Information Theory/Application to Digital Signal Transmission - Revision history

Guenter at 14:10, 28 February 2023

2023-02-28T14:10:47Z

2023-02-28T14:09:36Z

2023-02-28T14:08:28Z

2023-02-28T14:07:09Z

2023-02-19T15:45:19Z

2023-02-19T15:43:00Z

2023-02-16T15:49:09Z

2023-02-16T15:48:10Z

2023-02-16T15:47:49Z

2023-02-16T15:47:11Z

@@ Line 458: / Line 458: @@
 :$$ \text{Pr(bit error)} \equiv \text{Pr(block error)}  \hspace{0.05cm}.$$
-[[File:EN_Inf_T_3_3_S7c_vers2.png|frame|Definition of different error probabilities '''KORREKTUR: faults''']]
+[[File:EN_Inf_T_3_3_S7c.png|frame|Definition of different error probabilities]]
 {{GraueBox|TEXT=
 $\text{Example 7:}$&nbsp; The upper graph shows the first eight  received blocks&nbsp; $\underline{y}_j^{(n)}$&nbsp; with&nbsp; $n = 4$&nbsp; bits each.&nbsp;  Here channel errors are shaded green.

@@ Line 400: / Line 400: @@
 In order to interpret the channel coding theorem correctly, some basics of&nbsp; &raquo;'''channel coding'''&laquo;.&nbsp;  This extremely important area of Communications Engineering is covered in our learning tutorial&nbsp; $\rm LNTwww$&nbsp; in a separate book called&nbsp;  [[Channel_Coding|"Channel Coding"]].
-[[File:EN_Inf_T_3_3_S7a_vers2.png|center|frame|Model for binary&ndash;coded communication '''KORREKTUR: syndrome''']]
+[[File:EN_Inf_T_3_3_S7a.png|center|frame|Model for binary&ndash;coded communication]]
 The following description refers to the highly simplified model for&nbsp; [[Channel_Coding/Allgemeine_Beschreibung_linearer_Blockcodes|$\text{binary block codes}$]]:

@@ Line 183: / Line 183: @@
   \hspace{0.05cm},$$
-[[File:EN_Inf_T_3_3_S3b_vers2.png|right|frame|Information theoretic model of the binary channel under consideration '''KORREKTUR: ""''']]
+[[File:EN_Inf_T_3_3_S3b_vers2.png|right|frame|Information theoretic model of the binary channel under consideration]]
 *the&nbsp; &raquo;'''equivocation'''&laquo;:

@@ Line 271: / Line 271: @@
   \hspace{0.05cm}.\end{align*}$$
-[[File:P_ID2788__Inf_T_3_3_S4a.png|right|frame|Mutual information for the <br>"asymmetrical binary channel" '''KORREKTUR: Alle Angaben...''']]
+[[File:EN_Inf_T_3_3_S4a.png|right|frame|Mutual information for the <br>"asymmetrical binary channel"]]
 {{GraueBox|TEXT=
 $\text{Example 3}$:&nbsp;

@@ Line 514: / Line 514: @@
 Since the probability of the block errors can never be smaller than that of the bit errors,&nbsp; the block error probability "zero" is also not possible for&nbsp; $R > C$&nbsp;.&nbsp; <br>From the given bounds for the bit errors,
-:$$ {1}/{k} \cdot {\rm Pr}({\rm block error}) \le  {\rm Pr}({\rm bit error}) \le   {\rm Pr}({\rm block error})\hspace{0.05cm},$$
+:$$ {1}/{k} \cdot {\rm Pr}({\rm block\ error}) \le  {\rm Pr}({\rm bit\ error}) \le   {\rm Pr}({\rm block\ error})\hspace{0.05cm},$$
 a range for the block error probability can also be given:
-:$$ {\rm Pr}({\rm bit error})  \le   {\rm Pr}({\rm block error}) \le  k \cdot  {\rm Pr}({\rm bit error})\hspace{0.05cm}.$$
+:$$ {\rm Pr}({\rm bit\ error})  \le   {\rm Pr}({\rm block\ error}) \le  k \cdot  {\rm Pr}({\rm bit\ error})\hspace{0.05cm}.$$
 {{GraueBox|TEXT=

@@ Line 460: / Line 460: @@
 [[File:EN_Inf_T_3_3_S7c_vers2.png|frame|Definition of different error probabilities '''KORREKTUR: faults''']]
 {{GraueBox|TEXT=
-$\text{Example 7:}$&nbsp; The upper graph shows the first eight reception '''KORREKTUR: received?''' blocks&nbsp; $\underline{y}_j^{(n)}$&nbsp; with&nbsp; $n = 4$&nbsp; bits each.&nbsp;  Here channel errors are shaded green.
+$\text{Example 7:}$&nbsp; The upper graph shows the first eight  received blocks&nbsp; $\underline{y}_j^{(n)}$&nbsp; with&nbsp; $n = 4$&nbsp; bits each.&nbsp;  Here channel errors are shaded green.
 Below, the initial sequence&nbsp; $\underline{v}$&nbsp; after decoding is sketched, divided into blocks&nbsp; $\underline{v}_j^{(k)}$&nbsp; with&nbsp; $k = 3$&nbsp; bits each. Note:
@@ Line 469: / Line 469: @@
 For this, some&nbsp; (due to the short sequence)&nbsp; only very vague information about the error probabilities:
-*Half of the reception '''KORREKTUR: received''' bits are shaded green.&nbsp; From this follows:  &nbsp;
+*Half of the received bits are shaded green.&nbsp; From this follows:  &nbsp;
 :$$\text{Pr(channel error)} = 16/32 = 1/2.$$
@@ Line 476: / Line 476: @@
 *In contrast, with uncoded transmission would be: &nbsp;
-:$$\text{Pr(bit error)} = {\rm Pr(channel error)}  = 1/2.$$
+:$$\text{Pr(bit error)} = \text{Pr(channel error)}  = 1/2.$$
 *Half of the decoded blocks are outlined in blue. From this follows:   &nbsp;

← Older revision		Revision as of 15:49, 16 February 2023
Line 512:		Line 512:


−	Since the probability of the block errors can never be smaller than that of the bit errors,  the block error probability ~~„zero”~~ is also not possible for  $R > C$ .  <br>From the given bounds for the bit errors,	+	Since the probability of the block errors can never be smaller than that of the bit errors,  the block error probability "zero" is also not possible for  $R > C$ .  <br>From the given bounds for the bit errors,

	:$$ {1}/{k} \cdot {\rm Pr}({\rm block error}) \le {\rm Pr}({\rm bit error}) \le {\rm Pr}({\rm block error})\hspace{0.05cm},$$		:$$ {1}/{k} \cdot {\rm Pr}({\rm block error}) \le {\rm Pr}({\rm bit error}) \le {\rm Pr}({\rm block error})\hspace{0.05cm},$$