chapter 2 error-detecting codes. outline 2.1 why error-detecting codes? 2.2 simple parity checks 2.3...
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Outline
• 2.1 Why Error-Detecting Codes?• 2.2 Simple Parity Checks• 2.3 Error-Detecting Codes• 2.4 Independent Errors: White Noise• 2.5 Retransmission of Message• 2.6 Simple Burst Error-Detecting Codes• 2.7 Weighted Codes• 2.8 Review of Modular Arithmetic• 2.9 ISBN Book Numbers
2.1 Why Error-Detecting Codes?
sender receiverchannel
memory(channel)
write read
• We require very reliable transmission through the channel, whether it be through space when signaling from here to there (transmission), or through time when signaling from now to then (storage).
• Experience shows that it is not easy to build equipment that is highly reliable.
• If repetition is possible, then it is frequently sufficient merely to detect the presence of an error.
• It is possible to catch error only if there are some restrictions on what is a proper message.
• The problem is to keep these restrictions on the possible messages down to ones that are simple.
Encode Decode
feedback (only detected)
2.2 Simple Parity Checks
• (n-1) message + nth parity-check position
m1m2 ... mn-1p
even-parity check: decide p to make the number of 1’s in the message even
odd-parity check: decide p to make the number of 1’s in the message odd
• A single error or odd number of errors can be detected.
• A double error cannot be detected. Nor can any even number of errors be detected.
• Assumption– (1)The probability of an error in any one binary
position is a definite number p.– (2)Error in different positions are independent.
• Then– The probability of a single error is – The probability of a double error is
• Optimal length of message to be checked depends on both the reliability desired.
1)1( npnp
22 )1(2
)1( nppnn
2.3 Error-Detecting Codes
• a long message is
n -1 digits + 1 digit n digits • Redundancy: the number of binary digits actually
used divided by the minimum necessary.
• The excess redundancy is 1 / (n - 1).– For low redundancy use long message.– For high reliability use short messages
1
11
1
nn
n
2.4 Independent Errors: White Noise
• White Noise– (1) an equal probability p of an error in each
position.– (2) an independence of error in different
positions.
• Burst error– Errors occur in successive position.
• For white noise: no error : (1 – p)n
1 error : np(1 - p) n -1
2 error : even number of errors:
The probability of no errors is the first term (m=0)of the
series.
22 )1(2
)1( nppnn
]2/[
0
22
0
0
)1()2,(2
)21(1
)1(),()1(])1[(
)1(),(])1[(1
n
m
mnmn
n
k
knkk
n
k
knkn
ppmnCp
ppknCnpp
ppknCpp
2.6 Simple Burst Error-Detecting Codes
• Assumed that any burst length k was.
(0 ≤ k ≤ L)
lmmm 21
k
L
2.7 Weighted Codes
• People have a tendency to interchange adjacent digits of number; for example, 67 becomes 76 or 667 becomes 677.
• How to overcome these human errors, and we can detect easily.
• A rather frequent situation is to have an alphabet, plus space, plus the 10 decimal digits as the complete set of symbols to be used. This amounts to 26+1+10=37 symbols in the sending message.
• We weight the symbols with weights 1, 2, 3, . . . beginning with the check digit of the message.
m1 m2 m3 . .
w1=1 w2=2 w3=3 . ..• We reduce the sum modulo 37 so that a
check symbol can selected that will make the sum 0 modulo 37.
• If there are interchanged digits, their sums will different from original sums.
• If the interchanged digits are the kth and (k+1)st
kkkk
kk
kk
SSSS
KSSk
SKKS
11
1
1
otherwise,0)2()1(
)2()1(
)1()1(
2.8 Review of Modular Arithmetic
XOR AND
011
101
110
100
111
001
010
000
mbaba
mbb
maa
mod''
mod'
mod'