Aufgaben:Exercise 3.2Z: Sinc-Squared Spectrum with Diracs - Revision history

Guenter: Guenter moved page Aufgaben:Exercise 3.2Z: Sinc-Squared-Spectrum with Diracs to Aufgaben:Exercise 3.2Z: Sinc-Squared Spectrum with Diracs

2021-05-24T13:11:21Z

Guenter moved page Exercise 3.2Z: Sinc-Squared-Spectrum with Diracs to Exercise 3.2Z: Sinc-Squared Spectrum with Diracs

Guenter at 14:53, 21 April 2021

2021-04-21T14:53:38Z

Guenter at 14:50, 21 April 2021

2021-04-21T14:50:49Z

Noah at 13:45, 26 March 2021

2021-03-26T13:45:41Z

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

2021-03-23T12:39:18Z

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

Noah at 20:06, 20 January 2021

2021-01-20T20:06:31Z

Noah at 20:06, 20 January 2021

2021-01-20T20:06:10Z

Noah at 19:52, 20 January 2021

2021-01-20T19:52:35Z

Noah at 19:33, 20 January 2021

2021-01-20T19:33:27Z

Oezdemir: Oezdemir moved page Aufgaben:Exercis 3.2Z: Sinc-Squared-Spectrum with Diracs to Aufgaben:Exercise 3.2Z: Sinc-Squared-Spectrum with Diracs

2020-11-02T14:30:01Z

Oezdemir moved page Exercis 3.2Z: Sinc-Squared-Spectrum with Diracs to Exercise 3.2Z: Sinc-Squared-Spectrum with Diracs

@@ Line 63: / Line 63: @@
 '''(1)'''&nbsp;  The one-sided duration of the symmetrical triangular pulse is&nbsp; $T = 1/f_0\hspace{0.15 cm}\underline{ = 5 \,{\rm &micro; s}}$.
-*The spectral value&nbsp; $X_0 = X_1(f = 0)$&nbsp; indicates the pulse area of&nbsp; $x_1(t)$&nbsp; an.
+*The spectral value&nbsp; $X_0 = X_1(f = 0)$&nbsp; indicates the pulse area of&nbsp; $x_1(t)$.
 *This is equal to&nbsp; ${A} \cdot {T}$.&nbsp; From this follows:
 :$$A = \frac{X_0 }{T}  = \frac{ 10^{-5}\rm V/Hz }{5 \cdot 10^{-6}{\rm s}}\hspace{0.15 cm}\underline{= 2\;{\rm V}}.$$
-'''(2)'''&nbsp;  The DC component is given by the Dirac weight at&nbsp; $f = 0$&nbsp;. One obtains&nbsp; ${B} \hspace{0.15 cm}\underline{= -1 \,\text{V}}$.
+'''(2)'''&nbsp;  The DC component is given by the Dirac weight at&nbsp; $f = 0$.&nbsp; One obtains&nbsp; ${B} \hspace{0.15 cm}\underline{= -1 \,\text{V}}$.

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID496__Sig_Z_3_2_neu.png|right|frame|$\rm si$-squared spectrum with diracs]]
+[[File:P_ID496__Sig_Z_3_2_neu.png|right|frame|$\rm sinc^2$&ndash; spectrum with Diracs]]
 The sketched spectrum&nbsp; ${X(f)}$&nbsp; of a time signal&nbsp; ${x(t)}$&nbsp; is composed of
 * a continuous component&nbsp; $X_1(f)$,
-* plus three dirac-shaped spectral lines.
+* plus three discrete spectral lines &nbsp; &rArr; &nbsp;   "Dirac functions".
 The continuous component with&nbsp; $f_0 = 200\, \text{kHz}$&nbsp; and&nbsp; $X_0 = 10^{–5} \text{ V/Hz}$ is as follows:
-:$$X_1( f ) = X_0  \cdot {\mathop{\rm si}\nolimits} ^2 ( {\pi {f}/{f_0}} ),\quad {\rm where is}\quad {\mathop{\rm si}\nolimits} (x) = {\sin (x)}/{x}.$$
+:$$X_1( f ) = X_0  \cdot {\mathop{\rm sinc}\nolimits} ^2 ( {{f}/{f_0}} ),\quad {\rm where is}\quad {\mathop{\rm sinc}\nolimits} (x) = {\sin (\pi x)}/(\pi x).$$
-The spectral line at&nbsp; $f = 0$&nbsp; has the weight&nbsp; $–\hspace{-0.08cm}1\,\text{V}$. In addition, there are two lines at frequencies&nbsp; $\pm f_0$, both with weight&nbsp; $0.5\,\text{V}$.
+*The spectral line at&nbsp; $f = 0$&nbsp; has the weight&nbsp; $–\hspace{-0.08cm}1\,\text{V}$.
@@ Line 22: / Line 23: @@
 ''Hints:''
-*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse|Fourier Transform and Its Inverse]].
+*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse|Fourier Transform and its Inverse]].
-*Further information on this topic can be found in the learning video&nbsp; [[Kontinuierliche_und_diskrete_Spektren_(Lernvideo)|Continuous and discrete spectra]].
+*Further information on this topic can be found in the (German language) learning video&nbsp; [[Kontinuierliche_und_diskrete_Spektren_(Lernvideo)|Kontinuierliche und diskrete Spektren]] &nbsp; &rArr; &nbsp; "Continuous and discrete spectra".
-*It can be assumed as known that a triangular pulse&nbsp; $y(t)$&nbsp; symmetrical about&nbsp; $t = 0$&nbsp; with the amplitude&nbsp; ${A}$&nbsp; and the absolute duration&nbsp; $2T$&nbsp; $($i.e.:&nbsp; the signal values are unequal to $ 0 $ only between&nbsp; $–T$&nbsp; and&nbsp; $+T$&nbsp;)&nbsp; has the following spectral function:
+*It can be assumed as known:&nbsp; A triangular pulse&nbsp; $y(t)$&nbsp; with amplitude&nbsp; ${A}$,&nbsp; the absolute duration&nbsp; $2T$&nbsp; and symmetrical about&nbsp; $t = 0$&nbsp; $($i.e.:&nbsp; the signal values are&nbsp; $\ne 0 $&nbsp; only between&nbsp; $–T$&nbsp; and&nbsp; $+T$&nbsp;)&nbsp; has the following spectral function:
-:$$Y( f ) = A  \cdot T \cdot {\rm si}^2 ( \pi f T ).$$
+:$$Y( f ) = A  \cdot T \cdot {\rm sinc}^2 ( f T ).$$
@@ Line 32: / Line 33: @@
 <quiz display=simple>
-{What are the values of the parameters&nbsp; ${A}$&nbsp; (amplitude) and&nbsp; ${T}$&nbsp; (one-sided duration) of the triangular signal component&nbsp; $x_1(t)$?
+{What are the values of the parameters&nbsp; ${A}$&nbsp; (maximum) and&nbsp; ${T}$&nbsp; (one-sided duration) of the triangular signal component&nbsp; $x_1(t)$?
 |type="{}"}
 $A\ = \ $ { 2 3% } &nbsp;$\text{V}$

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID496__Sig_Z_3_2_neu.png|right|frame|$\rm si$-Quadrat-Spektrum mit Diracs]]
+[[File:P_ID496__Sig_Z_3_2_neu.png|right|frame|$\rm si$-squared spectrum with diracs]]
 The sketched spectrum&nbsp; ${X(f)}$&nbsp; of a time signal&nbsp; ${x(t)}$&nbsp; is composed of
@@ Line 11: / Line 11: @@
-The continuous component with&nbsp; $f_0 = 200\, \text{kHz}$&nbsp; and&nbsp; $X_0 = 10^{–5} \text{ V/Hz}$is as follows:
+The continuous component with&nbsp; $f_0 = 200\, \text{kHz}$&nbsp; and&nbsp; $X_0 = 10^{–5} \text{ V/Hz}$ is as follows:
 :$$X_1( f ) = X_0  \cdot {\mathop{\rm si}\nolimits} ^2 ( {\pi {f}/{f_0}} ),\quad {\rm where is}\quad {\mathop{\rm si}\nolimits} (x) = {\sin (x)}/{x}.$$
 The spectral line at&nbsp; $f = 0$&nbsp; has the weight&nbsp; $–\hspace{-0.08cm}1\,\text{V}$. In addition, there are two lines at frequencies&nbsp; $\pm f_0$, both with weight&nbsp; $0.5\,\text{V}$.
@@ Line 23: / Line 23: @@
 ''Hints:''
 *This exercise belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse|Fourier Transform and Its Inverse]].
-*Further information on this topic can be found in the learning video&nbsp; [[Kontinuierliche_und_diskrete_Spektren_(Lernvideo)|Kontinuierliche und diskrete Spektren]].
+*Further information on this topic can be found in the learning video&nbsp; [[Kontinuierliche_und_diskrete_Spektren_(Lernvideo)|Continuous and discrete spectra]].
-*It can be assumed as known that a triangular pulse&nbsp; $y(t)$&nbsp; symmetrical about&nbsp; $t = 0$&nbsp; with the amplitude&nbsp; ${A}$&nbsp; and the absolute duration&nbsp; $2T$&nbsp; $($i.e.:&nbsp; he signal values are unequal to $ 0 $ only between&nbsp; $–T$&nbsp; and&nbsp; $+T$&nbsp;)&nbsp; has the following spectral function:
+*It can be assumed as known that a triangular pulse&nbsp; $y(t)$&nbsp; symmetrical about&nbsp; $t = 0$&nbsp; with the amplitude&nbsp; ${A}$&nbsp; and the absolute duration&nbsp; $2T$&nbsp; $($i.e.:&nbsp; the signal values are unequal to $ 0 $ only between&nbsp; $–T$&nbsp; and&nbsp; $+T$&nbsp;)&nbsp; has the following spectral function:
 :$$Y( f ) = A  \cdot T \cdot {\rm si}^2 ( \pi f T ).$$
@@ Line 59: / Line 59: @@
 ===Solution===
 {{ML-Kopf}}
-[[File:P_ID498__Sig_Z_3_2_a_neu.png|right|frame|Fläche des Dreieckimpulses]]
+[[File:P_ID498__Sig_Z_3_2_a_neu.png|right|frame|Area of the triangular pulse]]
 '''(1)'''&nbsp;  The one-sided duration of the symmetrical triangular pulse is&nbsp; $T = 1/f_0\hspace{0.15 cm}\underline{ = 5 \,{\rm &micro; s}}$.

← Older revision		Revision as of 12:39, 23 March 2021
Line 82:		Line 82:

	__NOEDITSECTION__		__NOEDITSECTION__
−	[[Category:~~Exercises for~~ Signal Representation\|^3.1 Fourier Transform and Its Inverse^]]	+	[[Category:Signal Representation: Exercises\|^3.1 Fourier Transform and Its Inverse^]]

@@ Line 76: / Line 76: @@
 :$$x_{\text{max}} = A + B + C \hspace{0.15 cm}\underline{= +2 \text{V}}.$$
-*The minimum values of&nbsp; ${x(t)}$&nbsp;  result when the triangular pulse has decayed and the cosine function delivers the value&nbsp; $–\hspace{-0.08 cm}1 \,\text{V}$&nbsp; liefert:
+*The minimum values of&nbsp; ${x(t)}$&nbsp;  result when the triangular pulse has decayed and the cosine function delivers the value&nbsp; $–\hspace{-0.08 cm}1 \,\text{V}$&nbsp;:
 :$$x_\text{min} = {B} - {C}\hspace{0.15 cm}\underline{ = -2\, \text{V}}.$$
 {{ML-Fuß}}

@@ Line 25: / Line 25: @@
 *Further information on this topic can be found in the learning video&nbsp; [[Kontinuierliche_und_diskrete_Spektren_(Lernvideo)|Kontinuierliche und diskrete Spektren]].
-*It can be assumed as known that a triangular pulse&nbsp; $y(t)$&nbsp; symmetrical about&nbsp; $t = 0$&nbsp; with the amplitude&nbsp; ${A}$&nbsp; and the absolute duration&nbsp; $2T$&nbsp; $($i.e.:&nbsp; he signal values are unequal to$0)$&nbsp; only between&nbsp; $–T$&nbsp; and&nbsp; $+T$&nbsp; has the following spectral function:
+*It can be assumed as known that a triangular pulse&nbsp; $y(t)$&nbsp; symmetrical about&nbsp; $t = 0$&nbsp; with the amplitude&nbsp; ${A}$&nbsp; and the absolute duration&nbsp; $2T$&nbsp; $($i.e.:&nbsp; he signal values are unequal to $ 0 $ only between&nbsp; $–T$&nbsp; and&nbsp; $+T$&nbsp;)&nbsp; has the following spectral function:
 :$$Y( f ) = A  \cdot T \cdot {\rm si}^2 ( \pi f T ).$$
@@ Line 32: / Line 32: @@
 <quiz display=simple>
-{Welche Werte besitzen die Parameter&nbsp; ${A}$&nbsp; (Amplitude) und&nbsp; ${T}$&nbsp; (einseitige Dauer) des dreieckförmigen Signalanteils&nbsp; $x_1(t)$?
+{What are the values of the parameters&nbsp; ${A}$&nbsp; (amplitude) and&nbsp; ${T}$&nbsp; (one-sided duration) of the triangular signal component&nbsp; $x_1(t)$?
 |type="{}"}
 $A\ = \ $ { 2 3% } &nbsp;$\text{V}$
@@ Line 38: / Line 38: @@
-{Wie groß ist der Gleichsignalanteil&nbsp; ${B}$&nbsp; des Signals?
+{What is the DC component&nbsp; ${B}$&nbsp; of the signal?
 |type="{}"}
 $B\ = \ $  { -1.03--0.97 } &nbsp;$\text{V}$
-{Wie groß ist die Amplitude&nbsp; $C$&nbsp; des periodischen Anteils von&nbsp; $x(t)$?
+{What is the amplitude&nbsp; $C$&nbsp; of the periodic component of&nbsp; $x(t)$?
 |type="{}"}
 $C\ = \ $ { 1 3% } &nbsp;$\text{V}$
-{Wie groß sind der Maximalwert und der Minimalwert des Signals&nbsp; $x(t)$?
+{What are the maximum and minimum values of the signal&nbsp; $x(t)$?
 |type="{}"}
 $x_\text{max}\ = \ $ { 2 3% } &nbsp;$\text{V}$
@@ Line 57: / Line 57: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
 [[File:P_ID498__Sig_Z_3_2_a_neu.png|right|frame|Fläche des Dreieckimpulses]]

@@ Line 4: / Line 4: @@
 [[File:P_ID496__Sig_Z_3_2_neu.png|right|frame|$\rm si$-Quadrat-Spektrum mit Diracs]]
-Das skizzierte Spektrum&nbsp; ${X(f)}$&nbsp; eines Zeitsignals&nbsp; ${x(t)}$&nbsp; setzt sich zusammen aus
+The sketched spectrum&nbsp; ${X(f)}$&nbsp; of a time signal&nbsp; ${x(t)}$&nbsp; is composed of
-* einem kontinuierlichen Anteil&nbsp; $X_1(f)$,
+* a continuous component&nbsp; $X_1(f)$,
-* dazu drei diracförmigen Spektrallinien.
+* plus three dirac-shaped spectral lines.
-Der kontinuierliche Anteil lautet mit&nbsp; $f_0 = 200\, \text{kHz}$&nbsp; und&nbsp; $X_0 = 10^{–5} \text{ V/Hz}$:
+The continuous component with&nbsp; $f_0 = 200\, \text{kHz}$&nbsp; and&nbsp; $X_0 = 10^{–5} \text{ V/Hz}$is as follows:
-:$$X_1( f ) = X_0  \cdot {\mathop{\rm si}\nolimits} ^2 ( {\pi {f}/{f_0}} ),\quad {\rm wobei}\quad {\mathop{\rm si}\nolimits} (x) = {\sin (x)}/{x}.$$
+:$$X_1( f ) = X_0  \cdot {\mathop{\rm si}\nolimits} ^2 ( {\pi {f}/{f_0}} ),\quad {\rm where is}\quad {\mathop{\rm si}\nolimits} (x) = {\sin (x)}/{x}.$$
-Die Spektrallinie bei&nbsp; $f = 0$&nbsp; hat das Gewicht&nbsp; $–\hspace{-0.08cm}1\,\text{V}$. Daneben gibt es noch zwei Linien bei den Frequenzen&nbsp; $\pm f_0$, beide mit dem Gewicht&nbsp; $0.5\,\text{V}$.
+The spectral line at&nbsp; $f = 0$&nbsp; has the weight&nbsp; $–\hspace{-0.08cm}1\,\text{V}$. In addition, there are two lines at frequencies&nbsp; $\pm f_0$, both with weight&nbsp; $0.5\,\text{V}$.
@@ Line 21: / Line 21: @@
-''Hinweise:''
+''Hints:''
-*Die Aufgabe gehört zum Kapitel&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse|Fouriertransformation und &ndash;rücktransformation]].
+*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse|Fourier Transform and Its Inverse]].
-*Weitere Informationen zu dieser Thematik liefert das Lernvideo&nbsp; [[Kontinuierliche_und_diskrete_Spektren_(Lernvideo)|Kontinuierliche und diskrete Spektren]].
+*Further information on this topic can be found in the learning video&nbsp; [[Kontinuierliche_und_diskrete_Spektren_(Lernvideo)|Kontinuierliche und diskrete Spektren]].
-*Als bekannt vorausgesetzt werden kann, dass ein um&nbsp; $t = 0$&nbsp; symmetrischer Dreieckimpuls&nbsp; $y(t)$&nbsp; mit der Amplitude&nbsp; ${A}$&nbsp; und der absoluten Dauer&nbsp; $2T$&nbsp; $($das heißt:&nbsp; die Signalwerte sind nur zwischen&nbsp; $–T$&nbsp; und&nbsp; $+T$&nbsp; ungleich $0)$&nbsp; folgende Spektralfunktion besitzt:
+*It can be assumed as known that a triangular pulse&nbsp; $y(t)$&nbsp; symmetrical about&nbsp; $t = 0$&nbsp; with the amplitude&nbsp; ${A}$&nbsp; and the absolute duration&nbsp; $2T$&nbsp; $($i.e.:&nbsp; he signal values are unequal to$0)$&nbsp; only between&nbsp; $–T$&nbsp; and&nbsp; $+T$&nbsp; has the following spectral function:
 :$$Y( f ) = A  \cdot T \cdot {\rm si}^2 ( \pi f T ).$$
-===Fragebogen===
+===Question===
 <quiz display=simple>