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An overview of various error detection techniques including parity, check sum, crc (cyclic redundancy check) and hamming code. Topics covered include the concept of redundant bits, 2-d parity, check sum algorithm, frame error estimation, and crc algorithm. Each technique is explained with examples and their respective advantages and limitations.
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simple case^ • two copies of data^ • receiver compares copies ‘equal’ then no error.^ • probability of same bits corrupted low.
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parity byte
parity bits
Example: 16 bit integers –treat data as 16 bit integers
Add using 16 bit one’s complement.
take one’s complement of result
Frame Error: A probabilistic
Estimate
10000
4 p(1-p)
99999
(^10000)
Agreed upon polynomial C(x), degree k
Message exchanged:
M(x) + k bits = P(x)
Make P(x) exactly divisible by C(x).
If no errors at receiver
P(x) / C(x) – zero remainder
=> no errors
B(x) of degree > C(x)
B(x) divisible by C(x)
B(x) of degree = C(x) => B(x) divisible once by C(x)
B(x) – C(x) = remainder
subtract C(x) from B(x)
EXOR on matching pair of coefficients.
k
equivalent to adding
k
zeros
example: M(x) =
1000, C(x) of degree 2
x
3
*** x**
2
= x
5
= T(x) (10000)
D(x) is exactly divisible by C(x)
10001010 00100 - Remainder
101010000 – Message padded with 3 zeros000000100 -- Remainder101010100 – Message xored with remainder
Example:
Hamming distance = 5
Example:
If 0000000111 received
- has to be 0000011111
provided double bit errors.
code
