Error Detection

1
Error Detection
2
Objectives

• Understand Transmission Errors
• Understand two dimensional parity
• Understand CRC and be able to work examples

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Error Control Overview

• Errors occur due to
• Noise or interference in the communication
  channel
• Congestion in the network where packets musts be
  dropped
• Error Control Strategies
• Error Correcting codes (Forward Error Correction
  (FEC))
• Error detection and retransmission Automatic
  Repeat Request (ARQ)

4
Two-Dimensional Parity
5
(No Transcript)
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Cyclic Redundancy Check

• Add k bits of redundant data to an n-bit message.
• Represent n-bit message as an n-1 degree
  polynomial e.g., MSG10011010 corresponds to
  M(x) x7 x4 x3 x1.
• Let k be the degree of some divisor polynomial
  C(x) e.g., C(x) x3 x2 1.

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CRC

• Transmit polynomial P(x) that is evenly divisible
  by C(x), and receive polynomial P(x) E(x)
  E(x)0 implies no errors.
• Recipient divides (P(x) E(x)) by C(x) the
  remainder will be zero in only two cases E(x)
  was zero (i.e. there was no error), or E(x) is
  exactly divisible by C(x). Choose C(x) to make
  second case extremely rare.

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Example

• Make all legal messages divisible by 3
• If you want to send 10
• First multiply by 4 to get 40
• Now add 2 to make it divisible by 3 42
• When the data is received ..
• Divide by 3, if there is no remainder there is no
  error
• If no error, divide by 4 to get sent message
• If we receive 43, 44, 41, 40, then error
• 45 would not be recognized as an error

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Mod 2 Arithmetic review
1111 1010 0101
11001 x 101
11001 11001
1111101
10

• Sender
• multiply M(x) x7 x4 x3 x1 by xk for our
  example, we get
• x10 x7 x6 x4 (10011010000)
• divide result by C(x) (1101)
• Send 10011010000 101 10011010101,
• since this must be exactly divisible by C(x)

11111001 10011010000 Message 1101
1001 1101 1000 1101 1011
1101 1100 1101
1000 1101 101
Remainder
Generator 1101
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• Want to ensure that C(x) does not divide evenly
  into polynomial E(x).
• All single-bit errors, as long as the xk and x0
  terms have non-zero coefficients.
• All double-bit errors, as long as C(x) has a
  factor with at least three terms.
• Any odd number of errors, as long as C(x)
  contains the factor (x 1).
• Any burst error (i.e sequence of consecutive
  errored bits) for which the length of the burst
  is less than k bits.
• Most burst errors of larger than k bits can also
  be detected.

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Even Parity
Actually consists of using x1 polynomial Given
message 0111, multiply by x to get 01110 Now
divide by x111 11 01110 11
0010 11
1remainder Message 01110101111 even parity
0101
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• Common polynomials for C(x)

CRC CRC-8 CRC-10 CRC-12 CRC-16 CRC-CCITT CRC-32
C(x) x8x2x11 x10x9x5x4x11 x12x11x3x2x1
1 x16x15x21 x16x12x51 x32x26x23x22x16x
12x11x10 x8x7x5x4x2x1
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Internet Checksum Algorithm

• Idea view message as a sequence of 16-bit
  integers.
• Add these integers together using 16-bit ones
  complement arithmetic, and then take the ones
  complement of the result. That 16-bit number is
  the checksum.
• It is difficult to determine a way of changing
  the data and still generating the same checksum

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• 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)

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Checksum Example

• Message 0xe123 0x2045, count2
• sum 0xE123
• sum 0x2045 0x10168
• the carry can be at most one, so mask the high
  bits and add one
• sum 0x016810x0169
• return sum0xFE96
• sent message 0xe123 0x2045 0xFE96
• on receive sum0x01690xFE960xFFFF
• return sum0 if there were no errors

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Light
For the word of the Lord is truth, and whatsoever
is truth is light, and whatsoever is light is
Spirit, even the Spirit of Jesus Christ. And the
Spirit giveth light to every man that cometh into
the world and the Spirit enlighteneth every man
through the world, that hearkeneth to the voice
of the Spirit. DC 844546  
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What is this spirit?
It is the light and the life of all things. It is
the light and the life of man. It is the life of
the animal creation. It is the life of the
vegetable creation. It is in the earth . . . it
is in the stars . . . it is in the moon . . .
it is in the sun, and is the light of the sun,
and the power by which it was made and these
grosser particles of light that illuminate the
heavens and enable us to behold the works of
nature, are from that same Spirit which
enlightens our minds and unfolds the things of
God. As that light comes forth from the sun, so
the light of God comes to us. Charles W.
Penrose, JD 2621  
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Wavelength Division Multiplexing

• The visible spectrum of light has a spiritual
  counterpart. The spiritual spectrum relates to
  various levels of intelligence, beginning with
  animal instinct and moving to more refined forms
  of light and truth. These gradations include
  man's reasoning ability and conscience, the light
  that comes through the Holy Ghost prior to
  baptism, and the light one receives through the
  gift of the Holy Ghost after entering the Lord's
  kingdom. Finally, a fullness of light is received
  when one has proven worthy of the Second
  Comforter and receives the "more sure word of
  prophecy" (2 Pet. 119 DC 1315). (The
  spiritual spectrum of light is based on
  statements by Parley P. Pratt in Key to the
  Science of Theology, 9th ed. Salt Lake City
  Deseret Book, 1965, 4647 and Charles W.
  Penrose, JD 262122.)

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WDM must focus on one frequency
And if your eye be single to my glory, your whole
bodies shall be filled with light, and there
shall be no darkness in you and that body which
is filled with light comprehendeth all things.
DC 8867  
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Forward Error Correction
There is also a kernel of light within each man
and woman that "was not created or made, neither
indeed can be" (DC 9329). This light or
intelligence is called "the light of truth" as
well and is coeternal with the Creator. Because
"light cleaveth unto light," there is a natural
link or affinity between our intelligence and the
Light of Christ that equips all people with a
basic discernment of good and evil--that part of
us we call our conscience (DC 8840 see also
Moroni 71516).