Channel Coding/Code Description with State and Trellis Diagram - Revision history

Guenter at 16:10, 7 December 2022

2022-12-07T16:10:29Z

Guenter at 16:09, 7 December 2022

2022-12-07T16:09:15Z

Guenter at 16:06, 7 December 2022

2022-12-07T16:06:36Z

Hwang at 15:10, 30 November 2022

2022-11-30T15:10:53Z

Hwang at 15:03, 30 November 2022

2022-11-30T15:03:57Z

Hwang at 15:03, 30 November 2022

2022-11-30T15:03:04Z

Hwang at 15:00, 30 November 2022

2022-11-30T15:00:00Z

Hwang at 14:57, 30 November 2022

2022-11-30T14:57:36Z

Hwang at 14:51, 30 November 2022

2022-11-30T14:51:41Z

Hwang at 14:50, 30 November 2022

2022-11-30T14:50:16Z

@@ Line 187: / Line 187: @@
 *This is given for the convolutional codes in detail only in the chapter&nbsp; [[Channel_Coding/Distance_Characteristics_and_Error_Probability_Bounds|"Distance Characteristics and Error Probability Bounds"]].
 <br clear=all>
-[[File:EN_KC_T_3_3_S4c.png|right|frame|Optimal convolutional codes of rate&nbsp; $1/2$ '''KORREKTUR: filter'''|class=fit]]
+[[File:EN_KC_T_3_3_S4cd_v3.png|right|frame|Optimal convolutional codes of rate&nbsp; $1/2$ |class=fit]]
 At the outset,&nbsp; just a few remarks:
 # &nbsp; The free distance&nbsp; $d_{\rm F}$&nbsp; increases with increasing memory&nbsp; $m$&nbsp; provided that one uses appropriate polynomials for the transfer function matrix&nbsp; $\mathbf{G}(D)$&nbsp;.

← Older revision		Revision as of 16:09, 7 December 2022
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	*the information sequence  $\underline{u} = (1, 1, 1, 0, 0, 0, 1, 0, 1, \ \text{. ..})$		*the information sequence  $\underline{u} = (1, 1, 1, 0, 0, 0, 1, 0, 1, \ \text{. ..})$
−	[[File:~~EN_KC_T_3_3_S3b_v2~~.png\|right\|frame\|Trellis diagram of a sample sequence ~~'''KORREKTUR: time, encoded sequence'''~~\|class=fit]]	+	[[File:EN_KC_T_3_3_S3b_v3.png\|right\|frame\|Trellis diagram of a sample sequence\|class=fit]]

@@ Line 36: / Line 36: @@
 {{GraueBox|TEXT=
 $\text{Example 1:}$&nbsp;  The graph shows for the convolutional encoder sketched above
-[[File:EN_KC_T_3_3_S1b_v2.png|right|frame|To illustrate the states&nbsp; $S_{\mu}$ '''KORREKTUR: time, state, encoded sequence'''|class=fit]]
+[[File:EN_KC_T_3_3_S1b_v3.png|right|frame|To illustrate the states&nbsp; $S_{\mu}$ |class=fit]]
 <br>
 *the information sequence&nbsp; $\underline{u} = (u_1,u_2, \text{...} \ ) $ <br>&nbsp; &#8658; &nbsp; information bits&nbsp; $u_i$,
@@ Line 67: / Line 67: @@
 <br>
 The graph shows the state transition diagram for the&nbsp; $(n = 2, \ k = 1, \ m = 2)$&nbsp; convolutional encoder.
-[[File:EN_KC_T_3_3_S2_v2.png|right|frame|State transition diagram&nbsp; $\rm A$&nbsp; for the&nbsp; $(n = 2, \ k = 1, \ m = 2)$&nbsp; convolutional encoder; <br> &nbsp; &nbsp; &nbsp; in the following:&nbsp; "our standard convolutional encoder" '''KORREKTUR: time, encoded sequence'''|class=fit]]
+[[File:EN_KC_T_3_3_S2_v3.png|right|frame|State transition diagram&nbsp; $\rm A$&nbsp; for the&nbsp; $(n = 2, \ k = 1, \ m = 2)$&nbsp; convolutional encoder; <br> &nbsp; &nbsp; &nbsp; in the following:&nbsp; "our standard convolutional encoder" |class=fit]]
 # &nbsp;This diagram  provides all information about this special convolutional encoder in compact form.

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 <u>Note:</u>
-*So each punctured bit in the received sequence&nbsp; $\underline{y}$&nbsp; $($marked by an underscore$)$# &nbsp; is replaced by an&nbsp; "Erasure"&nbsp; $\rm (E)$,&nbsp; see&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Erasure_Channel_.E2.80.93_BEC| "Binary Erasure Channel"]].
+*So each punctured bit in the received sequence&nbsp; $\underline{y}$&nbsp; $($marked by an underscore$)$ &nbsp; is replaced by an&nbsp; "Erasure"&nbsp; $\rm (E)$,&nbsp; see&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#Binary_Erasure_Channel_.E2.80.93_BEC| "Binary Erasure Channel"]].
 *Such an erasure created by the puncturing&nbsp;  is treated by the decoder in the same way as an erasure by the channel.

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 Using&nbsp; $\mathbf{G}(D) = (1 + D, \ 1 + D^2)$ &nbsp; &rArr; &nbsp; red change,&nbsp; the state transition diagram changes from the&nbsp; [[Channel_Coding/Code_Description_with_State_and_Trellis_Diagram#Representation_in_the_state_transition_diagram|$\text{state transition diagram $\rm A$}$]]&nbsp; little at first glance.&nbsp; However, the&nbsp; effects are serious:
 # &nbsp; The free distance is no longer&nbsp; $d_{\rm F} = 5$,&nbsp; but&nbsp; $d_{\rm F} = 3$,&nbsp; where the encoded sequence &nbsp; $\underline{x} = (11, \ 01, \ 00, \ 00, \text{. ..})$ &nbsp; with only three&nbsp; '''ones'''&nbsp; belonging to the information sequence &nbsp; $\underline{u} = \underline{1} = (1, \ 1, \ 1, \ 1, \ \text{...})$.<br>
-# &nbsp; That is:&nbsp; By only three transmission errors at positions&nbsp; '''1''',&nbsp; '''2''',&nbsp; and&nbsp; '''4''',&nbsp; one corrupts the&nbsp; "one sequence"&nbsp; $(\underline{1})$&nbsp; into the&nbsp; "zero sequence"&nbsp; $(\underline{0})$&nbsp; and thus produces infinitely many decoding errors in state&nbsp; $S_3$.<br>
+# &nbsp; That is:&nbsp; By only three transmission errors at positions&nbsp; '''1''',&nbsp; '''2''',&nbsp; and&nbsp; '''4''',&nbsp; one falsifies the&nbsp; "one sequence"&nbsp; $(\underline{1})$&nbsp; into the&nbsp; "zero sequence"&nbsp; $(\underline{0})$&nbsp; and thus produces infinitely many decoding errors in state&nbsp; $S_3$.<br>
 # &nbsp; A convolutional encoder with these properties is called&nbsp; "catastrophic".&nbsp; The reason for this extremely unfavorable behavior is that here the two polynomials &nbsp; $1 + D$ &nbsp; and &nbsp; $1 + D^2$ &nbsp; are not divisor-remote.}}<br>

← Older revision		Revision as of 15:00, 30 November 2022
Line 127:		Line 127:

	*the information sequence  $\underline{u} = (1, 1, 1, 0, 0, 0, 1, 0, 1, \ \text{. ..})$		*the information sequence  $\underline{u} = (1, 1, 1, 0, 0, 0, 1, 0, 1, \ \text{. ..})$
−	[[File:EN_KC_T_3_3_S3b_v2.png\|right\|frame\|Trellis diagram of a sample sequence\|class=fit]]	+	[[File:EN_KC_T_3_3_S3b_v2.png\|right\|frame\|Trellis diagram of a sample sequence '''KORREKTUR: time, encoded sequence'''\|class=fit]]

← Older revision		Revision as of 14:57, 30 November 2022
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	*Also the information sequence  $\underline{u}$  can be read from the selected trellis path based on the colors of the individual branches.		*Also the information sequence  $\underline{u}$  can be read from the selected trellis path based on the colors of the individual branches.

−	*A red branch stands for the information bit  $u_i = 0$,#   a blue one for  $u_i = 1$.<br>	+	*A red branch stands for the information bit  $u_i = 0$,   a blue one for  $u_i = 1$.<br>

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 *Similarly,&nbsp; the successor state&nbsp; $s_{i+1}$&nbsp; is determined by&nbsp; $s_i$&nbsp; and&nbsp; $u_i$&nbsp; alone.
-*These properties are considered in the so-called&nbsp; "state transition diagram"&nbsp; in the next section}}.<br>
+*These properties are considered in the so-called&nbsp; "state transition diagram"&nbsp; in the next section.}}<br>
 == Representation in the state transition diagram ==