Aufgaben:Exercise 5.3: Mean Square Error - Revision history

Guenter at 15:14, 17 May 2021

2021-05-17T15:14:13Z

Guenter at 15:03, 17 May 2021

2021-05-17T15:03:08Z

Guenter at 15:01, 17 May 2021

2021-05-17T15:01:53Z

Guenter at 15:01, 17 May 2021

2021-05-17T15:01:07Z

Guenter at 15:00, 17 May 2021

2021-05-17T15:00:27Z

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"

Noah at 12:45, 22 March 2021

2021-03-22T12:45:32Z

Noah at 20:05, 21 March 2021

2021-03-21T20:05:16Z

Noah at 19:24, 21 March 2021

2021-03-21T19:24:59Z

Noah at 19:12, 21 March 2021

2021-03-21T19:12:32Z

@@ Line 111: / Line 111: @@
 '''(1)'''&nbsp; With the DFT parameters&nbsp; $N = 512$&nbsp; and&nbsp; $f_{\rm A} \cdot T = 1/8$&nbsp; the following follows after multiplying the two quantities:
 :$$f_{\rm P} \cdot T = N \cdot (f_{\rm A} \cdot T) = 64.$$
-*This covers the frequency range&nbsp; $–f_{\rm P}/2 \leq f < f_{\rm P}/2$&nbsp;:
+*This covers the frequency range&nbsp; $-f_{\rm P}/2 \leq f < +f_{\rm P}/2$:
 :$$f_{\rm max }\cdot T \hspace{0.15 cm}\underline{= 32}\hspace{0.05cm}.$$
@@ Line 120: / Line 120: @@
-'''(3)'''&nbsp; Correct is the <u>proposed solution 1 &nbsp; &rArr; &nbsp;  increase of the termination error</u>:
+'''(3)'''&nbsp; Correct is the <u>proposed solution 1 &nbsp; &rArr; &nbsp;  increase of the truncation error</u>:
-*This measure simultaneously halves&nbsp; $T_{\rm P}$&nbsp; from&nbsp; $8T$&nbsp; to&nbsp; $4T$&nbsp;.*Thus, only samples in the range&nbsp; $–2T \leq t < 2T$, are taken into account, which increases the termination error.
+*This measure simultaneously halves&nbsp; $T_{\rm P}$&nbsp; from&nbsp; $8T$&nbsp; to&nbsp; $4T$&nbsp;.
-*The mean square error&nbsp; $(\rm MQF)$&nbsp; increases from&nbsp; $0.15 \cdot 10^{-15}$&nbsp; to&nbsp; $8 \cdot 10^{-15}$ for the Gaussian pulse&nbsp; $x_1(t)$&nbsp;, although the aliasing error actually decreases slightly by this measure.
+*Thus, only samples in the range&nbsp; $–2T \leq t < 2T$ are taken into account, which increases the truncation error.
 '''(4)'''&nbsp; Correct is the <u>proposed solution 2 &nbsp; &rArr; &nbsp;  increase of the aliasing error:</u>:
-*By halving&nbsp; $f_{\rm A}$&nbsp; wird auch&nbsp; $f_{\rm P}$&nbsp; is also halved.
+*By halving&nbsp; $f_{\rm A}$&nbsp;  &rArr; &nbsp; $f_{\rm P}$&nbsp; 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&nbsp; $x_1(t)$&nbsp;, the mean square error&nbsp; $(\rm MQF)$&nbsp; increases from&nbsp; $1.5 \cdot 10^{-16}$&nbsp; to&nbsp; $3.3 \cdot 10^{-16}$.
+*Overall, for the Gaussian pulse&nbsp; $x_1(t)$, the mean square error&nbsp; $(\rm MQF)$&nbsp; increases from&nbsp; $1.5 \cdot 10^{-16}$&nbsp; to&nbsp; $3.3 \cdot 10^{-16}$.
@@ Line 135: / Line 137: @@
 '''(5)'''&nbsp;  <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.
-*Due to the slow,&nbsp; $\rm si$–shaped decay of the spectral function, the aliasing error dominates.
+*Due to the slow&nbsp; ($\rm sinc$–shaped)&nbsp; decay of the spectral function, the aliasing error dominates.
 *The&nbsp; $\rm MQF$ value at&nbsp; $f_{\rm A} \cdot T = 1/8$&nbsp; with&nbsp; $1.4 \cdot 10^{-5}$&nbsp; is therefore significantly larger than for the Gaussian pulse&nbsp; $(1.5 \cdot 10^{-16})$.
@@ Line 142: / Line 144: @@
 '''(6)'''&nbsp;  <u>Proposed solution 3</u> is correct:
 *The spectral function&nbsp; $X_3(f)$&nbsp; 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&nbsp; $\rm si$–shaped time function. This leads to the large&nbsp; $\rm MQF$ values given.
+*On the other hand, a truncation error is unavoidable with this&nbsp; $\rm sinc$–shaped time function.&nbsp; This leads to the large&nbsp; $\rm MQF$ values given.
 {{ML-Fuß}}
 __NOEDITSECTION__
 [[Category:Signal Representation: Exercises|^5.3 Possible DFT Errors^]]

@@ Line 35: / Line 35: @@
 For all spectral functions,&nbsp; $X(f = 0) = A \cdot T$.
-If the discrete-frequency spectrum is determined by the&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Discrete Fourier Transform]]&nbsp; with the DFT parameters
+If the discrete-frequency spectrum is determined by the&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Discrete Fourier Transform]]&nbsp; $\rm (DFT)$&nbsp; with the DFT parameters
 * $N = 512$ &nbsp; &rArr; &nbsp; number of samples considered in the time and frequency domain,
 *$f_{\rm A}$  &nbsp; &rArr; &nbsp; interpolation distance in the frequency domain,

@@ Line 10: / Line 10: @@
 :$$x_1(t) = A \cdot {\rm e}^{- \pi (t/T)^2} \hspace{0.05cm},$$
-*a&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Rectangular_Impulse|rectangular pulse]]&nbsp; $x_2(t)$&nbsp; with amplitude&nbsp; $A$&nbsp; and (equivalent) duration&nbsp; $T$:
+*a&nbsp; [[Signal_Representation/Special_Cases_of_Pulses#Rectangular_pulse|rectangular pulse]]&nbsp; $x_2(t)$&nbsp; with amplitude&nbsp; $A$&nbsp; and (equivalent) duration&nbsp; $T$:
 :$$x_2(t)   = \left\{ \begin{array}{c} A \\

@@ Line 6: / Line 6: @@
 We consider three pulses, namely
-*a&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Gaussian_Impulse|Gaussian pulse]]&nbsp; with amplitude&nbsp; $A$&nbsp; and  equivalent duration&nbsp; $T$:
+*a&nbsp; [[Signal_Representation/Special_Cases_of_Pulses#Gaussian_pulse|Gaussian pulse]]&nbsp; with amplitude&nbsp; $A$&nbsp; and  equivalent duration&nbsp; $T$:
 :$$x_1(t) = A \cdot {\rm e}^{- \pi (t/T)^2} \hspace{0.05cm},$$

@@ Line 3: / Line 3: @@
 }}
-[[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&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Gaussian_Impulse|Gaussian pulse]]&nbsp; with amplitude&nbsp; $A$&nbsp; and  equivalent duration&nbsp; $T$:
 :$$x_1(t) = A \cdot {\rm e}^{- \pi (t/T)^2} \hspace{0.05cm},$$
-*a&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Rectangular_Impulse|Rectangular pulse]]&nbsp; $x_2(t)$&nbsp; with amplitude&nbsp; $A$&nbsp; and (equivalent) duration&nbsp; $T$:
+*a&nbsp; [[Signal_Representation/Special_Cases_of_Impulse_Signals#Rectangular_Impulse|rectangular pulse]]&nbsp; $x_2(t)$&nbsp; with amplitude&nbsp; $A$&nbsp; and (equivalent) duration&nbsp; $T$:
 :$$x_2(t)   = \left\{ \begin{array}{c} A \\
@@ Line 20: / Line 20: @@
 \end{array}$$
-*a so called&nbsp; ''Sinc pulse''&nbsp; according to the following definition:
+*a so called&nbsp; "sinc pulse"&nbsp; 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&nbsp; $A = 1\ {\rm V}$&nbsp;  and&nbsp; $T = 1\ {\rm ms}$ in each case.
-The conventional&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse|Fourier Transform]]&nbsp;  leads to the following spectral functions:
+The conventional&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse|Fourier transform]]&nbsp;  leads to the following spectral functions:
 * $X_1(f)$&nbsp; is also Gaussian,
-* $X_2(f)$&nbsp; runs according to the&nbsp; $\rm si$–function,
+* $X_2(f)$&nbsp; runs according to the&nbsp; $\rm sinc$ function,
 * $X_3(f)$&nbsp; is constant for&nbsp; $|f| < 1/(2 T)$&nbsp; and outside zero.
@@ Line 35: / Line 35: @@
 For all spectral functions,&nbsp; $X(f = 0) = A \cdot T$.
-If the discrete-frequency spectrum is determined by the&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Discrete Fourier Transform(DFT)]]&nbsp; with the DFT parameters
+If the discrete-frequency spectrum is determined by the&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Discrete Fourier Transform]]&nbsp; with the DFT parameters
-* $N = 512$ &nbsp; &rArr; &nbsp; number of samples considered in the time and frequency domain,*$f_{\rm A}$  &nbsp; &rArr; &nbsp; interpolation distance in the frequency domain,
+* $N = 512$ &nbsp; &rArr; &nbsp; number of samples considered in the time and frequency domain,
@@ Line 42: / Line 43: @@
-The other DFT parameters are clearly fixed withn&bsp; $N$&nbsp; uan&nbsp; $f_{\rm A}$&nbsp; .The following applies to these:
+The other DFT parameters are clearly fixed withn&nbsp; $N$&nbsp; and&nbsp; $f_{\rm A}$.&nbsp; The following applies to these:
 :$$f_{\rm P} = N \cdot f_{\rm A},\hspace{0.3cm}T_{\rm P} = 1/f_{\rm A},\hspace{0.3cm}T_{\rm A} = T_{\rm
   P}/N
    \hspace{0.05cm}.$$
-The accuracy of the respective DFT approximation is captured by thenbsp; ''mean square error''&nbsp; (MSE, here MQF):
+The accuracy of the respective DFT approximation is captured by the&nbsp; "mean square error"&nbsp; $\rm (MSE)$.&nbsp; <br>Here, we use the designation&nbsp; $\rm MQF$ &nbsp; &rArr; &nbsp; (German:&nbsp; "Mittlerer Quadratischer Fehler"):
 :$${\rm MQF} =  \frac{1}{N}\cdot \sum_{\mu = 0 }^{N-1}
   \left|X(\mu \cdot f_{\rm A})-\frac{D(\mu)}{f_{\rm A}}\right|^2 \hspace{0.05cm}.$$
@@ Line 64: / Line 65: @@
 ''Hints:''
-*This task belongs to the chapter&nbsp; [[Signal_Representation/Possible_Errors_When_Using_DFT|Possible Errors when Using DFT]].
+*This task belongs to the chapter&nbsp; [[Signal_Representation/Possible_Errors_When_Using_DFT|Possible errors when using DFT]].
-*The theory for this chapter is summarised in the learning video&nbsp; [[Fehlermöglichkeiten_bei_Anwendung_der_DFT_(Lernvideo)|Possible Errors when Using DFT]]&nbsp;.
+*The theory for this chapter is summarised in the (German language) learning video <br> &nbsp; &nbsp; &nbsp;[[Fehlermöglichkeiten_bei_Anwendung_der_DFT_(Lernvideo)|Fehlermöglichkeiten bei Anwendung der DFT]] &nbsp; &rArr; &nbsp; "Possible errors when using DFT".
@@ Line 82: / Line 83: @@
 $T_{\rm A}/T\ = \ $ { 0.01562 3% }
-{Due to which effects does the MQF value for the Gaussian pulse increase when using &nbsp;  $f_{\rm A} \cdot T = 1/4$&nbsp; instead of&nbsp; $f_{\rm A} \cdot T = 1/8$&nbsp; verwendet?
+{Due to which effect does the MQF value for the Gaussian pulse increase when using &nbsp;  $f_{\rm A} \cdot T = 1/4$&nbsp; instead of&nbsp; $f_{\rm A} \cdot T = 1/8$?
 |type="()"}
 + The truncation 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&nbsp; $f_{\rm A} \cdot T = 1/16$&nbsp; instead of  $f_{\rm A} \cdot T = 1/4$&nbsp; verwendet?
+{Due to what effect does the MQF value for the Gaussian pulse increase when using&nbsp; $f_{\rm A} \cdot T = 1/16$&nbsp; instead of  $f_{\rm A} \cdot T = 1/4$?
 |type="()"}
-- The termination error is significantly increased.termination
+- The truncation error is significantly increased.
 + The aliasing error is significantly increased.
-{Compare the MQF(MSE) values of the rectangular pulse&nbsp; $x_2(t)$&nbsp; with those of the Gaussian pulse&nbsp; $x_1(t)$. Which of the following statements are true?
+{Compare the&nbsp; $\rm MQF$&nbsp; values of the rectangular pulse&nbsp; $x_2(t)$&nbsp; with those of the Gaussian pulse&nbsp; $x_1(t)$.&nbsp; Which of the following statements are true?
 |type="[]"}
 + $\rm MQF$&nbsp; becomes larger because the spectral function&nbsp; $X_2(f)$&nbsp; decays asymptotically slower than&nbsp; $X_1(f)$.
 + The aliasing error dominates.
-- The termination error dominates.
+- The truncation error dominates.
-{Compare the MQF values of the slit pulse&nbsp; $x_3(t)$&nbsp; with those of the Gaussian pulse&nbsp; $x_1(t)$. Which of the following statements are true?
+{Compare the&nbsp; $\rm MQF$&nbsp; values of the "sinc pulse"&nbsp; $x_3(t)$&nbsp; with those of the Gaussian pulse&nbsp; $x_1(t)$.&nbsp; Which of the following statements are true?
 |type="[]"}
 - $\rm MQF$&nbsp; becomes larger because the spectral function&nbsp; $X_3(f)$&nbsp; decays asymptotically slower than&nbsp; $X_1(f)$.
 - The aliasing error dominates.
-+ The termination error dominates.
++ The truncation error dominates.
 </quiz>

← Older revision		Revision as of 12:39, 23 March 2021
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−	[[Category:~~Exercises for~~ Signal Representation\|^5.3 Possible DFT Errors^]]	+	[[Category:Signal Representation: Exercises\|^5.3 Possible DFT Errors^]]

@@ Line 3: / Line 3: @@
 }}
-[[File:P_ID1145__Sig_A_5_3.png|250px|right|frame|Gaußimpuls, Rechteckimpuls, Spaltimpuls und einige Kenngrößen]]
+[[File:P_ID1145__Sig_A_5_3.png|250px|right|frame|Gaussian pulse, square pulse, sinc pulse and some parameters]]
 We consider three pulse-like signals, namely
@@ Line 74: / Line 74: @@
 <quiz display=simple>
-{Welcher Bereich&nbsp; $|f| \leq  f_{\text{max}}$&nbsp; wird mit&nbsp; $N = 512$&nbsp; und&nbsp; $f_{\rm A} \cdot T = 1/8$&nbsp; erfasst?
+{Which range&nbsp; $|f| \leq  f_{\text{max}}$&nbsp; is covered with&nbsp; $N = 512$&nbsp; and&nbsp; $f_{\rm A} \cdot T = 1/8$&nbsp;?
 |type="{}"}
 $f_{\text{max}} \cdot T\ = \ $  { 32 3% }
-{In welchem Zeitabstand&nbsp; $T_{\rm A}$&nbsp; liegen die Abtastwerte von&nbsp; $x(t)$&nbsp; vor?
+{At what time interval&nbsp; $T_{\rm A}$&nbsp; are the sampled values of&nbsp; $x(t)$&nbsp; available?
 |type="{}"}
 $T_{\rm A}/T\ = \ $ { 0.01562 3% }
-{Aufgrund welcher Effekte erhöht sich der MQF–Wert für den Gaußimpuls, wenn man&nbsp;  $f_{\rm A} \cdot T = 1/4$&nbsp; anstelle von&nbsp; $f_{\rm A} \cdot T = 1/8$&nbsp; verwendet?
+{Due to which effects does the MSE value for the Gaussian pulse increase when using &nbsp;  $f_{\rm A} \cdot T = 1/4$&nbsp; instead of&nbsp; $f_{\rm A} \cdot T = 1/8$&nbsp; verwendet?
 |type="()"}
-+ Der Abbruchfehler wird signifikant vergrößert.
++ The truncation error is significantly increased.
-- Der Aliasingfehler wird signifikant vergrößert.
+- The aliasing error is significantly increased.
-{Aufgrund welcher Effekte erhöht sich der MQF–Wert für den Gaußimpuls, wenn man&nbsp; $f_{\rm A} \cdot T = 1/16$&nbsp; anstelle von $f_{\rm A} \cdot T = 1/4$&nbsp; verwendet?
+{Due to what effects does the MSE value for the Gaussian momentum increase when using&nbsp; $f_{\rm A} \cdot T = 1/16$&nbsp; instead of  $f_{\rm A} \cdot T = 1/4$&nbsp; verwendet?
 |type="()"}
-- Der Abbruchfehler wird signifikant vergrößert.
+- The termination error is significantly increased.termination
-+ Der Aliasingfehler wird signifikant vergrößert.
++ The aliasing error is significantly increased.
-{Vergleichen Sie die MQF–Werte des Rechteckimpulses&nbsp; $x_2(t)$&nbsp; mit denen des Gaußimpulses&nbsp; $x_1(t)$. Welche der folgenden Aussagen treffen zu?
+{Compare the MQF(MSE) values of the rectangular pulse&nbsp; $x_2(t)$&nbsp; with those of the Gaussian pulse&nbsp; $x_1(t)$. Which of the following statements are true?
 |type="[]"}
-+ $\rm MQF$&nbsp; wird größer, da die Spektralfunktion&nbsp; $X_2(f)$&nbsp; asymptotisch langsamer abfällt als&nbsp; $X_1(f)$.
++ $\rm MQF$&nbsp; becomes larger because the spectral function&nbsp; $X_2(f)$&nbsp; decays asymptotically slower than&nbsp; $X_1(f)$.
-+ Es dominiert der Aliasingfehler.
++ The aliasing error dominates.
-- Es dominiert der Abbruchfehler.
+- The termination error dominates.
-{Vergleichen Sie die MQF–Werte des Spaltimpulses&nbsp; $x_3(t)$&nbsp; mit denen des Gaußimpulses&nbsp; $x_1(t)$. Welche der folgenden Aussagen treffen zu?
+{Compare the MQF values of the slit pulse&nbsp; $x_3(t)$&nbsp; with those of the Gaussian pulse&nbsp; $x_1(t)$. Which of the following statements are true?
 |type="[]"}
-- $\rm MQF$&nbsp; wird größer, da die Spektralfunktion&nbsp; $X_3(f)$&nbsp; asymptotisch langsamer abfällt als&nbsp; $X_1(f)$.
+- $\rm MQF$&nbsp; becomes larger because the spectral function&nbsp; $X_3(f)$&nbsp; decays asymptotically slower than&nbsp; $X_1(f)$.
-- Es dominiert der Aliasingfehler.
+- The aliasing error dominates.
-+ Es dominiert der Abbruchfehler.
++ The termination error dominates.
 </quiz>
-===Musterlösung===
+===Solution===
 {{ML-Kopf}}
-'''(1)'''&nbsp; Mit den DFT–Parametern&nbsp; $N = 512$&nbsp; und&nbsp; $f_{\rm A} \cdot T = 1/8$&nbsp; folgt nach Multiplikation der beiden Größen:
+'''(1)'''&nbsp; With the DFT parameters&nbsp; $N = 512$&nbsp; and&nbsp; $f_{\rm A} \cdot T = 1/8$&nbsp; the following follows after multiplying the two quantities:
 :$$f_{\rm P} \cdot T = N \cdot (f_{\rm A} \cdot T) = 64.$$
-*Dadurch wird der Frequenzbereich&nbsp; $–f_{\rm P}/2 \leq f < f_{\rm P}/2$&nbsp; erfasst:
+*This covers the frequency range&nbsp; $–f_{\rm P}/2 \leq f < f_{\rm P}/2$&nbsp;:
 :$$f_{\rm max }\cdot T \hspace{0.15 cm}\underline{= 32}\hspace{0.05cm}.$$
-'''(2)'''&nbsp; Die Periodifizierung der Zeitfunktion basiert auf dem Parameter&nbsp; $T_{\rm P} = 1/f_{\rm A} = 8T$.
+'''(2)'''&nbsp; The periodisation of the time function is based on the parameter&nbsp; $T_{\rm P} = 1/f_{\rm A} = 8T$.
-*Der Abstand zweier Abtastwerte beträgt somit
+*The distance between two samples is therefore
 :$$T_{\rm A}/T =  \frac{T_{\rm P}/T}{N} = \frac{8}{512}\hspace{0.15 cm}\underline{ = 0.015625}\hspace{0.05cm}.$$
-'''(3)'''&nbsp; Richtig ist der <u>Lösungsvorschlag 1 &nbsp; &rArr; &nbsp;  Erhöhung des Abbruchfehlers</u>:
+'''(3)'''&nbsp; Correct is the <u>proposed solution 1 &nbsp; &rArr; &nbsp;  increase of the termination error</u>:
-*Mit dieser Maßnahme wird gleichzeitig&nbsp; $T_{\rm P}$&nbsp; von&nbsp; $8T$&nbsp; auf&nbsp; $4T$&nbsp; halbiert.
+*This measure simultaneously halves&nbsp; $T_{\rm P}$&nbsp; from&nbsp; $8T$&nbsp; to&nbsp; $4T$&nbsp;.*Thus, only samples in the range&nbsp; $–2T \leq t < 2T$, are taken into account, which increases the termination error.
-*Berücksichtigt werden somit nur noch Abtastwerte im Bereich&nbsp; $–2T \leq t < 2T$, wodurch der Abbruchfehler erhöht wird.
+*The mean square error&nbsp; $(\rm MQF)$&nbsp; steigt dadurch beim Gaußimpuls&nbsp; $x_1(t)$&nbsp; von&nbsp; $0.15 \cdot 10^{-15}$&nbsp; auf&nbsp; $8 \cdot 10^{-15}$, obwohl der Aliasingfehler durch diese Maßnahme sogar etwas kleiner wird.

@@ Line 27: / Line 27: @@
 Let the signal parameters be&nbsp; $A = 1\ {\rm V}$&nbsp;  and&nbsp; $T = 1\ {\rm ms}$ in each case.
-Die konventionelle&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse|Fouriertransformation]]&nbsp;  führt zu folgenden Spektralfunktionen:
+The conventional&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse|Fourier Transform]]&nbsp;  leads to the following spectral functions:
-* $X_1(f)$&nbsp; ist ebenfalls gaußförmig,
+* $X_1(f)$&nbsp; is also Gaussian,
-* $X_2(f)$&nbsp; verläuft entsprechend der&nbsp; $\rm si$–Funktion,
+* $X_2(f)$&nbsp; runs according to the&nbsp; $\rm si$–function,
-* $X_3(f)$&nbsp; ist für&nbsp; $|f| < 1/(2 T)$&nbsp; konstant und außerhalb Null.
+* $X_3(f)$&nbsp; is constant for&nbsp; $|f| < 1/(2 T)$&nbsp; and outside zero.
-Für alle Spektralfunktionen gilt&nbsp; $X(f = 0) = A \cdot T$.
+For all spectral functions,&nbsp; $X(f = 0) = A \cdot T$.
-Ermittelt man das frequenzdiskrete Spektrum durch die&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Diskrete Fouriertransformation]]&nbsp; (DFT) mit den DFT-Parametern
+If the discrete-frequency spectrum is determined by the&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|Discrete Fourier Transform(DFT)]]&nbsp; (DFT) with the DFT parameters
-* $N = 512$ &nbsp; &rArr; &nbsp; Anzahl der berücksichtigten Abtastwerte im Zeit– und Frequenzbereich,
+* $N = 512$ &nbsp; &rArr; &nbsp; number of samples considered in the time and frequency domain,*$f_{\rm A}$  &nbsp; &rArr; &nbsp; interpolation distance in the frequency domain,
-so wird dies aufgrund von Abbruch– und/oder Aliasingfehler zu Verfälschungen führen.
+this will lead to distortions due to truncation and/or aliasing errors.
-Die weiteren DFT–Parameter liegen mit&nbsp; $N$&nbsp; und&nbsp; $f_{\rm A}$&nbsp; eindeutig fest. Für diese gilt:
+DThe other DFT parameters are clearly fixed withn&bsp; $N$&nbsp; uan&nbsp; $f_{\rm A}$&nbsp; .Fhe following applies to these:
 :$$f_{\rm P} = N \cdot f_{\rm A},\hspace{0.3cm}T_{\rm P} = 1/f_{\rm A},\hspace{0.3cm}T_{\rm A} = T_{\rm
   P}/N
    \hspace{0.05cm}.$$
-Die Genauigkeit der jeweiligen DFT–Approximation wird durch den&nbsp; ''mittleren quadratischen Fehler''&nbsp; (MQF) erfasst:
+DThe accuracy of the respective DFT approximation is captured by thenbsp; ''mean square error''&nbsp; (MSE):
 :$${\rm MQF} =  \frac{1}{N}\cdot \sum_{\mu = 0 }^{N-1}
   \left|X(\mu \cdot f_{\rm A})-\frac{D(\mu)}{f_{\rm A}}\right|^2 \hspace{0.05cm}.$$