Digital Signal Transmission/Error Probability for Baseband Transmission - Revision history

Hwang at 13:48, 23 January 2023

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Hwang at 23:38, 12 November 2022

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2022-04-28T14:05:32Z

@@ Line 250: / Line 250: @@
 :$$ p_{\rm B, \hspace{0.05cm}min} = {\rm Q} \left( \sqrt{{2 \cdot E_{\rm B}}/{N_0}}\right)
   = {1}/{2} \cdot {\rm erfc} \left( \sqrt{{ E_{\rm B}}/{N_0}}\right) \hspace{0.05cm}.$$
-The resulting configuration is a special case of the so-called&nbsp; '''maximum a posteriori receiver'''&nbsp; $\rm (MAP)$,&nbsp; which is discussed in the section &nbsp;[[Digital_Signal_Transmission/Optimale_Empfängerstrategien|"Optimal Receiver Strategies"]]&nbsp;  in the third main chapter of this book.
+The resulting configuration is a special case of the so-called&nbsp; '''maximum a&ndash;posteriori receiver'''&nbsp; $\rm (MAP)$,&nbsp; which is discussed in the section &nbsp;[[Digital_Signal_Transmission/Optimale_Empfängerstrategien|"Optimal Receiver Strategies"]]&nbsp;  in the third main chapter of this book.
 However,&nbsp; for the above equation to be valid,&nbsp; a number of conditions must be met:

@@ Line 200: / Line 200: @@
 &nbsp;$H_{\rm E}(f)$&nbsp; is a matched filter adapted to the basic transmitter pulse &nbsp;$g_s(t)$:&nbsp;
 :$$H_{\rm E}(f) = H_{\rm MF}(f) = K_{\rm MF}\cdot G_s^*(f)\hspace{0.05cm}.$$
-Compared to the &nbsp;[[Theory_of_Stochastic_Signals/Matched_Filter#Matched_filter_optimization| "matched filter optimization"]]&nbsp; page,&nbsp; the following modifications are considered here:
+Compared to the &nbsp;[[Theory_of_Stochastic_Signals/Matched_Filter#Matched_filter_optimization| "matched filter optimization"]]&nbsp; section,&nbsp; the following modifications are considered here:
 *The matched filter constant is here set to &nbsp;$K_{\rm MF} = 1/(s_0 \cdot T)$.&nbsp; Thus the frequency response &nbsp;$ H_{\rm MF}(f)$&nbsp; is dimensionless.
 *The in general freely selectable detection time is chosen here to &nbsp;$T_{\rm D} = 0$.&nbsp; However,&nbsp; this results in an acausal filter.
@@ Line 215: / Line 215: @@
 :$$p_{\rm B} = {\rm Q} \left( \sqrt{ {2 \cdot E_{\rm B} }/{N_0} }\right)=  {1}/{2} \cdot{\rm erfc} \left( \sqrt{ {E_{\rm B} }/{N_0} }\right)
 \hspace{0.05cm}.$$
-This equation is valid for the realization with matched filter as well as for the realization form&nbsp; "Integrate & Dump"&nbsp; (see next page).}}
+This equation is valid for the realization with matched filter as well as for the realization form&nbsp; "Integrate & Dump"&nbsp; (see next section).}}
@@ Line 234: / Line 234: @@
 One can see from these signal waveforms:
 *The signal component &nbsp;$d_{\rm S}(t)$&nbsp; of the detection signal at the detection times &nbsp; &rArr; &nbsp; yellow markers $\rm (MF$: &nbsp; at &nbsp;$\nu \cdot T$, &nbsp; $\rm I\&D$: &nbsp; at &nbsp;$\nu \cdot T +T/2)$&nbsp; is equal to&nbsp; $\pm s_0$&nbsp; in both cases.
-*The different detection times are due to the fact that the matched filter was assumed to be acausal&nbsp; (see last page),&nbsp; in contrast to&nbsp; "Integrate & Dump".
+*The different detection times are due to the fact that the matched filter was assumed to be acausal&nbsp; (see last section),&nbsp; in contrast to&nbsp; "Integrate & Dump".
 *For the matched filter receiver,&nbsp; the variance of the detection noise component is the same at all times &nbsp;$t$:&nbsp; &nbsp; ${\rm E}\big[d_{\rm N}^2(t)\big]= {\sigma _d ^2} = {\rm const.}$&nbsp;  In contrast,&nbsp; for the I&D&nbsp; receiver,&nbsp; the variance increases from symbol start to symbol end.
 *At the times marked in yellow,&nbsp; the detection noise power is the same in both cases,&nbsp; resulting in the same bit error probability.&nbsp; With &nbsp;$E_{\rm B} = s_0^2 \cdot T$&nbsp; holds again:

@@ Line 76: / Line 76: @@
 *However,&nbsp; according to &nbsp;[https://en.wikipedia.org/wiki/Abraham_de_Moivre "Moivre"]&nbsp; and &nbsp;[https://en.wikipedia.org/wiki/Pierre-Simon_Laplace "Laplace"]: &nbsp; The binomial distribution can be approximated by a Gaussian distribution:
 :$$f_{h{\rm B}}({h_{\rm B}}) \approx \frac{1}{\sqrt{2\pi}\cdot\sigma_{h{\rm B}}}\cdot {\rm e}^{-(h_{\rm B}-p_{\rm B})^2/(2 \hspace{0.05cm}\cdot \hspace{0.05cm}\sigma_{h{\rm B}}^2)}.$$
-*Using the &nbsp;[[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability|"Gaussian error integral"]]&nbsp; ${\rm Q}(x)$,&nbsp; the probability &nbsp;$p_\varepsilon$&nbsp;  can be calculated that the bit error rate &nbsp;$h_{\rm B}(N)$&nbsp; determined by simulation/measurement over &nbsp;$N$&nbsp; symbols differs in amount by less than a value &nbsp;$\varepsilon$&nbsp; from the actual bit error probability &nbsp;$p_{\rm B}$:&nbsp;
+*Using the &nbsp;[[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability|"Gaussian error integral"]]&nbsp; ${\rm Q}(x)$,&nbsp; the probability &nbsp;$p_\varepsilon$&nbsp;  can be calculated that the bit error rate &nbsp;$h_{\rm B}(N)$&nbsp; determined by simulation/measurement over &nbsp;$N$&nbsp; symbols differs in magnitude by less than a value &nbsp;$\varepsilon$&nbsp; from the actual bit error probability &nbsp;$p_{\rm B}$:&nbsp;
 :$$p_{\varepsilon}= {\rm Pr} \left( |h_{\rm B}(N) - p_{\rm B}| < \varepsilon \right)
    = 1 -2 \cdot {\rm Q} \left( \frac{\varepsilon}{\sigma_{h{\rm B}}} \right)=

@@ Line 156: / Line 156: @@
 We further assume the &nbsp;[[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System#Block_diagram_and_prerequisites_for_the_first_main_chapter|"conditions defined in the previous section"]].&nbsp;
-[[File:EN_Dig_T_1_2_S4_v2.png|right|frame|Optimal binary receiver (matched filter variant) '''KORREKTUR''': receiver filter]]
+[[File:EN_Dig_T_1_2_S4_v23.png|right|frame|Optimal binary receiver (matched filter variant) ]]
 *Then we can assume for the frequency response and the impulse response of the receiver filter:

@@ Line 39: / Line 39: @@
 *The above definition of the bit error probability applies whether or not there are statistical bindings within the error sequence &nbsp;$〈e_ν〉$.&nbsp; Depending on this,&nbsp;  one has to use different digital channel models in a system simulation.&nbsp; The complexity of the bit error probability calculation depends on this.
 <br>
-In the fifth main chapter it will be shown that the so-called &nbsp;[[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)|BSC model]]&nbsp; ("Binary Symmetrical Channel")&nbsp; provides statistically independent errors,&nbsp; while for the description of bundle error channels one has to resort to the models of &nbsp;[[Digital_Signal_Transmission/Burst_Error_Channels#Channel_model_according_to_Gilbert-Elliott|"Gilbert&ndash;Elliott"]]&nbsp; [Gil60]<ref>Gilbert, E. N.:&nbsp; Capacity of Burst–Noise Channel,&nbsp; In: Bell Syst. Techn. J. Vol. 39, 1960, pp. 1253–1266.</ref> and of &nbsp;[[Digital_Signal_Transmission/Burst_Error_Channels#Channel_model_according_to_McCullough|"McCullough"]]&nbsp; [McC68]<ref>McCullough, R.H.:&nbsp; The Binary Regenerative Channel,&nbsp; In: Bell Syst. Techn. J. (47), 1968.</ref>.
+In the fifth main chapter it will be shown that the so-called &nbsp;[[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)|"BSC model"]]&nbsp; ("Binary Symmetrical Channel")&nbsp; provides statistically independent errors,&nbsp; while for the description of bundle error channels one has to resort to the models of &nbsp;[[Digital_Signal_Transmission/Burst_Error_Channels#Channel_model_according_to_Gilbert-Elliott|"Gilbert&ndash;Elliott"]]&nbsp; [Gil60]<ref>Gilbert, E. N.:&nbsp; Capacity of Burst–Noise Channel,&nbsp; In: Bell Syst. Techn. J. Vol. 39, 1960, pp. 1253–1266.</ref> and of &nbsp;[[Digital_Signal_Transmission/Burst_Error_Channels#Channel_model_according_to_McCullough|"McCullough"]]&nbsp; [McC68]<ref>McCullough, R.H.:&nbsp; The Binary Regenerative Channel,&nbsp; In: Bell Syst. Techn. J. (47), 1968.</ref>.
@@ Line 60: / Line 60: @@
 == Bit error probability and bit error rate in the BSC model==
 <br>
-The following derivations are based on the BSC model&nbsp; ("Binary Symmetric Channel"),&nbsp; which is described in detail in &nbsp;[[Digitalsignal%C3%BCbertragung/Binary_Symmetric_Channel_(BSC)#Fehlerkorrelationsfunktion_des_BSC.E2.80.93Modells|chapter 5.2]].&nbsp;
+The following derivations are based on the BSC model&nbsp; ("Binary Symmetric Channel"),&nbsp; which is described in detail in &nbsp;[[Digitalsignal%C3%BCbertragung/Binary_Symmetric_Channel_(BSC)#Fehlerkorrelationsfunktion_des_BSC.E2.80.93Modells|"chapter 5.2"]].&nbsp;
 *Each bit is distorted with probability &nbsp;$p = {\rm Pr}(v_{\nu} \ne q_{\nu}) = {\rm Pr}(e_{\nu} = 1)$,&nbsp; independent of the error probabilities of the neighboring symbols.
 *Thus,&nbsp; the&nbsp; (average)&nbsp; bit error probability &nbsp;$p_{\rm B}$&nbsp; is also equal to &nbsp;$p$.
@@ Line 69: / Line 69: @@
 *The number of bit errors in the transmission of &nbsp;$N$&nbsp; symbols is a discrete random quantity:
 :$$n_{\rm B}(N) = \sum\limits_{\it \nu=\rm 1}^{\it N} e_{\nu} \hspace{0.2cm} \in \hspace{0.2cm} \{0, 1, \hspace{0.05cm}\text{...} \hspace{0.05cm} , N \}\hspace{0.05cm}.$$
-*In the case of statistically independent errors&nbsp; (BSC model),&nbsp; $n_{\rm B}(N)$&nbsp; is [[Theory_of_Stochastic_Signals/Binomial_Distribution#General_description_of_the_binomial_distribution|binomially distributed]].&nbsp; Consequently,&nbsp; mean and standard deviation of this random variable are:
+*In the case of statistically independent errors&nbsp; (BSC model),&nbsp; $n_{\rm B}(N)$&nbsp; is [[Theory_of_Stochastic_Signals/Binomial_Distribution#General_description_of_the_binomial_distribution|"binomially distributed"]].&nbsp; Consequently,&nbsp; mean and standard deviation of this random variable are:
 :$$m_{n{\rm B}}=N \cdot p_{\rm B},\hspace{0.2cm}\sigma_{n{\rm B}}=\sqrt{N\cdot p_{\rm B}\cdot (\rm 1- \it p_{\rm B})}\hspace{0.05cm}.$$
 *Therefore,&nbsp; for mean and standard deviation of the bit error rate &nbsp;$h_{\rm B}(N)= n_{\rm B}(N)/N$&nbsp; holds:
 <math>m_{h{\rm B}}= \frac{m_{n{\rm B}}}{N} = p_{\rm B}\hspace{0.05cm},\hspace{0.2cm}\sigma_{h{\rm B}}= \frac{\sigma_{n{\rm B}}}{N}=
    \sqrt{\frac{ p_{\rm B}\cdot (\rm 1- \it p_{\rm B})}{N}}\hspace{0.05cm}.</math>
-*However,&nbsp; according to &nbsp;[https://en.wikipedia.org/wiki/Abraham_de_Moivre Moivre]&nbsp; and &nbsp;[https://en.wikipedia.org/wiki/Pierre-Simon_Laplace Laplace]: &nbsp; The binomial distribution can be approximated by a Gaussian distribution:
+*However,&nbsp; according to &nbsp;[https://en.wikipedia.org/wiki/Abraham_de_Moivre "Moivre"]&nbsp; and &nbsp;[https://en.wikipedia.org/wiki/Pierre-Simon_Laplace "Laplace"]: &nbsp; The binomial distribution can be approximated by a Gaussian distribution:
 :$$f_{h{\rm B}}({h_{\rm B}}) \approx \frac{1}{\sqrt{2\pi}\cdot\sigma_{h{\rm B}}}\cdot {\rm e}^{-(h_{\rm B}-p_{\rm B})^2/(2 \hspace{0.05cm}\cdot \hspace{0.05cm}\sigma_{h{\rm B}}^2)}.$$
-*Using the &nbsp;[[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability|Gaussian error integral]]&nbsp; ${\rm Q}(x)$,&nbsp; the probability &nbsp;$p_\varepsilon$&nbsp;  can be calculated that the bit error rate &nbsp;$h_{\rm B}(N)$&nbsp; determined by simulation/measurement over &nbsp;$N$&nbsp; symbols differs in amount by less than a value &nbsp;$\varepsilon$&nbsp; from the actual bit error probability &nbsp;$p_{\rm B}$:&nbsp;
+*Using the &nbsp;[[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability|"Gaussian error integral"]]&nbsp; ${\rm Q}(x)$,&nbsp; the probability &nbsp;$p_\varepsilon$&nbsp;  can be calculated that the bit error rate &nbsp;$h_{\rm B}(N)$&nbsp; determined by simulation/measurement over &nbsp;$N$&nbsp; symbols differs in amount by less than a value &nbsp;$\varepsilon$&nbsp; from the actual bit error probability &nbsp;$p_{\rm B}$:&nbsp;
 :$$p_{\varepsilon}= {\rm Pr} \left( |h_{\rm B}(N) - p_{\rm B}| < \varepsilon \right)
    = 1 -2 \cdot {\rm Q} \left( \frac{\varepsilon}{\sigma_{h{\rm B}}} \right)=
@@ Line 97: / Line 97: @@
 == Error probability with Gaussian noise==
 <br>
-According to the &nbsp;[[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System#Block_diagram_and_prerequisites_for_the_first_main_chapter|prerequisites to this chapter]],&nbsp; we make the following assumptions:
+According to the &nbsp;[[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System#Block_diagram_and_prerequisites_for_the_first_main_chapter|"prerequisites to this chapter"]],&nbsp; we make the following assumptions:
-[[File:P_ID1259__Dig_T_1_2_S3_v2.png|right|frame|Error probability with Gaussian noise|class=fit]]
+[[File:P_ID1259__Dig_T_1_2_S3_v2.png|right|frame|"Error probability with Gaussian noise"|class=fit]]
 *The detection signal at the detection times can be represented as follows: &nbsp;
@@ Line 154: / Line 154: @@
 == Optimal binary receiver &ndash; "Matched Filter" realization ==
 <br>
-We further assume the &nbsp;[[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System#Block_diagram_and_prerequisites_for_the_first_main_chapter|conditions defined in the previous section]].&nbsp;
+We further assume the &nbsp;[[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System#Block_diagram_and_prerequisites_for_the_first_main_chapter|"conditions defined in the previous section"]].&nbsp;
 [[File:EN_Dig_T_1_2_S4_v2.png|right|frame|Optimal binary receiver (matched filter variant) '''KORREKTUR''': receiver filter]]
@@ Line 197: / Line 197: @@
-A comparison of this result with the section&nbsp;[[Theory_of_Stochastic_Signals/Matched_Filter#Optimization_criterion_of_the_matched_filter| Optimization Criterion of Matched Filter]]&nbsp; in the book&nbsp; "Theory of Stochastic Signals"&nbsp; shows that the receiver filter
+A comparison of this result with the section&nbsp;[[Theory_of_Stochastic_Signals/Matched_Filter#Optimization_criterion_of_the_matched_filter|"Optimization Criterion of Matched Filter"]]&nbsp; in the book&nbsp; "Theory of Stochastic Signals"&nbsp; shows that the receiver filter
 &nbsp;$H_{\rm E}(f)$&nbsp; is a matched filter adapted to the basic transmitter pulse &nbsp;$g_s(t)$:&nbsp;
 :$$H_{\rm E}(f) = H_{\rm MF}(f) = K_{\rm MF}\cdot G_s^*(f)\hspace{0.05cm}.$$
-Compared to the &nbsp;[[Theory_of_Stochastic_Signals/Matched_Filter#Matched_filter_optimization| matched filter optimization]]&nbsp; page,&nbsp; the following modifications are considered here:
+Compared to the &nbsp;[[Theory_of_Stochastic_Signals/Matched_Filter#Matched_filter_optimization| "matched filter optimization"]]&nbsp; page,&nbsp; the following modifications are considered here:
 *The matched filter constant is here set to &nbsp;$K_{\rm MF} = 1/(s_0 \cdot T)$.&nbsp; Thus the frequency response &nbsp;$ H_{\rm MF}(f)$&nbsp; is dimensionless.
 *The in general freely selectable detection time is chosen here to &nbsp;$T_{\rm D} = 0$.&nbsp; However,&nbsp; this results in an acausal filter.
-*The detection SNR can be represented for any basic transmitter pulse &nbsp;$g_s(t)$&nbsp; with spectrum &nbsp;$G_s(f)$&nbsp; as follows,&nbsp; where the right identity results from &nbsp;[https://en.wikipedia.org/wiki/Parseval%27s_theorem Parseval's theorem]:&nbsp;
+*The detection SNR can be represented for any basic transmitter pulse &nbsp;$g_s(t)$&nbsp; with spectrum &nbsp;$G_s(f)$&nbsp; as follows,&nbsp; where the right identity results from &nbsp;[https://en.wikipedia.org/wiki/Parseval%27s_theorem "Parseval's theorem"]:&nbsp;
 :$$\rho_d = \frac{2 \cdot E_{\rm B}}{N_0}\hspace{0.4cm}{\rm with}\hspace{0.4cm}
   E_{\rm B} =    \int^{+\infty} _{-\infty} g_s^2(t)\,{\rm
@@ Line 250: / Line 250: @@
 :$$ p_{\rm B, \hspace{0.05cm}min} = {\rm Q} \left( \sqrt{{2 \cdot E_{\rm B}}/{N_0}}\right)
   = {1}/{2} \cdot {\rm erfc} \left( \sqrt{{ E_{\rm B}}/{N_0}}\right) \hspace{0.05cm}.$$
-The resulting configuration is a special case of the so-called&nbsp; '''maximum a posteriori receiver'''&nbsp; $\rm (MAP)$,&nbsp; which is discussed in the section &nbsp;[[Digital_Signal_Transmission/Optimale_Empfängerstrategien|Optimal Receiver Strategies]]&nbsp;  in the third main chapter of this book.
+The resulting configuration is a special case of the so-called&nbsp; '''maximum a posteriori receiver'''&nbsp; $\rm (MAP)$,&nbsp; which is discussed in the section &nbsp;[[Digital_Signal_Transmission/Optimale_Empfängerstrategien|"Optimal Receiver Strategies"]]&nbsp;  in the third main chapter of this book.
 However,&nbsp; for the above equation to be valid,&nbsp; a number of conditions must be met:
@@ Line 263: / Line 263: @@
 *The above equation is valid independent of the basic transmitter pulse &nbsp;$g_s(t)$.&nbsp; Only the energy &nbsp;$E_{\rm B}$&nbsp; spent for the transmission of a binary symbol is decisive for the bit error probability &nbsp;$p_{\rm B}$ in addition to the noise power density &nbsp;$N_0$.&nbsp;
-*A prerequisite for the applicability of the above equation is that the detection of a symbol is not interfered with by other symbols. Such &nbsp;[[Digital_Signal_Transmission/Ursachen_und_Auswirkungen_von_Impulsinterferenzen|intersymbol interferences]]&nbsp; increase the bit error probability &nbsp;$p_{\rm B}$&nbsp; enormously.
+*A prerequisite for the applicability of the above equation is that the detection of a symbol is not interfered with by other symbols. Such &nbsp;[[Digital_Signal_Transmission/Ursachen_und_Auswirkungen_von_Impulsinterferenzen|"intersymbol interferences"]]&nbsp; increase the bit error probability &nbsp;$p_{\rm B}$&nbsp; enormously.
 *If the absolute duration &nbsp;$T_{\rm S}$&nbsp; of the basic transmitter pulse is less than or equal to the symbol spacing &nbsp;$T$,&nbsp;  the above equation is always applicable if the matched filter criterion is fulfilled.

@@ Line 158: / Line 158: @@
 [[File:EN_Dig_T_1_2_S4_v2.png|right|frame|Optimal binary receiver (matched filter variant) '''KORREKTUR''': receiver filter]]
-*Then we can assume for the frequency response and the pulse response of the receiver filter:
+*Then we can assume for the frequency response and the impulse response of the receiver filter:
 :$$H_{\rm E}(f) =  {\rm sinc}(f T)\hspace{0.05cm},$$
 :$$H_{\rm E}(f)  \hspace{0.4cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ \hspace{0.4cm} h_{\rm E}(t)  =   \left\{ \begin{array}{c} 1/T  \\

@@ Line 224: / Line 224: @@
 For rectangular NRZ transmission pulses,&nbsp; the matched filter can also be implemented as an integrator&nbsp; $($in each case over a symbol duration &nbsp;$T)$.&nbsp; Thus,&nbsp; the following applies to the detection signal at the detection times:
-[[File:EN_Dig_T_1_2_S6.png|right|frame|Signals at the receivers&nbsp;  "MF"&nbsp; and&nbsp; "I&D"]]
+[[File:EN_Dig_T_1_2_S6_alt_kontrast.png|right|frame|Signals at the receivers&nbsp;  "MF"&nbsp; and&nbsp; "I&D"]]
 :$$d(\nu \cdot T + T/2) = \frac {1}{T} \cdot \int^{\nu \cdot T + T/2} _{\nu \cdot T - T/2} r(t)\,{\rm
   d}t \hspace{0.05cm}.$$