Error Detection and Correction Techniques in Digital Communications, Slides of Computer Networks

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• Lecture No.

2

Error Detection and Correction

• Possible binary voltage encoding symbol
• Neighborhoods and erasure region

voltage

? (erasure)

• Possible QAM symbol
• Neighborhoods in green
• All other space results in erasure Input to digital level: valid symbols or erasures

4

Error Detection: How?

• We want only k redundant bits for an

n-bit message, where k < < n

• In Ethernet, 32-bit CRC for 12,000 bits (1500 bytes)
• k bits are derived from the original

message

• Both the sender and receiver know the

algorithm

5

1-bit Error Detection with Parity

• Every code has even number of 1’s

Valid Codes Parity encoding: gray dots are invalid and indicate errors

If only 1 bit flips, it can be detected

7

2-bit Erasure Correction With Voting

• Every code is copied three times

Remaining bit in a 2-erasure plane is not ambiguous

Cannot correct 1-error and 1-erasure

8

Hamming Distance (1950 Paper)

• Minimum number of bit flips between

code words

• 2 flips for parity
• 3 flips for voting
• n-bit error detection
• No code word changed into another code word
• Requires Hamming distance of n+

10

Hamming Codes (1950 Paper)

• Construction for 1-bit error-correcting codes
• Minimal number of check bits required
• Construction
  • Number of bits from 1 upward
  • Powers of 2 are check bits
  • All others are data bits
  • Check bit j is XOR of all bits k such that (j

AND k) = j

• – Example: 4 bits of data, 3 check bits Docsity.com

11

Error Bits or Error Burst

• Common model of errors
  • Probability of error per bit
  • Error in each bit independent of others
  • Value of incorrect bit independent of others
• Burst model
  • Probability of back-to-back bit errors
  • Error probability dependent on adjacent bits
  • Value of errors may have structure
• Why assume bursts?
  • Appropriate for some media (e.g ., Radio)
  • Faster signaling rate enhances such phenomena

13

Two-Dimensional Parity

• Adding one extra bit to a 7- bit code to balance 1s
• Extra parity byte for the entire frame
• Catches all 1, 2 and 3 bit errors and most 4-bit errors
• 14 redundant bits for a 42- bit message, in the example

1011110 1

1101001 0

(^0101001 )

1011111 0

0110100 1

0001110 1

1111011 0

Parity bits

Parity byte

Data

14

Two-Dimensional Parity

16

Internet Checksum Algorithm

View message as a sequence of 16-bit integers; sum using 16-bit ones-complement arithmetic ; take ones-complement of the result.

**u_short cksum(u_short buf, int count) { register u_long sum = 0; while (count--) { sum += buf++; if (sum & 0xFFFF0000) { / carry occurred, so wrap around / sum &= 0xFFFF; sum++; } } return ~(sum & 0xFFFF); }

17

Cyclic Redundancy Check

Theory

• Based on finite-field (binary-valued)

arithmetic

• Bit string represented as polynomial
• Coefficients are binary-valued
• Divide bit string polynomial by generator

polynomial to generate CRC

Practice

• Bitwise XOR’s Docsity.com

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Cyclic Redundancy Check

• Transmit polynomial P ( x ) that is evenly divisible

by C ( x )

• Shift left k bits, i.e., M ( x ) xk
• Add remainder of M ( x ) xk^ / C ( x ) into M ( x ) xk
• Receiver receives polynomial P ( x ) + E ( x )
• E ( x ) = 0 implies no errors
• Receiver divides ( P ( x ) + E ( x )) by C ( x ); remainder

will be zero ONLY if:

• E ( x ) was zero (no error), or
• E ( x ) is exactly divisible by C ( x )

20

CRC Example - Sender

• C(x) = x 3 +x 2 +1 = 1101  generator
• M(x) = x 8 + x 6 + x 5 + x 4 +1 = 101110001  message

101 Remainder