Digital Signal Transmission/Basics of Coded Transmission - Revision history

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@@ Line 79: / Line 79: @@
-== Source coding &ndash; Channel coding &ndash;  Transmission coding ==
+== Source coding &ndash; Channel coding &ndash;  Line coding ==
 <br>
 "Coding"&nbsp; is the conversion of the source symbol sequence &nbsp;$\langle q_\nu \rangle$&nbsp; with symbol set size &nbsp;$M_q$&nbsp; into an encoder symbol sequence &nbsp;$\langle c_\nu \rangle$&nbsp; with symbol set size &nbsp;$M_c$.&nbsp; Usually,&nbsp; coding manipulates the redundancy contained in a digital signal.&nbsp; Often &ndash; but not always &ndash; &nbsp;$M_q$&nbsp; and &nbsp;$M_c$&nbsp; are different.<br>
@@ Line 88: / Line 88: @@
 *On the other hand,&nbsp; with&nbsp; '''channel coding'''&nbsp; a noticeable improvement in the transmission behavior is achieved by using a redundancy specifically added at the transmitter to detect and correct transmission errors at the receiver end.&nbsp; Such codes,&nbsp; the most important of which are block codes,&nbsp; convolutional codes and turbo codes,&nbsp; are particularly important in the case of heavily disturbed channels.&nbsp; The greater the relative redundancy of the encoded signal,&nbsp; the better the correction properties of the code,&nbsp; albeit at a reduced user data rate.
-*'''Transmission coding''' &ndash; often referred to as&nbsp; "line coding"&nbsp; &ndash; is used to adapt the transmitted signal to the spectral characteristics of the transmission channel and the receiving equipment by recoding the source symbols.&nbsp; For example,&nbsp; in the case of a channel with the frequency response characteristic &nbsp;$H_{\rm K}(f=0) = 0$,&nbsp; over which consequently no DC signal can be transmitted,&nbsp; transmission coding must ensure that the encoder symbol sequence contains neither a long &nbsp;$\rm L$ sequence nor a long &nbsp;$\rm H$ sequence.<br>
+*'''Line coding'''&nbsp; is used to adapt the transmitted signal to the spectral characteristics of the transmission channel and the receiving equipment by recoding the source symbols.&nbsp; For example,&nbsp; in the case of a channel with the frequency response characteristic &nbsp;$H_{\rm K}(f=0) = 0$,&nbsp; over which consequently no DC signal can be transmitted,&nbsp; transmission coding must ensure that the encoder symbol sequence contains neither a long &nbsp;$\rm L$ sequence nor a long &nbsp;$\rm H$ sequence.<br>

@@ Line 94: / Line 94: @@
 *[[Channel_Coding|"Channel Coding"]]&nbsp; has its own book dedicated to it in our learning tutorial.
 *Source coding is covered in detail in the book&nbsp; [[Information_Theory|"Information Theory"]]&nbsp; (main chapter 2).
-*[[Examples_of_Communication_Systems/Sprachcodierung|"Speech coding"]]&nbsp; &ndash; described in the book "Examples of Communication Systems" &ndash;&nbsp;  is a special form of source coding.<br>
+*[[Examples_of_Communication_Systems/Speech_Coding|"Speech coding"]]&nbsp; &ndash; described in the book "Examples of Communication Systems" &ndash;&nbsp;  is a special form of source coding.<br>

@@ Line 94: / Line 94: @@
 *[[Channel_Coding|"Channel Coding"]]&nbsp; has its own book dedicated to it in our learning tutorial.
 *Source coding is covered in detail in the book&nbsp; [[Information_Theory|"Information Theory"]]&nbsp; (main chapter 2).
-*[[Examples_of_Communication_Systems/Sprachcodierung|"Voice coding"]]&nbsp; &ndash; described in the book "Examples of Communication Systems" &ndash;&nbsp;  is a special form of source coding.<br>
+*[[Examples_of_Communication_Systems/Sprachcodierung|"Speech coding"]]&nbsp; &ndash; described in the book "Examples of Communication Systems" &ndash;&nbsp;  is a special form of source coding.<br>

@@ Line 86: / Line 86: @@
 *The task of&nbsp; '''source coding'''&nbsp; is redundancy reduction for data compression,&nbsp; as applied for example in image coding.&nbsp; By exploiting statistical ties between the individual points of an image or between the brightness values of a point at different times&nbsp; (in the case of moving image sequences),&nbsp; methods can be developed that lead to a noticeable reduction in the amount of data&nbsp; (measured in&nbsp; "bit"&nbsp; or "byte"),&nbsp; while maintaining virtually the same&nbsp; (subjective)&nbsp; image quality.&nbsp; A simple example of this is "differential pulse code modulation"&nbsp; $\rm (DPCM)$.<br>
-*On the other hand,&nbsp; with&nbsp; '''channel coding'''&nbsp; a noticeable improvement in the transmission behavior is achieved by using a redundancy specifically added at the transmitter to detect and correct transmission errors at the receiver end.&nbsp; Such codes,&nbsp; the most important of which are block codes,&nbsp; convolutional codes and turbo codes,&nbsp; are particularly important in the case of heavily disturbed channels.&nbsp; The greater the relative redundancy of the coded signal,&nbsp; the better the correction properties of the code,&nbsp; albeit at a reduced user data rate.
+*On the other hand,&nbsp; with&nbsp; '''channel coding'''&nbsp; a noticeable improvement in the transmission behavior is achieved by using a redundancy specifically added at the transmitter to detect and correct transmission errors at the receiver end.&nbsp; Such codes,&nbsp; the most important of which are block codes,&nbsp; convolutional codes and turbo codes,&nbsp; are particularly important in the case of heavily disturbed channels.&nbsp; The greater the relative redundancy of the encoded signal,&nbsp; the better the correction properties of the code,&nbsp; albeit at a reduced user data rate.
 *'''Transmission coding''' &ndash; often referred to as&nbsp; "line coding"&nbsp; &ndash; is used to adapt the transmitted signal to the spectral characteristics of the transmission channel and the receiving equipment by recoding the source symbols.&nbsp; For example,&nbsp; in the case of a channel with the frequency response characteristic &nbsp;$H_{\rm K}(f=0) = 0$,&nbsp; over which consequently no DC signal can be transmitted,&nbsp; transmission coding must ensure that the encoder symbol sequence contains neither a long &nbsp;$\rm L$ sequence nor a long &nbsp;$\rm H$ sequence.<br>

@@ Line 8: / Line 8: @@
 == # OVERVIEW OF THE SECOND MAIN CHAPTER # ==
 <br>
-The second main chapter deals with so-called&nbsp; '''transmission coding''',&nbsp; which is sometimes also referred to as&nbsp; "line coding"&nbsp; in literature.&nbsp; In this process,&nbsp; an adaptation of the digital transmission signal to the characteristics of the transmission channel is achieved through the targeted addition of redundancy.&nbsp; In detail,&nbsp; the following are dealt with:
+The second main chapter deals with so-called&nbsp; '''transmission coding''',&nbsp; which is sometimes also referred to as&nbsp; "line coding"&nbsp; in literature.&nbsp; In this process,&nbsp; an adaptation of the digital transmitted signal to the characteristics of the transmission channel is achieved through the targeted addition of redundancy.&nbsp; In detail,&nbsp; the following are dealt with:
 #&nbsp; Some basic concepts of information theory such as&nbsp; &raquo;information content&laquo;&nbsp; and&nbsp; &raquo;entropy&laquo;,

@@ Line 10: / Line 10: @@
 The second main chapter deals with so-called&nbsp; '''transmission coding''',&nbsp; which is sometimes also referred to as&nbsp; "line coding"&nbsp; in literature.&nbsp; In this process,&nbsp; an adaptation of the digital transmission signal to the characteristics of the transmission channel is achieved through the targeted addition of redundancy.&nbsp; In detail,&nbsp; the following are dealt with:
-*some basic concepts of information theory such as&nbsp; &raquo;information content&laquo;&nbsp; and&nbsp; &raquo;entropy&laquo;,
+#&nbsp; Some basic concepts of information theory such as&nbsp; &raquo;information content&laquo;&nbsp; and&nbsp; &raquo;entropy&laquo;,
-*the&nbsp; &raquo;auto-correlation function&laquo;&nbsp; and the&nbsp; &raquo;power-spectral densities&laquo;&nbsp; of digital signals,
+#&nbsp; the&nbsp; &raquo;auto-correlation function&laquo;&nbsp; and the&nbsp; &raquo;power-spectral densities&laquo;&nbsp; of digital signals,
-*the&nbsp; &raquo;redundancy-free coding&laquo;&nbsp; which leads to a non-binary transmitted signal,
+#&nbsp; the&nbsp; &raquo;redundancy-free coding&laquo;&nbsp; which leads to a non-binary transmitted signal,
-*the calculation of&nbsp; &raquo;symbol and bit error probability&laquo;&nbsp; for&nbsp; &raquo;multilevel systems&laquo;&nbsp;,
+#&nbsp; the calculation of&nbsp; &raquo;symbol and bit error probability&laquo;&nbsp; for&nbsp; &raquo;multilevel systems&laquo;&nbsp;,
-*the so-called&nbsp; &raquo;4B3T codes&laquo;&nbsp; as an important example of&nbsp; &raquo;block-wise coding&laquo;,&nbsp; and
+#&nbsp; the so-called&nbsp; &raquo;4B3T codes&laquo;&nbsp; as an important example of&nbsp; &raquo;block-wise coding&laquo;,&nbsp; and
-*the&nbsp; &raquo;pseudo-ternary codes&laquo;,&nbsp; each of which realizes symbol-wise coding.
+#&nbsp; the&nbsp; &raquo;pseudo-ternary codes&laquo;,&nbsp; each of which realizes symbol-wise coding.

@@ Line 100: / Line 100: @@
 <br>
 In the following we always assume the block diagram sketched on the right and the following agreements:
-[[File:EN_Dig_T_2_1_S3_v2.png|right|frame|Block diagram for the description of multilevel and coded transmission systems '''Korrektur''' bzw.|class=fit]]
+[[File:EN_Dig_T_2_1_S3_v23.png|right|frame|Block diagram for the description of multilevel and coded transmission systems|class=fit]]
 *Let the digital source signal &nbsp;$q(t)$&nbsp; be binary &nbsp;$(M_q = 2)$&nbsp; and redundancy-free &nbsp;$(H_q = 1 \ \rm bit/symbol)$.

@@ Line 195: / Line 195: @@
 \cdot \varphi_a(\lambda)\cdot\varphi^{^{\bullet} }_{gs}(\tau -
 \lambda \cdot T)\hspace{0.05cm}.$$
-:&rArr; &nbsp; $s(t)$&nbsp; can be binary or multi-level '''KORREKTUR: multilevel''',&nbsp; unipolar or bipolar,&nbsp; redundancy-free or redundant (line-coded). The pulse shape is taken into account by the energy ACF.}}
+:&rArr; &nbsp; $s(t)$&nbsp; can be binary or multilevel,&nbsp; unipolar or bipolar,&nbsp; redundancy-free or redundant (line-coded). The pulse shape is taken into account by the energy ACF.}}

@@ Line 81: / Line 81: @@
 == Source coding &ndash; Channel coding &ndash;  Transmission coding ==
 <br>
-"Coding"&nbsp; is the conversion of the source symbol sequence &nbsp;$\langle q_\nu \rangle$&nbsp; with symbol set size &nbsp;$M_q$&nbsp; into a code symbol sequence '''KORREKTUR: encoder symbol sequence'''&nbsp;$\langle c_\nu \rangle$&nbsp; with symbol set size &nbsp;$M_c$.&nbsp; Usually,&nbsp; coding manipulates the redundancy contained in a digital signal.&nbsp; Often &ndash; but not always &ndash; &nbsp;$M_q$&nbsp; and &nbsp;$M_c$&nbsp; are different.<br>
+"Coding"&nbsp; is the conversion of the source symbol sequence &nbsp;$\langle q_\nu \rangle$&nbsp; with symbol set size &nbsp;$M_q$&nbsp; into an encoder symbol sequence &nbsp;$\langle c_\nu \rangle$&nbsp; with symbol set size &nbsp;$M_c$.&nbsp; Usually,&nbsp; coding manipulates the redundancy contained in a digital signal.&nbsp; Often &ndash; but not always &ndash; &nbsp;$M_q$&nbsp; and &nbsp;$M_c$&nbsp; are different.<br>
 A distinction is made between different types of coding depending on the target direction:
@@ Line 88: / Line 88: @@
 *On the other hand,&nbsp; with&nbsp; '''channel coding'''&nbsp; a noticeable improvement in the transmission behavior is achieved by using a redundancy specifically added at the transmitter to detect and correct transmission errors at the receiver end.&nbsp; Such codes,&nbsp; the most important of which are block codes,&nbsp; convolutional codes and turbo codes,&nbsp; are particularly important in the case of heavily disturbed channels.&nbsp; The greater the relative redundancy of the coded signal,&nbsp; the better the correction properties of the code,&nbsp; albeit at a reduced user data rate.
-*'''Transmission coding''' &ndash; often referred to as&nbsp; "line coding"&nbsp; &ndash; is used to adapt the transmitted signal to the spectral characteristics of the transmission channel and the receiving equipment by recoding the source symbols.&nbsp; For example,&nbsp; in the case of a channel with the frequency response characteristic &nbsp;$H_{\rm K}(f=0) = 0$,&nbsp; over which consequently no DC signal can be transmitted,&nbsp; transmission coding must ensure that the code symbol sequence '''KORREKTUR: encoder symbol sequence'''contains neither a long &nbsp;$\rm L$ sequence nor a long &nbsp;$\rm H$ sequence.<br>
+*'''Transmission coding''' &ndash; often referred to as&nbsp; "line coding"&nbsp; &ndash; is used to adapt the transmitted signal to the spectral characteristics of the transmission channel and the receiving equipment by recoding the source symbols.&nbsp; For example,&nbsp; in the case of a channel with the frequency response characteristic &nbsp;$H_{\rm K}(f=0) = 0$,&nbsp; over which consequently no DC signal can be transmitted,&nbsp; transmission coding must ensure that the encoder symbol sequence contains neither a long &nbsp;$\rm L$ sequence nor a long &nbsp;$\rm H$ sequence.<br>
 In the current book&nbsp; "Digital Signal Transmission"&nbsp; we deal exclusively with this last,&nbsp; transmission-technical aspect.
-*&nbsp;[[Channel_Coding|"Channel Coding"]]&nbsp; has its own book dedicated to it in our learning tutorial.
+*[[Channel_Coding|"Channel Coding"]]&nbsp; has its own book dedicated to it in our learning tutorial.
 *Source coding is covered in detail in the book&nbsp; [[Information_Theory|"Information Theory"]]&nbsp; (main chapter 2).
 *[[Examples_of_Communication_Systems/Sprachcodierung|"Voice coding"]]&nbsp; &ndash; described in the book "Examples of Communication Systems" &ndash;&nbsp;  is a special form of source coding.<br>
@@ Line 109: / Line 109: @@
 *For the comparison of transmission systems with different coding, &nbsp;$T_q$&nbsp; and &nbsp;$R_q$&nbsp; are always assumed to be constant.&nbsp; Note:&nbsp; In later chapters we use &nbsp;$T_{\rm B}=T_q$&nbsp; and &nbsp;$R_{\rm B}=R_q$ for this purpose.
-*The encoder signal '''KORREKTUR: encoded signal'''&nbsp;$c(t)$&nbsp; and also the transmitted signal &nbsp;$s(t)$&nbsp; after pulse shaping with &nbsp;$g_s(t)$&nbsp; have the step number '''KORREKTUR: level number'''&nbsp;$M_c$,&nbsp; the symbol duration &nbsp;$T_c$&nbsp; and the symbol rate &nbsp;$1/T_c$.&nbsp; The equivalent bit rate is
+*The encoded signal &nbsp;$c(t)$&nbsp; and also the transmitted signal &nbsp;$s(t)$&nbsp; after pulse shaping with &nbsp;$g_s(t)$&nbsp; have the level number &nbsp;$M_c$,&nbsp; the symbol duration &nbsp;$T_c$&nbsp; and the symbol rate &nbsp;$1/T_c$.&nbsp; The equivalent bit rate is
 :$$R_c = {{\rm log_2} (M_c)}/{T_c} \ge R_q \hspace{0.05cm}.$$
 *The equal sign is only valid for the &nbsp;[[Digital_Signal_Transmission/Redundancy-Free_Coding#Blockwise_coding_vs._symbolwise_coding|"redundancy-free codes"]]&nbsp; $(r_c = 0)$.&nbsp; <br>Otherwise, we obtain for the relative code redundancy:
@@ Line 116: / Line 116: @@
 Notes on nomenclature:
-#In the context of transmission codes, &nbsp;$R_c$&nbsp; always indicates in our tutorial the equivalent bit rate of the encoder signal '''KORREKTUR: encoded signal'''with unit&nbsp; "bit/s".&nbsp;
+#In the context of transmission codes, &nbsp;$R_c$&nbsp; always indicates in our tutorial the equivalent bit rate of the encoded signal with unit&nbsp; "bit/s".&nbsp;
 #In the literature on channel coding,&nbsp; $R_c$&nbsp; is often used to denote the dimensionless code rate &nbsp;$1 - r_c$&nbsp;.
 #$R_c = 1 $&nbsp; then indicates a redundancy-free code,&nbsp; while &nbsp;$R_c = 1/3 $&nbsp; indicates a code with the relative redundancy &nbsp;$r_c = 2/3 $.&nbsp;

@@ Line 94: / Line 94: @@
 *&nbsp;[[Channel_Coding|"Channel Coding"]]&nbsp; has its own book dedicated to it in our learning tutorial.
 *Source coding is covered in detail in the book&nbsp; [[Information_Theory|"Information Theory"]]&nbsp; (main chapter 2).
-*[[Examples_of_Communication_Systems/Sprachcodierung|Voice coding]]&nbsp; &ndash; described in the book "Examples of Communication Systems" &ndash;&nbsp;  is a special form of source coding.<br>
+*[[Examples_of_Communication_Systems/Sprachcodierung|"Voice coding"]]&nbsp; &ndash; described in the book "Examples of Communication Systems" &ndash;&nbsp;  is a special form of source coding.<br>
@@ Line 111: / Line 111: @@
 *The encoder signal '''KORREKTUR: encoded signal'''&nbsp;$c(t)$&nbsp; and also the transmitted signal &nbsp;$s(t)$&nbsp; after pulse shaping with &nbsp;$g_s(t)$&nbsp; have the step number '''KORREKTUR: level number'''&nbsp;$M_c$,&nbsp; the symbol duration &nbsp;$T_c$&nbsp; and the symbol rate &nbsp;$1/T_c$.&nbsp; The equivalent bit rate is
 :$$R_c = {{\rm log_2} (M_c)}/{T_c} \ge R_q \hspace{0.05cm}.$$
-*The equal sign is only valid for the &nbsp;[[Digital_Signal_Transmission/Redundancy-Free_Coding#Blockwise_coding_vs._symbolwise_coding|redundancy-free codes]]&nbsp; $(r_c = 0)$.&nbsp; <br>Otherwise, we obtain for the relative code redundancy:
+*The equal sign is only valid for the &nbsp;[[Digital_Signal_Transmission/Redundancy-Free_Coding#Blockwise_coding_vs._symbolwise_coding|"redundancy-free codes"]]&nbsp; $(r_c = 0)$.&nbsp; <br>Otherwise, we obtain for the relative code redundancy:
 :$$r_c =({R_c - R_q})/{R_c} = 1 - R_q/{R_c} \hspace{0.05cm}.$$
@@ Line 133: / Line 133: @@
   \hspace{0.3cm}r_c =3/4\cdot {\rm log_2} (3) \hspace{-0.05cm}- \hspace{-0.05cm}1 \approx 15.9\, \%
   \hspace{0.05cm}.$$
-Detailed information about the 4B3T codes can be found in the &nbsp;[[Digital_Signal_Transmission/Blockweise_Codierung_mit_4B3T-Codes|chapter of the same name ]].}}<br>
+Detailed information about the 4B3T codes can be found in the &nbsp;[[Digital_Signal_Transmission/Blockweise_Codierung_mit_4B3T-Codes|"chapter of the same name"]].}}<br>
@@ Line 145: / Line 145: @@
 It can be seen from this figure that a digital signal is generally non-stationary:
-*For the transmitted signal &nbsp;$s_{\rm G}(t)$&nbsp; with narrow Gaussian pulses,&nbsp; the &nbsp;[[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Stationary_random_processes|non-stationarity]]&nbsp; is obvious,&nbsp; since,&nbsp; for example,&nbsp; at multiples of &nbsp;$T$&nbsp; the variance is &nbsp;$\sigma_s^2 = s_0^2$,&nbsp; while exactly in between &nbsp; $\sigma_s^2 \approx 0$&nbsp; holds.<br>
+*For the transmitted signal &nbsp;$s_{\rm G}(t)$&nbsp; with narrow Gaussian pulses,&nbsp; the &nbsp;[[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Stationary_random_processes|"non-stationarity"]]&nbsp; is obvious,&nbsp; since,&nbsp; for example,&nbsp; at multiples of &nbsp;$T$&nbsp; the variance is &nbsp;$\sigma_s^2 = s_0^2$,&nbsp; while exactly in between &nbsp; $\sigma_s^2 \approx 0$&nbsp; holds.<br>
 *Also the signal &nbsp;$s_{\rm R}(t)$&nbsp; with NRZ rectangular pulses is non&ndash;stationary in a strict sense,&nbsp; because here the moments at the bit boundaries differ with respect to all other instants.&nbsp; For example, &nbsp;$s_{\rm R}(t = \pm T/2)=0$.
@@ Line 155: / Line 155: @@
-Many of the rules valid for &nbsp;[[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Ergodic_random_processes|ergodic processes]]&nbsp; can also be applied to&nbsp; "cycloergodic"&nbsp; (and hence to&nbsp; "cyclostationary")&nbsp; processes with only minor restrictions.
+Many of the rules valid for &nbsp;[[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Ergodic_random_processes|"ergodic processes"]]&nbsp; can also be applied to&nbsp; "cycloergodic"&nbsp; (and hence to&nbsp; "cyclostationary")&nbsp; processes with only minor restrictions.
-*In particular,&nbsp; for the &nbsp;[[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Random_processes|auto-correlation function]]&nbsp; $\rm (ACF)$&nbsp; of such random processes with sample signal &nbsp;$s(t)$ holds:
+*In particular,&nbsp; for the &nbsp;[[Theory_of_Stochastic_Signals/Auto-Correlation_Function#Random_processes|"auto-correlation function"]]&nbsp; $\rm (ACF)$&nbsp; of such random processes with sample signal &nbsp;$s(t)$ holds:
 :$$\varphi_s(\tau) = {\rm E}\big [s(t) \cdot s(t + \tau)\big ] \hspace{0.05cm}.$$
 *With the above equation of the transmitted signal,&nbsp; the ACF as a time average can also be written as follows:
@@ Line 189: / Line 189: @@
 \int_{-\infty}^{+\infty} g_s ( t ) \cdot  g_s ( t +
 \tau)\,{\rm d} t \hspace{0.05cm}.$$
-:&rArr; &nbsp; Since &nbsp;$g_s(t)$ is &nbsp; [[Signal_Representation/Signal_classification#Energy.E2.80.93Limited_and_Power.E2.80.93Limited_Signals| energy-limited]],&nbsp; the division by &nbsp;$T_{\rm M}$&nbsp; and the boundary transition can be omitted.
+:&rArr; &nbsp; Since &nbsp;$g_s(t)$ is &nbsp; [[Signal_Representation/Signal_classification#Energy.E2.80.93Limited_and_Power.E2.80.93Limited_Signals|"energy-limited"]],&nbsp; the division by &nbsp;$T_{\rm M}$&nbsp; and the boundary transition can be omitted.
 *For the&nbsp; '''auto-correlation function of a digital signal''' &nbsp;$s(t)$&nbsp; holds in general:
@@ Line 218: / Line 218: @@
 == PSD calculation of a digital signal ==
 <br>
-The corresponding quantity to the auto-correlation function&nbsp; $\rm (ACF)$&nbsp; of a random signal &nbsp; &rArr; &nbsp; $\varphi_s(\tau)$&nbsp; in the frequency domain is the&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density#Wiener-Khintchine_Theorem|power-spectral density]]&nbsp; $\rm (PSD)$ &nbsp; &rArr; &nbsp; ${\it \Phi}_s(f)$,&nbsp; which is in a fixed relation with the ACF via the Fourier integral:<br>
+The corresponding quantity to the auto-correlation function&nbsp; $\rm (ACF)$&nbsp; of a random signal &nbsp; &rArr; &nbsp; $\varphi_s(\tau)$&nbsp; in the frequency domain is the&nbsp; [[Theory_of_Stochastic_Signals/Power-Spectral_Density#Wiener-Khintchine_Theorem|"power-spectral density"]]&nbsp; $\rm (PSD)$ &nbsp; &rArr; &nbsp; ${\it \Phi}_s(f)$,&nbsp; which is in a fixed relation with the ACF via the Fourier integral:<br>
 :$$\varphi_s(\tau) \hspace{0.4cm}\circ\!\!-\!\!\!-\!\!\!-\!\!\bullet \hspace{0.4cm}
    {\it \Phi}_s(f)  =   \int_{-\infty}^{+\infty} \varphi_s(\tau) \cdot
@@ Line 227: / Line 227: @@
    {\it \Phi}^{^{\hspace{0.08cm}\bullet}}_{gs}(f)  = |G_s(f)|^2
    \hspace{0.05cm},$$
-:and the &nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|shifting theorem]],&nbsp; the&nbsp; '''power-spectral density of the digital signal''' &nbsp;$s(t)$&nbsp; can be represented in the following way:
+:and the &nbsp; [[Signal_Representation/Fourier_Transform_Theorems#Shifting_Theorem|"shifting theorem"]],&nbsp; the&nbsp; '''power-spectral density of the digital signal''' &nbsp;$s(t)$&nbsp; can be represented in the following way:
 :$${\it \Phi}_s(f)   =    \sum_{\lambda =
 -\infty}^{+\infty}{1}/{T} \cdot \varphi_a(\lambda)\cdot {\it
@@ Line 301: / Line 301: @@
 \end{array}$$
-Assumed here are equal probability amplitude coefficients &nbsp; &#8658; &nbsp; ${\rm Pr}(a_\nu =0) = {\rm Pr}(a_\nu =1) = 0.5$&nbsp; with no statistical ties, so that both the &nbsp;[[Theory_of_Stochastic_Signals/Moments_of_a_Discrete_Random_Variable#Second_order_moment_.E2.80.93_power_.E2.80.93_variance_.E2.80.93_standard_deviation|power]]&nbsp; $m_2$  and the &nbsp;[[Theory_of_Stochastic_Signals/Moments_of_a_Discrete_Random_Variable#First_order_moment_.E2.80.93_linear_mean_.E2.80.93_DC_component|linear mean]]&nbsp; $m_1$&nbsp; $($DC component$)$&nbsp; are &nbsp;$0.5$,&nbsp; respectively.<br>
+Assumed here are equal probability amplitude coefficients &nbsp; &#8658; &nbsp; ${\rm Pr}(a_\nu =0) = {\rm Pr}(a_\nu =1) = 0.5$&nbsp; with no statistical ties, so that both the &nbsp;[[Theory_of_Stochastic_Signals/Moments_of_a_Discrete_Random_Variable#Second_order_moment_.E2.80.93_power_.E2.80.93_variance_.E2.80.93_standard_deviation|"power"]]&nbsp; $m_2$  and the &nbsp;[[Theory_of_Stochastic_Signals/Moments_of_a_Discrete_Random_Variable#First_order_moment_.E2.80.93_linear_mean_.E2.80.93_DC_component|"linear mean"]]&nbsp; $m_1$&nbsp; $($DC component$)$&nbsp; are &nbsp;$0.5$,&nbsp; respectively.<br>
 The graph shows a signal section, the ACF and the PSD with unipolar amplitude coefficients,
@@ Line 308: / Line 308: @@
-There are the following differences compared to &nbsp;[[Digital_Signal_Transmission/Basics_of_Coded_Transmission#ACF_and_PSD_for_bipolar_binary_signals|bipolar signaling]]:
+There are the following differences compared to &nbsp;[[Digital_Signal_Transmission/Basics_of_Coded_Transmission#ACF_and_PSD_for_bipolar_binary_signals|"bipolar signaling"]]:
 *Adding the infinite number of triangular functions at distance &nbsp;$T$&nbsp; (all with the same height)&nbsp; results in a constant DC component &nbsp;$s_0^2/4$&nbsp; for the ACF in the left graph&nbsp; (NRZ).
@@ Line 320: / Line 320: @@
 *This unique triangle function leads to the continuous &nbsp;$\rm sinc^2$&ndash;shaped component (blue curve) of &nbsp;${\it \Phi}_s(f)$&nbsp; with the first zero at &nbsp;$1/T_{\rm S} = 2/T$.
-*In contrast,&nbsp; the periodic triangular function leads to an infinite sum of Dirac delta functions with different weights at the distance &nbsp;$1/T$&nbsp; (drawn in red)&nbsp; according to the laws of the &nbsp;[[Signal_Representation/Fourier_Series#General_description| Fourier series]].&nbsp;<br>
+*In contrast,&nbsp; the periodic triangular function leads to an infinite sum of Dirac delta functions with different weights at the distance &nbsp;$1/T$&nbsp; (drawn in red)&nbsp; according to the laws of the &nbsp;[[Signal_Representation/Fourier_Series#General_description|"Fourier series"]].&nbsp;<br>
 *The weights of the Dirac delta functions are proportional to the continuous (blue) PSD component.&nbsp; The Dirac delta line at &nbsp;$f = 0$ has the maximum weight &nbsp;$s_0^2/8$.&nbsp; In contrast,&nbsp; the Dirac delta lines at &nbsp;$\pm 2/T$&nbsp; and multiples thereof do not exist or have the weight &nbsp;$0$ in each case,&nbsp; since the continuous PSD component also has zeros here.<br>