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Error control coding for wireless communication technologies

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This paper outlines the design and implementation of error control codes, specifically focusing on Hamming codes, which effectively correct single errors in communication systems. We explore the construction of the parity check matrix and generator matrix, detailing the methodologies for syndromedecoding and matching systems. Discussion includes code parameters, error probability analysis in communication channels, and practical implementations of Hamming codes, as mandated by the EU-USA Atlantis Programme and Budapest University of Technology and Economics.

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Error control coding for wireless communication technologies

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  1. Errorcontrolcodingforwireless communicationtechnologies Backgroundmaterialfor Hamming codes EU-USA Atlantis Programme FIT & Budapest University of Technology and Economics

  2. Objective Design a codewhichcancorrecteverysingleerror. Motivation: Ifthechannel is goodthenitisenoughtohave a limited errorcorrectingcapability.

  3. Hamming codes Capable of correctingeverysingleerror, theyareperfectcodes: Construction of C(n,k) Hamming code: Constructthecolumnvectors of theparitycheckmatrixH byfulfillingthatallcolumnvectors must be differentformeachother and none of themcan be theallzerovector Constructthegeneratormatrix Design thematchinggatessyndromedecoding Implementthefullscheme

  4. The C(7,4) Hamming code Constructing the parity check matrix H Constructing the generator matrix G

  5. Step 2: Constructing the parity check matrix H Step 3: Constructing the generator matrix G

  6. Constructingthe matching system for decoding Match (1)/Mismatch (0) Matching systems Match (1)/Mismatch (0)

  7. E.g. Match (1)/Mismatch (0) Matching systems Match (1)/Mismatch (0)

  8. E.g. Match (1)/Mismatch (0) Matching systems Match (1)/Mismatch (0)

  9. E.g. Match (1)/Mismatch (0) Matching systems Match (1)/Mismatch (0)

  10. E.g. Match (1)/Mismatch (0) Matching systems Match (1)/Mismatch (0)

  11. E.g. Match (1)/Mismatch (0) Matching systems Match (1)/Mismatch (0)

  12. E.g. Match (1)/Mismatch (0) Matching systems Match (1)/Mismatch (0)

  13. E.g. Match (1)/Mismatch (0) Matching systems Match (1)/Mismatch (0)

  14. Implementation Mathcing system Mathcing system e u v s Mathcing system H G BSC with e Mathcing system u c Trunc Mathcing system Mathcing system Mathcing system

  15. The coding scheme 00100 100 01011 01111 01111 01 00100 01 Trunc

  16. Bit errorprobabilityanalysis BSC’ u' u Array off masking gates or LUT u H G Trunc Given a BSC with it tells us how to choose n and k parameters to fulfill a given BSC with

  17. The modifiedbit-errorprobabilityinthecaseas a function of codeparametersinthecaseof P_b=0.01 Lossindataspeed: 1/3; 4/7; 11/15; 57/63; 120/127; 247/255

  18. The modifiedbit-errorprobabilityinthecaseas a function of codeparametersinthecaseofP_b=0.001 Lossindataspeed: 1/3; 4/7; 11/15; 57/63; 120/127; 247/255

  19. Code design from the point of communication engineering Given a BSC with and a required level of QoS , design a code which can achieve 1. Evaluate if n-k is too large or if then there is no solution with correcting only single errors (then you need a more powerful code capable of correcting more than a single error) 2. Construct the parity check matrix obeying the rules: (i) each column vector is different; (ii) none of the column vector is the all-zero vector; (iii) the code is systematic 3. Implement the coding scheme Array of masking gates or LUT u H G Trunc BSC with

  20. Design an errorcorrectingcodefor a BSC (BER=0.01) toachieve BER’=0.00001 Step 1: indetifying the code parameters n=7, k=4

  21. Homeworkassignment Design thecodingscheme of a linearbinarycodewithgeneratormatrix Determinethecodewords GivetheparitycheckmatrixH Determinetheerrorgroups (givethe Standard Array of thecode) Determinethesyndromedecodingtable Evaluatetheprobability of theerrorvectorsoccuringinthesyndromedecodingtable Givethescheme of thesystem Describeallthebinaryvectorsintheschemewhensendingallthepossiblemessagevectors (ineachcaseslettheerrorvectorhave 1 inthefirstcomponent and rest of thecomponentsarezeros).

  22. Helpforthehomework

  23. Helpforthehomework Pick areknown

  24. Suggestedreadings D. Costello: Errorcontrolcodes, Wiley, 2005, Chapter 3

  25. ExpectedQuizquestion Given a generator matrix of a linear binary systematic code, determine the error group belonging to a given error vector Give the parity check matrix of the C(15,11) Hamming code

  26. Thank you for your attention !

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