Aufgaben:Exercise 2.3Z: Oscillation Parameters - Revision history

Hwang at 13:14, 18 January 2023

2023-01-18T13:14:53Z

Guenter at 14:31, 13 April 2021

2021-04-13T14:31:27Z

Guenter at 14:27, 13 April 2021

2021-04-13T14:27:21Z

Noah at 09:23, 26 March 2021

2021-03-26T09:23:02Z

Javier: Text replacement - "Category:Exercises for Signal Representation" to "Category:Signal Representation: Exercises"

2021-03-23T12:38:46Z

Text replacement - "Category:Exercises for Signal Representation" to "Category:Signal Representation: Exercises"

Noah at 21:23, 13 January 2021

2021-01-13T21:23:45Z

Noah at 21:00, 13 January 2021

2021-01-13T21:00:27Z

Noah at 20:59, 13 January 2021

2021-01-13T20:59:32Z

Noah at 20:36, 13 January 2021

2021-01-13T20:36:28Z

Noah at 20:30, 13 January 2021

2021-01-13T20:30:15Z

@@ Line 83: / Line 83: @@
 :$$X(f)={C}/{2}\cdot{\rm e}^{-{\rm j}\varphi}\cdot\delta(f-f_0)+{C}/{2}\cdot{\rm e}^{{\rm j}\varphi}\cdot\delta(f+f_0).$$
-*The weight of the Dirac line at &nbsp; $f = f_0$&nbsp; (first term) is &nbsp; ${C}/2 \cdot {\rm e}^{–\text{j}\varphi} = 3 \,\text{V} \cdot \cos(60^\circ)- 3 \,\text{V} \cdot \sin(60^\circ)\hspace{0.05cm}\approx \underline{1.5 \,\text{V} - \text{j} \cdot 2.6 \,\text{V}}$.
+*The weight of the Dirac delta line at &nbsp; $f = f_0$&nbsp; (first term) is &nbsp; ${C}/2 \cdot {\rm e}^{–\text{j}\varphi} = 3 \,\text{V} \cdot \cos(60^\circ)- 3 \,\text{V} \cdot \sin(60^\circ)\hspace{0.05cm}\approx \underline{1.5 \,\text{V} - \text{j} \cdot 2.6 \,\text{V}}$.
 {{ML-Fuß}}
 [[Category:Signal Representation: Exercises|^2.3 Harmonic Oscillation^]]

@@ Line 48: / Line 48: @@
 |type="{}"}
 $\tau\hspace{0.25cm} = \ $  { 2 3% } &nbsp;$\text{ms}$
-$\varphi\hspace{0.2cm} = \ $  { 60 3% } &nbsp;$\text{degree}$
+$\varphi\hspace{0.2cm} = \ $  { 60 3% } &nbsp;$\text{deg}$

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID316__Sig_Z_2_3.png|right|frame|Definition of&nbsp; $x_0$,&nbsp; $t_1$&nbsp; und&nbsp; $t_2$]]
+[[File:P_ID316__Sig_Z_2_3.png|right|frame|Definitions of&nbsp; $x_0$,&nbsp; $t_1$&nbsp; und&nbsp; $t_2$]]
 Every harmonic oscillation can also be written in the form
 :$$x(t)=C\cdot\cos\bigg(2\pi \cdot \frac{t-\tau}{T_0}\bigg).$$
@@ Line 10: / Line 10: @@
 :* the amplitude&nbsp; $C$,
-:* the period duration &nbsp; $T_0$,
+:* the period duration&nbsp; $T_0$,
 :* the shift&nbsp; $\tau$&nbsp; with respect to a cosine signal.
-A second form of representation is with the base frequency&nbsp; $f_0$&nbsp; and the phase&nbsp; $\varphi$:
+A second form of representation is with the basic frequency&nbsp; $f_0$&nbsp; and the phase&nbsp; $\varphi$:
 :$$x(t)=C \cdot\cos(2\pi f_0t-\varphi).$$
 From a harmonic oscillation it is now known that
-:* the first signal maximum occurs at&nbsp; $t_1 = 2 \,\text{ms}$&nbsp; auftritt,
+:* the first signal maximum occurs at&nbsp; $t_1 = 2 \,\text{ms}$,
-:* the second signal maximum occurs at&nbsp; $t_2 = 14 \,\text{ms}$&nbsp; auftritt,
+:* the second signal maximum occurs at&nbsp; $t_2 = 14 \,\text{ms}$,
-:* the value&nbsp; $x_0 ={x(t = 0)} = 3 \,\text{V}$&nbsp;.
+:* the value&nbsp; $x_0 ={x(t = 0)} = 3 \,\text{V}$.
@@ Line 48: / Line 48: @@
 |type="{}"}
 $\tau\hspace{0.25cm} = \ $  { 2 3% } &nbsp;$\text{ms}$
-$\varphi\hspace{0.2cm} = \ $  { 60 3% } &nbsp;$\text{Grad}$
+$\varphi\hspace{0.2cm} = \ $  { 60 3% } &nbsp;$\text{degree}$
-{What is the amplitude of the harmonic oscillation??
+{What is the amplitude of the harmonic oscillation?
 |type="{}"}
 ${C}\ = \ $  { 6 3% } &nbsp;$\text{V}$

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID316__Sig_Z_2_3.png|right|frame|Definition von&nbsp; $x_0$,&nbsp; $t_1$&nbsp; und&nbsp; $t_2$]]
+[[File:P_ID316__Sig_Z_2_3.png|right|frame|Definition of&nbsp; $x_0$,&nbsp; $t_1$&nbsp; und&nbsp; $t_2$]]
 Every harmonic oscillation can also be written in the form
 :$$x(t)=C\cdot\cos\bigg(2\pi \cdot \frac{t-\tau}{T_0}\bigg).$$

← Older revision		Revision as of 12:38, 23 March 2021
Line 87:		Line 87:


−	[[Category:~~Exercises for~~ Signal Representation\|^2.3 Harmonic Oscillation^]]	+	[[Category:Signal Representation: Exercises\|^2.3 Harmonic Oscillation^]]

@@ Line 67: / Line 67: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp;  Es gilt&nbsp; $T_0 = t_2 - t_1 = 12\, \text{ms}$&nbsp; und&nbsp; $f_0 = 1/T_0 \hspace{0.15cm} \underline{\approx 83.33\, \text{Hz}}$.
+'''(1)'''&nbsp;  It is&nbsp; $T_0 = t_2 - t_1 = 12\, \text{ms}$&nbsp; and&nbsp; $f_0 = 1/T_0 \hspace{0.15cm} \underline{\approx 83.33\, \text{Hz}}$.
-'''(2)'''&nbsp; Die Verschiebung beträgt&nbsp; $\tau \hspace{0.1cm} \underline{= 2\, \text{ms}}$&nbsp; und die Phase ist&nbsp; $\varphi = 2\pi \cdot \tau/T_0 = \pi/3$&nbsp; entsprechend&nbsp; $\varphi =\hspace{0.15cm} \underline{60^{\circ}}$.
+'''(2)'''&nbsp; The shift is&nbsp; $\tau \hspace{0.1cm} \underline{= 2\, \text{ms}}$&nbsp; and the phase is&nbsp; $\varphi = 2\pi \cdot \tau/T_0 = \pi/3$&nbsp; corresponding to&nbsp; $\varphi =\hspace{0.15cm} \underline{60^{\circ}}$.
-'''(3)'''&nbsp; Aus dem Wert zum Zeitpunkt&nbsp; $t = 0$&nbsp; folgt für die Amplitude&nbsp; ${C}$:
+'''(3)'''&nbsp; From the value at timet&nbsp; $t = 0$&nbsp; it follows for the amplitude&nbsp; ${C}$:
 :$$x_0=x(t=0)=C\cdot\cos(-60\,^\circ)={C}/{2}=\rm 3\,V
 \hspace{0.3 cm} \Rightarrow \hspace{0.3 cm}\hspace{0.15cm}\underline{\it C=\rm 6\,V}.$$
-'''(4)'''&nbsp; Die dazugehörige Spektralfunktion lautet:
+'''(4)'''&nbsp; The corresponding spectral function is:
 :$$X(f)={C}/{2}\cdot{\rm e}^{-{\rm j}\varphi}\cdot\delta(f-f_0)+{C}/{2}\cdot{\rm e}^{{\rm j}\varphi}\cdot\delta(f+f_0).$$
-*Das Gewicht der Diraclinie bei&nbsp; $f = f_0$&nbsp; (erster Term) ist &nbsp; ${C}/2 \cdot {\rm e}^{–\text{j}\varphi} = 3 \,\text{V} \cdot \cos(60^\circ)- 3 \,\text{V} \cdot \sin(60^\circ)\hspace{0.05cm}\approx \underline{1.5 \,\text{V} - \text{j} \cdot 2.6 \,\text{V}}$.
+*The weight of the Dirac line at &nbsp; $f = f_0$&nbsp; (first term) is &nbsp; ${C}/2 \cdot {\rm e}^{–\text{j}\varphi} = 3 \,\text{V} \cdot \cos(60^\circ)- 3 \,\text{V} \cdot \sin(60^\circ)\hspace{0.05cm}\approx \underline{1.5 \,\text{V} - \text{j} \cdot 2.6 \,\text{V}}$.
 {{ML-Fuß}}
 [[Category:Exercises for Signal Representation|^2.3 Harmonic Oscillation^]]

@@ Line 15: / Line 15: @@
-Eine zweite Darstellungsform lautet mit der Grundfrequenz&nbsp; $f_0$&nbsp; und der Phase&nbsp; $\varphi$:
+A second form of representation is with the base frequency&nbsp; $f_0$&nbsp; and the phase&nbsp; $\varphi$:
 :$$x(t)=C \cdot\cos(2\pi f_0t-\varphi).$$
-Von einer harmonischen Schwingung ist nun bekannt, dass
+From a harmonic oscillation it is now known that
-:* das erste Signalmaximum bei&nbsp; $t_1 = 2 \,\text{ms}$&nbsp; auftritt,
+:* the second signal maximum occurs at&nbsp; $t_2 = 14 \,\text{ms}$&nbsp; auftritt,
-:* das zweite Signalmaximum bei&nbsp; $t_2 = 14 \,\text{ms}$&nbsp; auftritt,
+:* the value&nbsp; $x_0 ={x(t = 0)} = 3 \,\text{V}$&nbsp;.
@@ Line 30: / Line 30: @@
+''Hint:''
-''Hinweis:''
+*This exercise belongs to the chapter&nbsp; [[ Signal_Representation/Harmonic_Oscillation|Harmonic Oscillation]].
-===Fragebogen===
+===Questions===
 <quiz display=simple>
-{Wie groß ist die Periodendauer&nbsp; $T_0$&nbsp; und die Grundfrequenz&nbsp; $f_0$?
+What is the period duration&nbsp; $T_0$&nbsp; and the base frequency&nbsp; $f_0$?
 |type="{}"}
 $T_0\hspace{0.2cm} = \ $  { 12 3% } &nbsp;$\text{ms}$
@@ Line 46: / Line 45: @@
-{Welchen Wert haben hier die Verschiebung&nbsp; $\tau$&nbsp; und die Phase&nbsp; $\varphi$&nbsp; $($in&nbsp; $\text{Grad})$&nbsp;?
+{What is the value of the shift&nbsp; $\tau$&nbsp; and the phase&nbsp; $\varphi$&nbsp; $($in&nbsp; $\text{degrees})$&nbsp;?
 |type="{}"}
 $\tau\hspace{0.25cm} = \ $  { 2 3% } &nbsp;$\text{ms}$
@@ Line 52: / Line 51: @@
-{Wie groß ist die Amplitude der harmonischen Schwingung?
+{What is the amplitude of the harmonic oscillation??
 |type="{}"}
 ${C}\ = \ $  { 6 3% } &nbsp;$\text{V}$
-{Wie lautet das Spektrum&nbsp; $X(f)$?&nbsp; Welches Gewicht hat die Spektrallinie bei&nbsp; $+f_0$&nbsp;?
+{What is the spectrum&nbsp; $X(f)$?&nbsp; What is the weight of the spectral line at&nbsp; $+f_0$&nbsp;?
 |type="{}"}
 $\text{Re}\big[X(f = f_0)\big]\ = \ $  { 1.5 3% } &nbsp;$\text{V}$

@@ Line 6: / Line 6: @@
 Every harmonic oscillation can also be written in the form
 :$$x(t)=C\cdot\cos\bigg(2\pi \cdot \frac{t-\tau}{T_0}\bigg).$$
-Die Schwingung ist somit durch drei Parameter vollständig bestimmt:
+The oscillation is thus completely determined by three parameters:
-:* die Amplitude&nbsp; $C$,
+:* the amplitude&nbsp; $C$,
-:* die Periodendauer&nbsp; $T_0$,
+:* the period duration &nbsp; $T_0$,
-:* die Verschiebung&nbsp; $\tau$&nbsp; gegenüber einem Cosinussignal.
+:* the shift&nbsp; $\tau$&nbsp; with respect to a cosine signal.