Digital Signal Transmission/Optimal Receiver Strategies - Revision history

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@@ Line 186: / Line 186: @@
 #In the presence of statistical bindings &nbsp;$\text{(Example 3)}$,&nbsp; the joint decision of &nbsp;$N$&nbsp; symbols noticeably reduces the error probability,&nbsp; since the maximum likelihood receiver takes the bindings into account.
 #Such bindings can be either deliberately created by transmission-side coding&nbsp; $($see the &nbsp;$\rm LNTwww$ book&nbsp; [[Channel_Coding|"Channel Coding"]])&nbsp; or unintentionally caused by&nbsp; (linear)&nbsp; channel distortions.<br>
-#In the presence of such&nbsp; "intersymbol interferences",&nbsp; the calculation of the error probability is much more difficult.&nbsp; However,&nbsp; comparable approximations as for the Viterbi receiver can be used,&nbsp; which are given at the &nbsp;[[Digital_Signal_Transmission/Viterbi_Receiver#Error_probability_in_maximum_likelihood_decision|end of the next chapter]].&nbsp; }}<br>
+#In the presence of such&nbsp; "intersymbol interferences",&nbsp; the calculation of the error probability is much more difficult.&nbsp; However,&nbsp; comparable approximations as for the Viterbi receiver can be used,&nbsp; which are given at the &nbsp;[[Digital_Signal_Transmission/Viterbi_Receiver#Bit_error_probability_with_maximum_likelihood_decision|end of the next chapter]].&nbsp; }}<br>
 == Correlation receiver with unipolar signaling ==

@@ Line 20: / Line 20: @@
 *In this model,&nbsp; any channel distortions are added to the transmitter and are thus already included in the basic transmission pulse &nbsp;$g_s(t)$&nbsp; and the signal &nbsp;$s(t)$.&nbsp; This measure is only for a simpler representation and is not a restriction.<br>
-*Knowing the currently applied received signal &nbsp;$s(t)$,&nbsp; the optimal receiver searches from the set &nbsp;$\{Q_0$, ... , $Q_{M-1}\}$&nbsp; of the possible source symbol sequences, the receiver searches for the most likely transmitted sequence &nbsp;$Q_j$&nbsp; and outputs this as a sink symbol sequence &nbsp;$V$.&nbsp; <br>
+*Knowing the currently applied received signal &nbsp;$r(t)$,&nbsp; the optimal receiver searches from the set &nbsp;$\{Q_0$, ... , $Q_{M-1}\}$&nbsp; of the possible source symbol sequences, the receiver searches for the most likely transmitted sequence &nbsp;$Q_j$&nbsp; and outputs this as a sink symbol sequence &nbsp;$V$.&nbsp; <br>
 *Before the actual decision algorithm,&nbsp; a numerical value &nbsp;$W_i$&nbsp; must be derived from the received signal &nbsp;$r(t)$&nbsp; for each possible sequence &nbsp;$Q_i$&nbsp; by suitable signal preprocessing.&nbsp; The larger &nbsp;$W_i$&nbsp; is,&nbsp; the greater the inference probability that &nbsp;$Q_i$&nbsp; was transmitted.<br>
@@ Line 69: / Line 69: @@
 &rArr; &nbsp; The received signal can &ndash; for whatever reason &ndash; take three different values, namely &nbsp;$r = 0$, &nbsp;$r = 1$&nbsp; and additionally &nbsp;$r = 0.5$.
 <br><br>
-<u>Note.</u>
+<u>Note:</u>
 *The received values &nbsp;$r = 0$&nbsp; and &nbsp;$r = 1$&nbsp; will be assigned to the transmitter values &nbsp;$s = 0 \ (Q_0)$&nbsp; resp. &nbsp;$s = 1 \ (Q_1)$,&nbsp;   by both,&nbsp; the ML and MAP decisions.
@@ Line 183: / Line 183: @@
 {{BlaueBox|TEXT=
 $\text{Conclusions:}$&nbsp;
-#If the input signal does not have statistical bindings &nbsp;$\text{(Example 2)}$,&nbsp; there is no improvement by joint decision of &nbsp;$N$&nbsp; symbols over symbol-wise decision &nbsp; <br>&rArr; &nbsp; $p_{\rm S} = {\rm Q} \left( \sqrt{ {2 \cdot E_{\rm B} }/{N_0} }\right)$&
+#If the input signal does not have statistical bindings &nbsp;$\text{(Example 2)}$,&nbsp; there is no improvement by joint decision of &nbsp;$N$&nbsp; symbols over symbol-wise decision &nbsp; <br>&rArr; &nbsp; $p_{\rm S} = {\rm Q} \left( \sqrt{ {2 \cdot E_{\rm B} }/{N_0} }\right)$.
 #In the presence of statistical bindings &nbsp;$\text{(Example 3)}$,&nbsp; the joint decision of &nbsp;$N$&nbsp; symbols noticeably reduces the error probability,&nbsp; since the maximum likelihood receiver takes the bindings into account.
 #Such bindings can be either deliberately created by transmission-side coding&nbsp; $($see the &nbsp;$\rm LNTwww$ book&nbsp; [[Channel_Coding|"Channel Coding"]])&nbsp; or unintentionally caused by&nbsp; (linear)&nbsp; channel distortions.<br>

@@ Line 86: / Line 86: @@
 == Maximum likelihood decision for Gaussian noise ==
 <br>
-We now assume that the received signal &nbsp;$r(t)$&nbsp; is additively composed of a useful signal &nbsp;$s(t)$&nbsp; and a noise component &nbsp;$n(t)$,&nbsp; where the noise is assumed to be Gaussian distributed and white &nbsp; &rArr; &nbsp; &nbsp;[[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System#Transmission_channel_and_interference|"AWGN noise"]]:
+We now assume that the received signal &nbsp;$r(t)$&nbsp; is additively composed of a useful component &nbsp;$s(t)$&nbsp; and a noise component &nbsp;$n(t)$,&nbsp; where the noise is assumed to be Gaussian distributed and white &nbsp; &rArr; &nbsp; &nbsp;[[Digital_Signal_Transmission/System_Components_of_a_Baseband_Transmission_System#Transmission_channel_and_interference|"AWGN noise"]]:
 :$$r(t) = s(t) + n(t) \hspace{0.05cm}.$$
 Any channel distortions are already applied to the signal &nbsp;$s(t)$&nbsp; for simplicity.<br>
-The necessary noise power limitation is realized by an integrator; this corresponds to an averaging of the noise values in the time domain. If one limits the integration interval to the range &nbsp;$t_1$&nbsp; to &nbsp;$t_2$, one can derive a quantity &nbsp;$W_i$&nbsp; for each source symbol sequence &nbsp;$Q_i$,&nbsp; which is a measure for the conditional probability &nbsp;${\rm Pr}\big [ r(t)\hspace{0.05cm} \vert \hspace{0.05cm}
+The necessary noise power limitation is realized by an integrator;&nbsp; this corresponds to an averaging of the noise values in the time domain.&nbsp; If one limits the integration interval to the range &nbsp;$t_1$&nbsp; to &nbsp;$t_2$,&nbsp; one can derive a quantity &nbsp;$W_i$&nbsp; for each source symbol sequence &nbsp;$Q_i$,&nbsp; which is a measure for the conditional probability &nbsp;${\rm Pr}\big [ r(t)\hspace{0.05cm} \vert \hspace{0.05cm}
   Q_i\big ] $:&nbsp;
 :$$W_i  =  \int_{t_1}^{t_2} r(t) \cdot s_i(t) \,{\rm d} t -
@@ Line 97: / Line 97: @@
 I_i - {E_i}/{2} \hspace{0.05cm}.$$
-This decision variable &nbsp;$W_i$&nbsp; can be derived using the &nbsp;$k$&ndash;dimensionial [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#Joint_probability_density_function|"joint probability density"]]&nbsp; of the noise $($with &nbsp;$k \to \infty)$&nbsp; and some boundary crossings. The result can be interpreted as follows:
+This decision variable &nbsp;$W_i$&nbsp; can be derived using the &nbsp;$k$&ndash;dimensionial&nbsp; [[Theory_of_Stochastic_Signals/Two-Dimensional_Random_Variables#Joint_probability_density_function|"joint probability density"]]&nbsp; of the noise&nbsp; $($with &nbsp;$k \to \infty)$&nbsp; and some boundary crossings.&nbsp; The result can be interpreted as follows:
-*Integration is used for noise power reduction by averaging. If &nbsp;$N$&nbsp; binary symbols are decided simultaneously by the maximum likelihood detector, set &nbsp;$t_1 = 0 $&nbsp; and &nbsp;$t_2 = N \cdot T$&nbsp; for distortion-free channel.
+*Integration is used for noise power reduction by averaging.&nbsp; If &nbsp;$N$&nbsp; binary symbols are decided simultaneously by the maximum likelihood detector,&nbsp; set &nbsp;$t_1 = 0 $&nbsp; and &nbsp;$t_2 = N \cdot T$&nbsp; for distortion-free channel.
 *The first term of the above decision variable &nbsp;$W_i$&nbsp; is equal to the &nbsp; [[Theory_of_Stochastic_Signals/Cross-Correlation_Function_and_Cross_Power-Spectral_Density#Definition_of_the_cross-correlation_function| "energy cross-correlation function"]]&nbsp; formed over the finite time interval &nbsp;$NT$&nbsp; between &nbsp;$r(t)$&nbsp; and &nbsp;$s_i(t)$&nbsp; at the point &nbsp;$\tau = 0$:
 :$$I_i  = \varphi_{r, \hspace{0.08cm}s_i} (\tau = 0) =  \int_{0}^{N \cdot T}r(t) \cdot s_i(t) \,{\rm d} t
 \hspace{0.05cm}.$$
-*The second term gives the half energy of the considered useful signal &nbsp;$s_i(t)$&nbsp; to be subtracted. The energy is equal to the ACF of the useful signal at the point &nbsp;$\tau = 0$:
+*The second term gives half the energy of the considered useful signal &nbsp;$s_i(t)$&nbsp; to be subtracted.&nbsp; The energy is equal to the auto-correlation function&nbsp; $\rm (ACF)$&nbsp; of &nbsp;$s_i(t)$&nbsp; at the time point &nbsp;$\tau = 0$:
 ::<math>E_i  =  \varphi_{s_i} (\tau = 0) = \int_{0}^{N \cdot T}
 s_i^2(t) \,{\rm d} t \hspace{0.05cm}.</math>
-*In case of distorting channel the impulse response &nbsp;$h_{\rm K}(t)$&nbsp; is not Dirac-shaped, but for example extended to the range &nbsp;$-T_{\rm K} \le t \le +T_{\rm K}$.&nbsp; In this case, &nbsp;$t_1 = -T_{\rm K}$&nbsp; and &nbsp;$t_2 = N \cdot T +T_{\rm K}$&nbsp; must be used for the two integration limits.<br>
+*In the case of a distorting channel,&nbsp; the channel impulse response &nbsp;$h_{\rm K}(t)$&nbsp; is not Dirac-shaped,&nbsp; but for example extended to the range &nbsp;$-T_{\rm K} \le t \le +T_{\rm K}$.&nbsp; In this case,&nbsp; $t_1 = -T_{\rm K}$&nbsp; and &nbsp;$t_2 = N \cdot T +T_{\rm K}$&nbsp; must be used for the integration limits.<br>
 == Matched filter receiver vs. correlation receiver ==
 <br>
-There are various circuit implementations of the maximum likelihood receiver.
+There are various circuit implementations of the maximum likelihood&nbsp; $\rm (ML)$&nbsp; receiver.
-[[File:EN_Dig_T_3_7_S4.png|center|frame|Correlation receiver for &nbsp;$N = 3$, &nbsp;$t_1 = 0$, &nbsp;$t_2 = 3T$ &nbsp; and &nbsp; $M = 2^3 = 8$ |class=fit]]
+&rArr; &nbsp; A second realization form is provided by the &nbsp;'''correlation receiver'''&nbsp; according to the following graph.&nbsp; One recognizes from this block diagram for the indicated parameters:
 *The drawn correlation receiver forms a total of &nbsp;$M = 8$&nbsp; cross-correlation functions between the received signal &nbsp;$r(t) = s_k(t) + n(t)$&nbsp; and the possible transmitted signals &nbsp;$s_i(t), \ i = 0$, ... , $M-1$. The following description assumes that the useful signal &nbsp;$s_k(t)$&nbsp; has been transmitted.<br>
-*The correlation receiver now searches for the maximum value &nbsp;$W_j$&nbsp; of all correlation values and outputs the corresponding sequence &nbsp;$Q_j$&nbsp; as a sink symbol sequence &nbsp;$V$.&nbsp; Formally, the ML decision rule can be expressed as follows:
+*This receiver searches for the maximum value &nbsp;$W_j$&nbsp; of all correlation values and outputs the corresponding sequence &nbsp;$Q_j$&nbsp; as sink symbol sequence &nbsp;$V$.&nbsp; Formally,&nbsp; the&nbsp; $\rm ML$&nbsp; decision rule can be expressed as follows:
-:$$V = Q_j, \hspace{0.2cm}{\rm falls}\hspace{0.2cm} W_i < W_j
+:$$V = Q_j, \hspace{0.2cm}{\rm if}\hspace{0.2cm} W_i < W_j
 \hspace{0.2cm}{\rm for}\hspace{0.2cm} {\rm
 all}\hspace{0.2cm} i \ne j \hspace{0.05cm}.$$
-*If we further assume that all transmitted signals &nbsp;$s_i(t)$&nbsp; have exactly the same energy, we can dispense with the subtraction of &nbsp;$E_i/2$&nbsp; in all branches. In this case, the following correlation values are compared &nbsp;$(i = 0$, ... , $M-1)$:
+*If we further assume that all transmitted signals &nbsp;$s_i(t)$&nbsp; have same energy,&nbsp; we can dispense with the subtraction of &nbsp;$E_i/2$&nbsp; in all branches.&nbsp; In this case,&nbsp; the following correlation values are compared &nbsp;$(i = 0$, ... , $M-1)$:
 ::<math>I_i  =  \int_{0}^{NT} s_j(t) \cdot s_i(t) \,{\rm d} t +
 \int_{0}^{NT} n(t) \cdot s_i(t) \,{\rm d} t
 \hspace{0.05cm}.</math>
-*With high probability, &nbsp;$I_j = I_k$&nbsp; is larger than all other comparison values &nbsp;$I_{j \ne k}$. However, if the noise &nbsp;$n(t)$&nbsp; is too large, the correlation receiver will also make a wrong decision.<br>
+*With high probability, &nbsp;$I_j = I_k$&nbsp; is larger than all other comparison values&nbsp; $I_{j \ne k}$ &nbsp; &rArr; &nbsp;  right decision.&nbsp; However,&nbsp; if the noise &nbsp;$n(t)$&nbsp; is too large,&nbsp; also the correlation receiver will make wrong decisions.<br>
 == Representation of the correlation receiver in the tree diagram==
 <br>
-Let us illustrate the operation of the correlation receiver in the tree diagram, where the &nbsp;$2^3 = 8$&nbsp; possible source symbol sequences &nbsp;$Q_i$&nbsp; of length &nbsp;$N = 3$&nbsp; are represented by bipolar rectangular transmitted signals &nbsp;$s_i(t)$:&nbsp;
+Let us illustrate  the correlation receiver operation in the tree diagram,&nbsp; where the &nbsp;$2^3 = 8$&nbsp; possible source symbol sequences &nbsp;$Q_i$&nbsp; of length &nbsp;$N = 3$&nbsp; are represented by bipolar rectangular transmitted signals &nbsp;$s_i(t)$.
-[[File:P ID1458 Dig T 3 7 S5a version1.png|center|frame|Possible bipolar transmitted signals for &nbsp;$N = 3$|class=fit]]
+[[File:P ID1458 Dig T 3 7 S5a version1.png|right|frame|All&nbsp; $2^3=8$&nbsp; possible bipolar transmitted signals&nbsp; $s_i(t)$&nbsp; for &nbsp;$N = 3$|class=fit]]
-The possible symbol sequences &nbsp;$Q_0 = \rm LLL$, ... , $Q_7 = \rm HHH$&nbsp; and the associated transmitted signals &nbsp;$s_0(t)$, ... , $s_7(t)$&nbsp; are listed above.
+*Due to bipolar amplitude coefficients and the rectangular shape &nbsp; &rArr; &nbsp; all signal energies are equal:&nbsp; $E_0 =  \text{...} = E_7 = N \cdot E_{\rm B}$, where &nbsp;$E_{\rm B}$&nbsp; indicates the energy of a single pulse of duration $T$.
-*Due to the bipolar amplitude coefficients and the rectangular shape all signal energies are equal: &nbsp; $E_0 =  \text{...} = E_7 = N \cdot E_{\rm B}$, where &nbsp;$E_{\rm B}$&nbsp; indicates the energy of a single pulse of duration $T$.
-*Therefore, subtraction of the &nbsp;$E_i/2$&nbsp; term in all branches can be omitted &nbsp; &rArr; &nbsp; a decision based on the correlation values &nbsp;$I_i$&nbsp; gives equally reliable results as maximizing the corrected values &nbsp;$W_i$.<br>
+*Therefore,&nbsp; the subtraction of the &nbsp;$E_i/2$&nbsp; term in all branches can be omitted &nbsp; &rArr; &nbsp; the decision based on the correlation values &nbsp;$I_i$&nbsp; gives equally reliable results as maximizing the corrected values &nbsp;$W_i$.

@@ Line 65: / Line 65: @@
 $\text{Example 1:}$&nbsp; To illustrate the&nbsp; "ML"&nbsp; and the&nbsp; "MAP"&nbsp; decision rule,&nbsp; we now construct a very simple example with only two source symbols &nbsp;$(M = 2)$.
 [[File:EN_Dig_T_3_7_S2.png|right|frame|For clarification of MAP and ML receiver|class=fit]]
-<br><br>&rArr; &nbsp; The two possible symbols &nbsp;$Q_0$&nbsp; and &nbsp;$Q_1$&nbsp; are represented by the transmitted signals &nbsp;$s = 0$&nbsp; and &nbsp;$s = 1$.&nbsp;
+<br><br>&rArr; &nbsp; The two possible symbols &nbsp;$Q_0$&nbsp; and &nbsp;$Q_1$&nbsp; are represented by the transmitted signals &nbsp;$s = 0$&nbsp; and &nbsp;$s = 1$.
+<br><br>
 &rArr; &nbsp; The received signal can &ndash; for whatever reason &ndash; take three different values, namely &nbsp;$r = 0$, &nbsp;$r = 1$&nbsp; and additionally &nbsp;$r = 0.5$.
+<br><br>
-*The received values &nbsp;$r = 0$&nbsp; and &nbsp;$r = 1$&nbsp; will be assigned to the transmitter values &nbsp;$s = 0 \ (Q_0)$&nbsp; and &nbsp;$s = 1 \ (Q_1)$,&nbsp; resp.,&nbsp;  by both,&nbsp; the ML and MAP decisions.
+<u>Note.</u>
 *In contrast, the decisions will give a different result with respect to the received value &nbsp;$r = 0.5$:&nbsp;
-*The maximum likelihood decision rule leads to the source symbol &nbsp;$Q_0$, because of
+:*The maximum likelihood&nbsp; $\rm (ML)$&nbsp; decision rule leads to the source symbol &nbsp;$Q_0$,&nbsp; because of:
-:$${\rm Pr}\big [ r= 0.5\hspace{0.05cm}\vert\hspace{0.05cm}
+::$${\rm Pr}\big [ r= 0.5\hspace{0.05cm}\vert\hspace{0.05cm}
   Q_0\big ] = 0.4 > {\rm Pr}\big [ r= 0.5\hspace{0.05cm} \vert \hspace{0.05cm}
   Q_1\big ] = 0.2 \hspace{0.05cm}.$$
-*The MAP decision, on the other hand, leads to the source symbol &nbsp;$Q_1$, since according to the secondary calculation in the graph:
+:*The maximum&ndash;a&ndash;posteriori&nbsp; $\rm (MAP)$&nbsp; decision rule leads to the source symbol &nbsp;$Q_1$,&nbsp; since according to the incidental calculation in the graph:
-:$${\rm Pr}\big [Q_1 \hspace{0.05cm}\vert\hspace{0.05cm}
+::$${\rm Pr}\big [Q_1 \hspace{0.05cm}\vert\hspace{0.05cm}
   r= 0.5\big ] = 0.6 > {\rm Pr}\big [Q_0 \hspace{0.05cm}\vert\hspace{0.05cm}
   r= 0.5\big ] = 0.4 \hspace{0.05cm}.$$}}<br>

@@ Line 8: / Line 8: @@
 == Considered scenario and prerequisites==
 <br>
-All digital receivers described so far always make symbolwise decisions. If, on the other hand, several symbols are decided simultaneously, statistical bindings between the received signal samples can be taken into account during detection, which results in a lower error probability &ndash; but at the cost of an additional runtime.<br>
+All digital receivers described so far always make symbol-wise decisions.&nbsp; If,&nbsp; on the other hand,&nbsp; several symbols are decided simultaneously,&nbsp; statistical bindings between the received signal samples can be taken into account during detection,&nbsp; which results in a lower error probability &ndash; but at the cost of an additional delay time.<br>
-In this &ndash; partly also in the next chapter &ndash; the following transmission model is assumed:<br>
+In this&nbsp; $($partly also in the next chapter$)$&nbsp; the following transmission model is assumed.&nbsp; Compared to the last two chapters,&nbsp; the following differences arise: <br>
-[[File:EN_Dig_T_3_7_S1.png|center|frame|Transmission system with optimal receiver|class=fit]]
+[[File:EN_Dig_T_3_7_S1.png|right|frame|Transmission system with optimal receiver|class=fit]]
-Compared to the last two chapters, the following differences arise:
+*$Q \in \{Q_i\}$&nbsp; with&nbsp; $i = 0$, ... , $M-1$&nbsp; denotes a time-constrained source symbol sequence &nbsp;$\langle q_\nu \rangle$ whose symbols are to be jointly decided by the receiver.<br>
-*If &nbsp;$Q$&nbsp; describes a sequence of &nbsp;$N$&nbsp; redundancy-free binary symbols, set &nbsp;$M = 2^N$.&nbsp; On the other hand, if the decision is symbolwise, &nbsp;$M$&nbsp; specifies the level number of the digital source.<br>
+*If the source &nbsp;$Q$&nbsp; describes a sequence of &nbsp;$N$&nbsp; redundancy-free binary symbols, set &nbsp;$M = 2^N$.&nbsp; On the other hand,&nbsp; if the decision is symbol-wise, &nbsp;$M$&nbsp; specifies the level number of the digital source.<br>
-*In the above model, any channel distortions are added to the transmitter and are thus already included in the basic pulse &nbsp;$g_s(t)$&nbsp; and the signal &nbsp;$s(t)$.&nbsp; This measure is only for a simpler representation and is not a restriction.<br>
+*In this model,&nbsp; any channel distortions are added to the transmitter and are thus already included in the basic transmission pulse &nbsp;$g_s(t)$&nbsp; and the signal &nbsp;$s(t)$.&nbsp; This measure is only for a simpler representation and is not a restriction.<br>
 *Knowing the currently applied received signal &nbsp;$s(t)$,&nbsp; the optimal receiver searches from the set &nbsp;$\{Q_0$, ... , $Q_{M-1}\}$&nbsp; of the possible source symbol sequences, the receiver searches for the most likely transmitted sequence &nbsp;$Q_j$&nbsp; and outputs this as a sink symbol sequence &nbsp;$V$.&nbsp; <br>
-*Before the actual decision algorithm, a numerical value &nbsp;$W_i$&nbsp; must be derived from the received signal &nbsp;$r(t)$&nbsp; for each possible sequence &nbsp;$Q_i$&nbsp; by suitable signal preprocessing. The larger &nbsp;$W_i$&nbsp; is, the greater the inference probability that &nbsp;$Q_i$&nbsp; was transmitted.<br>
+*Before the actual decision algorithm,&nbsp; a numerical value &nbsp;$W_i$&nbsp; must be derived from the received signal &nbsp;$r(t)$&nbsp; for each possible sequence &nbsp;$Q_i$&nbsp; by suitable signal preprocessing.&nbsp; The larger &nbsp;$W_i$&nbsp; is,&nbsp; the greater the inference probability that &nbsp;$Q_i$&nbsp; was transmitted.<br>
-*Signal preprocessing must provide for the necessary noise power limitation and &ndash; in the case of strong channel distortions &ndash; for sufficient pre-equalization of the resulting intersymbol interferences. In addition, preprocessing also includes sampling for time discretization.<br>
+*Signal preprocessing must provide for the necessary noise power limitation and &ndash; in the case of strong channel distortions &ndash; for sufficient pre-equalization of the resulting intersymbol interferences.&nbsp; In addition,&nbsp; preprocessing also includes sampling for time discretization.<br>
-== MAP and Maximum–Likelihood decision rule==
+== Maximum-a-posteriori and maximum–likelihood decision rule==
 <br>
-The (unconstrained) optimal receiver is called the MAP receiver, where "MAP" stands for "Maximum&ndash;a&ndash;posteriori".<br>
+The&nbsp; (unconstrained)&nbsp; optimal receiver is called the&nbsp; "MAP receiver",&nbsp; where&nbsp; "MAP"&nbsp; stands for&nbsp; "maximum&ndash;a&ndash;posteriori".<br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; The &nbsp;'''MAP receiver'''&nbsp; determines the &nbsp;$M$&nbsp; inference probabilities &nbsp;${\rm Pr}\big[Q_i \hspace{0.05cm}\vert \hspace{0.05cm}r(t)\big]$&nbsp; and sets the output sequence &nbsp;$V$&nbsp; according to the decision rule, where the index is: &nbsp; $i = 0$, ... , $M-1$&nbsp; as well as &nbsp;$i \ne j$:
+$\text{Definition:}$&nbsp; The&nbsp; '''maximum&ndash;a&ndash;posteriori receiver'''&nbsp; $($abbreviated&nbsp; $\rm MAP)$&nbsp; determines the &nbsp;$M$&nbsp; inference probabilities &nbsp;${\rm Pr}\big[Q_i \hspace{0.05cm}\vert \hspace{0.05cm}r(t)\big]$,&nbsp; and sets the output sequence &nbsp;$V$&nbsp; according to the decision rule,&nbsp; where the index is &nbsp; $i = 0$, ... , $M-1$&nbsp; as well as &nbsp;$i \ne j$:
 :$${\rm Pr}\big[Q_j \hspace{0.05cm}\vert \hspace{0.05cm} r(t)\big] > {\rm Pr}\big[Q_i \hspace{0.05cm}\vert \hspace{0.05cm} r(t)\big]
   \hspace{0.05cm}.$$}}<br>
-The &nbsp;[[Theory_of_Stochastic_Signals/Statistical_Dependence_and_Independence#Inference_probability|"inference probability"]]&nbsp; ${\rm Pr}\big[Q_i \hspace{0.05cm}\vert \hspace{0.05cm} r(t)\big]$&nbsp; indicates the probability with which the sequence &nbsp;$Q_i$&nbsp; was sent when the received signal &nbsp;$r(t)$&nbsp; is present at the decision. Using &nbsp;[[Theory_of_Stochastic_Signals/Statistical_Dependence_and_Independence#Conditional_Probability|"Bayes' theorem"]],&nbsp; this probability can be calculated as follows:
+*The &nbsp;[[Theory_of_Stochastic_Signals/Statistical_Dependence_and_Independence#Inference_probability|"inference probability"]]&nbsp; ${\rm Pr}\big[Q_i \hspace{0.05cm}\vert \hspace{0.05cm} r(t)\big]$&nbsp; indicates the probability with which the sequence &nbsp;$Q_i$&nbsp; was sent when the received signal &nbsp;$r(t)$&nbsp; is present at the decision.&nbsp; Using &nbsp;[[Theory_of_Stochastic_Signals/Statistical_Dependence_and_Independence#Conditional_Probability|"Bayes' theorem"]],&nbsp; this probability can be calculated as follows:
 :$${\rm Pr}\big[Q_i \hspace{0.05cm}|\hspace{0.05cm} r(t)\big] = \frac{ {\rm Pr}\big[ r(t)\hspace{0.05cm}|\hspace{0.05cm}
   Q_i \big] \cdot {\rm Pr}\big[Q_i]}{{\rm Pr}[r(t)\big]}
   \hspace{0.05cm}.$$
-The MAP decision rule can thus be reformulated or simplified as follows: &nbsp; Let the sink symbol sequence &nbsp;$V = Q_j$ if for all &nbsp;$i \ne j$&nbsp; holds:
+*The MAP decision rule can thus be reformulated or simplified as follows: &nbsp; Let the sink symbol sequence &nbsp;$V = Q_j$,&nbsp; if for all &nbsp;$i \ne j$&nbsp; holds:
 :$$\frac{ {\rm Pr}\big[ r(t)\hspace{0.05cm}|\hspace{0.05cm}
   Q_j \big] \cdot {\rm Pr}\big[Q_j)}{{\rm Pr}\big[r(t)\big]} > \frac{ {\rm Pr}\big[ r(t)\hspace{0.05cm}|\hspace{0.05cm}
@@ Line 49: / Line 48: @@
   Q_i \big] \cdot {\rm Pr}\big[Q_i\big] \hspace{0.05cm}.$$
-A further simplification of this MAP decision rule leads to the ML receiver, where "ML" stands for "maximum likelihood".<br>
+A further simplification of this MAP decision rule leads to the&nbsp; "ML receiver",&nbsp; where&nbsp; "ML"&nbsp; stands for&nbsp; "maximum likelihood".<br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; The &nbsp;'''maximum likelihood receiver'''&nbsp; &ndash; abbreviated ML &ndash; decides according to the conditional forward probabilities &nbsp;${\rm Pr}\big[r(t)\hspace{0.05cm} \vert \hspace{0.05cm}Q_i \big]$&nbsp; and sets the output sequence &nbsp;$V = Q_j$ if for all &nbsp;$i \ne j$&nbsp; holds:
+$\text{Definition:}$&nbsp; The &nbsp;'''maximum likelihood receiver'''&nbsp; $($abbreviated&nbsp; $\rm ML)$ &nbsp; decides according to the conditional forward probabilities &nbsp;${\rm Pr}\big[r(t)\hspace{0.05cm} \vert \hspace{0.05cm}Q_i \big]$,&nbsp; and sets the output sequence &nbsp;$V = Q_j$,&nbsp; if for all &nbsp;$i \ne j$&nbsp; holds:
 :$${\rm Pr}\big[ r(t)\hspace{0.05cm} \vert\hspace{0.05cm}
   Q_j \big] > {\rm Pr}\big[ r(t)\hspace{0.05cm} \vert \hspace{0.05cm}
@@ Line 58: / Line 57: @@
 A comparison of these two definitions shows:
-* For equally probable source symbols, the ML receiver and the MAP receiver use the same decision rules; thus, they are equivalent.
+* For equally probable source symbols,&nbsp; the&nbsp; "ML receiver"&nbsp; and the&nbsp; "MAP receiver"&nbsp; use the same decision rules.&nbsp; Thus,&nbsp; they are equivalent.
 *For symbols that are not equally probable, the ML receiver is inferior to the MAP receiver because it does not use all the available information for detection.<br>
 {{GraueBox|TEXT=
-$\text{Example 1:}$&nbsp; To illustrate ML and MAP decision rule, we now construct a very simple example with only two source symbols &nbsp;$(M = 2)$.
+$\text{Example 1:}$&nbsp; To illustrate the&nbsp; "ML"&nbsp; and the&nbsp; "MAP"&nbsp; decision rule,&nbsp; we now construct a very simple example with only two source symbols &nbsp;$(M = 2)$.
-*The two possible symbols &nbsp;$Q_0$&nbsp; and &nbsp;$Q_1$&nbsp; are represented by the transmitted signals &nbsp;$s = 0$&nbsp; and &nbsp;$s = 1$.&nbsp;
+[[File:EN_Dig_T_3_7_S2.png|right|frame|For clarification of MAP and ML receiver|class=fit]]
-*The received signal can &ndash; for whatever reason &ndash; take three different values, namely &nbsp;$r = 0$, &nbsp;$r = 1$&nbsp; and additionally &nbsp;$r = 0.5$.
+<br><br>&rArr; &nbsp; The two possible symbols &nbsp;$Q_0$&nbsp; and &nbsp;$Q_1$&nbsp; are represented by the transmitted signals &nbsp;$s = 0$&nbsp; and &nbsp;$s = 1$.&nbsp;
-[[File:EN_Dig_T_3_7_S2.png|center|frame|For clarification of MAP and ML receiver|class=fit]]
+*The received values &nbsp;$r = 0$&nbsp; and &nbsp;$r = 1$&nbsp; will be assigned to the transmitter values &nbsp;$s = 0 \ (Q_0)$&nbsp; and &nbsp;$s = 1 \ (Q_1)$,&nbsp; resp.,&nbsp;  by both,&nbsp; the ML and MAP decisions.
-The received values &nbsp;$r = 0$&nbsp; and &nbsp;$r = 1$&nbsp; will be assigned to the transmitter values &nbsp;$s = 0 \ (Q_0)$&nbsp; and &nbsp;$s = 1 \ (Q_1)$,&nbsp; respectively, by both the ML and MAP decisions. In contrast, the decisions will give a different result with respect to the received value &nbsp;$r = 0.5$:&nbsp;
+*In contrast, the decisions will give a different result with respect to the received value &nbsp;$r = 0.5$:&nbsp;
 *The maximum likelihood decision rule leads to the source symbol &nbsp;$Q_0$, because of

@@ Line 12: / Line 12: @@
 In this &ndash; partly also in the next chapter &ndash; the following transmission model is assumed:<br>
-[[File:P ID1455 Dig T 3 7 S1 version1.png|center|frame|Transmission system with optimal receiver|class=fit]]
+[[File:EN_Dig_T_3_7_S1.png|center|frame|Transmission system with optimal receiver|class=fit]]
 Compared to the last two chapters, the following differences arise:
@@ Line 68: / Line 68: @@
-[[File:P ID1461 Dig T 3 7 S2 version1.png|center|frame|For clarification of MAP and ML receiver|class=fit]]
+[[File:EN_Dig_T_3_7_S2.png|center|frame|For clarification of MAP and ML receiver|class=fit]]
 The received values &nbsp;$r = 0$&nbsp; and &nbsp;$r = 1$&nbsp; will be assigned to the transmitter values &nbsp;$s = 0 \ (Q_0)$&nbsp; and &nbsp;$s = 1 \ (Q_1)$,&nbsp; respectively, by both the ML and MAP decisions. In contrast, the decisions will give a different result with respect to the received value &nbsp;$r = 0.5$:&nbsp;
@@ Line 119: / Line 119: @@
 A second form of realization is provided by the &nbsp;'''correlation receiver'''&nbsp; according to the following graph.
-[[File:P ID1457 Dig T 3 7 S4 version1.png|center|frame|Correlation receiver for &nbsp;$N = 3$, &nbsp;$t_1 = 0$, &nbsp;$t_2 = 3T$ &nbsp; and &nbsp; $M = 2^3 = 8$ |class=fit]]
+[[File:EN_Dig_T_3_7_S4.png|center|frame|Correlation receiver for &nbsp;$N = 3$, &nbsp;$t_1 = 0$, &nbsp;$t_2 = 3T$ &nbsp; and &nbsp; $M = 2^3 = 8$ |class=fit]]
 One recognizes from this block diagram for the indicated parameters:
@@ Line 149: / Line 149: @@
 {{GraueBox|TEXT=
 $\text{Example 2:}$&nbsp; The graph shows the continuous integral values, assuming the actually transmitted signal &nbsp;$s_5(t)$&nbsp; and the noise-free case. For this case, the time-dependent integral values and the integral end values are valid:
-[[File:Dig_T_3_7_S5B_version2.png|right|frame|Correlation receiver: &nbsp; tree diagram in the noise-free case|class=fit]]
+[[File:EN_Dig_T_3_7_S5b.png|right|frame|Correlation receiver: &nbsp; tree diagram in the noise-free case|class=fit]]
 :$$i_i(t)  =  \int_{0}^{t} r(\tau) \cdot s_i(\tau) \,{\rm d}
 \tau =  \int_{0}^{t} s_5(\tau) \cdot s_i(\tau) \,{\rm d}
@@ Line 163: / Line 163: @@
 {{GraueBox|TEXT=
 $\text{Example 3:}$&nbsp; The graph for this example describes the same situation as &nbsp;$\text{Example 2}$, but now the received signal &nbsp;$r(t) = s_5(t)+ n(t)$&nbsp; is assumed. The variance of the AWGN noise &nbsp;$n(t)$&nbsp; here is &nbsp;$\sigma_n^2 = 4 \cdot E_{\rm B}/T$.
-[[File:Dig_T_3_7_S5C_version2.png|right|frame|Correlation receiver: tree diagram with noise |class=fit]]
+[[File:EN_Dig_T_3_7_S5c_neu.png|right|frame|Correlation receiver: tree diagram with noise |class=fit]]
 One can see from this graph compared to the noise-free case:
 *The function curves are now no longer straight due to the noise component &nbsp;$n(t)$&nbsp; and there are also slightly different final values than without noise.
@@ Line 212: / Line 212: @@
 $\text{Example 4:}$&nbsp; The graph shows the continuous integral values, again assuming the actual transmitted signal &nbsp;$s_5(t)$&nbsp; and the noise-free case. The corresponding bipolar equivalent was considered in [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Representation_of_the_correlation_receiver_in_the_tree_diagram|"Example 2"]].
-[[File:Dig_T_3_7_S5D_version2.png|right|frame|Tree diagram of the correlation receiver (unipolar)|class=fit]]
+[[File:EN_Dig_T_3_7_S5d.png|right|frame|Tree diagram of the correlation receiver (unipolar)|class=fit]]
 For this example, the following comparison values result, each normalized to &nbsp;$E_{\rm B}$:
 :$$I_5 = I_7 = 2, \hspace{0.2cm}I_1 = I_3 = I_4= I_6 = 1 \hspace{0.2cm},

@@ Line 177: / Line 177: @@
 *If the input signal does not have statistical bindings as in &nbsp;$\text{Example 2}$, there is no improvement by joint decision of &nbsp;$N$&nbsp; symbols over symbolwise decision.
 *In the presence of statistical bindings, the joint decision of &nbsp;$N$&nbsp; symbols noticeably reduces the error probability compared to &nbsp;$p_{\rm S} = {\rm Q} \left( \sqrt{ {2 \cdot E_{\rm B} }/{N_0} }\right)$&nbsp; (valid for symbolwise decision), since the maximum likelihood receiver takes the bindings into account.
-*Such bindings can be either deliberately created by transmission-side coding (see &nbsp;$\rm LNTwww$ book [[Channel_Coding|"Channel Coding"]]) oder durch (lineare) Kanalverzerrungen ungewollt entstehen.<br>
+*Such bindings can be either deliberately created by transmission-side coding (see &nbsp;$\rm LNTwww$ book [[Channel_Coding|"Channel Coding"]]) or unintentionally caused by (linear) channel distortions.<br>
-*Bei Vorhandensein solcher Impulsinterferenzen ist die Berechnung der Fehlerwahrscheinlichkeit deutlich schwieriger. Es können jedoch vergleichbare Näherungen wie beim Viterbi&ndash;Empfänger angegeben werden, die am &nbsp;[[Digital_Signal_Transmission/Viterbi–Empfänger#Fehlerwahrscheinlichkeit_bei_Maximum.E2.80.93Likelihood.E2.80.93Entscheidung|Ende des nächsten Kapitels ]]&nbsp; angegeben sind.}}<br>
+*In the presence of such intersymbol interference, the calculation of the error probability is much more difficult. However, comparable approximations as for the Viterbi receiver can be given, which are given at the &nbsp;[[Digital_Signal_Transmission/Viterbi_Receiver#Error_probability_in_maximum_likelihood_decision|"end of the next chapter"]].&nbsp; }}<br>
-== Korrelationsempfänger bei unipolarer Signalisierung ==
+== Correlation receiver with unipolar signaling ==
 <br>
-Bisher sind wir bei der Beschreibung des Korrelationsempfänger stets von binärer ''bipolarer''&nbsp; Signalisierung ausgegangen:
+So far, we have always assumed binary ''bipolar''&nbsp; signaling when describing the correlation receiver:
 :$$a_\nu  =   \left\{ \begin{array}{c} +1  \\
   -1 \\  \end{array} \right.\quad
 \begin{array}{*{1}c} {\rm{f\ddot{u}r}}
-\\  {\rm{f\ddot{u}r}}  \\ \end{array}\begin{array}{*{20}c}
+\\  {\rm{for}}  \\ \end{array}\begin{array}{*{20}c}
 q_\nu = \mathbf{H} \hspace{0.05cm}, \\
 q_\nu = \mathbf{L} \hspace{0.05cm}.  \\
 \end{array}$$
-Nun betrachten wir den Fall der binären ''unipolaren''&nbsp; Digitalsignalübertragung gilt:
+Now we consider the case of binary ''unipolar''&nbsp; digital signaling holds:
 :$$a_\nu  =   \left\{ \begin{array}{c} 1  \\
 \\  \end{array} \right.\quad
-\begin{array}{*{1}c} {\rm{f\ddot{u}r}}
+\begin{array}{*{1}c} {\rm{for}}
-\\  {\rm{f\ddot{u}r}}  \\ \end{array}\begin{array}{*{20}c}
+\\  {\rm{for}}  \\ \end{array}\begin{array}{*{20}c}
 q_\nu = \mathbf{H} \hspace{0.05cm}, \\
 q_\nu = \mathbf{L} \hspace{0.05cm}.  \\
 \end{array}$$
-Die &nbsp;$2^3 = 8$&nbsp; möglichen Quellensymbolfolgen &nbsp;$Q_i$&nbsp; der Länge &nbsp;$N = 3$&nbsp; werden nun durch unipolare rechteckförmige Sendesignale &nbsp;$s_i(t)$&nbsp; repräsentiert. Nachfolgend  aufgeführt sind die Symbolfolgen &nbsp;$Q_0 = \rm LLL$, ... , $Q_7 = \rm HHH$&nbsp; und die  Sendesignale &nbsp;$s_0(t)$, ... , $s_7(t)$.
+The &nbsp;$2^3 = 8$&nbsp; possible source symbol sequences &nbsp;$Q_i$&nbsp; of length &nbsp;$N = 3$&nbsp; are now represented by unipolar rectangular transmitted signals &nbsp;$s_i(t)$.&nbsp; Listed below are the symbol sequences &nbsp;$Q_0 = \rm LLL$, ... , $Q_7 = \rm HHH$&nbsp; and the transmitted signals &nbsp;$s_0(t)$, ... , $s_7(t)$.
-[[File:P ID1462 Dig T 3 7 S5c version1.png|center|frame|Mögliche unipolare Sendesignale für &nbsp;$N = 3$|class=fit]]
+[[File:P ID1462 Dig T 3 7 S5c version1.png|center|frame|Possible unipolar transmitted signals for &nbsp;$N = 3$|class=fit]]
-Durch Vergleich mit der &nbsp;[[Digital_Signal_Transmission/Optimale_Empfängerstrategien#Darstellung_des_Korrelationsempf.C3.A4ngers_im_Baumdiagramm|entsprechenden Tabelle]]&nbsp; für bipolare Signalisierung erkennt man:
+By comparing with the &nbsp;[[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Representation_of_the_correlation_receiver_in_the_tree_diagram|"corresponding table"]]&nbsp; for bipolar signaling, one can see:
-*Aufgrund der unipolaren Amplitudenkoeffizienten sind nun die Signalenergien &nbsp;$E_i$&nbsp; unterschiedlich, zum Beispiel gilt &nbsp;$E_0 =  0$&nbsp; und  &nbsp;$E_7 = 3 \cdot E_{\rm B}$.
+*Due to the unipolar amplitude coefficients, the signal energies &nbsp;$E_i$&nbsp; are now different, for example &nbsp;$E_0 =  0$&nbsp; and  &nbsp;$E_7 = 3 \cdot E_{\rm B}$.
-*Hier führt die auf den Integralendwerten &nbsp;$I_i$&nbsp; basierende Entscheidung nicht zum richtigen Ergebnis.
+*Here the decision based on the integral end values &nbsp;$I_i$&nbsp; does not lead to the correct result.
-*Vielmehr muss nun auf die korrigierten Vergleichswerte &nbsp;$W_i = I_i- E_i/2$&nbsp; zurückgegriffen werden.<br>
+*Instead, the corrected comparison values &nbsp;$W_i = I_i- E_i/2$&nbsp; must now be used.<br>
 {{GraueBox|TEXT=
-$\text{Beispiel 4:}$&nbsp; In der Grafik sind die fortlaufenden Integralwerte dargestellt, wobei wieder vom tatsächlich gesendeten Signal &nbsp;$s_5(t)$&nbsp; und dem rauschfreien Fall ausgegangen wird. Das entsprechende bipolare Äquivalent wurde im [[Digital_Signal_Transmission/Optimale_Empfängerstrategien#Darstellung_des_Korrelationsempf.C3.A4ngers_im_Baumdiagramm|Beispiel 2]] betrachtet.
+$\text{Example 4:}$&nbsp; The graph shows the continuous integral values, again assuming the actual transmitted signal &nbsp;$s_5(t)$&nbsp; and the noise-free case. The corresponding bipolar equivalent was considered in [[Digital_Signal_Transmission/Optimal_Receiver_Strategies#Representation_of_the_correlation_receiver_in_the_tree_diagram|"Example 2"]].
-[[File:Dig_T_3_7_S5D_version2.png|right|frame|Baumdiagramm des Korrelationsempfängers (unipolar)|class=fit]]
+[[File:Dig_T_3_7_S5D_version2.png|right|frame|Tree diagram of the correlation receiver (unipolar)|class=fit]]
-Für dieses Beispiel ergeben sich folgende Vergleichswerte, jeweils normiert auf &nbsp;$E_{\rm B}$:
+For this example, the following comparison values result, each normalized to &nbsp;$E_{\rm B}$:
 :$$I_5 = I_7 = 2, \hspace{0.2cm}I_1 = I_3 = I_4= I_6 = 1 \hspace{0.2cm},
   \hspace{0.2cm}I_0 = I_2 = 0
@@ Line 221: / Line 221: @@
   \hspace{0.05cm}.$$
-Das bedeutet:
+This means:
-*Bei einem Vergleich hinsichtlich der maximalen &nbsp;$I_i$&ndash;Werte wären die Quellensymbolfolgen &nbsp;$Q_5$&nbsp; und &nbsp;$Q_7$&nbsp; gleichwertig.
+*When compared in terms of maximum &nbsp;$I_i$ values, the source symbol sequences &nbsp;$Q_5$&nbsp; and &nbsp;$Q_7$&nbsp; would be equivalent.
-*Berücksichtigt man die unterschiedlichen Energien &nbsp;$(E_5 = 2, \ E_7 = 3)$, so  wird dagegen wegen &nbsp;$W_5 > W_7$&nbsp; eindeutig für die Folge &nbsp;$Q_5$&nbsp; entschieden.
+*On the other hand, if the different energies &nbsp;$(E_5 = 2, \ E_7 = 3)$ are taken into account, the decision is clearly in favor of the sequence &nbsp;$Q_5$&nbsp; because of &nbsp;$W_5 > W_7$.&nbsp;
-*Der Korrelationsempfänger gemäß &nbsp;$W_i = I_i- E_i/2$&nbsp; entscheidet also auch bei unipolarer Signalisierung richtig auf &nbsp;$s(t) = s_5(t)$. }}<br>
+*The correlation receiver according to &nbsp;$W_i = I_i- E_i/2$&nbsp; therefore decides correctly on &nbsp;$s(t) = s_5(t)$ even with unipolar signaling. }}<br>
-== Aufgaben zum Kapitel==
+== Exercises for the chapter==
 <br>
-[[Aufgaben:Aufgabe_3.09:_Korrelationsempfänger_für_unipolare_Signalisierung|Aufgabe 3.9: Korrelationsempfänger für unipolare Signalisierung]]
+[[Aufgaben:Exercise_3.09:_Correlation_Receiver_for_Unipolar_Signaling|Exercise 3.09: Correlation Receiver for Unipolar Signaling]]
-[[Aufgaben:Aufgabe_3.10:_Baumdiagramm_bei_Maximum-Likelihood|Aufgabe 3.10: Baumdiagramm bei Maximum-Likelihood]]
+[[Aufgaben:Exercise_3.10:_Maximum_Likelihood_Tree_Diagram|Exercise 3.10: Maximum Likelihood Tree Diagram]]
 {{Display}}

@@ Line 1: / Line 1: @@
 {{Header
-|Untermenü=Impulsinterferenzen und Entzerrungsverfahren
+|Untermenü=Intersymbol Interfering and Equalization Methods
 |Vorherige Seite=Entscheidungsrückkopplung
 |Nächste Seite=Viterbi–Empfänger
 }}
-== Betrachtetes Szenario und Voraussetzungen==
+== Considered scenario and prerequisites==
 <br>
-Alle bisher beschriebenen Digitalempfänger treffen stets symbolweise Entscheidungen. Werden dagegen mehrere Symbole gleichzeitig entschieden, so können bei der Detektion statistische Bindungen zwischen den Empfangssignalabtastwerten berücksichtigt werden, was eine geringere Fehlerwahrscheinlichkeit zur Folge hat &ndash; allerdings auf Kosten einer zusätzlichen Laufzeit.<br>
+All digital receivers described so far always make symbolwise decisions. If, on the other hand, several symbols are decided simultaneously, statistical bindings between the received signal samples can be taken into account during detection, which results in a lower error probability &ndash; but at the cost of an additional runtime.<br>
-In diesem &ndash; teilweise auch im nächsten Kapitel &ndash; wird von folgendem Übertragungsmodell ausgegangen:<br>
+In this &ndash; partly also in the next chapter &ndash; the following transmission model is assumed:<br>
-[[File:P ID1455 Dig T 3 7 S1 version1.png|center|frame|Übertragungssystem mit optimalem Empfänger|class=fit]]
+[[File:P ID1455 Dig T 3 7 S1 version1.png|center|frame|Transmission system with optimal receiver|class=fit]]
-Gegenüber den letzten beiden Kapiteln ergeben sich folgende Unterschiede:
+Compared to the last two chapters, the following differences arise:
-*$Q \in \{Q_i\}$&nbsp; mit&nbsp; $i = 0$, ... , $M-1$&nbsp; bezeichnet eine zeitlich begrenzte Quellensymbolfolge &nbsp;$\langle q_\nu \rangle$, deren Symbole vom optimalen Empfänger gemeinsam entschieden werden sollen.<br>
+*$Q \in \{Q_i\}$&nbsp; with&nbsp; $i = 0$, ... , $M-1$&nbsp; denotes a time-constrained source symbol sequence &nbsp;$\langle q_\nu \rangle$ whose symbols are to be jointly decided by the optimal receiver.<br>
-*Beschreibt &nbsp;$Q$&nbsp; eine Folge von &nbsp;$N$&nbsp; redundanzfreien Binärsymbolen, so ist &nbsp;$M = 2^N$&nbsp; zu setzen. Dagegen gibt &nbsp;$M$&nbsp; bei symbolweiser Entscheidung die Stufenzahl der digitalen Quelle an.<br>
+*If &nbsp;$Q$&nbsp; describes a sequence of &nbsp;$N$&nbsp; redundancy-free binary symbols, set &nbsp;$M = 2^N$.&nbsp; On the other hand, if the decision is symbolwise, &nbsp;$M$&nbsp; specifies the level number of the digital source.<br>
-*Im obigen Modell werden eventuelle Kanalverzerrungen dem Sender hinzugefügt und sind somit bereits im Grundimpuls &nbsp;$g_s(t)$&nbsp; und im Signal &nbsp;$s(t)$&nbsp; enthalten. Diese Maßnahme dient lediglich einer einfacheren Darstellung und stellt keine Einschränkung dar.<br>
+*In the above model, any channel distortions are added to the transmitter and are thus already included in the basic pulse &nbsp;$g_s(t)$&nbsp; and the signal &nbsp;$s(t)$.&nbsp; This measure is only for a simpler representation and is not a restriction.<br>
-*Der optimale Empfänger sucht unter Kenntnis des aktuell anliegenden Empfangssignals &nbsp;$s(t)$&nbsp; aus der Menge &nbsp;$\{Q_0$, ... , $Q_{M-1}\}$&nbsp; der möglichen Quellensymbolfolgen die am wahrscheinlichsten gesendete Folge &nbsp;$Q_j$&nbsp; und gibt diese als Sinkensymbolfolge &nbsp;$V$&nbsp; aus.<br>
+*Knowing the currently applied received signal &nbsp;$s(t)$,&nbsp; the optimal receiver searches from the set &nbsp;$\{Q_0$, ... , $Q_{M-1}\}$&nbsp; of the possible source symbol sequences, the receiver searches for the most likely transmitted sequence &nbsp;$Q_j$&nbsp; and outputs this as a sink symbol sequence &nbsp;$V$.&nbsp; <br>
-*Vor dem eigentlichen Entscheidungsalgorithmus muss durch eine geeignete Signalvorverarbeitung aus dem Empfangssignal &nbsp;$r(t)$&nbsp; für jede mögliche Folge &nbsp;$Q_i$&nbsp; ein Zahlenwert &nbsp;$W_i$&nbsp; abgeleitet werden. Je größer &nbsp;$W_i$&nbsp; ist, desto größer ist die Rückschlusswahrscheinlichkeit, dass &nbsp;$Q_i$&nbsp; gesendet wurde.<br>
+*Before the actual decision algorithm, a numerical value &nbsp;$W_i$&nbsp; must be derived from the received signal &nbsp;$r(t)$&nbsp; for each possible sequence &nbsp;$Q_i$&nbsp; by suitable signal preprocessing. The larger &nbsp;$W_i$&nbsp; is, the greater the inference probability that &nbsp;$Q_i$&nbsp; was transmitted.<br>
-*Die Signalvorverarbeitung muss für die erforderliche Rauschleistungsbegrenzung und &ndash; bei starken Kanalverzerrungen &ndash; für eine ausreichende Vorentzerrung der entstandenen Impulsinterferenzen sorgen. Außerdem beinhaltet die Vorverarbeitung auch die Abtastung zur Zeitdiskretisierung.<br>
+*Signal preprocessing must provide for the necessary noise power limitation and &ndash; in the case of strong channel distortions &ndash; for sufficient pre-equalization of the resulting intersymbol interferences. In addition, preprocessing also includes sampling for time discretization.<br>
 == MAP and Maximum–Likelihood decision rule==
 <br>
-Man bezeichnet den (uneingeschränkt) optimalen Empfänger als MAP&ndash;Empfänger, wobei "MAP" für "Maximum&ndash;a&ndash;posteriori" steht.<br>
+The (unconstrained) optimal receiver is called the MAP receiver, where "MAP" stands for "Maximum&ndash;a&ndash;posteriori".<br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; Der &nbsp;'''MAP&ndash;Empfänger'''&nbsp; ermittelt die &nbsp;$M$&nbsp; Rückschlusswahrscheinlichkeiten &nbsp;${\rm Pr}\big[Q_i \hspace{0.05cm}\vert \hspace{0.05cm}r(t)\big]$&nbsp; und setzt die Ausgangsfolge &nbsp;$V$&nbsp; gemäß der Entscheidungsregel, wobei für den Index gilt: &nbsp; $i = 0$, ... , $M-1$&nbsp; sowie &nbsp;$i \ne j$:
+$\text{Definition:}$&nbsp; The &nbsp;'''MAP receiver'''&nbsp; determines the &nbsp;$M$&nbsp; inference probabilities &nbsp;${\rm Pr}\big[Q_i \hspace{0.05cm}\vert \hspace{0.05cm}r(t)\big]$&nbsp; and sets the output sequence &nbsp;$V$&nbsp; according to the decision rule, where the index is: &nbsp; $i = 0$, ... , $M-1$&nbsp; as well as &nbsp;$i \ne j$:
 :$${\rm Pr}\big[Q_j \hspace{0.05cm}\vert \hspace{0.05cm} r(t)\big] > {\rm Pr}\big[Q_i \hspace{0.05cm}\vert \hspace{0.05cm} r(t)\big]
   \hspace{0.05cm}.$$}}<br>
-Die &nbsp;[[Theory_of_Stochastic_Signals/Statistische_Abhängigkeit_und_Unabhängigkeit#R.C3.BCckschlusswahrscheinlichkeit|Rückschlusswahrscheinlichkeit]]&nbsp; ${\rm Pr}\big[Q_i \hspace{0.05cm}\vert \hspace{0.05cm} r(t)\big]$&nbsp; gibt an, mit welcher Wahrscheinlichkeit die Folge &nbsp;$Q_i$&nbsp; gesendet wurde, wenn das Empfangssignal &nbsp;$r(t)$&nbsp;  am Entscheider anliegt. Mit dem &nbsp;[[Theory_of_Stochastic_Signals/Statistische_Abhängigkeit_und_Unabhängigkeit#Bedingte_Wahrscheinlichkeit|Satz von Bayes]]&nbsp; kann diese Wahrscheinlichkeit wie folgt berechnet werden:
+The &nbsp;[[Theory_of_Stochastic_Signals/Statistical_Dependence_and_Independence#Inference_probability|"inference probability"]]&nbsp; ${\rm Pr}\big[Q_i \hspace{0.05cm}\vert \hspace{0.05cm} r(t)\big]$&nbsp; indicates the probability with which the sequence &nbsp;$Q_i$&nbsp; was sent when the received signal &nbsp;$r(t)$&nbsp; is present at the decision. Using &nbsp;[[Theory_of_Stochastic_Signals/Statistical_Dependence_and_Independence#Conditional_Probability|"Bayes' theorem"]],&nbsp; this probability can be calculated as follows:
 :$${\rm Pr}\big[Q_i \hspace{0.05cm}|\hspace{0.05cm} r(t)\big] = \frac{ {\rm Pr}\big[ r(t)\hspace{0.05cm}|\hspace{0.05cm}
   Q_i \big] \cdot {\rm Pr}\big[Q_i]}{{\rm Pr}[r(t)\big]}
   \hspace{0.05cm}.$$
-Die MAP&ndash;Entscheidungsregel lässt sich somit wie folgt umformulieren bzw. vereinfachen: &nbsp; Man setze die Sinkensymbolfolge &nbsp;$V = Q_j$, falls für alle &nbsp;$i \ne j$&nbsp; gilt:
+The MAP decision rule can thus be reformulated or simplified as follows: &nbsp; Let the sink symbol sequence &nbsp;$V = Q_j$ if for all &nbsp;$i \ne j$&nbsp; holds:
 :$$\frac{ {\rm Pr}\big[ r(t)\hspace{0.05cm}|\hspace{0.05cm}
   Q_j \big] \cdot {\rm Pr}\big[Q_j)}{{\rm Pr}\big[r(t)\big]} > \frac{ {\rm Pr}\big[ r(t)\hspace{0.05cm}|\hspace{0.05cm}
@@ Line 49: / Line 49: @@
   Q_i \big] \cdot {\rm Pr}\big[Q_i\big] \hspace{0.05cm}.$$
-Eine weitere Vereinfachung dieser MAP&ndash;Entscheidungsregel führt zum ML&ndash;Empfänger, wobei "ML" für "Maximum&ndash;Likelihood" steht.<br>
+A further simplification of this MAP decision rule leads to the ML receiver, where "ML" stands for "maximum likelihood".<br>
 {{BlaueBox|TEXT=
-$\text{Definition:}$&nbsp; Der &nbsp;'''Maximum&ndash;Likelihood&ndash;Empfänger'''&nbsp; &ndash; abgekürzt ML &ndash; entscheidet nach den bedingten Vorwärtswahrscheinlichkeiten &nbsp;${\rm Pr}\big[r(t)\hspace{0.05cm} \vert \hspace{0.05cm}Q_i \big]$&nbsp; und setzt die Ausgangsfolge &nbsp;$V = Q_j$, falls für alle &nbsp;$i \ne j$&nbsp; gilt:
+$\text{Definition:}$&nbsp; The &nbsp;'''maximum likelihood receiver'''&nbsp; &ndash; abbreviated ML &ndash; decides according to the conditional forward probabilities &nbsp;${\rm Pr}\big[r(t)\hspace{0.05cm} \vert \hspace{0.05cm}Q_i \big]$&nbsp; and sets the output sequence &nbsp;$V = Q_j$ if for all &nbsp;$i \ne j$&nbsp; holds:
 :$${\rm Pr}\big[ r(t)\hspace{0.05cm} \vert\hspace{0.05cm}
   Q_j \big] > {\rm Pr}\big[ r(t)\hspace{0.05cm} \vert \hspace{0.05cm}