Aufgaben:Exercise 3.4: Trapezoidal Spectrum and Pulse - Revision history

Guenter at 13:17, 24 May 2021

2021-05-24T13:17:21Z

Guenter at 14:27, 26 April 2021

2021-04-26T14:27:36Z

Guenter at 14:19, 26 April 2021

2021-04-26T14:19:35Z

Guenter at 14:15, 26 April 2021

2021-04-26T14:15:35Z

Guenter at 13:37, 26 April 2021

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Guenter at 13:34, 26 April 2021

2021-04-26T13:34:31Z

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 19:11, 23 January 2021

2021-01-23T19:11:58Z

Noah at 19:11, 23 January 2021

2021-01-23T19:11:34Z

Noah at 18:48, 23 January 2021

2021-01-23T18:48:51Z

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID508__Sig_A_3_4.png|250px|right|frame|Trapezspektrum & Trapezimpuls]]
+[[File:P_ID508__Sig_A_3_4.png|250px|right|frame|Trapezoidal spectrum & trapezoidal pulse]]
 We consider here a trapezoidal spectral function&nbsp; $X(f)$&nbsp; according to the upper graph, which is completely described by the three parameters&nbsp; $X_0$,&nbsp; $f_1$&nbsp; and&nbsp; $f_2$.&nbsp; For the two corner frequencies,&nbsp;  $f_2 > 0$&nbsp; and&nbsp; $0 \leq f_1 \leq f_2$ always apply.

@@ Line 83: / Line 83: @@
 {{ML-Kopf}}
-'''(1)'''&nbsp; The equivalent bandwidth is by definition equal to the width of the equal-area rectangle:
+'''(1)'''&nbsp; The equivalent bandwidth is (by definition) equal to the width of the equal-area rectangle:
 :$$\Delta f = f_1  + f_2  \hspace{0.15 cm}\underline{= 4\;{\rm{kHz}}}{\rm{.}}$$
 *For the rolloff factor holds:
-:$${ {r_f = }}\frac{ {f_2  - f_1 }}{ {f_2  + f_1 }}\hspace{0.15 cm}\underline{ = 0.5}.$$
+:$${ {r_{\hspace{-0.05cm}f} = }}\frac{ {f_2  - f_1 }}{ {f_2  + f_1 }}\hspace{0.15 cm}\underline{ = 0.5}.$$
@@ Line 99: / Line 99: @@
 :$$x( {t = T} ) = x_0  \cdot {\mathop{\rm si}\nolimits} ( {\rm{\pi }} ) \cdot {\mathop{\rm si}\nolimits} ( { { {\rm{\pi }}}/{2}} )\hspace{0.15 cm}\underline{ = 0}.$$
-*Also at all multiples of&nbsp; $T$&nbsp; &nbsp; $x(t)$&nbsp; exhibits zero crossings. At time&nbsp; $t = T/2$&nbsp; holds:
+*Also at all multiples of&nbsp; $T$:&nbsp; &nbsp; $x(t)$&nbsp; exhibits zero crossings.&nbsp; At time&nbsp; $t = T/2$&nbsp; holds:
 :$$x( {t = T/2} ) = x_0  \cdot {\mathop{\rm si}\nolimits} ( { { {\rm{\pi }}}/{2}} ) \cdot {\mathop{\rm si}\nolimits}( { { {\rm{\pi }}}/{4}} ) = x_0  \cdot \frac{ { 1 \cdot \sqrt 2 /2}}{ { {\rm{\pi /}}2 \cdot {\rm{\pi /4}}}} = x_0  \cdot \frac{ {4 \cdot \sqrt 2 }}{ { {\rm{\pi }}^{\rm{2}} }} \hspace{0.15 cm}\underline{= 2.293\;{\rm{V}}}{\rm{.}}$$
@@ Line 108: / Line 108: @@
 :$$x( t ) = X_0  \cdot \Delta f \cdot {\mathop{\rm si}\nolimits} ( { {\rm{\pi }} \cdot \Delta f \cdot t} ) \cdot {\mathop{\rm si}\nolimits} ( { {\rm{\pi }} \cdot r_f \cdot \Delta f \cdot t} ).$$
-*Since both&nbsp; $X(f)$&nbsp; and&nbsp; $x(t)$&nbsp; are real and, moreover,&nbsp; $y(t)$&nbsp; is of the same form as&nbsp; $X(f)$&nbsp; , we obtain, taking into account all equivalences for the spectral function of the trapezoidal pulse:
+*Since both&nbsp; $X(f)$&nbsp; and&nbsp; $x(t)$&nbsp; are real and, moreover,&nbsp; $y(t)$&nbsp; is of the same form as&nbsp; $X(f),$&nbsp; we obtain, taking into account all equivalences for the spectral function of the trapezoidal pulse:
 :$$Y( f ) = y_0  \cdot \Delta t \cdot {\mathop{\rm si}\nolimits} ( { {\rm{\pi }} \cdot \Delta t \cdot f} ) \cdot {\mathop{\rm si}\nolimits} ( { {\rm{\pi }} \cdot r_t \cdot \Delta t \cdot f} ).$$

@@ Line 11: / Line 11: @@
 :$$\Delta f = f_1  + f_2,$$
 *the so-called&nbsp; [[Linear_and_Time_Invariant_Systems/Einige_systemtheoretische_Tiefpassfunktionen#Trapez.E2.80.93Tiefpass|rolloff factor]]&nbsp; (in the frequency domain):
-:$$r_f = \frac{ {f_2  - f_1 }}{ {f_2  + f_1 }}.$$
+:$$r_{\hspace{-0.05cm}f} = \frac{ {f_2  - f_1 }}{ {f_2  + f_1 }}.$$
 With these quantities, the associated time function (see middle graph) is:
-:$$x( t ) = X_0  \cdot \Delta f \cdot {\mathop{\rm si}\nolimits} ( { {\rm \pi}   \cdot \Delta f \cdot t} ) \cdot {\mathop{\rm si}\nolimits} ( { {\rm \pi}  \cdot  r_f \cdot \Delta f\cdot t} ).$$
+:$$x( t ) = X_0  \cdot \Delta f \cdot {\mathop{\rm si}\nolimits} ( { {\rm \pi}   \cdot \Delta f \cdot t} ) \cdot {\mathop{\rm si}\nolimits} ( { {\rm \pi}  \cdot  r_{\hspace{-0.05cm}f} \cdot \Delta f\cdot t} ).$$
 Here&nbsp; $\text{si}(x) = \text{sin}(x)/x$&nbsp; is the so-called&nbsp; "splitting function".
@@ Line 31: / Line 31: @@
 *the rolloff factor (in the time domain) with comparable definition as&nbsp; $r_f$:
-:$$r_t = \frac{ {t_2  - t_1 }}{ {t_2  + t_1 }}.$$
+:$$r_{\hspace{-0.05cm}t} = \frac{ {t_2  - t_1 }}{ {t_2  + t_1 }}.$$
 Let be&nbsp; $y_0 = 4\,\text{V}$,&nbsp; $\Delta t = 1\,\text{ms}$&nbsp; and&nbsp; $r_t = 0.5$.
@@ Line 59: / Line 59: @@
 |type="{}"}
 $\Delta f \ = \ $ { 4 3% } &nbsp;$\text{kHz}$
-$r_f \hspace{0.35cm} = \ $ { 0.5 3% }
+$r_{\hspace{-0.05cm}f} \hspace{0.35cm} = \ $ { 0.5 3% }
 {What are the signal values of&nbsp; $x(t)$&nbsp; at&nbsp; $t = 0$,&nbsp; $t = T$&nbsp; and&nbsp; $t = T/2$?

@@ Line 45: / Line 45: @@
 *You can check your results using the two interactive applets &nbsp;
-::[[Applets:Pulses_and_Spectra|Pulses and Spectra]],
+:[[Applets:Pulses_and_Spectra|Pulses and Spectra]],
-::[[Applets:Frequency_%26_Impulse_Responses|Frequency & Impulse Responses]].
+:[[Applets:Frequency_%26_Impulse_Responses|Frequency & Impulse Responses]].
@@ Line 56: / Line 56: @@
 <quiz display=simple>
-{For the given parameters, what is the equivalent bandwidth and rolloff factor of the spectrum&nbsp; $X(f)$?
+{What are the equivalent bandwidth and the rolloff factor of the spectrum&nbsp; $X(f)$&nbsp; for the given parameters?
 |type="{}"}
 $\Delta f \ = \ $ { 4 3% } &nbsp;$\text{kHz}$
@@ Line 67: / Line 67: @@
 $x(t=T/2)\ = \ $ { 2.293 3% } &nbsp;$\text{V}$
-{What is the spectrum&nbsp; $Y(f)$&nbsp; of the trapezoidal pulse witht&nbsp; $y_0 = 4\,\text{V}$,&nbsp; $\Delta t = 1\,\text{ms}$&nbsp;  and&nbsp; $r_t = 0.5$? <br>What are the spectral values at the given frequencies?
+{What is the spectrum&nbsp; $Y(f)$&nbsp; of the trapezoidal pulse with&nbsp; $y_0 = 4\,\text{V}$,&nbsp; $\Delta t = 1\,\text{ms}$&nbsp;  and&nbsp; $r_t = 0.5$? <br>What are the spectral values at the given frequencies?
 |type="{}"}
 $Y(f = 0)\hspace{0.2cm} = \ $ { 4 3% }  &nbsp;$\text{mV/Hz}$

@@ Line 45: / Line 45: @@
 *You can check your results using the two interactive applets &nbsp;
-::[[Applets:Impulse_und_Spektren|Impulses and Spectra]],
+::[[Applets:Pulses_and_Spectra|Pulses and Spectra]],
 ::[[Applets:Frequency_%26_Impulse_Responses|Frequency & Impulse Responses]].

@@ Line 5: / Line 5: @@
 [[File:P_ID508__Sig_A_3_4.png|250px|right|frame|Trapezspektrum & Trapezimpuls]]
-We consider here a trapezoidal spectral function&nbsp; $X(f)$&nbsp; according to the upper graph, which is completely described by the three parameters&nbsp; $X_0$,&nbsp; $f_1$&nbsp; and&nbsp; $f_2$&nbsp;. For the two corner frequencies,&nbsp;  $f_2 > 0$&nbsp; and&nbsp; $0 \leq f_1 \leq f_2$ always apply.
+We consider here a trapezoidal spectral function&nbsp; $X(f)$&nbsp; according to the upper graph, which is completely described by the three parameters&nbsp; $X_0$,&nbsp; $f_1$&nbsp; and&nbsp; $f_2$.&nbsp; For the two corner frequencies,&nbsp;  $f_2 > 0$&nbsp; and&nbsp; $0 \leq f_1 \leq f_2$ always apply.
 Instead of the corner frequencies&nbsp; $f_1$&nbsp; and&nbsp; $f_2$&nbsp;, the following two descriptive variables can also be used:
-*the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Reciprocity_Theorem_of_Time_duration_and_Bandwidth|equivalent bandwidth]]:
+*the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Reciprocity_Theorem_of_time_duration_and_bandwidth|equivalent bandwidth]]:
 :$$\Delta f = f_1  + f_2,$$
 *the so-called&nbsp; [[Linear_and_Time_Invariant_Systems/Einige_systemtheoretische_Tiefpassfunktionen#Trapez.E2.80.93Tiefpass|rolloff factor]]&nbsp; (in the frequency domain):
 :$$r_f = \frac{ {f_2  - f_1 }}{ {f_2  + f_1 }}.$$
-With these quantities, the associated time function (see middle graph) is:-
+With these quantities, the associated time function (see middle graph) is:
 :$$x( t ) = X_0  \cdot \Delta f \cdot {\mathop{\rm si}\nolimits} ( { {\rm \pi}   \cdot \Delta f \cdot t} ) \cdot {\mathop{\rm si}\nolimits} ( { {\rm \pi}  \cdot  r_f \cdot \Delta f\cdot t} ).$$
-Here&nbsp; $\text{si}(x) = \text{sin}(x)/x$&nbsp; is the so-called splitting function.
+Here&nbsp; $\text{si}(x) = \text{sin}(x)/x$&nbsp; is the so-called&nbsp; "splitting function".
-In this example, the numerical values&nbsp; $X_0 = 10^{–3}\,\text{V/Hz}$,&nbsp; $f_1 = 1\,\text{kHz}$&nbsp; and&nbsp; $f_2 = 3\,\text{kHz}$&nbsp; are to be used. The time&nbsp; $T = 1/\Delta f$&nbsp; is only used for standardisation purposes.
+In this example, the numerical values&nbsp; $X_0 = 10^{–3}\,\text{V/Hz}$,&nbsp; $f_1 = 1\,\text{kHz}$&nbsp; and&nbsp; $f_2 = 3\,\text{kHz}$&nbsp; are to be used.&nbsp; The time&nbsp; $T = 1/\Delta f$&nbsp; is only used for normative purpose.
-From subtask&nbsp; '''(3)'''&nbsp;  a trapezoidal signal&nbsp; $y(t)$&nbsp; is considered, which is identical in shape to the spectrum&nbsp; $X(f)$&nbsp;.
 The following can be used here as descriptive variables:
 *the pulse amplitude&nbsp; $y_0 = y(t = 0)$,
-*the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Reciprocity_Theorem_of_Time_duration_and_Bandwidth|equivalent pulse duration]]&nbsp; (defined via the rectangle with the same area):
+*the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Reciprocity_Theorem_of_time_duration_and_bandwidth|equivalent pulse duration]]&nbsp; (defined via the rectangle&ndash;in&ndash;time with the same area):
 :$$\Delta t = t_1  + t_2,$$
@@ Line 32: / Line 33: @@
 :$$r_t = \frac{ {t_2  - t_1 }}{ {t_2  + t_1 }}.$$
-Let&nbsp; $y_0 = 4\,\text{V}$,&nbsp; $\Delta t = 1\,\text{ms}$&nbsp; and&nbsp; $r_t = 0.5$.
+Let be&nbsp; $y_0 = 4\,\text{V}$,&nbsp; $\Delta t = 1\,\text{ms}$&nbsp; and&nbsp; $r_t = 0.5$.
@@ Line 42: / Line 41: @@
 ''Hints:''
-*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Transform_Laws|Fourier Transform Laws]].
+*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Transform_Theorems|Fourier Transform Theorems]].
-*Use the&nbsp;  [[Signal_Representation/Fourier_Transform_Laws#Vertauschungssatz|Vertauschungssatz]]&nbsp; and the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Similarity_Theorem|Similarity Theorem]].
+*Use the&nbsp;  [[Signal_Representation/Fourier_Transform_Theorems#Duality_Theorem|Duality Theorem]]&nbsp; and the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Similarity_Theorem|Similarity Theorem]].
-*You can check your results using the two interactive applets &nbsp; [[Applets:Impulse_und_Spektren|Impulses and Spectra]]&nbsp; sowie&nbsp;  [[Applets:Frequenzgang_und_Impulsantwort|Frequenzgang und Impulsantwort]]&nbsp;.
+*You can check your results using the two interactive applets &nbsp;

← Older revision		Revision as of 12:38, 23 March 2021
Line 131:		Line 131:

	__NOEDITSECTION__		__NOEDITSECTION__
−	[[Category:~~Exercises for~~ Signal Representation\|^3.3 Fourier Transform Theorems^]]	+	[[Category:Signal Representation: Exercises\|^3.3 Fourier Transform Theorems^]]

@@ Line 127: / Line 127: @@
 :$$Y( {f = 1\;{\rm{kHz}}} ) = Y_0  \cdot {\mathop{\rm si}\nolimits} ( {{ {\rm{\pi }}}/{2}} ) \cdot {\mathop{\rm si}\nolimits} ( {{ {\rm{\pi }}}/{4}} ) \hspace{0.15 cm}\underline{= 2.293\,{\rm{mV/Hz}}}{\rm{.}}$$
-*In subtask&nbsp; '''(3)'''&nbsp; this spectral value occurred at the frequency&nbsp; $f = 0.5\,\rm{kHz}$&nbsp;  aufgetreten.
+*In subtask&nbsp; '''(3)'''&nbsp; this spectral value occurred at the frequency&nbsp; $f = 0.5\,\rm{kHz}$&nbsp;.
 {{ML-Fuß}}
 __NOEDITSECTION__
 [[Category:Exercises for Signal Representation|^3.3 Fourier Transform Theorems^]]

@@ Line 55: / Line 55: @@
 <quiz display=simple>
-{Wie groß sind bei den gegebenen Parametern die äquivalente Bandbreite und der Rolloff-Faktor des Spektrums&nbsp; $X(f)$?
+{For the given parameters, what is the equivalent bandwidth and rolloff factor of the spectrum&nbsp; $X(f)$?
 |type="{}"}
 $\Delta f \ = \ $ { 4 3% } &nbsp;$\text{kHz}$
 $r_f \hspace{0.35cm} = \ $ { 0.5 3% }
-{Wie groß sind die Signalwerte von&nbsp; $x(t)$&nbsp; bei&nbsp; $t = 0$,&nbsp; $t = T$&nbsp; und&nbsp; $t = T/2$?
+{What are the signal values of&nbsp; $x(t)$&nbsp; at&nbsp; $t = 0$,&nbsp; $t = T$&nbsp; and&nbsp; $t = T/2$?
 |type="{}"}
 $x(t=0)\hspace{0.2cm} = \ $ { 4 3% } &nbsp;$\text{V}$
@@ Line 66: / Line 66: @@
 $x(t=T/2)\ = \ $ { 2.293 3% } &nbsp;$\text{V}$
-{Wie lautet das Spektrum&nbsp; $Y(f)$&nbsp; des Trapezimpulses mit&nbsp; $y_0 = 4\,\text{V}$,&nbsp; $\Delta t = 1\,\text{ms}$&nbsp;  und&nbsp; $r_t = 0.5$? <br>Wie groß sind die Spektralwerte bei den angegebenen Frequenzen?
+{What is the spectrum&nbsp; $Y(f)$&nbsp; of the trapezoidal pulse witht&nbsp; $y_0 = 4\,\text{V}$,&nbsp; $\Delta t = 1\,\text{ms}$&nbsp;  and&nbsp; $r_t = 0.5$? <br>What are the spectral values at the given frequencies?
 |type="{}"}
 $Y(f = 0)\hspace{0.2cm} = \ $ { 4 3% }  &nbsp;$\text{mV/Hz}$
@@ Line 72: / Line 72: @@
 $Y(f = 1.0 \,\text{kHz})\ = \ $ { 0. } &nbsp;$\text{mV/Hz}$
-{Welche Spektralwerte ergeben sich mit&nbsp; $y_0 = 8\,\text{V}$,&nbsp; $\Delta t = 0.5\,\text{ms }$&nbsp; und&nbsp; $r_t = 0.5$?
+{Which spectral values result with&nbsp; $y_0 = 8\,\text{V}$,&nbsp; $\Delta t = 0.5\,\text{ms }$&nbsp; and&nbsp; $r_t = 0.5$?
 |type="{}"}
 $Y(f=0)\hspace{0.2cm}= \ $ { 4 3% } &nbsp;$\text{mV/Hz}$
@@ Line 79: / Line 79: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Die äquivalente Bandbreite ist per Definition gleich der Breite des flächengleichen Rechtecks:
+'''(1)'''&nbsp; The equivalent bandwidth is by definition equal to the width of the equal-area rectangle:
 :$$\Delta f = f_1  + f_2  \hspace{0.15 cm}\underline{= 4\;{\rm{kHz}}}{\rm{.}}$$
-*Für den Rolloff-Faktor gilt:
+*For the rolloff factor holds:
 :$${ {r_f = }}\frac{ {f_2  - f_1 }}{ {f_2  + f_1 }}\hspace{0.15 cm}\underline{ = 0.5}.$$
-'''(2)'''&nbsp; Der Maximalwert des Impulses&nbsp; $x(t)$&nbsp; tritt zum Zeitpunkt&nbsp; $t = 0$&nbsp; auf:
+'''(2)'''&nbsp; The maximum value of the pulse&nbsp; $x(t)$&nbsp; occurs at time&nbsp; $t = 0$&nbsp;:
 :$$x_0 = x(t = 0) = X_0 \cdot \Delta f \hspace{0.15 cm}\underline{= 4\, \text{V}}.$$
-*Zum Zeitpunkt&nbsp; $t = T = 1/\Delta f$&nbsp; gilt aufgrund von&nbsp; $\text{si}(\pi) = 0$:
+*At time&nbsp; $t = T = 1/\Delta f$&nbsp; applies due to&nbsp; $\text{si}(\pi) = 0$:
 :$$x( {t = T} ) = x_0  \cdot {\mathop{\rm si}\nolimits} ( {\rm{\pi }} ) \cdot {\mathop{\rm si}\nolimits} ( { { {\rm{\pi }}}/{2}} )\hspace{0.15 cm}\underline{ = 0}.$$
-*Auch bei allen Vielfachen von&nbsp; $T$&nbsp; weist&nbsp; $x(t)$&nbsp; Nulldurchgänge auf. Zum Zeitpunkt&nbsp; $t = T/2$&nbsp; gilt:
+*Also at all multiples of&nbsp; $T$&nbsp; &nbsp; $x(t)$&nbsp; exhibits zero crossings. At time&nbsp; $t = T/2$&nbsp; holds:
 :$$x( {t = T/2} ) = x_0  \cdot {\mathop{\rm si}\nolimits} ( { { {\rm{\pi }}}/{2}} ) \cdot {\mathop{\rm si}\nolimits}( { { {\rm{\pi }}}/{4}} ) = x_0  \cdot \frac{ { 1 \cdot \sqrt 2 /2}}{ { {\rm{\pi /}}2 \cdot {\rm{\pi /4}}}} = x_0  \cdot \frac{ {4 \cdot \sqrt 2 }}{ { {\rm{\pi }}^{\rm{2}} }} \hspace{0.15 cm}\underline{= 2.293\;{\rm{V}}}{\rm{.}}$$
-'''(3)'''&nbsp; Die zum trapezförmigen Spektrum&nbsp; $X(f)$&nbsp; zugehörige Zeitfunktion lautet entsprechend der Angabe:
+'''(3)'''&nbsp; The time function associated with the trapezoidal spectrum&nbsp; $X(f)$&nbsp; is according to the specification:
 :$$x( t ) = X_0  \cdot \Delta f \cdot {\mathop{\rm si}\nolimits} ( { {\rm{\pi }} \cdot \Delta f \cdot t} ) \cdot {\mathop{\rm si}\nolimits} ( { {\rm{\pi }} \cdot r_f \cdot \Delta f \cdot t} ).$$
-*Da sowohl&nbsp; $X(f)$&nbsp; als auch&nbsp; $x(t)$&nbsp; reell sind und zudem&nbsp; $y(t)$&nbsp; formgleich mit&nbsp; $X(f)$&nbsp; ist, erhält man unter Berücksichtigung aller Äquivalenzen für die Spektralfunktion des Trapezimpulses:
+*Since both&nbsp; $X(f)$&nbsp; and&nbsp; $x(t)$&nbsp; are real and, moreover,&nbsp; $y(t)$&nbsp; is of the same form as&nbsp; $X(f)$&nbsp; , we obtain, taking into account all equivalences for the spectral function of the trapezoidal pulse:
 :$$Y( f ) = y_0  \cdot \Delta t \cdot {\mathop{\rm si}\nolimits} ( { {\rm{\pi }} \cdot \Delta t \cdot f} ) \cdot {\mathop{\rm si}\nolimits} ( { {\rm{\pi }} \cdot r_t \cdot \Delta t \cdot f} ).$$
-*Insbesondere gilt:
+*In particular, holds:
 :$$Y( {f = 0} ) = y_0  \cdot \Delta t \hspace{0.15 cm}\underline{= 4 \;{\rm{mV/Hz}}}{\rm{,}}$$
@@ Line 120: / Line 120: @@
-'''(4)'''&nbsp; Der Spektralwert bei der Frequenz&nbsp; $f = 0$&nbsp; wird nicht verändert: &nbsp;
+'''(4)'''&nbsp; The spectral value at frequency&nbsp; $f = 0$&nbsp; is not changed: &nbsp;
 :$$Y_0 = y_0 \cdot \Delta t \hspace{0.15 cm}\underline{= 4 \,\rm{mV/Hz}}.$$
-*Da nun aber die Zeitfunktion nur halb so breit ist, verbreitert sich das Spektrum um den Faktor&nbsp; $2$:
+*But since the time function is now only half as wide, the spectrum widens by a factor of&nbsp; $2$:
 :$$Y( {f = 1\;{\rm{kHz}}} ) = Y_0  \cdot {\mathop{\rm si}\nolimits} ( {{ {\rm{\pi }}}/{2}} ) \cdot {\mathop{\rm si}\nolimits} ( {{ {\rm{\pi }}}/{4}} ) \hspace{0.15 cm}\underline{= 2.293\,{\rm{mV/Hz}}}{\rm{.}}$$
-*In der Teilaufgabe&nbsp; '''(3)'''&nbsp; ist dieser Spektralwert bei der Frequenz&nbsp; $f = 0.5\,\rm{kHz}$&nbsp;  aufgetreten.
+*In subtask&nbsp; '''(3)'''&nbsp; this spectral value occurred at the frequency&nbsp; $f = 0.5\,\rm{kHz}$&nbsp;  aufgetreten.
 {{ML-Fuß}}
 __NOEDITSECTION__
 [[Category:Exercises for Signal Representation|^3.3 Fourier Transform Theorems^]]

@@ Line 41: / Line 41: @@
-''Hinweise:''
+''Hints:''
-*Die Aufgabe gehört zum  Kapitel&nbsp; [[Signal_Representation/Fourier_Transform_Laws|Gesetzmäßigkeiten der Fouriertransformation]].
+*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Transform_Laws|Fourier Transform Laws]].
-*Verwenden Sie zur Lösung den&nbsp;  [[Signal_Representation/Fourier_Transform_Laws#Vertauschungssatz|Vertauschungssatz]]&nbsp; und den&nbsp; [[Signal_Representation/Fourier_Transform_Laws#.C3.84hnlichkeitssatz|Ähnlichkeitssatz]].
+*Use the&nbsp;  [[Signal_Representation/Fourier_Transform_Laws#Vertauschungssatz|Vertauschungssatz]]&nbsp; and the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Similarity_Theorem|Similarity Theorem]].
-*Sie können Ihre Ergebnisse anhand der beiden  interaktiven Applets&nbsp; [[Applets:Impulse_und_Spektren|Impulse und Spektren]]&nbsp; sowie&nbsp;  [[Applets:Frequenzgang_und_Impulsantwort|Frequenzgang und Impulsantwort]]&nbsp; überprüfen.
+*You can check your results using the two interactive applets &nbsp; [[Applets:Impulse_und_Spektren|Impulses and Spectra]]&nbsp; sowie&nbsp;  [[Applets:Frequenzgang_und_Impulsantwort|Frequenzgang und Impulsantwort]]&nbsp;.
@@ Line 51: / Line 51: @@
-===Fragebogen===
+===Questions===
 <quiz display=simple>