1. Error Detection Neil Tang 9/26/2008
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2. Outline Basic Idea Objectives Two-Dimensional Parity
Checksum Algorithm
Cyclic Redundancy Check (CRC)
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3. Basic Idea
Add redundant information to a frame that can be used to determine if errors have been introduced.
Sender: It applies the algorithm to the message to generate the redundant bits and then transmits the original message along with the extra bits.
Receiver: It applies the same algorithm on the received message to come up with a result then compares it with the expected results.
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4. Basic Idea Trivial Solution: always transmitting two complete copies.
Real Solutions: k redundant bits for n-bit messages, k<<n, e.g., CRC, k = 32bit, n = 12,000bits.
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5. Objectives Maximize the probability of detecting errors.
Minimize the number of redundant bits.
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6. Two-Dimensional Parity
It can catch 1-, 2-, 3- and most 4-bits errors. Why?
14 redundant bits for 42-bit message. Good enough?
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7. Checksum Algorithm
Basic Ideas: The checksum is obtained by adding all words in the original message.
Internet Checksum Algorithms: Add 16-bit subsequences together using the ones complement arithmetic and then take the ones complement of the result as the checksum.
E.g., +)
1000 +)
1001 →0110
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8. Checksum Algorithm Cost Effective: 16 bits for a message of any length
Inefficiency for Error Detection: a pair of single-bit errors may cause trouble.
Easy Implementation: Several lines of codes
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9. CRC
(n+1) bit message can be represented as a polynomial of degree n. E.g., → M(x) = 1x7+ 1x4+1x3+1x1
The sender and the receiver agree on a divisor polynomial C(x) with a degree of k, e.g., C(x)=x3+x2+1. C(x) is usually specified by the standards, e.g., in Ethernet, k=32, CRC-32.
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10. Polynomial Arithmetic
B(x) can be divided by C(x) if same degree
The remainder can be obtained by performing XOR on each pair of matching coefficients. E.g., (x3+1) can be divided by (x3+x2+1) and the remainder is x2. How? 1001 XOR 1101 = 0100
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11. Algorithm to Obtain CRC
Multiply M(x) by xk , i.e., attach k 0s at the end of the message. Call this extended message T(x).
Divide T(x) by C(x) and find the remainder (CRC).
Attach CRC to M(x) and send the new message.
Generator C(x)
1101
Message T(x)
1001
1000
1011
1100
101
Remainder
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12. CRC Algorithm
Sender: It applies the algorithm to obtain CRC (remainder). Attach CRC to the end of the original message (e.g., ) and send it.
Receiver: divides the received polynomial by C(x). If 0, no error; otherwise, corrupted.
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13. Obtain C(x)
Basic Idea: Select C(x) so that it is very unlikely to divide evenly into a message with errors.
Frequently used C(X) in Table 2.5
Provable Results:
- Single bit error detectable – the coefficients of the first and last term
of C(x) are not 0.
- Double bit error detectable – C(x) contain at least three 1
coefficients
- Odd number of errors detectable - C(x) contains the factor (x+1)
- Any “burst” (consecutive) error detectable – its length less than k
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14. CRC Algorithm Cost Effective Efficient for Error Detection
Easy Implementation
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15. Error Detection Vs. Error Correction
Error Correction is used only if
Errors happen frequently: wireless environment
The cost of retransmission is too high: satellite link, acoustic link.
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