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Access Codes

Access Codes. A.J. Han Vinck. content. Permutation Codes Random Access bounds 3. Random Access Codes 4. Optical Orthogonal Codes. PERMUTATION CODES. Impulsive noise, broadcast, background Random access codes. Typical noise situation. Narrow band (permanent) Broad band (single impulse)

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Access Codes

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  1. Access Codes A.J. Han Vinck

  2. content • Permutation Codes • Random Access bounds • 3. Random Access Codes • 4. Optical Orthogonal Codes A.J. Han Vinck

  3. PERMUTATION CODES • Impulsive noise, broadcast, background • Random access codes A.J. Han Vinck

  4. Typical noise situation Narrow band (permanent) Broad band (single impulse) Broad + narrow band ••• ••• ••• A.J. Han Vinck

  5. idea Transmit messages as:  sequences (code words) of length M where all M symbols are different  minimum distance (# of differences) D Example: M = 3 D = 2 Code: 123 312 231 132 321 213 f time A.J. Han Vinck

  6. Communication structure example ( 3,2,1 ) message encoder modulator t M-FSK or PPM f Time and frequency diversity ! t > T > T > T 3 Energy detectors f A.J. Han Vinck

  7. Non-coherent detection (FFT) Envelope detector y1 filter matched to f1 1 Quantize > Th = 1 < Th = 0 Envelope detector y2 X filter matched to f2 0   Envelope detector yM 0 filter matched to fM sample 1 0 0 0 Detect Presence of code sequence 0 0 1 0 0 1 0 0 0 0 0 1 A.J. Han Vinck

  8. Non-coherent detection Quadrature receiver using correlators (*)2 cos2fit  sin2fit (*)2 A.J. Han Vinck

  9. Non-coherent detection: performance • Decoder: outputs sequence at minimum distance • Error if noise generates valid sequence • Advantage: time and frequency diversity • robusts against: Broad- and narrowband noise 1 0 0 0 1 0 1 0 Detect Presence of code sequence 0 0 1 0 0 0 1 0 0 1 0 0 1 1 1 1 0 0 0 1 0 0 1 1 A.J. Han Vinck

  10. 2 Code words M = 5 at D = 3 C1 = 1,2,4,3,5 C2 = 1,2,3,5,4 frequency time A.J. Han Vinck

  11. + narrow band noise C1 = 1,2,4,3,5 frequency Narrow band time A.J. Han Vinck

  12. + broad band noise (impulse) broad band C1 = 1,2,4,3,5 frequency time A.J. Han Vinck

  13. + background noise C1 = 1,2,4,3,- C2 = 1,2,-,5,4 Both code words 4 agreements! frequency insert delete time A.J. Han Vinck

  14. Performance minimum distance decoding NOTE: Sequences have minimum distance D • Errors agree with sequences in only 1 positions • > D-1 errors are needed to create another sequence • If E correct symbols disappear due to background noise • > D-1-E errors are needed to create decoding errors A.J. Han Vinck

  15. Upperbound on cardinality Order codewords specified by set of M-D symbols • Set 1: 1, 2, x, x, x 2, 1, x, x, x • Set 2: x, 1, x, 2, x x, 2, x, 1, x • Set 3: x. 1, 2, x, x x, 2, 1, x, x etc. • For distance D, set constains  (M-D)! * D codewords • There are different sets. • Hence: A.J. Han Vinck

  16. Code parameters (1) • We showed that • Q1: when do we achieve equality? • Q2: if not, what is the upperbound • References: -Ian Blake, Permutation codes for discrete channels (1975, IT) -P. Frankl and M. Deza, On the max. # of Permutations with given Max. Or Min. Distance (19977, Jrnl of Comb. Th.) A.J. Han Vinck

  17. Code parameters (2) • Simple code constructions: D = 2, 3, M • all cases M < 7 solved • Interesting cases left ? • D = 2 3 4 5 6 7 • 6 x x x 18 x |C| = 18 is the Klöve (2000) result • 7 x x ? ? x x • 8 x x ? ? x x • 9 x x ? ?? x • 10 x x ? ?? ? A.J. Han Vinck

  18. Simple codes D = M • Cyclic permutation of M integers has D = M • |C| = M! / (M-1)! = M • Example: 1 2 3 3 1 2 2 3 1 A.J. Han Vinck

  19. Simple codes D = 2 • The code with all M! permutation has D = 2 • |C| = M! / (2-1)! = M! • Proof: • All symbols are different • Codewords differ in at least 2 positions x x a x x x b x x x b x x x a x A.J. Han Vinck

  20. Simple codes D = M-1 • Construction for prime P: example M = 3 - starting sets B = 0 1 2 2B = 0 2 1 - add constant vector B + 0  0 1 2 2B + 0  0 2 1 B + 1  1 2 0 2B + 1  1 0 2 B + 2  2 0 1 2B + 2  2 1 0 In general: |C| = M!/(M-2)! = M*(M-1) 2 3 A.J. Han Vinck

  21. Random access codes A.J. Han Vinck

  22. A.J. Han Vinck

  23. Optical access model tr 1 rec 1 tr 2 rec 2 OR   rec T tr T We want: „Uncoordinated and Random Access“ A.J. Han Vinck

  24. Time division: central control N users  inefficient, when small # active users synchronous easy A.J. Han Vinck

  25. Code division synchronized 0 Code division efficient, but complex 1 signature A.J. Han Vinck

  26. Several possibilities A or 0 B or shifted C or another A.J. Han Vinck

  27. Superimposed codes  T code words should not produce a valid code word T words Valid word N n ?  N  ? A.J. Han Vinck

  28. bounds Lower bound: # combinations for large N: superimposed signatures exist s.t. T log2 N < n < 3 T2 log2 N Obvious for T out of N items A.J. Han Vinck

  29. Example: T  2, n = 9, N = 12 User signature 1 001 001 010 2 001 010 100 3 001 100 001 4 010 001 100 5 010 010 001 6 010 100 010 7 100 001 001 8 100 010 010 9 100 100 100 10 000 000 111 11 000 111 000 12 111 000 000 R = 2/9 TDMA gives R = 2/12 Example: 011 101 101 = x OR y ? A.J. Han Vinck

  30. Example: packet transmission A.J. Han Vinck

  31. A.J. Han Vinck

  32. (sync) Binary access model (cont‘d) In Out OR A.J. Han Vinck

  33. A.J. Han Vinck

  34. For PPM: make access model M-ary tr 1 rec 1 tr 2 rec 2 OR   rec T tr T A.J. Han Vinck

  35. Maximum throughput Normalized SUM throughput 0.69 bits/channel use Note: we have M-channels available Hence: PPM does not reduce efficiency! -”On the Capacity of the Asynchronous T-User M-frequency noislesss Multiple Access Channel” IEEE Trans. on Information Theory, pp. 2235-2238, November 1996. (A.J. Han Vinck and Jeroen Keuning) A.J. Han Vinck

  36. Low density signaling A.J. Han Vinck

  37. M-FSK multi-access (cont) • Sender 1 {1,2,   , M} • Sender 2 {1,2,   , M} •    |Y| = 2M -1 • Sender N {1,2,   , M} SUM CAPACITY  M-1 bits/transm. Example for M = 3: input{ 1, 2, 3} output { (1), (2), (3), (1,3), (1,2), (2,3), (1,2,3) } Simple time sharing gives R = log2 M bits/transm. A.J. Han Vinck

  38. M-FSK multi-access (cont) • Capacity obtaining group time sharing! User M=2 M = 3 (2bits/tr) M = 4 (3 bits/tr.) I 0 1 0 1 0 1 I+1 0 2 0 2 I+2 0 3 Output(0),(1) (0,1),(0,2),(1,2) (0,1),(0,2),(1,2) Y (0) (0), (1,2,3) (0,1,2),(0,1,3),(0,2,3) Group I A.J. Han Vinck

  39. Frequency hopping 1 (many variations) 1 0 f Symbol time Hopping period A.J. Han Vinck

  40. Frequency hopping 2 0 1 f t Symbol time Hopping period Different hopping patterns A.J. Han Vinck

  41. Frequency hopping 3 f Slow hopping t 01 11 10 00 f fast hopping t 01 11 10 00 A.J. Han Vinck

  42. advantages • FH avoids tone jammers • FH applies usually noncoherent modulation • the hopping span can be very large • FH is an avoidance system: does not suffer on near-far effect! A.J. Han Vinck

  43. A.J. Han Vinck

  44. A.J. Han Vinck

  45. A.J. Han Vinck

  46. Non-coherent detection Envelope detector y1 filter matched to f1 1 Quantize > Th = 1 < Th = 0 Envelope detector y2 X filter matched to f2 0   Envelope detector yM 0 filter matched to fM sample 1 0 0 0 Detect Presence of signature 0 0 1 0 0 1 0 0 0 0 0 1 A.J. Han Vinck

  47. A.J. Han Vinck

  48. Transmission model Every user has a signature of length n Example: n = 4, M = 3 3 2 2 1 1: send signature 0: send no pulses A.J. Han Vinck

  49. Channel model tr 1 rec 1 tr 2 rec 2 OR   rec T tr T A.J. Han Vinck

  50. M-ary random access codes T words Valid word T N n OR of  T signatures does not produce a non transmitting valid signature A.J. Han Vinck

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