Aufgaben:Exercise 5.3: Mean Square Error - Revision history

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	The other DFT parameters are clearly fixed withn  $N$  and  $f_{\rm A}$.  The following applies to these:		The other DFT parameters are clearly fixed withn  $N$  and  $f_{\rm A}$.  The following applies to these:
	:$$f_{\rm P} = N \cdot f_{\rm A},\hspace{0.3cm}T_{\rm P} = 1/f_{\rm A},\hspace{0.3cm}T_{\rm A} = T_{\~~rmP~~}/N\hspace{0.05cm}.$$		:$$f_{\rm P} = N \cdot f_{\rm A},\hspace{0.3cm}T_{\rm P} = 1/f_{\rm A},\hspace{0.3cm}T_{\rm A} = T_{\rm P}/N\hspace{0.05cm}.$$

	The accuracy of the respective DFT approximation is captured by the  "mean square error"  $\rm (MSE)$.  <br>Here, we use the designation  $\rm MQF$   ⇒   (German:  "Mittlerer Quadratischer Fehler"):		The accuracy of the respective DFT approximation is captured by the  "mean square error"  $\rm (MSE)$.  <br>Here, we use the designation  $\rm MQF$   ⇒   (German:  "Mittlerer Quadratischer Fehler"):
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	[[de:Aufgaben:5.~~3_Mittlerer_Quadratischer_Fehler~~]]		[[de:Aufgaben:Aufgabe 5.3: Mittlerer Quadratischer Fehler]]

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	*a  [[Signal_Representation/Special_Cases_of_Pulses#Rectangular_pulse\|rectangular pulse]]  $x_2(t)$  with amplitude  $A$  and (equivalent) duration  $T$:		*a  [[Signal_Representation/Special_Cases_of_Pulses#Rectangular_pulse\|rectangular pulse]]  $x_2(t)$  with amplitude  $A$  and (equivalent) duration  $T$:

	:$$x_2(t) = \left\{ \begin{array}{c} A \\		:$$x_2(t) = \left\{ \begin{array}{c} A \\0 \\ \end{array} \right.\quad\begin{array}{{10}c} {\rm{f\ddot{u}r}}\\ {\rm{f\ddot{u}r}} \\ \end{array}\begin{array}{{20}c}\|t\| < T/2 \hspace{0.05cm}, \\\|t\| > T/2 \hspace{0.05cm}, \\\end{array}$$*a so called  "sinc pulse"  according to the following definition::$$x_3(t) = A \cdot {\rm sinc}(t/ T) ,\hspace{0.15cm}{\rm sinc}(x) =\sin(\pi x)/(\pi x)\hspace{0.05cm}.$$
	0 \\ \end{array} \right.\quad
	\begin{array}{*{10}c} {\rm{f\ddot{u}r}}
	\\ {\rm{f\ddot{u}r}} \\ \end{array}\begin{array}{*{20}c}
	\|t\| < T/2 \hspace{0.05cm}, \\
	\|t\| > T/2 \hspace{0.05cm}, \\
	\end{array}$$*a so called  "sinc pulse"  according to the following definition::$$x_3(t) = A \cdot {\rm sinc}(t/ T) ,\hspace{0.15cm}{\rm sinc}(x) =
	\sin(\pi x)/(\pi x)\hspace{0.05cm}.$$

	Let the signal parameters be  $A = 1\ {\rm V}$  and  $T = 1\ {\rm ms}$ in each case.		Let the signal parameters be  $A = 1\ {\rm V}$  and  $T = 1\ {\rm ms}$ in each case.

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	\|t\| < T/2 \hspace{0.05cm}, \\		\|t\| < T/2 \hspace{0.05cm}, \\
	\|t\| > T/2 \hspace{0.05cm}, \\		\|t\| > T/2 \hspace{0.05cm}, \\
	\end{array}$$		\end{array}$$*a so called  "sinc pulse"  according to the following definition::$$x_3(t) = A \cdot {\rm sinc}(t/ T) ,\hspace{0.15cm}{\rm sinc}(x) =

	*a so called  "sinc pulse"  according to the following definition:

	:$$x_3(t) = A \cdot {\rm sinc}(t/ T) ,\hspace{0.15cm}{\rm sinc}(x) =
	\sin(\pi x)/(\pi x)\hspace{0.05cm}.$$		\sin(\pi x)/(\pi x)\hspace{0.05cm}.$$

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	The other DFT parameters are clearly fixed withn  $N$  and  $f_{\rm A}$.  The following applies to these:		The other DFT parameters are clearly fixed withn  $N$  and  $f_{\rm A}$.  The following applies to these:
	:$$f_{\rm P} = N \cdot f_{\rm A},\hspace{0.3cm}T_{\rm P} = 1/f_{\rm A},\hspace{0.3cm}T_{\rm A} = T_{\rm		:$$f_{\rm P} = N \cdot f_{\rm A},\hspace{0.3cm}T_{\rm P} = 1/f_{\rm A},\hspace{0.3cm}T_{\rm A} = T_{\rmP}/N\hspace{0.05cm}.$$
	P}/N
	\hspace{0.05cm}.$$

	The accuracy of the respective DFT approximation is captured by the  "mean square error"  $\rm (MSE)$.  <br>Here, we use the designation  $\rm MQF$   ⇒   (German:  "Mittlerer Quadratischer Fehler"):		The accuracy of the respective DFT approximation is captured by the  "mean square error"  $\rm (MSE)$.  <br>Here, we use the designation  $\rm MQF$   ⇒   (German:  "Mittlerer Quadratischer Fehler"):
	:$${\rm MQF} = \frac{1}{N}\cdot \sum_{\mu = 0 }^{N-1}		:$${\rm MQF} = \frac{1}{N}\cdot \sum_{\mu = 0 }^{N-1}\left\|X(\mu \cdot f_{\rm A})-\frac{D(\mu)}{f_{\rm A}}\right\|^2 \hspace{0.05cm}.$$
	\left\|X(\mu \cdot f_{\rm A})-\frac{D(\mu)}{f_{\rm A}}\right\|^2 \hspace{0.05cm}.$$

	The resulting MQF values are given in the graph above, valid for  $N = 512$  as well as for		The resulting MQF values are given in the graph above, valid for  $N = 512$  as well as for

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	[[Category:Signal Representation: Exercises\|^5.3 Possible DFT Errors^]]		[[Category:Signal Representation: Exercises\|^5.3 Possible DFT Errors^]]
			[[de:Aufgaben:5.3_Mittlerer_Quadratischer_Fehler]]

Guenter at 15:14, 17 May 2021

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	'''(1)'''  With the DFT parameters  $N = 512$  and  $f_{\rm A} \cdot T = 1/8$  the following follows after multiplying the two quantities:		'''(1)'''  With the DFT parameters  $N = 512$  and  $f_{\rm A} \cdot T = 1/8$  the following follows after multiplying the two quantities:
	:$$f_{\rm P} \cdot T = N \cdot (f_{\rm A} \cdot T) = 64.$$		:$$f_{\rm P} \cdot T = N \cdot (f_{\rm A} \cdot T) = 64.$$
	*This covers the frequency range  $~~–f_~~{\rm P}/2 \leq f < f_{\rm P}/2$~~ ~~:		*This covers the frequency range  $-f_{\rm P}/2 \leq f < +f_{\rm P}/2$:
	:$$f_{\rm max }\cdot T \hspace{0.15 cm}\underline{= 32}\hspace{0.05cm}.$$		:$$f_{\rm max }\cdot T \hspace{0.15 cm}\underline{= 32}\hspace{0.05cm}.$$

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	'''(3)'''  Correct is the <u>proposed solution 1   ⇒   increase of the ~~termination~~ error</u>:		'''(3)'''  Correct is the <u>proposed solution 1   ⇒   increase of the truncation error</u>:
	This measure simultaneously halves  $T_{\rm P}$  from  $8T$  to  $4T$ .Thus, only samples in the range  $–2T \leq t < 2T$, are taken into account, which increases the ~~termination~~ error.		*This measure simultaneously halves  $T_{\rm P}$  from  $8T$  to  $4T$ .
	*The mean square error  $(\rm MQF)$  increases from  $0.15 \cdot 10^{-15}$  to  $8 \cdot 10^{-15}$ for the Gaussian pulse  $x_1(t)$ , although the aliasing error actually decreases slightly by this measure.		*Thus, only samples in the range  $–2T \leq t < 2T$ are taken into account, which increases the truncation error.
			*The mean square error  $(\rm MQF)$  increases from  $0.15 \cdot 10^{-15}$  to  $8 \cdot 10^{-15}$ for the Gaussian pulse  $x_1(t)$,
			*although the aliasing error actually decreases slightly by this measure.



	'''(4)'''  Correct is the <u>proposed solution 2   ⇒   increase of the aliasing error:</u>:		'''(4)'''  Correct is the <u>proposed solution 2   ⇒   increase of the aliasing error:</u>:
	*By halving  $f_{\rm A}$  ~~wird auch~~  $f_{\rm P}$  is also halved.		*By halving  $f_{\rm A}$  ⇒   $f_{\rm P}$  is also halved.
	*As a result, the aliasing error becomes somewhat larger with a smaller ~~termination~~ error at the same time.		*As a result, the aliasing error becomes somewhat larger with a smaller truncation error at the same time.
	*Overall, for the Gaussian pulse  $x_1(t)$~~ ~~, the mean square error  $(\rm MQF)$  increases from  $1.5 \cdot 10^{-16}$  to  $3.3 \cdot 10^{-16}$.		*Overall, for the Gaussian pulse  $x_1(t)$, the mean square error  $(\rm MQF)$  increases from  $1.5 \cdot 10^{-16}$  to  $3.3 \cdot 10^{-16}$.


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	'''(5)'''  <u>Proposed solutions 1 and 2</u> are correct:		'''(5)'''  <u>Proposed solutions 1 and 2</u> are correct:
	*As can be seen from the graph, the last statement is not true in contrast to the first two.		*As can be seen from the graph, the last statement is not true in contrast to the first two.
	*Due to the slow,  $\rm si$–shaped decay of the spectral function, the aliasing error dominates.		*Due to the slow  ($\rm sinc$–shaped)  decay of the spectral function, the aliasing error dominates.
	*The  $\rm MQF$ value at  $f_{\rm A} \cdot T = 1/8$  with  $1.4 \cdot 10^{-5}$  is therefore significantly larger than for the Gaussian pulse  $(1.5 \cdot 10^{-16})$.		*The  $\rm MQF$ value at  $f_{\rm A} \cdot T = 1/8$  with  $1.4 \cdot 10^{-5}$  is therefore significantly larger than for the Gaussian pulse  $(1.5 \cdot 10^{-16})$.

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	'''(6)'''  <u>Proposed solution 3</u> is correct:		'''(6)'''  <u>Proposed solution 3</u> is correct:
	*The spectral function  $X_3(f)$  here has a rectangular lead, so that the first two statements do not apply.		*The spectral function  $X_3(f)$  here has a rectangular lead, so that the first two statements do not apply.
	*On the other hand, a ~~termination~~ error is unavoidable with this  $\rm si$–shaped time function. This leads to the large  $\rm MQF$ values given.		*On the other hand, a truncation error is unavoidable with this  $\rm sinc$–shaped time function.  This leads to the large  $\rm MQF$ values given.
	{{ML-Fuß}}		{{ML-Fuß}}

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	[[Category:Signal Representation: Exercises\|^5.3 Possible DFT Errors^]]		[[Category:Signal Representation: Exercises\|^5.3 Possible DFT Errors^]]

Guenter at 15:03, 17 May 2021

2021-05-17T15:03:08Z

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	For all spectral functions,  $X(f = 0) = A \cdot T$.		For all spectral functions,  $X(f = 0) = A \cdot T$.

	If the discrete-frequency spectrum is determined by the  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)\|Discrete Fourier Transform]]  with the DFT parameters		If the discrete-frequency spectrum is determined by the  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)\|Discrete Fourier Transform]]  $\rm (DFT)$  with the DFT parameters
	* $N = 512$   ⇒   number of samples considered in the time and frequency domain,		* $N = 512$   ⇒   number of samples considered in the time and frequency domain,
	*$f_{\rm A}$   ⇒   interpolation distance in the frequency domain,		*$f_{\rm A}$   ⇒   interpolation distance in the frequency domain,

Guenter at 15:01, 17 May 2021

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	:$$x_1(t) = A \cdot {\rm e}^{- \pi (t/T)^2} \hspace{0.05cm},$$		:$$x_1(t) = A \cdot {\rm e}^{- \pi (t/T)^2} \hspace{0.05cm},$$

	*a  [[Signal_Representation/~~Special_Cases_of_Impulse_Signals~~#~~Rectangular_Impulse~~\|rectangular pulse]]  $x_2(t)$  with amplitude  $A$  and (equivalent) duration  $T$:		*a  [[Signal_Representation/Special_Cases_of_Pulses#Rectangular_pulse\|rectangular pulse]]  $x_2(t)$  with amplitude  $A$  and (equivalent) duration  $T$:

	:$$x_2(t) = \left\{ \begin{array}{c} A \\		:$$x_2(t) = \left\{ \begin{array}{c} A \\

Guenter at 15:01, 17 May 2021

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	We consider three pulses, namely		We consider three pulses, namely
	*a  [[Signal_Representation/~~Special_Cases_of_Impulse_Signals~~#~~Gaussian_Impulse~~\|Gaussian pulse]]  with amplitude  $A$  and equivalent duration  $T$:		*a  [[Signal_Representation/Special_Cases_of_Pulses#Gaussian_pulse\|Gaussian pulse]]  with amplitude  $A$  and equivalent duration  $T$:

	:$$x_1(t) = A \cdot {\rm e}^{- \pi (t/T)^2} \hspace{0.05cm},$$		:$$x_1(t) = A \cdot {\rm e}^{- \pi (t/T)^2} \hspace{0.05cm},$$

Guenter at 15:00, 17 May 2021

2021-05-17T15:00:27Z

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	}}		}}

	[[File:P_ID1145__Sig_A_5_3.png\|250px\|right\|frame\|Gaussian pulse, square pulse, sinc pulse and some parameters]]		[[File:P_ID1145__Sig_A_5_3.png\|250px\|right\|frame\|Gaussian pulse, square pulse, <br>sinc pulse and some parameters]]

	We consider three ~~pulse-like signals~~, namely		We consider three pulses, namely
	*a  [[Signal_Representation/Special_Cases_of_Impulse_Signals#Gaussian_Impulse\|Gaussian pulse]]  with amplitude  $A$  and equivalent duration  $T$:		*a  [[Signal_Representation/Special_Cases_of_Impulse_Signals#Gaussian_Impulse\|Gaussian pulse]]  with amplitude  $A$  and equivalent duration  $T$:

	:$$x_1(t) = A \cdot {\rm e}^{- \pi (t/T)^2} \hspace{0.05cm},$$		:$$x_1(t) = A \cdot {\rm e}^{- \pi (t/T)^2} \hspace{0.05cm},$$

	*a  [[Signal_Representation/Special_Cases_of_Impulse_Signals#Rectangular_Impulse\|~~Rectangular~~ pulse]]  $x_2(t)$  with amplitude  $A$  and (equivalent) duration  $T$:		*a  [[Signal_Representation/Special_Cases_of_Impulse_Signals#Rectangular_Impulse\|rectangular pulse]]  $x_2(t)$  with amplitude  $A$  and (equivalent) duration  $T$:

	:$$x_2(t) = \left\{ \begin{array}{c} A \\		:$$x_2(t) = \left\{ \begin{array}{c} A \\
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	\end{array}$$		\end{array}$$

	*a so called  ~~''Sinc~~ pulse''  according to the following definition:		*a so called  "sinc pulse"  according to the following definition:

	:$$x_3(t) = A \cdot {\rm si}(~~\pi \cdot~~ t/ T) ,\hspace{0.15cm}{\rm si}(x) =		:$$x_3(t) = A \cdot {\rm sinc}(t/ T) ,\hspace{0.15cm}{\rm sinc}(x) =
	\sin(x)/x\hspace{0.05cm}.$$		\sin(\pi x)/(\pi x)\hspace{0.05cm}.$$

	Let the signal parameters be  $A = 1\ {\rm V}$  and  $T = 1\ {\rm ms}$ in each case.		Let the signal parameters be  $A = 1\ {\rm V}$  and  $T = 1\ {\rm ms}$ in each case.

	The conventional  [[Signal_Representation/Fourier_Transform_and_Its_Inverse\|Fourier ~~Transform~~]]  leads to the following spectral functions:		The conventional  [[Signal_Representation/Fourier_Transform_and_Its_Inverse\|Fourier transform]]  leads to the following spectral functions:
	* $X_1(f)$  is also Gaussian,		* $X_1(f)$  is also Gaussian,
	* $X_2(f)$  runs according to the  $\rm si$~~–function~~,		* $X_2(f)$  runs according to the  $\rm sinc$ function,
	* $X_3(f)$  is constant for  $\|f\| < 1/(2 T)$  and outside zero.		* $X_3(f)$  is constant for  $\|f\| < 1/(2 T)$  and outside zero.

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	For all spectral functions,  $X(f = 0) = A \cdot T$.		For all spectral functions,  $X(f = 0) = A \cdot T$.

	If the discrete-frequency spectrum is determined by the  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)\|Discrete Fourier Transform~~(DFT)~~]]  with the DFT parameters		If the discrete-frequency spectrum is determined by the  [[Signal_Representation/Discrete_Fourier_Transform_(DFT)\|Discrete Fourier Transform]]  with the DFT parameters
	* $N = 512$   ⇒   number of samples considered in the time and frequency domain,*$f_{\rm A}$   ⇒   interpolation distance in the frequency domain,		* $N = 512$   ⇒   number of samples considered in the time and frequency domain,
			*$f_{\rm A}$   ⇒   interpolation distance in the frequency domain,


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	The other DFT parameters are clearly fixed withn&~~bsp~~; $N$  ~~uan~~  $f_{\rm A}$  .The following applies to these:		The other DFT parameters are clearly fixed withn  $N$  and  $f_{\rm A}$.  The following applies to these:
	:$$f_{\rm P} = N \cdot f_{\rm A},\hspace{0.3cm}T_{\rm P} = 1/f_{\rm A},\hspace{0.3cm}T_{\rm A} = T_{\rm		:$$f_{\rm P} = N \cdot f_{\rm A},\hspace{0.3cm}T_{\rm P} = 1/f_{\rm A},\hspace{0.3cm}T_{\rm A} = T_{\rm
	P}/N		P}/N
	\hspace{0.05cm}.$$		\hspace{0.05cm}.$$

	The accuracy of the respective DFT approximation is captured by ~~thenbsp~~; ''mean square error''  (MSE, ~~here~~ MQF):		The accuracy of the respective DFT approximation is captured by the  "mean square error"  $\rm (MSE)$.  <br>Here, we use the designation  $\rm MQF$   ⇒   (German:  "Mittlerer Quadratischer Fehler"):
	:$${\rm MQF} = \frac{1}{N}\cdot \sum_{\mu = 0 }^{N-1}		:$${\rm MQF} = \frac{1}{N}\cdot \sum_{\mu = 0 }^{N-1}
	\left\|X(\mu \cdot f_{\rm A})-\frac{D(\mu)}{f_{\rm A}}\right\|^2 \hspace{0.05cm}.$$		\left\|X(\mu \cdot f_{\rm A})-\frac{D(\mu)}{f_{\rm A}}\right\|^2 \hspace{0.05cm}.$$
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	''Hints:''		''Hints:''
	*This task belongs to the chapter  [[Signal_Representation/Possible_Errors_When_Using_DFT\|Possible ~~Errors~~ when ~~Using~~ DFT]].		*This task belongs to the chapter  [[Signal_Representation/Possible_Errors_When_Using_DFT\|Possible errors when using DFT]].

	*The theory for this chapter is summarised in the learning video  [[Fehlermöglichkeiten_bei_Anwendung_der_DFT_(Lernvideo)\|~~Possible Errors when Using~~ DFT]] .		*The theory for this chapter is summarised in the (German language) learning video <br>      [[Fehlermöglichkeiten_bei_Anwendung_der_DFT_(Lernvideo)\|Fehlermöglichkeiten bei Anwendung der DFT]]   ⇒   "Possible errors when using DFT".


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	$T_{\rm A}/T\ = \ $ { 0.01562 3% }		$T_{\rm A}/T\ = \ $ { 0.01562 3% }

	{Due to which ~~effects~~ does the MQF value for the Gaussian pulse increase when using   $f_{\rm A} \cdot T = 1/4$  instead of  $f_{\rm A} \cdot T = 1/8$~~  verwendet~~?		{Due to which effect does the MQF value for the Gaussian pulse increase when using   $f_{\rm A} \cdot T = 1/4$  instead of  $f_{\rm A} \cdot T = 1/8$?
	\|type="()"}		\|type="()"}
	+ The truncation error is significantly increased.		+ The truncation error is significantly increased.
	- The aliasing error is significantly increased.		- The aliasing error is significantly increased.

	{Due to what ~~effects~~ does the MQF value for the Gaussian ~~momentum~~ increase when using  $f_{\rm A} \cdot T = 1/16$  instead of $f_{\rm A} \cdot T = 1/4$~~  verwendet~~?		{Due to what effect does the MQF value for the Gaussian pulse increase when using  $f_{\rm A} \cdot T = 1/16$  instead of $f_{\rm A} \cdot T = 1/4$?
	\|type="()"}		\|type="()"}
	- The ~~termination~~ error is significantly increased.~~termination~~		- The truncation error is significantly increased.
	+ The aliasing error is significantly increased.		+ The aliasing error is significantly increased.

	{Compare the MQF~~(MSE)~~ values of the rectangular pulse  $x_2(t)$  with those of the Gaussian pulse  $x_1(t)$. Which of the following statements are true?		{Compare the  $\rm MQF$  values of the rectangular pulse  $x_2(t)$  with those of the Gaussian pulse  $x_1(t)$.  Which of the following statements are true?
	\|type="[]"}		\|type="[]"}
	+ $\rm MQF$  becomes larger because the spectral function  $X_2(f)$  decays asymptotically slower than  $X_1(f)$.		+ $\rm MQF$  becomes larger because the spectral function  $X_2(f)$  decays asymptotically slower than  $X_1(f)$.
	+ The aliasing error dominates.		+ The aliasing error dominates.
	- The ~~termination~~ error dominates.		- The truncation error dominates.

	{Compare the MQF values of the ~~slit~~ pulse  $x_3(t)$  with those of the Gaussian pulse  $x_1(t)$. Which of the following statements are true?		{Compare the  $\rm MQF$  values of the "sinc pulse"  $x_3(t)$  with those of the Gaussian pulse  $x_1(t)$.  Which of the following statements are true?
	\|type="[]"}		\|type="[]"}
	- $\rm MQF$  becomes larger because the spectral function  $X_3(f)$  decays asymptotically slower than  $X_1(f)$.		- $\rm MQF$  becomes larger because the spectral function  $X_3(f)$  decays asymptotically slower than  $X_1(f)$.
	- The aliasing error dominates.		- The aliasing error dominates.
	+ The ~~termination~~ error dominates.		+ The truncation error dominates.

	</quiz>		</quiz>

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

2021-03-23T12:39:08Z

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

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