Aufgaben:Exercise 2.4Z: Triangular Function - Revision history

Guenter at 12:48, 15 April 2021

2021-04-15T12:48:40Z

Guenter at 12:44, 15 April 2021

2021-04-15T12:44:31Z

Guenter at 12:42, 15 April 2021

2021-04-15T12:42:05Z

Noah at 09:25, 26 March 2021

2021-03-26T09:25:10Z

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

2021-03-23T12:39:19Z

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

Noah at 18:21, 16 January 2021

2021-01-16T18:21:17Z

Noah at 18:15, 16 January 2021

2021-01-16T18:15:29Z

Noah at 18:07, 16 January 2021

2021-01-16T18:07:31Z

Oezdemir: Oezdemir moved page Aufgaben:Aufgabe 2.4Z: Triangular Function to Aufgaben:Exercise 2.4Z: Triangular Function

2020-11-02T14:11:40Z

Oezdemir moved page Aufgabe 2.4Z: Triangular Function to Exercise 2.4Z: Triangular Function

Oezdemir: Oezdemir moved page Aufgaben:Aufgabe 2.4Z: Dreiecksignal to Aufgaben:Aufgabe 2.4Z: Triangular Function

2020-11-02T14:11:17Z

Oezdemir moved page Aufgabe 2.4Z: Dreiecksignal to Aufgabe 2.4Z: Triangular Function

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 {{ML-Kopf}}
 '''(1)'''&nbsp;  Proposed <u>solutions 1 and 2</u> are correct:
-*The DC component is actually&nbsp; $T_1/T_0$. Since&nbsp; ${x(t)}$&nbsp; is an even function, all sine coefficients&nbsp; $B_n = 0$.
+*The DC coefficient is actually&nbsp; $T_1/T_0$.&nbsp; Since&nbsp; ${x(t)}$&nbsp; is an even function, all sine coefficients&nbsp; $B_n = 0$.
-*The even cosine coefficients&nbsp; $A_{2n}$&nbsp; only disappear if &nbsp; $T_1 = T_0/2$&nbsp;.
+*The even cosine coefficients&nbsp; $A_{2n}$&nbsp; only disappear if &nbsp; $T_1 = T_0/2$.&nbsp;
-*In this case the condition&nbsp; ${x(t)} = 2A_0 - x(t - T_0/2)$&nbsp; is fulfilled $($mit $A_0 = 0.5)$.
+*In this case the condition&nbsp; ${x(t)} = 2A_0 - x(t - T_0/2)$&nbsp; is fulfilled&nbsp; $($with $A_0 = 0.5)$.
@@ Line 84: / Line 84: @@
 *At time&nbsp; $t = 0$&nbsp; this gives:
 :$$x_3(t=0)=\frac{1}{4}+\frac{4}{\pi^2}\cdot \frac{29}{18}\approx 0.9 \hspace{0.5cm}\Rightarrow  \hspace{0.5cm}\varepsilon_3(t=0)=x_3(t=0)-x(t=0)\hspace{0.15cm}\underline{=-0.1}.$$
-* For the time&nbsp; $t = 0$&nbsp; and for multiples of the period&nbsp; $T_0$&nbsp; (peak of the triangular functions in each case)&nbsp; the deviation is greatest in terms of magnitude.
+* For the time&nbsp; $t = 0$&nbsp; and for multiples of the period&nbsp; $T_0$&nbsp; $($peak of the triangular functions in each case$)$&nbsp; the deviation is greatest in terms of magnitude.
 {{ML-Fuß}}

@@ Line 19: / Line 19: @@
 ''Hints:''
 *This exercise belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Series|Fourier Series]].
-*You can find a compact summary of the topic in the two learning videos
+*You can find a compact summary of the topic in the two (German language) learning videos
 ::[[Zur_Berechnung_der_Fourierkoeffizienten_(Lernvideo)|Zur Berechnung der Fourierkoeffizienten]] &nbsp; &rArr; &nbsp; "To calculate the Fourier coefficients"
 :: [[Eigenschaften_der_Fourierreihendarstellung_(Lernvideo)|Eigenschaften der Fourierreihendarstellung]] &nbsp; &rArr; &nbsp;   "Properties of the Fourier series representation"
@@ Line 33: / Line 33: @@
 {Which of the following statements are true for all permissible&nbsp; $T_0$&nbsp; and&nbsp; $T_1$&nbsp;?
 |type="[]"}
-+ The DC component is&nbsp; $A_0 = T_1/T_0$.
++ The DC coefficient is&nbsp; $A_0 = T_1/T_0$.
-+ All sine coefficients&nbsp; $B_n$&nbsp; are zeroo.
++ All sine coefficients&nbsp; $B_n$&nbsp; are zero.
 - All cosine coefficients&nbsp; $A_n$&nbsp; with even &nbsp; $n$&nbsp; are zero.
-{Calculate the Fourier coefficients&nbsp; $A_n$&nbsp; in general form. What are the values for&nbsp; $A_1$,&nbsp; $A_2$&nbsp; and&nbsp; $A_3$&nbsp; with&nbsp; $T_1/T_0 = 0.25$?
+{Calculate the Fourier coefficients&nbsp; $A_n$&nbsp; in general form.&nbsp; What are the values for&nbsp; $A_1$,&nbsp; $A_2$&nbsp; and&nbsp; $A_3$&nbsp; with&nbsp; $T_1/T_0 = 0.25$?
 |type="{}"}
 $A_1\ = \ $ { 0.405 3% }
@@ Line 45: / Line 45: @@
-{Write the function&nbsp; ${x(t)}$&nbsp; as a Fourier series and break it off after&nbsp; $N = 3$&nbsp; coefficients. How large is the error &nbsp; $\varepsilon_3(t = 0)$?
+{Write the function&nbsp; ${x(t)}$&nbsp; as a Fourier series and break it off after&nbsp; $N = 3$&nbsp; coefficients.&nbsp; How large is the error &nbsp; $\varepsilon_3(t = 0)$?
 |type="{}"}
 $\varepsilon_3(t = 0)\ = \ $  { -0.11--0.09 }

@@ Line 20: / Line 20: @@
 *This exercise belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Series|Fourier Series]].
 *You can find a compact summary of the topic in the two learning videos
-::[[Zur_Berechnung_der_Fourierkoeffizienten_(Lernvideo)|Zur Berechnung der Fourierkoeffizienten]]
+::[[Zur_Berechnung_der_Fourierkoeffizienten_(Lernvideo)|Zur Berechnung der Fourierkoeffizienten]] &nbsp; &rArr; &nbsp; "To calculate the Fourier coefficients"
-:: [[Eigenschaften_der_Fourierreihendarstellung_(Lernvideo)|Eigenschaften der Fourierreihendarstellung]]
+:: [[Eigenschaften_der_Fourierreihendarstellung_(Lernvideo)|Eigenschaften der Fourierreihendarstellung]] &nbsp; &rArr; &nbsp;   "Properties of the Fourier series representation"
 *To solve the problem, you can use the following definite integral &nbsp; (let $n$ be an integer$)$:

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID317__Sig_Z_2_4.png|right|frame|Vorgegebenes Dreiecksignal]]
+[[File:P_ID317__Sig_Z_2_4.png|right|frame|Preset triangular signal]]
 We consider the signal&nbsp; ${x(t)}$&nbsp; with&nbsp; $T_0$&nbsp; according to the adjacent sketch, where the second signal parameter&nbsp;  $T_1 ≤ T_0/2$&nbsp; is to apply. This signal is dimensionless and limited to&nbsp; $1$&nbsp;.

← Older revision		Revision as of 12:39, 23 March 2021
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−	[[Category:~~Exercises for~~ Signal Representation\|^2.4 Fourier Series^]]	+	[[Category:Signal Representation: Exercises\|^2.4 Fourier Series^]]

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 '''(2)'''&nbsp; Taking advantage of the symmetry property&nbsp; ${x(-t)} = {x(t)}$&nbsp; one obtains:
 :$$A_n=2 \cdot \frac{2}{T_0}\cdot \hspace{-0.1cm}\int_0^{T_1}(1-\frac{t}{T_1})\cos(2\pi n\frac{t}{T_0})\, {\rm d}t.$$
-*Dies führt zu zwei Teilintegralen&nbsp; $I_1$&nbsp; und&nbsp; $I_2$. Das erste lautet:
+*This leads to two partial integrals&nbsp; $I_1$&nbsp; and&nbsp; $I_2$. The first is:
 :$$I_1=\frac{4}{T_0} \cdot \hspace{-0.1cm} \int_0^{T_1}\cos(2\pi n\frac{t}{T_0})\,{\rm d}t=\frac{2}{\pi n}\sin(2\pi n\frac{T_1}{T_0}).$$
-*Für das zweite Integral gilt mit dem Integral auf der Angabenseite:
+*For the second integral, with the integral on the statement side:
 :$$I_2=\frac{-4}{T_0\cdot T_1}\cdot \hspace{-0.1cm}\int_0^{T_1}t\cdot\cos(2\pi n\frac{t}{T_0})\,{\rm d}t=\frac{-4}{T_0\cdot T_1}\cdot \hspace{0.1cm}\left[\frac{T^2_0 \cdot \cos(2\pi nt/T_0)}{4\pi^2n^2}+\frac{T_0 \cdot t \cdot \sin(2\pi nt/T_0)}{2\pi n}\right]^{T_1}_0.$$
-*Dieses letzte Integral kann wie folgt zusammengefasst werden:
+*This last integral can be summarised as follows:
 :$$I_2=\frac{-\cos(2\pi nT_1/T_0)}{\pi^2 n^2T_1/T_0}+\frac{1}{\pi^2 n^2 T_1/T_0}-I_1.$$
-*Daraus folgt mit&nbsp; $1 - \cos(2\alpha) = 2 \cdot \sin^2(\alpha)$:
+*From this follows with&nbsp; $1 - \cos(2\alpha) = 2 \cdot \sin^2(\alpha)$:
 :$$A_n=I_1+I_2=\frac{1-\cos(2\pi nT_1/T_0)}{\pi^2 n^2 T_1/T_0}=\frac{2\sin^2 (\pi nT_1/T_0)}{\pi^2 n^2 T_1/T_0}.$$
-*Für&nbsp; $T_1/T_0 = 0.25$&nbsp; erhält man:
+*For&nbsp; $T_1/T_0 = 0.25$&nbsp; one obtains:
 :$$A_n=\frac{8\sin^2 (\pi n/4)}{\pi^2 n^2}.$$
-*Insbesondere gilt:
+*In particular:
 :$$A_1=\frac{8}{\pi^2}\sin^2(\pi/4)=\frac{4}{\pi^2}\hspace{0.15cm}\underline{\approx 0.405},\hspace{0.5cm}
 A_2=\frac{2}{\pi^2}\sin^2(\pi/2)=\frac{2}{\pi^2}\hspace{0.15cm}\underline{\approx 0.202},\hspace{0.5cm}
@@ Line 80: / Line 80: @@
-'''(3)'''&nbsp;  Es gilt:
+'''(3)'''&nbsp;  It holds:
 :$$x_3(t)=\frac{1}{4}+\frac{4}{\pi^2}\left[\cos(\omega_0 t)+\frac{1}{2}\cos(2\omega_0 t)+\frac{1}{9}\cos(3\omega_0 t)\right].$$
-*Zum Zeitpunkt&nbsp; $t = 0$&nbsp; ergibt sich hieraus:
+*At time&nbsp; $t = 0$&nbsp; this gives:
 :$$x_3(t=0)=\frac{1}{4}+\frac{4}{\pi^2}\cdot \frac{29}{18}\approx 0.9 \hspace{0.5cm}\Rightarrow  \hspace{0.5cm}\varepsilon_3(t=0)=x_3(t=0)-x(t=0)\hspace{0.15cm}\underline{=-0.1}.$$
-* Für die Zeit&nbsp; $t = 0$&nbsp; und bei Vielfachen der Periodendauer&nbsp; $T_0$&nbsp; (jeweils Spitze der Dreiecksfunktionen)&nbsp; ist die Abweichung betragsmäßig am größten.
+* For the time&nbsp; $t = 0$&nbsp; and for multiples of the period&nbsp; $T_0$&nbsp; (peak of the triangular functions in each case)&nbsp; the deviation is greatest in terms of magnitude.
 {{ML-Fuß}}

@@ Line 4: / Line 4: @@
 [[File:P_ID317__Sig_Z_2_4.png|right|frame|Vorgegebenes Dreiecksignal]]
-Wir betrachten das mit&nbsp; $T_0$&nbsp; periodische Signal&nbsp; ${x(t)}$&nbsp; entsprechend nebenstehender Skizze, wobei für den zweiten Signalparameter&nbsp;  $T_1 ≤ T_0/2$&nbsp; gelten soll. Dieses Signal ist dimensionslos und auf&nbsp; $1$&nbsp; begrenzt.
+We consider the signal&nbsp; ${x(t)}$&nbsp; with&nbsp; $T_0$&nbsp; according to the adjacent sketch, where the second signal parameter&nbsp;  $T_1 ≤ T_0/2$&nbsp; is to apply. This signal is dimensionless and limited to&nbsp; $1$&nbsp;.
-In der Teilaufgabe&nbsp; '''(3)'''&nbsp; wird die auf nur&nbsp; $N = 3$&nbsp; Koeffizienten basierende Fourierreihendarstellung&nbsp; $x_3(t)$&nbsp; verwendet.
+In subtask&nbsp; '''(3)'''&nbsp; the Fourier series representation&nbsp; $x_3(t)$&nbsp;based on only&nbsp; $N = 3$&nbsp;  coefficients is used.
-Die Differenz zwischen der abgebrochenen Fourierreihe und dem tatsächlichen Signal lautet:
+The difference between the truncated Fourier series and the actual signal is:
 :$$\varepsilon_3(t)=x_3(t)-x(t).$$
@@ Line 17: / Line 17: @@
-''Hinweise:''
+''Hints:''
-*Die Aufgabe gehört zum Kapitel&nbsp; [[Signal_Representation/Fourier_Series|Fourierreihe]].
+*This exercise belongs to the chapter&nbsp; [[Signal_Representation/Fourier_Series|Fourier Series]].
-*Eine kompakte Zusammenfassung der Thematik finden Sie in den beiden Lernvideos
+*You can find a compact summary of the topic in the two learning videos
 ::[[Zur_Berechnung_der_Fourierkoeffizienten_(Lernvideo)|Zur Berechnung der Fourierkoeffizienten]]
 :: [[Eigenschaften_der_Fourierreihendarstellung_(Lernvideo)|Eigenschaften der Fourierreihendarstellung]]
-*Zur Lösung der Aufgabe können Sie das folgende bestimmte Integral benutzen&nbsp; $(n$ sei ganzzahlig$)$:
+*To solve the problem, you can use the following definite integral &nbsp; (let $n$ be an integer$)$:
 :$$\int u \cdot \cos(au)\,{\rm d}u  =  \frac{\cos(au)}{a^2} + \frac{u \cdot \sin(au)}{a}.$$
-===Fragebogen===
+===Questions===
 <quiz display=simple>
-{Welche der folgenden Aussagen treffen für alle zulässigen&nbsp; $T_0$&nbsp; und&nbsp; $T_1$&nbsp; zu?
+{Which of the following statements are true for all permissible&nbsp; $T_0$&nbsp; and&nbsp; $T_1$&nbsp;?
 |type="[]"}
 + Der Gleichanteil beträgt&nbsp; $A_0 = T_1/T_0$.

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