Aufgaben:Exercise 2.5Z: Square Wave - Revision history

Hwang at 12:30, 18 January 2023

2023-01-18T12:30:19Z

Javier: Text replacement - "analyse" to "analyze"

2021-09-22T11:47:37Z

Text replacement - "analyse" to "analyze"

Guenter at 09:12, 16 April 2021

2021-04-16T09:12:48Z

Guenter at 09:10, 16 April 2021

2021-04-16T09:10:14Z

Guenter at 09:04, 16 April 2021

2021-04-16T09:04:45Z

Noah at 09:26, 26 March 2021

2021-03-26T09:26:04Z

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

2021-03-23T12:39:24Z

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

Noah at 19:42, 16 January 2021

2021-01-16T19:42:45Z

Noah at 19:41, 16 January 2021

2021-01-16T19:41:57Z

Noah at 19:31, 16 January 2021

2021-01-16T19:31:55Z

@@ Line 34: / Line 34: @@
 {Which statements are true for the signal&nbsp; $x(t)$&nbsp; with&nbsp; $\Delta t/T_0 = 0.5$?
 |type="[]"}
-+ The spectral function&nbsp; ${X(f)}$&nbsp; contains a Dirac function at&nbsp; $f = 0$&nbsp; with the weight&nbsp; $0.5$.
++ The spectral function&nbsp; ${X(f)}$&nbsp; contains a Dirac delta function at&nbsp; $f = 0$&nbsp; with the weight&nbsp; $0.5$.
-- The spectral function&nbsp; ${X(f)}$&nbsp; contains Dirac lines at all multiples of the base frequency&nbsp; $f_0 = 1/T_0$.
+- The spectral function&nbsp; ${X(f)}$&nbsp; contains Dirac delta lines at all multiples of the base frequency&nbsp; $f_0 = 1/T_0$.
-+ The spectral function&nbsp; ${X(f)}$&nbsp; contains Dirac lines at odd multiples of the base frequency &nbsp; $f_0$.
++ The spectral function&nbsp; ${X(f)}$&nbsp; contains Dirac delta lines at odd multiples of the base frequency &nbsp; $f_0$.
 - The spectral line at&nbsp; $f_0$&nbsp; has the weight&nbsp; $2/\pi = 0.636$.
 + The spectral line at&nbsp; $–\hspace{-0.1cm}f_0$&nbsp; has the weight&nbsp; $1/\pi = 0.318$.
@@ Line 43: / Line 43: @@
 {Which statements are true for the signal&nbsp; $x(t)$&nbsp; with&nbsp; $\Delta t/T_0 = 0.25$?
 |type="[]"}
-+ The spectral function&nbsp; ${X(f)}$&nbsp; contains Dirac lines at all odd multiples of the base frequency&nbsp; $f_0$.
++ The spectral function&nbsp; ${X(f)}$&nbsp; contains Dirac delta lines at all odd multiples of the base frequency&nbsp; $f_0$.
-+ ${X(f)}$&nbsp; has Dirac lines at&nbsp; $\pm2f_0$,&nbsp; $\pm6f_0$,&nbsp; $\pm10f_0$, etc.
++ ${X(f)}$&nbsp; has Dirac delta lines at&nbsp; $\pm2f_0$,&nbsp; $\pm6f_0$,&nbsp; $\pm10f_0$, etc.
-- ${X(f)}$&nbsp; has Dirac lines at&nbsp; $\pm4f_0$,&nbsp; $\pm8f_0$,&nbsp; $\pm12f_0$, etc.
+- ${X(f)}$&nbsp; has Dirac delta lines at&nbsp; $\pm4f_0$,&nbsp; $\pm8f_0$,&nbsp; $\pm12f_0$, etc.
-+ The Dirac line at&nbsp; $2f_0$&nbsp; has the weight&nbsp; $1/(2\pi) = 0.159$.
++ The Dirac delta line at&nbsp; $2f_0$&nbsp; has the weight&nbsp; $1/(2\pi) = 0.159$.
@@ Line 72: / Line 72: @@
 {{ML-Kopf}}
 '''(1)'''&nbsp; <u>Statements 1, 3 and 5</u> are correct:
-*The spectral function contains a Dirac line at&nbsp; $f = 0$&nbsp; with weight&nbsp; $0.5$&nbsp; (DC component)&nbsp; as well as further spectral lines at odd multiples&nbsp; ($n = \pm1, \pm3, \pm5,\text{...}$ of $f_0$.
+*The spectral function contains a Dirac delta line at&nbsp; $f = 0$&nbsp; with weight&nbsp; $0.5$&nbsp; (DC component)&nbsp; as well as further spectral lines at odd multiples&nbsp; ($n = \pm1, \pm3, \pm5,\text{...}$ of $f_0$.
 *The weights at&nbsp; $\pm f_0$ are $A_1/2 = 1/\pi = 0.318$&nbsp; in each case.
@@ Line 78: / Line 78: @@
 '''(2)'''&nbsp;  <u>Statements 1, 2 and 4</u> are correct:
 *Spectral lines exist at all odd multiples of the basic frequency, and additionally at the&nbsp; $2–{\rm fold}$,&nbsp; $6–{\rm fold}$&nbsp; and&nbsp; $10–{\rm fold}$.
-*For example&nbsp; $A_1 = 1/\pi = 0.450$.&nbsp; The Dirac line at&nbsp; $2f_0$&nbsp; thus has the weight&nbsp; $A_2/2 = 1/(2\pi) = 0.159$.
+*For example&nbsp; $A_1 = 1/\pi = 0.450$.&nbsp; The Dirac delta line at&nbsp; $2f_0$&nbsp; thus has the weight&nbsp; $A_2/2 = 1/(2\pi) = 0.159$.
 *For&nbsp; $n = 4$,&nbsp; $n = 8$, etc., on the other hand, the coefficients&nbsp; $A_n = 0$, since the following holds for the sine function: &nbsp; $\sin(\pi) = \sin(2\pi) =\text{ ...} = 0$.

@@ Line 9: / Line 9: @@
 The DC signal coefficient is&nbsp; $A_0 = \Delta t/T_0$&nbsp; and the following applies to the cosine coefficients:
 :$$A_n=\frac{2}{n\pi}\cdot \sin(n\pi \Delta t/T_0).$$
-In subtasks&nbsp; '''(1)'''&nbsp; and&nbsp; '''(2)'''&nbsp; the signal&nbsp; $x(t)$&nbsp; is analysed for the two parameter values&nbsp; $\Delta t/T_0 = 0.5$&nbsp; and&nbsp; $\Delta t/T_0 = 0.25$&nbsp; respectively.
+In subtasks&nbsp; '''(1)'''&nbsp; and&nbsp; '''(2)'''&nbsp; the signal&nbsp; $x(t)$&nbsp; is analyzed for the two parameter values&nbsp; $\Delta t/T_0 = 0.5$&nbsp; and&nbsp; $\Delta t/T_0 = 0.25$&nbsp; respectively.
 Then we consider the two signals&nbsp; $y(t)$&nbsp; and&nbsp; $z(t)$, each with&nbsp; $\Delta t/T_0 = 0.25$. There is a fixed relationship between these signals and&nbsp; $x(t)$&nbsp; which can be exploited for the calculation.

@@ Line 72: / Line 72: @@
 {{ML-Kopf}}
 '''(1)'''&nbsp; <u>Statements 1, 3 and 5</u> are correct:
-*The spectral function contains a Dirac line at&nbsp; $f = 0$&nbsp; with the weight&nbsp; $0.5$&nbsp; (DC component)&nbsp; as well as further spectral lines at odd multiples&nbsp; ($n = \pm1, \pm3, \pm5,\text{...}$ of $f_0$.
+*The spectral function contains a Dirac line at&nbsp; $f = 0$&nbsp; with weight&nbsp; $0.5$&nbsp; (DC component)&nbsp; as well as further spectral lines at odd multiples&nbsp; ($n = \pm1, \pm3, \pm5,\text{...}$ of $f_0$.
-The weights at&nbsp; $\pm f_0$ are $A_1/2 = 1/\pi = 0.318$&nbsp; in each case.
+*The weights at&nbsp; $\pm f_0$ are $A_1/2 = 1/\pi = 0.318$&nbsp; in each case.
@@ Line 96: / Line 96: @@
 Thus one obtains:
 :$$A_1^{(z)}=-A_1^{(y)}={\sqrt2}/{\pi}\hspace{0.15cm}\underline{=+0.450}, \hspace {0.5cm} A_2^{(z)}=A_2^{(y)}=-{1}/{\pi}\hspace{0.15cm}\underline{=-0.318}.$$
-he same result is obtained starting from the given coefficients with&nbsp; $\Delta t/T_0 = 0.75$:
+The same result is obtained starting from the given coefficients with&nbsp; $\Delta t/T_0 = 0.75$:
 :$$A_1^{(z)}={2}/{\pi} \cdot \sin({3}/{4}\cdot \pi)={\sqrt2}/{\pi},
 \hspace {0.5cm}A_2^{(z)}=

@@ Line 72: / Line 72: @@
 {{ML-Kopf}}
 '''(1)'''&nbsp; <u>Statements 1, 3 and 5</u> are correct:
-*The spectral function contains a Dirac function at $f = 0$ with the weight $0.5$ (DC component) as well as further spectral lines at odd multiples ($n = \pm1, \pm3, \pm5,\text{...}$ of $f_0$.
+*The spectral function contains a Dirac line at&nbsp; $f = 0$&nbsp; with the weight&nbsp; $0.5$&nbsp; (DC component)&nbsp; as well as further spectral lines at odd multiples&nbsp; ($n = \pm1, \pm3, \pm5,\text{...}$ of $f_0$.
-The weights at $\pm f_0$ are $A_1/2 = 1/\pi = 0.318$ in each case.
+The weights at&nbsp; $\pm f_0$ are $A_1/2 = 1/\pi = 0.318$&nbsp; in each case.
 '''(2)'''&nbsp;  <u>Statements 1, 2 and 4</u> are correct:
-*Spectral lines exist at all odd multiples of the fundamental frequency, and additionally at the $2–{\rm fold}$, $6–{\rm fold}$ and $10–{\rm fold}$.
+*Spectral lines exist at all odd multiples of the basic frequency, and additionally at the&nbsp; $2–{\rm fold}$,&nbsp; $6–{\rm fold}$&nbsp; and&nbsp; $10–{\rm fold}$.
-*For example $A_1 = 1/\pi = 0.450$. The spectral line at $2f_0$ thus has the weight $A_2/2 = 1/(2\pi) = 0.159$.
+*For example&nbsp; $A_1 = 1/\pi = 0.450$.&nbsp; The Dirac line at&nbsp; $2f_0$&nbsp; thus has the weight&nbsp; $A_2/2 = 1/(2\pi) = 0.159$.
-*For $n = 4$, $n = 8$, etc., on the other hand, the coefficients $A_n = 0$, since the following holds for the sine function: &nbsp; $\sin(\pi) = \sin(2\pi) =\text{ ...} = 0$.
+*For&nbsp; $n = 4$,&nbsp; $n = 8$, etc., on the other hand, the coefficients&nbsp; $A_n = 0$, since the following holds for the sine function: &nbsp; $\sin(\pi) = \sin(2\pi) =\text{ ...} = 0$.
-'''(3)'''&nbsp;  From the graphical representation of the signal ${y(t)}$ it is clear that $A_0 = 0.75$ must apply. The same result can be obtained using the relationship:
+'''(3)'''&nbsp;  From the graphical representation of the signal&nbsp; ${y(t)}$&nbsp; it is clear that&nbsp; $A_0 = 0.75$&nbsp; must apply.&nbsp; The same result can be obtained using the relationship:
 :$$A_0^{(y)}=1-A_0^{(x)}=1-0.25\hspace{0.15cm}\underline{=0.75}.$$
-'''(4)'''&nbsp;  The following applies: ${y(t)} = 1 - x(t)$. For $n \neq 0$ the Fourier coefficients are the same as for the signal $x(t)$, but with negative signs. In particular:
+'''(4)'''&nbsp;  The following applies:&nbsp; ${y(t)} = 1 - x(t)$.&nbsp; For&nbsp; $n \neq 0$&nbsp; the Fourier coefficients are the same as for the signal&nbsp; $x(t)$, but with negative signs.&nbsp; In particular:
 :$$A_1^{(y)} = -A_1^{(x)}=-{2}/{\pi} \cdot \sin({\pi}/{4})= -{\sqrt2}/{\pi}\hspace{0.15cm}\underline{\approx -0.450},$$
 :$$A_2^{(y)} = -A_2^{(x)}=-{1}/{\pi}\hspace{0.15cm}\underline{ \approx - 0.318}.$$
-'''(5)'''&nbsp;  ${z(t)} = y(t - T_0/2)$ applies. With the Fourier series representation of ${y(t)}$ it follows:
+'''(5)'''&nbsp;  ${z(t)} = y(t - T_0/2)$ applies.&nbsp; With the Fourier series representation of&nbsp; ${y(t)}$&nbsp; it follows:
 :$$z(t)=A_0+A_1^{(y)}\cos(\omega_0(t-\frac{T_0}{2}))+A_2^{(y)}\cos(2\omega_0(t-\frac{T_0}{2}))+A_3^{(y)}\cos(3\omega_0(t-\frac{T_0}{2}))+\ldots$$
 :$$\Rightarrow \quad z(t)=A_0-A_1^{(y)}\cos(\omega_0 t)+A_2^{(y)}\cos(2\omega_0 t)-A_3^{(y)}\cos(3\omega_0 t)+\text{...}$$
 Thus one obtains:
 :$$A_1^{(z)}=-A_1^{(y)}={\sqrt2}/{\pi}\hspace{0.15cm}\underline{=+0.450}, \hspace {0.5cm} A_2^{(z)}=A_2^{(y)}=-{1}/{\pi}\hspace{0.15cm}\underline{=-0.318}.$$
-he same result is obtained starting from the given coefficients with $\Delta t/T_0 = 0.75$:
+he same result is obtained starting from the given coefficients with&nbsp; $\Delta t/T_0 = 0.75$:
 :$$A_1^{(z)}={2}/{\pi} \cdot \sin({3}/{4}\cdot \pi)={\sqrt2}/{\pi},
 \hspace {0.5cm}A_2^{(z)}=

@@ Line 4: / Line 4: @@
 [[File:P_ID323__Sig_Z_2_5.png|right|frame|Various square wave signals]]
-The signal&nbsp; $x(t)$&nbsp; periodic with time&nbsp; $T_0$&nbsp; is described by the single parameter&nbsp; $\Delta t$&nbsp;; let the amplitude of the square-wave pulses be&nbsp; $1$ in each case. Since&nbsp; $x(t)$&nbsp; is even, all sine coefficients&nbsp; $B_n = 0$.
+The signal&nbsp; $x(t)$&nbsp; periodic with time&nbsp; $T_0$&nbsp; is described by the single parameter&nbsp; $\Delta t$;&nbsp;
 The DC signal coefficient is&nbsp; $A_0 = \Delta t/T_0$&nbsp; and the following applies to the cosine coefficients:
@@ Line 20: / Line 21: @@
 ''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
-::[[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".
@@ Line 35: / Line 36: @@
 + The spectral function&nbsp; ${X(f)}$&nbsp; contains a Dirac function at&nbsp; $f = 0$&nbsp; with the weight&nbsp; $0.5$.
 - The spectral function&nbsp; ${X(f)}$&nbsp; contains Dirac lines at all multiples of the base frequency&nbsp; $f_0 = 1/T_0$.
-+ The spectral function&nbsp; ${X(f)}$&nbsp; contains diraclines at odd multiples of the base frequency &nbsp; $f_0$.
++ The spectral function&nbsp; ${X(f)}$&nbsp; contains Dirac lines at odd multiples of the base frequency &nbsp; $f_0$.
 - The spectral line at&nbsp; $f_0$&nbsp; has the weight&nbsp; $2/\pi = 0.636$.
 + The spectral line at&nbsp; $–\hspace{-0.1cm}f_0$&nbsp; has the weight&nbsp; $1/\pi = 0.318$.
@@ Line 42: / Line 43: @@
 {Which statements are true for the signal&nbsp; $x(t)$&nbsp; with&nbsp; $\Delta t/T_0 = 0.25$?
 |type="[]"}
-+ The spectral function&nbsp; ${X(f)}$&nbsp; contains diraclines at all odd multiples of the base frequency&nbsp; $f_0$.
++ The spectral function&nbsp; ${X(f)}$&nbsp; contains Dirac lines at all odd multiples of the base frequency&nbsp; $f_0$.
-+ ${X(f)}$&nbsp; has diraclines at&nbsp; $\pm2f_0$,&nbsp; $\pm6f_0$,&nbsp; $\pm10f_0$, etc.
++ ${X(f)}$&nbsp; has Dirac lines at&nbsp; $\pm2f_0$,&nbsp; $\pm6f_0$,&nbsp; $\pm10f_0$, etc.
-- ${X(f)}$&nbsp; has diraclines at&nbsp; $\pm4f_0$,&nbsp; $\pm8f_0$,&nbsp; $\pm12f_0$, etc.
+- ${X(f)}$&nbsp; has Dirac lines at&nbsp; $\pm4f_0$,&nbsp; $\pm8f_0$,&nbsp; $\pm12f_0$, etc.
-+ The spectral line at&nbsp; $2f_0$&nbsp; has the weight&nbsp; $1/(2\pi) = 0.159$.
++ The Dirac line at&nbsp; $2f_0$&nbsp; has the weight&nbsp; $1/(2\pi) = 0.159$.
-{What is the DC component of the signal&nbsp; ${y(t)}$?
+{What is the DC coefficient of the signal&nbsp; ${y(t)}$?
 |type="{}"}
 $y(t)$: &nbsp; $A_0 \ = \ $ { 0.75 3% }
-{What is the relationship between the signals&nbsp; $x(t)$&nbsp; and&nbsp; ${y(t)}$? With the help of these considerations, give the Fourier coefficients of&nbsp; ${y(t)}$. <br>What are the coefficients&nbsp; $A_1$&nbsp; and&nbsp; $A_2$&nbsp; of this signal?
+{What is the relationship between the signals&nbsp; $x(t)$&nbsp; and&nbsp; ${y(t)}$?&nbsp; With the help of these considerations, give the Fourier coefficients of&nbsp; ${y(t)}$. <br>What are the coefficients&nbsp; $A_1$&nbsp; and&nbsp; $A_2$&nbsp; of this signal?
 |type="{}"}
 $y(t)$: &nbsp; $A_1\ = \ $ { -0.46--0.44 }
@@ Line 59: / Line 60: @@
-{What is the relationship between the signals&nbsp; ${y(t)}$&nbsp; and&nbsp; ${z(t)}$? What are the coefficients&nbsp; $A_1$&nbsp; and&nbsp; $A_2$&nbsp; of the signal&nbsp; ${z(t)}$? <br>Check the result using the given coefficients of the signal&nbsp; $x(t)$.
+{What is the relationship between the signals&nbsp; ${y(t)}$&nbsp; and&nbsp; ${z(t)}$?&nbsp; What are the coefficients&nbsp; $A_1$&nbsp; and&nbsp; $A_2$&nbsp; of the signal&nbsp; ${z(t)}$? <br>Check the result using the given coefficients of the signal&nbsp; $x(t)$.
 |type="{}"}
 $z(t)$: &nbsp; $A_1 \ = \ $ { 0.45 3% }

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

@@ Line 31: / Line 31: @@
 <quiz display=simple>
-{Whcih statements are true for the signal&nbsp; $x(t)$&nbsp; with&nbsp; $\Delta t/T_0 = 0.5$?
+{Which statements are true for the signal&nbsp; $x(t)$&nbsp; with&nbsp; $\Delta t/T_0 = 0.5$?
 |type="[]"}
 + The spectral function&nbsp; ${X(f)}$&nbsp; contains a Dirac function at&nbsp; $f = 0$&nbsp; with the weight&nbsp; $0.5$.
@@ Line 70: / Line 70: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp;  Richtig sind die <u>Aussagen 1, 3 und 5</u>:
+'''(1)'''&nbsp; <u>Statements 1, 3 and 5</u> are correct:
-*Die Spektralfunktion beinhaltet eine Diracfunktion bei $f = 0$ mit dem Gewicht $0.5$ (Gleichanteil) sowie weitere Spektrallinien bei ungeradzahligen Vielfachen ($n = \pm1, \pm3, \pm5,\text{...}$) von $f_0$.
+*The spectral function contains a Dirac function at $f = 0$ with the weight $0.5$ (DC component) as well as further spectral lines at odd multiples ($n = \pm1, \pm3, \pm5,\text{...}$ of $f_0$.
-*Die Gewichte bei $\pm f_0$ sind jeweils $A_1/2 = 1/\pi = 0.318$.
+The weights at $\pm f_0$ are $A_1/2 = 1/\pi = 0.318$ in each case.
-'''(2)'''&nbsp;  Richtig sind die <u>Aussagen 1, 2 und 4</u>:
+'''(2)'''&nbsp;  <u>Statements 1, 2 and 4</u> are correct:
-*Bei allen ungeradzahligen Vielfachen der Grundfrequenz existieren Spektrallinien, zusätzlich noch bei den $2–{\rm fachen}$, $6–{\rm fachen}$ und $10–{\rm fachen}$.
+*Spectral lines exist at all odd multiples of the fundamental frequency, and additionally at the $2–{\rm fold}$, $6–{\rm fold}$ and $10–{\rm fold}$.
-*Beispielsweise gilt $A_1 = 1/\pi = 0.450$. Die Spektrallinie bei $2f_0$ hat somit das Gewicht $A_2/2 = 1/(2\pi) = 0.159$.
+*For example $A_1 = 1/\pi = 0.450$. The spectral line at $2f_0$ thus has the weight $A_2/2 = 1/(2\pi) = 0.159$.
-*Für $n = 4$, $n = 8$, usw. sind dagegen die Koeffizienten $A_n = 0$, da für die Sinusfunktion gilt: &nbsp; $\sin(\pi) = \sin(2\pi) =\text{ ...} = 0$.
+*For $n = 4$, $n = 8$, etc., on the other hand, the coefficients $A_n = 0$, since the following holds for the sine function: &nbsp; $\sin(\pi) = \sin(2\pi) =\text{ ...} = 0$.