Aufgaben:Exercise 4.10: Signal Waveforms of the 16-QAM - Revision history

Hwang at 15:20, 23 January 2023

2023-01-23T15:20:24Z

Guenter at 14:29, 16 April 2022

2022-04-16T14:29:52Z

Guenter at 13:45, 16 April 2022

2022-04-16T13:45:33Z

Reed at 14:28, 19 March 2022

2022-03-19T14:28:39Z

Reed at 14:27, 19 March 2022

2022-03-19T14:27:59Z

Reed at 14:27, 19 March 2022

2022-03-19T14:27:31Z

Reed at 14:26, 19 March 2022

2022-03-19T14:26:57Z

Reed at 14:26, 19 March 2022

2022-03-19T14:26:17Z

Reed at 14:22, 19 March 2022

2022-03-19T14:22:51Z

Guenter at 13:39, 19 March 2022

2022-03-19T13:39:45Z

@@ Line 13: / Line 13: @@
 :$$s_{\rm cos}(t)  =  a_{\rm I}\cdot g_0 \cdot \cos(2 \pi f_{\rm T} t)\hspace{0.05cm},$$
 :$$ s_{\rm -sin}(t) =  -a_{\rm Q} \cdot g_0 \cdot \sin(2 \pi f_{\rm T} t)\hspace{0.05cm}.$$
-* The 16-QAM transmit signal is then the sum of these two component signals:
+* The 16-QAM transmitted signal is then the sum of these two component signals:
 :$$s(t) = s_{\rm cos}(t)+ s_{\rm -sin}(t) \hspace{0.05cm}.$$

@@ Line 74: / Line 74: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; From the (red) inphase signal  &nbsp; &rArr; &nbsp; the real part follows (first equation according to the definition, second equation according to the graph):
+'''(1)'''&nbsp; From the&nbsp; (red)&nbsp; inphase signal  &nbsp; &rArr; &nbsp; the real part follows&nbsp; (first equation according to the definition,&nbsp; second equation according to the graph):
 :$$ s_{\rm cos}(t)= a_{\rm I}\cdot g_0 \cdot \cos(2 \pi f_{\rm T} t)= g_0 \cdot \cos(2 \pi f_{\rm T} t)\hspace{0.3cm}\Rightarrow \hspace{0.3cm}a_{\rm I}\hspace{0.15cm}\underline {= +1} \hspace{0.05cm}.$$
-*Accordingly, we recognize from the quadrature signal ⇒ imaginary part:
+*Accordingly,&nbsp; we recognize from the quadrature signal &nbsp; ⇒ &nbsp; imaginary part:
 :$$ s_{\rm -sin}(t)= -a_{\rm Q}\cdot g_0 \cdot \sin(2 \pi f_{\rm T} t)= -g_0 \cdot \sin(2 \pi f_{\rm T} t)\hspace{0.3cm}\Rightarrow \hspace{0.3cm}a_{\rm Q}\hspace{0.15cm}\underline {= +1} \hspace{0.05cm}.$$
-'''(2)'''&nbsp; The two partial signals each have the (maximum) envelope &nbsp; $g_0$, while &nbsp; $s_0$&nbsp;characterizes the transmitted signal&nbsp; $s(t)$&nbsp;.
+'''(2)'''&nbsp; The two partial signals each have the&nbsp; (maximum)&nbsp; envelope&nbsp; $g_0$,&nbsp; while &nbsp;$s_0$&nbsp;characterizes the transmitted signal&nbsp; $s(t)$.
 *As can be seen from the signal space allocation&nbsp; (see Exercise 4.10Z)&nbsp;:
 :$${s_0}/{ g_0 }= \sqrt{2}\hspace{0.15cm}\underline { = 1.414} \hspace{0.05cm}.$$
@@ Line 87: / Line 87: @@
-'''(3)'''&nbsp; The amplitude coefficients &nbsp; $a_{\rm I}$&nbsp; and&nbsp; $a_{\rm Q}$&nbsp; have the same signs as in subtask &nbsp; '''(1)''', but with smaller magnitude:
+'''(3)'''&nbsp; The amplitude coefficients&nbsp; $a_{\rm I}$&nbsp; and&nbsp; $a_{\rm Q}$&nbsp; have the same signs as in subtask&nbsp; '''(1)''',&nbsp; but with smaller magnitude:
 :$$a_{\rm I} = + 1/3\hspace{0.15cm}\underline {= +0.333} \hspace{0.05cm},\hspace{0.5cm}a_{\rm Q} = + 1/3\hspace{0.15cm}\underline {= +0.333} \hspace{0.05cm}.$$
-'''(4)'''&nbsp;  In the third (green) interval, we can see a minus cosine signal with amplitude&nbsp; $g_0$&nbsp; and a minus sine signal with amplitude&nbsp; $g_0/3$:
+'''(4)'''&nbsp;  In the third&nbsp; (green)&nbsp; interval,&nbsp; we can see a minus-cosine signal with amplitude&nbsp; $g_0$&nbsp; and a minus-sine signal with amplitude&nbsp; $g_0/3$:
 :$$a_{\rm I} = \hspace{0.15cm}\underline {= -1} \hspace{0.05cm},\hspace{0.5cm}a_{\rm Q} = + 1/3\hspace{0.15cm}\underline {= +0.333} \hspace{0.05cm}.$$
-As is yet to be calculated in sub-task&nbsp; '''(4)'''&nbsp; of Exercise 4.10Z, here the magnitude is equal to&nbsp; $|a| =1.054$&nbsp; and the phase angle is &nbsp; ${\rm arc} \ a \approx 161^\circ$.
+*As is yet to be calculated in sub-task&nbsp; '''(4)'''&nbsp; of Exercise 4.10Z,&nbsp; here the magnitude is equal to&nbsp; $|a| =1.054$&nbsp; and the phase angle is &nbsp; ${\rm arc} \ a \approx 161^\circ$.
@@ Line 105: / Line 105: @@
 '''(6)'''&nbsp; The maximum signal energy occurs when one of the four outer vertices is occupied. In that case:
 :$$ E_{\rm S, \hspace{0.05cm}max} = {1}/{2}\cdot s_0^2 \cdot T = {1}/{2}\cdot \left (\sqrt{2} \cdot g_0 \right )^2 \cdot T = g_0^2 \cdot T = (1\,{\rm V})^2 \cdot (1\,{\rm \mu s}) \hspace{0.15cm}\underline {= 10^{-6}\,{\rm V^2s}}\hspace{0.05cm}.$$
-*The average signal energy is equal to the maximum value when only the corner points of the signal space mapping are occupied and the "inner symbols" are excluded from the encoding.
+*The average signal energy is equal to the maximum value when only the corner points of the signal space mapping are occupied and the&nbsp; "inner symbols"&nbsp; are excluded from the encoding.
-'''(7)'''&nbsp; Four bits are transmitted per symbol. From this it follows that:
+'''(7)'''&nbsp; Four bits are transmitted per symbol.&nbsp; From this it follows that:
 :$$ E_{\rm B, \hspace{0.05cm}max} = {E_{\rm S, \hspace{0.05cm}max}}/{4}\hspace{0.15cm}\underline {= 0.25 \cdot 10^{-6}\,{\rm V^2s}}\hspace{0.05cm}.$$
-'''(8)'''&nbsp;  The minimum signal energy is obtained at one of the inner signal space points and is smaller by a factor of $9$&nbsp; than in subtask &nbsp; '''(7)''':
+'''(8)'''&nbsp;  The minimum signal energy is obtained at one of the inner signal space points and is smaller by a factor of $9$&nbsp; than in subtask&nbsp; '''(7)''':
 :$$E_{\rm B, \hspace{0.05cm}min} = \frac{E_{\rm B, \hspace{0.05cm}max}}{9} = \frac{g_0^2 \cdot T}{36} \hspace{0.15cm}\underline { \approx 0.028 \cdot 10^{-6}\,{\rm V^2s}}\hspace{0.05cm}.$$
-*In the theory section, it was shown that, assuming all symbols are equally likely, the average signal energy per bit for 16-QAM is approximately:
+*In the theory section,&nbsp; it was shown that,&nbsp; assuming all symbols are equally likely,&nbsp; the average signal energy per bit for 16-QAM is approximately:
 :$$E_{\rm B} \approx 0.139 · g_0^2 \cdot T = 0.035 \cdot 10^{-6}\,{\rm V^2s}.$$
 {{ML-Fuß}}

@@ Line 7: / Line 7: @@
 Let us consider the 16–QAM method according to the [[Modulation_Methods/Quadrature_Amplitude_Modulation#General_description_and_signal_space_allocation|block diagram]]&nbsp; in the theory section.
-Very briefly, it can be described as follows:
+Very briefly,&nbsp; it can be described as follows:
-*After serial-parallel conversion and subsequent signal space assignment, four bits of the binary redundancy-free source signal &nbsp;$q(t)$&nbsp; at the input each result in a complex-valued amplitude coefficient
+*After serial-parallel conversion and subsequent signal space assignment,&nbsp; four bits of the binary redundancy-free source signal &nbsp;$q(t)$&nbsp; at the input each result in a complex-valued amplitude coefficient
 :$$a = a_{\rm I} +{\rm j} · a_{\rm Q}.$$
-*With the basic rectangular transmission pulse &nbsp;$g_s(t)$&nbsp; in the range from &nbsp;$0$&nbsp; to &nbsp;$T$&nbsp; and of height &nbsp;$g_0$&nbsp;, after multiplication with the cosine function or minus-sine function in the given time interval, we obtain:
+*With the rectangular transmission pulse &nbsp;$g_s(t)$&nbsp; in the range from &nbsp;$0$&nbsp; to &nbsp;$T$&nbsp; and of height &nbsp;$g_0$,&nbsp; after multiplication with the cosine function or minus-sine function in the given time interval,&nbsp; we obtain:
 :$$s_{\rm cos}(t)  =  a_{\rm I}\cdot g_0 \cdot \cos(2 \pi f_{\rm T} t)\hspace{0.05cm},$$
 :$$ s_{\rm -sin}(t) =  -a_{\rm Q} \cdot g_0 \cdot \sin(2 \pi f_{\rm T} t)\hspace{0.05cm}.$$
@@ Line 16: / Line 16: @@
 :$$s(t) = s_{\rm cos}(t)+ s_{\rm -sin}(t) \hspace{0.05cm}.$$
-The graph shows the signals &nbsp;$s_{\rm cos}(t)$, &nbsp;$s_{\rm –sin}(t)$&nbsp; and &nbsp;$s(t)$ for four selected symbols. Using these, the amplitude coefficients are to be determined.
+The graph shows the signals &nbsp;$s_{\rm cos}(t)$, &nbsp;$s_{\rm –sin}(t)$&nbsp; and &nbsp;$s(t)$ for four selected symbols.&nbsp; Using these,&nbsp; the amplitude coefficients are to be determined.
 Hints:
+*This exercise belongs to the chapter&nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation|"Quadrature Amplitude Modulation"]].
+*The page&nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation#Signal_waveforms_for_4.E2.80.93QAM|"Signal waveforms for 4–QAM"]]&nbsp; is helpful for completing this exercise.
 *The signal space allocation considered can be seen in &nbsp;[[Aufgaben:Exercise_4.10Z:_Signal_Space_Constellation_of_the_16-QAM|Exercise 4.10Z]].&nbsp; The color highlights also correspond.
-''Hints:''
+*From question &nbsp; '''(6)'''&nbsp; onwards,&nbsp; use the parameter values&nbsp;$g_0 = 1 \ \rm V$&nbsp; and &nbsp;$T = 1 \ \rm &micro; s$.
-*This exercise belongs to the chapter&nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation|Quadrature Amplitude Modulation]].
+*Energy values are to be given in &nbsp;$\rm V^2s$&nbsp;; like this,&nbsp; they refer to the reference resistance &nbsp;$R = 1 \ \rm \Omega$.
@@ Line 39: / Line 33: @@
 <quiz display=simple>
-{What are the real and imaginary parts of the amplitude coefficient &nbsp;$a$&nbsp; in the red time interval &nbsp;$(0 < t < T)$?
+{What are the real and imaginary part of the amplitude coefficient &nbsp;$a$&nbsp; in the red time interval &nbsp;$(0 < t < T)$?
 |type="{}"}
 $a_{\rm I} \ = \ $ { 1 3% }
@@ Line 45: / Line 39: @@
-{What is the relationship between &nbsp;$s_0$&nbsp;(maximum envelope of the transmitted signal) and &nbsp;$g_0$&nbsp; (maximum envelope of the partial signals)?
+{What is the relationship between &nbsp;$s_0$&nbsp; (maximum envelope of the transmitted signal) and &nbsp;$g_0$&nbsp; (maximum envelope of the partial signals)?
 |type="{}"}
 $s_0/g_0 \ = \ $ { 1.414 3% }
@@ Line 54: / Line 48: @@
 $a_{\rm Q} \ = \ $ { 0.333 3% }
-{What is the amplitude coefficient &nbsp;$a$&nbsp; in the green time interval &nbsp;$(2T < t < 3T)$?&nbsp; Also determine its magnitude and phase.
+{What is the amplitude coefficient &nbsp;$a$&nbsp; in the green time interval &nbsp;$(2T < t < 3T)$?&nbsp; Determine also its magnitude and phase.
 |type="{}"}
 $a_{\rm I} \ = \ $ { -1.03--0.97 }

@@ Line 28: / Line 28: @@
 *This exercise belongs to the chapter&nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation|Quadrature Amplitude Modulation]].
 *The page&nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation#Signal_waveforms_for_4.E2.80.93QAM|Signal waveforms for 4–QAM]]&nbsp; is helpful for completing this exercise.
-*The signal space allocation considered can be seen in the exercise page &nbsp;[[Aufgaben:4.10Z:_Signal_Space_Constellation_of_the_16-QAM–QAM|Exercise 4.10Z]].&nbsp; <br>The color highlights also correspond.
+*The signal space allocation considered can be seen in the exercise page &nbsp;[[Aufgaben:Exercise_4.10Z:_Signal_Space_Constellation_of_the_16-QAM|Exercise 4.10Z]].&nbsp; <br>The color highlights also correspond.
 *From question &nbsp; '''(6)'''&nbsp; onwards, use the parameter values&nbsp;$g_0 = 1 \ \rm V$&nbsp; and &nbsp;$T = 1 \ \rm &micro; s$.

@@ Line 28: / Line 28: @@
 *This exercise belongs to the chapter&nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation|Quadrature Amplitude Modulation]].
 *The page&nbsp; [[Modulation_Methods/Quadrature_Amplitude_Modulation#Signal_waveforms_for_4.E2.80.93QAM|Signal waveforms for 4–QAM]]&nbsp; is helpful for completing this exercise.
-*The signal space allocation considered can be seen in the exercise page &nbsp;[[4.10Z:_Signal_Space_Constellation_of_the_16-QAM–QAM|Exercise 4.10Z]].&nbsp; <br>The color highlights also correspond.
+*The signal space allocation considered can be seen in the exercise page &nbsp;[[Aufgaben:4.10Z:_Signal_Space_Constellation_of_the_16-QAM–QAM|Exercise 4.10Z]].&nbsp; <br>The color highlights also correspond.
 *From question &nbsp; '''(6)'''&nbsp; onwards, use the parameter values&nbsp;$g_0 = 1 \ \rm V$&nbsp; and &nbsp;$T = 1 \ \rm &micro; s$.

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