Aufgaben:Exercise 3.6: Even and Odd Time Signals - Revision history

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	*This exercise belongs to the chapter  [[Signal_Representation/Fourier_Transform_Theorems\|Fourier Transform Theorems]].		*This exercise belongs to the chapter  [[Signal_Representation/Fourier_Transform_Theorems\|Fourier Transform Theorems]].
	*All the theorems presented here are illustrated with examples in the (German language) learning video<br>      [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)\|Gesetzmäßigkeiten der Fouriertransformation]]   ⇒   "Regularities to the Fourier transform".		*All the theorems presented here are illustrated with examples in the (German language) learning video<br>      [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)\|Gesetzmäßigkeiten der Fouriertransformation]]   ⇒   "Regularities to the Fourier transform".
	*Solve this task with the help of the  [[Signal_Representation/~~Fourier_Transform_Theorems~~#Assignment_Theorem\|Assignment Theorem]].		*Solve this task with the help of the  [[Signal_Representation/The_Fourier_Transform_Theorems#Assignment_Theorem\|Assignment Theorem]].
	*For the first two subtasks use the signal parameters  $A_u = 1\,\text{V}$  and  $T = 1\,\text{ms}$.		*For the first two subtasks use the signal parameters  $A_u = 1\,\text{V}$  and  $T = 1\,\text{ms}$.

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	[[de:Aufgaben:3.~~6_Gerades/ungerades_Zeitsignal~~]]		[[de:Aufgaben:Aufgabe 3.6: Gerades und ungerades Zeitsignal]]

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			[[de:Aufgaben:3.6_Gerades/ungerades_Zeitsignal]]

Guenter at 13:22, 27 April 2021

2021-04-27T13:22:40Z

Guenter at 13:12, 27 April 2021

2021-04-27T13:12:06Z

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	*This exercise belongs to the chapter  [[Signal_Representation/Fourier_Transform_Theorems\|Fourier Transform Theorems]].		*This exercise belongs to the chapter  [[Signal_Representation/Fourier_Transform_Theorems\|Fourier Transform Theorems]].
	*All the theorems presented here are illustrated with examples in the (German language) learning video<br>      [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)\|Gesetzmäßigkeiten der Fouriertransformation]]   ⇒   "Regularities to the Fourier transform".		*All the theorems presented here are illustrated with examples in the (German language) learning video<br>      [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)\|Gesetzmäßigkeiten der Fouriertransformation]]   ⇒   "Regularities to the Fourier transform".
	*Solve this task with the help of the  [[Signal_Representation/~~Fourier_Transform_Laws~~#~~Mapping_theorem~~\|~~mapping theorem~~]].		*Solve this task with the help of the  [[Signal_Representation/Fourier_Transform_Theorems#Assignment_Theorem\|Assignment Theorem]].
	*~~Use~~ the signal parameters  $A_u = 1\,\text{ V}$  and  $T = 1\,\text{ ms}$ ~~for the first two subtasks~~.		*For the first two subtasks use the signal parameters  $A_u = 1\,\text{V}$  and  $T = 1\,\text{ms}$.


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	<quiz display=simple>		<quiz display=simple>
	{Calculate the (~~purely~~ imaginary) spectral values of the ~~unbalanced~~ signal  $u(t)$  at the frequencies  $f = 0.5\,\text{kHz}$  and  $f = 1\,\text{kHz}$.		{Calculate the (imaginary) spectral values of the odd signal  $u(t)$  at the frequencies  $f = 0.5\,\text{kHz}$  and  $f = 1\,\text{kHz}$.
	\|type="{}"}		\|type="{}"}
	${\rm Im}\big[U(f=0.5 \,\text{kHz})\big] \ = \ $ { -0.205--0.195 }  $\text{mV/Hz}$		${\rm Im}\big[U(f=0.5 \,\text{kHz})\big] \ = \ $ { -0.205--0.195 }  $\text{mV/Hz}$

Guenter at 13:07, 27 April 2021

2021-04-27T13:07:33Z

Guenter: Guenter moved page Aufgaben:Exercise 3.6: Even/Odd Time Signal to Aufgaben:Exercise 3.6: Even and Odd Time Signals

2021-04-27T12:55:28Z

Guenter moved page Aufgaben:Exercise 3.6: Even/Odd Time Signal to Aufgaben:Exercise 3.6: Even and Odd Time Signals

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

2021-03-23T12:39:40Z

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

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

Noah at 17:36, 27 January 2021

2021-01-27T17:36:37Z

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Oezdemir: Oezdemir moved page Aufgaben:Aufgabe 3.6: Gerades/ungerades Zeitsignal to Aufgaben:Exercise 3.6: Even/Odd Time Signal

2020-11-02T15:30:01Z

Oezdemir moved page Aufgaben:Aufgabe 3.6: Gerades/ungerades Zeitsignal to Aufgaben:Exercise 3.6: Even/Odd Time Signal

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Oezdemir at 15:24, 2 November 2020

2020-11-02T15:24:30Z

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	[[Category:Exercises for Signal Representation\|^3. ~~Aperiodische Signale - Impulse~~^]]		[[Category:Exercises for Signal Representation\|^3.3 Fourier Transform Theorems^]]

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	:$$U( {f = 0.5\;{\rm{kHz}}} ) = - {\rm{j}} \cdot \frac{ {A_u \cdot T}}{ {\rm{\pi }}} \cdot {\mathop{\rm si}\nolimits} ( {{ {\rm{\pi }}}/{2}} ) = - {\rm{j}} \cdot \frac{2}{ { {\rm{\pi }}^{\rm{2}} }} \cdot A_{\rm u} \cdot T.$$		:$$U( {f = 0.5\;{\rm{kHz}}} ) = - {\rm{j}} \cdot \frac{ {A_u \cdot T}}{ {\rm{\pi }}} \cdot {\mathop{\rm si}\nolimits} ( {{ {\rm{\pi }}}/{2}} ) = - {\rm{j}} \cdot \frac{2}{ { {\rm{\pi }}^{\rm{2}} }} \cdot A_{\rm u} \cdot T.$$

	*The imaginary part is ~~numerically~~  ${\rm Im}[U(f=0.5 \,\text{kHz})]\; \underline{\approx 0.2 \,\text{mV/Hz}}$.		*The imaginary part is  ${\rm Im}[U(f=0.5 \,\text{kHz})]\; \underline{\approx 0.2 \,\text{mV/Hz}}$.
	*In contrast, the si function at  $f \cdot T = 1$  yields the value zero, while the cosine is equal to  $-1$ . Thus, with  $A_u = 1\,\text{V}$  and  $T = 1\,\text{ms}$ one obtains:		*In contrast, the  $\rm si$–function at  $f \cdot T = 1$  yields the value zero, while the cosine is equal to  $-1$.  Thus with  $A_u = 1\,\text{V}$  and  $T = 1\,\text{ms}$  one obtains:

	:$$U( {f = 1\;{\rm{kHz}}} ) = {\rm{j}} \cdot \frac{ {A_{\rm u} \cdot T}}{ { {\rm{2\pi }}}} \hspace{0.3 cm} \Rightarrow \hspace{0.3 cm} {\rm Re} [\text{...}] \hspace{0.15 cm}\underline{ = 0}, \hspace{0.3 cm}{\rm Im} [\text{...}] \hspace{0.15 cm}\underline{\approx 0.159 \;{\rm{mV/Hz}}}.$$		:$$U( {f = 1\;{\rm{kHz}}} ) = {\rm{j}} \cdot \frac{ {A_{\rm u} \cdot T}}{ { {\rm{2\pi }}}} \hspace{0.3 cm} \Rightarrow \hspace{0.3 cm} {\rm Re} [\text{...}] \hspace{0.15 cm}\underline{ = 0}, \hspace{0.3 cm}{\rm Im} [\text{...}] \hspace{0.15 cm}\underline{\approx 0.159 \;{\rm{mV/Hz}}}.$$
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	'''(2)'''  According to the ~~mapping theorem~~, an odd time function  $u(t)$  always has an imaginary and at the same time odd spectrum		'''(2)'''  According to the  "Assignment Theorem", an odd time function  $u(t)$  always has an imaginary and at the same time odd spectrum  $U( { - f} ) = - U( f )$.  With the boundary transition  $f \rightarrow \infty$  follows from the given equation
	$U( { - f} ) = - U( f ).$ With the boundary transition  $f \rightarrow \infty$  follows from the given equation

	:$$U( f ) = - {\rm{j}} \cdot \frac{ {A_u \cdot T}}{ {2{\rm{\pi }}fT}}\big[ { {\mathop{\rm si}\nolimits} ( {{\rm{\pi }}fT} ) - \cos ( { {\rm{\pi }}fT} )} \big]$$		:$$U( f ) = - {\rm{j}} \cdot \frac{ {A_u \cdot T}}{ {2{\rm{\pi }}fT}}\big[ { {\mathop{\rm si}\nolimits} ( {{\rm{\pi }}fT} ) - \cos ( { {\rm{\pi }}fT} )} \big]$$

	the result  $U(f = 0) = 0$. Formally, one could confirm this result by applying l'Hospital's rule.		the result  $U(f = 0) = 0$.  Formally, one could confirm this result by applying l'Hospital's rule.

	We proceed a little more pragmatically.		We proceed a little more pragmatically.
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	*For even smaller frequency values, the result also becomes smaller and smaller.		*For even smaller frequency values, the result also becomes smaller and smaller.
	*You get to the result  $U(f = 0)\;\underline{ = 0}$, more quickly if you take into account that the integral over  $u(t)$  disappears.		*You get to the result  $U(f = 0)\;\underline{ = 0}$  more quickly if you take into account that the integral over  $u(t)$  disappears.
	*So you don't have to calculate at all.		*So you don't have to calculate at all.



	'''(3)'''  The signal  $x(t)$  can be divided into ~~the~~ even and ~~the~~ odd part, which lead to the even real part and the odd imaginary part of  $X(f)$ :		'''(3)'''  The signal  $x(t)$  can be divided into an even and an odd part, which lead to the even real part and the odd imaginary part of  $X(f)$ :
	*The even part is equal to the function  $g(t)$  with  $A_g = 3\,\text{V}$. From this follows for the real part of the spectral value at  $f \cdot T = 0.5$:		*The even part is equal to the function  $g(t)$  with  $A_g = 3\,\text{V}$.  From this follows for the real part of the spectral value at  $f \cdot T = 0.5$:

	:$${\mathop{\rm Re}\nolimits} \left[ {X( {f \cdot T = 0.5} )} \right] = A_{\rm g} \cdot T \cdot {\mathop{\rm si}\nolimits} ( {{ {\rm{\pi }}}/{2}} ) \hspace{0.15 cm}\underline{= 1.91 \;{\rm{mV/Hz}}}{\rm{.}}$$		:$${\mathop{\rm Re}\nolimits} \left[ {X( {f \cdot T = 0.5} )} \right] = A_{\rm g} \cdot T \cdot {\mathop{\rm si}\nolimits} ( {{ {\rm{\pi }}}/{2}} ) \hspace{0.15 cm}\underline{= 1.91 \;{\rm{mV/Hz}}}{\rm{.}}$$

	*The imaginary part results from the spectral function  $U(f)$ with $A_u = 1\,\text{V}$. This was already calculated in subtask  '''(1)'''~~ ~~:		*The imaginary part results from the spectral function  $U(f)$  with $A_u = 1\,\text{V}$.  This was already calculated in subtask  '''(1)''':

	:$${\mathop{\rm Im}\nolimits} \left[ {X( {f \cdot T = 0.5} )} \right] \hspace{0.15 cm}\underline{\approx - 0.2 \;{\rm{mV/Hz}}}{\rm{.}}$$		:$${\mathop{\rm Im}\nolimits} \left[ {X( {f \cdot T = 0.5} )} \right] \hspace{0.15 cm}\underline{\approx - 0.2 \;{\rm{mV/Hz}}}{\rm{.}}$$

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	}}		}}

	[[File:P_ID516__Sig_A_3_6_neu.png\|250px\|right\|frame\|~~&bdquo;Keilfunktion” sowie ein gerades und ein ungerades Zeitsignal~~]]		[[File:P_ID516__Sig_A_3_6_neu.png\|250px\|right\|frame\|"Wedge function" as well as an even and an odd time signal]]

	We are looking for the spectrum  $X(f)$  of the pulse~~-shaped signal~~  $x(t)$ sketched opposite, which rises linearly from  $2\,\text{ V}$  to  $4\,\text{ V}$  in the range from  $–T/2$  to  $+T/2$  and is zero outside.		We are looking for the spectrum  $X(f)$  of the pulse  $x(t)$ sketched opposite, which rises linearly from  $2\hspace{0.05cm} \text{V}$  to  $4\hspace{0.05cm} \text{V}$  in the range from  $–T/2$  to  $+T/2$  and is zero outside.

	The spectral functions of the signals  $g(t)$  and  $u(t)$  shown below are assumed to be known:		The spectral functions of the signals  $g(t)$  and  $u(t)$  shown below are assumed to be known:
	*The even, rectangular time function  $g(t)$  has the spectrum		*The even (German:  gerade) rectangular time function  $g(t)$  has the spectrum

	:$$G( f ) = A_g \cdot T \cdot {\mathop{\rm si}\nolimits}( { {\rm{\pi }}fT} ) \hspace{0.3cm} {\rm{~~mit~~}}\hspace{0.3cm} {\mathop{\rm si}\nolimits}( x ) = {\sin ( x )}/{x}.$$		:$$G( f ) = A_g \cdot T \cdot {\mathop{\rm si}\nolimits}( { {\rm{\pi }}fT} ) \hspace{0.3cm} {\rm{with}}\hspace{0.3cm} {\mathop{\rm si}\nolimits}( x ) = {\sin ( x )}/{x}.$$

	*The spectrum of the ~~asymmetric~~ function  $u(t)$  is:		*The spectrum of the odd (German:  ungerade) function  $u(t)$  is:

	:$$U( f ) = - {\rm{j}} \cdot \frac{ {A_u \cdot T}}{ {2{\rm{\pi }}fT}}\big[ {{\mathop{\rm si}\nolimits} ( { {\rm{\pi }}fT} ) - \cos ( { {\rm{\pi }}fT} )} \big].$$		:$$U( f ) = - {\rm{j}} \cdot \frac{ {A_u \cdot T}}{ {2{\rm{\pi }}fT}}\big[ {{\mathop{\rm si}\nolimits} ( { {\rm{\pi }}fT} ) - \cos ( { {\rm{\pi }}fT} )} \big].$$





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	''Hints:''		''Hints:''
	*This exercise belongs to the chapter  [[Signal_Representation/~~Fourier_Transform_Laws~~\|Fourier Transform ~~Laws~~]].		*This exercise belongs to the chapter  [[Signal_Representation/Fourier_Transform_Theorems\|Fourier Transform Theorems]].
	*All ~~of these laws~~ are illustrated with examples in the learning video  [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)\|Gesetzmäßigkeiten der Fouriertransformation]] .		*All the theorems presented here are illustrated with examples in the (German language) learning video<br>      [[Gesetzmäßigkeiten_der_Fouriertransformation_(Lernvideo)\|Gesetzmäßigkeiten der Fouriertransformation]]   ⇒   "Regularities to the Fourier transform".
	*Solve this task with the help of the  [[Signal_Representation/Fourier_Transform_Laws#Mapping_theorem\|mapping theorem]].		*Solve this task with the help of the  [[Signal_Representation/Fourier_Transform_Laws#Mapping_theorem\|mapping theorem]].
	*Use the signal parameters  $A_u = 1\,\text{ V}$  and  $T = 1\,\text{ ms}$ for the first two subtasks.		*Use the signal parameters  $A_u = 1\,\text{ V}$  and  $T = 1\,\text{ ms}$ for the first two subtasks.