Applets:Discrete Fouriertransform and Inverse - Revision history

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 This interactive calculation tool was designed and implemented at the&nbsp; [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]&nbsp; at the&nbsp; [https://www.tum.de/en Technical University of Munich].
-*The first version was created in 2008 by&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|&raquo;Thomas Großer&laquo;]]&nbsp; as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|Günter Söder]]).
+*The first version was created in 2008 by&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|&raquo;Thomas Großer&laquo;]]&nbsp; as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_from_1974-2024.29|Günter Söder]]).
 *Last revision and English version 2020/2021 by&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|&raquo;Carolin Mirschina&laquo;]]&nbsp; in the context of a working student activity.&nbsp;

← Older revision		Revision as of 16:02, 6 July 2023
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	*The first version was created in 2008 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29\|»Thomas Großer«]]  as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\|Günter Söder]]).		*The first version was created in 2008 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29\|»Thomas Großer«]]  as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29\|Günter Söder]]).

−	*Last revision and English version 2020/2021 by  [[~~Biografien_und_Bibliografien~~/~~An_LNTwww_beteiligte_Studierende~~#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|»Carolin Mirschina«]]  in the context of a working student activity.	+	*Last revision and English version 2020/2021 by  [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29\|»Carolin Mirschina«]]  in the context of a working student activity.

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 This interactive calculation tool was designed and implemented at the&nbsp; [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]&nbsp; at the&nbsp; [https://www.tum.de/en Technical University of Munich].
-*The first version was created in 2008 by&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|&raquo;Thomas Großer&laquo;]]&nbsp; as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
+*The first version was created in 2008 by&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|&raquo;Thomas Großer&laquo;]]&nbsp; as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/LNTwww_members_from_LNT#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28at_LNT_since_1974.29|Günter Söder]]).
 *Last revision and English version 2020/2021 by&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|&raquo;Carolin Mirschina&laquo;]]&nbsp; in the context of a working student activity.&nbsp;

@@ Line 320: / Line 320: @@
 <br>
 [[File:Aufgaben_2D-Gauss.png|right]]
-* First select the number (1,...) of the exercise. <br>
+* First select the number&nbsp; $($1,&nbsp;...$)$&nbsp; of the exercise. <br>
 * A description of the exercise will be displayed.
 *The parameter values are adjusted. <br>
-*Solution after pressing "Show solution". <br>
+*Solution after pressing&nbsp; &raquo;Show solution&laquo;. <br>
-*The number 0 corresponds to a "Reset":
+*The number&nbsp; "$0$"&nbsp; corresponds to a&nbsp; &raquo;Reset&laquo;:
 #Same setting as at program start.
-#Output of a "reset text" with further explanations about the applet.
+#Output of a&nbsp; &raquo;reset text&laquo;&nbsp; with further explanations about the applet.
@@ Line 333: / Line 333: @@
 :*&nbsp;All coefficients in the time domain are&nbsp; $d(\nu)=1$.&nbsp; Thus all&nbsp; $D(\mu)=0$&nbsp; with the exception of&nbsp; $\textrm{Re}[D(0)]=1$.  <br>
-:*&nbsp;This corresponds to the conventional (time-continuous) Fourier Transform:&nbsp; $x(t)=A\hspace{0.15cm} \circ\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm} X(f)=A\cdot \delta (f=0)$&nbsp; with&nbsp; $A=1$.
+:*&nbsp;This corresponds to the conventional&nbsp; $($time-continuous$)$&nbsp; Fourier Transform:&nbsp; $x(t)=A\hspace{0.15cm} \circ\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm} X(f)=A\cdot \delta (f=0)$&nbsp; with&nbsp; $A=1$.
 {{BlaueBox|TEXT=
@@ Line 366: / Line 366: @@
 }}
-:*&nbsp;A shift in the time domain changes the cosine signal to a&nbsp; &raquo;harmonic oscillation&raquo;&nbsp; with arbitrary phase.
+:*&nbsp;A shift in the time domain changes the cosine signal to a&nbsp; "harmonic oscillation"&nbsp; with arbitrary phase.
 :*&nbsp;The&nbsp; $D(\mu)$&nbsp; field is still zero except for&nbsp; $D(1)$&nbsp; and&nbsp; $D(15)$.&nbsp; The absolute values&nbsp; $|D(1)|$&nbsp; and&nbsp; $|D(15)|$&nbsp; also remain the same.
 :*&nbsp;The only change concerns the phase,&nbsp; i.e. the different distribution of the absolute values between the real and imaginary part.
 {{BlaueBox|TEXT=
-'''(7)'''&nbsp; New setting:&nbsp; DFT of signal&nbsp; $\rm (f)$:&nbsp; Sinusoidal signal.&nbsp; Interpret the result in the frequency domain.&nbsp; What is the analogon of the conventional Fourier transform?
+'''(7)'''&nbsp; New setting:&nbsp; DFT of signal&nbsp; $\rm (f)$:&nbsp; Sinusoidal signal.&nbsp; Interpret the result in the frequency domain.&nbsp; What is the analogon of the conventional Fourier Transform?
 }}
 :*&nbsp;The sine signal results from the cosine signal by applying four time shifts.&nbsp; Therefore all statements of&nbsp; '''(6)'''&nbsp; are still valid.
-:*&nbsp;For the conventional&nbsp; $($time continuous$)$&nbsp; Fourier transform it holds that&nbsp; $x(t)= \sin(2\pi \cdot f_0 \cdot t)\hspace{0.15cm}\circ\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm}
+:*&nbsp;For the conventional (time continuous) Fourier transform it holds that&nbsp; $x(t)= \sin(2\pi \cdot f_0 \cdot t)\hspace{0.15cm}\circ\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm}
 X(f)=j/2 \cdot [\delta (f+f_0)-\delta (f-f_0)]$.
 :*&nbsp;The coefficient&nbsp; $D(1)$ &nbsp; $\Rightarrow$ &nbsp;$($frequency:&nbsp; $+f_0)$&nbsp; is imaginary and has the imaginary part&nbsp; $-0.5$.&nbsp; Accordingly,&nbsp; $\textrm{Im}[D(15)]=+0.5$ &nbsp; &rArr; &nbsp; $($frequency:&nbsp; $-f_0)$&nbsp; applies.
 {{BlaueBox|TEXT=
-'''(8)'''&nbsp; New setting:&nbsp; DFT of signal&nbsp; $\rm (g)$:&nbsp; Cosine signal&nbsp; $($two periods$)$.&nbsp; Interpret the result in comparison to exercise&nbsp; '''(5)'''.
+'''(8)'''&nbsp; New setting:&nbsp; DFT of signal&nbsp; $\rm (g)$:&nbsp; Cosine signal (two periods).&nbsp; Interpret the result in comparison to exercise&nbsp; '''(5)'''.
 }}
@@ Line 387: / Line 387: @@
 {{BlaueBox|TEXT=
-'''(9)'''&nbsp; Now examine the case DFT of a sinodial signal&nbsp; $($two periods$)$.&nbsp; Which modifications do you need to make in the time domain?&nbsp; Interpret the result.
+'''(9)'''&nbsp; Now examine the case DFT of a sinodial signal (two periods).&nbsp; Which modifications do you need to make in the time domain?&nbsp; Interpret the result.
 }}
-:*&nbsp;The desired signal can be obtained from the DFT of signal&nbsp; $\rm (g)$:&nbsp; Cosine signal (two periods) with two shifts.&nbsp; For&nbsp; '''(7)''':&nbsp; there were four shifts.
+:*&nbsp;The desired signal can be obtained from the DFT of signal&nbsp; $\rm (g)$:&nbsp; Cosine signal (two periods) with two shifts.&nbsp; With the result of&nbsp; '''(7)''':&nbsp; Four shifts.
 :*&nbsp;The DFT result is accordingly&nbsp; $\textrm{Im}[D(2)]=-0.5$&nbsp; and&nbsp; $\textrm{Im}[D(14)]=+0.5$.
 {{BlaueBox|TEXT=
-'''(10)'''&nbsp; New setting:&nbsp; DFT of signal&nbsp; $\rm (h)$:&nbsp; Alternating time coefficients.&nbsp; Interpret the DFT result.}}
+'''(10)'''&nbsp; New setting: DFT of signal&nbsp; $\rm (h)$:&nbsp; Alternating time coefficients.&nbsp; Interpret the DFT result.}}
-:*&nbsp;Here,&nbsp; the time continuous Fourier transform is given by:&nbsp; $x(t)=\cos(2\pi \cdot (8 f_0) \cdot t)\hspace{0.15cm}\circ\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm}
+:*&nbsp;Here, the time continuous Fourier transform is given by:&nbsp; $x(t)=\cos(2\pi \cdot (8 f_0) \cdot t)\hspace{0.15cm}\circ\!\!\!-\!\!\!-\!\!\!-\!\!\bullet\hspace{0.15cm}
 X(f)=0.5 \cdot \delta (f- 8 f_0)+0.5 \cdot \delta (f+ 8 f_0)$.
-:*&nbsp;$8 f_0$&nbsp; is the highest frequency that can be displayed with&nbsp; $N=16$&nbsp; in the DFT.&nbsp; There are only two sampled values per period,&nbsp; namely&nbsp; $+1$&nbsp; and&nbsp; $-1$.
+:*&nbsp;$8 f_0$ is the highest frequency that can be displayed with&nbsp; $N=16$&nbsp; in the DFT.&nbsp; There are only two sampled values per period, namely $+1$ and $-1$.
-:*&nbsp;Difference to exercise&nbsp; '''(5)''':&nbsp; $D(1)=0.5$&nbsp; now becomes&nbsp; $D(8)=0.5$.&nbsp; Likewise,&nbsp; $D(15)=0.5$&nbsp; is also shifted to&nbsp; $D(8)=0.5$.&nbsp; Final result:&nbsp; $D(8)=1$.
+:*&nbsp;Difference to exercise&nbsp; '''(5)''':&nbsp; $D(1)=0.5$&nbsp; now becomes&nbsp; $D(8)=0.5$.&nbsp; Likewise,&nbsp; $D(15)=0.5$&nbsp; is shifted to&nbsp; $D(8)=0.5$.&nbsp; Final result:&nbsp; $D(8)=1$.
 {{BlaueBox|TEXT=
-'''(11)'''&nbsp; What are the differences between the two settings&nbsp; &raquo;DFT from signal&nbsp; $\rm (i)$:&nbsp; Dirac delta impulse&laquo;&nbsp;  and &nbsp;  &raquo;IDFT from spectrum&nbsp; $\rm (I)$:&nbsp; Dirac delta spectrum&laquo;?
+'''(11)'''&nbsp; What are the differences between the two settings DFT from signal&nbsp; $\rm (i)$:&nbsp; Dirac delta impulse  and &nbsp;  IDFT from spectrum&nbsp; $\rm (I)$:&nbsp; Dirac delta spectrum?
 }}
-:*&nbsp;Only a few!&nbsp; In the first case,&nbsp; all coefficients are&nbsp; $D(\mu) = 1/N=0.0625$&nbsp; $($real$)$;&nbsp; whereas in the second case,&nbsp; all coefficients are&nbsp; $d(\nu)=1$&nbsp; $($real$)$.
+:*&nbsp;None! In the first case, all coefficients are&nbsp; $D(\mu)=1$ (real);&nbsp; in the second case, however, equivalently&nbsp; $d(\nu)=1$ (real).
 {{BlaueBox|TEXT=
-'''(12)'''&nbsp; Are there differences in shifting the real&nbsp; &raquo;$1$&laquo;&nbsp; in the according input fields by one place at a time,&nbsp; that is for&nbsp; &raquo;$d(\nu = 1)=1$&laquo;&nbsp; and&nbsp; &raquo;$D(\mu = 1)=1$&laquo;?
+'''(12)'''&nbsp; Are there differences in shifting the real&nbsp; "$1$"&nbsp; in the according input fields by one place at a time, that is for&nbsp; $d(\nu = 1)=1$&nbsp; and&nbsp; $D(\mu = 1)=1$?
 }}
@@ Line 492: / Line 492: @@
 ==About the Authors==
-This interactive applet was designed and implemented at the&nbsp; [https://www.ce.cit.tum.de/en/lnt/home/ &raquo;Institute for Communications Engineering&laquo;]&nbsp; at the&nbsp; [https://www.tum.de/en &raquo;Technical University of Munich&laquo;].
+This interactive calculation tool was designed and implemented at the&nbsp; [https://www.ei.tum.de/en/lnt/home/ Institute for Communications Engineering]&nbsp; at the&nbsp; [https://www.tum.de/en Technical University of Munich].
-*The first version was created in 2008 by&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|&raquo;Thomas Großer&laquo;]]&nbsp; as part of his diploma thesis with “FlashMX – Actionscript”&nbsp; $($Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|&raquo;Günter Söder&laquo;]]$)$.
+*The first version was created in 2008 by&nbsp; [[Biographies_and_Bibliographies/Students_involved_in_LNTwww#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|&raquo;Thomas Großer&laquo;]]&nbsp; as part of his diploma thesis with “FlashMX – Actionscript” (Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|Günter Söder]]).
 *Last revision and English version 2020/2021 by&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Carolin_Mirschina_.28Ingenieurspraxis_Math_2019.2C_danach_Werkstudentin.29|&raquo;Carolin Mirschina&laquo;]]&nbsp; in the context of a working student activity.&nbsp;
-The conversion of this applet to HTML 5 was financially supported by&nbsp; [https://www.ei.tum.de/studium/studienzuschuesse/ &raquo;Studienzuschüsse&laquo;]&nbsp; ("study grants") of the TUM Faculty EI. We thank them.
+The conversion of this applet to HTML 5 was financially supported by&nbsp; [https://www.ei.tum.de/studium/studienzuschuesse/ Studienzuschüsse]&nbsp; ("study grants") of the TUM Faculty EI. We thank them.

@@ Line 492: / Line 492: @@
 ==About the Authors==
-This interactive applet was designed and implemented at the&nbsp; [https://www.ei.tum.de/en/lnt/home/ &raquo;Institute for Communications Engineering&laquo;]&nbsp; at the&nbsp; [https://www.tum.de/en &raquo;Technical University of Munich&laquo;].
+This interactive applet was designed and implemented at the&nbsp; [https://www.ce.cit.tum.de/en/lnt/home/ &raquo;Institute for Communications Engineering&laquo;]&nbsp; at the&nbsp; [https://www.tum.de/en &raquo;Technical University of Munich&laquo;].
 *The first version was created in 2008 by&nbsp; [[Biografien_und_Bibliografien/An_LNTwww_beteiligte_Studierende#Thomas_Gro.C3.9Fer_.28Diplomarbeit_LB_2006.2C_danach_freie_Mitarbeit_bis_2010.29|&raquo;Thomas Großer&laquo;]]&nbsp; as part of his diploma thesis with “FlashMX – Actionscript”&nbsp; $($Supervisor: [[Biographies_and_Bibliographies/An_LNTwww_beteiligte_Mitarbeiter_und_Dozenten#Prof._Dr.-Ing._habil._G.C3.BCnter_S.C3.B6der_.28am_LNT_seit_1974.29|&raquo;Günter Söder&laquo;]]$)$.

@@ Line 294: / Line 294: @@
 The upper graph shows the asymmetric triangular pulse&nbsp; $x(t)$&nbsp; whose absolute width is smaller than&nbsp; $T_{\rm P} = N \cdot T_{\rm A}$.
-[[File:P_ID1139__Sig_T_5_1_S7b_neu.png|right|frame|For the assignment of the DFT coefficients with&nbsp; $N=8$]]
+[[File:EN_Sig_T_5_1_S7b_neu.png|right|frame|On assigning of the DFT coefficients with&nbsp; $N=8$]]
 The sketch below shows the assigned DFT coefficients valid for&nbsp;  $N = 8$:

@@ Line 262: / Line 262: @@
 ===Interpretation of DFT and IDFT===
-<br>
 The graph shows the discrete coefficients in the time and frequency domain together with the periodized continuous-time functions.
-[[File:P_ID1136__Sig_T_5_1_S7_neu.png|center|frame|Time and frequency domain coefficients of the DFT]]
+[[File:P_ID1136__Sig_T_5_1_S7_neu.png|right|frame|Time and frequency domain coefficients of the DFT]]
 *The spectral coefficients&nbsp; $D(\mu)$&nbsp; must always be set complex to be able to consider also odd time functions.
-*In order to be able to transform band&ndash;pass signals in the equivalent lown&dash;pass range, complex time coefficients&nbsp; $d(\nu)$ are usually used.
-*The basic interval for&nbsp; $\nu$&nbsp; and&nbsp;  $\mu$&nbsp; is usually defined as the range from&nbsp; $0$&nbsp; to&nbsp; $N - 1$, as in the above diagram.
+*In order to be able to transform band&ndash;pass signals in the equivalent low&ndash;pass range,&nbsp; complex time coefficients&nbsp; $d(\nu)$ are usually used.
-*With the complex-valued number sequences&nbsp; $\langle \hspace{0.1cm}d(\nu)\hspace{0.1cm}\rangle  = \langle \hspace{0.1cm}d(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , d(N-1) \hspace{0.1cm}\rangle$  &nbsp; as well as &nbsp; $\langle \hspace{0.1cm}D(\mu)\hspace{0.1cm}\rangle  =   \langle \hspace{0.1cm}D(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , D(N-1) \hspace{0.1cm}\rangle$&nbsp; DFT and IDFT are symbolized similar to the conventional Fourier transform:
-:$$\langle \hspace{0.1cm} D(\mu)\hspace{0.1cm}\rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle \hspace{0.1cm} d(\nu) \hspace{0.1cm}\rangle  \hspace{0.05cm}.$$
+*The basic interval for&nbsp; $\nu$&nbsp; and&nbsp;  $\mu$&nbsp; is usually defined as the range from&nbsp; $0$&nbsp; to&nbsp; $N - 1$,&nbsp; as in the above diagram.
-*If the time function&nbsp; $x(t)$&nbsp; is already limited to the range&nbsp; $0 \le t \lt N \cdot T_{\rm A}$,&nbsp; then the time coefficients output by the IDFT directly indicate the samples of the time function:  &nbsp; $d(\nu) = x(\nu \cdot T_{\rm A}).$
-*If&nbsp; $x(t)$&nbsp; is shifted with respect to the basic interval, one has to choose the assignment between&nbsp; $x(t)$&nbsp; and the coefficients&nbsp; $d(\nu)$&nbsp; shown in&nbsp; $\text{Example 3}$.&nbsp;
+*With the complex-valued number sequences&nbsp;
 {{GraueBox|TEXT=
 $\text{Example 3:}$&nbsp;
-The upper graph shows the asymmetric triangular pulse&nbsp; $x(t)$ whose absolute width is smaller than&nbsp; $T_{\rm P} = N \cdot T_{\rm A}$.
+The upper graph shows the asymmetric triangular pulse&nbsp; $x(t)$&nbsp; whose absolute width is smaller than&nbsp; $T_{\rm P} = N \cdot T_{\rm A}$.
 [[File:P_ID1139__Sig_T_5_1_S7b_neu.png|right|frame|For the assignment of the DFT coefficients with&nbsp; $N=8$]]
-The sketch below shows the assigned DFT coefficients valid for&nbsp;  $N = 8$
+The sketch below shows the assigned DFT coefficients valid for&nbsp;  $N = 8$:
-*For&nbsp;  $\nu = 0,\hspace{0.05cm}\text{...} \hspace{0.05cm} , N/2 = 4$,&nbsp; &nbsp; $d(\nu) = x(\nu \cdot T_{\rm A})$ is valid:
+*For&nbsp;  $\nu = 0,\hspace{0.05cm}\text{...} \hspace{0.05cm} , N/2 = 4$,&nbsp; &nbsp; $d(\nu) = x(\nu \cdot T_{\rm A})$ &nbsp; is valid:
 :$$d(0) = x (0)\hspace{0.05cm}, \hspace{0.15cm}
@@ Line 294: / Line 305: @@
 :$$d(3) = x (3T_{\rm A})\hspace{0.05cm}, \hspace{0.15cm}
 d(4) = x (4T_{\rm A})\hspace{0.05cm}.$$
-*On the other hand, the coefficients&nbsp; $d(5)$,&nbsp; $d(6)$&nbsp; and&nbsp; d$(7)$&nbsp; are to be set as follows:
+*The coefficients&nbsp; $d(5)$,&nbsp; $d(6)$&nbsp; and&nbsp; d$(7)$&nbsp; are to be set as follows:
 :$$d(\nu) = x \big ((\nu\hspace{-0.05cm} - \hspace{-0.05cm} N ) \cdot T_{\rm   A}\big )  $$

@@ Line 239: / Line 239: @@
   D(\mu) \cdot  {w}^{-\nu \hspace{0.03cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}.$$
-Then holds with the indexing variables&nbsp;
+With the indexing variables&nbsp;
 *$\nu = 0,  \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1$,
-*$\mu = 0,  \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1$:
+*$\mu = 0,  \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1$,:
-:$$d(\nu) =
    {\rm P}\left\{x(t)\right\}{\big \vert}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm
-   A} }\hspace{0.01cm},$$
+   A} }\hspace{0.01cm},$
 #$D(\mu) = f_{\rm A} \cdot
    {\rm P}\left\{X(f)\right\}{\big \vert}_{f \hspace{0.05cm}= \hspace{0.05cm}\mu \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm A} }
-   \hspace{0.01cm},$$
+   \hspace{0.01cm},$
+#$w  = {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N}
-:$$w  = {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N}
+  \hspace{0.01cm}.$}}
 A comparison between DFT and IDFT shows that exactly the same algorithm can be used. The only differences of the IDFT compared to the DFT are:
-*The exponent of the rotation factor must be applied with a different sign.
+*The exponent of the rotation factor must be applied with different sign.
 *With the IDFT the division by&nbsp; $N$ is omitted.