Aufgaben:Exercise 4.8: Different Error Probabilities - Revision history

Guenter at 14:42, 15 April 2022

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Guenter at 14:26, 15 April 2022

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Guenter at 13:28, 15 April 2022

2022-04-15T13:28:58Z

Hwang at 16:26, 21 March 2022

2022-03-21T16:26:08Z

Hwang at 16:19, 21 March 2022

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Guenter at 14:02, 16 March 2022

2022-03-16T14:02:33Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.8: Fehlerwahrscheinlichkeiten to Aufgaben:Exercise 4.8: Different Error Probabilities

2022-03-16T13:58:18Z

Guenter moved page Aufgabe 4.8: Fehlerwahrscheinlichkeiten to Exercise 4.8: Different Error Probabilities

Javier: Text replacement - "Category:Aufgaben zu Modulationsverfahren" to "Category:Modulation Methods: Exercises"

2021-03-23T12:48:43Z

Text replacement - "Category:Aufgaben zu Modulationsverfahren" to "Category:Modulation Methods: Exercises"

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

2020-07-09T12:56:25Z

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

Javier: Text replacement - "[[Modulationsverfahren" to "[[Modulation_Methods"

2020-07-09T12:43:00Z

Text replacement - "[[Modulationsverfahren" to "[[Modulation_Methods"

@@ Line 28: / Line 28: @@
 *The derivations can be found in the chapter &nbsp;[[Digital_Signal_Transmission/Lineare_digitale_Modulation_–_Kohärente_Demodulation|Linear Digital Modulation - Coherent Demodulation]]&nbsp; of the book "Digital Signal Transmission".
 *For numerical evaluations,&nbsp; you can use the following upper bound:
-:$$ {\rm Q}_{\rm S} (x) = \frac{\rm 1}{\sqrt{\rm 2\pi} \cdot x}\cdot \rm e^{\it -x^{\rm 2}/\rm 2} \ge {\rm Q} (x)\hspace{0.05cm}.$$
+:$$ {\rm Q}_{\rm S} (x) = \frac{\rm 1}{\sqrt{\rm 2\pi} \cdot x}\cdot \rm e^{\it -x^{\rm 2}/\rm 2} \ge {\rm Q} ({\it x})\hspace{0.05cm}.$$

@@ Line 69: / Line 69: @@
 :$$p_{\rm B} = {\rm Q}\left ( \sqrt{10} \right ) \approx {\rm Q_{\rm S}}\left ( \sqrt{10} \right )= \frac{\rm 1}{\sqrt{\rm 20\pi} }\cdot \rm e^{-5 }\hspace{0.15cm}\underline {= 85 \cdot 10^{-5}}\hspace{0.05cm}.$$
 *The actual value according to the specification sheet is&nbsp; $78.3 · 10^{–5}$.&nbsp;
-*Thus, the given equation&nbsp; ${\rm Q_S}(x)$&nbsp; is actually an upper bound for&nbsp; ${\rm Q}(x)$.
+*Thus,&nbsp; the given equation&nbsp; ${\rm Q_S}(x)$&nbsp; is actually an upper bound for&nbsp; ${\rm Q}(x)$.
 *The relative error of using&nbsp;  ${\rm Q_S}(x)$&nbsp; instead of&nbsp;  ${\rm Q}(x)$&nbsp; in this case is less than&nbsp; $10\%$.
-'''(2)'''&nbsp; For BPSK, the corresponding equation is:
+'''(2)'''&nbsp; For BPSK,&nbsp; the corresponding equation is:
 :$$ p_{\rm B} = {\rm Q}\left ( \sqrt{20} \right ) \approx {\rm Q_{\rm S}}\left ( \sqrt{20} \right )= \frac{\rm 1}{\sqrt{\rm 40\pi} }\cdot \rm e^{-10 }\hspace{0.15cm}\underline {= 0.405 \cdot 10^{-5}}\hspace{0.05cm}.$$
 *Now, by using&nbsp; ${\rm Q_S}(x)$,&nbsp; the relative error is only&nbsp; $5 \%$.
@@ Line 81: / Line 81: @@
-'''(3)'''&nbsp; For BPSK, according to the specification, a (logarithmized) value of&nbsp; $9.6\ \rm  dB$&nbsp; is required for this.
+'''(3)'''&nbsp; For BPSK,&nbsp; according to the specification,&nbsp; a (logarithmized) value of&nbsp; $9.6\ \rm  dB$&nbsp; is required for this.
-*For ASK, the logarithmized value must be increased by about&nbsp; $3\ \rm  dB$&nbsp; &nbsp; ⇒ &nbsp; $10 · \lg E_{\rm B}/N_0 \hspace{0.15cm}\underline {= 12.6 \ \rm dB}$.
+*For ASK,&nbsp; the logarithmized value must be increased by about&nbsp; $3\ \rm  dB$&nbsp; &nbsp; ⇒ &nbsp; $10 · \lg E_{\rm B}/N_0 \hspace{0.15cm}\underline {= 12.6 \ \rm dB}$.
-'''(4)'''&nbsp; According to the given DPSK equation, with&nbsp; $ E_{\rm B}/N_0 = 10 $:
+'''(4)'''&nbsp; According to the given DPSK equation,&nbsp; with&nbsp; $ E_{\rm B}/N_0 = 10 $:
 :$$p_{\rm B} = {\rm 1}/{2 }\cdot \rm e^{-10 }\hspace{0.15cm}\underline {\approx 2.27 \cdot 10^{-5}}\hspace{0.05cm}.$$
-*As can already be seen from the diagram on the specification page, DPSK with differential coherent demodulation lies between binary phase modulation&nbsp; (BPSK)&nbsp; and binary amplitude modulation&nbsp; (ASK) when coherent demodulation is provided for both.
+*As can already be seen from the diagram on the specification page,&nbsp; DPSK with differential coherent demodulation lies between binary phase modulation&nbsp; (BPSK)&nbsp; and binary amplitude modulation&nbsp; (ASK) when coherent demodulation is provided for both.
-'''(5)'''&nbsp; From the inverse function of the given equation, we obtain:
+'''(5)'''&nbsp; From the inverse function of the given equation,&nbsp; we obtain:
 :$$ \frac{E_{\rm B}} {N_{\rm 0}}= {\rm ln}\hspace{0.1cm}\frac{1}{2 p_{\rm B}}= {\rm ln}(50000)\approx 10.82 \hspace{0.3cm}\Rightarrow \hspace{0.3cm} 10 \cdot {\rm lg}\hspace{0.1cm}\frac{E_{\rm B}} {N_{\rm 0}}\hspace{0.15cm}\underline {\approx 10.4\,\,{\rm dB}}\hspace{0.05cm}.$$

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID1703__Mod_A_4_7.png|right|frame|AWGN error probability curves of ASK, BPSK and DPSK]]
+[[File:P_ID1703__Mod_A_4_7.png|right|frame|AWGN error probability curves of <br>ASK, BPSK and DPSK]]
-Here, the bit error probabilities &nbsp;$p_{\rm B}$&nbsp; of the digital modulation methods ASK and BPSK are given without further derivation.&nbsp; For example, with the so-called Q function, one obtains
+Here,&nbsp; the bit error probabilities &nbsp;$p_{\rm B}$&nbsp; of the digital modulation methods ASK and BPSK are given without further derivation.&nbsp; For example,&nbsp; with the so-called Q function,
-:$$ {\rm Q} (x) = \frac{\rm 1}{\sqrt{\rm 2\pi}}\cdot \int_{\it x}^{+\infty}\rm e^{\it -u^{\rm 2}/\rm 2}\,d \it u$$
+:$$ {\rm Q} (x) = \frac{\rm 1}{\sqrt{\rm 2\pi}}\cdot \int_{\it x}^{+\infty}\rm e^{\it -u^{\rm 2}/\rm 2}\,d \it u,$$
-for the AWGN channel&nbsp; –&nbsp; characterized by the quotient&nbsp;  $E_{\rm B}/N_0$&nbsp; –&nbsp; and further optimal conditions&nbsp; (for example coherent demodulation)
+one obtains for the AWGN channel&nbsp; –&nbsp; characterized by the quotient&nbsp;  $E_{\rm B}/N_0$&nbsp; –&nbsp; and further optimal conditions&nbsp; (for example coherent demodulation)
-* for &nbsp;''Amplitude Shift Keying''&nbsp; $\rm (ASK)$:
+* for &nbsp;"Amplitude Shift Keying"&nbsp; $\rm (ASK)$:
 :$$ p_{\rm B} = {\rm Q}\left ( \sqrt{{E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ) \hspace{0.05cm},$$
-* for &nbsp;''Binary Phase Shift Keying''&nbsp; $\rm (BPSK)$:
+* for &nbsp;"Binary Phase Shift Keying"&nbsp; $\rm (BPSK)$:
 :$$ p_{\rm B} = {\rm Q}\left ( \sqrt{{2 \cdot E_{\rm B}}/{N_0 }} \hspace{0.1cm}\right ) \hspace{0.05cm}.$$
+* for &nbsp;"Differential Phase Shift Keying"&nbsp;  $\rm (DPSK)$&nbsp; with differential coherent demodulation is:
 :$$p_{\rm B} ={1}/{2} \cdot {\rm e}^{- E_{\rm B}/{N_0 }}\hspace{0.05cm}.$$
-However, ASK could also be demodulated non-coherently.&nbsp; In this case the following would apply:
+:However,&nbsp; ASK could also be demodulated non-coherently.&nbsp; In this case the following would apply:
 :$$ p_{\rm B} = {1}/{2} \cdot {\rm e}^{- E_{\rm B}/(2{N_0 })}\hspace{0.05cm}.$$
-The first three error probabilities are shown in the diagram.&nbsp; For example, for &nbsp;$10 · \lg E_{\rm B}/N_0 = 10 \ \rm dB$&nbsp;  corresponding to the exact functions, one obtains:
+The first three error probabilities are shown in the diagram.&nbsp; For example,&nbsp; for &nbsp;$10 · \lg E_{\rm B}/N_0 = 10 \ \rm dB$&nbsp;  corresponding to the exact functions,&nbsp; one obtains:
 :$$ p_{\rm B} = 7.83 \cdot 10^{-4}\,\,{\rm (ASK)}\hspace{0.05cm},\hspace{0.3cm} p_{\rm B} = 3.87 \cdot 10^{-6}\,\,{\rm (BPSK)}\hspace{0.05cm},$$
-For BPSK to reach or fall below the bit error probability &nbsp;$p_{\rm B} = 10^{–5}$&nbsp; must be &nbsp;$10 · \lg E_{\rm B}/N_0 \ge 9.6 \ \rm dB$.&nbsp;
+For BPSK to reach or fall below the bit error probability &nbsp;$p_{\rm B} = 10^{–5}$ &nbsp; &rArr; &nbsp; $10 · \lg E_{\rm B}/N_0 \ge 9.6 \ \rm dB$.&nbsp;
@@ Line 27: / Line 23: @@
-''Notes:''
+Notes:
 *The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Linear_Digital_Modulation|Linear Digital Modulation]].
 *Reference is made in particular to the section&nbsp; [[Modulation_Methods/Linear_Digital_Modulation#Error_probabilities_-_a_brief_overview|Error probabilities - a brief overview]].
 *The derivations can be found in the chapter &nbsp;[[Digital_Signal_Transmission/Lineare_digitale_Modulation_–_Kohärente_Demodulation|Linear Digital Modulation - Coherent Demodulation]]&nbsp; of the book "Digital Signal Transmission".
+*For numerical evaluations,&nbsp; you can use the following upper bound:
 :$$ {\rm Q}_{\rm S} (x) = \frac{\rm 1}{\sqrt{\rm 2\pi} \cdot x}\cdot \rm e^{\it -x^{\rm 2}/\rm 2} \ge {\rm Q} (x)\hspace{0.05cm}.$$
@@ Line 40: / Line 35: @@
 <quiz display=simple>
-{Calculate the &nbsp;<u>ASK</u> bit error probability for &nbsp;$10 · \lg E_{\rm B}/N_0 = 10 \ \rm dB$&nbsp; using the upper bound&nbsp; ${\rm Q_S}(x)$.
+{Calculate the &nbsp;'''ASK''' bit error probability for &nbsp;$10 · \lg E_{\rm B}/N_0 = 10 \ \rm dB$&nbsp; using the upper bound&nbsp; ${\rm Q_S}(x)$.
 |type="{}"}
 $p_{\rm B} \ = \ $  { 85 3% } $\ \cdot 10^{-5}$
-{Calculate the &nbsp;<u>BPSK</u> bit error probability for &nbsp;$10 · \lg E_{\rm B}/N_0 = 10 \ \rm dB$&nbsp; using the upper bound&nbsp; ${\rm Q_S}(x)$.
+{Calculate the &nbsp;'''BPSK''' bit error probability for &nbsp;$10 · \lg E_{\rm B}/N_0 = 10 \ \rm dB$&nbsp; using the upper bound&nbsp; ${\rm Q_S}(x)$.
 |type="{}"}
 $p_{\rm B} \ = \ $ { 0.405 3% } $\ \cdot 10^{-5}$
-{Specify the minimum value for &nbsp;$E_{\rm B}/N_0$&nbsp; (in dB) for &nbsp;<u>ASK</u>&nbsp; to achieve the bit error probability &nbsp;$p_{\rm B} = 10^{–5}$.&nbsp;
+{Specify the minimum value for &nbsp;$E_{\rm B}/N_0$&nbsp; (in dB) for &nbsp;'''ASK'''&nbsp; to achieve the bit error probability &nbsp;$p_{\rm B} = 10^{–5}$.&nbsp;
 |type="{}"}
 $10 · \lg E_{\rm B}/N_0 \ = \ $ { 12.6 3% } $\ \rm dB$
-{Calculate the &nbsp;<u>DPSK</u> bit error probability for &nbsp;$10 · \lg E_{\rm B}/N_0 = 10 \ \rm dB$.
+{Calculate the &nbsp;'''DPSK''' bit error probability for &nbsp;$10 · \lg E_{\rm B}/N_0 = 10 \ \rm dB$.
 |type="{}"}
 $p_{\rm B} \ = \ $ { 2.27 3% } $\ \cdot 10^{-5}$
-{For &nbsp;<u>DPSK</u>,&nbsp; specify the minimum value for &nbsp;$E_{\rm B}/N_0$&nbsp; (in dB) to achieve the bit error probability &nbsp;$p_{\rm B} = 10^{–5}$.&nbsp;
+{For &nbsp;'''DPSK''',&nbsp; specify the minimum value for &nbsp;$E_{\rm B}/N_0$&nbsp; (in dB) to achieve the bit error probability &nbsp;$p_{\rm B} = 10^{–5}$.&nbsp;
 |type="{}"}
 $10 · \lg E_{\rm B}/N_0 \ = \ $ { 10.4 3% } $\ \rm dB$
-{On the other hand, what &nbsp;$E_{\rm B}/N_0$&nbsp; (in dB)  is needed for &nbsp;<u>incoherent ASK</u> to achieve &nbsp;$p_{\rm B} = 10^{–5}$?&nbsp;
+{On the other hand,&nbsp; what &nbsp;$E_{\rm B}/N_0$&nbsp; (in dB)&nbsp;  is needed for &nbsp;'''incoherent ASK'''&nbsp; to achieve &nbsp;$p_{\rm B} = 10^{–5}$?&nbsp;
 |type="{}"}
 $10 · \lg E_{\rm B}/N_0 \ = \ $ { 13.4 3% }  $\ \rm dB$

@@ Line 29: / Line 29: @@
 ''Notes:''
 *The exercise belongs to the chapter&nbsp; [[Modulation_Methods/Linear_Digital_Modulation|Linear Digital Modulation]].
-*Reference is made in particular to the page&nbsp; [[Modulation_Methods/Linear_Digital_Modulation#Error_probabilities_-_a_brief_overview|Error probabilities - a brief overview]].
+*Reference is made in particular to the section&nbsp; [[Modulation_Methods/Linear_Digital_Modulation#Error_probabilities_-_a_brief_overview|Error probabilities - a brief overview]].
 *The derivations can be found in the chapter &nbsp;[[Digital_Signal_Transmission/Lineare_digitale_Modulation_–_Kohärente_Demodulation|Linear Digital Modulation - Coherent Demodulation]]&nbsp; of the book "Digital Signal Transmission".

← Older revision		Revision as of 14:02, 16 March 2022
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−	[[Category:Modulation Methods: Exercises\|^4.2 ~~Lineare digitale~~ Modulation^]]	+	[[Category:Modulation Methods: Exercises\|^4.2 Linear Digital Modulation^]]

← Older revision	Revision as of 13:58, 16 March 2022
(No difference)

← Older revision		Revision as of 12:48, 23 March 2021
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−	[[Category:~~Aufgaben zu Modulationsverfahren~~\|^4.2 Lineare digitale Modulation^]]	+	[[Category:Modulation Methods: Exercises\|^4.2 Lineare digitale Modulation^]]

@@ Line 30: / Line 30: @@
 *Die Aufgabe gehört zum  Kapitel&nbsp; [[Modulation_Methods/Lineare_digitale_Modulation|Lineare digitale Modulation]].
 *Bezug genommen wird insbesondere auf die Seite&nbsp; [[Modulation_Methods/Lineare_digitale_Modulation#Fehlerwahrscheinlichkeiten_-_ein_kurzer_.C3.9Cberblick|Fehlerwahrscheinlichkeiten - ein kurzer Überblick]].
-*Die Herleitungen finden Sie im Kapitel &nbsp;[[Digitalsignalübertragung/Lineare_digitale_Modulation_–_Kohärente_Demodulation|Lineare digitale Modulation – Kohärente Demodulation]]&nbsp; des Buches „Digitalsignalübertragung”.
+*Die Herleitungen finden Sie im Kapitel &nbsp;[[Digital_Signal_Transmission/Lineare_digitale_Modulation_–_Kohärente_Demodulation|Lineare digitale Modulation – Kohärente Demodulation]]&nbsp; des Buches „Digitalsignalübertragung”.
 *Für die numerischen Auswertungen können Sie die folgende obere Schranke verwenden: