Nets3(Error Detection) - โ€ซโ€ช2.4 Error Detection and...

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Unformatted text preview: โ€ซโ€ช2.4. Error Detection and Correction:โ€ฌโ€ฌ โ€ซ)ืžืงื‘ื™ืœ ืœืžืฆื’ืช โ€ช(13โ€ฌโ€ฌ โ€ซื‘ืคืจืงื™ื ืงื•ื“ืžื™ื ื“ื ื• ื‘ โ€ช bit errorsโ€ฌืฉื™ื›ื•ืœื™ื ืœืงืจื•ืช ื‘ื–ืžืŸ ืžืขื‘ืจ ื”ื ืชื•ื ื™ื ื‘ืจืฉืชโ€ช.โ€ฌโ€ฌ โ€ซื‘ืคืจืง ื–ื” ื ืขืกื•ืง ื‘ื–ื™ื”ื•ื™ ื”ืฉื’ื™ืื•ืช ื”ืืœื” )ื›ื›ื” ื ืงืจื ื‘ืขื‘ืจื™ืช ืœ โ€ช(Bit Errorsโ€ฌโ€ฌ โ€ซื•ื“ืจื›ื™ ื”ื˜ื™ืคื•ืœ ื‘ื”ืŸโ€ช.โ€ฌโ€ฌ โ€ซโ€ชโ€ฌโ€ฌ โ€ซื”ืฉื’ื™ืื•ืช ื™ื›ื•ืœื•ืช ืœื”ืชืจื—ืฉ ื‘ื›ืœ ืื—ื“ ืžื”ืงื•ื“ืงื•ื“ื™ืโ€ช ,โ€ฌื‘ืงื™ืฉื•ืจ ืื• ื‘ืื—ืชโ€ฌ โ€ซื”ืชื—ื ื•ืช ื‘ื“ืจืšโ€ช.โ€ฌโ€ฌ โ€ซื™ื“ื•ืข ืฉื‘ื“"ื› ื”ืฉื’ื™ืื•ืช ืงื•ืจื•ืช ื‘ืจืฆืคื™ื ืฉืœ ื‘ื™ื˜ื™ื ืœื ื ื›ื•ื ื™ื )โ€ช(burstsโ€ฌโ€ฌ โ€ซืื•ืœื ื”ืŸ ื™ื›ื•ืœื•ืช ืœื”ื™ื•ืช ื’ื ืฉืœ ื‘ื™ื˜ ื‘ื•ื“ื“โ€ช.โ€ฌโ€ฌ โ€ซืžืจื’ืข ืฉื–ื™ื”ื™ื ื• ืฉื’ื™ืืช ื”ืขื‘ืจื” ื™ืฉ ืœื ื• โ€ช 2โ€ฌืืคืฉืจื•ื™ื•ืชโ€ช:โ€ฌโ€ฌ โ€ซโ€ช .1โ€ฌืœื”ื—ื–ื™ืจ ืœืฉื•ืœื— ื”ื•ื“ืขื” ืขืœ ื”ืฉื’ื™ืื” ืขืœ ืžื ืช ืฉื™ืฉืœื— ืืช ื”ื ืชื•ื ื™ื ืžื—ื“ืฉโ€ช.โ€ฌโ€ฌ โ€ซโ€ช .2โ€ฌืœื”ืฉืชืžืฉ ื‘ืื—ื“ ืžื›ืžื” ืืœื’ื•ืจื™ืชืžื™ ืชื™ืงื•ืŸ ืฉื™ืคื•ืจื˜ื• ื‘ื”ืžืฉืšโ€ช.โ€ฌโ€ฌ โ€ซโ€ชโ€ฌโ€ฌ โ€ซืžื•ืฉื’ื™ืโ€ช:โ€ฌโ€ฌ โ€ซโ€ช โ€“ BER โ€“ Bit Error Rate oโ€ฌืจืžืช ื”ืกื‘ื™ืจื•ืช ืฉื‘ื™ื˜ ืžืกื•ื™ื ื™ืฉืชื ื”โ€ฌ โ€ซื‘ื”ืขื‘ืจื”โ€ช.โ€ฌโ€ฌ โ€ซโ€ช:PER โ€“ Packet Error Rate oโ€ฌโ€ฌ โ€ซโ€ชโ€ฌโ€ฌ โ€ซโ€ชโ€ฌโ€ฌ โ€ซโ€ชPER = 1 โ€“ (1 โ€“ BER)^Nโ€ฌโ€ฌ โ€ซโ€ชPER ~= N (BER) if N (BER) << 1โ€ฌโ€ฌ โ€ซโ€ช(e.g. N = 10^4, BER = 10^(-7โ€ฌโ€ฌ โ€ซโ€ช(PER = 10^(-3โ€ฌโ€ฌ โ€ซโ€ชoโ€ฌโ€ฌ โ€ซโ€ช:Hamming Distanceโ€ฌโ€ฌ โ€ซืžืงื“ื ื”ืžืฆื™ื™ืŸ ืืช ืžืกืคืจ ื”ื‘ื™ื˜ื™ื ื”ืฉื•ื ื™ื ื‘ื™ืŸ ืฉื ื™ ืงื˜ืขื™ ืžื™ื“ืขโ€ช.โ€ฌโ€ฌ โ€ซืœื“ื•ื’ืžื” โ€“ ื‘ื“ื•ื’ืžื” ื”ื–ื• โ€ช.HD = 3โ€ฌโ€ฌ โ€ซโ€ช10010101โ€ฌโ€ฌ โ€ซโ€ช10111001โ€ฌโ€ฌ โ€ซโ€ช00101100โ€ฌโ€ฌ โ€ซโ€ชโ€ฌโ€ฌ โ€ซื–ื™ื”ื•ื™ ืฉื’ื™ืื•ืช โ€“ ืจืขื™ื•ืŸ ื›ืœืœื™โ€ช:โ€ฌโ€ฌ โ€ซื”ืฉื™ื˜ื” ื”ืขื™ืงืจื™ืช ืœื–ื™ื”ื•ื™ ืฉื’ื™ืื•ืช ื”ื™ื ืœืฉืœื•ื— ืขื ื”ืžื™ื“ืข ื”ืžืงื•ืจื™ ืื™ื–ืฉื”ื•โ€ฌ โ€ซืžื™ื“ืข ืขื•ื“ืฃ ื”ืžื›ื™ืœ ื ืชื•ื ื™ื ื›ืœืฉื”ื ืขืœ ื”ืžื™ื“ืข ื”ื ืฉืœื— )โ€ชredundantโ€ฌโ€ฌ โ€ซโ€ช .(informationโ€ฌืžื™ื“ืข ืขื•ื“ืฃ ื–ื” ื™ืขื–ื•ืจ ืœื ื• ืœื–ื”ื•ืช ื˜ืขื•ื™ื•ืช ื‘ืงืœื˜ ืฉื”ื’ื™ืขโ€ช.โ€ฌโ€ฌ โ€ซื‘ืžืงืจื” ื”ืงื™ืฆื•ื ื™ ื ืฉืœื— ื›ืœ ื‘ื™ื˜ ืคืขืžื™ื™ืโ€ช:โ€ฌโ€ฌ โ€ซืขื‘ื•ืจ ื”ืงืœื˜ โ€ช 1011001โ€ฌื ืฉืœื— ืืช ื”ื”ื•ื“ืขื” โ€ช .11001111000011โ€ฌืื ื”ืžืงื‘ืœโ€ฌ โ€ซื™ื–ื”ื” ืื™ื–ืฉื”ื• ื‘ื™ื˜ ืฉืœื ื”ื’ื™ืข ื‘"ื–ื•ื’" ื”ื•ื ื™ื™ื“ืข ืฉื™ืฉ ืฉื’ื™ืื”โ€ช.โ€ฌโ€ฌ โ€ซื–ื”ื• ื›ืžื•ื‘ืŸ ืคื™ืชืจื•ืŸ ืžืกื•ืจื‘ืœ ื•ื™ืงืจ ื‘ื–ืžืŸ ืฉืœื™ื—ื” ื•ื–ื™ื›ืจื•ืŸ ื•ืœื ื™ื–ื”ื” ืฉื’ื™ืื”โ€ฌ โ€ซืฉื”ืชืจื—ืฉื” ื‘ืฉื ื™ ื‘ื™ื˜ื™ื ืจืฆื•ืคื™ื! ื›ืžื• ื›ืŸ ื”ื•ื ืœื ื™ื›ื•ืœ ืœืชืงืŸ ืฉื’ื™ืื” ืฉื”ื•ืโ€ฌ โ€ซื™ื–ื”ื” ื›ื™ ื”ื•ื ืœื ื™ื“ืข ืื™ื–ื” ื‘ื™ื˜ ื ื›ื•ืŸ ื•ืื™ื–ื” ืฉื’ื•ื™โ€ช.โ€ฌโ€ฌ โ€ซื ื™ืชืŸ ืœื›ืชื•ื‘ ื›ืœ ื‘ื™ื˜ โ€ช 3โ€ฌืคืขืžื™ื ื•ืื– ื‘ื–ื™ื”ื•ื™ ืฉืœืฉื” ืœื ื–ื”ื” ื ื“ืข ืฉื™ืฉ ื˜ืขื•ืชโ€ฌ โ€ซื•ืืฃ ื ื•ื›ืœ ืœืชืงืŸ ืื•ืชื” ื‘ืกื‘ื™ืจื•ืช ื˜ื•ื‘ื”โ€ช .โ€ฌื’ื ืฉื™ื˜ื” ื–ื• ื™ืงืจื” ืžืื•ื“โ€ช.โ€ฌโ€ฌ โ€ซืื ื• ื ืฆื™ื’ ืฉื™ื˜ื•ืช ื‘ื”ืŸ ื”ืžื™ื“ืข ื”ืขื•ื“ืฃ ื™ื”ื™ื” ืื•ืคื˜ื™ืžืœื™ โ€“ ื›ืœื•ืžืจ ื›ืžื” ืฉื™ื•ืชืจโ€ฌ โ€ซืงื˜ืŸโ€ช:โ€ฌโ€ฌ โ€ซโ€ช:One/Two Dimensional Parityโ€ฌโ€ฌ โ€ซื‘ืžื™ืžื“ ืื—ื“ ื”ืจืขื™ื•ืŸ ื”ื•ื ืœื”ื•ืกื™ืฃ ื‘ื™ื˜ ืื—ื“ ืœื›ืœ ื‘ื™ืช )โ€ช 7โ€ฌื‘ื™ื˜ื™ื( ื‘ืื•ืคืŸโ€ฌ โ€ซืฉื™ืฉืœื™ื ืœืžืกืคืจ ื–ื•ื’ื™ ืฉืœ ืื—ื“ื•ืชโ€ช .โ€ฌืฉื™ื˜ื” ื–ื• ืžืืคืฉืจืช ืœื ื• ืœื–ื”ื•ืช ื›ืœ ืžืกืคืจโ€ฌ โ€ซืื™ ื–ื•ื’ื™ ืฉืœ ืฉื’ื™ืื•ืช ื‘ื›ืœ ื‘ื™ืชโ€ช) .โ€ฌืื ื”ืžืงื‘ืœ ืจื•ืื” ืฉื™ืฉ ืœื• ืžืก ืื™ ื–ื•ื’ื™ ืฉืœโ€ฌ โ€ซืื—ื“ื•ืช ืื– ื”ื•ื ื™ื•ื“ืข ืฉื™ืฉ โ€ช ...1/3/5โ€ฌืฉื’ื™ืื•ืช ื‘ืžื™ื“ืขโ€ช(.โ€ฌโ€ฌ โ€ซืžืก ื–ื•ื’ื™ ืฉืœ ืฉื’ื™ืื™ื•ืช ืœื ื™ื–ื•ื”ื” ืžื›ื™ื•ื•ืŸ ืฉืžืก' ื”ืื—ื“ื•ืช ื™ื™ืฉืืจ ื–ื•ื’ื™โ€ฌ โ€ซื‘ืฉื ื™ ืžื™ืžื“ื™ื ืื ื—ื ื• ืžื•ืกื™ืคื™ื ื‘ื™ืช ื ื•ืกืฃ ืขืœ ื›ืœ โ€ช Xโ€ฌื‘ืชื™ืโ€ช .โ€ฌื‘ื‘ื™ืช ื–ื” ืื ื—ื ื•โ€ฌ โ€ซืจื•ืฉืžื™ื ื‘ื™ื˜ื™ื ื”ืžืฉืœื™ืžื™ื ืœืžืกืคืจ ื–ื•ื’ื™ ืฉืœ ืื—ื“ื•ืช ื‘ืกื›ื•ื ื”ืขืžื•ื“ื•ืชโ€ฌ โ€ซื‘ื‘ืชื™ื ืฉืœืคื ื™ื”ืโ€ช.โ€ฌโ€ฌ โ€ซโ€ช1โ€ฌโ€ฌ โ€ซโ€ช0โ€ฌโ€ฌ โ€ซโ€ช1โ€ฌโ€ฌ โ€ซโ€ช1โ€ฌโ€ฌ โ€ซโ€ช1โ€ฌโ€ฌ โ€ซโ€ช0โ€ฌ โ€ซโ€ช0โ€ฌโ€ฌ โ€ซโ€ช0101001โ€ฌโ€ฌ โ€ซโ€ช1101001โ€ฌโ€ฌ โ€ซโ€ช1011110โ€ฌโ€ฌ โ€ซโ€ช0001110โ€ฌโ€ฌ โ€ซโ€ช0110100โ€ฌโ€ฌ โ€ซโ€ช1011111โ€ฌโ€ฌ โ€ซโ€ชParity byte 1111011โ€ฌโ€ฌ โ€ซโ€ชDATAโ€ฌโ€ฌ โ€ซื‘ืฉื™ื˜ื” ื–ื• ื ื™ืชืŸ ื’ื ืœืชืงืŸ ืฉื’ื™ืื•ืช ืฉืœ ื‘ื™ื˜ ื‘ื•ื“ื“ ื›ื™ ื‘ืืžืฆืขื•ืช ื—ื™ืชื•ืš ืฉืœโ€ฌ โ€ซืขืžื•ื“ื” ื•ืฉื•ืจื” ืฉื’ื•ื™ื•ืช ื ื™ืชืŸ ืœื“ืขืช ืื™ื–ื” ื‘ื™ื˜ ืฉื’ื•ื™ ื•ืœืชืงืŸ ืื•ืชื•โ€ช.โ€ฌโ€ฌ โ€ซโ€ช) โ€“ Internet Checksum Algorithmโ€ฌืงื™ื™ื ื‘ืกืคืจ ืื‘ืœ ืœืโ€ฌ โ€ซื‘ืžืฆื’ืชโ€ช(.โ€ฌโ€ฌ โ€ซื”ืจืขื™ื•ืŸโ€ช :โ€ฌืกื•ื›ืžื™ื ืืช ื›ืœ ื”ืžื™ืœื™ื ืฉืจื•ืฆื™ื ืœื”ืขื‘ื™ืจ ื•ืžืขื‘ื™ืจื™ื ื’ื ืืช ื”ืชื•ืฆืื”โ€ฌ โ€ซ)ื”ืชื•ืฆืื” ืฉืœ ื—ื™ื‘ื•ืจ ื›ืœ ื”ืžื™ืœื™ื ื ืงืจืืช โ€ช .(checksumโ€ฌื”ืžืงื‘ืœ ืžื‘ืฆืข ืืช ืื•ืชื•โ€ฌ โ€ซื—ื™ืฉื•ื‘ ืขืœ ื”ืžื™ื“ืข ื•ืžืฉื•ื•ื” ืืช ื”ืชื•ืฆืื” ืฉืงื™ื‘ืœ ืœโ€ช .checksum-โ€ฌืื ืงื™ื™ืžืช ืฉื’ื™ืื”โ€ฌ โ€ซื‘ืžื™ื“ืข ืื• ืืคื™ืœื• ื‘โ€ช checksum-โ€ฌืฉื ืฉืœื—โ€ช ,โ€ฌืฉื’ื™ืื” ื–ื• ืชืชื’ืœื” ื›ื™ ื”ืชื•ืฆืื•ืช ื™ื”ื™ื•โ€ฌ โ€ซืฉื•ื ื•ืชโ€ช.โ€ฌโ€ฌ โ€ซืงื™ื™ืžื•ืช ื”ืžื•ืŸ ื“ืจื›ื™ื ืœืžื™ืžื•ืฉโ€ช ,โ€ฌื”ืกื›ืžื” ื‘ื” ื”ืื™ื ื˜ืจื ื˜ ืขื•ื‘ื“โ€ช:โ€ฌโ€ฌ โ€ซื”ืžื™ืœื™ื ื‘ืื•ืจืš โ€ช.bit16โ€ฌโ€ฌ โ€ซืžื—ื‘ืจื™ื ืื•ืชื ื™ื—ื“ ื‘ืฉื™ื˜ืช ื”ืžืฉืœื™ื ืœืื—ื“โ€ช.โ€ฌโ€ฌ โ€ซืชื–ื›ื•ืจืชโ€ช :โ€ฌื‘ืฉื™ื˜ืช ื”ืžืฉืœื™ื ืœืื—ื“ โ€“โ€ช xโ€ฌื ื—ืฉื‘ ื”ืžืฉืœื™ื ืฉืœ โ€ช xโ€ฌื›ืœื•ืžืจ ื›ืœ ื‘ื™ื˜ ืฉืœ โ€ชxโ€ฌโ€ฌ โ€ซืžืชื”ืคืšโ€ฌ โ€ซืื ื‘ื—ื™ื‘ื•ืจ ื”ื™ื” โ€ช carryโ€ฌื‘โ€ช msb-โ€ฌืื–ื™ ื”ื•ื ืžืชื•ื•ืกืฃ ืœืชื•ืฆืื”โ€ช.โ€ฌโ€ฌ โ€ซื“ื•ื’ืžืโ€ช (3-)+(5-) :โ€ฌื‘ื™ื™ืฆื•ื’ ืฉืœ โ€ช 4โ€ฌื‘ื™ื˜ื™ื )ืจืง ืœืฆื•ืจืš ื”ื“ื•ื’ืžื ื‘ืžืงืจื” ื”ื›ืœืœื™ ื–ื”โ€ฌ โ€ซโ€ช 16โ€ฌื‘ื™ื˜ื™ื(โ€ช.โ€ฌโ€ฌ โ€ซโ€ช1010=5- <- 0101=5+โ€ฌโ€ฌ โ€ซโ€ช1100=3- <- 0011=3+โ€ฌโ€ฌ โ€ซโ€ช-----โ€ฌโ€ฌโ€ซโ€ช 0110โ€ฌื•ื™ืฉ ืœื ื• โ€ช carryโ€ฌืฉืœ ืื—ื“โ€ฌ โ€ซโ€ช1โ€ฌโ€ฌ โ€ซโ€ช-----โ€ฌโ€ฌโ€ซโ€ช0111โ€ฌโ€ฌ โ€ซื ื‘ืฆืข ืžืฉืœื™ื ืœืื—ื“โ€ช 1000 :โ€ฌื•ืœื›ืŸ ื”ืžืก' ื”ื•ื โ€ช.-8โ€ฌโ€ฌ โ€ซ)ื›ืœ ื”ื ื•ืฉื ืฉืœ โ€ช checksumโ€ฌืœื ื ืžืฆื ื‘ืžืฆื’ื•ืช(โ€ช.โ€ฌโ€ฌ โ€ซโ€ช:CRC โ€“ Cyclic Redundancy Checkโ€ฌโ€ฌ โ€ซืคืจื•ื˜ื•ืงื•ืœ ืงืฆืช ืžื•ืจื›ื‘ ืื‘ืœ ืžืื•ื“ ื˜ื›ื ื™ ื‘ื™ืŸ ื”ืฉื•ืœื— ืœืžืงื‘ืœโ€ช.โ€ฌโ€ฌ โ€ซืžื•ืฉื’ื™ืโ€ช:โ€ฌโ€ฌ โ€ซโ€ช โ€“ (M(X oโ€ฌืคื•ืœื™ื ื•ื ืžื“ืจื’ื” โ€ช nโ€ฌื”ืžื™ื™ืฆื’ โ€ช n+1โ€ฌื‘ื™ื˜ื™ื ืžื”ืงืœื˜โ€ช .โ€ฌืืชโ€ฌ โ€ซื”ืคื•ืœื™ื ื•ื ื‘ื•ื ื™ื ืข"ื™ ื”ื›ืคืœื” ืฉืœ ื›ืœ ื‘ื™ื˜ ื‘ืขืจืš โ€ช 1โ€ฌื‘ื—ื–ืงืช ืžืงื“ืโ€ฌ โ€ซื”ื‘ื™ื˜ ื‘ืงืœื˜โ€ช .โ€ฌื“ื•ื’ืžื”โ€ช:โ€ฌโ€ฌ โ€ซโ€ชM ( x) X 7 X 4 X 3 X 1โ€ฌโ€ฌ โ€ซโ€ช10011010 โ€ฌโ€ฌ โ€ซโ€ช โ€“ (C(X oโ€ฌืžื—ืœืง ืคื•ืœื™ื ื•ืžื™ืืœื™ ืžื“ืจื’ื” โ€ช.kโ€ฌโ€ฌ โ€ซื™ืฉื ืŸ ืชื‘ื ื™ื•ืช ืงื‘ื•ืขื•ืช ืฉืœ ืžื—ืœืงื™ื ื›ืืœื” ื•ื›ืœ ืื—ืช ืžื”ืŸ ื‘ืขืœืชโ€ฌ โ€ซืชื›ื•ื ื•ืช ืžืกื•ื™ืžื•ืช ืฉื˜ื•ื‘ื•ืช ืœื–ื™ื”ื•ื™ ื˜ืขื•ื™ื•ืช ืžืกื•ื’ื™ื ืฉื•ื ื™ืโ€ช.โ€ฌโ€ฌ โ€ซืœื“ื•ื’ืžื” โ€“ ื” โ€ช Ethernetโ€ฌืžืฉืชืžืฉ ื‘ โ€ช (C(Xโ€ฌืžื“ืจื’ื” โ€ช.32โ€ฌโ€ฌ โ€ซื“ื•ื’ืžืื•ืช ื ื•ืกืคื•ืชโ€ช:โ€ฌโ€ฌ โ€ซโ€ชCRC-8:โ€ฌโ€ฌ โ€ซโ€ชx8 + x2 + x1 + 1โ€ฌโ€ฌ โ€ซโ€ชCRC-10:โ€ฌโ€ฌ โ€ซโ€ชx10 + x9 + x5 + x4 + x1 + 1โ€ฌโ€ฌ โ€ซโ€ชCRC-12:โ€ฌโ€ฌ โ€ซโ€ชx12 + x11 + x3 +x2 +1โ€ฌโ€ฌ โ€ซโ€ชCRC-16:โ€ฌโ€ฌ โ€ซโ€ชx16 + x15 + x2 +1โ€ฌโ€ฌ โ€ซโ€ชCRC-CCITT: x16 + x12 + x5 + 1โ€ฌโ€ฌ โ€ซโ€ชCRC-32:โ€ฌโ€ฌ โ€ซโ€ชx32 + x26 + x23 + x22 + x16 + x12โ€ฌโ€ฌ โ€ซโ€ช+ x11 + x10 + x8 + x7 + x5 + x4 + x2 + x + 1โ€ฌโ€ฌ โ€ซโ€ช โ€“ (T(X oโ€ฌื”ื”ื•ื“ืขื” ืื•ืชื” ืฉื•ืœื— ื”ืฉื•ืœื—โ€ช.โ€ฌโ€ฌ โ€ซื”ื•ื“ืขื” ื–ื• ื”ื™ื ื‘ื’ื•ื“ืœ )โ€ช n+1) + kโ€ฌื‘ื™ื˜โ€ช.โ€ฌโ€ฌ โ€ซโ€ช (T(Xโ€ฌืžืชื—ืœืง ื‘ื“ื™ื•ืง ื‘ืคื•ืœื™ื ื•ื โ€ช ,(C(Xโ€ฌื•ืžื›ื™ืœ ื‘ืชื•ื›ื• ืœืžืขืฉื” ืืชโ€ฌ โ€ซโ€ช ,(M(Xโ€ฌืžืžื ื” ื™ืคืขื ื— ื”ืžืงื‘ืœ ืืช ื”ื”ื•ื“ืขื” ื”ืžืงื•ืจื™ืชโ€ช.โ€ฌโ€ฌ โ€ซืชื–ื›ื•ืจืช โ€“ โ€ช (T(Xโ€ฌืžืชื—ืœืง ื‘ โ€ช (C(Xโ€ฌืื ื”ืžืขืœื” ืฉืœ โ€ช=< (T(Xโ€ฌโ€ฌ โ€ซืžื”ืžืขืœื” ืฉืœ โ€ช.(C(Xโ€ฌโ€ฌ โ€ซโ€ช โ€“ (R(X oโ€ฌื”ืฉืืจื™ืช ื”ืžืชืงื‘ืœืช ืžื—ืœื•ืงืช โ€ช (T(Xโ€ฌื‘ โ€ช) .(C(Xโ€ฌืื ื™ืฉ ื›ื–ื•(โ€ฌ โ€ซืžื•ืฆืื™ื ืื•ืชื” ื‘ืืžืฆืขื•ืช ื‘ื™ืฆื•ืข ืคืขื•ืœืช โ€ช XORโ€ฌื‘ื™ืŸ โ€ช (T(Xโ€ฌืœ โ€ช.(C(Xโ€ฌโ€ฌ โ€ซโ€ชC ( x) X 3 X 2 1โ€ฌโ€ฌ โ€ซโ€ช .1โ€ฌื™ืฆื™ืจืช ื”ื”ื•ื“ืขื” ืข"ื™ ื”ืฉื•ืœื—โ€ช:โ€ฌโ€ฌ โ€ซโ€ช(adds k zeros at the message end .1โ€ฌโ€ฌ โ€ซโ€ช kโ€ฌื”ื•ื ื”ืžืขืœื” ืฉืœ โ€ช.(C(Xโ€ฌโ€ฌ โ€ซโ€ชExampโ€ฌโ€ฌ โ€ซโ€ช3โ€ฌโ€ฌ โ€ซโ€ชTโ€ฌโ€ฌ โ€ซ(โ€ฌ โ€ซโ€ชxโ€ฌโ€ฌ โ€ซ)โ€ฌ โ€ซโ€ชXโ€ฌโ€ฌ โ€ซโ€ช1โ€ฌโ€ฌ โ€ซโ€ชleโ€ฌโ€ฌ โ€ซโ€ช(C(Xโ€ฌโ€ฌ โ€ซโ€ช011โ€ฌโ€ฌ โ€ซ~โ€ฌ โ€ซโ€ช(T(Xโ€ฌโ€ฌ โ€ซโ€ช0 0 11โ€ฌโ€ฌ โ€ซ~โ€ฌ โ€ซโ€ช0 1 10โ€ฌโ€ฌ โ€ซโ€ชXORโ€ฌโ€ฌ โ€ซโ€ช0โ€ฌโ€ฌ โ€ซ)โ€ช(T(X) = M(Xโ€ฌโ€ฌ โ€ซโ€ช:(Divide T(X) by C(X) and find the remainder R(X .2โ€ฌโ€ฌ โ€ซื”ืžืขืœื” ืฉืœ โ€ช (C(Xโ€ฌื”ื™ื โ€ช 3โ€ฌื•ืœื›ืŸ ืžื•ืกื™ืคื™ื โ€ช 3โ€ฌืืคืกื™ื ืžื™ืžื™ืŸ ืœ โ€ช(M(Xโ€ฌโ€ฌ โ€ซื•ื™ื•ืฆืจื™ื ืืช ื”ืจืฆืฃ ื”ืžื—ื•ืœืงโ€ช:โ€ฌโ€ฌ โ€ซโ€ช: (T(X)=T(X) โ€“ R(X .3โ€ฌโ€ฌ โ€ซืžื—ืกื™ืจื™ื ืืช ื”ืฉืืจื™ืช ื•ืžืงื‘ืœื™ื ืืช โ€ช (T(Xโ€ฌืžื•ื›ืŸ ืœืฉืœื™ื—ื”โ€ช.โ€ฌโ€ฌ โ€ซโ€ช (T(Xโ€ฌื”ื–ื” ืžืชื—ืœืง ื‘ื“ื™ื•ืง ื‘ โ€ช (C(Xโ€ฌื•ืžื›ื™ืœ ืืช โ€ช (M(Xโ€ฌืฉืืœื™ื• ืžืฉื•ืจืฉืจโ€ฌ โ€ซืžื™ืžื™ืŸ โ€ช.(R(Xโ€ฌโ€ฌ โ€ซโ€ช(2โ€ฌโ€ฌ โ€ซืงื‘ืœืช ื”ื”ื•ื“ืขื”โ€ช:โ€ฌโ€ฌ โ€ซื”ืžืงื‘ืœ ืžื›ื™ืจ ืืช โ€ช (C(Xโ€ฌื›ื™ ื”ื•ื ืงื‘ื•ืข ื•ืžื•ืกื›ื ืžืจืืฉโ€ช.โ€ฌโ€ฌ โ€ซื”ื•ื ืžื—ืœืง ืืช โ€ช (T(Xโ€ฌื‘โ€ช (C(X -โ€ฌื•ืžื•ืฆื ืืช โ€ช โ€“ (R(Xโ€ฌื”ืฉืืจื™ืชโ€ช.โ€ฌโ€ฌ โ€ซืื ื”ืฉืืจื™ืช ื”ื™ื โ€ช โ€“ 0โ€ฌืžืฉืžืข ืฉืื™ืŸ ืฉื’ื™ืื•ืช ื‘ื”ืขื‘ืจื”โ€ช.โ€ฌโ€ฌ โ€ซืื ื”ืฉืืจื™ืช ื”ื™ื ืœื โ€ช โ€“ 0โ€ฌืžืฉืžืข ืฉื™ืฉ ืฉื’ื™ืื•ืช ื‘ื”ืขื‘ืจื”โ€ช.โ€ฌโ€ฌ โ€ซืข"ืž ืœื”ื•ืฆื™ื ืžื”ื”ื•ื“ืขื” ืฉื”ืชืงื‘ืœื” ืืชื” ื”ื”ื•ื“ืขื” ื”ืžืงื•ืจื™ืช ืขืœ ื”ืฉื•ืœื—โ€ฌ โ€ซืœื”ืกื™ืจ ืืช โ€ช kโ€ฌื”ื‘ื™ื˜ื™ื ื”ื™ืžื ื™ื™ื )ืžื” ืฉื”ื™ื” โ€ช ,((R(Xโ€ฌื•ื”ื•ื ื ื•ืชืจ ืขื ื” โ€“ โ€ช(M(Xโ€ฌโ€ฌ โ€ซื”ืžืงื•ืจื™ โ€“ ื”ื”ื•ื“ืขื”!โ€ฌ โ€ซื–ื™ื”ื•ื™ ืกื•ื’ื™ื ืฉื•ื ื™ื ืฉืœ ื˜ืขื•ื™ื•ืชโ€ช:โ€ฌโ€ฌ โ€ซโ€ช:(T(X) + E(Xโ€ฌโ€ฌ โ€ซื ื™ืชืŸ ืœื—ืฉื•ื‘ ืขืœ ืงืœื˜ ืขื ืฉื’ื™ืื•ืช ื›ืขืœโ€ฌ โ€ซื›ืืฉืจ โ€ช (T(Xโ€ฌื”ื•ื ื”ื”ื•ื“ืขื” ื”ืžืงื•ืจื™ืช ื• โ€ช (E(Xโ€ฌื”ื•ื ื”ืฉื’ื™ืื”โ€ช:โ€ฌโ€ฌ โ€ซโ€ช E(X) = X^iโ€ฌื›ืœื•ืžืจ ื™ืฉ ืฉื’ื™ืื” ื‘ืžืงื•ื ื” โ€ช.iโ€ฌโ€ฌ โ€ซื”ื“ืจืš ื”ื™ื—ื™ื“ื” ืฉื”ืฉื’ื™ืื” ืชืขื‘ื•ืจ ืืช ื”ื‘ื“ื™ืงื” ืฉืœ โ€ช CRCโ€ฌื”ื™ื ืื โ€ช (E(Xโ€ฌื™ืชื—ืœืง ื‘โ€ฌ โ€ซโ€ช (C(Xโ€ฌืœืœื ืฉืืจื™ืชโ€ช .โ€ฌืœื›ืŸ ืขืœื™ื ื• ืœืžืฆื•ื โ€ช (C(Xโ€ฌื›ื–ื” ืฉืœื ืกื‘ื™ืจ ืฉื™ื—ืœืง ืกื•ื’ื™ืโ€ฌ โ€ซืฉื›ื™ื—ื™ื ืฉืœ โ€ช.(E(Xโ€ฌโ€ฌ โ€ซโ€ชโ€ฌโ€ฌ โ€ซโ€ชโ€ฌโ€ฌ โ€ซโ€ชโ€ฌโ€ฌ โ€ซโ€ชโ€ฌโ€ฌ โ€ซโ€ช:Single Bit Errorโ€ฌโ€ฌ โ€ซื™ืชื’ืœื” ืข"ื™ โ€ช (C(Xโ€ฌืฉื‘ื• ื™ืฉ ืžืงื“ื โ€ช 1โ€ฌื‘ื”ืชื—ืœื” ื•โ€ช 1-โ€ฌื‘ืกื•ืฃ ื‘ื˜ื•ื— ืœื ื™ื—ืœืง โ€ช(E(Xโ€ฌโ€ฌ โ€ซืขื โ€ช.single bit errorโ€ฌโ€ฌ โ€ซโ€ช:Double Bit Errorโ€ฌโ€ฌ โ€ซื™ืชื’ืœื” ืข"ื™ โ€ช (C(Xโ€ฌืฉื‘ื• ื™ืฉ โ€ช 3โ€ฌืชื ืื™ืโ€ช(terms factor 3) .โ€ฌโ€ฌ โ€ซื›ืœ ืžืก ืœื ื–ื•ื’ื™ ืฉืœ ืฉื’ื™ืื•ืช ื™ืชื’ืœื” ืข"ื™ โ€ช (C(Xโ€ฌืขื โ€ช x+1โ€ฌื‘ืกื•ืฃโ€ช.โ€ฌโ€ฌ โ€ซื›ืœ ืจืฆืฃ ืฉืœ ืฉื’ื™ืื•ืช ื‘ืื•ืจืš ืงื˜ืŸ ืžโ€ช k-โ€ฌื™ืชื’ืœื” ืข"ื™ โ€ช (C(Xโ€ฌืขื ื“ืจื’ื” โ€ช.kโ€ฌโ€ฌ โ€ซืชื™ืงื•ืŸ ืฉื’ื™ืื•ืช โ€“ โ€ชCorrectionโ€ฌโ€ฌ โ€ซโ€ช:Errorโ€ฌโ€ฌ โ€ซโ€ช โ€ฌืœื–ื”ื•ืช ืื• ืœืชืงืŸ?โ€ฌ โ€ซืจืื™ื ื• ืฉืื ืื ื—ื ื• ืจืง ืžื–ื”ื™ื ืฉื’ื™ืื•ืช ื•ืœื ืžืชืงื ื™ื ืื•ืชืŸ ืื ื—ื ื• ื™ื›ื•ืœื™ืโ€ฌ โ€ซืœื”ืกืชืคืง ื‘ืฉืœื™ื—ืช ืคื—ื•ืช ืžื™ื“ืขโ€ช ,โ€ฌื•ื›ืŸ ืœื”ืฉืชืžืฉ ื‘ืชื”ืœื™ื›ื™ื ืคืฉื•ื˜ื™ื ื™ื•ืชืจโ€ช.โ€ฌโ€ฌ โ€ซืชื™ืงื•ืŸ ื”ืžื™ื“ืข ืœืขื•ืžืช ื–ืืช ื™ื—ืกื•ืš ืœื ื• ืฉืœื™ื—ื•ืช ื—ื•ื–ืจื•ืช ืฉืœ ืžื™ื“ืข ืฉื ื™ื–ื•ืงโ€ช.โ€ฌโ€ฌ โ€ซืื ื›ืš ืžื” ืขื“ื™ืฃ? ืœืชืงืŸ ืื• ืคืฉื•ื˜ ืœืฉืœื•ื— ืžื—ื“ืฉ?โ€ฌ โ€ซืจื•ื‘ ื”ืจืฉืชื•ืช ื”ืงื™ื™ืžื•ืช ื›ื™ื•ื ืžื‘ืฆืขื•ืช ื–ื™ื”ื•ื™ ืฉืœ ืฉื’ื™ืื•ืช ืœืœื ืชื™ืงื•ืŸ ืฉืœื”ืŸโ€ช.โ€ฌโ€ฌ โ€ซื‘ืขืžื•ื“ ื”ื‘ื ื ืจืื” ื“ื•ื’ืžื” ื”ืžืจืื” ืœืžื” ืขื“ื™ืฃ ืœื ืœืชืงืŸโ€ช...โ€ฌโ€ฌ ...
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    As a current student on this bumpy collegiate pathway, I stumbled upon Course Hero, where I can find study resources for nearly all my courses, get online help from tutors 24/7, and even share my old projects, papers, and lecture notes with other students.

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    Kiran Temple University Fox School of Business โ€˜17, Course Hero Intern

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    I cannot even describe how much Course Hero helped me this summer. Itโ€™s truly become something I can always rely on and help me. In the end, I was not only able to survive summer classes, but I was able to thrive thanks to Course Hero.

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    Dana University of Pennsylvania โ€˜17, Course Hero Intern

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    The ability to access any universityโ€™s resources through Course Hero proved invaluable in my case. I was behind on Tulane coursework and actually used UCLAโ€™s materials to help me move forward and get everything together on time.

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    Jill Tulane University โ€˜16, Course Hero Intern