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

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	{{Display}}		{{Display}}
			[[de:Digitalsignalübertragung/Fehlerwahrscheinlichkeit_bei_Basisbandübertragung]]

Hwang at 13:48, 23 January 2023

2023-01-23T13:48:53Z

← Older revision		Revision as of 15:48, 23 January 2023
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	:$$ p_{\rm B, \hspace{0.05cm}min} = {\rm Q} \left( \sqrt{{2 \cdot E_{\rm B}}/{N_0}}\right)		:$$ 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}.$$		= {1}/{2} \cdot {\rm erfc} \left( \sqrt{{ E_{\rm B}}/{N_0}}\right) \hspace{0.05cm}.$$
	The resulting configuration is a special case of the so-called  '''maximum a posteriori receiver'''  $\rm (MAP)$,  which is discussed in the section  [[Digital_Signal_Transmission/Optimale_Empfängerstrategien\|"Optimal Receiver Strategies"]]  in the third main chapter of this book.		The resulting configuration is a special case of the so-called  '''maximum a–posteriori receiver'''  $\rm (MAP)$,  which is discussed in the section  [[Digital_Signal_Transmission/Optimale_Empfängerstrategien\|"Optimal Receiver Strategies"]]  in the third main chapter of this book.

	However,  for the above equation to be valid,  a number of conditions must be met:		However,  for the above equation to be valid,  a number of conditions must be met:

Hwang at 23:38, 12 November 2022

2022-11-12T23:38:49Z

Hwang at 05:34, 18 September 2022

2022-09-18T05:34:16Z

← Older revision		Revision as of 07:34, 18 September 2022
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	*However,  according to  [https://en.wikipedia.org/wiki/Abraham_de_Moivre "Moivre"]  and  [https://en.wikipedia.org/wiki/Pierre-Simon_Laplace "Laplace"]:   The binomial distribution can be approximated by a Gaussian distribution:		*However,  according to  [https://en.wikipedia.org/wiki/Abraham_de_Moivre "Moivre"]  and  [https://en.wikipedia.org/wiki/Pierre-Simon_Laplace "Laplace"]:   The binomial distribution can be approximated by a Gaussian distribution:
	:$$f_{h{\rm B}}({h_{\rm B}}) \approx \frac{1}{\sqrt{2\pi}\cdot\sigma_{h{\rm B}}}\cdot {\rm e}^{-(h_{\rm B}-p_{\rm B})^2/(2 \hspace{0.05cm}\cdot \hspace{0.05cm}\sigma_{h{\rm B}}^2)}.$$		:$$f_{h{\rm B}}({h_{\rm B}}) \approx \frac{1}{\sqrt{2\pi}\cdot\sigma_{h{\rm B}}}\cdot {\rm e}^{-(h_{\rm B}-p_{\rm B})^2/(2 \hspace{0.05cm}\cdot \hspace{0.05cm}\sigma_{h{\rm B}}^2)}.$$
	*Using the  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability\|"Gaussian error integral"]]  ${\rm Q}(x)$,  the probability  $p_\varepsilon$  can be calculated that the bit error rate  $h_{\rm B}(N)$  determined by simulation/measurement over  $N$  symbols differs in ~~amount~~ by less than a value  $\varepsilon$  from the actual bit error probability  $p_{\rm B}$:		*Using the  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability\|"Gaussian error integral"]]  ${\rm Q}(x)$,  the probability  $p_\varepsilon$  can be calculated that the bit error rate  $h_{\rm B}(N)$  determined by simulation/measurement over  $N$  symbols differs in magnitude by less than a value  $\varepsilon$  from the actual bit error probability  $p_{\rm B}$:
	:$$p_{\varepsilon}= {\rm Pr} \left( \|h_{\rm B}(N) - p_{\rm B}\| < \varepsilon \right)		:$$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)=		= 1 -2 \cdot {\rm Q} \left( \frac{\varepsilon}{\sigma_{h{\rm B}}} \right)=

Guenter at 15:08, 24 August 2022

2022-08-24T15:08:23Z

← Older revision		Revision as of 17:08, 24 August 2022
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	We further assume the  [[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"]].		We further assume the  [[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"]].

	[[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:		*Then we can assume for the frequency response and the impulse response of the receiver filter:

Hwang at 14:46, 9 May 2022

2022-05-09T14:46:58Z

Hwang at 14:24, 9 May 2022

2022-05-09T14:24:46Z

← Older revision		Revision as of 16:24, 9 May 2022
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	*The above definition of the bit error probability applies whether or not there are statistical bindings within the error sequence  $〈e_ν〉$.  Depending on this,  one has to use different digital channel models in a system simulation.  The complexity of the bit error probability calculation depends on this.		*The above definition of the bit error probability applies whether or not there are statistical bindings within the error sequence  $〈e_ν〉$.  Depending on this,  one has to use different digital channel models in a system simulation.  The complexity of the bit error probability calculation depends on this.
	<br>		<br>
	In the fifth main chapter it will be shown that the so-called  [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)\|BSC model]]  ("Binary Symmetrical Channel")  provides statistically independent errors,  while for the description of bundle error channels one has to resort to the models of  [[Digital_Signal_Transmission/~~Bündelfehlerkanäle~~#~~Kanalmodell_nach_Gilbert.E2.80.93Elliott~~\|Gilbert–Elliott]]  [Gil60]<ref>Gilbert, E. N.:  Capacity of Burst–Noise Channel,  In: Bell Syst. Techn. J. Vol. 39, 1960, pp. 1253–1266.</ref> and of  [[Digital_Signal_Transmission/~~Bündelfehlerkanäle~~#~~Kanalmodell_nach_McCullough~~\|McCullough]]  [McC68]<ref>McCullough, R.H.:  The Binary Regenerative Channel,  In: Bell Syst. Techn. J. (47), 1968.</ref>.		In the fifth main chapter it will be shown that the so-called  [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)\|BSC model]]  ("Binary Symmetrical Channel")  provides statistically independent errors,  while for the description of bundle error channels one has to resort to the models of  [[Digital_Signal_Transmission/Burst_Error_Channels#Channel_model_according_to_Gilbert-Elliott\|"Gilbert–Elliott"]]  [Gil60]<ref>Gilbert, E. N.:  Capacity of Burst–Noise Channel,  In: Bell Syst. Techn. J. Vol. 39, 1960, pp. 1253–1266.</ref> and of  [[Digital_Signal_Transmission/Burst_Error_Channels#Channel_model_according_to_McCullough\|"McCullough"]]  [McC68]<ref>McCullough, R.H.:  The Binary Regenerative Channel,  In: Bell Syst. Techn. J. (47), 1968.</ref>.

Hwang at 14:02, 9 May 2022

2022-05-09T14:02:49Z

← Older revision		Revision as of 16:02, 9 May 2022
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	[[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_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:
	:$$H_{\rm E}(f) = {\rm sinc}(f T)\hspace{0.05cm},$$		:$$H_{\rm E}(f) = {\rm sinc}(f T)\hspace{0.05cm},$$
	:$$H_{\rm E}(f) \hspace{0.4cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ \hspace{0.4cm} h_{\rm E}(t) = \left\{ \begin{array}{c} 1/T \\		:$$H_{\rm E}(f) \hspace{0.4cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ \hspace{0.4cm} h_{\rm E}(t) = \left\{ \begin{array}{c} 1/T \\

← Older revision		Revision as of 16:15, 16 March 2026
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	:This means:   For a binary system with threshold  $E = 0$,  the bit error probability  $p_{\rm B}$  does not depend on the symbol probabilities  $p_{\rm L} $  and  $p_{\rm H} = 1- p_{\rm L}$.		:This means:   For a binary system with threshold  $E = 0$,  the bit error probability  $p_{\rm B}$  does not depend on the symbol probabilities  $p_{\rm L} $  and  $p_{\rm H} = 1- p_{\rm L}$.
	*The probability that the AWGN noise term  $d_{\rm N}$  with standard deviation  $\sigma_d$  is larger than the amplitude  $s_0$  of the NRZ transmission pulse is thus given by:		*The probability that the AWGN noise term  $d_{\rm N}$  with standard deviation  $\sigma_d$  is larger than the amplitude  $s_0$  of the NRZ transmission pulse is thus given by:
	:$$p_{\rm B} = \int_{s_0}^{+\infty}f_{d{\rm N}}(d_{\rm N})\,{\rm d} d_{\rm N} =\frac{\rm 1}{\sqrt{2\pi} \cdot \sigma_d}\int_{s_0}^{+\infty}{\rm e} ^{-d_{\rm N}^2/(2\sigma_d^2) }\,{\rm d} d_{\~~rmN~~}\hspace{0.05cm}.$$		:$$p_{\rm B} = \int_{s_0}^{+\infty}f_{d{\rm N}}(d_{\rm N})\,{\rm d} d_{\rm N} =\frac{\rm 1}{\sqrt{2\pi} \cdot \sigma_d}\int_{s_0}^{+\infty}{\rm e} ^{-d_{\rm N}^2/(2\sigma_d^2) }\,{\rm d} d_{\rm N}\hspace{0.05cm}.$$
	*Using the complementary Gaussian error integral  ${\rm Q}(x)$,  the result is:		*Using the complementary Gaussian error integral  ${\rm Q}(x)$,  the result is:
	:$$p_{\rm B} = {\rm Q} \left( \frac{s_0}{\sigma_d}\right)\hspace{0.4cm}{\rm with}\hspace{0.4cm}\rm Q (\it x) = \frac{\rm 1}{\sqrt{\rm 2\pi}}\int_{\~~itx~~}^{+\infty}\rm e^{\it -u^{\rm 2}/\rm 2}\,d \it u \hspace{0.05cm}.$$		:$$p_{\rm B} = {\rm Q} \left( \frac{s_0}{\sigma_d}\right)\hspace{0.4cm}{\rm with}\hspace{0.4cm}\rm Q (\it x) = \frac{\rm 1}{\sqrt{\rm 2\pi}}\int_{\it x}^{+\infty}\rm e^{\it -u^{\rm 2}/\rm 2}\,d \it u \hspace{0.05cm}.$$
	*Often  – especially in the English-language literature –  the comparable  "complementary error function"  ${\rm erfc}(x)$  is used instead of  ${\rm Q}(x)$.  With this applies:		*Often  – especially in the English-language literature –  the comparable  "complementary error function"  ${\rm erfc}(x)$  is used instead of  ${\rm Q}(x)$.  With this applies:
	:$$p_{\rm B} = {1}/{2} \cdot {\rm erfc} \left( \frac{s_0}{\sqrt{2}\cdot \sigma_d}\right)\hspace{0.4cm}{\rm with}\hspace{0.4cm}{\rm erfc} (\it x) = \frac{\rm 2}{\sqrt{\rm \pi}}\int_{\~~itx~~}^{+\infty}\rm e^{\it -u^{\rm 2}}\,d \it u \hspace{0.05cm}.$$		:$$p_{\rm B} = {1}/{2} \cdot {\rm erfc} \left( \frac{s_0}{\sqrt{2}\cdot \sigma_d}\right)\hspace{0.4cm}{\rm with}\hspace{0.4cm}{\rm erfc} (\it x) = \frac{\rm 2}{\sqrt{\rm \pi}}\int_{\it x}^{+\infty}\rm e^{\it -u^{\rm 2}}\,d \it u \hspace{0.05cm}.$$
	*Both functions can be found in formula collections in tabular form.  However,  you can also use our HTML 5/JavaScript applet  [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen\|"Complementary Gaussian Error Functions"]]  to calculate the function values of  ${\rm Q}(x)$  and  $1/2 \cdot {\rm erfc}(x)$.		*Both functions can be found in formula collections in tabular form.  However,  you can also use our HTML 5/JavaScript applet  [[Applets:Komplementäre_Gaußsche_Fehlerfunktionen\|"Complementary Gaussian Error Functions"]]  to calculate the function values of  ${\rm Q}(x)$  and  $1/2 \cdot {\rm erfc}(x)$.

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	*Then we can assume for the frequency response and the impulse response of the receiver filter:		*Then we can assume for the frequency response and the impulse response of the receiver filter:
	:$$H_{\rm E}(f) = {\rm sinc}(f T)\hspace{0.05cm},$$		:$$H_{\rm E}(f) = {\rm sinc}(f T)\hspace{0.05cm},$$
	:$$H_{\rm E}(f) \hspace{0.4cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ \hspace{0.4cm} h_{\rm E}(t) = \left\{ \begin{array}{c} 1/T \\		:$$H_{\rm E}(f) \hspace{0.4cm}\bullet\!\!-\!\!\!-\!\!\!-\!\!\circ \hspace{0.4cm} h_{\rm E}(t) = \left\{ \begin{array}{c} 1/T \\1/(2T) \\ 0 \\ \end{array} \right.\quad\begin{array}{{1}c} {\rm{for}}\\ {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{{20}c}\|\hspace{0.05cm}t\hspace{0.05cm}\|< T/2 \hspace{0.05cm},\\\|\hspace{0.05cm}t\hspace{0.05cm}\|= T/2 \hspace{0.05cm},\\\|\hspace{0.05cm}t\hspace{0.05cm}\|>T/2 \hspace{0.05cm}. \\\end{array}$$
	1/(2T) \\ 0 \\ \end{array} \right.\quad
	\begin{array}{*{1}c} {\rm{for}}
	\\ {\rm{for}} \\ {\rm{for}} \\ \end{array}\begin{array}{*{20}c}
	\|\hspace{0.05cm}t\hspace{0.05cm}\|< T/2 \hspace{0.05cm},\\
	\|\hspace{0.05cm}t\hspace{0.05cm}\|= T/2 \hspace{0.05cm},\\
	\|\hspace{0.05cm}t\hspace{0.05cm}\|>T/2 \hspace{0.05cm}. \\
	\end{array}$$

	*Because of linearity,  it can be written for the signal component of the detection signal  $d(t)$:		*Because of linearity,  it can be written for the signal component of the detection signal  $d(t)$:
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	*The in general freely selectable detection time is chosen here to  $T_{\rm D} = 0$.  However,  this results in an acausal filter.		*The in general freely selectable detection time is chosen here to  $T_{\rm D} = 0$.  However,  this results in an acausal filter.
	*The detection SNR can be represented for any basic transmitter pulse  $g_s(t)$  with spectrum  $G_s(f)$  as follows,  where the right identity results from  [https://en.wikipedia.org/wiki/Parseval%27s_theorem "Parseval's theorem"]:		*The detection SNR can be represented for any basic transmitter pulse  $g_s(t)$  with spectrum  $G_s(f)$  as follows,  where the right identity results from  [https://en.wikipedia.org/wiki/Parseval%27s_theorem "Parseval's theorem"]:
	:$$\rho_d = \frac{2 \cdot E_{\rm B}}{N_0}\hspace{0.4cm}{\rm with}\hspace{0.4cm}E_{\rm B} = \int^{+\infty} _{-\infty} g_s^2(t)\,{\~~rmd~~}t = \int^{+\infty} _{-\infty} \|G_s(f)\|^2\,{\~~rmd~~}f\hspace{0.05cm}.$$		:$$\rho_d = \frac{2 \cdot E_{\rm B}}{N_0}\hspace{0.4cm}{\rm with}\hspace{0.4cm}E_{\rm B} = \int^{+\infty} _{-\infty} g_s^2(t)\,{\rm d}t = \int^{+\infty} _{-\infty} \|G_s(f)\|^2\,{\rm d}f\hspace{0.05cm}.$$
	*$E_{\rm B}$  is often referred to as  "energy per bit"  and  $E_{\rm B}/N_0$ – incorrectly – as  $\rm SNR$.  Indeed,  as can be seen from the last equation,  for binary baseband transmission  $E_{\rm B}/N_0$  differs from the detection SNR  $\rho_d$  by a factor of  $2$.		*$E_{\rm B}$  is often referred to as  "energy per bit"  and  $E_{\rm B}/N_0$ – incorrectly – as  $\rm SNR$.  Indeed,  as can be seen from the last equation,  for binary baseband transmission  $E_{\rm B}/N_0$  differs from the detection SNR  $\rho_d$  by a factor of  $2$.

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	[[File:EN_Dig_T_1_2_S6_alt_kontrast.png\|right\|frame\|Signals at the receivers  "MF"  and  "I&D"]]		[[File:EN_Dig_T_1_2_S6_alt_kontrast.png\|right\|frame\|Signals at the receivers  "MF"  and  "I&D"]]
	:$$d(\nu \cdot T + T/2) = \frac {1}{T} \cdot \int^{\nu \cdot T + T/2} _{\nu \cdot T - T/2} r(t)\,{\~~rmd~~}t \hspace{0.05cm}.$$		:$$d(\nu \cdot T + T/2) = \frac {1}{T} \cdot \int^{\nu \cdot T + T/2} _{\nu \cdot T - T/2} r(t)\,{\rm d}t \hspace{0.05cm}.$$
	The diagram illustrates the differences in the realization of the optimal binary receiver		The diagram illustrates the differences in the realization of the optimal binary receiver
	*with matched filter $\rm (MF)$   ⇒   middle figure,  and		*with matched filter $\rm (MF)$   ⇒   middle figure,  and

← Older revision		Revision as of 01:38, 13 November 2022
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	$H_{\rm E}(f)$  is a matched filter adapted to the basic transmitter pulse  $g_s(t)$:		$H_{\rm E}(f)$  is a matched filter adapted to the basic transmitter pulse  $g_s(t)$:
	:$$H_{\rm E}(f) = H_{\rm MF}(f) = K_{\rm MF}\cdot G_s^*(f)\hspace{0.05cm}.$$		:$$H_{\rm E}(f) = H_{\rm MF}(f) = K_{\rm MF}\cdot G_s^*(f)\hspace{0.05cm}.$$
	Compared to the  [[Theory_of_Stochastic_Signals/Matched_Filter#Matched_filter_optimization\| "matched filter optimization"]]  ~~page~~,  the following modifications are considered here:		Compared to the  [[Theory_of_Stochastic_Signals/Matched_Filter#Matched_filter_optimization\| "matched filter optimization"]]  section,  the following modifications are considered here:
	*The matched filter constant is here set to  $K_{\rm MF} = 1/(s_0 \cdot T)$.  Thus the frequency response  $ H_{\rm MF}(f)$  is dimensionless.		*The matched filter constant is here set to  $K_{\rm MF} = 1/(s_0 \cdot T)$.  Thus the frequency response  $ H_{\rm MF}(f)$  is dimensionless.
	*The in general freely selectable detection time is chosen here to  $T_{\rm D} = 0$.  However,  this results in an acausal filter.		*The in general freely selectable detection time is chosen here to  $T_{\rm D} = 0$.  However,  this results in an acausal filter.
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	:$$p_{\rm B} = {\rm Q} \left( \sqrt{ {2 \cdot E_{\rm B} }/{N_0} }\right)= {1}/{2} \cdot{\rm erfc} \left( \sqrt{ {E_{\rm B} }/{N_0} }\right)		:$$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}.$$		\hspace{0.05cm}.$$
	This equation is valid for the realization with matched filter as well as for the realization form  "Integrate & Dump"  (see next ~~page~~).}}		This equation is valid for the realization with matched filter as well as for the realization form  "Integrate & Dump"  (see next section).}}


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	One can see from these signal waveforms:		One can see from these signal waveforms:
	*The signal component  $d_{\rm S}(t)$  of the detection signal at the detection times   ⇒   yellow markers $\rm (MF$:   at  $\nu \cdot T$,   $\rm I\&D$:   at  $\nu \cdot T +T/2)$  is equal to  $\pm s_0$  in both cases.		*The signal component  $d_{\rm S}(t)$  of the detection signal at the detection times   ⇒   yellow markers $\rm (MF$:   at  $\nu \cdot T$,   $\rm I\&D$:   at  $\nu \cdot T +T/2)$  is equal to  $\pm s_0$  in both cases.
	*The different detection times are due to the fact that the matched filter was assumed to be acausal  (see last ~~page~~),  in contrast to  "Integrate & Dump".		*The different detection times are due to the fact that the matched filter was assumed to be acausal  (see last section),  in contrast to  "Integrate & Dump".
	*For the matched filter receiver,  the variance of the detection noise component is the same at all times  $t$:    ${\rm E}\big[d_{\rm N}^2(t)\big]= {\sigma _d ^2} = {\rm const.}$  In contrast,  for the I&D  receiver,  the variance increases from symbol start to symbol end.		*For the matched filter receiver,  the variance of the detection noise component is the same at all times  $t$:    ${\rm E}\big[d_{\rm N}^2(t)\big]= {\sigma _d ^2} = {\rm const.}$  In contrast,  for the I&D  receiver,  the variance increases from symbol start to symbol end.
	*At the times marked in yellow,  the detection noise power is the same in both cases,  resulting in the same bit error probability.  With  $E_{\rm B} = s_0^2 \cdot T$  holds again:		*At the times marked in yellow,  the detection noise power is the same in both cases,  resulting in the same bit error probability.  With  $E_{\rm B} = s_0^2 \cdot T$  holds again:

← Older revision		Revision as of 16:46, 9 May 2022
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	*The above definition of the bit error probability applies whether or not there are statistical bindings within the error sequence  $〈e_ν〉$.  Depending on this,  one has to use different digital channel models in a system simulation.  The complexity of the bit error probability calculation depends on this.		*The above definition of the bit error probability applies whether or not there are statistical bindings within the error sequence  $〈e_ν〉$.  Depending on this,  one has to use different digital channel models in a system simulation.  The complexity of the bit error probability calculation depends on this.
	<br>		<br>
	In the fifth main chapter it will be shown that the so-called  [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)\|BSC model]]  ("Binary Symmetrical Channel")  provides statistically independent errors,  while for the description of bundle error channels one has to resort to the models of  [[Digital_Signal_Transmission/Burst_Error_Channels#Channel_model_according_to_Gilbert-Elliott\|"Gilbert–Elliott"]]  [Gil60]<ref>Gilbert, E. N.:  Capacity of Burst–Noise Channel,  In: Bell Syst. Techn. J. Vol. 39, 1960, pp. 1253–1266.</ref> and of  [[Digital_Signal_Transmission/Burst_Error_Channels#Channel_model_according_to_McCullough\|"McCullough"]]  [McC68]<ref>McCullough, R.H.:  The Binary Regenerative Channel,  In: Bell Syst. Techn. J. (47), 1968.</ref>.		In the fifth main chapter it will be shown that the so-called  [[Digital_Signal_Transmission/Binary_Symmetric_Channel_(BSC)\|"BSC model"]]  ("Binary Symmetrical Channel")  provides statistically independent errors,  while for the description of bundle error channels one has to resort to the models of  [[Digital_Signal_Transmission/Burst_Error_Channels#Channel_model_according_to_Gilbert-Elliott\|"Gilbert–Elliott"]]  [Gil60]<ref>Gilbert, E. N.:  Capacity of Burst–Noise Channel,  In: Bell Syst. Techn. J. Vol. 39, 1960, pp. 1253–1266.</ref> and of  [[Digital_Signal_Transmission/Burst_Error_Channels#Channel_model_according_to_McCullough\|"McCullough"]]  [McC68]<ref>McCullough, R.H.:  The Binary Regenerative Channel,  In: Bell Syst. Techn. J. (47), 1968.</ref>.


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	== Bit error probability and bit error rate in the BSC model==		== Bit error probability and bit error rate in the BSC model==
	<br>		<br>
	The following derivations are based on the BSC model  ("Binary Symmetric Channel"),  which is described in detail in  [[Digitalsignal%C3%BCbertragung/Binary_Symmetric_Channel_(BSC)#Fehlerkorrelationsfunktion_des_BSC.E2.80.93Modells\|chapter 5.2]].		The following derivations are based on the BSC model  ("Binary Symmetric Channel"),  which is described in detail in  [[Digitalsignal%C3%BCbertragung/Binary_Symmetric_Channel_(BSC)#Fehlerkorrelationsfunktion_des_BSC.E2.80.93Modells\|"chapter 5.2"]].
	*Each bit is distorted with probability  $p = {\rm Pr}(v_{\nu} \ne q_{\nu}) = {\rm Pr}(e_{\nu} = 1)$,  independent of the error probabilities of the neighboring symbols.		*Each bit is distorted with probability  $p = {\rm Pr}(v_{\nu} \ne q_{\nu}) = {\rm Pr}(e_{\nu} = 1)$,  independent of the error probabilities of the neighboring symbols.
	*Thus,  the  (average)  bit error probability  $p_{\rm B}$  is also equal to  $p$.		*Thus,  the  (average)  bit error probability  $p_{\rm B}$  is also equal to  $p$.
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	*The number of bit errors in the transmission of  $N$  symbols is a discrete random quantity:		*The number of bit errors in the transmission of  $N$  symbols is a discrete random quantity:
	:$$n_{\rm B}(N) = \sum\limits_{\it \nu=\rm 1}^{\it N} e_{\nu} \hspace{0.2cm} \in \hspace{0.2cm} \{0, 1, \hspace{0.05cm}\text{...} \hspace{0.05cm} , N \}\hspace{0.05cm}.$$		:$$n_{\rm B}(N) = \sum\limits_{\it \nu=\rm 1}^{\it N} e_{\nu} \hspace{0.2cm} \in \hspace{0.2cm} \{0, 1, \hspace{0.05cm}\text{...} \hspace{0.05cm} , N \}\hspace{0.05cm}.$$
	*In the case of statistically independent errors  (BSC model),  $n_{\rm B}(N)$  is [[Theory_of_Stochastic_Signals/Binomial_Distribution#General_description_of_the_binomial_distribution\|binomially distributed]].  Consequently,  mean and standard deviation of this random variable are:		*In the case of statistically independent errors  (BSC model),  $n_{\rm B}(N)$  is [[Theory_of_Stochastic_Signals/Binomial_Distribution#General_description_of_the_binomial_distribution\|"binomially distributed"]].  Consequently,  mean and standard deviation of this random variable are:
	:$$m_{n{\rm B}}=N \cdot p_{\rm B},\hspace{0.2cm}\sigma_{n{\rm B}}=\sqrt{N\cdot p_{\rm B}\cdot (\rm 1- \it p_{\rm B})}\hspace{0.05cm}.$$		:$$m_{n{\rm B}}=N \cdot p_{\rm B},\hspace{0.2cm}\sigma_{n{\rm B}}=\sqrt{N\cdot p_{\rm B}\cdot (\rm 1- \it p_{\rm B})}\hspace{0.05cm}.$$
	*Therefore,  for mean and standard deviation of the bit error rate  $h_{\rm B}(N)= n_{\rm B}(N)/N$  holds:		*Therefore,  for mean and standard deviation of the bit error rate  $h_{\rm B}(N)= n_{\rm B}(N)/N$  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}=		<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>		\sqrt{\frac{ p_{\rm B}\cdot (\rm 1- \it p_{\rm B})}{N}}\hspace{0.05cm}.</math>
	*However,  according to  [https://en.wikipedia.org/wiki/Abraham_de_Moivre Moivre]  and  [https://en.wikipedia.org/wiki/Pierre-Simon_Laplace Laplace]:   The binomial distribution can be approximated by a Gaussian distribution:		*However,  according to  [https://en.wikipedia.org/wiki/Abraham_de_Moivre "Moivre"]  and  [https://en.wikipedia.org/wiki/Pierre-Simon_Laplace "Laplace"]:   The binomial distribution can be approximated by a Gaussian distribution:
	:$$f_{h{\rm B}}({h_{\rm B}}) \approx \frac{1}{\sqrt{2\pi}\cdot\sigma_{h{\rm B}}}\cdot {\rm e}^{-(h_{\rm B}-p_{\rm B})^2/(2 \hspace{0.05cm}\cdot \hspace{0.05cm}\sigma_{h{\rm B}}^2)}.$$		:$$f_{h{\rm B}}({h_{\rm B}}) \approx \frac{1}{\sqrt{2\pi}\cdot\sigma_{h{\rm B}}}\cdot {\rm e}^{-(h_{\rm B}-p_{\rm B})^2/(2 \hspace{0.05cm}\cdot \hspace{0.05cm}\sigma_{h{\rm B}}^2)}.$$
	*Using the  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability\|Gaussian error integral]]  ${\rm Q}(x)$,  the probability  $p_\varepsilon$  can be calculated that the bit error rate  $h_{\rm B}(N)$  determined by simulation/measurement over  $N$  symbols differs in amount by less than a value  $\varepsilon$  from the actual bit error probability  $p_{\rm B}$:		*Using the  [[Theory_of_Stochastic_Signals/Gaussian_Distributed_Random_Variables#Exceedance_probability\|"Gaussian error integral"]]  ${\rm Q}(x)$,  the probability  $p_\varepsilon$  can be calculated that the bit error rate  $h_{\rm B}(N)$  determined by simulation/measurement over  $N$  symbols differs in amount by less than a value  $\varepsilon$  from the actual bit error probability  $p_{\rm B}$:
	:$$p_{\varepsilon}= {\rm Pr} \left( \|h_{\rm B}(N) - p_{\rm B}\| < \varepsilon \right)		:$$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)=		= 1 -2 \cdot {\rm Q} \left( \frac{\varepsilon}{\sigma_{h{\rm B}}} \right)=
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	== Error probability with Gaussian noise==		== Error probability with Gaussian noise==
	<br>		<br>
	According to the  [[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System#Block_diagram_and_prerequisites_for_the_first_main_chapter\|prerequisites to this chapter]],  we make the following assumptions:		According to the  [[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System#Block_diagram_and_prerequisites_for_the_first_main_chapter\|"prerequisites to this chapter"]],  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:		*The detection signal at the detection times can be represented as follows:
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	== Optimal binary receiver – "Matched Filter" realization ==		== Optimal binary receiver – "Matched Filter" realization ==
	<br>		<br>
	We further assume the  [[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]].		We further assume the  [[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"]].

	[[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_v2.png\|right\|frame\|Optimal binary receiver (matched filter variant) '''KORREKTUR''': receiver filter]]
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	A comparison of this result with the section [[Theory_of_Stochastic_Signals/Matched_Filter#Optimization_criterion_of_the_matched_filter\| Optimization Criterion of Matched Filter]]  in the book  "Theory of Stochastic Signals"  shows that the receiver filter		A comparison of this result with the section [[Theory_of_Stochastic_Signals/Matched_Filter#Optimization_criterion_of_the_matched_filter\|"Optimization Criterion of Matched Filter"]]  in the book  "Theory of Stochastic Signals"  shows that the receiver filter
	$H_{\rm E}(f)$  is a matched filter adapted to the basic transmitter pulse  $g_s(t)$:		$H_{\rm E}(f)$  is a matched filter adapted to the basic transmitter pulse  $g_s(t)$:
	:$$H_{\rm E}(f) = H_{\rm MF}(f) = K_{\rm MF}\cdot G_s^*(f)\hspace{0.05cm}.$$		:$$H_{\rm E}(f) = H_{\rm MF}(f) = K_{\rm MF}\cdot G_s^*(f)\hspace{0.05cm}.$$
	Compared to the  [[Theory_of_Stochastic_Signals/Matched_Filter#Matched_filter_optimization\| matched filter optimization]]  page,  the following modifications are considered here:		Compared to the  [[Theory_of_Stochastic_Signals/Matched_Filter#Matched_filter_optimization\| "matched filter optimization"]]  page,  the following modifications are considered here:
	*The matched filter constant is here set to  $K_{\rm MF} = 1/(s_0 \cdot T)$.  Thus the frequency response  $ H_{\rm MF}(f)$  is dimensionless.		*The matched filter constant is here set to  $K_{\rm MF} = 1/(s_0 \cdot T)$.  Thus the frequency response  $ H_{\rm MF}(f)$  is dimensionless.
	*The in general freely selectable detection time is chosen here to  $T_{\rm D} = 0$.  However,  this results in an acausal filter.		*The in general freely selectable detection time is chosen here to  $T_{\rm D} = 0$.  However,  this results in an acausal filter.
	*The detection SNR can be represented for any basic transmitter pulse  $g_s(t)$  with spectrum  $G_s(f)$  as follows,  where the right identity results from  [https://en.wikipedia.org/wiki/Parseval%27s_theorem Parseval's theorem]:		*The detection SNR can be represented for any basic transmitter pulse  $g_s(t)$  with spectrum  $G_s(f)$  as follows,  where the right identity results from  [https://en.wikipedia.org/wiki/Parseval%27s_theorem "Parseval's theorem"]:
	:$$\rho_d = \frac{2 \cdot E_{\rm B}}{N_0}\hspace{0.4cm}{\rm with}\hspace{0.4cm}		:$$\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		E_{\rm B} = \int^{+\infty} _{-\infty} g_s^2(t)\,{\rm
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	:$$ p_{\rm B, \hspace{0.05cm}min} = {\rm Q} \left( \sqrt{{2 \cdot E_{\rm B}}/{N_0}}\right)		:$$ 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}.$$		= {1}/{2} \cdot {\rm erfc} \left( \sqrt{{ E_{\rm B}}/{N_0}}\right) \hspace{0.05cm}.$$
	The resulting configuration is a special case of the so-called  '''maximum a posteriori receiver'''  $\rm (MAP)$,  which is discussed in the section  [[Digital_Signal_Transmission/Optimale_Empfängerstrategien\|Optimal Receiver Strategies]]  in the third main chapter of this book.		The resulting configuration is a special case of the so-called  '''maximum a posteriori receiver'''  $\rm (MAP)$,  which is discussed in the section  [[Digital_Signal_Transmission/Optimale_Empfängerstrategien\|"Optimal Receiver Strategies"]]  in the third main chapter of this book.

	However,  for the above equation to be valid,  a number of conditions must be met:		However,  for the above equation to be valid,  a number of conditions must be met:
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	*The above equation is valid independent of the basic transmitter pulse  $g_s(t)$.  Only the energy  $E_{\rm B}$  spent for the transmission of a binary symbol is decisive for the bit error probability  $p_{\rm B}$ in addition to the noise power density  $N_0$.		*The above equation is valid independent of the basic transmitter pulse  $g_s(t)$.  Only the energy  $E_{\rm B}$  spent for the transmission of a binary symbol is decisive for the bit error probability  $p_{\rm B}$ in addition to the noise power density  $N_0$.

	*A prerequisite for the applicability of the above equation is that the detection of a symbol is not interfered with by other symbols. Such  [[Digital_Signal_Transmission/Ursachen_und_Auswirkungen_von_Impulsinterferenzen\|intersymbol interferences]]  increase the bit error probability  $p_{\rm B}$  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  [[Digital_Signal_Transmission/Ursachen_und_Auswirkungen_von_Impulsinterferenzen\|"intersymbol interferences"]]  increase the bit error probability  $p_{\rm B}$  enormously.

	*If the absolute duration  $T_{\rm S}$  of the basic transmitter pulse is less than or equal to the symbol spacing  $T$,  the above equation is always applicable if the matched filter criterion is fulfilled.		*If the absolute duration  $T_{\rm S}$  of the basic transmitter pulse is less than or equal to the symbol spacing  $T$,  the above equation is always applicable if the matched filter criterion is fulfilled.