Aufgaben:Exercise 4.14Z: Offset QPSK vs. MSK - Revision history

Reed at 15:49, 21 March 2022

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Reed at 15:45, 21 March 2022

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Guenter at 13:51, 19 March 2022

2022-03-19T13:51:33Z

Guenter: Guenter moved page Aufgaben:Aufgabe 4.14Z: Offset–QPSK vs. MSK to Aufgaben:Exercise 4.14Z: Offset QPSK vs. MSK

2022-03-19T13:45:36Z

Guenter moved page Aufgabe 4.14Z: Offset–QPSK vs. MSK to Exercise 4.14Z: Offset QPSK vs. MSK

Javier: Text replacement - "„" to """

2021-05-28T14:39:15Z

Text replacement - "„" to """

← Older revision		Revision as of 15:49, 21 March 2022
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	*The associated phase function  $ϕ(t)$  is determined in  [[Aufgaben:Exercise_4.14:_Phase_Progression_of_the_MSK \|Exercise 4.14]] , and is also based on the  (normalized)  MSK fundamental pulse:		*The associated phase function  $ϕ(t)$  is determined in  [[Aufgaben:Exercise_4.14:_Phase_Progression_of_the_MSK \|Exercise 4.14]] , and is also based on the  (normalized)  MSK fundamental pulse:
−	:$$g_{\rm MSK}(t) = \left\{ \begin{array}{l} \cos (\pi/2 \cdot t/T ) \\ 0 \\ \end{array} \right.\quad \begin{array}{{5}c}{\rm{for}} \\{\rm{ }} \\ \end{array}\begin{array}{{10}c} -T \le t \le +T \hspace{0.05cm}, \\ {\rm ~~otherwise~~}. \\ \end{array}$$	+	:$$g_{\rm MSK}(t) = \left\{ \begin{array}{l} \cos (\pi/2 \cdot t/T ) \\ 0 \\ \end{array} \right.\quad \begin{array}{{5}c}{\rm{for}} \\{\rm{otherwise}} \\ \end{array}\begin{array}{{10}c} -T \le t \le +T \hspace{0.05cm}, \\ {\rm }. \\ \end{array}$$

← Older revision		Revision as of 15:49, 21 March 2022
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	*The associated phase function  $ϕ(t)$  is determined in  [[Aufgaben:Exercise_4.14:_Phase_Progression_of_the_MSK \|Exercise 4.14]] , and is also based on the  (normalized)  MSK fundamental pulse:		*The associated phase function  $ϕ(t)$  is determined in  [[Aufgaben:Exercise_4.14:_Phase_Progression_of_the_MSK \|Exercise 4.14]] , and is also based on the  (normalized)  MSK fundamental pulse:
−	:$$g_{\rm MSK}(t) = \left\{ \begin{array}{l} \cos (\pi/2 \cdot t/T ) \\ 0 \\ \end{array} \right.\quad \begin{array}{{5}c}{\rm{for}} \\{\rm{~~for~~}} \\ \end{array}\begin{array}{{10}c} -T \le t \le +T \hspace{0.05cm}, \\ {\rm otherwise}. \\ \end{array}$$	+	:$$g_{\rm MSK}(t) = \left\{ \begin{array}{l} \cos (\pi/2 \cdot t/T ) \\ 0 \\ \end{array} \right.\quad \begin{array}{{5}c}{\rm{for}} \\{\rm{ }} \\ \end{array}\begin{array}{{10}c} -T \le t \le +T \hspace{0.05cm}, \\ {\rm otherwise}. \\ \end{array}$$

@@ Line 38: / Line 38: @@
 *The associated phase function &nbsp;$ϕ(t)$&nbsp; is determined in &nbsp;[[Aufgaben:Exercise_4.14:_Phase_Progression_of_the_MSK |Exercise 4.14]]&nbsp;, and is also based on the&nbsp; (normalized)&nbsp; MSK fundamental pulse:
-:$$g_{\rm MSK}(t) = \left\{ \begin{array}{l} \cos (\pi/2 \cdot t/T ) \\ 0 \\ \end{array} \right.\quad \begin{array}{*{5}c}{\rm{for}} \\{\rm{for}} \\ \end{array}\begin{array}{*{10}c} -T \le t \le +T \hspace{0.05cm}, \\ {\rm sonst}. \\ \end{array}$$
+:$$g_{\rm MSK}(t) = \left\{ \begin{array}{l} \cos (\pi/2 \cdot t/T ) \\ 0 \\ \end{array} \right.\quad \begin{array}{*{5}c}{\rm{for}} \\{\rm{for}} \\ \end{array}\begin{array}{*{10}c} -T \le t \le +T \hspace{0.05cm}, \\ {\rm otherwise}. \\ \end{array}$$
@@ Line 76: / Line 76: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Aus der oberen Skizze kann man&nbsp; $T_{\rm B} \hspace{0.15cm}\underline{ = 1 \ \rm &micro; s}$&nbsp; ablesen.
+'''(1)'''&nbsp; It can be seen from the upper plot that &nbsp; $T_{\rm B} \hspace{0.15cm}\underline{ = 1 \ \rm &micro; s}$&nbsp;.
-'''(2)'''&nbsp; Bei QPSK bzw. Offset–QPSK ist aufgrund der Seriell–Parallel–Wandlung die Symboldauer&nbsp; $T$&nbsp; doppelt so groß wie die Bitdauer&nbsp;  $T_{\rm B}$:
+'''(2)'''&nbsp; For QPSK or offset QPSK , the symbol duration $T$&nbsp; is twice the bit duration&nbsp;  $T_{\rm B}$ due to serial-to-parallel conversion:
 :$$ T = 2 \cdot T_{\rm B} \hspace{0.15cm}\underline {= 2\,{\rm &micro;  s}} \hspace{0.05cm}.$$
-'''(3)'''&nbsp; Entsprechend der aus der Skizze für die ersten Bit erkennbaren Zuordnung gilt:
+'''(3)'''&nbsp; According to the allocation evident in the plot for the first bits:
 :$$ a_{\rm I3} = q_5  \hspace{0.15cm}\underline {= +1},$$
 :$$a_{\rm Q3} = q_6 \hspace{0.15cm}\underline {= +1},$$
@@ Line 90: / Line 90: @@
-'''(4)'''&nbsp; Bei der MSK ist die Symboldauer&nbsp; $T$&nbsp; gleich der Bitdauer&nbsp; $T_{\rm B}$:
+'''(4)'''&nbsp; In MSK, the symbol duration&nbsp; $T$&nbsp;is equal to the bit duration &nbsp; $T_{\rm B}$:
 :$$T = T_{\rm B}\hspace{0.15cm}\underline { = 1\,{\rm &micro;  s}} \hspace{0.05cm}.$$
-'''(5)'''&nbsp; Entsprechend der angegebenen Umcodiervorschrift gilt mit&nbsp; $a_4 = –1$:
+'''(5)'''&nbsp; According to the given recoding rule, when &nbsp; $a_4 = –1$, we get:
 :$$q_5 = +1 \hspace{0.3cm}  \Rightarrow  \hspace{0.3cm}a_5 = a_4 \cdot q_5 \hspace{0.15cm}\underline {= -1},$$
 :$$q_6 = +1 \hspace{0.3cm}  \Rightarrow  \hspace{0.3cm}a_6 = -a_5 \cdot q_6 \hspace{0.15cm}\underline {= +1},$$

← Older revision		Revision as of 15:45, 21 March 2022
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	*The associated phase function  $ϕ(t)$  is determined in  [[Aufgaben:Exercise_4.14:_Phase_Progression_of_the_MSK \|Exercise 4.14]] , and is also based on the  (normalized)  MSK fundamental pulse:		*The associated phase function  $ϕ(t)$  is determined in  [[Aufgaben:Exercise_4.14:_Phase_Progression_of_the_MSK \|Exercise 4.14]] , and is also based on the  (normalized)  MSK fundamental pulse:
−	:$$g_{\rm MSK}(t) = \left\{ \begin{array}{l} \cos (\pi/2 \cdot t/T ) \\ 0 \\ \end{array} \right.\quad \begin{array}{{5}c}{\rm{~~f\ddot{u}r~~}} \\{\rm{~~f\ddot{u}r~~}} \\ \end{array}\begin{array}{{10}c} -T \le t \le +T \hspace{0.05cm}, \\ {\rm sonst}. \\ \end{array}$$	+	:$$g_{\rm MSK}(t) = \left\{ \begin{array}{l} \cos (\pi/2 \cdot t/T ) \\ 0 \\ \end{array} \right.\quad \begin{array}{{5}c}{\rm{for}} \\{\rm{for}} \\ \end{array}\begin{array}{{10}c} -T \le t \le +T \hspace{0.05cm}, \\ {\rm sonst}. \\ \end{array}$$

@@ Line 44: / Line 44: @@
 <quiz display=simple>
-{Wie groß ist die Bitdauer &nbsp;$T_{\rm B}$&nbsp; des Quellensignals?
+{What is the bit duration &nbsp;$T_{\rm B}$&nbsp; of the source signal?
 |type="{}"}
 $T_{\rm B} \ = \ $ { 1 3% } $\ \rm &micro; s$
-{Wie groß ist die Symboldauer &nbsp;$T$&nbsp; der Offset–QPSK?
+{What is the symbol duration &nbsp;$T$&nbsp; of the offset QPSK?
 |type="{}"}
 $T \ = \ $ { 2 3%  } $\ \rm  &micro; s$
-{Geben Sie die genannten Amplitudenkoeffizienten der Offset–QPSK an.
+{Give the above amplitude coefficients of the offset QPSK.
 |type="{}"}
 $a_{\rm I3} \hspace{0.25cm} = \ $ { 1 3% }
@@ Line 60: / Line 60: @@
 $a_{\rm Q4} \ = \ $ { 1 3% }
-{Wie groß ist die Symboldauer &nbsp;$T$&nbsp; der &nbsp;MSK?
+{What is the symbol duration &nbsp;$T$&nbsp; of the &nbsp;MSK?
 |type="{}"}
 $T \ = \ $ { 1 3% } $\ \rm &micro; s$
-{Geben Sie die genannten  Amplitudenkoeffizienten der MSK an.
+{Give the above amplitude coefficients of the MSK.
 |type="{}"}
 $a_5 \ = \ $ { -1.03--0.97 }

@@ Line 9: / Line 9: @@
 The graph shows this serial-to-parallel conversion in the top three plots for the first four bits of the source signal &nbsp;$q(t)$.&nbsp; It should be noted:
-* Die Darstellung der Offset–QPSK gilt für einen rechteckigförmigen Grundimpuls.&nbsp; Die Koeffizienten &nbsp;$a_{{\rm I}ν}$&nbsp; und &nbsp;$a_{{\rm Q}ν}$&nbsp; können die Werte &nbsp;$±1$&nbsp; annehmen.
+* The Offset–QPSK plot is for for a rectangular-shaped fundamental pulse.&nbsp; The coefficients &nbsp;$a_{{\rm I}ν}$&nbsp; and &nbsp;$a_{{\rm Q}ν}$&nbsp;  can take the values &nbsp;$±1$&nbsp;.
-* Durchläuft der Zeitindex der Quellensymbole die Werte &nbsp;$k =1,$ ... $, 8$, so nimmt die Zeitvariable &nbsp;$ν$&nbsp; nur die Werte &nbsp;$1,$ ... $, 4$&nbsp; an.
+* If the time index of the source symbols passes through the values &nbsp;$k =1,$ ... $, 8$, then the time variable &nbsp;$ν$&nbsp; only takes on the values &nbsp;$1,$ ... $, 4$&nbsp; an.
-* Die Skizze berücksichtigt auch den Zeitversatz (Offset) für den Quadraturzweig.
+* The sketch also takes the time offset for the quadrature branch into account.
-Bei der&nbsp; "MSK–Realisierung mittels Offset–QPSK"&nbsp; ist eine Umcodierung erforderlich.&nbsp; Hierbei gilt mit &nbsp;$q_k ∈ \{+1, –1\}$&nbsp; und &nbsp;$a_k ∈ \{+1, –1\}$:
+For a&nbsp; "MSK–implementation using Offset–QPSK"&nbsp; a recoding is required.&nbsp; Here, with &nbsp;$q_k ∈ \{+1, –1\}$&nbsp; and &nbsp;$a_k ∈ \{+1, –1\}$, it holds that:
 :$$a_k = (-1)^{k+1} \cdot a_{k-1} \cdot q_k \hspace{0.05cm}.$$
-Beispielsweise erhält man unter der Annahme &nbsp;$a_0 = +1$:
+For example, by assuming &nbsp;$a_0 = +1$ one gets:
 :$$a_1 =  a_0 \cdot q_1 = +1,\hspace{0.4cm}a_2 = -a_1 \cdot q_2 = +1,\hspace{0.4cm}
 a_3  =  a_2 \cdot q_3 = -1,\hspace{0.4cm}a_4 = -a_3 \cdot q_4 = -1 \hspace{0.05cm}.$$
-Weiter ist zu berücksichtigen:
+Additionally, one must take into account:
-* Die Koeffizienten &nbsp;$a_0 = +1$, &nbsp;$a_2 = +1$, &nbsp;$a_4 = -1$&nbsp; sowie die noch zu berechnenden Koeffizienten &nbsp;$a_6$&nbsp; und &nbsp;$a_8$&nbsp; werden dem Signal &nbsp;$s_{\rm I}(t)$&nbsp; zugeordnet.
+* The coefficients &nbsp;$a_0 = +1$, &nbsp;$a_2 = +1$, &nbsp;$a_4 = -1$&nbsp; and the coefficients &nbsp;$a_6$&nbsp; and &nbsp;$a_8$&nbsp; which are yet to be calculated, are assigned to the signal &nbsp;$s_{\rm I}(t)$&nbsp;.
-* Dagegen werden die Koeffizienten &nbsp;$a_1 = +1$&nbsp; und &nbsp;$a_3 = -1$&nbsp; sowie alle weiteren Koeffizienten mit ungeradem Index dem Signal &nbsp;$s_{\rm Q}(t)$&nbsp; beaufschlagt.
+* On the other hand, the coefficients &nbsp;$a_1 = +1$&nbsp; and &nbsp;$a_3 = -1$&nbsp; as well as all other coefficients with an odd index are applied to the signal &nbsp;$s_{\rm Q}(t)$&nbsp;.
@@ Line 33: / Line 33: @@
-''Hinweise:''
+''Hints:''
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Modulation_Methods/Nichtlineare_digitale_Modulation|Nichtlineare digitale Modulation]].
+*This exercise belongs to the chapter&nbsp; [[Modulation_Methods/Nonlinear_Digital_Modulation|Nonlinear Digital Modulation]].
-*Bezug genommen wird insbesondere auf den Abschnitt&nbsp; [[Modulation_Methods/Nichtlineare_digitale_Modulation#Realisierung_der_MSK_als_Offset.E2.80.93QPSK|Realisierung der MSK als Offset-QPSK]].
+*Particular reference is made to the section&nbsp; [[Modulation_Methods/Nonlinear_Digital_Modulation#Realizing_MSK_as_Offset.E2.80.93QPSK|Realizing MSK as Offset–QPSK]].
-*In &nbsp;[[Aufgaben:4.14_Phasenverlauf_der_MSK |Aufgabe 4.14]]&nbsp; wird die zugehörige Phasenfunktion &nbsp;$ϕ(t)$&nbsp; ermittelt, wobei ebenfalls der&nbsp; (normierte)&nbsp; MSK–Grundimpuls zugrunde liegt:
+*The associated phase function &nbsp;$ϕ(t)$&nbsp; is determined in &nbsp;[[Aufgaben:Exercise_4.14:_Phase_Progression_of_the_MSK |Exercise 4.14]]&nbsp;, and is also based on the&nbsp; (normalized)&nbsp; MSK fundamental pulse:
 :$$g_{\rm MSK}(t) = \left\{ \begin{array}{l} \cos (\pi/2 \cdot t/T ) \\ 0 \\ \end{array} \right.\quad \begin{array}{*{5}c}{\rm{f\ddot{u}r}} \\{\rm{f\ddot{u}r}} \\ \end{array}\begin{array}{*{10}c} -T \le t \le +T \hspace{0.05cm}, \\ {\rm sonst}. \\ \end{array}$$
-===Fragebogen===
+===Questions===
 <quiz display=simple>
@@ Line 74: / Line 74: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
 '''(1)'''&nbsp; Aus der oberen Skizze kann man&nbsp; $T_{\rm B} \hspace{0.15cm}\underline{ = 1 \ \rm &micro; s}$&nbsp; ablesen.

@@ Line 4: / Line 4: @@
 [[File:P_ID1742__Mod_Z_4_13.png|right|frame|Koeffizientenzuordnung bei O-QPSK und MSK]]
-Eine Realisierungsmöglichkeit für die&nbsp; $\rm MSK$&nbsp; bietet die&nbsp; "Offset–QPSK"&nbsp; $\rm (O–QPSK)$, wie aus den &nbsp;[[Modulation_Methods/Nichtlineare_digitale_Modulation#Realisierung_der_MSK_als_Offset.E2.80.93QPSK|Blockschaltbildern]]&nbsp; im Theorieteil hervorgeht.
+One possible implementation fordie&nbsp; $\rm MSK$&nbsp; is offered by&nbsp; "Offset–QPSK"&nbsp; $\rm (O–QPSK)$, as can be seen from the &nbsp;[[Modulation_Methods/Nonlinear_Digital_Modulation#Realizing_MSK_as_Offset.E2.80.93QPSK|block diagrams]]&nbsp; in the theory section.
-Beim&nbsp; "normalen Offset–QPSK–Betrieb"&nbsp; werden jeweils zwei Bit der Quellensymbolfolge &nbsp;$〈q_k〉$&nbsp; einem Bit &nbsp;$a_{{\rm I}ν}$&nbsp; im Inphasezweig und sowie einem Bit &nbsp;$a_{{\rm Q}ν}$&nbsp; im Quadraturzweig zugeordnet.
+In "normal offset QPSK operation", two bits of the source symbol sequence&nbsp;$〈q_k〉$&nbsp;are assigned to one bit 𝑎Iν&nbsp;$a_{{\rm I}ν}$&nbsp; in the in-phase branch and one bit &nbsp;$a_{{\rm Q}ν}$&nbsp; in the quadrature branch, respectively.
-Die Grafik zeigt diese Seriell–Parallel–Wandlung in den drei oberen Diagrammen für die ersten vier Bit des grün gezeichneten Quellensignals &nbsp;$q(t)$.&nbsp; Dabei ist zu beachten:
+The graph shows this serial-to-parallel conversion in the top three plots for the first four bits of the source signal &nbsp;$q(t)$.&nbsp; It should be noted:
 * Die Darstellung der Offset–QPSK gilt für einen rechteckigförmigen Grundimpuls.&nbsp; Die Koeffizienten &nbsp;$a_{{\rm I}ν}$&nbsp; und &nbsp;$a_{{\rm Q}ν}$&nbsp; können die Werte &nbsp;$±1$&nbsp; annehmen.
 * Durchläuft der Zeitindex der Quellensymbole die Werte &nbsp;$k =1,$ ... $, 8$, so nimmt die Zeitvariable &nbsp;$ν$&nbsp; nur die Werte &nbsp;$1,$ ... $, 4$&nbsp; an.

← Older revision		Revision as of 13:51, 19 March 2022
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−	[[Category:Modulation Methods: Exercises\|^4.4 ~~Nichtlineare digitale~~ Modulation^]]	+	[[Category:Modulation Methods: Exercises\|^4.4 Non-linear Digital Modulation^]]

← Older revision	Revision as of 13:45, 19 March 2022
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