Signal Representation/Discrete-Time Signal Representation - Revision history

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← Older revision		Revision as of 12:51, 22 June 2023
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	*At the output of an ideal transmission system,  the  likewise discrete-time signal  $y_{\rm A}(t) = x_{\rm A}(t)$  is present.		*At the output of an ideal transmission system,  the  likewise discrete-time signal  $y_{\rm A}(t) = x_{\rm A}(t)$  is present.

−	*The question now is how the block  ~~$\text{~~signal reconstruction}$  is to be designed so that  $y(t) = x(t)$  applies.	+	*The question now is how the block  »'''signal reconstruction'''«  is to be designed so that  $y(t) = x(t)$  applies.

@@ Line 82: / Line 82: @@
 <br>
 {{BlaueBox|TEXT=
-$\text{Theorem:}$&nbsp;  Developing the&nbsp; &raquo;Dirac comb&laquo;&nbsp; into a&nbsp; [[Signal_Representation/Fourier_Series|&raquo;Fourier series&raquo;]]&nbsp; and transforming it into the frequency domain using the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|&raquo;shifting theorem&laquo;]]&nbsp; gives the following Fourier correspondence:
+$\text{Theorem:}$&nbsp;  Developing the&nbsp; &raquo;Dirac comb&laquo;&nbsp; into a&nbsp; [[Signal_Representation/Fourier_Series|&raquo;Fourier series&laquo;]] &nbsp; and transforming it into the frequency domain using the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|&raquo;shifting theorem&laquo;]]&nbsp; gives the following Fourier correspondence:
 :$$p_{\delta}(t) =  \sum_{\nu = - \infty }^{+\infty} T_{\rm A} \cdot

@@ Line 223: / Line 223: @@
 ==Sampling theorem==
 <br>
-The complete reconstruction of the analog signal&nbsp; $y(t)$&nbsp; from the sampled signal&nbsp; $y_{\rm A}(t) = x_{\rm A}(t)$&nbsp; is only possible if the sampling rate&nbsp; $f_{\rm A}$&nbsp; corresponding to the bandwidth&nbsp; $B_{\rm NF}$&nbsp; of the source signal has been chosen correctly.
+The complete reconstruction of the analog signal&nbsp; $y(t)$&nbsp; from the sampled signal&nbsp; $y_{\rm A}(t) = x_{\rm A}(t)$&nbsp; is only possible if the sampling rate&nbsp; $f_{\rm A}$&nbsp;  has been chosen correctly corresponding to the bandwidth&nbsp; $B_{\rm NF}$&nbsp; of the source signal.
-From the graph in the&nbsp; [[Signal_Representation/Discrete-Time_Signal_Representation#Signal_reconstruction|"last section"]]&nbsp;, it can be seen that the following condition must be fulfilled:
+From the graph in the&nbsp; [[Signal_Representation/Discrete-Time_Signal_Representation#Signal_reconstruction|&raquo;last section&laquo;]]&nbsp;, it can be seen that the following condition must be fulfilled:
 :$$f_{\rm A} - B_{\rm   NF} > B_{\rm   NF} \hspace{0.3cm}\Rightarrow \hspace{0.3cm}f_{\rm A} > 2 \cdot  B_{\rm   NF}\hspace{0.05cm}.$$
 {{BlaueBox|TEXT=
-$\text{Sampling Theorem:}$&nbsp; If an analog signal&nbsp; $x(t)$&nbsp; has spectral components in the range&nbsp; $\vert f \vert < B_{\rm NF}$, it only can be completely reconstructed from its sampled signal&nbsp; $x_{\rm A}(t)$&nbsp; if the sampling rate is sufficiently large:
+$\text{Sampling Theorem:}$&nbsp;
 :$$f_{\rm A} ≥ 2 \cdot B_{\rm NF}.$$
+*Accordingly, the following must apply to the distance between two samples:
 :$$T_{\rm A} \le \frac{1}{ 2 \cdot B_{\rm   NF} }\hspace{0.05cm}.$$}}
-If the largest possible value &nbsp; ⇒ &nbsp; $T_{\rm A} = 1/(2B_{\rm NF})$&nbsp; is used for sampling,
+If the largest possible value &nbsp; ⇒ &nbsp; $T_{\rm A} = 1/(2B_{\rm NF})$&nbsp; is used for sampling,&nbsp; then,&nbsp; in order to reconstruct the analog signal from its sampled values,&nbsp; one must use an ideal,&nbsp; rectangular low-pass filter with cut-off frequency&nbsp;
-*then,&nbsp; in order to reconstruct the analog signal from its sampled values,
+:$$f_{\rm G} = f_{\rm A}/2 = 1/(2T_{\rm A}).$$
 {{GraueBox|TEXT=
-$\text{Example 3:}$&nbsp; The graph above shows the spectrum&nbsp; $X(f)$&nbsp;  of an analog signal limited to&nbsp; $\pm\text{ 5 kHz}$.&nbsp; Below you see the spectrum&nbsp; $X_{\rm A}(f)$&nbsp; of the sampled signal with&nbsp; $T_{\rm A} =\,\text{ 100 &micro;s}$&nbsp; ⇒ &nbsp; $f_{\rm A}=\,\text{ 10 kHz}$.
+$\text{Example 3:}$&nbsp; The upper graph shows the spectrum&nbsp; $X(f)$&nbsp;  of an analog signal limited to&nbsp; $\pm\text{ 5 kHz}$.&nbsp; Below you see the spectrum&nbsp; $X_{\rm A}(f)$&nbsp; of the sampled signal with&nbsp;
 [[File:P_ID1125__Sig_T_5_1_S6_neu.png|right|frame|Sampling theorem in the frequency domain]]
-<br>Additionally drawn is the frequency response&nbsp; $H(f)$&nbsp; of the low-pass filter for signal reconstruction, whose cut-off frequency must be &nbsp; $f_{\rm G} = f_{\rm A}/2 = 5\,\text{ kHz}$.
+:$$T_{\rm A} =\,\text{ 100 &micro;s} &nbsp; \ ⇒ \ &nbsp;  f_{\rm A}=\,\text{ 10 kHz}.$$
-*With any other&nbsp; $f_{\rm G}$ value, the result would be&nbsp; $Y(f) \neq X(f)$.
+If the sampling at the transmitter had been done with sampling rate
-*For&nbsp; $f_{\rm G} < 5\,\text{ kHz}$&nbsp; the upper&nbsp; $X(f)$ parts are missing.
+:$$f_{\rm A} < 10\,\text{ kHz}&nbsp; &nbsp;\ ⇒ \ &nbsp; T_{\rm A} >100 \,{\rm &micro;  s},$$
-''Note'': &nbsp;  There is an interactive applet on the topic covered here: &nbsp; [[Applets:Sampling_of_Analog_Signals_and_Signal_Reconstruction|"Sampling of Analog Signals and Signal Reconstruction"]]
+<u>Note:</u> &nbsp;  There is an interactive applet on the topic covered here: &nbsp; [[Applets:Sampling_of_Analog_Signals_and_Signal_Reconstruction|&raquo;Sampling of Analog Signals and Signal Reconstruction&laquo;]].}}

@@ Line 153: / Line 153: @@
 ==Frequency domain representation==
 <br>
-The spectrum of the sampled signal&nbsp; $x_{\rm A}(t)$&nbsp; is obtained by applying the&nbsp; [[Signal_Representation/The_Convolution_Theorem_and_Operation#Convolution_in_the_frequency_domain|"Convolution Theorem in the frequency domain"]].&nbsp; This states that multiplication in the time domain corresponds to convolution in the spectral domain:
+The spectrum of the sampled signal&nbsp; $x_{\rm A}(t)$&nbsp; is obtained by applying the&nbsp; [[Signal_Representation/The_Convolution_Theorem_and_Operation#Convolution_in_the_frequency_domain|&raquo;Convolution Theorem in the frequency domain&laquo;]].&nbsp; This states that multiplication in the time domain corresponds to convolution in the spectral domain:
 :$$x_{\rm A}(t) = x(t) \cdot p_{\delta}(t)\hspace{0.2cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet\, \hspace{0.2cm}
   X_{\rm A}(f) = X(f) \star P_{\delta}(f)\hspace{0.05cm}.$$
-From the spectrum&nbsp; $X(f)$&nbsp; by convolution with the Dirac delta line shifted by&nbsp; $\mu \cdot f_{\rm A}$&nbsp; we get:
+*From the spectrum&nbsp; $X(f)$&nbsp; by convolution with the Dirac delta line shifted by&nbsp; $\mu \cdot f_{\rm A}$&nbsp; we get:
 :$$X(f) \star \delta
@@ Line 165: / Line 165: @@
   )\hspace{0.05cm}.$$
-Applying this result to all Dirac delta lines of the Dirac comb, we finally obtain:
+*Applying this result to all Dirac delta lines of the Dirac comb,&nbsp; we finally obtain:
 :$$X_{\rm A}(f) = X(f) \star \sum_{\mu = - \infty }^{+\infty} \delta
@@ Line 173: / Line 173: @@
 {{BlaueBox|TEXT=
-$\text{Conclusion:}$&nbsp; The sampling of the analog time signal&nbsp; $x(t)$&nbsp; at equidistant intervals&nbsp; $T_{\rm A}$&nbsp; leads in the spectral domain to a&nbsp; $\text{periodic continuation}$&nbsp; of&nbsp; $X(f)$&nbsp; with frequency spacing&nbsp; $f_{\rm A} = 1/T_{\rm A}$. }}
+$\text{Conclusion:}$&nbsp; The sampling of the analog time signal&nbsp; $x(t)$&nbsp; at equidistant intervals&nbsp; $T_{\rm A}$&nbsp; leads in the spectral domain to a&nbsp; '''periodic continuation'''&nbsp; of&nbsp; $X(f)$&nbsp; with frequency spacing&nbsp; $f_{\rm A} = 1/T_{\rm A}$. }}
 {{GraueBox|TEXT=
 $\text{Example 2:}$&nbsp;
-The upper graph shows&nbsp; (schematically!)&nbsp; the spectrum&nbsp; $X(f)$&nbsp; of an analog signal&nbsp; $x(t)$, which includes frequencies up to&nbsp; $5 \text{ kHz}$&nbsp;.
+The upper graph shows&nbsp; $($schematically!$)$&nbsp; the spectrum&nbsp; $X(f)$&nbsp; of an analog signal&nbsp; $x(t)$,&nbsp; which includes frequencies up to&nbsp; $5 \text{ kHz}$&nbsp;.
 [[File:P_ID1122__Sig_T_5_1_S4_neu.png|right|frame|Spectrum of the sampled signal]]
 Sampling the signal at the sampling rate&nbsp; $f_{\rm A}\,\text{ = 20 kHz}$, i.e. at the respective distance&nbsp; $T_{\rm A}\, = {\rm 50 \, &micro;s}$&nbsp; we obtain the periodic spectrum&nbsp; $X_{\rm A}(f)$&nbsp; sketched below.
-*Since the Dirac delta functions are infinitely narrow, the sampled signal&nbsp; $x_{\rm A}(t)$&nbsp; also contains arbitrary high-frequency components.
+Sampling the signal at the sampling rate&nbsp; $f_{\rm A}\,\text{ = 20 kHz}$,&nbsp; i.e. at the respective distance&nbsp; $T_{\rm A}\, = {\rm 50 \, &micro;s}$&nbsp; we obtain the periodic spectrum&nbsp; $X_{\rm A}(f)$&nbsp; sketched below.
 *Accordingly, the spectral function&nbsp; $X_{\rm A}(f)$&nbsp; of the sampled signal is extended to infinity.}}
 ==Signal reconstruction==
 <br>
-Signal sampling is not an end in itself in a digital transmission system; it must be reversed at some point.&nbsp; Consider, for example, the following system:
+Signal sampling is not an end in itself in a digital transmission system;&nbsp; it must be reversed at some point.&nbsp; Consider,&nbsp; for example,&nbsp; the following system:
-[[File:EN_Sig_T_5_1_S2_v2.png|left|frame|Sampling and reconstruction of a signal]]
+[[File:P_ID1124__Sig_T_5_1_S5b_neu.png|right|frame|Frequency domain representation of the signal reconstruction process]]
 *The analog signal&nbsp; $x(t)$&nbsp; with bandwidth&nbsp; $B_{\rm NF}$&nbsp; is sampled as described above.
 *At the output of an ideal transmission system, the likewise discrete-time signal&nbsp; $y_{\rm A}(t) = x_{\rm A}(t)$&nbsp; is present.
 *The question now is how the block&nbsp; $\text{signal reconstruction}$&nbsp; is to be designed so that&nbsp; $y(t) = x(t)$&nbsp; applies.
-<br clear=all>
-The solution is simple if one considers the spectral functions:&nbsp; One obtains from&nbsp; $Y_{\rm A}(f)$&nbsp; the spectrum&nbsp; $Y(f) = X(f)$&nbsp; by a low-pass with&nbsp; [[Linear_and_Time_Invariant_Systems/System_Description_in_Frequency_Domain#Frequency_response_.E2.80.93_Transfer_function|$\text{frequency response}$]]&nbsp; $H(f)$, which&nbsp;
-[[File:P_ID1124__Sig_T_5_1_S5b_neu.png|right|frame|Frequency domain representation of the signal reconstruction process]]
+The solution is simple if one considers the spectral functions:&nbsp;
 *passes the low frequencies unaltered:
 :$$H(f) = 1 \hspace{0.3cm}{\rm{for}} \hspace{0.3cm} |f| \le B_{\rm
@@ Line 206: / Line 215: @@
    NF}\hspace{0.05cm}.$$
-Further it can be seen from the graph that the frequency response&nbsp; $H(f)$&nbsp;  can be arbitrarily shaped in the range of&nbsp; $B_{\rm NF}$&nbsp; to&nbsp; $f_{\rm A}-B_{\rm NF}$,&nbsp; as long as both of the above conditions are met,
+Further it can be seen from the graph that the frequency response&nbsp; $H(f)$&nbsp;  can be arbitrarily shaped in the range of&nbsp; $B_{\rm NF}$&nbsp; to&nbsp; $f_{\rm A}-B_{\rm NF}$,&nbsp; <br>as long as both of the above conditions are met,
-*for example, linearly sloping (dashed line)
+*e.g.,&nbsp; linearly sloping&nbsp; $($dashed line$)$
-*or also rectangular-in-frequency.
+*or rectangular-in-frequency.

@@ Line 53: / Line 53: @@
-It should be noted that other description forms are found in the literature.&nbsp; However,&nbsp; to the authors,&nbsp; the form chosen here appears to be the most appropriate in terms of spectral representation and derivation of the&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|$\text{Discrete Fourier Transform}$]]&nbsp;  $\rm (DFT)$.
+It should be noted that other description forms are found in the literature.&nbsp; However,&nbsp; to the authors,&nbsp; the form chosen here appears to be the most appropriate in terms of spectral representation and derivation of the&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|&raquo;Discrete Fourier Transform&laquo;]]&nbsp;  $\rm (DFT)$.
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; The&nbsp; $\text{Dirac comb}$&nbsp; (in the time domain)&nbsp; consists of infinitely many Dirac deltas, each equally spaced&nbsp; $T_{\rm A}$&nbsp; and all with equal impulse weight&nbsp; $T_{\rm A}$:
+$\text{Definition:}$&nbsp; The&nbsp; &raquo;'''Dirac comb'''&laquo;&nbsp; $($in the time domain$)$&nbsp; consists of infinitely many Dirac deltas,&nbsp; each equally spaced&nbsp; $T_{\rm A}$&nbsp; and all with equal impulse weight&nbsp; $T_{\rm A}$:
 :$$p_{\delta}(t) =  \sum_{\nu = - \infty }^{+\infty} T_{\rm A} \cdot
@@ Line 62: / Line 62: @@
   )\hspace{0.05cm}.$$
-Sometimes&nbsp; $p_{\delta}(t)$&nbsp; is also called&nbsp; "Dirac delta train".}}
+*Sometimes&nbsp; $p_{\delta}(t)$&nbsp; is also called&nbsp; &raquo;impulse train&laquo;&nbsp; or&nbsp; &raquo;sampling function&laquo;.
 Based on this definition, the sampled signal&nbsp; $x_{\rm A}(t)$&nbsp; has the following properties:
-*The sampled signal at the considered time&nbsp; $(\nu \cdot T_{\rm A})$&nbsp; is equal to&nbsp; $T_{\rm A} \cdot x(\nu \cdot T_{\rm A}) \cdot \delta (0)$.
+Based on this definition,&nbsp; the sampled signal&nbsp; $x_{\rm A}(t)$&nbsp; has the following properties:
-*Since the Dirac delta function&nbsp; $\delta (t)$&nbsp; is infinite at time&nbsp; $t = 0$&nbsp; all signal values&nbsp; $x_{\rm A}(\nu \cdot T_{\rm A})$&nbsp; are also infinite.
+#The sampled signal at the considered time&nbsp; $(\nu \cdot T_{\rm A})$&nbsp; is equal to&nbsp; $T_{\rm A} \cdot x(\nu \cdot T_{\rm A}) \cdot \delta (0)$.
-*Thus, the factor&nbsp; $K$&nbsp; introduced in the last section is actually infinite as well.
+#Since the Dirac delta function&nbsp; $\delta (t)$&nbsp; is infinite at time&nbsp; $t = 0$&nbsp; all signal values&nbsp; $x_{\rm A}(\nu \cdot T_{\rm A})$&nbsp; are also infinite.
-*However, two samples&nbsp; $x_{\rm A}(\nu_1 \cdot T_{\rm A})$&nbsp; and&nbsp; $x_{\rm A}(\nu_2 \cdot T_{\rm A})$&nbsp; differ in the same proportion as the signal values&nbsp; $x(\nu_1 \cdot T_{\rm A})$&nbsp; and&nbsp; $x(\nu_2 \cdot T_{\rm A})$.
+#Thus,&nbsp; the factor&nbsp; $K$&nbsp; introduced in the last section is actually infinite as well.
-*The samples of&nbsp; $x(t)$&nbsp; appear in the weights of the Dirac delta functions:
+#However,&nbsp; two samples&nbsp; $x_{\rm A}(\nu_1 \cdot T_{\rm A})$&nbsp; and&nbsp; $x_{\rm A}(\nu_2 \cdot T_{\rm A})$&nbsp; differ in the same proportion as the signal values&nbsp; $x(\nu_1 \cdot T_{\rm A})$&nbsp; and&nbsp; $x(\nu_2 \cdot T_{\rm A})$.
-:$$x_{\rm A}(t) =  \sum_{\nu = - \infty }^{+\infty} T_{\rm A} \cdot x(\nu \cdot T_{\rm A})\cdot
+::$$x_{\rm A}(t) =  \sum_{\nu = - \infty }^{+\infty} T_{\rm A} \cdot x(\nu \cdot T_{\rm A})\cdot
   \delta (t- \nu \cdot T_{\rm A}
   )\hspace{0.05cm}.$$
+The following sections will show that these equations,&nbsp; which take some getting used to,&nbsp; do lead to reasonable results,&nbsp; if they are applied consistently.
 ==Dirac comb in time and frequency domain==
 <br>
 {{BlaueBox|TEXT=
-$\text{Theorem:}$&nbsp;  Developing the&nbsp; $\text{Dirac comb}$&nbsp; into a&nbsp; [[Signal_Representation/Fourier_Series|$\text{Fourier series}$]]&nbsp; and transforming it into the frequency domain using the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|$\text{Shifting Theorem}$]]&nbsp; gives the following Fourier correspondence:
+$\text{Theorem:}$&nbsp;  Developing the&nbsp; &raquo;Dirac comb&laquo;&nbsp; into a&nbsp; [[Signal_Representation/Fourier_Series|&raquo;Fourier series&raquo;]]&nbsp; and transforming it into the frequency domain using the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|&raquo;shifting theorem&laquo;]]&nbsp; gives the following Fourier correspondence:
 :$$p_{\delta}(t) =  \sum_{\nu = - \infty }^{+\infty} T_{\rm A} \cdot
@@ Line 91: / Line 89: @@
   (f- \mu \cdot f_{\rm A} ).$$
-Here&nbsp; $f_{\rm A} = 1/T_{\rm A}$&nbsp; gives the distance between two adjacent Dirac delta lines in the frequency domain. }}
+Here,&nbsp; $f_{\rm A} = 1/T_{\rm A}$&nbsp; is the distance between two adjacent Dirac delta lines in the frequency domain. }}
@@ Line 97: / Line 95: @@
 $\text{Proof:}$&nbsp; The derivation of the spectral function&nbsp; $P_{\delta}(f)$&nbsp;  given here is done in several steps:
-'''(1)'''&nbsp;&nbsp; Since&nbsp; $p_{\delta}(t)$&nbsp; is periodic with the constant distance&nbsp; $T_{\rm A}$&nbsp; between two Dirac delta lines, the&nbsp; [[Signal_Representation/Fourier_Series#Complex_Fourier_series|$\text{complex Fourier series}$]]&nbsp; can be applied:
+'''(1)'''&nbsp;&nbsp; Since&nbsp; $p_{\delta}(t)$&nbsp; is periodic with the constant distance&nbsp; $T_{\rm A}$&nbsp; between two Dirac delta lines, the&nbsp; [[Signal_Representation/Fourier_Series#Complex_Fourier_series|&raquo;complex Fourier series&laquo;]]&nbsp; can be applied:
 :$$p_{\delta}(t) =  \sum_{\mu = - \infty }^{+\infty} D_{\mu} \cdot
@@ Line 110: / Line 108: @@
   }^{+T_{\rm A}/2}{\delta}(t) \cdot {\rm e}^{- {\rm j} \hspace{0.05cm}\cdot \hspace{0.05cm} 2 \pi \hspace{0.05cm}\cdot \hspace{0.05cm}\mu \hspace{0.05cm}
   \cdot \hspace{0.05cm}t/T_{\rm A} }\hspace{0.1cm} {\rm d}t\hspace{0.05cm}.$$
-'''(3)'''&nbsp;&nbsp; Considering that for&nbsp; $t \neq 0$&nbsp; the Dirac delta is zero and for&nbsp; $t = 0$&nbsp; the complex rotation factor is equal to&nbsp; $1$, it holds further:
+'''(3)'''&nbsp;&nbsp; Considering that for&nbsp; $t \neq 0$&nbsp; the Dirac delta is zero and for&nbsp; $t = 0$&nbsp; the complex rotation factor is equal to&nbsp; $1$,&nbsp; it holds further:
 :$$D_{\mu} = \int_{- T_{\rm A}/2
   }^{+T_{\rm A}/2}{\delta}(t) \hspace{0.1cm} {\rm d}t = 1\hspace{0.5cm}{\Rightarrow}\hspace{0.5cm}
@@ Line 116: / Line 114: @@
   \cdot 2 \hspace{0.05cm} \pi \cdot \hspace{0.05cm}\mu \hspace{0.05cm}\cdot \hspace{0.05cm}t/T_{\rm A} }\hspace{0.05cm}.
   $$
-'''(4)'''&nbsp;&nbsp; The &nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|$\text{Shifting Theorem in the frequency domain}$]]&nbsp; with&nbsp; $f_{\rm A} = 1/T_{\rm A}$:
+'''(4)'''&nbsp;&nbsp; The&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|&raquo;shifting theorem in the frequency domain&laquo;]]&nbsp; with&nbsp; $f_{\rm A} = 1/T_{\rm A}$:
 :$${\rm e}^{ {\rm j} \hspace{0.05cm}
 \hspace{0.05cm} \cdot 2 \hspace{0.05cm} \pi \hspace{0.05cm}\cdot \hspace{0.05cm}\mu \hspace{0.05cm}\cdot \hspace{0.05cm}
@@ Line 123: / Line 121: @@
   (f- \mu \cdot f_{\rm A}
   )\hspace{0.05cm}.$$
-'''(5)'''&nbsp;&nbsp; If you apply this result to each individual summand, you finally get:
+'''(5)'''&nbsp;&nbsp; If you apply this result to each individual summand,&nbsp; you finally get:
 :$$P_{\delta}(f) =  \sum_{\mu = - \infty }^{+\infty} \delta
@@ Line 132: / Line 130: @@
 The result states:
-*The Dirac comb&nbsp; $p_{\delta}(t)$&nbsp; in the time domain consists of infinitely many Dirac deltas, each at the same distance&nbsp; $T_{\rm A}$&nbsp; and all with the same impulse weight&nbsp; $T_{\rm A}$.
+#The Dirac comb&nbsp; $p_{\delta}(t)$&nbsp; in the time domain consists of infinitely many Dirac deltas,&nbsp; each at the same distance&nbsp; $T_{\rm A}$&nbsp; and all with the same impulse weight&nbsp; $T_{\rm A}$.
-*The Fourier transform of&nbsp; $p_{\delta}(t)$&nbsp; gives again a Dirac comb, but now in the frequency range &nbsp; ⇒ &nbsp; $P_{\delta}(f)$.
+#The Fourier transform of&nbsp; $p_{\delta}(t)$&nbsp; gives again a Dirac comb,&nbsp; but now in the frequency range &nbsp; ⇒ &nbsp; $P_{\delta}(f)$.
-*$P_{\delta}(f)$&nbsp; also consists of infinitely many Dirac deltas, but now in the respective distance&nbsp; $f_{\rm A} = 1/T_{\rm A}$&nbsp; and all with impulse weights&nbsp; $1$.
+#$P_{\delta}(f)$&nbsp; also consists of infinitely many Dirac deltas,&nbsp; but now in the respective distance&nbsp; $f_{\rm A} = 1/T_{\rm A}$&nbsp; and all with impulse weights&nbsp; $1$.
-*The distances of the Dirac delta lines in the time and frequency domain representation thus follow the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Reciprocity_Theorem_of_time_duration_and_bandwidth|$\text{Reciprocity Theorem}$]]: &nbsp;
+#The distances of the Dirac delta lines in the time and frequency domain representation thus follow the&nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Reciprocity_Theorem_of_time_duration_and_bandwidth|&raquo;reciprocity theorem&raquo;]]: &nbsp; $T_{\rm A} \cdot f_{\rm A} = 1 \hspace{0.05cm}.$
 {{GraueBox|TEXT=
 $\text{Example 1:}$&nbsp; The graph illustrates the above statements for
 *$T_{\rm A} = 50\,{\rm &micro;s}$,
 *$f_{\rm A} = 1/T_{\rm A} = 20\,\text{kHz}$ .
 One can also see from this sketch the different impulse weights of&nbsp; $p_{\delta}(t)$&nbsp; and&nbsp; $P_{\delta}(f)$.}}

@@ Line 37: / Line 37: @@
 *Let the continuous-time signal be&nbsp; $x(t)$.&nbsp; Let the signal sampled at equidistant intervals&nbsp; $T_{\rm A}$&nbsp; be&nbsp; $x_{\rm A}(t)$.
-*Let the control variable&nbsp; $\nu$&nbsp; of the sample be an&nbsp; [[Signal_Representation/Calculating_with_Complex_Numbers#The_set_of_real_numbers|$\text{integer}$]]:
+*Let the run variable&nbsp; $\nu$&nbsp; of the sample be an&nbsp; [[Signal_Representation/Calculating_with_Complex_Numbers#The_set_of_real_numbers|$\text{integer}$]]:
 :$$\nu \in \mathbb{Z} =  \{\hspace{0.05cm} \text{...}\hspace{0.05cm} , –3, –2, –1, \hspace{0.2cm}0, +1, +2, +3, \text{...} \hspace{0.05cm}\} .$$
-*Outside the sampling time points&nbsp; $\nu \cdot T_{\rm A}$&nbsp; always holds&nbsp; $x_{\rm A}(t) = 0$.
+*Outside the sampling time points&nbsp; $\nu \cdot T_{\rm A}$&nbsp; always holds&nbsp; $x_{\rm A}(t) = 0$.&nbsp; At the equidistant sampling times,&nbsp; the result is  with the constant&nbsp; $K$:
 :$$x_{\rm A}(\nu \cdot T_{\rm A}) = K \cdot x(\nu \cdot T_{\rm A})\hspace{0.05cm}.$$
@@ Line 50: / Line 48: @@
 <br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; Here,&nbsp; $\text{"sampling"}$&nbsp; shall be understood as the multiplication of the continuous-time signal&nbsp; $x(t)$&nbsp; by the&nbsp; Dirac comb&nbsp; $p_{\delta}(t)$:
+$\text{Definition:}$&nbsp; &raquo;'''sampling'''&laquo;&nbsp; shall be understood as the multiplication of the continuous-time signal&nbsp; $x(t)$&nbsp; by the&nbsp; &raquo;Dirac comb&laquo;&nbsp; $p_{\delta}(t)$:
 :$$x_{\rm A}(t) = x(t) \cdot p_{\delta}(t)\hspace{0.05cm}.$$}}
-It should be noted that other description forms are found in the literature. However, to the authors, the form chosen here appears to be the most appropriate in terms of spectral representation and derivation of the&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|$\text{Discrete Fourier Transform}$]]&nbsp;  $\rm (DFT)$.
+It should be noted that other description forms are found in the literature.&nbsp; However,&nbsp; to the authors,&nbsp; the form chosen here appears to be the most appropriate in terms of spectral representation and derivation of the&nbsp; [[Signal_Representation/Discrete_Fourier_Transform_(DFT)|$\text{Discrete Fourier Transform}$]]&nbsp;  $\rm (DFT)$.
 {{BlaueBox|TEXT=

@@ Line 10: / Line 10: @@
 A prerequisite for the system-theoretical investigation of digital systems or for their computer simulation is a suitable discrete-time signal description.&nbsp;
-This chapter clarifies the mathematical transition from continuous-time to discrete-time signals,&nbsp; starting from the&nbsp;  [[Signal_Representation/Fourier_Transform_and_Its_Inverse|&raquo;Fourier transform theorems&laquo;]].
+This chapter clarifies the mathematical transition from continuous-time to discrete-time signals,&nbsp; starting from the&nbsp;  [[Signal_Representation/The_Fourier_Transform_and_its_Inverse|&raquo;Fourier transform theorems&laquo;]].
 The chapter includes in detail:

@@ Line 10: / Line 10: @@
 A prerequisite for the system-theoretical investigation of digital systems or for their computer simulation is a suitable discrete-time signal description.&nbsp;
-This chapter clarifies the mathematical transition from continuous-time to discrete-time signals, starting from the&nbsp;  [[Signal_Representation/Fourier_Transform_and_Its_Inverse|$\text{Fourier transform theorems}$]].
+This chapter clarifies the mathematical transition from continuous-time to discrete-time signals,&nbsp; starting from the&nbsp;  [[Signal_Representation/Fourier_Transform_and_Its_Inverse|&raquo;Fourier transform theorems&laquo;]].
 The chapter includes in detail:
 #The&nbsp; &raquo;time and frequency domain representation&laquo;&nbsp; of discrete-time signals,
-#the&nbsp; &raquo;sampling theorem&laquo;, which must be strictly observed in time discretization,
+#the&nbsp; &raquo;sampling theorem&laquo;,&nbsp; which must be strictly observed in time discretization,
 #the&nbsp; &raquo;reconstruction of the analog signal&laquo;&nbsp; from the discrete-time representation,
-#the&nbsp; &raquo;Discrete Fourier Transform&laquo;&nbsp; $\rm (DFT)$&nbsp; and its inverse&nbsp; $\rm (IDFT)$,
+#the&nbsp; &raquo;discrete Fourier transform&laquo;&nbsp; $\rm (DFT)$&nbsp; and its inverse&nbsp; $\rm (IDFT)$,
 #the&nbsp; &raquo;possibilities for error&laquo;&nbsp; when applying DFT and IDFT,
-#the application of&nbsp; &raquo;spectral analysis&laquo;&nbsp; to the improvement of metrological procedures, and
+#the application of&nbsp; &raquo;spectral analysis&laquo;&nbsp; to the improvement of metrological procedures,&nbsp; and
 #the&nbsp; &raquo;FFT algorithm&laquo;&nbsp; particularly suitable for computer implementation.
@@ Line 24: / Line 24: @@
 ==Principle and motivation==
 <br>
-Many source signals are analog and thus simultaneously&nbsp; [[Signal_Representation/Signal_classification#Continuous-time_and_discrete-time_signals|$\text{continuous-time}$]]&nbsp; and&nbsp; [[Signal_Representation/Signal_classification#Continuous-valued_and_discrete-valued_signals|$\text{continuous}$]].&nbsp; If such an analog signal is to be transmitted by means of a digital system, the following preprocessing steps are required:
+Many source signals are analog and thus simultaneously&nbsp; [[Signal_Representation/Signal_classification#Continuous-time_and_discrete-time_signals|&raquo;continuous-time&laquo;]]&nbsp; and&nbsp; [[Signal_Representation/Signal_classification#Continuous-valued_and_discrete-valued_signals|&raquo;continuous&ndash;valued&laquo;]].&nbsp; If such an analog signal is to be transmitted by means of a digital system,&nbsp; the following preprocessing steps are required:
-*the&nbsp; $\text{sampling}$&nbsp; of the source signal&nbsp; $x(t)$, which is expediently - but not necessarily - performed at equidistant times &nbsp; &rArr; &nbsp; $\text{time discretization}$,
+*the&nbsp; &raquo;'''sampling'''&laquo;&nbsp; of the source signal&nbsp; $x(t)$,&nbsp; which is expediently &ndash; but not necessarily &ndash; performed at equidistant times &nbsp; &rArr; &nbsp; &raquo;time discretization&laquo;,
-Quantization is not discussed in detail until the chapter&nbsp; [[Modulation_Methods/Pulscodemodulation|"Pulse Code Modulation"]] &nbsp;of the book&nbsp; "Modulation Methods".
 [[File:P_ID1120__Sig_T_5_1_S1_neu.png|right|frame|On time discretization of the continuous-time signal&nbsp; $x(t)$]]
-In the following, we use the following nomenclature to describe the sampling:
+In the following,&nbsp; we use the following nomenclature to describe the sampling:
-*Let the continuous-time signal be&nbsp; $x(t)$.
+*Let the continuous-time signal be&nbsp; $x(t)$.&nbsp; Let the signal sampled at equidistant intervals&nbsp; $T_{\rm A}$&nbsp; be&nbsp; $x_{\rm A}(t)$.
-*Let the sampled  signal sampled at equidistant intervals&nbsp; $T_{\rm A}$&nbsp; be&nbsp; $x_{\rm A}(t)$.
-*Let the run variable&nbsp; $\nu$&nbsp; of the sample be an&nbsp; [[Signal_Representation/Calculating_with_Complex_Numbers#The_set_of_real_numbers|$\text{integer}$]]:
+*Let the control variable&nbsp; $\nu$&nbsp; of the sample be an&nbsp; [[Signal_Representation/Calculating_with_Complex_Numbers#The_set_of_real_numbers|$\text{integer}$]]:
 :$$\nu \in \mathbb{Z} =  \{\hspace{0.05cm} \text{...}\hspace{0.05cm} , –3, –2, –1, \hspace{0.2cm}0, +1, +2, +3, \text{...} \hspace{0.05cm}\} .$$
 *Outside the sampling time points&nbsp; $\nu \cdot T_{\rm A}$&nbsp; always holds&nbsp; $x_{\rm A}(t) = 0$.