Aufgaben:Exercise 1.08Z: BPSK Error Probability - Revision history

Hwang at 15:12, 10 May 2022

2022-05-10T15:12:58Z

Guenter at 15:24, 6 May 2022

2022-05-06T15:24:40Z

Guenter at 14:25, 6 May 2022

2022-05-06T14:25:25Z

Hwang at 13:56, 29 March 2022

2022-03-29T13:56:04Z

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Guenter at 14:04, 19 March 2022

2022-03-19T14:04:52Z

Guenter: Guenter moved page Aufgaben:Aufgabe 1.08Z: BPSK-Fehlerwahrscheinlichkeit to Aufgaben:Exercise 1.08Z: BPSK Error Probability

2022-03-19T14:00:54Z

Guenter moved page Aufgabe 1.08Z: BPSK-Fehlerwahrscheinlichkeit to Exercise 1.08Z: BPSK Error Probability

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

2021-03-23T12:49:26Z

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

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

2020-07-09T12:44:20Z

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

Tasnad: Text replacement - "Komplementäre_Gaußsche_Fehlerfunktionen_(neues_Applet)" to "Komplementäre_Gaußsche_Fehlerfunktionen"

2020-06-10T15:55:25Z

Text replacement - "Komplementäre_Gaußsche_Fehlerfunktionen_(neues_Applet)" to "Komplementäre_Gaußsche_Fehlerfunktionen"

Tasnad: Text replacement - "[File:" to "[File:"

2020-05-26T14:09:44Z

Text replacement - "[File:" to "[File:"

@@ Line 33: / Line 33: @@
 Notes:
-*The exercise belongs to the chapter&nbsp;  [[Digital_Signal_Transmission/Linear_Digital_Modulation_-_Coherent_Demodulation|Linear Digital Modulation - Coherent Demodulation]].
+*The exercise belongs to the chapter&nbsp;  [[Digital_Signal_Transmission/Linear_Digital_Modulation_-_Coherent_Demodulation|"Linear Digital Modulation - Coherent Demodulation"]].
 *Reference is also made to the chapter&nbsp; [[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission|"Error Probability for Baseband Transmission"]].
@@ Line 91: / Line 91: @@
   ) =  {\rm Q}(4)= 0.317 \cdot 10^{-4}.$$
-*A comparison with question&nbsp; '''(4)'''&nbsp; of&nbsp; [[Aufgaben:Exercise_1.08:_Comparison_of_ASK_and_BPSK|Exercise 1.8]]&nbsp; shows that &nbsp;$E_{\rm B}/N_{0} = 8$&nbsp; is not&nbsp; (exactly)&nbsp; equal to &nbsp;$10 \cdot \lg E_{\rm B}/N_{0} = 9 \ \rm dB$.&nbsp;
+*A comparison with question&nbsp; '''(4)'''&nbsp; of&nbsp; [[Aufgaben:Exercise_1.08:_Comparison_of_ASK_and_BPSK|"Exercise 1.8"]]&nbsp; shows that &nbsp;$E_{\rm B}/N_{0} = 8$&nbsp; is not&nbsp; (exactly)&nbsp; equal to &nbsp;$10 \cdot \lg E_{\rm B}/N_{0} = 9 \ \rm dB$.&nbsp;
 *In the first case &nbsp;$p_{\rm BB} = 0.317 \cdot 10^{–4}$ is obtained,&nbsp; in the second &nbsp;$p_{\rm BB} = 0.336 \cdot 10^{-4}$.

@@ Line 77: / Line 77: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; The noise rms value here is given by
+'''(1)'''&nbsp; The noise rms value is given here by
 :$$\sigma_d = \sqrt{\frac{N_0}{2 \cdot T_{\rm B}}}= \sqrt{\frac{2 \cdot 10^{-9}\,{\rm V^2/Hz}}{2 \cdot 1\,{\rm ns}}}= 1\,{\rm V}\hspace{0.3cm}\Rightarrow \hspace{0.3cm}p_{\rm BB} = {\rm Q}\left ({s_0}/{\sigma_d } \right
   )=  {\rm Q}(4)= 0.317 \cdot 10^{-4}\hspace{0.1cm}\underline {= 0.00317 \%}.$$
@@ Line 85: / Line 85: @@
 :$$E_{\rm B} = s_0^2 \cdot T_{\rm B}= (4\,{\rm V})^2 \cdot
 ^{-9}\,{\rm s}\hspace{0.1cm}\underline {= 1.6 \cdot 10^{-8}\,{\rm V^2s}}.$$
-Of course, the additional equation given gives the exact same error probability:
+*Of course, the additional given equation gives the exact same error probability:
 :$$p_{\rm BB} = {\rm Q}\left ( \sqrt{\frac{2 \cdot E_{\rm B}}{N_0 }} \hspace{0.1cm}\right
   ) = {\rm Q}\left ( \sqrt{\frac{2 \cdot 16 \cdot 10^{-9}\,{\rm V^2s}}{2 \cdot 10^{-9}\, {\rm
@@ Line 91: / Line 91: @@
   ) =  {\rm Q}(4)= 0.317 \cdot 10^{-4}.$$
-A comparison with question '''(4)''' of [[Aufgaben:Exercise_1.08:_Comparison_of_ASK_and_BPSK|Exercise 1.8]] shows that &nbsp;$E_{\rm B}/N_{0} = 8$&nbsp; is not (exactly) equal to &nbsp;$10 \cdot \lg E_{\rm B}/N_{0} = 9 \ \rm dB$.&nbsp; In the first case &nbsp;$p_{\rm BB} = 0.317 \cdot 10^{–4}$ is obtained, in the second &nbsp;$p_{\rm BB} = 0.336 \cdot 10^{-4}$.
+*A comparison with question&nbsp; '''(4)'''&nbsp; of&nbsp; [[Aufgaben:Exercise_1.08:_Comparison_of_ASK_and_BPSK|Exercise 1.8]]&nbsp; shows that &nbsp;$E_{\rm B}/N_{0} = 8$&nbsp; is not&nbsp; (exactly)&nbsp; equal to &nbsp;$10 \cdot \lg E_{\rm B}/N_{0} = 9 \ \rm dB$.&nbsp;
-'''(3)'''&nbsp; At half the transmission amplitude $s_{0} = 2 \ \rm V$, the energy per bit decreases to a quarter and the following equations apply:
+'''(3)'''&nbsp; At half the transmission amplitude&nbsp; $s_{0} = 2 \ \rm V$,&nbsp; the energy per bit decreases to a quarter and the following equations apply:
 :$$p_{\rm BB} = {\rm Q}\left ( \frac{s_0}{\sigma_d } \right )= {\rm Q}\left ( \frac{2\,{\rm V}}{1\,{\rm V}} \right )\hspace{0.1cm}\underline {= {\rm Q}(2)= 2.27 \%},$$
 :$$p_{\rm BB} = {\rm Q}\left ( \sqrt{\frac{2 \cdot E_{\rm B}}{N_0 }} \hspace{0.1cm}\right ) = {\rm Q}\left ( \sqrt{\frac{2 \cdot 4 \cdot 10^{-9}\,{\rm V^2s}}{2 \cdot 10^{-9}\, {\rm V^2/Hz} }} \hspace{0.1cm}\right ) = {\rm Q}(2)= 2.27 \%.$$
-'''(4)'''&nbsp; Considering only half the energy $E_{\rm B} = s^{2}_{0} \cdot T_{\rm B}/2$, we obtain with $\sigma^{2}_{d} = N_{0}/T_{\rm B}$ and
+'''(4)'''&nbsp; Considering only half the energy&nbsp; $E_{\rm B} = s^{2}_{0} \cdot T_{\rm B}/2$,&nbsp; we obtain with&nbsp; $\sigma^{2}_{d} = N_{0}/T_{\rm B}$&nbsp; and
 :$$p_{\rm BPSK} = {\rm Q}\left ( {s_0}/{\sigma_d } \right )= {\rm Q}\left ( \sqrt{{s_0^2 \cdot T_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ) = {\rm Q}\left ( \sqrt{{2 \cdot E_{\rm B}}/{N_0 }}\hspace{0.1cm}\right )$$
 the exact same result as for the optimal baseband system &nbsp; &rArr; &nbsp; <u>solution 2</u>.
-'''(5)'''&nbsp; Of course, this also gives the exact same results as for the baseband transmission:
+'''(5)'''&nbsp; Of course,&nbsp; this also gives the exact same results as for the baseband transmission:
 :$${ E_{\rm B}}/{N_0 }= 8{\rm :} \hspace{0.2cm}p_{\rm BPSK} = {\rm Q}(\sqrt{16}) =  {\rm Q}(4)\hspace{0.1cm}\underline {= 0.00317 \%},$$
 :$${ E_{\rm B}}/{N_0 }= 2{\rm :} \hspace{0.2cm}p_{\rm BPSK} = {\rm Q}(\sqrt{4}) =  {\rm Q}(2) \hspace{0.1cm}\underline {= 2.27 \%}.$$

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID1681__Dig_Z_4_1.png|right|frame|Numerical values of the function &nbsp;${\rm Q}(x)$]]
+[[File:P_ID1681__Dig_Z_4_1.png|right|frame|Numerical values of function &nbsp;${\rm Q}(x)$]]
 We assume the optimal baseband transmission system for binary signals with
 *bipolar amplitude coefficients &nbsp;$a_{\nu} \in \{–1, +1\}$,
 *rectangular transmitted signal with signal values &nbsp;$\pm s_{0}$&nbsp; and bit duration &nbsp;$T_{\rm B}$,
 *AWGN noise with noise power density &nbsp;$N_{0}$,
 *receiver filter according to the matched filter principle,
 *decision with the optimal threshold &nbsp;$E = 0$.
-Unless otherwise specified, you should also assume the following numerical values:
+Unless otherwise specified,&nbsp; you should also assume the following numerical values:
 :$$ s_0 = 4\,{\rm V},\hspace{0.2cm} T_{\rm B} = 1\,{\rm ns},\hspace{0.2cm}N_0 = 2 \cdot 10^{-9}\, {\rm V^2/Hz} \hspace{0.05cm}.$$
-The bit error probability of this "baseband system" has already been given in the chapter &nbsp;[[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission|Error Probability for Baseband Transmission]]&nbsp; (Index: &nbsp;BB):
+The bit error probability of this&nbsp; "baseband system"&nbsp; has already been given in the chapter &nbsp;[[Digital_Signal_Transmission/Error_Probability_for_Baseband_Transmission|"Error Probability for Baseband Transmission"]]&nbsp; $($Index: &nbsp;$\rm BB)$:
 :$$p_{\rm BB} = {\rm Q}\left ( \frac{s_0}{\sigma_d } \right )\hspace{0.2cm}{\rm with}\hspace{0.2cm}\sigma_d = \sqrt{\frac{N_0}{2 \cdot T_{\rm B}}}.$$
-Here, &nbsp;$\sigma_{d}$&nbsp; denotes the noise rms value at the decision and &nbsp;${\rm Q}(x)$&nbsp; denotes the complementary Gaussian error function, which is given here in tabular form.
+Here, &nbsp;$\sigma_{d}$&nbsp; denotes the noise rms value at the decision device and &nbsp;${\rm Q}(x)$&nbsp; denotes the complementary Gaussian error function,&nbsp; which is given here in tabular form.&nbsp; This error probability can also be expressed in the form
-This error probability can also be expressed in the form
+The error probability of a comparable transmission system with&nbsp; "Binary Phase Shift Keying" is&nbsp;  $($Index: &nbsp;$\rm BPSK)$:
 :$$p_{\rm BPSK} = {\rm Q}\left ( {s_0}/{\sigma_d } \right )\hspace{0.2cm}{\rm with}\hspace{0.2cm}\sigma_d = \sqrt{{N_0}/{T_{\rm B}}}.$$
@@ Line 30: / Line 32: @@
-''Notes:''
+*Since the signal value &nbsp;$s_{0}$&nbsp; is specified here in&nbsp; "volts"&nbsp; and no specification is made for the reference resistance, &nbsp;$E_{\rm B}$&nbsp; has the unit "$\rm V^{2}/Hz$".
@@ Line 46: / Line 48: @@
 <quiz display=simple>
-{Let &nbsp;$s_{0} = 4 \, \rm V$. What is the error probability &nbsp;$p_{\rm BB}$&nbsp; of the baseband system?
+{Let &nbsp;$s_{0} = 4 \, \rm V$.&nbsp; What is the error probability &nbsp;$p_{\rm BB}$&nbsp; of the baseband system?
 |type="{}"}
 $p_{\rm BB} \ = \ $ { 0.00317 3% } $\ \% $
@@ Line 54: / Line 56: @@
 $E_{\rm B} \ = \ $ { 1.6 3% } $\ \cdot 10^{-8}\  \rm V^{2}s $
-{What is the error probability &nbsp;$p_{\rm BB}$&nbsp; at half the transmission amplitude &nbsp;$(s_{0} = 2 \, \rm V)$?
+{What is the error probability &nbsp;$p_{\rm BB}$&nbsp; at half the transmission amplitude &nbsp; $(s_{0} = 2 \, \rm V)$?
 |type="{}"}
 $p_{\rm BB} \ = \ $ {  2.27 3% } $\ \% $
@@ Line 64: / Line 66: @@
 -$p_{\rm BPSK} =  {\rm Q}\big [(4\cdot E_{\rm B}/N_{0})^{1/2}\big ]$.
-{What are the error probabilities for the BPSK and &nbsp;$E_{\rm B}/N_{0} = 8$&nbsp; and  &nbsp;$E_{\rm B}/N_{0} = 2$?
+{What are the error probabilities for the BPSK and &nbsp;$E_{\rm B}/N_{0} = 8$&nbsp; resp.  &nbsp;$E_{\rm B}/N_{0} = 2$?
 |type="{}"}
 $E_{\rm B}/N_{0} = 8\text{:}\hspace{0.4cm}  p_{\rm BPSK} \ = \ $ { 0.00317 3% } $\ \% $

← Older revision		Revision as of 14:04, 19 March 2022
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−	[[Category:Digital Signal Transmission: Exercises\|^1.5 ~~Lineare digitale~~ Modulation^]]	+	[[Category:Digital Signal Transmission: Exercises\|^1.5 Linear Digital Modulation^]]

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−	[[Category:~~Aufgaben zu Digitalsignalübertragung~~\|^1.5 Lineare digitale Modulation^]]	+	[[Category:Digital Signal Transmission: Exercises\|^1.5 Lineare digitale Modulation^]]

← Older revision	Revision as of 14:00, 19 March 2022
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@@ Line 15: / Line 15: @@
 :$$ s_0 = 4\,{\rm V},\hspace{0.2cm} T_{\rm B} = 1\,{\rm ns},\hspace{0.2cm}N_0 = 2 \cdot 10^{-9}\, {\rm V^2/Hz} \hspace{0.05cm}.$$
-Die Bitfehlerwahrscheinlichkeit dieses „Basisbandsystems” wurde bereits im Kapitel &nbsp;[[Digitalsignalübertragung/Fehlerwahrscheinlichkeit_bei_Basisbandübertragung|Fehlerwahrscheinlichkeit bei Basisbandübertragung]]&nbsp; angegeben (Index: &nbsp;BB):
+Die Bitfehlerwahrscheinlichkeit dieses „Basisbandsystems” wurde bereits im Kapitel &nbsp;[[Digital_Signal_Transmission/Fehlerwahrscheinlichkeit_bei_Basisbandübertragung|Fehlerwahrscheinlichkeit bei Basisbandübertragung]]&nbsp; angegeben (Index: &nbsp;BB):
 :$$p_{\rm BB} = {\rm Q}\left ( \frac{s_0}{\sigma_d } \right )\hspace{0.2cm}{\rm mit}\hspace{0.2cm}\sigma_d = \sqrt{\frac{N_0}{2 \cdot T_{\rm B}}}.$$
@@ Line 33: / Line 33: @@
 ''Hinweise:''
-*Die Aufgabe gehört zum  Kapitel&nbsp;  [[Digitalsignalübertragung/Lineare_digitale_Modulation_–_Kohärente_Demodulation|Lineare digitale Modulation – Kohärente Demodulation]].
+*Die Aufgabe gehört zum  Kapitel&nbsp;  [[Digital_Signal_Transmission/Lineare_digitale_Modulation_–_Kohärente_Demodulation|Lineare digitale Modulation – Kohärente Demodulation]].
-*Bezug genommen wird auch auf das Kapitel&nbsp; [[Digitalsignalübertragung/Fehlerwahrscheinlichkeit_bei_Basisbandübertragung|Fehlerwahrscheinlichkeit bei Basisbandübertragung]].
+*Bezug genommen wird auch auf das Kapitel&nbsp; [[Digital_Signal_Transmission/Fehlerwahrscheinlichkeit_bei_Basisbandübertragung|Fehlerwahrscheinlichkeit bei Basisbandübertragung]].
 *Sie können die Ergebnisse mit dem Applet&nbsp; [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen|Komplementäre Gaußsche Fehlerfunktionen]] überprüfen.

@@ Line 36: / Line 36: @@
 *Bezug genommen wird auch auf das Kapitel&nbsp; [[Digitalsignalübertragung/Fehlerwahrscheinlichkeit_bei_Basisbandübertragung|Fehlerwahrscheinlichkeit bei Basisbandübertragung]].
-*Sie können die Ergebnisse mit dem Applet&nbsp; [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen_(neues_Applet)|Komplementäre Gaußsche Fehlerfunktionen]] überprüfen.
+*Sie können die Ergebnisse mit dem Applet&nbsp; [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen|Komplementäre Gaußsche Fehlerfunktionen]] überprüfen.
 *Da hier der Signalwert &nbsp;$s_{0}$&nbsp; in „Volt” angegeben ist und keine Angabe zum Bezugswiderstand gemacht wird, hat &nbsp;$E_{\rm B}$&nbsp; die Einheit „$\rm V^{2}/Hz$”.

← Older revision	Revision as of 14:09, 26 May 2020
(No difference)