Aufgaben:Exercise 3.7Z: Regenerator Field Length - Revision history

Guenter at 14:51, 28 June 2022

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Guenter at 12:13, 23 June 2022

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Hwang at 08:49, 26 May 2022

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Guenter at 15:16, 7 May 2022

2022-05-07T15:16:04Z

Guenter: Guenter moved page Aufgaben:Aufgabe 3.7Z: Regeneratorfeldlänge to Aufgaben:Exercise 3.7Z: Regenerator Field Length

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Guenter moved page Aufgabe 3.7Z: Regeneratorfeldlänge to Exercise 3.7Z: Regenerator Field Length

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2021-05-28T14:41:23Z

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Javier: Text replacement - "Category:Aufgaben zu Digitalsignalübertragung" to "Category:Digital Signal Transmission: Exercises"

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Text replacement - "Category:Aufgaben zu Digitalsignalübertragung" to "Category:Digital Signal Transmission: Exercises"

Javier: Text replacement - "[[Digitalsignalübertragung" to "[[Digital_Signal_Transmission"

2020-07-09T12:53:33Z

Text replacement - "[[Digitalsignalübertragung" to "[[Digital_Signal_Transmission"

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Digital_Signal_Transmission/Linear_Nyquist_Equalization}}
-[[File:P_ID1438__Dig_Z_3_7.png|right|frame|Results of a system simulation '''Korrektur''': GLP]]
+[[File:EN_Dig_z_3_7.png|right|frame|Results of a system simulation]]
 By simulation,&nbsp; it was shown that there is approximately a linear relationship between the so-called&nbsp; "system efficiency" &nbsp; $\eta$&nbsp; and the&nbsp; "characteristic cable attenuation" &nbsp; $a_*$&nbsp; of a coaxial cable &ndash; both plotted in &nbsp;$\rm dB$&nbsp; &ndash; if the characteristic cable attenuation is sufficiently large &nbsp;$(a_* &#8805; 40 \ \rm dB)$:
 :$$10 \cdot {\rm lg}\hspace{0.1cm}\eta  \hspace{0.15cm} {\rm (in \hspace{0.15cm}dB)}= A + B \cdot a_{\star}

@@ Line 68: / Line 68: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Calculating the system efficiency under the assumption $a_* = 40 \ \rm dB$, we obtain for the four system variants:
+'''(1)'''&nbsp; Calculating the system efficiency under the assumption&nbsp; $a_* = 40 \ \rm dB$,&nbsp; we obtain for the four system variants:
-:$$({\rm GTP},\hspace{0.1cm}M=2) \text{:}\hspace{0.3cm} 10 \cdot {\rm
+:$$({\rm GLP},\hspace{0.1cm}M=2) \text{:}\hspace{0.3cm} 10 \cdot {\rm
 lg}\hspace{0.1cm}\eta
 = +9.4\,{\rm dB} -1.10 \cdot 40\,{\rm dB} = -34.6\,{\rm dB}\hspace{0.05cm},$$
-:$$({\rm GTP},\hspace{0.1cm}M=8) \text{:}\hspace{0.3cm}10 \cdot {\rm
+:$$({\rm GLP},\hspace{0.1cm}M=8) \text{:}\hspace{0.3cm}10 \cdot {\rm
 lg}\hspace{0.1cm}\eta
 = -1.3\,{\rm dB} -0.91 \cdot 40\,{\rm dB} = -37.7\,{\rm dB}\hspace{0.05cm},$$
@@ Line 83: / Line 83: @@
 From this it follows:
-*The <u>first statement</u> is true because the system $({\rm ONE},\hspace{0.1cm} M = 8)$ is already best at $40 \ \rm dB$ cable attenuation and has the most favorable $\rm B$ coefficient.
+*The&nbsp; <u>first statement</u>&nbsp; is true because the system&nbsp; $({\rm ONE},\hspace{0.1cm} M = 8)$&nbsp; is already best at&nbsp; $40 \ \rm dB$&nbsp; cable attenuation and has the most favorable&nbsp; $\rm B$&nbsp; coefficient.
 '''(2)'''&nbsp; As a determination equation, we use here:
 :$$-1.3\,{\rm dB} -0.91 \cdot a_{\star} =  +4.5 \,{\rm dB}-0.96 \cdot a_{\star}\hspace{0.3cm}
 \Rightarrow \hspace{0.3cm} 0.05 \cdot a_{\star} = 5.8\,{\rm dB}
@@ Line 93: / Line 94: @@
 That is:
-*Up to the characteristic cable attenuation $a_* = 116 \ \rm dB$ (note: this is an unrealistically large value for currently realized systems), the binary Nyquist system is superior to the system $({\rm GTP},\hspace{0.1cm} M = 8)$.
+*Up to the characteristic cable attenuation&nbsp; $a_* = 116 \ \rm dB$&nbsp; $($note:&nbsp; this is an unrealistically large value for currently realized systems$)$,&nbsp; the binary Nyquist system is superior to the system $({\rm GLP},\hspace{0.1cm} M = 8)$.
 *Only for larger values than $a_{\rm *, \ limit} = 116 \ \rm dB$ does the advantage of the latter $(M = 8$ and thus significantly lower symbol rate$)$ outweigh the disadvantage $($octal decision and thus greater weight of intersymbol interference$)$.
-'''(3)'''&nbsp; The sink SNR should be at least $16.1 \ \rm dB$, which means that it must be valid:
+'''(3)'''&nbsp; The sink SNR should be at least&nbsp; $16.1 \ \rm dB$,&nbsp; which means that it must be valid:
 :$$10 \cdot {\rm lg}\hspace{0.1cm}\rho = 10 \cdot {\rm lg}
 \hspace{0.1cm}\frac{s_0^2 }{N_0 \cdot R_{\rm B}} + 10 \cdot {\rm
@@ Line 108: / Line 110: @@
   \ 16.1- 100\hspace{0.15cm}\underline {= -83.9\,{\rm dB} = 10 \cdot {\rm
 lg}\hspace{0.1cm}\eta_{\hspace{0.05cm} \rm min}}\hspace{0.05cm}.$$
 '''(4)'''&nbsp; For the system considered here: &nbsp; $10 \cdot {\rm lg}\hspace{0.1cm}\eta = -9.3\,{\rm dB} -0.54 \cdot a_{\star}.$
-*Thus, from the system efficiency condition &nbsp; &rArr; &nbsp; $10 \cdot {\rm lg} \, \eta > \hspace{0.1cm}&ndash;83.9 \ \rm dB $, the characteristic cable attenuation condition is:
+*Thus,&nbsp; from the system efficiency condition &nbsp; &rArr; &nbsp; $10 \cdot {\rm lg} \, \eta > \hspace{0.1cm}&ndash;83.9 \ \rm dB $,&nbsp; the characteristic cable attenuation condition is:
 :$$a_{\star} <  \frac{-83.9\,{\rm dB} + 9.3\,{\rm dB}} {-0.54} \approx
 .1\,{\rm dB}   \hspace{0.05cm}.$$
@@ Line 121: / Line 122: @@
     \hspace{0.05cm}.$$
-:thus the maximum cable length (regenerator field length) can be specified:
+:thus the maximum cable length&nbsp; ("regenerator field length")&nbsp; can be specified:
 :$$l_{\rm max} =  \frac{138.1\,{\rm dB} } {2.36\,{\rm dB}/{\rm
 km} \cdot \sqrt{\rm MHz})\cdot \sqrt{500\,{\rm MHz}}} \hspace{0.15cm}\underline {\approx 2.62\,
@@ Line 127: / Line 128: @@
-'''(5)'''&nbsp; Following the same procedure, but in a more compact notation, results in a smaller regenerator field length for this "worse" system:
+'''(5)'''&nbsp; Following the same procedure,&nbsp; but in a more compact notation,&nbsp; results in a smaller&nbsp; "regenerator field length"&nbsp; for this&nbsp; "worse"&nbsp; system:
 :$$l_{\rm max} =  \frac{-(83.9\,{\rm dB}+A)/B } {2.36\,{\rm dB}/{\rm
 km} \cdot \sqrt{500}} =  \frac{+(83.9+9.4)/1.10 } {2.36\cdot

@@ Line 1: / Line 1: @@
 {{quiz-Header|Buchseite=Digital_Signal_Transmission/Linear_Nyquist_Equalization}}
-[[File:P_ID1438__Dig_Z_3_7.png|right|frame|Results of a system simulation]]
+[[File:P_ID1438__Dig_Z_3_7.png|right|frame|Results of a system simulation '''Korrektur''': GLP]]
-By simulation, it was shown that there is approximately a linear relationship between the so-called system efficiency &nbsp;$\eta$&nbsp; and the characteristic cable attenuation &nbsp;$a_*$&nbsp; of a coaxial cable &ndash; both plotted in &nbsp;$\rm dB$&nbsp; &ndash; if the characteristic cable attenuation is sufficiently large &nbsp;$(a_* &#8805; 40 \ \rm dB)$:
+By simulation,&nbsp; it was shown that there is approximately a linear relationship between the so-called&nbsp; "system efficiency" &nbsp; $\eta$&nbsp; and the&nbsp; "characteristic cable attenuation" &nbsp; $a_*$&nbsp; of a coaxial cable &ndash; both plotted in &nbsp;$\rm dB$&nbsp; &ndash; if the characteristic cable attenuation is sufficiently large &nbsp;$(a_* &#8805; 40 \ \rm dB)$:
 :$$10 \cdot {\rm lg}\hspace{0.1cm}\eta  \hspace{0.15cm} {\rm (in \hspace{0.15cm}dB)}= A + B \cdot a_{\star}
   \hspace{0.05cm}.$$
-In the table, the empirically found coefficients &nbsp;$A$&nbsp; and &nbsp;$B$&nbsp; are given for four exemplary system variants:
+In the table,&nbsp; the empirically found coefficients &nbsp;$A$&nbsp; and &nbsp;$B$&nbsp; are given for four exemplary system variants:
-* for the intersymbol interference binary system &nbsp;$(M = 2)$&nbsp; with Gaussian low-pass filter &nbsp;$\rm (GTP)$, see chapter&nbsp; <br>[[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference|"Error Probability with Intersymbol Interference"]],
+* for the intersymbol interference binary system &nbsp;$(M = 2)$&nbsp; with Gaussian low-pass filter &nbsp;$\rm (GLP)$,&nbsp; see chapter&nbsp; <br>[[Digital_Signal_Transmission/Error_Probability_with_Intersymbol_Interference|"Error Probability with Intersymbol Interference"]],
-* for the intersymbol interference octal system &nbsp;$(M = 8)$&nbsp; with Gaussian low-pass filter &nbsp;$\rm (GTP)$, see chapter&nbsp; <br>[[Digital_Signal_Transmission/Intersymbol_Interference_for_Multi-Level_Transmission|"Intersymbol Interference for Multi-Level Transmission"]],
-*for optimal intersymbol interference free systems &nbsp;$\rm (ONE)$, see chapter&nbsp; [[Digital_Signal_Transmission/Linear_Nyquist_Equalization|"Linear Nyquist Equalization"]]; &nbsp;$M = 2$&nbsp; and&nbsp; $M = 8$.
+* for the intersymbol interference octal system &nbsp;$(M = 8)$&nbsp; with Gaussian low-pass filter &nbsp;$\rm (GLP)$,&nbsp; see chapter&nbsp; <br>[[Digital_Signal_Transmission/Intersymbol_Interference_for_Multi-Level_Transmission|"Intersymbol Interference for Multi-Level Transmission"]],
 The larger the system efficiency &nbsp;$\eta$,&nbsp; the better a system is for a given value &nbsp;$a_*$&nbsp; (and thus a fixed cable length).
+For the calculation of the&nbsp; '''regenerator field length'''&nbsp; (distance between two repeaters),&nbsp; it should be noted:
-For the calculation of the regenerator field length (distance between two repeaters), it should be noted that
+* The worst-case error probability should not be larger than &nbsp;$10^{-10}$, which results in the minimum sink SNR:
 :$$10 \cdot {\rm lg}\hspace{0.1cm}\rho_{\rm min} \approx 16.1\,{\rm
 dB}  \hspace{0.05cm},$$
-* the logarithmized ratio of transmit energy (per bit) and AWGN noise power density is about &nbsp;$100 \ \rm dB$,&nbsp; for example:
+* The logarithmized ratio of transmit energy&nbsp; (per bit)&nbsp; and AWGN noise power density is about &nbsp;$100 \ \rm dB$,&nbsp; for example:
 :$$s_0 = 3\,{\rm V},\hspace{0.2cm}R_{\rm B} = 1\,{\rm
 Gbit/s},\hspace{0.2cm}N_{\rm 0} = 9 \cdot 10^{-19}\,{\rm V^2/Hz}$$
@@ Line 27: / Line 28: @@
 \cdot 10^{-9}\,{\rm 1/s}}
   = 100\,{\rm
-dB}  \hspace{0.05cm},$$
+dB}  \hspace{0.05cm}.$$
-* a standard coaxial cable with dimensions &nbsp;$2.6 \ \rm mm$&nbsp; (inside) and &nbsp;$9.5 \ \rm mm$&nbsp; (outside) is to be used, for which the following relationship is valid:
+* A standard coaxial cable with dimensions &nbsp;$2.6 \ \rm mm$&nbsp; (inside)&nbsp; and &nbsp;$9.5 \ \rm mm$&nbsp; (outside)&nbsp; is to be used,&nbsp; for which the following relationship is valid:
 :$$a_{\star} =  \frac{2.36\,{\rm dB} } {{\rm km} \cdot \sqrt{{\rm
 MHz}}} \cdot l \cdot \sqrt{{R_{\rm B}}/{2}}
     \hspace{0.05cm}.$$
-:Here, &nbsp;$a_*$&nbsp; denotes the characteristic attenuation at half the bit rate &ndash; at &nbsp;$500 \ \rm MHz$&nbsp; in the example &ndash; and &nbsp;$l$&nbsp; denotes the cable length.
+:Here, &nbsp;$a_*$&nbsp; denotes the characteristic attenuation at half the bit rate &ndash; in the example at &nbsp;$500 \ \rm MHz$&nbsp; &ndash; and &nbsp;$l$&nbsp; denotes the cable length.
-''Note:''
+Note:&nbsp;  The exercise belongs to the chapter&nbsp; [[Digital_Signal_Transmission/Linear_Nyquist_Equalization|"Linear Nyquist Equalization"]].
@@ Line 47: / Line 47: @@
 |type="[]"}
 + The system &nbsp;$({\rm ONE}, \ M = 8)$&nbsp; is best for any &nbsp;$a_*$.&nbsp;
-- The system &nbsp;$({\rm GTP}, \ M = 2)$&nbsp; is worst for &nbsp;$a_* &#8805; 40 \ \rm dB$.&nbsp;
+- The system &nbsp;$({\rm GLP}, \ M = 2)$&nbsp; is worst for &nbsp;$a_* &#8805; 40 \ \rm dB$.&nbsp;
-{Starting from which cable attenuation is &nbsp;$({\rm GTP}, \ M = 8)$&nbsp; better than &nbsp;$({\rm ONE}, \ M = 2)$?
+{Starting from which cable attenuation is the system &nbsp;$({\rm GLP}, \ M = 8)$&nbsp; better than the system &nbsp;$({\rm ONE}, \ M = 2)$?
 |type="{}"}
 $a_{\rm *, \ limit}\ = \ $ { 116 3% } $\ \rm dB$
@@ Line 57: / Line 57: @@
 $10 \cdot {\rm lg} \ \eta_{\hspace{0.05cm}\rm min} \ = \ $ { -86.417--81.383 } $\ \rm dB$
-{What is the maximum length of the coaxial cable for &nbsp;$({\rm ONE}, \ M = 8)$?&nbsp;
+{What is the maximum length of the coaxial cable for the system  &nbsp;$({\rm ONE}, \ M = 8)$?&nbsp;
 |type="{}"}
 $l_{\hspace{0.05cm}\rm max}\ = \ $ { 2.62 3% } $\ \rm km$
-{What is the maximum length of the coaxial cable at &nbsp;$({\rm GTP}, \ M = 2)$?&nbsp;
+{What is the maximum length of the coaxial cable for the system &nbsp;$({\rm GTP}, \ M = 2)$?&nbsp;
 |type="{}"}
 $l_{\hspace{0.05cm}\rm max}\ = \ $ { 1.61 3% } $\ \rm km$

← Older revision		Revision as of 15:16, 7 May 2022
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−	[[Category:Digital Signal Transmission: Exercises\|^3.5 ~~Lineare Nyquistentzerrung~~^]]	+	[[Category:Digital Signal Transmission: Exercises\|^3.5 Linear Nyquist Equalization^]]

@@ Line 128: / Line 128: @@
-'''(5)'''&nbsp; Nach gleichem Vorgehen, aber in kompakterer Schreibweise, ergibt sich für dieses &bdquo;schlechtere" System eine kleinere Regeneratorfeldlänge:
+'''(5)'''&nbsp; Nach gleichem Vorgehen, aber in kompakterer Schreibweise, ergibt sich für dieses "schlechtere" System eine kleinere Regeneratorfeldlänge:
 :$$l_{\rm max} =  \frac{-(83.9\,{\rm dB}+A)/B } {2.36\,{\rm dB}/{\rm
 km} \cdot \sqrt{500}} =  \frac{+(83.9+9.4)/1.10 } {2.36\cdot

← Older revision		Revision as of 12:51, 23 March 2021
Line 137:		Line 137:


−	[[Category:~~Aufgaben zu Digitalsignalübertragung~~\|^3.5 Lineare Nyquistentzerrung^]]	+	[[Category:Digital Signal Transmission: Exercises\|^3.5 Lineare Nyquistentzerrung^]]

@@ Line 7: / Line 7: @@
 In der Tabelle sind für vier beispielhafte Systemvarianten die empirisch gefundenen Koeffizienten &nbsp;$A$&nbsp; und &nbsp;$B$&nbsp; angegeben:
-* für  das impulsinterferenzbehaftete Binärsystem &nbsp;$(M = 2)$&nbsp; mit Gaußtiefpass &nbsp;$\rm (GTP)$, siehe Kapitel&nbsp; <br>[[Digitalsignalübertragung/Fehlerwahrscheinlichkeit_unter_Berücksichtigung_von_Impulsinterferenzen|Fehlerwahrscheinlichkeit unter Berücksichtigung von Impulsinterferenzen]],
+* für  das impulsinterferenzbehaftete Binärsystem &nbsp;$(M = 2)$&nbsp; mit Gaußtiefpass &nbsp;$\rm (GTP)$, siehe Kapitel&nbsp; <br>[[Digital_Signal_Transmission/Fehlerwahrscheinlichkeit_unter_Berücksichtigung_von_Impulsinterferenzen|Fehlerwahrscheinlichkeit unter Berücksichtigung von Impulsinterferenzen]],
-* für  das impulsinterferenzbehaftete Oktalsystem &nbsp;$(M = 8)$&nbsp; mit Gaußtiefpass &nbsp;$\rm (GTP)$, siehe Kapitel&nbsp; <br>[[Digitalsignalübertragung/Impulsinterferenzen_bei_mehrstufiger_Übertragung|Impulsinterferenzen bei mehrstufiger Übertragung]],
+* für  das impulsinterferenzbehaftete Oktalsystem &nbsp;$(M = 8)$&nbsp; mit Gaußtiefpass &nbsp;$\rm (GTP)$, siehe Kapitel&nbsp; <br>[[Digital_Signal_Transmission/Impulsinterferenzen_bei_mehrstufiger_Übertragung|Impulsinterferenzen bei mehrstufiger Übertragung]],
-*für  optimale impulsinterferenzfreie Systeme  &nbsp;$\rm (ONE)$, siehe Kapitel&nbsp; [[Digitalsignalübertragung/Lineare_Nyquistentzerrung|Lineare Nyquistentzerrung]]; &nbsp;$M = 2$&nbsp; und&nbsp; $M = 8$.
+*für  optimale impulsinterferenzfreie Systeme  &nbsp;$\rm (ONE)$, siehe Kapitel&nbsp; [[Digital_Signal_Transmission/Lineare_Nyquistentzerrung|Lineare Nyquistentzerrung]]; &nbsp;$M = 2$&nbsp; und&nbsp; $M = 8$.