Aufgaben:Exercise 4.10: Binary and Quaternary - Revision history

Hwang at 12:20, 18 January 2023

2023-01-18T12:20:13Z

Guenter at 17:11, 19 March 2022

2022-03-19T17:11:13Z

Guenter at 17:00, 19 March 2022

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Guenter at 16:57, 19 March 2022

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Noah at 20:52, 28 February 2022

2022-02-28T20:52:30Z

Noah at 20:52, 28 February 2022

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Noah at 20:49, 28 February 2022

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Guenter at 08:56, 28 January 2022

2022-01-28T08:56:47Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.10: Binär und quaternär to Aufgaben:Exercise 4.10: Binary and Quaternary

2022-01-28T08:42:20Z

Guenter moved page Aufgabe 4.10: Binär und quaternär to Exercise 4.10: Binary and Quaternary

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

2021-03-23T12:43:31Z

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

@@ Line 70: / Line 70: @@
 '''(3)'''&nbsp; The&nbsp; ACF $\varphi_b(\tau)$&nbsp; of the binary signal is also identically zero due to the statistically independent symbols in the range&nbsp; $| \tau| > T$.&nbsp;
 *For&nbsp; $-T \le \tau \le +T$&nbsp; itnalso results in a triangular shape.
-*For the quadratic mean,&nbsp; one obtains:
+*For the second moment,&nbsp; one obtains:
 :$$\varphi_b (\tau = 0) = b_{\rm 0}^{\rm 2}.$$

@@ Line 55: / Line 55: @@
 {{ML-Kopf}}
 '''(1)'''&nbsp; The ACF value at the point&nbsp; $\tau = 0$&nbsp; corresponds to the mean signal power, i.e. the variance of&nbsp; $q(t)$.&nbsp; For this holds:
-[[File:P_ID385__Sto_A_4_10_b_neu.png|right|frame|Triangular ACF]]
+[[File:P_ID385__Sto_A_4_10_b_neu.png|right|frame|Triangular auto-correlation function]]
 :$$\varphi_q(\tau = 0)= {1}/{6 } \cdot ({\rm 3\,V})^2 + {2}/{6 } \cdot ({\rm 1\,V})^2 + {2}/{6 } \cdot (-{\rm 1\,V})^2 + {1}/{6 } \cdot (-{\rm 3\,V})^2= \rm {22}/{6 }\, \rm V^2\hspace{0.15cm}\underline{= \rm 3.667 \,V^2}.$$
 '''(2)'''&nbsp; The individual symbols were assumed to be statistically independent.
-*Therefore, and because of the lack of a DC component, for any integer value of&nbsp; $\nu$, the following applies here:
+*Therefore,&nbsp; and because of the lack of a DC component,&nbsp; for any integer value of&nbsp; $\nu$,&nbsp; the following applies here:
 :$${\rm E} \big [ q(t) \cdot q ( t + \nu T) \big ] = {\rm E}  \big [ q(t) \big ] \cdot {\rm E} \big [ q ( t + \nu T) \big ]\hspace{0.15cm}\underline{ = 0}.$$
-*Thus, the ACF we are looking for has the shape sketched on the right.
+*Thus,&nbsp; the ACF we are looking for has the shape sketched on the right.
-*In the range&nbsp; $-T \le \tau \le +T$&nbsp; the ACF is sectionwise linear, i.e. triangular, due to the rectangular pulse shape.
+*In the range&nbsp; $-T \le \tau \le +T$&nbsp; the ACF is sectionwise linear,&nbsp; i.e. triangular,&nbsp; due to the rectangular pulse shape.
 <br clear=all>
-'''(3)'''&nbsp; The&nbsp; ACF $\varphi_b(\tau)$&nbsp; of the binary signal is also identically zero due to the statistically independent symbols in the range&nbsp; $| \tau| > T$&nbsp; and for&nbsp; $-T \le \tau \le +T$&nbsp; also results in a triangular shape.
+'''(3)'''&nbsp; The&nbsp; ACF $\varphi_b(\tau)$&nbsp; of the binary signal is also identically zero due to the statistically independent symbols in the range&nbsp; $| \tau| > T$.&nbsp;
-*For the quadratic mean, one obtains:
+*For&nbsp; $-T \le \tau \le +T$&nbsp; itnalso results in a triangular shape.
 :$$\varphi_b (\tau = 0) = b_{\rm 0}^{\rm 2}.$$
-*With&nbsp; $b_0\hspace{0.15cm}\underline{= 1.915\, \rm V}$&nbsp; the two autocorrelation functions&nbsp; $\varphi_q(\tau)$&nbsp; and&nbsp; $\varphi_b(\tau)$ are identical.
+*With&nbsp; $b_0\hspace{0.15cm}\underline{= 1.915\, \rm V}$&nbsp; the two auto-correlation functions&nbsp; $\varphi_q(\tau)$&nbsp; and&nbsp; $\varphi_b(\tau)$ are identical.
-'''(4)'''&nbsp; Correct are <u>the proposed solutions 1, 3, and 4</u>.
+'''(4)'''&nbsp; Correct are&nbsp; <u>the proposed solutions 1, 3, and 4</u>.
 From the autocorrelation function we can actually determine:
-*the period&nbsp; $T_0$: &nbsp; this is the same for the pattern signals and the ACF;
+*The period&nbsp; $T_0$: &nbsp; this is the same for the pattern signals and the ACF;
-* the linear mean: &nbsp; root of the final value of the ACF for&nbsp; $\tau \to \infty$&nbsp; and
+* the linear mean: &nbsp; root of the final value of the ACF for&nbsp; $\tau \to \infty$;&nbsp;
 * the variance: &nbsp;difference of the ACF values of&nbsp; $\tau = 0$&nbsp; and&nbsp; $\tau \to \infty$.
 Cannot be determined:
-* the probability density function: &nbsp;despite&nbsp; $\varphi_q(\tau) =\varphi_b(\tau)$&nbsp; is&nbsp; $f_q(q) \ne f_b(b)$;
+* The probability density function&nbsp; $\rm (PDF)$: <br> &nbsp; &nbsp; &nbsp; despite&nbsp; $\varphi_q(\tau) =\varphi_b(\tau)$ &nbsp; &rArr; &nbsp; $f_q(q) \ne f_b(b)$;
-* the moments of higher order: &nbsp;for their calculation one needs the PDF;
+* the moments of higher order: <br> &nbsp; &nbsp; &nbsp; for their calculation one needs the PDF;
-* All phase relations and symmetry properties are not recognizable from the ACF.
+* all phase relations and symmetry properties are not recognizable from the ACF.
 {{ML-Fuß}}

@@ Line 54: / Line 54: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The ACF value at the point&nbsp; $\tau = 0$&nbsp; corresponds to the mean signal power, i.e. the root mean square value of&nbsp; $q(t)$. For this holds:
+'''(1)'''&nbsp; The ACF value at the point&nbsp; $\tau = 0$&nbsp; corresponds to the mean signal power, i.e. the variance of&nbsp; $q(t)$.&nbsp; For this holds:
 [[File:P_ID385__Sto_A_4_10_b_neu.png|right|frame|Triangular ACF]]
 :$$\varphi_q(\tau = 0)= {1}/{6 } \cdot ({\rm 3\,V})^2 + {2}/{6 } \cdot ({\rm 1\,V})^2 + {2}/{6 } \cdot (-{\rm 1\,V})^2 + {1}/{6 } \cdot (-{\rm 3\,V})^2= \rm {22}/{6 }\, \rm V^2\hspace{0.15cm}\underline{= \rm 3.667 \,V^2}.$$

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID384__Sto_A_4_10.png|right|300px|frame|Binary signal&nbsp; $b(t)$&nbsp; and quaternary signal&nbsp; $q(t)$]]
+[[File:P_ID384__Sto_A_4_10.png|right|300px|frame|Binary signal&nbsp; $b(t)$,&nbsp; quaternary signal&nbsp; $q(t)$]]
-We consider here a binay signal&nbsp; $b(t)$&nbsp; and a quaternary&auml;rsignal&nbsp; $q(t)$, where:
+We consider here a binay signal&nbsp; $b(t)$&nbsp; and a quaternary signal&nbsp; $q(t)$.
-*The two signals are rectangular in shape, and the duration of each rectangle is&nbsp; $T$&nbsp; (symbol duration).
+*The two signals are rectangular in shape.&nbsp; The duration of each rectangle is&nbsp; $T$&nbsp; (symbol duration).
 *The symbols represented by the pulse heights of the individual rectangular pulses&nbsp; $($with step number&nbsp; $M = 2$ &nbsp;or&nbsp; $M = 4)$&nbsp; are statistically independent.
-*Because of the bipolar signal constellation, both signals are have no DC values if the symbol probabilities are chosen appropriately (symmetrically).
+*Because of the bipolar signal constellation,&nbsp; both signals have no DC component if the symbol probabilities are chosen appropriately&nbsp; (symmetrically).
-*Because of the latter property, it follows for the probabilities of the binary symbols:
+*Because of the latter property,&nbsp; it follows for the probabilities of the binary symbols:
 :$${\rm Pr}\big[b(t) = +b_0\big] = {\rm Pr}\big[b(t) = -b_0\big] ={1}/{2}.$$
-*In contrast, for the quarternary signal:
+*In contrast,&nbsp; for the quarternary signal:
 :$${\rm Pr}\big[q(t) = +3 \hspace{0.05cm}{\rm V}\big] = {\rm Pr}\big[q(t) = -3 \hspace{0.05cm}{\rm V}\big]= {1}/{6},$$
 :$${\rm Pr}\big[q(t) = +1 \hspace{0.05cm}{\rm V}\big] = {\rm Pr}\big[q(t) = -1 \hspace{0.05cm}{\rm V}\big]= {2}/{6}.$$
+'''Hint''':&nbsp; This exercise belongs to the chapter&nbsp; [[Theory_of_Stochastic_Signals/Auto-Correlation_Function|Auto-Correlation Function]].
@@ Line 27: / Line 23: @@
 <quiz display=simple>
-{Calculate the ACF&ndash;value&nbsp; $\varphi_q(\tau = 0)$&nbsp; of the quaternary signal.
+{Calculate the ACF value&nbsp; $\varphi_q(\tau = 0)$&nbsp; of the quaternary signal.
 |type="{}"}
 $\varphi_q(\tau = 0) \ = \ $ { 3.667 3% } $\ \rm V^2$
-{What is the magnitude of the ACF&ndash;value when&nbsp; $\tau = T$&nbsp;? Justify why the ACF&ndash;values for&nbsp; $\tau| > T$&nbsp; are the same size.&nbsp; Sketch the ACF&ndash;progression.
+{What is the magnitude of the ACF value when&nbsp; $\tau = T$&nbsp;? Justify why the ACF values for&nbsp; $|\tau| > T$&nbsp; are of the same size.&nbsp; Sketch the ACF diagram.
 |type="{}"}
 $\varphi_q(\tau = T) \ = \ $ { 0. } $\ \rm V^2$
@@ Line 38: / Line 34: @@
-{With which amplitude values&nbsp; $(\pm b_0)$&nbsp; does the binary signal&nbsp; $b(t)$&nbsp; have exactly the same ASF?
+{With which amplitude values&nbsp; $(\pm b_0)$&nbsp; does the binary signal&nbsp; $b(t)$&nbsp; have exactly the same ACF?
 |type="{}"}
 $b_0\ = \ $ { 1.915 3% } $\ \rm V$
-{Which of the following descriptive quantities of a stochastic process can be determined from the AKF?
+{Which of the following descriptive quantities of a stochastic process can be determined from the ACF?
 |type="[]"}
 + Period duration.

@@ Line 45: / Line 45: @@
 {Which of the following descriptive quantities of a stochastic process can be determined from the AKF?
 |type="[]"}
-+ period duration.
++ Period duration.
 - Probability density function.
 + Linear mean value.
 + Variance.
 - 3rd order moment.
--phase relations.
+- Phase relations.

← 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 08:42, 28 January 2022
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← Older revision		Revision as of 12:43, 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)^]]