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Forward Error Correction Steven Marx CSC45712/04/2001

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

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

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What is FEC? (2)

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Why FEC? Alternatives: ARQ (Automatic Repeat reQuest) requires feedback bad for multicast tolerance only suitable for some applications

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

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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

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How is this possible? (2) Could be represented as matrix multiplication To encode: To decode, use subset of rows.

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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.

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A Problem a and b are 8-bit numbers a + b may require more bits loss of precision means loss of data

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

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A Solution (2) Specific example: “prime field” or “Galois Field” - GF(p) elements 0 to p-1 modulo p arithmetic Problem: with the exception of p = 2, log(p) > log(p) bits required requires modulo operations

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Extension Fields q = p r elements with p prime, r > 1 “extension field”, or GF(p r ) 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

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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 α q-1 = α 0 = 1

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Multiplication and Division (2) for all x, x = α l l is x’s “logarithm”

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

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Vandermonde Matrices g i,j = x i j-1 x i ’s are elements of GF(p r ) called “Vandermonde Matrices” invertible if all x i ’s different y = Gx G -1 y = G -1 Gx = x can be extended with the identity matrix for systematic codes

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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

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Swarmcast (2) Star Wars: Episode Two Trailer 300Mb/s 100Mb/s 50Mb/s 100Mb/s 50Mb/s

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Other useful applications multicast streaming media: less “I” frames in MPEGS one-way communication high delay pathways storage

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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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