Aufgaben:Exercise 3.3Z: Optimization of a Coaxial Cable System - Revision history

Hwang at 13:06, 28 June 2022

2022-06-28T13:06:14Z

Hwang at 08:57, 27 June 2022

2022-06-27T08:57:17Z

Guenter at 12:43, 19 June 2022

2022-06-19T12:43:40Z

Guenter at 11:18, 19 June 2022

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Guenter at 11:17, 19 June 2022

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Guenter at 11:13, 19 June 2022

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Hwang at 15:56, 16 May 2022

2022-05-16T15:56:13Z

Guenter at 15:00, 7 May 2022

2022-05-07T15:00:07Z

Guenter: Guenter moved page Aufgaben:Aufgabe 3.3Z: Optimierung eines Koaxialkabelsystems to Aufgaben:Exercise 3.3Z: Optimization of a Coaxial Cable System

2022-05-07T14:58:20Z

Guenter moved page Aufgabe 3.3Z: Optimierung eines Koaxialkabelsystems to Exercise 3.3Z: Optimization of a Coaxial Cable System

Hwang at 13:26, 2 May 2022

2022-05-02T13:26:07Z

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID1409__Dig_Z_3_3.png|right|frame|Normalized system parameters for different cut-off frequencies]]
+[[File:P_ID1409__Dig_Z_3_3.png|right|frame|Normalized system parameters for different cutoff frequencies]]
 We consider a redundancy-free binary transmission system with the following specifications:
 * The transmission pulses are NRZ rectangular and have energy &nbsp;$E_{\rm B} = s_0^2 \cdot T$.

← Older revision		Revision as of 08:57, 27 June 2022
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	*The exercise belongs to the chapter  [[Digital_Signal_Transmission/Consideration_of_Channel_Distortion_and_Equalization\|"Consideration of Channel Distortion and Equalization"]].		*The exercise belongs to the chapter  [[Digital_Signal_Transmission/Consideration_of_Channel_Distortion_and_Equalization\|"Consideration of Channel Distortion and Equalization"]].

−	* Use the interaction module  [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen\|"Complementary Gaussian Error Functions"]]  for numerical evaluation of the Q-function..	+	* Use the interaction module  [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen\|"Complementary Gaussian Error Functions"]]  for numerical evaluation of the Q-function.

@@ Line 55: / Line 55: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; For the optimization it is sufficient to maximize the quotient $\ddot{o}(T_{\rm D})/\sigma_d$:
+'''(1)'''&nbsp; For the optimization it is sufficient to maximize the quotient&nbsp; $\ddot{o}(T_{\rm D})/\sigma_d$:
-*This is maximized from the values given in the table for the cutoff frequency &nbsp;$f_{\rm G, opt} \cdot T \underline {= 0.4}$&nbsp; with $0.735/0.197 \approx 3.73$.
+*This is maximized from the values given in the table for the cutoff frequency &nbsp;$f_{\rm G, opt} \cdot T \underline {= 0.4}$&nbsp; with&nbsp; $0.735/0.197 \approx 3.73$.
-*As a comparison: &nbsp; For &nbsp;$f_{\rm G} \cdot T = 0.3$&nbsp; the result is $0.192/0.094 \approx 2.04$ due to the smaller eye opening and for &nbsp;$f_{\rm G} \cdot T = 0.5$&nbsp; the quotient is also smaller than for the optimum: $1.159/0.379 \approx 3.05$.
+*As a comparison: &nbsp; For &nbsp;$f_{\rm G} \cdot T = 0.3$&nbsp; the result is&nbsp; $0.192/0.094 \approx 2.04$&nbsp; due to the smaller eye opening.
 *An even larger cutoff frequency leads to a very large noise rms value without simultaneously increasing the vertical eye opening in the same way.
-'''(2)'''&nbsp; Using the result from '''(1)''', we further obtain:
+'''(2)'''&nbsp; Using the result from&nbsp; '''(1)''',&nbsp; we further obtain:
 :$$\rho_{\rm U} = \left ( {3.73}/{2} \right )^2 \approx 3.48 \hspace{0.3cm}\Rightarrow \hspace{0.3cm}
 \cdot {\rm
@@ Line 69: / Line 70: @@
-'''(3)'''&nbsp;
+'''(3)'''&nbsp; With the given&nbsp; $10 \cdot {\rm lg} \, E_{\rm B}/N_0 = 40 \ \rm dB$,&nbsp; i.e. $E_{\rm B}/N_0 = 10^4$,&nbsp; the worst-case signal-to-noise ratio has been found to be&nbsp; $10 \cdot {\rm lg} \, \rho_{\rm U} \approx 5.41 \, {\rm dB}$.
-*With the given $10 \cdot {\rm lg} \, E_{\rm B}/N_0 = 40 \ \rm dB$, i.e. $E_{\rm B}/N_0 = 10^4$, the worst-case signal-to-noise ratio has been found to be $10 \cdot {\rm lg} \, \rho_{\rm U} \approx 5.41 \, {\rm dB}$.
+*However,&nbsp; for the worst-case error probability&nbsp; $p_{\rm U} = 10^{\rm -6}$ &nbsp; &rArr; &nbsp;  $10 \cdot {\rm lg} \, \rho_{\rm U} > 13.55 \, {\rm dB}$&nbsp; must be obtained.
-*However, for the worst-case error probability $p_{\rm U} = 10^{\rm -6}$, $10 \cdot {\rm lg} \, \rho_{\rm U} > 13.55 \, {\rm dB}$ must be obtained.
+*This is achieved by increasing the quotient&nbsp; $E_{\rm B}/N_0$&nbsp; accordingly:
 :$$10 \cdot {\rm
 lg}\hspace{0.1cm}{E_{\rm B}}/{N_0} = 40\,{\rm dB}
@@ Line 84: / Line 84: @@
 '''(4)'''&nbsp;
-*The upper bound for $p_{\rm S}$ is equal to the worst-case error probability $p_{\rm U} = \underline {10^{\rm -6}}$.
+*The upper bound&nbsp; for $p_{\rm S}$&nbsp; is equal to the worst-case error probability&nbsp; $p_{\rm U} = \underline {10^{\rm -6}}$.
 *The lower bound is $\underline {0.25 \cdot 10^{\rm -6}}$, which is smaller by a factor of 4.
 {{ML-Fuß}}
 [[Category:Digital Signal Transmission: Exercises|^3.3 Channel Distortion and Equalization^]]

@@ Line 18: / Line 18: @@
    \right) \hspace{0.3cm}\Rightarrow \hspace{0.3cm} p_{\rm U} = {\rm Q} \left( \sqrt{\rho_{\rm U}}\right)$$
-*This quantity represents an upper bound for the mean error probability &nbsp;$p_{\rm S}$:&nbsp; &nbsp;  $p_{\rm S} \le p_{\rm U}$.
+*This quantity represents an upper bound for the mean error probability &nbsp;  $p_{\rm S} \le p_{\rm U}$.
 *For &nbsp;$f_{\rm G} \cdot T &#8805; 0.4$,&nbsp; a lower bound can also be given: &nbsp;  $p_{\rm S} \ge p_{\rm U}/4$.

@@ Line 18: / Line 18: @@
    \right) \hspace{0.3cm}\Rightarrow \hspace{0.3cm} p_{\rm U} = {\rm Q} \left( \sqrt{\rho_{\rm U}}\right)$$
 *This quantity represents an upper bound for the mean error probability &nbsp;$p_{\rm S}$:&nbsp; &nbsp;  $p_{\rm S} \le p_{\rm U}$.
 *For &nbsp;$f_{\rm G} \cdot T &#8805; 0.4$,&nbsp; a lower bound can also be given: &nbsp;  $p_{\rm S} \ge p_{\rm U}/4$.
+* Use the interaction module &nbsp;[[Applets:Komplementäre_Gaußsche_Fehlerfunktionen|"Complementary Gaussian Error Functions"]]&nbsp; for numerical evaluation of the Q-function..
@@ Line 34: / Line 34: @@
 ===Questions===
 <quiz display=simple>
-{Within the given grid, determine the optimal cutoff frequency with respect to the "worst-case error probability" criterion.
+{Within the given grid,&nbsp; determine the optimal cutoff frequency with respect to the&nbsp; "worst-case error probability"&nbsp; criterion.
 |type="{}"}
 $f_\text{G, opt} \cdot T \  = \ $  { 0.4 3% }
-{What values does this give for the worst-case signal-to-noise ratio and the worst-case error probability?
+{What values does this give for the&nbsp; "worst-case signal-to-noise ratio"&nbsp; and the worst-case error probability?
 |type="{}"}
 $f_\text{G} = \text{G, opt:}\hspace{0.4cm} 10 \cdot {\rm lg} \, \rho_{\rm U}  \  = \ $ { 5.41 3% } ${\ \rm dB}$
 $\hspace{4.07cm}p_{\rm U}  \  = \ $ { 3.1 3% } $\ \rm \%$
-{To what value would we need to reduce the noise power density &nbsp;$N_0$&nbsp; (with respect to signal energy) so that &nbsp;$p_{\rm U}$&nbsp; is no greater than &nbsp;$10^{\rm -6}$?
+{To what value would we need to reduce the noise power density &nbsp;$N_0$&nbsp; (with respect to signal energy)&nbsp; so that &nbsp;$p_{\rm U}$&nbsp; is not greater than &nbsp;$10^{\rm -6}$?
 |type="{}"}
 $N_0/E_{\rm B} \  = \ $ { 1.53 3% } $\ \cdot 10^{\rm -5}$
-{For the assumptions made in '''(3)''', give a lower bound and an upper bound for the average error probability &nbsp;$p_{\rm S}$.&nbsp;
+{For the assumptions made in&nbsp; '''(3)''',&nbsp; give a lower  and an upper bound for the&nbsp; "average error probability" &nbsp; $p_{\rm S}$.&nbsp;
 |type="{}"}
 $p_\text{ S, min}\hspace{0.02cm} \  = \ $ { 0.25 3% } $\ \cdot 10^{\rm -6}$

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID1409__Dig_Z_3_3.png|right|frame|Normalized system sizes for different cut-off frequencies]]
+[[File:P_ID1409__Dig_Z_3_3.png|right|frame|Normalized system parameters for different cut-off frequencies]]
 We consider a redundancy-free binary transmission system with the following specifications:
 * The transmission pulses are NRZ rectangular and have energy &nbsp;$E_{\rm B} = s_0^2 \cdot T$.
 * The channel is a coaxial cable with characteristic cable attenuation &nbsp;$a_* = 40 \, {\rm dB}$.
 * AWGN noise with noise power density &nbsp;$N_0 = 0.0001 \cdot E_{\rm B}$&nbsp; is present.
 * The receiver frequency response &nbsp;$H_{\rm E}(f)$&nbsp; includes an ideal channel equalizer &nbsp;$H_{\rm K}^{\rm -1}(f)$&nbsp; and a Gaussian low-pass filter &nbsp;$H_{\rm G}(f)$&nbsp; with cutoff frequency &nbsp;$f_{\rm G}$&nbsp; for noise power limitation.
-The table shows the eye opening &nbsp;$\ddot{o}(T_{\rm D})$&nbsp; as well as the detection noise rms value &nbsp;$\sigma_{\rm d}$&nbsp; &ndash; each normalized to the transmitted amplitude &nbsp;$s_0$&nbsp; &ndash; for different cutoff frequencies &nbsp;$f_{\rm G}$. The cutoff frequency is to be chosen such that the worst-case error probability is as small as possible, with the following definition:
+The table shows the eye opening &nbsp; $\ddot{o}(T_{\rm D})$ &nbsp; as well as the detection noise rms value &nbsp; $\sigma_{\rm d}$ &nbsp; &ndash; each normalized to the transmitted amplitude &nbsp;$s_0$&nbsp; &ndash; for different cutoff frequencies &nbsp;$f_{\rm G}$.&nbsp; The cutoff frequency is to be chosen such that the worst-case error probability is as small as possible,&nbsp; with the following definition:
 :$$p_{\rm U} = {\rm Q} \left( \frac{\ddot{o}(T_{\rm D})/2}{ \sigma_d}
    \right) \hspace{0.3cm}\Rightarrow \hspace{0.3cm} p_{\rm U} = {\rm Q} \left( \sqrt{\rho_{\rm U}}\right)$$

@@ Line 23: / Line 23: @@
 ''Notes:''
-*The exercise belongs to the chapter&nbsp; [[Digital_Signal_Transmission/Consideration_of_Channel_Distortion_and_Equalization|Consideration of Channel Distortion and Equalization]].
+*The exercise belongs to the chapter&nbsp; [[Digital_Signal_Transmission/Consideration_of_Channel_Distortion_and_Equalization|"Consideration of Channel Distortion and Equalization"]].
-* Use the interaction module &nbsp;[[Applets:Komplementäre_Gaußsche_Fehlerfunktionen|Complementary Gaussian Error Functions]] for numerical evaluation of the Q-function..
+* Use the interaction module &nbsp;[[Applets:Komplementäre_Gaußsche_Fehlerfunktionen|"Complementary Gaussian Error Functions"]] for numerical evaluation of the Q-function..

← Older revision		Revision as of 15:00, 7 May 2022
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−	[[Category:Digital Signal Transmission: Exercises\|^3.3 ~~Kanalverzerrungen und Entzerrung~~^]]	+	[[Category:Digital Signal Transmission: Exercises\|^3.3 Channel Distortion and Equalization^]]

← Older revision	Revision as of 14:58, 7 May 2022
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