Aufgaben:Exercise 4.14Z: 4-QAM and 4-PSK - Revision history

Guenter at 15:05, 1 October 2022

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Hwang at 19:42, 1 September 2022

2022-09-01T19:42:16Z

Guenter at 15:32, 23 August 2022

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Guenter at 15:02, 23 August 2022

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Guenter at 14:28, 23 August 2022

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Guenter at 14:25, 23 August 2022

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Hwang at 19:39, 15 August 2022

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Hwang at 19:17, 15 August 2022

2022-08-15T19:17:27Z

Guenter at 11:59, 13 July 2022

2022-07-13T11:59:07Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.14Z: 4-QAM und 4-PSK to Aufgaben:Exercise 4.14Z: 4-QAM and 4-PSK

2022-07-13T11:52:39Z

Guenter moved page Aufgabe 4.14Z: 4-QAM und 4-PSK to Exercise 4.14Z: 4-QAM and 4-PSK

@@ Line 91: / Line 91: @@
 :$$p_{\rm B} = \frac{p_{\rm S}}{2} \le
   {\rm Q} \left [ \sqrt{ { E_{\rm S}}/{ N_0}} \right ] = {\rm Q} \left [ \sqrt{ { 2E_{\rm B}}/{ N_0}} \right ] \hspace{0.05cm}.$$
-*As derived in the solution to&nbsp; [[Aufgaben:Exercise_4.13:_Four-level_QAM|"Exercise 4.13"]],&nbsp; even exactly '''KORREKTUR: it is even exactly valid''':
+*As derived in the solution to&nbsp; [[Aufgaben:Exercise_4.13:_Four-level_QAM|"Exercise 4.13"]],&nbsp; it is even exactly valid''':
 :$$p_{\rm B} =  {\rm Q} \left [ \sqrt{ { 2E_{\rm B}}/{ N_0}} \right ] \hspace{0.05cm}.$$
 *In this derivation,&nbsp; it was used that the&nbsp; "4&ndash;QAM"&nbsp; can be represented by two orthogonal BPSK modulations&nbsp; (with cosine and minus sinusoidal carriers,&nbsp; respectively).

@@ Line 91: / Line 91: @@
 :$$p_{\rm B} = \frac{p_{\rm S}}{2} \le
   {\rm Q} \left [ \sqrt{ { E_{\rm S}}/{ N_0}} \right ] = {\rm Q} \left [ \sqrt{ { 2E_{\rm B}}/{ N_0}} \right ] \hspace{0.05cm}.$$
-*As derived in the solution to&nbsp; [[Aufgaben:Exercise_4.13:_Four-level_QAM|"Exercise 4.13"]],&nbsp; even exactly:
+*As derived in the solution to&nbsp; [[Aufgaben:Exercise_4.13:_Four-level_QAM|"Exercise 4.13"]],&nbsp; even exactly '''KORREKTUR: it is even exactly valid''':
 :$$p_{\rm B} =  {\rm Q} \left [ \sqrt{ { 2E_{\rm B}}/{ N_0}} \right ] \hspace{0.05cm}.$$
 *In this derivation,&nbsp; it was used that the&nbsp; "4&ndash;QAM"&nbsp; can be represented by two orthogonal BPSK modulations&nbsp; (with cosine and minus sinusoidal carriers,&nbsp; respectively).

@@ Line 60: / Line 60: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; With&nbsp; $M = 4$,&nbsp; the signal space points are&nbsp; $\boldsymbol{s}_i = (s_{\rm I \it i}, s_{\rm Q \it i})$&nbsp; of digital phase modulation&nbsp; ($i = 0, \ \text{...} \ , 3$):
+'''(1)'''&nbsp; With&nbsp; $M = 4$,&nbsp; the signal space points are&nbsp; $\boldsymbol{s}_i = (s_{\rm I \it i}, s_{\rm Q \it i})$&nbsp; of digital phase modulation&nbsp; $(i = 0, \ \text{...} \ , 3)$:
 :$$s_{{\rm I}i} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \cos \left ( { 2\pi i}/{ M} + \phi_{\rm off} \right ) \hspace{0.05cm},$$
 :$$ s_{{\rm Q}i} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \sin \left ( { 2\pi i}/{ M} + \phi_{\rm off} \right ) \hspace{0.05cm}.$$

@@ Line 60: / Line 60: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; With $M = 4$, the signal space points are $\boldsymbol{s}_i = (s_{\rm I \it i}, s_{\rm Q \it i})$ of digital phase modulation ($i = 0, \ \text{...} \ , 3$):
+'''(1)'''&nbsp; With&nbsp; $M = 4$,&nbsp; the signal space points are&nbsp; $\boldsymbol{s}_i = (s_{\rm I \it i}, s_{\rm Q \it i})$&nbsp; of digital phase modulation&nbsp; ($i = 0, \ \text{...} \ , 3$):
 :$$s_{{\rm I}i} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \cos \left ( { 2\pi i}/{ M} + \phi_{\rm off} \right ) \hspace{0.05cm},$$
 :$$ s_{{\rm Q}i} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \sin \left ( { 2\pi i}/{ M} + \phi_{\rm off} \right ) \hspace{0.05cm}.$$
-With $\phi_{\rm off} \ \underline {= \pi/2 \ (45^°)}$, we obtain exactly the signal space points of the 4&ndash;QAM:
+*With&nbsp; $\phi_{\rm off} \ \underline {= \pi/2 \ (45^°)}$,&nbsp; we obtain exactly the signal space points of the 4&ndash;QAM:
 :$$\boldsymbol{ s}_{\rm 0} = (+\sqrt{2}, +\sqrt{2})\hspace{0.05cm},\hspace{0.2cm}\boldsymbol{ s}_{\rm 1} = (-\sqrt{2}, +\sqrt{2})\hspace{0.05cm},\hspace{0.2cm} \boldsymbol{ s}_{\rm 3} = (-\sqrt{2}, -\sqrt{2})\hspace{0.05cm},\hspace{0.2cm}\boldsymbol{ s}_{\rm 4} = (+\sqrt{2}, -\sqrt{2})
   \hspace{0.05cm}.$$
-'''(2)'''&nbsp; <u>Solution 2</u> is correct: For the 4&ndash;PSK, the equation given earlier gives:
+'''(2)'''&nbsp; <u>Solution 2</u>&nbsp; is correct: For the&nbsp; "4&ndash;PSK"&nbsp; holds:
 :$$p_{\rm S} \le  p_{\rm UB} \hspace{-0.15cm} \ = \ \hspace{-0.15cm}2 \cdot {\rm Q} \left [ \sin ({ \pi}/{ M}) \cdot \sqrt{ { 2E_{\rm S}}/{ N_0}} \right ] =  2 \cdot {\rm Q} \left [ { 1}/{ \sqrt{2}} \cdot \sqrt{ { 2E_{\rm S}}/{ N_0}} \right ]=
 \cdot {\rm Q} \left [ \sqrt{ { E_{\rm S}}/{ N_0}} \right ] \hspace{0.05cm}.$$
-'''(3)'''&nbsp; <u>Solution 2</u> is correct:
+'''(3)'''&nbsp; <u>Solution 2</u>&nbsp; is correct:
-*The 4&ndash;QAM is identical with the 4&ndash;PSK (regarding error probability even independent of the phase offset).
+*The&nbsp; "4&ndash;QAM"&nbsp; is identical with the&nbsp; "4&ndash;PSK"&nbsp; (regarding error probability even independent of the phase offset).
-'''(4)'''&nbsp; Here again the <u>second solution</u> is correct:
+*The resulting&nbsp; "Union Bound"&nbsp; is then twice as large as the 4&ndash;PSK bound.
 *In Gray coding, each symbol error results in a bit error if only adjacent regions are considered: &nbsp; $p_{\rm B} \approx p_{\rm S}/2$.
 *Furthermore, $E_{\rm S} = 2 \ E_{\rm B}$ is valid. It follows that
 :$$p_{\rm B} = \frac{p_{\rm S}}{2} \le
   {\rm Q} \left [ \sqrt{ { E_{\rm S}}/{ N_0}} \right ] = {\rm Q} \left [ \sqrt{ { 2E_{\rm B}}/{ N_0}} \right ] \hspace{0.05cm}.$$
-*As derived in the solution to [[Aufgaben:Exercise_4.13:_Four-level_QAM|"Exercise 4.13"]], even exactly:
+*As derived in the solution to&nbsp; [[Aufgaben:Exercise_4.13:_Four-level_QAM|"Exercise 4.13"]],&nbsp; even exactly:
 :$$p_{\rm B} =  {\rm Q} \left [ \sqrt{ { 2E_{\rm B}}/{ N_0}} \right ] \hspace{0.05cm}.$$
-*In this derivation, it was used that the 4&ndash;QAM can be represented by two orthogonal BPSK modulations (with cosine and minus sinusoidal carriers, respectively).
+*In this derivation,&nbsp; it was used that the&nbsp; "4&ndash;QAM"&nbsp; can be represented by two orthogonal BPSK modulations&nbsp; (with cosine and minus sinusoidal carriers,&nbsp; respectively).
-*Thus, the bit error probability of the 4&ndash;QAM and thus also of the 4&ndash;PSK as a function of $E_{\rm B}/N_0$ is the same as for BPSK.
+*Thus,&nbsp; the bit error probability of the&nbsp; "4&ndash;QAM"&nbsp; and thus also of the&nbsp; "4&ndash;PSK" as a function of&nbsp; $E_{\rm B}/N_0$&nbsp; is the same as for BPSK.

@@ Line 6: / Line 6: @@
 :$$ p_{\rm UB}  =  4 \cdot {\rm Q} \left [  \sqrt{ { E_{\rm S}}/{ N_0}} \hspace{0.05cm}\right ]  \ge p_{\rm S}  \hspace{0.05cm}.$$
-In the theory section,&nbsp; one can also find the&nbsp; "Union&ndash;Bound" for&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#M.E2.80.93level_amplitude_shift_keying_.28M.E2.80.93ASK.29| "M&ndash;level phase modulation"]]&nbsp; &nbsp; $\rm (M&ndash;PSK)$,&nbsp;
+In the theory section,&nbsp; one can also find the&nbsp; "Union&ndash;Bound" for&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Multi-level_phase.E2.80.93shift_keying_.28M.E2.80.93PSK.29| "M&ndash;level phase modulation"]]&nbsp; &nbsp; $\rm (M&ndash;PSK)$,&nbsp;
 :$$ p_{\rm UB}  =  2 \cdot {\rm Q} \left [ \sin ({ \pi}/{ M}) \cdot \sqrt{ { 2E_{\rm S}}/{ N_0}} \hspace{0.05cm}\right ]  \ge p_{\rm S} \hspace{0.05cm}.$$

@@ Line 2: / Line 2: @@
 {{quiz-Header|Buchseite=Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation}}
-[[File:P_ID2068__Dig_Z_4_14.png|right|frame|Signal space constellation of 4-QAM and 4-PSK]]
+[[File:P_ID2068__Dig_Z_4_14.png|right|frame|Signal space constellation of the&nbsp; "4-QAM"&nbsp; and&nbsp; "4-PSK"]]
-For&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Quadrature_amplitude_modulation_.28M-QAM.29|"quadrature amplitude modulation"]]&nbsp; ($M$&ndash;QAM), an upper bound&nbsp;("Union&ndash;Bound")&nbsp; on the symbol error probability was given in the theory section for&nbsp; $M &#8805; 16$:&nbsp;
+For&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Quadrature_amplitude_modulation_.28M-QAM.29|"quadrature amplitude modulation"]]&nbsp; $\rm (M&ndash;QAM)$,&nbsp; an upper bound&nbsp; ("Union&ndash;Bound")&nbsp; on the symbol error probability was given in the theory section for&nbsp; $M &#8805; 16$:&nbsp;
 :$$ p_{\rm UB}  =  4 \cdot {\rm Q} \left [  \sqrt{ { E_{\rm S}}/{ N_0}} \hspace{0.05cm}\right ]  \ge p_{\rm S}  \hspace{0.05cm}.$$
-In the theory section, one can also find the "Union&ndash;Bound" for&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#M.E2.80.93level_amplitude_shift_keying_.28M.E2.80.93ASK.29| "<i>M</i>&ndash;level phase modulation"]]&nbsp; (<i>M</i>&ndash;PSK)
+In the theory section,&nbsp; one can also find the&nbsp; "Union&ndash;Bound" for&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#M.E2.80.93level_amplitude_shift_keying_.28M.E2.80.93ASK.29| "M&ndash;level phase modulation"]]&nbsp; &nbsp; $\rm (M&ndash;PSK)$,&nbsp;
 :$$ p_{\rm UB}  =  2 \cdot {\rm Q} \left [ \sin ({ \pi}/{ M}) \cdot \sqrt{ { 2E_{\rm S}}/{ N_0}} \hspace{0.05cm}\right ]  \ge p_{\rm S} \hspace{0.05cm}.$$
-In both methods, each signal space point has exactly the same energy, namely&nbsp; $E_{\rm S}$.
+In both methods,&nbsp;  each signal space point has exactly the same energy,&nbsp; namely&nbsp; $E_{\rm S}$.
-From the graph, one can see that for the special case&nbsp; $M = 4$,&nbsp; the two modulation processes should actually be identical, which is not directly evident from the above equations.
+From the graph,&nbsp; one can see that for the special case&nbsp; $M = 4$,&nbsp; the two modulation processes should actually be identical,&nbsp; which is not directly evident from the above equations.
-The 4&ndash;PSK is shown here with the phase offset&nbsp; $\phi_{\rm off} = 0$.&nbsp; With a general phase offset, on the other hand, the in-phase and quadrature components of the signal space points are generally:&nbsp; $(i = 0, \ ... \ , M = 1)$:
+The 4&ndash;PSK is shown here with the phase offset&nbsp; $\phi_{\rm off} = 0$.&nbsp; With a general phase offset,&nbsp; on the other hand,&nbsp; the in-phase and quadrature components of the signal space points are generally:&nbsp; $(i = 0, \ ... \ , M = 1)$:
 :$$s_{{\rm I}i} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \cos \left ( { 2\pi i}/{ M} + \phi_{\rm off} \right ) \hspace{0.05cm},$$
 :$$ s_{{\rm Q}i} \hspace{-0.1cm} \ = \ \hspace{-0.1cm} \sin \left ( { 2\pi i}/{ M} + \phi_{\rm off} \right ) \hspace{0.05cm}.$$
@@ Line 21: / Line 21: @@
-''Notes:''
+Notes:
 * The exercise belongs to the chapter&nbsp;  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation| "Carrier Frequency Systems with Coherent Demodulation"]].
 * Reference is made in particular to the sections&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Quadrature_amplitude_modulation_.28M-QAM.29|"Quadrature amplitude modulation"]]&nbsp; and&nbsp;  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Multi-level_phase.E2.80.93shift_keying_.28M.E2.80.93PSK.29|"Multi-level phase modulation"]].
 * In the above diagram the Gray mapping of the symbols to bit-duples is shown in red.
-* All results of the exercise can be checked by simulation with the interactive applet [[Applets:MPSK_%26_Union-Bound(Applet)|"M&ndash;level Phase Shift Keying and Union Bound"]].
@@ Line 48: / Line 51: @@
 - $p_{\rm S} &#8804; 2 \cdot {\rm Q}[\sqrt{2E_{\rm S}/N_0}\hspace{0.05cm}]$.
-{What is the bit error probability bound for the 4&ndash;QAM, assuming Gray coding?
+{What is the bit error probability bound for the 4&ndash;QAM,&nbsp; assuming Gray coding?
 |type="[]"}
 - $p_{\rm B} &#8804; 2 \cdot {\rm Q}[\sqrt{2E_{\rm B}/N_0}\hspace{0.05cm}]$,

@@ Line 1: / Line 1: @@
-{{quiz-Header|Buchseite=Digitalsignalübertragung/Trägerfrequenzsysteme mit kohärenter Demodulation}}
+{{quiz-Header|Buchseite=Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation}}
-[[File:P_ID2068__Dig_Z_4_14.png|right|frame|Signalraumkonstellation von 4–QAM und 4-PSK]]
+[[File:P_ID2068__Dig_Z_4_14.png|right|frame|Signal space constellation of 4-QAM and 4-PSK]]
-Für die&nbsp; [[Digitalsignal%C3%BCbertragung/Tr%C3%A4gerfrequenzsysteme_mit_koh%C3%A4renter_Demodulation#Quadraturamplitudenmodulation_.28M.E2.80.93QAM.29| Quadraturamplitudenmodulation]]&nbsp; ($M$&ndash;QAM) wurde im Theorieteil für&nbsp; $M &#8805; 16$&nbsp; eine obere Schranke &nbsp;("Union&ndash;Bound")&nbsp; der Symbolfehlerwahrscheinlichkeit angegeben:
+For&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#Quadrature_amplitude_modulation_.28M-QAM.29|"quadrature amplitude modulation"]]&nbsp; ($M$&ndash;QAM), an upper bound&nbsp;("Union&ndash;Bound")&nbsp; on the symbol error probability was given in the theory section for&nbsp; $M &#8805; 16$:&nbsp;
 :$$ p_{\rm UB}  =  4 \cdot {\rm Q} \left [  \sqrt{ { E_{\rm S}}/{ N_0}} \hspace{0.05cm}\right ]  \ge p_{\rm S}  \hspace{0.05cm}.$$
-Im Theorieteil findet man ebenfalls die "Union&ndash;Bound" für die&nbsp; [[Digitalsignal%C3%BCbertragung/Tr%C3%A4gerfrequenzsysteme_mit_koh%C3%A4renter_Demodulation#Mehrstufiges_Phase.E2.80.93Shift_Keying_.28M.E2.80.93PSK.29| <i>M</i>&ndash;stufige Phasenmodulation]]&nbsp; (<i>M</i>&ndash;PSK)
+In the theory section, one can also find the "Union&ndash;Bound" for&nbsp; [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation#M.E2.80.93level_amplitude_shift_keying_.28M.E2.80.93ASK.29| "<i>M</i>&ndash;level phase modulation"]]&nbsp; (<i>M</i>&ndash;PSK)
 :$$ p_{\rm UB}  =  2 \cdot {\rm Q} \left [ \sin ({ \pi}/{ M}) \cdot \sqrt{ { 2E_{\rm S}}/{ N_0}} \hspace{0.05cm}\right ]  \ge p_{\rm S} \hspace{0.05cm}.$$
@@ Line 22: / Line 22: @@
 ''Hinweise:''
-* Die Aufgabe gehört zum Kapitel&nbsp;  [[Digitalsignal%C3%BCbertragung/Tr%C3%A4gerfrequenzsysteme_mit_koh%C3%A4renter_Demodulation| Trägerfrequenzsysteme mit kohärenter Demodulation]].
+* Die Aufgabe gehört zum Kapitel&nbsp;  [[Digital_Signal_Transmission/Carrier_Frequency_Systems_with_Coherent_Demodulation| "Carrier Frequency Systems with Coherent Demodulation"]].
 * Bezug genommen wird insbesondere auf die Seiten&nbsp; [[Digitalsignal%C3%BCbertragung/Tr%C3%A4gerfrequenzsysteme_mit_koh%C3%A4renter_Demodulation#Quadraturamplitudenmodulation_.28M.E2.80.93QAM.29| Quadraturamplitudenmodulation]]&nbsp; und&nbsp;  [[Digitalsignal%C3%BCbertragung/Tr%C3%A4gerfrequenzsysteme_mit_koh%C3%A4renter_Demodulation#Mehrstufiges_Phase.E2.80.93Shift_Keying_.28M.E2.80.93PSK.29|Mehrstufige Phasenmodulation]].
 * In der obigen Grafik rot eingezeichnet ist die Gray&ndash;Zuordnung der Symbole zu Bitdupeln.

← Older revision		Revision as of 11:59, 13 July 2022
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−	[[Category:Digital Signal Transmission: Exercises\|^4.4 ~~Kohärente~~ Demodulation^]]	+	[[Category:Digital Signal Transmission: Exercises\|^4.4 Coherent Demodulation^]]