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The Math Behind the Compact Disc

The Math Behind the Compact Disc. Linear Algebra and Error-Correcting Codes. william j. martin. mathematical sciences. wpi wednesday december 3. 2008 fairfield university. How the device works.

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The Math Behind the Compact Disc

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  1. The Math Behind the Compact Disc Linear Algebra and Error-Correcting Codes william j. martin. mathematical sciences. wpi wednesday december 3. 2008 fairfield university

  2. How the device works The compact disc is a complex system incorporating interesting ideas from engineering, physics, CS and math. We will focus only on the mathematics of the error- correction strategy. For more info on the CD, see Kelin Kuhn’s book “Laser Engineering”: W J Martin Mathematical Sciences WPI

  3. Borrowed from K J Kuhn’s book “Laser Engineering” W J Martin Mathematical Sciences WPI

  4. The Pits • Each pit is 0.5 microns wide… • and 0.83 to 3.56 microns long. • Tracks are separated by 1.6 microns of “land” • Wavelength of green light is about 0.5 micron • 40 tracks under one strand of human hair W J Martin Mathematical Sciences WPI

  5. Modelling a CommunicationsChannel Linear algebra model: r = m+e (vector add.) W J Martin Mathematical Sciences WPI

  6. Channel with Error Correction W J Martin Mathematical Sciences WPI

  7. A number system that the computer can understand: F = { 0, 1 } Ordinary multiplication Addition: 1+1=0 Now music is turned into binary vectors! Turn it into an algebra problem! W J Martin Mathematical Sciences WPI

  8. A bit (or a nibble?) of graph theory • The n-cube is a type of Hamming graph • Vertices are all binary n-tuples • n-tuples are adjacent if they differ in only one coordinate • Nice ‘eigenvalues’! W J Martin Mathematical Sciences WPI

  9. Binary Vector Spaces • The vectors are all possible binary n-tuples 0 0 1 0 1 1 1 0 1 0 1 1 0 0 0 + 0 0 1 1 1 1 0 0 0 0 0 0 0 0 1 = 0 0 0 1 0 0 1 0 1 0 1 1 0 0 1 W J Martin Mathematical Sciences WPI

  10. This is a metric: dist( x, y )  0 with dist( x, y ) = 0 iff x=y dist( x, y ) = dist( y, x ) Triangle inequality dist( x, z )  dist( x, y ) + dist( y, z ) Hamming Distance The distance between two binary n-tuples x and y is the number of coordinates in which they differ dist( 001100, 001011 ) = 3 W J Martin Mathematical Sciences WPI

  11. Theorem n Let C (the “code”) be a subset of F with minimum distance between any two codewords equal to d. Then there exists an algorithm which corrects up to t errors per transmitted codeword if and only if d  2t + 1. W J Martin Mathematical Sciences WPI

  12. Proof If x and y are distinct codewords, then the balls of radius t around them are disjoint. So if the received vector is within distance t of x, it must be at distance > t from any other codeword. So decoding is unique. W J Martin Mathematical Sciences WPI

  13. A Useful Extension of the Theorem The above (computationally infeasible) decoding algorithm also correctly recovers from any t symbol errors and any s symbol erasures provided d > 2t+s. transmit: 0 1 1 2 2 3 0 receive: 0 1 3 3 ? ? ? (here, t=2 errors and s=3 erasures) W J Martin Mathematical Sciences WPI

  14. Small Example • Let C denote the “rowspace” of the matrix Then C = { 000000, 110100, 011010, 101110, 001101, 111001, 010111, 100011 } and C has minimum distance 3 so C allows correction of any single-bit error in any transmitted codeword. W J Martin Mathematical Sciences WPI

  15. The binary Hamming code Codewords: 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 1 0 1 0 0 0 0 0 1 0 1 1 1 0 1 1 0 1 0 0 1 0 0 1 0 1 1 0 0 1 1 0 1 0 1 1 0 0 1 0 1 0 0 0 1 1 0 1 1 1 1 0 0 1 0 1 0 0 0 1 1 0 0 1 1 1 0 0 1 0 1 0 0 0 1 1 1 0 1 1 1 0 0 1 0 1 0 0 0 1 0 1 0 1 1 1 0 • Quadratic Residues! • In we have • = 1 6 = 1 • = 4 5 = 4 • 3 = 2 4 = 2 Z Z 7 2 2 2 2 2 2 W J Martin Mathematical Sciences WPI

  16. The Fano projective plane 3 Vector Space: F “Poynts”: 1-dim. subspaces “Lynes”: 2-dim. subspaces 2 W J Martin Mathematical Sciences WPI

  17. C = nullsp(H) where All codewords: 0 0 0 0 0 0 0 1 1 1 1 1 1 1 0 0 0 1 1 1 1 1 1 1 0 0 0 0 0 1 1 0 0 1 1 1 0 0 1 1 0 0 0 1 1 1 1 0 0 1 0 0 0 0 1 1 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 0 1 1 0 1 0 0 1 0 0 1 0 1 1 1 0 0 1 1 0 0 0 1 1 0 0 1 1 1 0 1 0 0 1 0 0 1 0 1 1 0 W J Martin Mathematical Sciences WPI

  18. Codes from polynomials Let’s replace F={0,1} with F={0,1,…,6} (with modular arithmetic). Now consider the vector space F[z] of all polynomials in z with coefficients in F. For any subset N of F, we have a linear transformation L: F[z]  F via f(z)  [ f(0), f(1), f(2), f(3), f(4), f(5) ] (Here, we use, N={0,1,2,3,4,5}.) This is a Reed-Solomon code. N W J Martin Mathematical Sciences WPI

  19. Polynomials to Codewords Example: • Let the message be [1, 2, 2] (working mod 7) • Polynomial is f(z) = z + 2 z + 2 • Codeword is [f(0), f(1), f(2), f(3), f(4), f(5)] = [ 2, 5, 3, 3, 5, 2] 2 W J Martin Mathematical Sciences WPI

  20. Reed-Solomon Codes FACT: Two polynomials of degree less than k having k points of intersection must be equal. SO: Reed-Solomon code of length n<q and dim k has min. dist. n-k+1 W J Martin Mathematical Sciences WPI

  21. Compact Disc Parameters SONY/Philips design (1980) • Music is sampled 44,100 times per second • Each sample consists of 32 bits, representing left and right channel signal magnitude 0—65535 (Pulse Code Modulation – PCM) • So chip must process 1,411,200 raw data bits per second • But it gets much worse! W J Martin Mathematical Sciences WPI

  22. Cross-Interleaved RS Codes • Inner code is a 28-dimensional subspace of a 32-dimensional vector space over a finite field of size 256. • Outer code is a 24-dimensional subspace of a 28-dimensional vector space. • Six 32-bit samples make up a 192-bit frame which is encoded as a 224-bit codeword. (Eventually, codewords have length 588 bits!) W J Martin Mathematical Sciences WPI

  23. Encoding – The numbers • The codewords from the first code are interleaved into a virtually infinite array of 28 rows of symbols over GF(256). • We pull out 8 binary columns (one symbol) to obtain a 28x8=224-bit frame which is then encoded using another Reed-Solomon code to obtain a codeword of length 256 bits. W J Martin Mathematical Sciences WPI

  24. Interleaving to disperse errors • Codewords of first code are stacked like bricks • 28 rows of vectors over GF(256) • Extract columns and re-encode using second Reed-Solomon code W J Martin Mathematical Sciences WPI

  25. Splitting Odd and Even Bits W J Martin Mathematical Sciences WPI

  26. Back to the Pits • Each pit is 0.5 microns wide… • and 0.83 to 3.56 microns long. • Tracks are separated by 1.6 microns of “land” • Not all 01-sequences can be recorded W J Martin Mathematical Sciences WPI

  27. EFM: Eight-to-Fourteen Modulation • This encoding scheme can only store sequences where each consecutive pair of ones is separated by at least 2 and at most 10 zeros • This is achieved by a mapping F  F which is given by a lookup table. 8 14 2 2 W J Martin Mathematical Sciences WPI

  28. Further Processing • Three more ‘merge bits’ are added to each of these 14 • So 256+8=264=33x8 bits, carrying six samples, or 192 information bits, gets encoded as 588 channel bits on the disk • This represents 0.000136 seconds of music W J Martin Mathematical Sciences WPI

  29. What actually goes on the disc? • We must do this 7,350 times per second • So CD player reads 4,321,800 bits per second of music produced • To get 74 minutes of music, we must store 74x60x4321800 = 19,188,792,000 bits of data on the compact disc! W J Martin Mathematical Sciences WPI

  30. When in doubt, erase • Inner code has minimum distance 5 (over GF(256)) • Rather than correct two-symbol errors, the CD just erases the entire received vector. W J Martin Mathematical Sciences WPI

  31. So…how good is it? • The two Reed-Solomon codes team up to correct ‘burst’ errors of up to 4000 consecutive data bits (2.5 mm scratch on disc) • If signal at time t cannot be recovered, interpolate • With smart data distribution, this allows for recovery from burst errors of up to 12,000 data bits (7.5 mm track length on disc) • If all else fails, mute, giving 0.00028 sec of silence. W J Martin Mathematical Sciences WPI

  32. Space communications (Mariner,Voyager,etc.) DVD, CD-R, CD-ROM Cell phones, internet packets Memory: chips, hard drives, USB sticks RAID disk arrays Quantum computing Other Applications W J Martin Mathematical Sciences WPI

  33. The Last Slide Thank You All! W J Martin Mathematical Sciences WPI

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