \cdot \big[(\rm 2V)^2 +(- \rm 1V)^2 \big]\hspace{0.15cm}\underline{ = \rm 2 \,V^2}.$$		:$$P_x = \frac{2 T}{5 T} \cdot \big[(\rm 2V)^2 +(- \rm 1V)^2 \big]\hspace{0.15cm}\underline{ = \rm 2 \,V^2}.$$


	'''(4)'''  The accompanying graph shows in each case in the range from  $0$  to  $T_0 = 5T$.		'''(4)'''  The accompanying graph shows in each case in the range from  $0$  to  $T_0 = 5T$.
	*above the product  $x(t) \cdot x(t+T)$,		:*above the product  $x(t) \cdot x(t+T)$,
	*down the product  $x(t) \cdot x(t+2T)$.		:*down the product  $x(t) \cdot x(t+2T)$.


	Note that  $x(t+T)$  means a shift of the signal  $x(t)$  by  $T$  to the left.		*Note that  $x(t+T)$  means a shift of the signal  $x(t)$  by  $T$  to the left.

	From these sketches follow the relations:		*From these sketches follow the relations:
	:$$\varphi_x (T)= \rm {1}/{5 } \cdot (\rm 4V^2 + \rm 1V^2 - \rm 2V^2) \hspace{0.15cm}\underline{= \rm 0.6\, V^2},$$		:$$\varphi_x (T)= \rm {1}/{5 } \cdot (\rm 4V^2 + \rm 1V^2 - \rm 2V^2) \hspace{0.15cm}\underline{= \rm 0.6\, V^2},$$
	:$$\varphi_x ( 2 T)= \rm {1}/{5 } \cdot(-\rm 2V^2 \cdot 3) \hspace{0.15cm}\underline{= - \rm 1.2 \,V^2}.$$		:$$\varphi_x ( 2 T)= \rm {1}/{5 } \cdot(-\rm 2V^2 \cdot 3) \hspace{0.15cm}\underline{= - \rm 1.2 \,V^2}.$$


	~~'''(5)'''  An auto-correlation function is always even:   $\varphi_x (-\tau)= \varphi_x (\tau)$. ~~
	*In addition, for periodic processes, the ACF is also periodic with the same periodic time  $T_0$  as the individual pattern functions. It follows that:
	[[File:P_ID383__Sto_Z_4_9_e.png\|right\|frame\|Wanted auto-correlation function]]		[[File:P_ID383__Sto_Z_4_9_e.png\|right\|frame\|Wanted auto-correlation function]]
			'''(5)'''  An auto-correlation function is always even:
			:$$\varphi_x (-\tau)= \varphi_x (\tau).$$
			*In addition,  for periodic processes,  the ACF is also periodic with the same period duration  $T_0$  as the individual pattern functions.  It follows that:

	:$$\varphi_x ( 0) = \varphi_x (5 T) = \varphi_x (10 T) = \ \text{...} \ = \it P_x = \rm 2 \,V^2,$$		:$$\varphi_x ( 0) = \varphi_x (5 T) = \varphi_x (10 T) = \ \text{...} \ = \it P_x = \rm 2 \,V^2,$$
Line 84:		Line 85:
	:$$\varphi_x (4 T) = \varphi_x (-4 T) =\varphi_x ( T) = \ \text{...} \ \hspace{0.15cm}\underline{= \rm 0.6 \,V^2}.$$		:$$\varphi_x (4 T) = \varphi_x (-4 T) =\varphi_x ( T) = \ \text{...} \ \hspace{0.15cm}\underline{= \rm 0.6 \,V^2}.$$

	*The calculated ACF values can be connected by straight line sections, since integration ~~ü~~over rectangular functions always yields linear subsections.		*The calculated ACF values can be connected by straight line sections,  since integration over rectangular functions always yields linear subsections.


Line 90:		Line 91:
	'''(6)'''  The five intervals  $(0$ to $T)$,  $(T$ to $2T)$, ... ,  $(4T$ to $5T)$  provide the contributions.		'''(6)'''  The five intervals  $(0$ to $T)$,  $(T$ to $2T)$, ... ,  $(4T$ to $5T)$  provide the contributions.
	:$$(+1.3; -0.3; -1.2; -0.3; +1.3) \cdot \rm V^2.$$		:$$(+1.3; -0.3; -1.2; -0.3; +1.3) \cdot \rm V^2.$$
	*This gives the expected value (linear mean):		*This gives the expected value  (linear mean):
	:$${\rm E}\big[\varphi_x(\tau)\big] = 1/5 \cdot (1.3-0.3 -1.2 -0.3 +1.3]\hspace{0.15cm}\underline{= \rm 0.16 \,V^2}.$$		:$${\rm E}\big[\varphi_x(\tau)\big] = 1/5 \cdot (1.3-0.3 -1.2 -0.3 +1.3]\hspace{0.15cm}\underline{= \rm 0.16 \,V^2}.$$

Guenter at 16:20, 19 March 2022

2022-03-19T16:20:51Z

← Older revision		Revision as of 18:20, 19 March 2022
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  $\{x_i(t)\}$, which is fully characterized by the presented pattern function  $x(t)$~~ ~~ .		We consider in this exercise a periodic and simultaneously ergodic stochastic process  $\{x_i(t)\}$,  which is fully characterized by the presented pattern function  $x(t)$.

	Further pattern signals of the random process  $\{x_i(t)\}$  are obtained by shifting by delays ~~of different sizes~~  $\tau_i$, where  $\tau_i$  is assumed to be uniformly distributed between  $0$  and the period  $T_0$~~ ~~ .		Further pattern signals of the random process  $\{x_i(t)\}$  are obtained by shifting by different delays  $\tau_i$,  where  $\tau_i$  is assumed to be uniformly distributed between  $0$  and the period  $T_0$.


			'''Hint:'''  The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function\|Auto-Correlation Function]].




	Hint:
	*The exercise belongs to the chapter  [[Theory_of_Stochastic_Signals/Auto-Correlation_Function\|Auto-Correlation Function]].


Line 22:		Line 16:

	<quiz display=simple>		<quiz display=simple>
	{Determine the period duration  $T_0$ normalized to the period duration  $T$ defined in the sketch.		{Determine the period duration  $T_0$  normalized to the period duration  $T$  defined in the sketch.
	\|type="{}"}		\|type="{}"}
	$T_0/T \ = \ $ { 5 3% }		$T_0/T \ = \ $ { 5 3% }


	{What is the size of the DC signal component (linear mean)  $m_x$  of the described process  $\{x_i(t)\}$?		{What is the size of the DC signal component   ⇒   linear mean  $m_x$  of the described process  $\{x_i(t)\}$?
	\|type="{}"}		\|type="{}"}
	$m_x \ = \ $ { 0.4 3% } $\ \rm V$		$m_x \ = \ $ { 0.4 3% } $\ \rm V$


	{What is the process power (related to the resistor  $1 \hspace{0.05cm} \rm \Omega$ )?		{What is the process power  (related to the resistor  $1 \hspace{0.05cm} \rm \Omega$ )?
	\|type="{}"}		\|type="{}"}
	$P_x \ = \ $ { 2 3% } $\ \rm V^2$		$P_x \ = \ $ { 2 3% } $\ \rm V^2$


	{Calculate the ~~ASF~~ values ~~für~~  $\tau = T$  and  $\tau = 2T$.		{Calculate the ACF values for  $\tau = T$  and  $\tau = 2T$.
	\|type="{}"}		\|type="{}"}
	$\varphi_x(\tau = T) \ = \ $ { 0.6 3% } $\ \rm V^2$		$\varphi_x(\tau = T) \ = \ $ { 0.6 3% } $\ \rm V^2$

Noah at 20:27, 28 February 2022

2022-02-28T20:27:57Z

← Older revision		Revision as of 22:27, 28 February 2022
Line 66:		Line 66:


	'''(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:
	:$$P_x = \frac{2 T}{5 T} \cdot \big[(\rm 2V)^2 +(- \rm 1V)^2 \big]\hspace{0.15cm}\underline{ = \rm 2 \,V^2}.$$		:$$P_x = \frac{2 T}{5 T} \cdot \big[(\rm 2V)^2 +(- \rm 1V)^2 \big]\hspace{0.15cm}\underline{ = \rm 2 \,V^2}.$$

Noah at 17:32, 28 February 2022

2022-02-28T17:32:30Z

← Older revision		Revision as of 19:32, 28 February 2022
Line 84:		Line 84:
	'''(5)'''  An auto-correlation function is always even:   $\varphi_x (-\tau)= \varphi_x (\tau)$.		'''(5)'''  An auto-correlation function is always even:   $\varphi_x (-\tau)= \varphi_x (\tau)$.
	*In addition, for periodic processes, the ACF is also periodic with the same periodic time  $T_0$  as the individual pattern functions. It follows that:		*In addition, for periodic processes, the ACF is also periodic with the same periodic time  $T_0$  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]]

	:$$\varphi_x ( 0) = \varphi_x (5 T) = \varphi_x (10 T) = \ \text{...} \ = \it P_x = \rm 2 \,V^2,$$		:$$\varphi_x ( 0) = \varphi_x (5 T) = \varphi_x (10 T) = \ \text{...} \ = \it P_x = \rm 2 \,V^2,$$

Noah at 17:31, 28 February 2022

2022-02-28T17:31:03Z

Show changes

Guenter at 08:56, 28 January 2022

2022-01-28T08:56:23Z

← Older revision		Revision as of 10:56, 28 January 2022
Line 105:		Line 105:


	[[Category:Theory of Stochastic Signals: Exercises\|^4.4 ~~Autokorrelationsfunktion (AKF)~~^]]		[[Category:Theory of Stochastic Signals: Exercises\|^4.4 Auto-Correlation Function^]]

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 Aufgaben:Aufgabe 4.09Z: Periodische AKF to Aufgaben:Exercise 4.09Z: Periodic ACF

← Older revision	Revision as of 10:40, 28 January 2022
(No difference)

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"

← Older revision		Revision as of 14:44, 23 March 2021
Line 105:		Line 105:


	[[Category:~~Aufgaben zu Stochastische Signaltheorie~~\|^4.4 Autokorrelationsfunktion (AKF)^]]		[[Category:Theory of Stochastic Signals: Exercises\|^4.4 Autokorrelationsfunktion (AKF)^]]