Aufgaben:Exercise 1.08: Comparison of ASK and BPSK - Revision history

Guenter at 13:12, 6 May 2022

2022-05-06T13:12:45Z

Guenter at 13:04, 6 May 2022

2022-05-06T13:04:33Z

Hwang at 13:22, 29 March 2022

2022-03-29T13:22:59Z

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Guenter at 14:04, 19 March 2022

2022-03-19T14:04:20Z

Guenter: Guenter moved page Aufgaben:Aufgabe 1.08: Vergleich ASK und BPSK to Aufgaben:Exercise 1.08: Comparison of ASK and BPSK

2022-03-19T13:59:46Z

Guenter moved page Aufgabe 1.08: Vergleich ASK und BPSK to Exercise 1.08: Comparison of ASK and BPSK

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

2021-03-23T12:51:03Z

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

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

2020-07-09T12:50:31Z

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

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

2020-06-10T15:55:58Z

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

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

2020-05-26T14:09:46Z

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

Guenter at 10:25, 6 February 2019

2019-02-06T10:25:51Z

@@ Line 62: / Line 62: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; It is already obvious from the equations on the information page that <u>solution 2</u> is correct. The defining equations are:
+'''(1)'''&nbsp; It is already obvious from the equations on the information page that&nbsp; <u>solution 2</u>&nbsp; is correct.&nbsp; The defining equations are:
 :$$\rm Q ({\it x}) = \ \frac{\rm 1}{\sqrt{\rm 2\pi}}\int_{\it
 x}^{+\infty}\rm e^{\it -u^{\rm 2}/\rm 2}\,d \it u
@@ Line 69: / Line 69: @@
 \pi}}\int_{\it x}^{+\infty}\rm e^{\it -u^{\rm 2}}\,d \it u
 \hspace{0.05cm}.$$
-By simple substitutions, the above relationship can be easily proved:
+*By simple substitutions,&nbsp; the above relationship can be easily proved:
 :$${\rm Q} ( x) = 1/2 \cdot {\rm erfc} (x/\sqrt{2}) \hspace{0.05cm}.$$
-'''(2)'''&nbsp; The <u>first two solutions</u> are correct:
+'''(2)'''&nbsp; The&nbsp; <u>first two solutions</u>&nbsp; are correct:
-*The equations are valid only for the AWGN channel and for an optimal binary receiver, for example, according to the matched filter approach.
+*The equations are valid only for the AWGN channel and for an optimal binary receiver,&nbsp; for example,&nbsp; according to the matched filter approach.
-*Intersymbol interfering &ndash; caused by the channel or the receiver filter &ndash; is not covered by this.
+*Intersymbol interfering&nbsp; &ndash; caused by the channel or the receiver filter &ndash; is not covered by this.
-*The exact transmission pulse shaping, on the other hand, does not matter as long as the receiver filter $H_{\rm E}(f)$ is matched to the transmission spectrum. Rather:
+*The exact transmission pulse shaping,&nbsp; on the other hand,&nbsp; does not matter as long as the receiver filter&nbsp; $H_{\rm E}(f)$&nbsp; is matched to the transmission spectrum.&nbsp; Rather:
-*Two different transmission pulse shapers $H_{\rm S}(f)$ lead to exactly the same error probability if they have the same energy per bit.
+*Two different transmission pulse shapers&nbsp; $H_{\rm S}(f)$&nbsp; lead to exactly the same error probability if they have the same energy per bit.
 '''(3)'''&nbsp; The results can be read directly from the table:
-:$$p_{\rm ASK} \hspace{0.1cm}\underline {= 0.343 \cdot 10^{-4}},\hspace{0.3cm}p_{\rm BPSK} \hspace{0.1cm}\underline {= 0.901 \cdot 10^{-8}}.$$
+:$$p_{\rm ASK} \hspace{0.1cm}\underline {= 0.343 \cdot 10^{-4}},$$
 '''(4)'''&nbsp; With &nbsp;$E_{\rm B}/N_{0} = 8\  \Rightarrow \ 10  \cdot \lg \ E_{\rm B}/N_{0} \approx 9 \ \rm dB$,&nbsp; the following error probabilities are obtained:
-:$$p_{\rm ASK} \hspace{0.1cm}\underline {= 0.241 \cdot 10^{-2}},\hspace{0.3cm}p_{\rm BPSK} \hspace{0.1cm}\underline {= 0.336 \cdot 10^{-4}}.$$
+:$$p_{\rm ASK} \hspace{0.1cm}\underline {= 0.241 \cdot 10^{-2}}$$
-'''(5)'''&nbsp; From question '''(3)''', it follows that for binary phase modulation, &nbsp;$10  \cdot \lg \ E_{\rm B}/N_{0} \approx 12 \ \rm dB$&nbsp; must be satisfied for &nbsp;$p_{\rm BPSK} \approx 10^{-8}$&nbsp; to be possible.
+'''(5)'''&nbsp; From question&nbsp; '''(3)''',&nbsp; it follows that for binary phase modulation,&nbsp; &nbsp;$10  \cdot \lg \ E_{\rm B}/N_{0} \approx 12 \ \rm dB$&nbsp; must be satisfied for &nbsp;$p_{\rm BPSK} \approx 10^{-8}$&nbsp; to be possible.
-*However, the given equations also show that the ASK curve is &nbsp;$3 \ \rm dB$ $($exactly $3.01 \ \rm dB)$&nbsp; to the right of the BPSK curve.
+*However,&nbsp; the given equations also show that the ASK curve is &nbsp;$3 \ \rm dB$ $($exactly $3.01 \ \rm dB)$&nbsp; to the right of the BPSK curve.
 *It follows that:
 :$$10 \cdot {\rm lg}\hspace{0.1cm}(E_{\rm B}/N_{\rm 0})_{\rm min}\hspace{0.1cm}\underline {\approx 15\,\,{\rm dB}} \hspace{0.05cm}.$$

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID1680__Dig_A_4_1.png|right|frame|Bit error probabilities <br>of ASK and BPSK]]
+[[File:P_ID1680__Dig_A_4_1.png|right|frame|Bit error probabilities&nbsp; (ASK and BPSK)]]
-The bit error probabilities of &nbsp;''Amplitude Shift Keying''&nbsp; (ASK) and &nbsp;''Binary Shift Keying''&nbsp; (BPSK) modulation modes are often given by the following two equations:
+The bit error probabilities of &nbsp;"Amplitude Shift Keying"&nbsp; $\rm (ASK)$&nbsp; and &nbsp;"Binary Shift Keying"&nbsp; $\rm (BPSK)$&nbsp; modulation are often given by the following two equations:
 :$$p_{\rm ASK}  = \ {\rm Q}\left ( \sqrt{\frac{E_{\rm B}}{N_0 }} \hspace{0.1cm}\right )  = \ {1}/{2}\cdot {\rm erfc}\left ( \sqrt{\frac{E_{\rm B}}{2 \cdot N_0 }} \right ),$$
 :$$ p_{\rm BPSK} = \ {\rm Q}\left ( \sqrt{\frac{2 \cdot E_{\rm B}}{N_0 }} \hspace{0.1cm}\right )  = \ {1}/{2}\cdot {\rm erfc}\left ( \sqrt{\frac{E_{\rm B}}{ N_0 }} \right ).$$
-These two equations are evaluated in the attached table. The following applies:
+These equations are evaluated in the attached table.&nbsp; The following applies:
 *$E_{\rm B}$&nbsp; indicates the average energy per bit.
 *$N_{0}$&nbsp; is the noise power density.
@@ Line 14: / Line 14: @@
-It should be noted that these equations do not apply in general, but only under certain idealized conditions. These conditions are to be worked out in this exercise.
+It should be noted that these equations do not apply in general,&nbsp; but only under certain idealized conditions.&nbsp; These conditions are to be worked out in this exercise.
 Notes:
+*The exercise belongs to the chapter&nbsp;  [[Digital_Signal_Transmission/Linear_Digital_Modulation_-_Coherent_Demodulation|"Linear Digital Modulation - Coherent Demodulation"]].
-''Notes:''
+*You can check the results with the HTML5/JavaScript applet &nbsp;[[Applets:Komplementäre_Gaußsche_Fehlerfunktionen|"Complementary Gaussian Error Functions"]].&nbsp;
@@ Line 32: / Line 29: @@
 <quiz display=simple>
 {What is the relationship between &nbsp;${\rm Q}(x)$&nbsp; and &nbsp;${\rm erfc}(x)$?
-|type="[]"}
+|type="()"}
-- &nbsp;${\rm Q}(x)= 2 \cdot{\rm erfc}(x)$ holds,
+- &nbsp;${\rm Q}(x)= 2 \cdot{\rm erfc}(x)$,
-+ &nbsp;${\rm Q}(x)= 0.5 \cdot{\rm erfc}(x)/\sqrt{2})$ holds,
++ &nbsp;${\rm Q}(x)= 0.5 \cdot{\rm erfc}(x)/\sqrt{2})$,
-- &nbsp;${\rm erfc}(x)= 0.5 \cdot{\rm Q}(x)/\sqrt{2})$ holds.
+- &nbsp;${\rm erfc}(x)= 0.5 \cdot{\rm Q}(x)/\sqrt{2})$.
 {When do the given equations for the error probability apply?
 |type="[]"}
 +  They apply only to the AWGN channel.
-+  They apply only to the matched filter receiver (or variants).
++  They apply only to the matched filter receiver&nbsp; (or variants).
 -  The equations take into account intersymbol interfering.
 -  The equations apply only to rectangular signals.
@@ Line 55: / Line 52: @@
 $ p_{\rm BPSK} \ = \ $ { 0.336 3% } $\ \cdot 10^{-4}$
-{The error probability should not exceed &nbsp;$10^{-8}$. What is the required &nbsp;$10 \cdot \lg \ E_{\rm B}/N_{0}$&nbsp; for ASK?
+{The error probability should not exceed &nbsp;$10^{-8}$.&nbsp; What is the required &nbsp;$10 \cdot \lg \ E_{\rm B}/N_{0}$&nbsp; for ASK?
 |type="{}"}
 $(E_{\rm B}/N_{0})_{\rm min} \ = \ $ { 15 3% } $\ \rm dB $

← 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^]]

← Older revision		Revision as of 12:51, 23 March 2021
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−	[[Category:~~Aufgaben zu Digitalsignalübertragung~~\|^1.5 Lineare digitale Modulation^]]	+	[[Category:Digital Signal Transmission: Exercises\|^1.5 Lineare digitale Modulation^]]

@@ Line 22: / Line 22: @@
 ''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]].
 *Sie können die Ergebnisse mit dem Applet &nbsp;[[Applets:Komplementäre_Gaußsche_Fehlerfunktionen|Komplementäre Gaußsche Fehlerfunktionen]]&nbsp; überprüfen.

← Older revision		Revision as of 15:55, 10 June 2020
Line 24:		Line 24:
	*Die Aufgabe gehört zum Kapitel  [[Digitalsignalübertragung/Lineare_digitale_Modulation_–_Kohärente_Demodulation\|Lineare digitale Modulation – Kohärente Demodulation]].		*Die Aufgabe gehört zum Kapitel  [[Digitalsignalübertragung/Lineare_digitale_Modulation_–_Kohärente_Demodulation\|Lineare digitale Modulation – Kohärente Demodulation]].

−	*Sie können die Ergebnisse mit dem Applet  [[Applets:~~Komplementäre_Gaußsche_Fehlerfunktionen_(neues_Applet)~~\|Komplementäre Gaußsche Fehlerfunktionen]]  überprüfen.	+	*Sie können die Ergebnisse mit dem Applet  [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen\|Komplementäre Gaußsche Fehlerfunktionen]]  überprüfen.

@@ Line 24: / Line 24: @@
 *Die Aufgabe gehört zum  Kapitel&nbsp;  [[Digitalsignalübertragung/Lineare_digitale_Modulation_–_Kohärente_Demodulation|Lineare digitale Modulation – Kohärente Demodulation]].
-*Sie können die Ergebnisse mit dem Applet [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen_(neues_Applet)|Komplementäre Gaußsche Fehlerfunktionen]] überprüfen.
+*Sie können die Ergebnisse mit dem Applet &nbsp;[[Applets:Komplementäre_Gaußsche_Fehlerfunktionen_(neues_Applet)|Komplementäre Gaußsche Fehlerfunktionen]]&nbsp; überprüfen.
@@ Line 37: / Line 37: @@
 -  Es gilt &nbsp;${\rm erfc}(x)= 0.5 \cdot{\rm Q}(x)/\sqrt{2})$.
-{Wann gelten die angegebenen Gleichungen für die Fehlerwahrscheinlichkeits?
+{Wann gelten die angegebenen Gleichungen für die Fehlerwahrscheinlichkeit?
 |type="[]"}
 +  Sie gelten nur für den AWGN–Kanal.
@@ Line 66: / Line 66: @@
 {{ML-Kopf}}
 '''(1)'''&nbsp; Bereits aus den Gleichungen auf der Angabenseite ist ersichtlich, dass der <u>Lösungsvorschlag 2</u> richtig ist. Die Definitionsgleichungen lauten:
-:$$\rm Q (\it x) = \ \frac{\rm 1}{\sqrt{\rm 2\pi}}\int_{\it
+:$$\rm Q ({\it x}) = \ \frac{\rm 1}{\sqrt{\rm 2\pi}}\int_{\it
 x}^{+\infty}\rm e^{\it -u^{\rm 2}/\rm 2}\,d \it u
 \hspace{0.05cm},$$
-:$$\rm erfc (\it x)  = \ \frac{\rm 2}{\sqrt{\rm
+:$$\rm erfc ({\it x})  = \ \frac{\rm 2}{\sqrt{\rm
 \pi}}\int_{\it x}^{+\infty}\rm e^{\it -u^{\rm 2}}\,d \it u
 \hspace{0.05cm}.$$
 Durch einfache Substitutionen kann der oben genannte Zusammenhang einfach nachgewiesen werden:
-:$$\rm Q ( x) = 1/2 \cdot \rm erfc (x/\sqrt{2}) \hspace{0.05cm}.$$
+:$${\rm Q} ( x) = 1/2 \cdot {\rm erfc} (x/\sqrt{2}) \hspace{0.05cm}.$$
 '''(2)'''&nbsp; Richtig sind die <u>beiden ersten Lösungsvorschläge:</u>
@@ Line 82: / Line 83: @@
-'''(3)'''&nbsp; Diese Ergebnisse können direkt aus der Tabelle abgelesen werden:
 :$$p_{\rm ASK} \hspace{0.1cm}\underline {= 0.343 \cdot 10^{-4}},\hspace{0.3cm}p_{\rm BPSK} \hspace{0.1cm}\underline {= 0.901 \cdot 10^{-8}}.$$
 '''(4)'''&nbsp; Mit $E_{\rm B}/N_{0} = 8\  \Rightarrow \ 10  \cdot \lg \ E_{\rm B}/N_{0} \approx 9 \ \rm dB$ erhält man folgende Fehlerwahrscheinlichkeiten:
 :$$p_{\rm ASK} \hspace{0.1cm}\underline {= 0.241 \cdot 10^{-2}},\hspace{0.3cm}p_{\rm BPSK} \hspace{0.1cm}\underline {= 0.336 \cdot 10^{-4}}.$$
-'''(5)'''&nbsp; Aus der Teilaufgabe (3) folgt, dass bei der binären Phasenmodulation $10  \cdot \lg \ E_{\rm B}/N_{0} \approx 12 \ \rm dB$ erfüllt sein muss, damit $p_{\rm BPSK} \approx 10^{-8}$ möglich ist. Die angegebenen Gleichungen zeigen aber auch, dass die ASK–Kurve um $3 \ \rm dB$ (exakt $3.01 \ \rm dB$) rechts von der BPSK–Kurve liegt. Daraus folgt:
 :$$10 \cdot {\rm lg}\hspace{0.1cm}(E_{\rm B}/N_{\rm 0})_{\rm min}\hspace{0.1cm}\underline {\approx 15\,\,{\rm dB}} \hspace{0.05cm}.$$