Aufgaben:Exercise 3.2Z: Laplace and Fourier - Revision history

Guenter at 12:20, 13 October 2021

2021-10-13T12:20:13Z

Guenter at 12:14, 13 October 2021

2021-10-13T12:14:59Z

Guenter at 12:09, 13 October 2021

2021-10-13T12:09:23Z

Oezer at 09:31, 11 October 2021

2021-10-11T09:31:11Z

Oezer at 22:53, 10 October 2021

2021-10-10T22:53:28Z

Oezer at 22:52, 10 October 2021

2021-10-10T22:52:00Z

Oezer at 22:49, 10 October 2021

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Oezer at 22:45, 10 October 2021

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Oezer at 22:33, 10 October 2021

2021-10-10T22:33:06Z

Oezer at 22:30, 10 October 2021

2021-10-10T22:30:46Z

@@ Line 99: / Line 99: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; <u>Suggested solutions 1 and 3</u> are correct:
+'''(1)'''&nbsp; <u>Suggested solutions 1 and 3</u>&nbsp; are correct:
-*Considering that the Dirac function is non-zero only at&nbsp; $t= 0$&nbsp; and that the integral over the Dirac yields the value&nbsp; $1$&nbsp; as long as the integration interval includes the time&nbsp; $t= 0$,&nbsp; the following is obtained:
+*Considering that the Dirac delta function is non-zero only at&nbsp; $t= 0$&nbsp; and that the integral over the Dirac delta function yields the value&nbsp; $1$&nbsp; as long as the integration interval includes the time&nbsp; $t= 0$,&nbsp; the following is obtained:
 :$$A(f) = 1, \hspace{0.2cm}A_{\rm
   L}(p) = 1  \hspace{0.05cm} .$$
@@ Line 106: / Line 106: @@
-'''(2)'''&nbsp; <u>Suggested solutions 1 and 3</u> are correct again:
+'''(2)'''&nbsp; <u>Suggested solutions 1 and 3</u>&nbsp; are correct again:
-*The step function&nbsp; $b(t) = \gamma(t)$&nbsp; is the integral over the Dirac function&nbsp; $a(t) = \delta(t)$ &nbsp; &rArr; &nbsp; the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Integration_Theorem|integration theorem]]&nbsp; can be applied:
+*The step function&nbsp; $b(t) = \gamma(t)$&nbsp; is the integral over the Dirac delta function&nbsp; $a(t) = \delta(t)$ &nbsp; &rArr; &nbsp; the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Integration_Theorem|integration theorem]]&nbsp; can be applied:
 :$$b(t) = \int_{-\infty}^t {a(\tau)} \hspace{0.1cm}{\rm
   d}\tau  \hspace{0.3cm}\Rightarrow \hspace{0.3cm} B_{\rm L}(p) =A_{\rm L}(p)\cdot
@@ Line 117: / Line 117: @@
-'''(3)'''&nbsp; <u>Suggested solutions 2 and 3</u> are correct:
+'''(3)'''&nbsp; <u>Suggested solutions 2 and 3</u>&nbsp; are correct:
-*Since the (causal) rectangular function can be represented as the difference of two step functions, the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|shifting theorem]] yields:
+*Since the&nbsp; (causal)&nbsp; rectangular function can be represented as the difference of two step functions,&nbsp; the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|shifting theorem]]&nbsp; yields:
 :$$c(t)= b(t) - b(t-T)  \hspace{0.3cm}
 \Rightarrow \hspace{0.3cm} C_{\rm L}(p) =B_{\rm L}(p)- B_{\rm L}(p)
@@ Line 124: / Line 124: @@
   \hspace{0.05cm} .$$
-*The following holds for the&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|Fourier spectrum]] since the rectangular function has finite energy:
+*The following holds for the&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|Fourier spectrum]]&nbsp; since the rectangular function has finite energy:
 :$$C(f) =  C_{\rm L}(p)\Bigg |_{\hspace{0.1cm} p\hspace{0.05cm}=\hspace{0.05cm}{\rm j \hspace{0.05cm}2\pi \it
   f}} =   \frac{1}{{\rm j} \cdot 2\pi f} \cdot \big [ 1- {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi f T} \big ]
@@ Line 134: / Line 134: @@
-'''(4)'''&nbsp; <u>Suggested solution 1</u> is correct because the following holds:
+'''(4)'''&nbsp; <u>Suggested solution 1</u>&nbsp; is correct because the following holds:
 :$$d(t) = \frac{1}{T} \cdot \int\limits_{-\infty}^t {c(\tau)} \hspace{0.1cm}{\rm
   d}\tau  \hspace{0.3cm}\Rightarrow \hspace{0.3cm} D_{\rm L}(p) =C_{\rm L}(p)\cdot
 \frac{1}{p \cdot  T} = \frac{1- {\rm e}^{-p \hspace{0.05cm}\cdot \hspace{0.05cm}T}}{p^2 \cdot
 T}\hspace{0.05cm} .$$
-*Since&nbsp; $d(t)$&nbsp; extends to infinity, the simple relation between&nbsp; $D_{\rm L}(p)$&nbsp; and&nbsp; $D(f)$&nbsp; according to proposed solution 3 is not valid.
+*Since&nbsp; $d(t)$&nbsp; extends to infinity,&nbsp; the simple relation between&nbsp; $D_{\rm L}(p)$&nbsp; and&nbsp; $D(f)$&nbsp; according to proposed solution 3 is not valid.
-*$D(f)$&nbsp; rather also includes a Dirac function at frequency&nbsp; $f = 0$.
+*$D(f)$&nbsp; rather also includes a Dirac delta function at frequency&nbsp; $f = 0$.
 {{ML-Fuß}}

@@ Line 59: / Line 59: @@
 Please note:
 *The exercise belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Laplace_Transform_and_p-Transfer_Function|Laplace Transform and p-Transfer Function]].
-*The educational video &nbsp; [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)|Fourier transform theorems]]&nbsp; might be helpful.
+*The&nbsp; (German language)&nbsp; learning video &nbsp; [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)|"Gesetzmäßigkeiten der Fouriertransformation"]]&nbsp; &rArr; &nbsp;  "Fourier transform theorems"&nbsp; might be helpful.
@@ Line 84: / Line 84: @@
 - $C_{\rm L}(p) = {\rm si}(pT)$.
 + $C_{\rm L}(p) =  \big [1-{\rm e}^{-p\hspace{0.05cm} \cdot \hspace{0.05cm}T} \big ]/p$.
-+ $C(f) = C_{\rm L}(p)$ &nbsp;mit&nbsp; $p = 2 \pi f$.
++ $C(f) = C_{\rm L}(p)$ &nbsp;with&nbsp; $p = 2 \pi f$.
@@ Line 91: / Line 91: @@
 + $D_{\rm L}(p) =  \big[1-{\rm e}^{-p\hspace{0.05cm} \cdot \hspace{0.05cm}T}\big]/(p^2T)$.
 - $D_{\rm L}(p) =  1-{\rm e}^{-p\hspace{0.05cm} \cdot \hspace{0.05cm}T}$.
-- $D(f) = D_{\rm L}(p)$ &nbsp;mit&nbsp; $p = 2 \pi f$.
+- $D(f) = D_{\rm L}(p)$ &nbsp;with&nbsp; $p = 2 \pi f$.

@@ Line 4: / Line 4: @@
 [[File:P_ID1764__LZI_Z_3_2.png|right|frame|Four causal time signals]]
-The Fourier transformation can be applied to any deterministic signal&nbsp; $x(t)$&nbsp;. Then, the following holds for the spectral function:
+The Fourier transformation can be applied to any deterministic signal&nbsp; $x(t)$.&nbsp; Then, the following holds for the spectral function:
 :$$X(f) =    \int_{-\infty}^{
 +\infty}
@@ Line 10: / Line 10: @@
   d}t\hspace{0.05cm}\hspace{0.05cm} .$$
-For power-limited signals &ndash; characteristics: &nbsp; infinite energy &ndash; $X(f)$&nbsp; also includes distributions (Dirac functions).
+For power-limited signals&nbsp; &ndash; characteristics: &nbsp; infinite energy &ndash;&nbsp; $X(f)$&nbsp; also includes distributions&nbsp; (Dirac delta  functions).
-For all causal signals (and only for these), the Laplace transformation is also applicable beside the Fourier transformation:
+For all causal signals&nbsp; (and only for these),&nbsp; the Laplace transformation is also applicable beside the Fourier transformation:
 :$$X_{\rm L}(p) =    \int_{0}^{
 \infty}
@@ Line 20: / Line 20: @@
 In the diagram you can see several causal time functions that will be covered in this exercise:
-* the Dirac function&nbsp; $a(t)$,
+* the Dirac delta function&nbsp; $a(t)$,
-* the step function&nbsp; $b(t)$,
+* the unit step function&nbsp; $b(t)$,
 * the rectangular function&nbsp; $c(t)$,
@@ Line 29: / Line 29: @@
-The&nbsp; [[Signal_Representation/Fourier_Transform_Theorems|Fourier transform theorems]]&nbsp; usually (though not always) also apply to the Laplace transformation where&nbsp; $p ={\rm j} \cdot 2 \pi f$&nbsp; is to be set:
+The&nbsp; [[Signal_Representation/Fourier_Transform_Theorems|Fourier transform theorems]]&nbsp; usually&nbsp; (though not always)&nbsp; also apply to the Laplace transformation where&nbsp; $p ={\rm j} \cdot 2 \pi f$&nbsp; is to be set:
-* For example, the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|shifting theorem]]&nbsp; in Laplace or Fourier representation is:
+* For example,&nbsp; the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|shifting theorem]]&nbsp; in Laplace or Fourier representation is:
 :$$x(t- \tau) \quad
 \circ\!\!-\!\!\!-^{\hspace{-0.25cm}\rm
@@ Line 40: / Line 40: @@
 X(f)\cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm}2\pi f \hspace{0.05cm} \cdot \hspace{0.05cm}\tau}\hspace{0.05cm} .$$
-* In contrast, there are differences in the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Integration_Theorem|integration theorem]]&nbsp;:
+* In contrast,&nbsp; there are differences in the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Integration_Theorem|integration theorem]]&nbsp;:
 :$$\int {x(\tau)} \hspace{0.1cm}{\rm
   d}\tau \quad
@@ Line 57: / Line 57: @@
 Please note:
 *The exercise belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Laplace_Transform_and_p-Transfer_Function|Laplace Transform and p-Transfer Function]].
 *The educational video &nbsp; [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)|Fourier transform theorems]]&nbsp; might be helpful.

@@ Line 29: / Line 29: @@
-The&nbsp; [[Signal_Representation/Fourier_Transform_Theorems|Fourier transform theorems]]&nbsp; gelten meist (allerdings nicht immer) auch für die Laplace&ndash;Transformation, wobei&nbsp; $p ={\rm j} \cdot 2 \pi f$&nbsp; zu setzen ist:
+The&nbsp; [[Signal_Representation/Fourier_Transform_Theorems|Fourier transform theorems]]&nbsp; usually (though not always) also apply to the Laplace transformation where&nbsp; $p ={\rm j} \cdot 2 \pi f$&nbsp; is to be set:
-* Zum Beispiel lautet der&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|shifting theorem]]&nbsp; in Laplace&ndash; bzw. Fourier&ndash;Darstellung:
+* For example, the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|shifting theorem]]&nbsp; in Laplace or Fourier representation is:
 :$$x(t- \tau) \quad
 \circ\!\!-\!\!\!-^{\hspace{-0.25cm}\rm
@@ Line 40: / Line 40: @@
 X(f)\cdot {\rm e}^{-{\rm j}\hspace{0.05cm} \cdot \hspace{0.05cm}2\pi f \hspace{0.05cm} \cdot \hspace{0.05cm}\tau}\hspace{0.05cm} .$$
-* Dagegen ergeben sich beim&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Integration_Theorem|integration theorem]]&nbsp; Unterschiede:
+* In contrast, there are differences in the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Integration_Theorem|integration theorem]]&nbsp;:
 :$$\int {x(\tau)} \hspace{0.1cm}{\rm
   d}\tau \quad
@@ Line 62: / Line 62: @@
 ''Please note:''
 *The exercise belongs to the chapter&nbsp; [[Linear_and_Time_Invariant_Systems/Laplace_Transform_and_p-Transfer_Function|Laplace Transform and p-Transfer Function]].
-*Das Lernvideo &nbsp; [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)|Fourier transform theorems]]&nbsp;  könnte hilfreich sein .
+*The educational video &nbsp; [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)|Fourier transform theorems]]&nbsp; might be helpful.
@@ Line 69: / Line 69: @@
 <quiz display=simple>
-{Wie lauten die Spektraltransformationen des Signals&nbsp; $a(t) = \delta(t)$?
+{What are the spectral transformations of the signal&nbsp; $a(t) = \delta(t)$?
 |type="[]"}
 + $A_{\rm L}(p) = 1$.
@@ Line 76: / Line 76: @@
-{Wie lauten die Spektraltransformationen der Sprungfunktion&nbsp; $b(t) = \gamma(t)$?
+{What are the spectral transformations of the step function&nbsp; $b(t) = \gamma(t)$?
 |type="[]"}
 + $B_{\rm L}(p) = 1/p$.
@@ Line 83: / Line 83: @@
-{Wie lauten die Spektraltransformationen der Rechteckfunktion&nbsp; $c(t)$?
+{What are the spectral transformations of the rectangular function&nbsp; $c(t)$?
 |type="[]"}
 - $C_{\rm L}(p) = {\rm si}(pT)$.
@@ Line 90: / Line 90: @@
-{Wie lauten die Spektraltransformationen der Rampenfunktion&nbsp; $d(t)$?
+{What are the spectral transformations of the ramp function&nbsp; $d(t)$?
 |type="[]"}
 + $D_{\rm L}(p) =  \big[1-{\rm e}^{-p\hspace{0.05cm} \cdot \hspace{0.05cm}T}\big]/(p^2T)$.
@@ Line 102: / Line 102: @@
 ===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Richtig sind die <u>Lösungsvorschläge 1 und 3</u>:
+'''(1)'''&nbsp; <u>Suggested solutions 1 and 3</u> are correct:
-*Berücksichtigt man, dass die Diracfunktion nur bei&nbsp; $t= 0$&nbsp; ungleich Null ist und das Integral über den Dirac den Wert&nbsp; $1$&nbsp; liefert, solange das Integrationsintervall den Zeitpunkt&nbsp; $t= 0$&nbsp; einschließt, so erhält man:
+*Considering that the Dirac function is non-zero only at&nbsp; $t= 0$&nbsp; and that the integral over the Dirac yields the value&nbsp; $1$&nbsp; as long as the integration interval includes the time&nbsp; $t= 0$,&nbsp; the following is obtained:
 :$$A(f) = 1, \hspace{0.2cm}A_{\rm
   L}(p) = 1  \hspace{0.05cm} .$$
@@ Line 109: / Line 109: @@
-'''(2)'''&nbsp; Richtig sind wieder die <u>Lösungsvorschläge 1 und 3</u>:
+'''(2)'''&nbsp; <u>Suggested solutions 1 and 3</u> are correct again:
-*Die Sprungfunktion&nbsp; $b(t) = \gamma(t)$&nbsp; ist das Integral über die Diracfunktion&nbsp; $a(t) = \delta(t)$ &nbsp; &rArr; &nbsp; man kann den&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Integration_Theorem|integration theorem]]&nbsp; anwenden:
+*The step function&nbsp; $b(t) = \gamma(t)$&nbsp; is the integral over the Dirac function&nbsp; $a(t) = \delta(t)$ &nbsp; &rArr; &nbsp; the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Integration_Theorem|integration theorem]]&nbsp; can be applied:
 :$$b(t) = \int_{-\infty}^t {a(\tau)} \hspace{0.1cm}{\rm
   d}\tau  \hspace{0.3cm}\Rightarrow \hspace{0.3cm} B_{\rm L}(p) =A_{\rm L}(p)\cdot
@@ Line 120: / Line 120: @@
-'''(3)'''&nbsp; Richtig sind die <u>Lösungsvorschläge 2 und 3</u>:
+'''(3)'''&nbsp; <u>Suggested solutions 2 and 3</u> are correct:
-*Nachdem die (kausale) Rechteckfunktion als Differenz zweier Sprungfunktionen dargestellt werden kann, erhält man mit dem&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|shifting theorem]]:
+*Since the (causal) rectangular function can be represented as the difference of two step functions, the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|shifting theorem]] yields:
 :$$c(t)= b(t) - b(t-T)  \hspace{0.3cm}
 \Rightarrow \hspace{0.3cm} C_{\rm L}(p) =B_{\rm L}(p)- B_{\rm L}(p)
@@ Line 127: / Line 127: @@
   \hspace{0.05cm} .$$
-*Da die Rechteckfunktion eine endliche Energie besitzt, gilt für das&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|Fourier spectrum]]:
+*The following holds for the&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|Fourier spectrum]] since the rectangular function has finite energy:
 :$$C(f) =  C_{\rm L}(p)\Bigg |_{\hspace{0.1cm} p\hspace{0.05cm}=\hspace{0.05cm}{\rm j \hspace{0.05cm}2\pi \it
   f}} =   \frac{1}{{\rm j} \cdot 2\pi f} \cdot \big [ 1- {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2\pi f T} \big ]
   \hspace{0.05cm}.$$
-*Nach einigen trigonometrischen Umformungen kann hierfür auch geschrieben werden:
+*The following can also be written for this using some trigonometric transformations:
 :$$C(f) =  T \cdot {\rm si} (2 \pi  f{T})+ {\rm j} \cdot \frac{{\rm cos} (2 \pi  f{T})-1}{2\pi f}
   \hspace{0.05cm}.$$
@@ Line 137: / Line 137: @@
-'''(4)'''&nbsp; Richtig ist der <u>Lösungsvorschlag 1</u>, da gilt:
+'''(4)'''&nbsp; <u>Suggested solution 1</u> is correct because the following holds:
 :$$d(t) = \frac{1}{T} \cdot \int\limits_{-\infty}^t {c(\tau)} \hspace{0.1cm}{\rm
   d}\tau  \hspace{0.3cm}\Rightarrow \hspace{0.3cm} D_{\rm L}(p) =C_{\rm L}(p)\cdot
 \frac{1}{p \cdot  T} = \frac{1- {\rm e}^{-p \hspace{0.05cm}\cdot \hspace{0.05cm}T}}{p^2 \cdot
 T}\hspace{0.05cm} .$$
-*Da sich&nbsp; $d(t)$&nbsp; bis ins Unendliche erstreckt, ist der einfache Zusammenhang zwischen&nbsp; $D_{\rm L}(p)$&nbsp; und&nbsp; $D(f)$&nbsp; gemäß Lösungsvorschlag 3 nicht gegeben.
+*Since&nbsp; $d(t)$&nbsp; extends to infinity, the simple relation between&nbsp; $D_{\rm L}(p)$&nbsp; and&nbsp; $D(f)$&nbsp; according to proposed solution 3 is not valid.
-*$D(f)$&nbsp; beinhaltet vielmehr auch eine Diracfunktion bei der Frequenz&nbsp; $f = 0$.
+*$D(f)$&nbsp; rather also includes a Dirac function at frequency&nbsp; $f = 0$.
 {{ML-Fuß}}

@@ Line 12: / Line 12: @@
 For power-limited signals &ndash; characteristics: &nbsp; infinite energy &ndash; $X(f)$&nbsp; also includes distributions (Dirac functions).
-For all causal signals (and only for these), the Laplace transformation is also applicable besides the Fourier transformation:
+For all causal signals (and only for these), the Laplace transformation is also applicable beside the Fourier transformation:
 :$$X_{\rm L}(p) =    \int_{0}^{
 \infty}