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Forward Error Correction

Forward Error Correction. Steven Marx CSC457 12/04/2001. Outline. What is FEC? Why do we need it? How does it work? Where is it used?. What is FEC?. Send k packets Reconstruct n packets Such that we can tolerate k-n losses Called an (n, k) FEC code. What is FEC? (2). Why FEC?.

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Forward Error Correction

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  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)> log(p) bits required • requires modulo operations

  13. Extension Fields • q = pr elements with p prime, r > 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 α whose powers generate all non-zero elements. • In GF(5), α = 2, whose powers are (1,2,4,3,1,…). • Powers of α repeat with period q - 1, so αq-1 = α0 = 1

  15. Multiplication and Division (2) • for all x, x = αl • l is x’s “logarithm”

  16. Multiplication and Division (3) An example: GF(5) -> α = 2 3 = 23 mod 5 4 = 22 mod 5 3 * 4 = 23+2 mod 5 = 32 mod 5 = 2 mod 5 3 * 4 = 12 mod 5 = 2 mod 5

  17. Vandermonde Matrices • gi,j = xij-1 • xi’s are elements of GF(pr) • called “Vandermonde Matrices” • invertible if all xi’s 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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