Signal Representation/Discrete Fourier Transform (DFT) - Revision history

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Guenter at 15:48, 22 June 2023

2023-06-22T15:48:34Z

@@ Line 248: / Line 248: @@
 <br clear=all>
 With the complex-valued number sequences&nbsp;
-:$$\langle \hspace{0.1cm}d(\nu)\hspace{0.1cm}\rangle = \langle \hspace{0.1cm}d(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , d(N-1) \hspace{0.1cm}\rangle,$$
+:$$\langle \hspace{0.03cm}d(\nu)\hspace{0.03cm}\rangle = \langle \hspace{0.03cm}d(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , d(N-1) \hspace{0.03cm}\rangle,$$
-:$$\langle \hspace{0.1cm}D(\mu)\hspace{0.1cm}\rangle = \langle \hspace{0.1cm}D(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , D(N-1) \hspace{0.1cm}\rangle,$$
+:$$\langle \hspace{0.03cm}D(\mu)\hspace{0.03cm}\rangle = \langle \hspace{0.03cm}D(0), \hspace{0.05cm}\text{...} \hspace{0.05cm} , D(N-1) \hspace{0.03cm}\rangle,$$
 DFT and IDFT are symbolized similarly to the conventional Fourier transform:
-:$$\langle \hspace{0.1cm} D(\mu)\hspace{0.1cm}\rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle \hspace{0.1cm} d(\nu) \hspace{0.1cm}\rangle \hspace{0.05cm}.$$
+:$$\langle \hspace{0.03cm} D(\mu)\hspace{0.03cm}\rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle \hspace{0.03cm} d(\nu) \hspace{0.03cm}\rangle \hspace{0.05cm}.$$
 #If the function&nbsp; $x(t)$&nbsp; is already limited to the range&nbsp; $0 \le t \lt N \cdot T_{\rm A}$&nbsp; then the IDFT output directly give 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,&nbsp; one has to choose the association shown in&nbsp; $\text{Example 3}$&nbsp; between&nbsp; $x(t)$&nbsp; and the coefficients&nbsp; $d(\nu)$.

@@ Line 174: / Line 174: @@
 For reasons of simplified notation, we now make the following substitutions:
 *The&nbsp; $N$&nbsp; &raquo;time-domain coefficients&laquo;&nbsp; are with the variable&nbsp; $\nu = 0$, ... , $N - 1$:
-:$$d(\nu) =.
+:$$d(\nu) =
    {\rm P}\left\{x(t)\right\}{\big|}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm A}}\hspace{0.05cm}.$$
 *Let&nbsp; $N$&nbsp; &raquo;frequency domain coefficients&laquo;&nbsp; be associated with the variable&nbsp; $\mu = 0,$ ... , $N-1$:

@@ Line 42: / Line 42: @@
 *Without limiting generality,&nbsp; $x(t)$&nbsp; and&nbsp; $X(f)$&nbsp; are each real and Gaussian.
-*According to the chapter&nbsp; [[Signal_Representation/Time_Discrete_Signal_Representation|&raquo;Discrete-Time Signal Representation&laquo;]]&nbsp; one can describe the sampling of the time signal&nbsp; $x(t)$&nbsp; by multiplying it by a Dirac delta train &nbsp; &rArr; &nbsp; &raquo;Dirac comb&nbsp; in the time domain&laquo;
+*According to the chapter&nbsp; [[Signal_Representation/Time_Discrete_Signal_Representation|&raquo;Discrete-Time Signal Representation&laquo;]]&nbsp; one can describe the sampling of the time signal&nbsp; $x(t)$&nbsp; by multiplying it by a Dirac delta train &nbsp; &rArr; &nbsp; <br>&raquo;Dirac comb&nbsp; in the time domain&laquo;
 [[File:P_ID1132__Sig_T_5_1_S2_neu.png|right|frame| Time discretization &nbsp; &rArr; &nbsp; Periodification in the frequency domain.&nbsp; <u>Notation:</u><br> &nbsp; &nbsp; ${\rm A}\{x(t)\}$:&nbsp; Signal&nbsp; $x(t)$&nbsp; after&nbsp; &raquo;sampling&laquo;&nbsp; $($German:&nbsp; "Abtastung"$)$ &nbsp; &rArr; &nbsp; $\rm A\{\text{...}\}$ <br> &nbsp; &nbsp; ${\rm P}\{X(f)\}$:&nbsp; Spectrum&nbsp; $X(f)$&nbsp; after&nbsp; &raquo;periodification&laquo;&nbsp;  &nbsp; &rArr; &nbsp; $\rm P\{\text{...}\}$]]

@@ Line 8: / Line 8: @@
 ==Arguments for the discrete implementation of the Fourier transform==
 <br>
-The&nbsp; &raquo;'''Fourier transform'''&laquo;&nbsp; according to the previous description in chapter&nbsp; [[Signal_Representation/The_Fourier_Transform_and_its_Inverse|&raquo;Aperiodic Signals &ndash; Pulses&laquo;]]&nbsp; has an infinitely high selectivity due to the unlimited extension of the integration interval and is therefore an ideal theoretical tool of spectral analysis.
+The&nbsp; &raquo;'''Fourier transform'''&laquo;&nbsp; according to the previous description in chapter&nbsp; [[Signal_Representation/The_Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|&raquo;Aperiodic Signals &ndash; Pulses&laquo;]]&nbsp; has an infinitely high selectivity due to the unlimited extension of the integration interval and is therefore an ideal theoretical tool of spectral analysis.
 If the spectral components&nbsp; $X(f)$&nbsp; of a time function&nbsp; $x(t)$&nbsp; are to be determined numerically,&nbsp; the general transformation equations

@@ Line 8: / Line 8: @@
 ==Arguments for the discrete implementation of the Fourier transform==
 <br>
-The&nbsp; &raquo;'''Fourier transform'''&laquo;&nbsp; according to the previous description in chapter&nbsp; [[Signal_Representation/Fourier_Transform_and_Its_Inverse|&raquo;Aperiodic Signals &ndash; Pulses&laquo;]]&nbsp; has an infinitely high selectivity due to the unlimited extension of the integration interval and is therefore an ideal theoretical tool of spectral analysis.
+The&nbsp; &raquo;'''Fourier transform'''&laquo;&nbsp; according to the previous description in chapter&nbsp; [[Signal_Representation/The_Fourier_Transform_and_its_Inverse|&raquo;Aperiodic Signals &ndash; Pulses&laquo;]]&nbsp; has an infinitely high selectivity due to the unlimited extension of the integration interval and is therefore an ideal theoretical tool of spectral analysis.
 If the spectral components&nbsp; $X(f)$&nbsp; of a time function&nbsp; $x(t)$&nbsp; are to be determined numerically,&nbsp; the general transformation equations

@@ Line 199: / Line 199: @@
 ==Inverse discrete Fourier transform==
 <br>
-The Inverse Discrete Fourier Transform (IDFT) describes the&nbsp;   [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_second_Fourier_integral|$\text{second Fourier integral}$]]
+The&nbsp; &raquo;inverse discrete Fourier transform&laquo;&nbsp;  describes the&nbsp;   [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_second_Fourier_integral|&raquo;second Fourier integral&laquo;]]:
 :$$\begin{align*}x(t) & =  \int_{-\infty
@@ Line 210: / Line 210: @@
    A}}\hspace{0.01cm}.$$
-[[File:P_ID2731__Sig_T_5_1_S6_neu.png|right|frame|On defining the&nbsp; "Inverse Discrete Fourier Transform"&nbsp; $\rm (IDFT)$&nbsp; with&nbsp; $N=8$]]
+{{BlaueBox|TEXT= $\text{Definition:}$&nbsp; The term&nbsp; &raquo;Inverse Discrete Fourier Transform&raquo; &nbsp; $\rm (IDFT)$&nbsp;&nbsp; means the calculation of the signal coefficients&nbsp; $d(\nu)$&nbsp; from the spectral coefficients&nbsp; $D(\mu)$:
-{{BlaueBox|TEXT=
+[[File:P_ID2731__Sig_T_5_1_S6_neu.png|right|frame|On defining the&nbsp; &raquo;Inverse Discrete Fourier Transform&laquo;&nbsp; $\rm (IDFT)$&nbsp; with&nbsp; $N=8$]]
 :$$d(\nu) = \sum_{\mu = 0 }^{N-1}
   D(\mu) \cdot {w}^{-\nu \hspace{0.07cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}.$$
-With the indices&nbsp; $\nu = 0, \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1, $&nbsp; &nbsp; $\mu = 0, \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1$&nbsp; holds again:
+*With indices&nbsp; $\nu = 0, \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1, $&nbsp; and &nbsp; $\mu = 0, \hspace{0.05cm}\text{...} \hspace{0.05cm}, N-1$&nbsp; holds:
 :$$d(\nu) =
    {\rm P}\left\{x(t)\right\}{\big \vert}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm
@@ Line 229: / Line 225: @@
 :$$w  = {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N}
-  \hspace{0.01cm}.$$}}
+  \hspace{0.01cm}.$$
+*The only differences between IDFT and DFT are:
-A comparison between the&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)#From_the_continuous_to_the_discrete_Fourier_transform|$\text{DFT}$]]&nbsp; and IDFT shows that exactly the same algorithm can be used.&nbsp; The only differences between IDFT and DFT are:
+#The exponent of the rotation factor is to be applied with different sign.
-*The exponent of the rotation factor is to be applied with different sign.
+#In the IDFT,&nbsp; the division by&nbsp; $N$&nbsp; is omitted.}}
@@ Line 243: / Line 240: @@
 [[File:P_ID1136__Sig_T_5_1_S7_neu.png|right|frame|Time and frequency coefficients of the DFT]]
-When using DFT or IDFT, please note:
+When using DFT or IDFT,&nbsp; please note:
 #According to the above definitions, the DFT coefficients&nbsp; $d(ν)$&nbsp; and&nbsp; $D(\mu)$&nbsp; always have the unit of the time function.
-#Dividing&nbsp; $D(\mu)$&nbsp; by&nbsp; $f_{\rm A}$, one obtains the spectral value&nbsp; $X(\mu \cdot f_{\rm A})$.
+#Dividing&nbsp; $D(\mu)$&nbsp; by&nbsp; $f_{\rm A}$,&nbsp; one obtains the spectral value&nbsp; $X(\mu \cdot f_{\rm A})$.
 #The spectral coefficients&nbsp; $D(\mu)$&nbsp; must always be complex in order to be able to consider odd time functions.
 #One also uses complex time coefficients&nbsp; $d(\nu)$ &nbsp; &rArr; &nbsp; DFT and IDFT are also applicable to band-pass signals.
-#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; (filled circles in the graph).
+#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; $($filled circles in the graph$)$.
 <br clear=all>
-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; and &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 similarly to the conventional Fourier transform:
+With the complex-valued number sequences&nbsp;
 :$$\langle \hspace{0.1cm} D(\mu)\hspace{0.1cm}\rangle \hspace{0.2cm}\bullet\!\!-\!\!\!-(N)\!-\!\!\!-\!\!\hspace{0.05cm}\circ\, \hspace{0.2cm} \langle \hspace{0.1cm} d(\nu) \hspace{0.1cm}\rangle \hspace{0.05cm}.$$
-*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 give the samples of the time function:  &nbsp;
+#If the function&nbsp; $x(t)$&nbsp; is already limited to the range&nbsp; $0 \le t \lt N \cdot T_{\rm A}$&nbsp; then the IDFT output directly give the samples of the time function:   &nbsp;  $d(\nu) = x(\nu \cdot T_{\rm A}).$
-:$$d(\nu) = x(\nu \cdot T_{\rm A}).$$
+#If&nbsp; $x(t)$&nbsp; is shifted with respect to the basic interval,&nbsp; one has to choose the association shown in&nbsp; $\text{Example 3}$&nbsp; between&nbsp; $x(t)$&nbsp; and the coefficients&nbsp; $d(\nu)$.
@@ Line 262: / Line 262: @@
 [[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; $N = 8)$.
-*For&nbsp; $\nu = 0,\hspace{0.05cm}\text{...} \hspace{0.05cm} , N/2 = 4$&nbsp; $d(\nu) = x(\nu \cdot T_{\rm A})$ holds:
+*For&nbsp; $\nu = 0,\hspace{0.05cm}\text{...} \hspace{0.05cm} , N/2 = 4$&nbsp; holds &nbsp; $d(\nu) = x(\nu \cdot T_{\rm A})$ :
 :$$d(0) = x (0)\hspace{0.05cm}, \hspace{0.15cm}
@@ Line 274: / Line 275: @@
 *The coefficients&nbsp; $d(5)$,&nbsp; $d(6)$&nbsp; and&nbsp; d$(7)$&nbsp; are to be set as follows:
-:$$d(\nu) = x \big ((\nu\hspace{-0.05cm} - \hspace{-0.05cm} N ) \cdot T_{\rm   A}\big )  $$
+:$$d(5) = x (-3T_{\rm A})\hspace{0.05cm}, \hspace{0.35cm}
-:$$ \Rightarrow \hspace{0.2cm}d(5) = x (-3T_{\rm A})\hspace{0.05cm}, \hspace{0.35cm}
+}}
 ==Exercises for the chapter==

@@ Line 114: / Line 114: @@
 ==Finite signal representation==
 <br>
-[[File:P_ID1135__Sig_T_5_1_S4_neu.png|right|frame|Finite signals of the Discrete Fourier Transform (DFT)]]
+[[File:P_ID1135__Sig_T_5_1_S4_neu.png|right|frame|Finite signals of the Discrete Fourier Transform&nbsp; $\rm (DFT)$]]
-One arrives at the so-called&nbsp; "finite signal representation"&nbsp;, if both
+One arrives at the so-called&nbsp; &raquo;finite signal representation&laquo;,&nbsp; if both
-*the time function&nbsp; $x(t)$
+*the time function&nbsp; $x(t)$,&nbsp;
-*and the spectral function&nbsp; $X(f)$
-are specified exclusively by their sample values.
+are specified exclusively by their sample values.&nbsp; This graph is to be interpreted as follows:
-The graph is to be interpreted as follows:
+*In the right graph,&nbsp; the function&nbsp; $\text{P}\{ \text{A}\{ X(f)\}\}$&nbsp; is drawn in green.&nbsp; This results from the periodification&nbsp; $($with&nbsp; $f_{\rm P})$&nbsp; of the sampled spectral function&nbsp; $\{ \text{A}\{ X(f)\}\}$.
-*In the left graph, the function&nbsp; $\text{A}\{ \text{P}\{ x(t)\}\}$ is drawn in blue.&nbsp; This results from sampling the periodified time function&nbsp; $\text{P}\{ x(t)\}$&nbsp; with equidistant Dirac deltas at a distance&nbsp; $T_{\rm A} = 1/f_{\rm P}$.
-*In the right graph, the function $\text{P}\{ \text{A}\{ X(f)\}\}$ is drawn in green.&nbsp; This results from the periodification&nbsp; $($with&nbsp; $f_{\rm P})$&nbsp; of the sampled spectral function&nbsp; $\{ \text{A}\{ X(f)\}\}$.
+*There is a Fourier correspondence between the blue finite signal&nbsp; $($in the left sketch$)$&nbsp; and the green finite signal&nbsp; $($in the right sketch$)$,&nbsp; as follows:
 :$${\rm A}\{{\rm P}\{x(t)\}\} \hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm} {\rm P}\{{\rm A}\{X(f)\}\} \hspace{0.05cm}.$$
-*The Dirac delta lines of the periodic continuation&nbsp; $\text{P}\{ \text{A}\{ X(f)\}\}$&nbsp; of the sampled spectral function, however, only fall into the same frequency grid as those of&nbsp; $\text{A}\{ X(f)\}$ if the frequency period&nbsp; $f_{\rm P}$&nbsp; is an integer multiple&nbsp; $(N)$&nbsp; of the frequency sampling interval&nbsp; $f_{\rm A}$&nbsp;.
+*The Dirac delta lines of the periodic continuation&nbsp; $\text{P}\{ \text{A}\{ X(f)\}\}$&nbsp; of the sampled spectral function,&nbsp; however,&nbsp; only fall into the same frequency grid as those of&nbsp; $\text{A}\{ X(f)\}$&nbsp; if the frequency period&nbsp; $f_{\rm P}$&nbsp; is an integer multiple&nbsp; $(N)$&nbsp; of the frequency sampling interval&nbsp; $f_{\rm A}$.
-*Therefore, when using the finite signal representation, the following condition must always be fulfilled, where in practice the natural number&nbsp; $N$&nbsp; is usually a power of two&nbsp; (the above graph is based on the value&nbsp; $N = 8$&nbsp;):
 :$$f_{\rm P} = N \cdot f_{\rm A} \hspace{0.5cm} \Rightarrow\hspace{0.5cm} {1}/{T_{\rm A}}= N \cdot f_{\rm A} \hspace{0.5cm} \Rightarrow\hspace{0.5cm}
@@ Line 141: / Line 143: @@
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp;
+$\text{Conclusions:}$&nbsp;
-*The time function&nbsp; $\text{P}\{ \text{A}\{ x(t)\}\}$&nbsp; has the period&nbsp; $T_{\rm P} = N \cdot T_{\rm A}$.
+#The time function&nbsp; $\text{P}\{ \text{A}\{ x(t)\}\}$&nbsp; has the period&nbsp; $T_{\rm P} = N \cdot T_{\rm A}$.
-*The period in the frequency domain is&nbsp; $f_{\rm P} = N \cdot f_{\rm A}$.
+#The period in the frequency domain is&nbsp; $f_{\rm P} = N \cdot f_{\rm A}$.
-*For the description of the discretised time and frequency response in each case&nbsp; $N$&nbsp; $\text{complex numerical values}$&nbsp; in the form of impulse weights are thus sufficient.}}
+#For the description of the discretized time and frequency response in each case&nbsp; $N$&nbsp; '''complex numerical values'''&nbsp; in the form of impulse weights are sufficient.}}
 {{GraueBox|TEXT=
 $\text{Example 2:}$&nbsp;
-A time-limited (pulse-like) signal&nbsp; $x(t)$&nbsp; is present in sampled form, where the distance between two samples is&nbsp; $T_{\rm A} = 1\, {\rm &micro; s}$:
+A time-limited&nbsp; $($pulse-like$)$&nbsp; signal&nbsp; $x(t)$&nbsp; is present in sampled form,&nbsp; where the distance between two samples is&nbsp; $T_{\rm A} = 1\, {\rm &micro; s}$:
 *After a discrete Fourier transform with&nbsp; $N = 512$&nbsp; the spectrum&nbsp; $X(f)$&nbsp; is available in form of frequency-samples at spacing&nbsp; $f_{\rm A} = (N \cdot T_{\rm A})^{-1} \approx 1.953\,\text{kHz} $.
 *Increasing the DFT parameter to&nbsp; $N= 2048$ results in a (four times) finer frequency grid with&nbsp; $f_{\rm A} \approx 488\,\text{Hz}$.}}
 ==From the continuous to the discrete Fourier transform==
 <br>
-From the conventional&nbsp;[[Signal_Representation/Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|$\text{first Fourier integral}$]]
+From the conventional&nbsp; [[Signal_Representation/Fourier_Transform_and_its_Inverse#The_first_Fourier_integral|&raquo;first Fourier integral&laquo;]]
 :$$X(f) =\int_{-\infty
   }^{+\infty}x(t) \cdot {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm} f  \hspace{0.05cm}\cdot \hspace{0.05cm}t}\hspace{0.1cm} {\rm d}t$$
-arises from discretization&nbsp; $(\text{d}t \to T_{\rm A}$,&nbsp; $t \to \nu \cdot T_{\rm A}$,&nbsp; $f \to \mu \cdot f_{\rm A}$,&nbsp; $T_{\rm A} \cdot f_{\rm A} = 1/N)$&nbsp; the sampled and periodized spectral function
+arises from discretization&nbsp; $(\text{d}t \to T_{\rm A}$,&nbsp; $t \to \nu \cdot T_{\rm A}$,&nbsp; $f \to \mu \cdot f_{\rm A}$,&nbsp; $T_{\rm A} \cdot f_{\rm A} = 1/N)$&nbsp; the sampled and periodified spectral function
 :$${\rm P}\{X(\mu \cdot f_{\rm A})\} = T_{\rm A} \cdot \sum_{\nu = 0 }^{N-1}
@@ Line 167: / Line 170: @@
   \cdot \hspace{0.05cm}\mu /N} \hspace{0.05cm}.$$
-It is taken into account that due to the discretization, the periodized functions are to be used in each case.
+It is taken into account that due to the discretization,&nbsp; the periodified functions are to be used in each case.
 For reasons of simplified notation, we now make the following substitutions:
-*The&nbsp; $N$&nbsp; $\text{time-domain coefficients}$&nbsp; are with the variable&nbsp; $\nu = 0$, ... , $N - 1$:
+*The&nbsp; $N$&nbsp; &raquo;time-domain coefficients&laquo;&nbsp; are with the variable&nbsp; $\nu = 0$, ... , $N - 1$:
 :$$d(\nu) =.
    {\rm P}\left\{x(t)\right\}{\big|}_{t \hspace{0.05cm}= \hspace{0.05cm}\nu \hspace{0.05cm}\cdot \hspace{0.05cm}T_{\rm A}}\hspace{0.05cm}.$$
-*Let&nbsp; $N$&nbsp; $\text{frequency domain coefficients}$&nbsp; be associated with the variable&nbsp; $\mu = 0,$ ... , $N$ – 1:
+*Let&nbsp; $N$&nbsp; &raquo;frequency domain coefficients&laquo;&nbsp; be associated with the variable&nbsp; $\mu = 0,$ ... , $N-1$:
 :$$D(\mu) = f_{\rm A} \cdot
    {\rm P}\left\{X(f)\right\}{\big|}_{f \hspace{0.05cm}= \hspace{0.05cm}\mu \hspace{0.05cm}\cdot \hspace{0.05cm}f_{\rm A}}\hspace{0.05cm}.$$
-*Abbreviation is written for the from&nbsp; $N$&nbsp; dependent&nbsp; $\text{complex rotation factor}$&nbsp;:
+*Abbreviation is written for the from&nbsp; $N$&nbsp; dependent&nbsp; &raquo;complex rotation factor&laquo;&nbsp;:
 :$$w = {\rm e}^{-{\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi /N}
   = \cos \left( {2 \pi}/{N}\right)-{\rm j} \cdot \sin \left( {2 \pi}/{N}\right)
   \hspace{0.05cm}.$$
 {{BlaueBox|TEXT=
 $\text{Definition:}$&nbsp;
-The term&nbsp; $\text{Discrete Fourier Transform   (DFT)}$&nbsp; means the calculation of the&nbsp; $N$&nbsp; spectral coefficients&nbsp; $D(\mu)$&nbsp; from the&nbsp; $N$&nbsp; signal coefficients&nbsp; $d(\nu)$:
+The term&nbsp; &raquo;Discrete Fourier Transform&laquo;&nbsp; $\rm   (DFT)$&nbsp; means the calculation of the&nbsp; $N$&nbsp; spectral coefficients&nbsp; $D(\mu)$&nbsp; from the&nbsp; $N$&nbsp; signal coefficients&nbsp; $d(\nu)$:
+[[File:P_ID2730__Sig_T_5_1_S5_neu.png|right|frame|On defining the&nbsp; &raquo;Discrete Fourier Transform&laquo;&nbsp; $\rm (DFT)$&nbsp; with&nbsp; $N=8$]]
 :$$D(\mu) = \frac{1}{N} \cdot \sum_{\nu = 0 }^{N-1}
    d(\nu)\cdot  {w}^{\hspace{0.05cm}\nu \hspace{0.07cm} \cdot \hspace{0.05cm}\mu} \hspace{0.05cm}. $$
 *the&nbsp; $N = 8$&nbsp; spectral coefficients&nbsp; $D(\mu)$&nbsp; at the green filling.}}

@@ Line 80: / Line 80: @@
 ==Frequency discretization &ndash; Periodification in the time domain ==
 <br>
-The discretization of&nbsp; $X(f)$&nbsp; can also be described by a multiplication with a Dirac comb in the frequency domain.&nbsp; The result is the sampled spectrum in the distance $f_{\rm A}$:
+The discretization of&nbsp; $X(f)$&nbsp; can also be described by a multiplication with a Dirac comb in the frequency domain.&nbsp; The result is the sampled spectrum with distance&nbsp; $f_{\rm A}$:
 :$${\rm A}\{X(f)\} =  X(f) \cdot   \sum_{\mu = - \infty }^{+\infty}
   f_{\rm A} \cdot \delta (f- \mu \cdot f_{\rm A } ) =  \sum_{\mu = - \infty }^{+\infty}
   f_{\rm A} \cdot X(\mu \cdot f_{\rm A } ) \cdot\delta (f- \mu \cdot f_{\rm A } )\hspace{0.05cm}.$$
-If one transforms the Dirac comb $($with impulse weights&nbsp; $f_{\rm A})$&nbsp; used here into the time domain, one obtains with&nbsp; $T_{\rm P} = 1/f_{\rm A}$:
+*If one transforms the Dirac comb $($with impulse weights&nbsp; $f_{\rm A})$&nbsp; into the time domain,&nbsp; one obtains with&nbsp; $T_{\rm P} = 1/f_{\rm A}$:
 :$$\sum_{\mu = - \infty }^{+\infty}
@@ Line 92: / Line 92: @@
    \delta (t- \nu \cdot T_{\rm P } ) \hspace{0.05cm}.$$
-The multiplication with&nbsp; $X(f)$&nbsp; corresponds in the time domain to the convolution with&nbsp; $x(t)$.&nbsp; One obtains the signal&nbsp; $\text{P}\{ x(t)\}$&nbsp; periodified in the distance&nbsp; $T_{\rm P}$:
+*The multiplication with&nbsp; $X(f)$&nbsp; corresponds in the time domain to the convolution with&nbsp; $x(t)$.&nbsp; One obtains the signal&nbsp; $\text{P}\{ x(t)\}$&nbsp; periodified with distance&nbsp; $T_{\rm P}$:
 :$${\rm A}\{X(f)\} \hspace{0.2cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ\, \hspace{0.2cm}
@@ Line 99: / Line 99: @@
    x (t- \nu \cdot T_{\rm P } ) \hspace{0.05cm}.$$
 [[File:P_ID1134__Sig_T_5_1_S3_neu.png|right|frame|Frequency discretization &ndash; Periodization in the time domain]]
 $\text{Example 1:}$&nbsp;
 This correlation is illustrated in the graph:
-*Due to the coarse frequency rastering, this example results in a relatively small value for the time period&nbsp; $T_{\rm P}$&nbsp;.
+*Due to the coarse frequency rastering,&nbsp; this example results in a relatively small value for the time period&nbsp; $T_{\rm P}$&nbsp;.
-* Therefore due to overlaps, the (blue) periodified time signal&nbsp; $\text{P}\{ x(t)\}$&nbsp; differs significantly from&nbsp; $x(t)$.
+* Therefore due to overlaps,&nbsp; the&nbsp; $($blue$)$&nbsp; periodified time signal&nbsp; $\text{P}\{ x(t)\}$&nbsp; differs significantly from&nbsp; $x(t)$.