Channel Coding/Error Probability and Application Areas - Revision history

Guenter at 13:51, 13 December 2022

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@@ Line 55: / Line 55: @@
 <br>
 The prerequisites for the following calculation of block error probability of a Reed&ndash;Solomon coding system and conversion to binary symbols are summarized in the diagram:
-[[File:EN_KC_T_2_6_S2a_neu_v3.png|right|frame|Application of Reed-Solomon coding for binary channels '''Korrektur'''|class=fit]]
+[[File:EN_KC_T_2_6_S2a_neu_v3.png|right|frame|Application of Reed-Solomon coding for binary channels.&nbsp; at point<br>'''(1)''' &nbsp;$k$&ndash;bit symbols; &nbsp;  '''(2)''' &nbsp;$n$&ndash;bit symbols; &nbsp; '''(3)''' &nbsp;$n$&ndash;bit symbols;  &nbsp; '''(4)''' &nbsp;$(n \cdot m)$&ndash;bit blocks;|class=fit]]
 # Assume&nbsp; $(n,\ k)$ Reed&ndash;Solomon coding with symbols of&nbsp; $c_i &#8712; {\rm GF}(2^m)$.&nbsp; The smaller the code rate&nbsp; $R=k/n$,&nbsp; the less information can be transmitted at a fixed data rate.<br>

@@ Line 55: / Line 55: @@
 <br>
 The prerequisites for the following calculation of block error probability of a Reed&ndash;Solomon coding system and conversion to binary symbols are summarized in the diagram:
-[[File:EN_KC_T_2_6_S2b_neu_v3.png|right|frame|Application of Reed-Solomon coding for binary channels '''Korrektur'''|class=fit]]
+[[File:EN_KC_T_2_6_S2a_neu_v3.png|right|frame|Application of Reed-Solomon coding for binary channels '''Korrektur'''|class=fit]]
 # Assume&nbsp; $(n,\ k)$ Reed&ndash;Solomon coding with symbols of&nbsp; $c_i &#8712; {\rm GF}(2^m)$.&nbsp; The smaller the code rate&nbsp; $R=k/n$,&nbsp; the less information can be transmitted at a fixed data rate.<br>

@@ Line 55: / Line 55: @@
 <br>
 The prerequisites for the following calculation of block error probability of a Reed&ndash;Solomon coding system and conversion to binary symbols are summarized in the diagram:
-[[File:EN_KC_T_2_6_S2b_neu.png|right|frame|Application of Reed-Solomon coding for binary channels '''Korrektur'''|class=fit]]
+[[File:EN_KC_T_2_6_S2b_neu_v3.png|right|frame|Application of Reed-Solomon coding for binary channels '''Korrektur'''|class=fit]]
 # Assume&nbsp; $(n,\ k)$ Reed&ndash;Solomon coding with symbols of&nbsp; $c_i &#8712; {\rm GF}(2^m)$.&nbsp; The smaller the code rate&nbsp; $R=k/n$,&nbsp; the less information can be transmitted at a fixed data rate.<br>

@@ Line 55: / Line 55: @@
 <br>
 The prerequisites for the following calculation of block error probability of a Reed&ndash;Solomon coding system and conversion to binary symbols are summarized in the diagram:
-[[File:P ID2565 KC T 2 6 S2a v2.png|right|frame|Application of Reed-Solomon coding for binary channels '''Korrektur'''|class=fit]]
+[[File:EN_KC_T_2_6_S2b_neu.png|right|frame|Application of Reed-Solomon coding for binary channels '''Korrektur'''|class=fit]]
 # Assume&nbsp; $(n,\ k)$ Reed&ndash;Solomon coding with symbols of&nbsp; $c_i &#8712; {\rm GF}(2^m)$.&nbsp; The smaller the code rate&nbsp; $R=k/n$,&nbsp; the less information can be transmitted at a fixed data rate.<br>

@@ Line 141: / Line 141: @@
 {{BlaueBox|TEXT=
 $\text{Conclusion:}$&nbsp;
-*Reed&ndash;Solomon codes are not very good at the memoryless AWGN channel.&nbsp;  Both codes are more than&nbsp; $4 \, \rm dB$&nbsp; away from the information-theoretical&nbsp; [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding#AWGN_channel_capacity_for_binary_input_signals| $\text{Shannon limit }$ '''KORREKTUR: bound''']].<br>
+*Reed&ndash;Solomon codes are not very good at the memoryless AWGN channel.&nbsp;  Both codes are more than&nbsp; $4 \, \rm dB$&nbsp; away from the information-theoretical&nbsp; [[Channel_Coding/Information_Theoretical_Limits_of_Channel_Coding#AWGN_channel_capacity_for_binary_input_signals| $\text{Shannon bound }$ ]].<br>
 *But Reed&ndash;Solomon codes are very effective with so-called&nbsp; [[Digital_Signal_Transmission/Burst_Error_Channels| $\text{Burst Error Channels}$]].&nbsp; They are therefore mainly used for fading channels,&nbsp; memory systems,&nbsp; etc.<br>

@@ Line 154: / Line 154: @@
 *The inner code is almost always a binary code and in satellite&ndash; and mobile communications often a convolutional code.
-*If one wishes to study only Reed&ndash;Solomon coding/decoding, one replaces the grayed components in a simulation with a single block called a&nbsp; "super channel".<br>
+*If one wishes to study only Reed&ndash;Solomon encoding/decoding, one replaces the grayed components in a simulation with a single block called a&nbsp; "super channel".<br>
 *Concatenated coding systems are very efficient if an interleaver is connected between the two encoders to further relax burst errors.<br>

@@ Line 104: / Line 104: @@
 Shown are the calculated curves &nbsp; ${ \rm Pr}( \underline{v} \ne \underline{u})$ &nbsp; for two different Reed&ndash;Solomon codes corresponding to the&nbsp; "Deep Space Standards"&nbsp; according to&nbsp; $\rm CCSDS$&nbsp; $($"Consultative Committee for Space Data Systems"$)$,&nbsp; namely
-[[File:EN_KC_T_2_6_S2b.png|right|frame|Block error probabilities of two Reed-Solomon codes of length&nbsp; $n = 255$ <br><u>Hint:</u> &nbsp; You are to perform completely the calculation in the&nbsp; [[Aufgaben:Exercise_2.15:_Block_Error_Probability_with_AWGN|"Exercise 2.15"]]&nbsp; for the&nbsp; $\text{RSC (7, 3, 5)}_{8}$&nbsp; &ndash; thus for somewhat clearer parameters.|class=fit]]
+[[File:EN_KC_T_2_6_S2b.png|right|frame|Block error probabilities of two Reed-Solomon codes of length&nbsp; $n = 255$ <br><u>Note:</u> &nbsp; You are to perform completely the calculation in the&nbsp; [[Aufgaben:Exercise_2.15:_Block_Error_Probability_with_AWGN|"Exercise 2.15"]]&nbsp; for the&nbsp; $\text{RSC (7, 3, 5)}_{8}$&nbsp; &ndash; thus for somewhat clearer parameters.|class=fit]]
 # the&nbsp; $\text{RSC (255, 239, 17)}_{256}$&nbsp; with code rate&nbsp; $R=0.9373$,&nbsp; which can correct up to&nbsp; $t = 8$&nbsp; errors,&nbsp; and<br>

@@ Line 59: / Line 59: @@
 # Assume&nbsp; $(n,\ k)$ Reed&ndash;Solomon coding with symbols of&nbsp; $c_i &#8712; {\rm GF}(2^m)$.&nbsp; The smaller the code rate&nbsp; $R=k/n$,&nbsp; the less information can be transmitted at a fixed data rate.<br>
 # Each symbol is represented by&nbsp; $m$&nbsp; bits &nbsp; &#8658; &nbsp; "binary mapping".&nbsp; A block&nbsp; $($code word &nbsp;$\underline {c} )$&nbsp; thus consists of&nbsp; $n$&nbsp; symbols or of&nbsp; $n \cdot m$&nbsp; binary characters&nbsp; $($bits$)$&nbsp; combined in&nbsp; $\underline {c}_{\rm bin} $.<br>
-# In addition,&nbsp; the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_binary_input|$\text{AWGN Channel}$]],&nbsp; identified by the parameter&nbsp; $E_{\rm B}/N_0 $ is assumed.&nbsp; According to this channel model the transmission happens bipolar: &nbsp; "0" &nbsp; &#8596; &nbsp; $+1$&nbsp; and&nbsp; "1" &nbsp; &#8596; $-1$.
+# In addition,&nbsp; the&nbsp; [[Channel_Coding/Channel_Models_and_Decision_Structures#AWGN_channel_at_binary_input|$\text{AWGN Channel}$]],&nbsp; identified by the parameter&nbsp; $E_{\rm B}/N_0 $ is assumed.&nbsp; According to this channel model the transmission is bipolar: &nbsp; "0" &nbsp; &#8596; &nbsp; $+1$&nbsp; and&nbsp; "1" &nbsp; &#8596; $-1$.
 # The receiver makes hard decisions on bit level.&nbsp; Before decoding including error correction,&nbsp; the binary symbols are converted back to&nbsp; ${\rm GF}(2^m)$&nbsp; symbols.<br>