Aufgaben:Exercise 3.1: Spectrum of the Exponential Pulse - Revision history

Guenter at 13:05, 21 April 2021

2021-04-21T13:05:05Z

Guenter at 12:57, 21 April 2021

2021-04-21T12:57:43Z

Noah at 13:10, 26 March 2021

2021-03-26T13:10:51Z

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

2021-03-23T12:38:57Z

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

Noah at 20:55, 16 January 2021

2021-01-16T20:55:49Z

Noah at 20:55, 16 January 2021

2021-01-16T20:55:33Z

Noah at 20:45, 16 January 2021

2021-01-16T20:45:33Z

Oezdemir: Oezdemir moved page Aufgaben:Aufgabe 3.1: Spektrum des Exponentialimpulses to Aufgaben:Exercise 3.1: Spectrum of the Exponential Pulse

2020-11-02T14:26:28Z

Oezdemir moved page Aufgabe 3.1: Spektrum des Exponentialimpulses to Exercise 3.1: Spectrum of the Exponential Pulse

Oezdemir at 14:23, 2 November 2020

2020-11-02T14:23:38Z

Javier: Text replacement - "Signaldarstellung/Fouriertransformation und -rücktransformation" to "Signal_Representation/Fourier_Transform_and_Its_Inverse"

2020-09-01T09:05:37Z

Text replacement - "Signaldarstellung/Fouriertransformation und -rücktransformation" to "Signal_Representation/Fourier_Transform_and_Its_Inverse"

@@ Line 74: / Line 74: @@
 X( {f = 0}) = A \cdot T{ = 3 \cdot 10^{ - 3}\; {\rm V/Hz}} \hspace{0.15 cm}\underline{ = 3 \; {\rm mV/Hz}}.$$
-*At the frequency&nbsp; $f = 0$&nbsp;, the spectrum is therefore purely real:
+*At the frequency $f = 0$&nbsp;, the spectrum is purely real:
 :$$\text{Re}[X(f=0)] \hspace{0.15 cm}\underline{ = 3 \; {\rm mV/Hz}}&nbsp; \hspace{1.15 cm}\text{Im}[X(f=0)] \hspace{0.15 cm}\underline{ =0}.$$
@@ Line 81: / Line 81: @@
 '''(2)'''&nbsp; With the abbreviations&nbsp; $X_0 = A \cdot T$&nbsp; and&nbsp; $f_0 = 1/(2\pi T)$&nbsp; the spectral function is:
 :$$X( f) = \frac{ {X_0 }}{ {1 +{\rm j} \cdot f/f_0 }} = \frac{ {X_0 }}{ {1 + \left( {f/f_0 } \right)^2 }} \cdot \left( {1 - {\rm j} \cdot f/f_0 } \right).$$
 Divided into real and imaginary parts, this gives:
-$${\mathop{\rm Re}\nolimits} [ {X(f)}] = \frac{ {X_0 }}{{1 + \left( {f/f_0 } \right)^2 }},
+:$${\mathop{\rm Re}\nolimits} [ {X(f)}] = \frac{ {X_0 }}{{1 + \left( {f/f_0 } \right)^2 }},
 \hspace{0.5 cm}{\mathop{\rm Im}\nolimits} [ {X(f)}] =  - \frac{ {X_0  \cdot f/f_0 }}{ {1 + \left( {f/f_0 } \right)^2 }}.$$
@@ Line 95: / Line 94: @@
 [[File:P_ID548__Sig_A_3_1_c_neu.png|right|frame|Magnitude spectrum of the exponential pulse]]
-'''(3)'''&nbsp; The magnitude of a complex-valued function, which is a quotient, is equal to the quotient of the magnitudes of the numerator and denominator.
 *Thus one obtains:
@@ Line 109: / Line 109: @@
 '''(4)'''&nbsp; The general rule for the phase function is:
-$$\varphi ( f ) = \arctan \left( {\frac{ { - {\mathop{\rm Im}\nolimits}[{X(f)} ]}}{{ {\mathop{\rm Re}\nolimits} [ {X(f)} ]}}} \right) = \arctan \left( {f/f_0 } \right).$$
+:$$\varphi ( f ) = \arctan \left( {\frac{ { - {\mathop{\rm Im}\nolimits}[{X(f)} ]}}{{ {\mathop{\rm Re}\nolimits} [ {X(f)} ]}}} \right) = \arctan \left( {f/f_0 } \right).$$
-*For $f = f_0$ this gives&nbsp; $\arctan(1)= \pi /4 \hspace{0.15 cm}\underline{\approx 0.785}$.
+*For $f = f_0$&nbsp; this gives&nbsp; $\varphi ( f_0 ) =\arctan(1)= \pi /4 \hspace{0.15 cm}\underline{\approx 0.785}$.
-*For very large values of&nbsp; $f$&nbsp; the phase function approaches&nbsp; $\arctan(\infty) = \pi /2 \hspace{0.15 cm}\underline{ \approx 1.571}$&nbsp; an.
+*For very large values of $f$&nbsp; the phase function approaches&nbsp; $\varphi ( f \to \infty ) =\arctan(\infty) = \pi /2 \hspace{0.15 cm}\underline{ \approx 1.571}$.
 *Both specifications are to be understood in radians.

@@ Line 6: / Line 6: @@
 In this task, a causal signal&nbsp; $x(t)$&nbsp;is considered
-*which rises abruptly from zero to&nbsp; $A$&nbsp; at time&nbsp; $t = 0$&nbsp;, and
+*which rises abruptly from zero to&nbsp; $A$&nbsp; at time&nbsp; $t = 0$,&nbsp; and
-*decreases exponentially with the time constant&nbsp; $T$&nbsp; for&nbsp; $t > 0$&nbsp;:
+*decreases exponentially with the time constant&nbsp; $T$&nbsp; for&nbsp; $t > 0$:
 :$$x(t) = A \cdot {\rm e}^{ - t/T} .$$
-At the saltus at time&nbsp; $t = 0$&nbsp;, &nbsp; $x(t = 0) = A/2$.
+At the jumping point at time&nbsp; $t = 0$,&nbsp; $x(t = 0) = A/2$.
 Use the following parameters for the numerical calculations:
@@ Line 17: / Line 17: @@
 :$$A = 3 \hspace{0.1cm} {\rm V}, \hspace{0.4cm} T = 1 \hspace{0.1cm} {\rm ms} .$$
-The spectral function&nbsp; $X(f)$&nbsp; to be calculated will be complex and therefore can be
+The spectral function&nbsp; $X(f)$&nbsp; to be calculated will be complex and therefore can be represented
 *by real and imaginary part, but also
-*by magnitude and phase
+*by magnitude and phase.
@@ Line 33: / Line 28: @@
 ''Hint:''
-*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]].
@@ Line 41: / Line 39: @@
 <quiz display=simple>
-{Calculate the spectral function&nbsp; $X(f)$. What spectral value results at the frequency&nbsp; $f = 0$?
+{Calculate the spectral function&nbsp; $X(f)$.&nbsp; What spectral value results at the frequency $f = 0$?
 |type="{}"}
 $\text{Re}[X(f=0)] \ = \ $ { 3 3% }  &nbsp;$\rm mV/Hz$
 $\text{Im}[X(f=0)] \ = \  $ { 0. } &nbsp;$\rm mV/Hz$
-{What are the real and imaginary parts of&nbsp; $X(f)$&nbsp; using&nbsp; $f_0 = 1/(2\pi T)$. What are the values when&nbsp; $f = f_0$?
+{What are the real and imaginary parts of&nbsp; $X(f)$&nbsp; using&nbsp; $f_0 = 1/(2\pi T)$.&nbsp; What are the values when $f = f_0$?
 |type="{}"}
 $\text{Re}[X(f=f_0)] \ = \  $ { 1.5 3% } &nbsp;$\rm mV/Hz$
 $\text{Im}[X(f=f_0)] \ = \  $ { -1.55--1.45 } &nbsp;$\rm mV/Hz$
-{Calculate the magnitude function&nbsp; $|X(f)|$. Which values result at the frequency&nbsp; $f = f_0$&nbsp; and at very high frequencies?
+{Calculate the magnitude function&nbsp; $|X(f)|$.&nbsp; Which values result at the frequency $f = f_0$&nbsp; and at very high frequencies?
 |type="{}"}
 $|X(f=f_0)| \hspace{0.25cm} = \  $ { 2.12 3% } &nbsp;$\rm mV/Hz$
 $|X(f\rightarrow \infty)| \ = \  $ { 0. } &nbsp;$\rm mV/Hz$
-{Calculate the phase function&nbsp; $\varphi(f)$. What values result for this at the frequency&nbsp; $f = f_0$&nbsp; and at very high frequencies?
+{Calculate the phase function&nbsp; $\varphi(f)$.&nbsp; What values result for this at the frequency $f = f_0$&nbsp; and at very high frequencies?
 |type="{}"}
 $\varphi(f=f_0) \hspace{0.25cm} = \  $ { 0.785 3% } &nbsp;$\rm rad$

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID494__Sig_A_3_1.png|right|frame|Exponentialimpuls]]
+[[File:P_ID494__Sig_A_3_1.png|right|frame|Exponential pulse]]
 In this task, a causal signal&nbsp; $x(t)$&nbsp;is considered
@@ Line 32: / Line 32: @@
-''Hints:''
+''Hint:''
 *This exercise belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse|Fourier Transform and Its Inverse]].
@@ Line 97: / Line 97: @@
-[[File:P_ID548__Sig_A_3_1_c_neu.png|right|frame|Betragsspektrum des Exponentialimpulses]]
+[[File:P_ID548__Sig_A_3_1_c_neu.png|right|frame|Magnitude spectrum of the exponential pulse]]
 '''(3)'''&nbsp; The magnitude of a complex-valued function, which is a quotient, is equal to the quotient of the magnitudes of the numerator and denominator.
 *Thus one obtains:

← Older revision		Revision as of 12:38, 23 March 2021
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−	[[Category:~~Exercises for~~ Signal Representation\|^3.1 Fourier Transform and Its Inverse^]]	+	[[Category:Signal Representation: Exercises\|^3.1 Fourier Transform and Its Inverse^]]

← Older revision		Revision as of 20:55, 16 January 2021
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	:$$\Rightarrow \hspace{0.5 cm} \left\| {X( {f = f_0} )} \right\| = { {X_0 }}/{ {\sqrt 2 }}\hspace{0.15 cm}\underline{ = 2.12 \;{\rm mV/Hz}}.$$		:$$\Rightarrow \hspace{0.5 cm} \left\| {X( {f = f_0} )} \right\| = { {X_0 }}/{ {\sqrt 2 }}\hspace{0.15 cm}\underline{ = 2.12 \;{\rm mV/Hz}}.$$

−	*At very high frequencies  $(f \rightarrow \infty)$  the magnitude is<u>almost zero</u> (see sketch).	+	*At very high frequencies  $(f \rightarrow \infty)$  the magnitude is <u>almost zero</u> (see sketch).

@@ Line 5: / Line 5: @@
 [[File:P_ID494__Sig_A_3_1.png|right|frame|Exponentialimpuls]]
-In dieser Aufgabe wird ein kausales Signal&nbsp; $x(t)$&nbsp; betrachtet,
+In this task, a causal signal&nbsp; $x(t)$&nbsp;is considered
-*das zum Zeitpunkt&nbsp; $t = 0$&nbsp; sprungartig von Null auf&nbsp; $A$&nbsp; ansteigt, und
+*which rises abruptly from zero to&nbsp; $A$&nbsp; at time&nbsp; $t = 0$&nbsp;, and
-*für Zeiten&nbsp; $t > 0$&nbsp; exponentiell mit der Zeitkonstanten&nbsp; $T$&nbsp; abfällt:
+*decreases exponentially with the time constant&nbsp; $T$&nbsp; for&nbsp; $t > 0$&nbsp;:
 :$$x(t) = A \cdot {\rm e}^{ - t/T} .$$
-An der Sprungstelle zum Zeitpunkt&nbsp; $t = 0$&nbsp; gilt&nbsp; $x(t = 0) = A/2$.
+At the saltus at time&nbsp; $t = 0$&nbsp;, &nbsp; $x(t = 0) = A/2$.
-Verwenden Sie für die numerischen Berechnungen folgende Parameter:
+Use the following parameters for the numerical calculations:
 :$$A = 3 \hspace{0.1cm} {\rm V}, \hspace{0.4cm} T = 1 \hspace{0.1cm} {\rm ms} .$$
-Die zu berechnende Spektralfunktion&nbsp; $X(f)$&nbsp; wird komplex sein und kann daher
+The spectral function&nbsp; $X(f)$&nbsp; to be calculated will be complex and therefore can be
-*nach Real– und Imaginärteil, aber auch
+*by real and imaginary part, but also
-*nach Betrag und Phase
+*by magnitude and phase
+can be represented. Use the notation
 :$$X( f ) = \left| {X( f )} \right| \cdot {\rm e}^{ - {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} \varphi( f )} .$$
@@ Line 33: / Line 32: @@
-''Hinweis:''
+''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]].
-===Fragebogen===
+===Questions===
 <quiz display=simple>

← Older revision	Revision as of 14:26, 2 November 2020
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← Older revision		Revision as of 14:23, 2 November 2020
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−	[[Category:Exercises for Signal Representation\|^3. ~~Aperiodische Signale - Impulse~~^]]	+	[[Category:Exercises for Signal Representation\|^3.1 Fourier Transform and Its Inverse^]]

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−	{{quiz-Header\|Buchseite=~~Signaldarstellung~~/~~Fouriertransformation und -rücktransformation~~	+	{{quiz-Header\|Buchseite=Signal_Representation/Fourier_Transform_and_Its_Inverse
	}}		}}