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Forward Error Correction

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Such that we can tolerate k-n losses. Called an (n, k) FEC code. What ... streaming media: less 'I' frames in MPEGS. one-way communication. high delay pathways ... – PowerPoint PPT presentation

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Title: Forward Error Correction


1
Forward Error Correction
  • Steven Marx
  • CSC457
  • 12/04/2001

2
Outline
  • What is FEC?
  • Why do we need it?
  • How does it work?
  • Where is it used?

3
What is FEC?
  • Send k packets
  • Reconstruct n packets
  • Such that we can tolerate k-n losses
  • Called an (n, k) FEC code

4
What is FEC? (2)
5
Why FEC?
  • Alternatives
  • ARQ (Automatic Repeat reQuest)
  • requires feedback
  • bad for multicast
  • tolerance
  • only suitable for some applications

6
Why FEC? (2)
  • Advantages
  • sometimes no feedback channel necessary
  • long delay path
  • one-way transmission
  • avoids multicast problems
  • Disadvantages
  • computationally expensive
  • requires over-transmission

7
How is this possible?
An easy example (n, k) (2, 3) FEC
code transmitting two numbers a and b Send
three packets 1. a 2. b 3. a b
8
How is this possible? (2)
Could be represented as matrix multiplication To
encode
To decode, use subset of rows.
9
How is this possible? (3)
More generally y Gx, where G is a generator
matrix G is constructed in such a way that any
subset of rows is linearly independent. A
systematic generator matrix includes the
identity matrix.
10
A Problem
  • a and b are 8-bit numbers
  • a b may require more bits
  • loss of precision means loss of data

11
A Solution
  • Finite fields
  • field
  • we can add, subtract, multiply, and divide as
    with integers
  • closed over these operations
  • finite finite number of elements

12
A Solution (2)
  • Specific example
  • prime field or Galois Field - GF(p)
  • elements 0 to p-1
  • modulo p arithmeticProblem
  • with the exception of p 2, ?log(p)?gt log(p)
    bits required
  • requires modulo operations

13
Extension Fields
  • q pr elements with p prime, r gt 1
  • extension field, or GF(pr)
  • elements can be considered polynomials of
    degree r - 1
  • sum just sum modulo p
  • extra simple with p 2
  • exactly r bits needed
  • sums and differences just XORs

14
Multiplication and Division
  • Exists an a whose powers generate all non-zero
    elements.
  • In GF(5), a 2, whose powers are (1,2,4,3,1,).
  • Powers of a repeat with period q - 1, so aq-1
    a0 1

15
Multiplication and Division (2)
  • for all x, x al
  • l is xs logarithm

16
Multiplication and Division (3)
An example GF(5) -gt a 2 3 23 mod 5 4 22
mod 5 3 4 232 mod 5 32 mod 5
2 mod 5 3 4 12 mod 5 2 mod 5
17
Vandermonde Matrices
  • gi,j xij-1
  • xis are elements of GF(pr)
  • called Vandermonde Matrices
  • invertible if all xis different
  • y Gx
  • G-1y G-1Gx x
  • can be extended with the identity matrix for
    systematic codes

18
Swarmcast - a real example
  • for media distribution
  • reduces bandwidth requirements of the server
  • server transmits to a small number of clients
  • while downloading, those clients also transmit
    packets to other clients
  • FEC used to maximize chances of getting useful
    packets

19
Swarmcast (2)
Star Wars Episode Two Trailer
300Mb/s
100Mb/s
100Mb/s
50Mb/s
100Mb/s
100Mb/s
50Mb/s
50Mb/s
100Mb/s
100Mb/s
20
Other useful applications
  • multicast
  • streaming media less I frames in MPEGS
  • one-way communication
  • high delay pathways
  • storage

21
Conclusion
  • FEC
  • allows error correction without retransmission
  • requires redundancy in transmission
  • useful for multicast
  • not extensively used at the packet level
  • more important with high bandwidth, high
    latency, as is the trend
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