CS412 Introduction to Computer Networking

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CS412 Introduction to Computer Networking
Telecommunication

• Error Correction/Detection

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Topics

• Introduction
• Error Correction
• Error Detection

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Introduction

• Transmission impairments (errors)
• Attenuation
• Loss of energy as signal propagates
• Delay Distortion
• Components travel at different speeds
• Noise
• Unwanted energy from other sources

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Figure 3.21 Attenuation
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Figure 3.23 Distortion
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Figure 3.24 Noise
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Isolated and Burst Error

• Isolated 0 -gt 1, or 1 -gt 0
• Burst errors in consecutive bits
• 2 or more bits in the data unit have changed

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Isolated and Burst Error

• Compare burst errors to isolated errors
• Fewer error blocks given a fixed error rate
• Example error rate 0.001 per bit
• (1 error out of 1000 bits)
• block size 1000 bits
• burst size 100
• isolated error an error contained in most
  blocks
• burst error 1 or 2 error blocks out of 100
  blocks
• Error correction/detection much harder

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Error Correcting/Detecting Codes

• Redundancy added to data
• Error correction
• Referred to as forward error correction
• Detect and correct error
• Error detection
• Detect error and request retransmission
• Codeword (n bits)
• data redundancy
• (m bits) (r bits)
• n m r

When to use which one?
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10.3 Redundancy
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Error Correction

• Hamming Code
• For m-bit data we need r-bit redundancy, where
  (mr1) ? 2r
• Redundancy bits are placed in position of 2s
  power
• Example If m 7, then r 4

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10.15 Redundancy bits calculation
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10.16 Example of redundancy bit calculation
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10.17 Error detection using Hamming code
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Correct Burst Errors

• Organize k codewords into a (k x n) matrix
• (k x r) check bits for (k x m) data
• Transmit one column at a time
• Reconstruct matrix after received
• Correct burst errors of length ? k

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Correct Burst Errors
n
m
k
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10.18 Burst error correction example
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Error Detection

• Parity bit
• Error detection
• Added to data so that number of 1 bits in
  codeword is
• Even (even parity)
• Odd (odd parity)
• E.g., ASCII of H is 1001000, its codeword is
• ________ if even parity is used
• ________ if odd parity is used

01001000
11001000
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10.5 Even-parity concept
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Error Detection vs. Correction

• Error-detecting and retransmission
• More efficient than error-correction when error
  rate is low
• Example
• error rate 0.000001 per bit
• block size 1000 bits
• check bits per block
• Hamming code error correction 10
• Parity code error detection 1
• overhead for 1M data e. c. 10000 bits
• e. d. 2001 bits

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Use Parity to Detect Burst Errors

• Organize a block into (k x n) matrix
• One parity for each column
• ? one row of parities at the bottom
• Transmit one row at a time
• Can detect burst errors of length ? n

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Example

• Original data
• 11100111 11011101 00111001 10101001
• Organized into 11100111
• 11011101
• 00111001
• 10101001
• 10101010 ? LRC
• Transmitted
• 11100111 11011101 00111001 10101001 10101010
• Burst error
• 11100111 11010000 01001001 10101001 10101010
• Error detected ? whole block is discarded
• 11100111 11010000 0100001 10101001 10101010

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Error-Detecting Code - CRC

• Bit stream is treated as polynomial w/
  coefficients 0 and 1
• Example
• data 10100111
• polynomial
• degree 7
• Modulo 2 arithmetic is used
• Example 10011011 11110000
• 11001010 -10100110
• 01010001 01010110
• XOR

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Error-Detecting Code - CRC

• Use generator polynomial G(x) to calculate
  checksum
• Frame P(x) generator G(x)
• degree of G(x) d
• Transmitted checksummed frame P(x)xd R(x)
• Its guaranteed that P(x)xd R(x) is divisible
  by G(x)!!

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Error-Detecting Code - CRC

• Receiver divides checksummed frame by G(x)
• If remainder is zero
• No error, CRC is removed
• Otherwise
• Error, the frame is discarded

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10.8 Binary division in a CRC generator

• Frame
• Generator
• Frame transmitted
• 100100001

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10.9 Binary division in CRC checker
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CRC - Example
1 1 0 0 0 0

• Frame
• Generator

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CRC Properties

• Single error detection
• Double error detection w/ carefully chosen G(x)
• Odd number error detection if (x 1) is a factor
  of G(x)
• Detect burst error length ? r for r check bits
• Can be implemented in hardware using simple shift
  register circuit

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Table 10.1 Standard polynomials
Name Polynomial Application
CRC-8 x8 x2 x 1 ATM header
CRC-10 x10 x9 x5 x4 x 2 1 ATM AAL
ITU-16 x16 x12 x5 1 HDLC
ITU-32 x32 x26 x23 x22 x16 x12 x11 x10 x8 x7 x5 x4 x2 x 1 LANs