Aufgaben:Exercise 3.3Z: Characteristics Determination - Revision history

Reed at 15:03, 9 April 2022

2022-04-09T15:03:39Z

Reed: /* Solution */

2022-04-09T15:03:02Z

Solution

Reed at 14:51, 9 April 2022

2022-04-09T14:51:48Z

Guenter at 08:33, 30 March 2022

2022-03-30T08:33:36Z

Reed: /* Solution */

2022-03-14T17:49:46Z

Solution

Reed at 17:44, 14 March 2022

2022-03-14T17:44:52Z

Reed: /* Questions */

2022-03-14T17:44:29Z

Questions

Reed at 17:15, 14 March 2022

2022-03-14T17:15:21Z

Reed at 17:14, 14 March 2022

2022-03-14T17:14:26Z

Guenter at 16:51, 8 January 2022

2022-01-08T16:51:13Z

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 '''(5)'''&nbsp; The given equation can be rewritten as follows:
 :$$\eta = \frac{2 \cdot {\rm J}_{1}{(\eta)}}{{\rm J}_{0}(\eta) + {\rm J}_{2}(\eta)} \hspace{0.05cm}.$$
-*Mit&nbsp; ${\rm J}_0(η) = 0.512$,&nbsp; ${\rm J}_1(η) = 0.558$&nbsp; und&nbsp; ${\rm J}_2(η) = 0.232$&nbsp; erhält man somit:
+*With&nbsp; ${\rm J}_0(η) = 0.512$,&nbsp; ${\rm J}_1(η) = 0.558$&nbsp; and&nbsp; ${\rm J}_2(η) = 0.232$&nbsp; we thus get:
 :$$ \eta = \frac{2 \cdot 0.558}{0.512 + 0.232}\hspace{0.15cm}\underline { = 1.5}\hspace{0.05cm}.$$

@@ Line 80: / Line 80: @@
 '''(4)'''&nbsp; When &nbsp; $n = 1$ &nbsp; &rArr; &nbsp; $f = 3 \ \rm kHz$&nbsp; as in question &nbsp; '''(2)''', the phase is:
 :$$ \phi_{\rm N} + 90^\circ = 60^\circ \hspace{0.3cm} \Rightarrow \hspace{0.3cm}\phi_{\rm N} = -30^\circ\hspace{0.05cm}.$$
-*Checkin this result when &nbsp; $n = 2$ &nbsp; &rArr; &nbsp; $f = 6 \ \rm kHz$&nbsp; as in question&nbsp; '''(3)'''&nbsp; yields the same value:
+*Checking this result when &nbsp; $n = 2$ &nbsp; &rArr; &nbsp; $f = 6 \ \rm kHz$&nbsp; as in question&nbsp; '''(3)'''&nbsp; yields the same value:
 :$$ 2\cdot (\phi_{\rm N} + 90^\circ) = 120^\circ \hspace{0.3cm} \Rightarrow \hspace{0.3cm}\phi_{\rm N} \hspace{0.15cm}\underline {= -30^\circ}\hspace{0.05cm}.$$

← Older revision		Revision as of 14:51, 9 April 2022
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−	{{quiz-Header\|Buchseite=~~Modulationsverfahren~~/~~Phasenmodulation~~ (PM)	+	{{quiz-Header\|Buchseite=Modulation_Methods/Phase_Modulation_(PM)
	}}		}}

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID1085__Mod_Z_3_3.png|right|frame|Spektrum des analytischen Signals]]
+[[File:P_ID1085__Mod_Z_3_3.png|right|frame|Spectrum of the analytical signal]]
 Let us consider the phase modulation of the harmonic oscillation
 :$$q(t) = A_{\rm N} \cdot \cos(2 \pi \cdot f_{\rm N} \cdot t + \phi_{\rm N}) \hspace{0.05cm},$$

@@ Line 62: / Line 62: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Bezüglich&nbsp; $|S_+(f)|$&nbsp; gibt es eine Symmetrie zur Trägerfrequenz&nbsp; $f_{\rm T}\hspace{0.15cm}\underline { = 40 \ \rm kHz}$.&nbsp;  Der Abstand zwischen den Spektrallinien beträgt&nbsp; $f_{\rm N}\hspace{0.15cm}\underline { = 3 \ \rm kHz}$.
+'''(1)'''&nbsp; Regarding &nbsp; $|S_+(f)|$&nbsp; there is a symmetry with respect to the carrier frequency&nbsp; $f_{\rm T}\hspace{0.15cm}\underline { = 40 \ \rm kHz}$.&nbsp;  The distance between the spectral lines is&nbsp; $f_{\rm N}\hspace{0.15cm}\underline { = 3 \ \rm kHz}$.
-'''(2)'''&nbsp; Unter Berücksichtigung von&nbsp; $S_{\rm TP}(f = 3{\ \rm kHz}) = S_+(f = 43 \ \rm kHz)$&nbsp; gilt:
+'''(2)'''&nbsp; Considering&nbsp; $S_{\rm TP}(f = 3{\ \rm kHz}) = S_+(f = 43 \ \rm kHz)$&nbsp;, it holds that:
 :$$|S_{\rm TP}(f = 3\,{\rm kHz})| = \sqrt{0.279^2 + 0.483^2} \hspace{0.15cm}\underline {= 0.558}\hspace{0.05cm},$$
 :$$ {\rm arc}\hspace{0.15cm} S_{\rm TP}(f = 3\,{\rm kHz}) = \arctan \frac{0.483}{0.279} = \arctan 1.732\hspace{0.15cm}\underline { = 60^\circ} \hspace{0.05cm}.$$
@@ Line 72: / Line 72: @@
-'''(3)'''&nbsp; In analoger Weise zur Teilaufgabe&nbsp; '''(2)'''&nbsp; erhält man für die Frequenz&nbsp; $f = 6 \ \rm kHz$:
+'''(3)'''&nbsp; Analogously to in question&nbsp; '''(2)'''&nbsp;, at frequency &nbsp; $f = 6 \ \rm kHz$ we obtain:
 :$$|S_{\rm TP}(f = 6\,{\rm kHz})| = \sqrt{(-0.116)^2 + 0.201^2} \hspace{0.15cm}\underline {= 0.232}\hspace{0.05cm},$$
 :$${\rm arc}\hspace{0.15cm} S_{\rm TP}(f = 6\,{\rm kHz}) = \arctan \frac{-0.116}{0.201} = 180^\circ - \arctan 1.732 \hspace{0.15cm}\underline {= 120^\circ} \hspace{0.05cm}.$$
@@ Line 78: / Line 78: @@
-'''(4)'''&nbsp; Die Phase lautet für&nbsp; $n = 1$ &nbsp; &rArr; &nbsp; $f = 3 \ \rm kHz$&nbsp; entsprechend Teilaufgabe&nbsp; '''(2)''':
+'''(4)'''&nbsp; When &nbsp; $n = 1$ &nbsp; &rArr; &nbsp; $f = 3 \ \rm kHz$&nbsp; as in question &nbsp; '''(2)''', the phase is:
 :$$ \phi_{\rm N} + 90^\circ = 60^\circ \hspace{0.3cm} \Rightarrow \hspace{0.3cm}\phi_{\rm N} = -30^\circ\hspace{0.05cm}.$$
-*Die Überprüfung dieses Ergebnisses mit&nbsp; $n = 2$ &nbsp; &rArr; &nbsp; $f = 6 \ \rm kHz$&nbsp; entsprechend Teilaufgabe&nbsp; '''(3)'''&nbsp; liefert den gleichen Wert:
+*Checkin this result when &nbsp; $n = 2$ &nbsp; &rArr; &nbsp; $f = 6 \ \rm kHz$&nbsp; as in question&nbsp; '''(3)'''&nbsp; yields the same value:
 :$$ 2\cdot (\phi_{\rm N} + 90^\circ) = 120^\circ \hspace{0.3cm} \Rightarrow \hspace{0.3cm}\phi_{\rm N} \hspace{0.15cm}\underline {= -30^\circ}\hspace{0.05cm}.$$
-'''(5)'''&nbsp; Die angegebene Gleichung kann wie folgt umgeformt werden:
+'''(5)'''&nbsp; The given equation can be rewritten as follows:
 :$$\eta = \frac{2 \cdot {\rm J}_{1}{(\eta)}}{{\rm J}_{0}(\eta) + {\rm J}_{2}(\eta)} \hspace{0.05cm}.$$
 *Mit&nbsp; ${\rm J}_0(η) = 0.512$,&nbsp; ${\rm J}_1(η) = 0.558$&nbsp; und&nbsp; ${\rm J}_2(η) = 0.232$&nbsp; erhält man somit:

@@ Line 33: / Line 33: @@
 <quiz display=simple>
-{Wie groß sind die Frequenzen &nbsp;$f_{\rm T}$&nbsp; und &nbsp;$f_{\rm N}$?
+{What are the frequencies &nbsp;$f_{\rm T}$&nbsp; and &nbsp;$f_{\rm N}$?
 |type="{}"}
 $f_{\rm T} \ = \ $  { 40 3% } $\ \rm kHz$
 $f_{\rm N} \ = \ $ { 3 3% } $\ \rm kHz$
-{Berechnen Sie den Betrag und die Phase von &nbsp;$S_{\rm TP}(f = 3 \ \rm kHz)$.
+{Calculate the magnitude and the phase of &nbsp;$S_{\rm TP}(f = 3 \ \rm kHz)$.
 |type="{}"}
 $|S_{\rm TP}(f = 3 \ \rm kHz)| \ = \ $  { 0.558 3% }
 ${\rm arc} \ S_{\rm TP}(f = 3\ \rm  kHz) \ = \ $ { 60 3% } $\ \rm Grad$
-{Berechnen Sie den Betrag und die Phase von &nbsp;$S_{\rm TP}(f = 6 \ \rm kHz)$.
+{Calculate the magnitude and the phase of &nbsp;$S_{\rm TP}(f = 6 \ \rm kHz)$.
 |type="{}"}
 $|S_{\rm TP}(f = 6 \ \rm kHz)| \ = \ $ { 0.232 3% }
 ${\rm arc} \ S_{\rm TP}(f = 6\ \rm  kHz) \ = \ $ { 120 3% } $\ \rm Grad$
-{Wie groß ist die Phase des Quellensignals &nbsp;$q(t)$?
+{What is the phase of the source signal &nbsp;$q(t)$?
 |type="{}"}
 $ϕ_{\rm N} \ = \ $ { -30.9--29.1 } $\ \rm Grad$
-{Wie groß ist der Modulationsindex &nbsp;$η$&nbsp;?
+{What is the magnitude of the modulation index &nbsp;$η$&nbsp;?
 |type="{}"}
 $η \ = \ $ { 1.5 3% }

@@ Line 24: / Line 24: @@
 ''Hints:''
 *This exercise belongs to the chapter&nbsp; [[Modulation_Methods/Phase_Modulation_(PM)|Phase Modulation]].
-*Particular reference is made to the page&nbsp;  [[Modulation_Methods/Phasenmodulation_(PM)#.C3.84quivalentes_TP.E2.80.93Signal_bei_Phasenmodulation|Äquivalentes Tiefpass-Signal bei Phasenmodulation]].
+*Particular reference is made to the page&nbsp;  [[Modulation_Methods/Phase_Modulation_(PM)#Equivalent_low-pass_signal_in_phase_modulation|Equivalent low-pass signal in phase modulation]].
 *For the calculation of the modulation index, you can take advantage of the following property of the Bessel function:

@@ Line 4: / Line 4: @@
 [[File:P_ID1085__Mod_Z_3_3.png|right|frame|Spektrum des analytischen Signals]]
-Wir betrachten die Phasenmodulation der harmonischen Schwingung
+Let us consider the phase modulation of the harmonic oscillation
 :$$q(t) = A_{\rm N} \cdot \cos(2 \pi \cdot f_{\rm N} \cdot t + \phi_{\rm N}) \hspace{0.05cm},$$
-die bei Voraussetzung einer normierten Trägeramplitude &nbsp;$(A_{\rm T} = 1)$&nbsp; zu folgendem Sendesignal führt:
+which, given a normalized carrier amplitude &nbsp;$(A_{\rm T} = 1)$&nbsp;, leads to the following transmitted signal:
 :$$ s(t) = \cos \hspace{-0.1cm}\big[\omega_{\rm T} \cdot t + K_{\rm PM} \cdot q(t) \big]\hspace{0.05cm}.$$
-Das Spektrum des dazugehörigen analytischen Signals &nbsp;$s_{\rm TP}(t)$&nbsp; lautet allgemein:
+The spectrum of the corresponding analytical signal&nbsp;$s_{\rm TP}(t)$&nbsp; is generally:
 :$$S_{\rm TP}(f) = \sum_{n = - \infty}^{+\infty}{\rm J}_n (\eta) \cdot {\rm e}^{\hspace{0.05cm}{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm}n\hspace{0.05cm}\cdot \hspace{0.05cm}(\phi_{\rm N}\hspace{0.05cm}+\hspace{0.05cm} 90^\circ) }\cdot \hspace{0.05cm} \delta (f - n \cdot f_{\rm N})\hspace{0.05cm}$$
-Hierbei bezeichnet man &nbsp;$η = K_{\rm PM} · A_{\rm N}$&nbsp; als den Modulationsindex.
+Here, &nbsp;$η = K_{\rm PM} · A_{\rm N}$&nbsp; is called the modulation index.
-In der Grafik ist das Spektrum &nbsp;$S_+(f)$&nbsp; des analytischen Signals &nbsp;$s_+(t)$&nbsp; getrennt nach Real- und Imaginärteil dargestellt. Aus diesem sollen die Kenngrößen &nbsp;$f_{\rm T}$, &nbsp;$f_{\rm N}$, &nbsp;$ϕ_{\rm N}$&nbsp; und &nbsp;$η$&nbsp; ermittelt werden.
+In the graph, the real and imaginary parts of the spectrum &nbsp;$S_+(f)$&nbsp; of the analytical signal &nbsp;$s_+(t)$&nbsp;are shown separately. This should be used to determine the characteristics &nbsp;$f_{\rm T}$, &nbsp;$f_{\rm N}$, &nbsp;$ϕ_{\rm N}$&nbsp; and &nbsp;$η$&nbsp;.
@@ Line 22: / Line 22: @@
-''Hinweise:''
+''Hints:''
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Modulation_Methods/Phasenmodulation_(PM)|Phasenmodulation]].
+*This exercise belongs to the chapter&nbsp; [[Modulation_Methods/Phase_Modulation_(PM)|Phase Modulation]].
-*Bezug genommen wird insbesondere auf die Seite&nbsp;  [[Modulation_Methods/Phasenmodulation_(PM)#.C3.84quivalentes_TP.E2.80.93Signal_bei_Phasenmodulation|Äquivalentes Tiefpass-Signal bei Phasenmodulation]].
+*Particular reference is made to the page&nbsp;  [[Modulation_Methods/Phasenmodulation_(PM)#.C3.84quivalentes_TP.E2.80.93Signal_bei_Phasenmodulation|Äquivalentes Tiefpass-Signal bei Phasenmodulation]].
-*Zur Berechnung des Modulationsindex können Sie folgende Eigenschaft der Besselfunktion ausnutzen:
+*For the calculation of the modulation index, you can take advantage of the following property of the Bessel function:
 :$${\rm J}_n (\eta) = \frac{2 \cdot (n-1)}{\eta} \cdot {\rm J}_{n-1} (\eta) - {\rm J}_{n-2} (\eta) \hspace{0.3cm}\Rightarrow \hspace{0.3cm}{\rm J}_{2} (\eta)= {2}/{\eta} \cdot {\rm J}_{1} (\eta) - {\rm J}_{0} (\eta) \hspace{0.05cm}.$$
-===Fragebogen===
+===Questions===
 <quiz display=simple>

← Older revision		Revision as of 16:51, 8 January 2022
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−	[[Category:Modulation Methods: Exercises\|^3.1 ~~Phasenmodulation (PM)~~^]]	+	[[Category:Modulation Methods: Exercises\|^3.1 Phase Modulation^]]