Aufgaben:Exercise 3.4Z: Trapezoid, Rectangle and Triangle - Revision history

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

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

Noah at 19:32, 23 January 2021

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Noah at 19:31, 23 January 2021

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Noah at 19:24, 23 January 2021

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Noah at 19:22, 23 January 2021

2021-01-23T19:22:47Z

Oezdemir: Oezdemir moved page Aufgaben:Aufgabe 3.4Z: Trapez, Rechteck und Dreieck to Aufgaben:Exercise 3.4Z: Trapezoid, Rectangle and Triangle

2020-11-02T15:27:52Z

Oezdemir moved page Aufgabe 3.4Z: Trapez, Rechteck und Dreieck to Exercise 3.4Z: Trapezoid, Rectangle and Triangle

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID510__Sig_Z_3_4.png|right|frame|Trapezoidal pulse and its limiting cases&nbsp; "Rectangle"&nbsp; and&nbsp; "Triangle&rdquo; ]]
+[[File:P_ID510__Sig_Z_3_4.png|right|frame|Trapezoidal pulse and its limiting cases&nbsp; "Rectangle"&nbsp; and&nbsp; "Triangle" ]]
 Three different pulse shapes are considered.&nbsp; The pulse&nbsp; ${x(t)}$&nbsp; is trapezoidal.&nbsp; For&nbsp; $| t | < t_1 = 4 \,\text{ms}$&nbsp;the time course is constant equal to&nbsp; ${A} = 1\, \text{V}$.&nbsp; Afterwards,&nbsp; ${x(t)}$&nbsp; drops linearly to the value zero until the time&nbsp; $t_2 = 6\, \text{ms}$.&nbsp;

@@ Line 69: / Line 69: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp;  The equivalent pulse duration is&nbsp; $\Delta t = t_1 + t_2 \;\underline{= 10 \,\text{ms}}$&nbsp; and the rolloff factor&nbsp; $r_t = 2/10 \;\underline{= 0.2}$.
+'''(1)'''&nbsp;  The equivalent pulse duration is&nbsp; $\Delta t = t_1 + t_2 \;\underline{= 10 \,\text{ms}}$&nbsp; and the rolloff factor is&nbsp; $r_t = 2/10 \;\underline{= 0.2}$.
@@ Line 76: / Line 76: @@
 *The spectral value at&nbsp; $f = 0$&nbsp; is&nbsp; $A \cdot \Delta t = 10 \,\text{mV/Hz}$.
 *Since&nbsp; ${X(f)}$&nbsp; is real and can assume both positive and negative values, only the two phase values&nbsp; $0$&nbsp; und&nbsp; $\pi$&nbsp; are possible.
-*Zeros exist due to the first si-function at all multiples of&nbsp; $1/\Delta t = 100\, \text{Hz}$.
+*Zeros exist due to the first si&ndash;function at all multiples of&nbsp; $1/\Delta t = 100\, \text{Hz}$.
-*The second si function leads to zero crossings at intervals of&nbsp; $1/(r_t \cdot \Delta t) = 500 \,\text{Hz}$. These coincide exactly with the zeros of the first si-function.
+*The second si&ndash;function leads to zero crossings at intervals of&nbsp; $1/(r_t \cdot \Delta t) = 500 \,\text{Hz}$.&nbsp; These coincide exactly with the zeros of the first si&ndash;function.
@@ Line 89: / Line 89: @@
 '''(4)'''&nbsp; Proposed <u>solutions 2 and 3</u> are correct:
 *For the triangular pulse, the rolloff factor is&nbsp; $r_t = 1$.
-*The equivalent pulse duration is&nbsp; $\Delta t = 10 \,\text{ms}$. It follows that &nbsp; $D( f ) = A \cdot \Delta t \cdot {\mathop{\rm si}\nolimits} ^2 ( {{\rm{\pi }} \cdot \Delta t \cdot f} )$&nbsp; and&nbsp; $D( f = 0) = A \cdot \Delta t  = X( f = 0)$.
+*The equivalent pulse duration is&nbsp; $\Delta t = 10 \,\text{ms}$.&nbsp; It follows that &nbsp; $D( f ) = A \cdot \Delta t \cdot {\mathop{\rm si}\nolimits} ^2 ( {{\rm{\pi }} \cdot \Delta t \cdot f} )$&nbsp; and&nbsp; $D( f = 0) = A \cdot \Delta t  = X( f = 0)$.
-*Since&nbsp; ${D(f)}$&nbsp; cannot become negative, the phase&nbsp; $[{\rm arc} \; {D(f)}]$&nbsp; is always zero. The phase value&nbsp; $\pi$&nbsp; $(180°)$&nbsp; is therefore not possible with the triangular form.
+*Since&nbsp; ${D(f)}$&nbsp; cannot become negative, the phase&nbsp; $[{\rm arc} \; {D(f)}]$&nbsp; is always zero.&nbsp; The phase value&nbsp; $\pi$&nbsp; $(180°)$&nbsp; is therefore not possible with the triangular pulse.
 {{ML-Fuß}}

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID510__Sig_Z_3_4.png|right|frame|Trapezimpuls und dessen Grenzfälle &bdquo;Rechteck&rdquo; und &bdquo;Dreieck&rdquo; ]]
+[[File:P_ID510__Sig_Z_3_4.png|right|frame|Trapezoidal pulse and its limiting cases&nbsp; "Rectangle"&nbsp; and&nbsp; "Triangle&rdquo; ]]
-Three different pulse shapes are considered. The pulse&nbsp; ${x(t)}$&nbsp; is trapezoidal. For&nbsp; $| t | < t_1 = 4 \,\text{ms}$&nbsp;the time course is constant equal to&nbsp; ${A} = 1\, \text{V}$. Afterwards,&nbsp; ${x(t)}$&nbsp; drops linearly to the value zero until the time&nbsp; $t_2 = 6\, \text{ms}$&nbsp;.
+Three different pulse shapes are considered.&nbsp; The pulse&nbsp; ${x(t)}$&nbsp; is trapezoidal.&nbsp; For&nbsp; $| t | < t_1 = 4 \,\text{ms}$&nbsp;the time course is constant equal to&nbsp; ${A} = 1\, \text{V}$.&nbsp; Afterwards,&nbsp; ${x(t)}$&nbsp; drops linearly to the value zero until the time&nbsp; $t_2 = 6\, \text{ms}$.&nbsp;
-* the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Reciprocity_Theorem_of_Time_duration_and_Bandwidth|equivalent bandwidth]]&nbsp;
+The spectral function of the trapezoidal pulse is
-* and the so-called roll-off factor (in the time domain)
+:$$X( f ) = A \cdot \Delta t \cdot {\mathop{\rm si}\nolimits}( {{\rm \pi}  \cdot \Delta t \cdot f} ) \cdot  \hspace{0.1cm}{\mathop{\rm si}\nolimits}( {{\rm \pi}\cdot \Delta t \cdot  r_t \cdot  f} ).$$
-is the spectral function of the trapezoidal pulse:
+with the two derived system quantities, namely
-:$$X( f ) = A \cdot \Delta t \cdot {\mathop{\rm si}\nolimits}( {{\rm \pi}  \cdot \Delta t \cdot f} ) \cdot  \hspace{0.1cm}{\mathop{\rm si}\nolimits}( {{\rm \pi}\cdot \Delta t \cdot  r_t \cdot  f} ).$$
-Furthermore, the rectangular momentum&nbsp; ${r(t)}$&nbsp; and the triangular momentum&nbsp; ${d(t)}$&nbsp; are also shown in the graph, both of which can be interpreted as limiting cases of the trapezoidal momentum&nbsp; ${x(t)}$&nbsp;.
+* the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Reciprocity_Theorem_of_Time_duration_and_Bandwidth|equivalent bandwidth]],&nbsp;
@@ Line 24: / Line 27: @@
 ''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]].
 *You can check your results using the two interactive applets &nbsp;
-*You can check your results using the two interactive applets &nbsp; [[Applets:Impulse_und_Spektren|Impulses and Spectra]]&nbsp; and&nbsp;  [[Applets:Frequenzgang_und_Impulsantwort|Frequenzgang und Impulsantwort]]&nbsp;.
+:[[Applets:Pulses_and_Spectra|Pulses and Spectra]],
@@ Line 33: / Line 37: @@
 <quiz display=simple>
-{What is the equivalent impulse duration and the rolloff factor of&nbsp; ${x(t)}$?
+{What is the equivalent pulse duration and the rolloff factor of&nbsp; ${x(t)}$?
 |type="{}"}
 $\Delta t \ = \ $ { 10 3% } &nbsp;$\text{ms}$
@@ Line 42: / Line 46: @@
 |type="[]"}
 - The spectral value at frequency&nbsp; $f = 0$&nbsp; is equal to&nbsp; $20 \,\text{mV/Hz}$.
-+ For the phase function the values&nbsp; $0$&nbsp; or&nbsp; $\pi$&nbsp; $(180^{\circ})$&nbsp; are possible.
++ For the phase function the values&nbsp; $0$&nbsp; and&nbsp; $\pi$&nbsp; $(180^{\circ})$&nbsp; are possible.
-+ ${X(f)}$&nbsp; only has zeros at all multiples of&nbsp; $100 \,\text{Hz}$&nbsp; auf.
++ ${X(f)}$&nbsp; only has zeros at all multiples of&nbsp; $100 \,\text{Hz}$.
-{Which statements are true regarding the spectral function&nbsp; ${R(f)}$&nbsp; ?
+{Which statements are true regarding the spectral function&nbsp; ${R(f)}$&nbsp;?
 |type="[]"}
 + The spectral value at frequency&nbsp; $f = 0$&nbsp; is equal to&nbsp; ${X(f = 0)}$.
-+ The values $0$ or&nbsp; $\pi$&nbsp; $(180^{\circ})$&nbsp; are possible for the phase function.
++ The values&nbsp; $0$&nbsp; and&nbsp; $\pi$&nbsp; $(180^{\circ})$&nbsp; are possible for the phase function.
-+ ${R(f)}$&nbsp; only has zeros at all multiples of &nbsp; $100 \,\text{Hz}$ auf..
++ ${R(f)}$&nbsp; only has zeros at all multiples of&nbsp; $100 \,\text{Hz}$.
-{Which statements are true regarding the spectral function&nbsp; ${D(f)}$&nbsp; ?
+{Which statements are true regarding the spectral function&nbsp; ${D(f)}$&nbsp;?
 |type="[]"}
 + The spectral value at frequency&nbsp; $f = 0$&nbsp; is equal to&nbsp; ${X(f = 0)}$.
-- The values $0$ or&nbsp; $\pi$&nbsp; $(180^{\circ})$&nbsp; are possible for the phase function.
+- The values&nbsp; $0$&nbsp; and&nbsp; $\pi$&nbsp; $(180^{\circ})$&nbsp; are possible for the phase function.
-+ ${D(f)}$&nbsp; only has zeros at all multiples of &nbsp; $100 \,\text{Hz}$ auf.
++ ${D(f)}$&nbsp; only has zeros at all multiples of&nbsp; $100 \,\text{Hz}$.

← Older revision		Revision as of 12:39, 23 March 2021
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	__NOEDITSECTION__		__NOEDITSECTION__
−	[[Category:~~Exercises for~~ Signal Representation\|^3.3 Fourier Transform Theorems^]]	+	[[Category:Signal Representation: Exercises\|^3.3 Fourier Transform Theorems^]]

← Older revision		Revision as of 19:32, 23 January 2021
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	*This exercise belongs to the chapter  [[Signal_Representation/Fourier_Transform_Laws\|Fourier Transform Laws]].		*This exercise belongs to the chapter  [[Signal_Representation/Fourier_Transform_Laws\|Fourier Transform Laws]].

−	*You can check your results using the two interactive applets   [[Applets:Impulse_und_Spektren\|Impulses and Spectra]]  ~~sowie~~  [[Applets:Frequenzgang_und_Impulsantwort\|Frequenzgang und Impulsantwort]] .	+	*You can check your results using the two interactive applets   [[Applets:Impulse_und_Spektren\|Impulses and Spectra]]  and  [[Applets:Frequenzgang_und_Impulsantwort\|Frequenzgang und Impulsantwort]] .

@@ Line 65: / Line 65: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp;  Die äquivalente Impulsdauer ist&nbsp; $\Delta t = t_1 + t_2 \;\underline{= 10 \,\text{ms}}$&nbsp; und der Rolloff-Faktor&nbsp; $r_t = 2/10 \;\underline{= 0.2}$.
+'''(1)'''&nbsp;  The equivalent pulse duration is&nbsp; $\Delta t = t_1 + t_2 \;\underline{= 10 \,\text{ms}}$&nbsp; and the rolloff factor&nbsp; $r_t = 2/10 \;\underline{= 0.2}$.
-'''(2)'''&nbsp; Richtig sind die <u>Lösungsvorschläge 2 und 3</u>:
+'''(2)'''&nbsp; Proposed <u>solutions 2 and 3</u> are correct:
-*Der Spektralwert bei&nbsp; $f = 0$&nbsp; beträgt&nbsp; $A \cdot \Delta t = 10 \,\text{mV/Hz}$.
+*The spectral value at&nbsp; $f = 0$&nbsp; is&nbsp; $A \cdot \Delta t = 10 \,\text{mV/Hz}$.
-*Da&nbsp; ${X(f)}$&nbsp; reell ist und sowohl positive als auch negative Werte annehmen kann, sind nur die zwei Phasenwerte&nbsp; $0$&nbsp; und&nbsp; $\pi$&nbsp; möglich.
+*Since&nbsp; ${X(f)}$&nbsp; is real and can assume both positive and negative values, only the two phase values&nbsp; $0$&nbsp; und&nbsp; $\pi$&nbsp; are possible.
-*Nullstellen gibt es aufgrund der ersten si-Funktion bei allen Vielfachen von&nbsp; $1/\Delta t = 100\, \text{Hz}$.
+*Zeros exist due to the first si-function at all multiples of&nbsp; $1/\Delta t = 100\, \text{Hz}$.
-*Die zweite si-Funktion führt zu Nulldurchgängen im Abstand&nbsp; $1/(r_t \cdot \Delta t) = 500 \,\text{Hz}$. Diese fallen exakt mit den Nullstellen der ersten si-Funktion zusammen.
+*The second si function leads to zero crossings at intervals of&nbsp; $1/(r_t \cdot \Delta t) = 500 \,\text{Hz}$. These coincide exactly with the zeros of the first si-function.
-'''(3)'''&nbsp;   <u>Alle Lösungsvorschläge</u> sind zutreffend:
+'''(3)'''&nbsp;   <u>All proposed solutions</u> are correct:
-*Mit der äquivalenten Impulsdauer&nbsp; $\Delta t = 10 \,\text{ms}$&nbsp; und dem Rolloff-Faktor&nbsp; $r_t = 0$&nbsp; erhält man: &nbsp; $R( f ) = A \cdot \Delta t \cdot {\mathop{\rm si}\nolimits} ( {{\rm{\pi }} \cdot \Delta t \cdot f} ).$
+*With the equivalent pulse duration&nbsp; $\Delta t = 10 \,\text{ms}$&nbsp; and the rolloff factor&nbsp; $r_t = 0$&nbsp; one obtains: &nbsp; $R( f ) = A \cdot \Delta t \cdot {\mathop{\rm si}\nolimits} ( {{\rm{\pi }} \cdot \Delta t \cdot f} ).$
-*Daraus folgt&nbsp; $R( f = 0) = A \cdot \Delta t = X( f = 0).$
+*It follows that&nbsp; $R( f = 0) = A \cdot \Delta t = X( f = 0).$
-'''(4)'''&nbsp; Hier sind die <u>Lösungsvorschläge 1 und 3</u>  zutreffend:
+'''(4)'''&nbsp; Proposed <u>solutions 2 and 3</u> are correct:
-*Beim Dreieckimpuls ist der Rolloff-Faktor&nbsp; $r_t = 1$.
+*For the triangular pulse, the rolloff factor is&nbsp; $r_t = 1$.
-*Die äquivalente Impulsdauer ist&nbsp; $\Delta t = 10 \,\text{ms}$. Daraus folgt &nbsp; $D( f ) = A \cdot \Delta t \cdot {\mathop{\rm si}\nolimits} ^2 ( {{\rm{\pi }} \cdot \Delta t \cdot f} )$&nbsp; und&nbsp; $D( f = 0) = A \cdot \Delta t  = X( f = 0)$.
+*The equivalent pulse duration is&nbsp; $\Delta t = 10 \,\text{ms}$. It follows that &nbsp; $D( f ) = A \cdot \Delta t \cdot {\mathop{\rm si}\nolimits} ^2 ( {{\rm{\pi }} \cdot \Delta t \cdot f} )$&nbsp; and&nbsp; $D( f = 0) = A \cdot \Delta t  = X( f = 0)$.
-*Da&nbsp; ${D(f)}$&nbsp; nicht negativ werden kann, ist die Phase&nbsp; $[{\rm arc} \; {D(f)}]$&nbsp; stets Null. Der Phasenwert&nbsp; $\pi$&nbsp; $(180°)$&nbsp; ist also bei der Dreieckform nicht möglich.
+*Since&nbsp; ${D(f)}$&nbsp; cannot become negative, the phase&nbsp; $[{\rm arc} \; {D(f)}]$&nbsp; is always zero. The phase value&nbsp; $\pi$&nbsp; $(180°)$&nbsp; is therefore not possible with the triangular form.
 {{ML-Fuß}}

@@ Line 4: / Line 4: @@
 [[File:P_ID510__Sig_Z_3_4.png|right|frame|Trapezimpuls und dessen Grenzfälle &bdquo;Rechteck&rdquo; und &bdquo;Dreieck&rdquo; ]]
-Betrachtet werden drei unterschiedliche Impulsformen. Der Impuls&nbsp; ${x(t)}$&nbsp; ist trapezförmig. Für&nbsp; $| t | < t_1 = 4 \,\text{ms}$&nbsp; ist der Zeitverlauf konstant gleich&nbsp; ${A} = 1\, \text{V}$. Danach fällt&nbsp; ${x(t)}$&nbsp; bis zum Zeitpunkt&nbsp; $t_2 = 6\, \text{ms}$&nbsp; linear bis auf den Wert Null ab.
+Three different pulse shapes are considered. The pulse&nbsp; ${x(t)}$&nbsp; is trapezoidal. For&nbsp; $| t | < t_1 = 4 \,\text{ms}$&nbsp;the time course is constant equal to&nbsp; ${A} = 1\, \text{V}$. Afterwards,&nbsp; ${x(t)}$&nbsp; drops linearly to the value zero until the time&nbsp; $t_2 = 6\, \text{ms}$&nbsp;.
-Mit den beiden abgeleiteten Systemgrößen, nämlich
+* the&nbsp; [[Signal_Representation/Fourier_Transform_Laws#Reciprocity_Theorem_of_Time_duration_and_Bandwidth|equivalent bandwidth]]&nbsp;
 :$$\Delta t = t_1  + t_2$$
-* und dem so genannten Rolloff-Faktor (im Zeitbereich)
+* and the so-called roll-off factor (in the time domain)
 :$$r_t = \frac{t_2  - t_1 }{t_2  + t_1 }$$
-lautet die Spektralfunktion des Trapezimpulses:
+is the spectral function of the trapezoidal pulse:
 :$$X( f ) = A \cdot \Delta t \cdot {\mathop{\rm si}\nolimits}( {{\rm \pi}  \cdot \Delta t \cdot f} ) \cdot  \hspace{0.1cm}{\mathop{\rm si}\nolimits}( {{\rm \pi}\cdot \Delta t \cdot  r_t \cdot  f} ).$$
-Weiter sind in der Grafik noch der Rechteckimpuls&nbsp; ${r(t)}$&nbsp; und der Dreieckimpuls&nbsp; ${d(t)}$&nbsp; dargestellt, die beide als Grenzfälle des Trapezimpulses&nbsp; ${x(t)}$&nbsp; interpretiert werden können.
+Furthermore, the rectangular momentum&nbsp; ${r(t)}$&nbsp; and the triangular momentum&nbsp; ${d(t)}$&nbsp; are also shown in the graph, both of which can be interpreted as limiting cases of the trapezoidal momentum&nbsp; ${x(t)}$&nbsp;.
@@ Line 24: / Line 23: @@
-''Hinweise:''
+*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;.
-===Fragebogen===
+===Questions===
 <quiz display=simple>
-{Wie groß sind die äquivalente Impulsdauer und der Rolloff-Faktor von&nbsp; ${x(t)}$?
+{What is the equivalent impulse duration and the rolloff factor of&nbsp; ${x(t)}$?
 |type="{}"}
 $\Delta t \ = \ $ { 10 3% } &nbsp;$\text{ms}$
@@ Line 41: / Line 39: @@
-{Welche Aussagen sind hinsichtlich der Spektralfunktion&nbsp; ${X(f)}$&nbsp; zutreffend?
+{Which statements are true regarding the spectral function&nbsp; ${X(f)}$&nbsp;?
 |type="[]"}
-- Der Spektralwert bei der Frequenz&nbsp; $f = 0$&nbsp; ist gleich&nbsp; $20 \,\text{mV/Hz}$.
+- The spectral value at frequency&nbsp; $f = 0$&nbsp; is equal to&nbsp; $20 \,\text{mV/Hz}$.
-+ Für die Phasenfunktion sind die Werte&nbsp; $0$&nbsp; oder&nbsp; $\pi$&nbsp; $(180^{\circ})$&nbsp; möglich.
++ For the phase function the values&nbsp; $0$&nbsp; or&nbsp; $\pi$&nbsp; $(180^{\circ})$&nbsp; are possible.
-+ ${X(f)}$&nbsp; weist nur Nullstellen bei allen Vielfachen von&nbsp; $100 \,\text{Hz}$&nbsp; auf.
++ ${X(f)}$&nbsp; only has zeros at all multiples of&nbsp; $100 \,\text{Hz}$&nbsp; auf.
-{Welche Aussagen sind hinsichtlich der Spektralfunktion&nbsp; ${R(f)}$&nbsp; zutreffend?
+{Which statements are true regarding the spectral function&nbsp; ${R(f)}$&nbsp; ?
 |type="[]"}
 + Der Spektralwert bei der Frequenz&nbsp; $f = 0$&nbsp; ist gleich ${X(f = 0)}$.
@@ Line 57: / Line 55: @@
 {Welche Aussagen sind hinsichtlich der Spektralfunktion&nbsp; ${D(f)}$&nbsp; zutreffend?
 |type="[]"}
-+ Der Spektralwert bei der Frequenz&nbsp; $f = 0$&nbsp; ist gleich&nbsp; ${X(f = 0)}$.
++ The spectral value at frequency&nbsp; $f = 0$&nbsp; is equal to&nbsp; ${X(f = 0)}$.
-- Für die Phasenfunktion sind die Werte $0$ oder&nbsp; $\pi$&nbsp; $(180^{\circ})$&nbsp; möglich.
+- The values $0$ or&nbsp; $\pi$&nbsp; $(180^{\circ})$&nbsp; are possible for the phase function.
-+ ${D(f)}$&nbsp; weist nur Nullstellen bei allen Vielfachen von&nbsp; $100 \,\text{Hz}$ auf.
++ ${D(f)}$&nbsp; only has zeros at all multiples of &nbsp; $100 \,\text{Hz}$ auf.
@@ Line 65: / Line 63: @@
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
 '''(1)'''&nbsp;  Die äquivalente Impulsdauer ist&nbsp; $\Delta t = t_1 + t_2 \;\underline{= 10 \,\text{ms}}$&nbsp; und der Rolloff-Faktor&nbsp; $r_t = 2/10 \;\underline{= 0.2}$.