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Wireless networks

Wireless networks. Philippe Jacquet INRIA Ecole Polytechnique France. How to measure complexity ? Deterministic entropy: A system Z with N states : N very large base 2 log for the minimum number of bits to describe the system It is the information quantity contained in the system

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Wireless networks

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  1. Wireless networks Philippe Jacquet INRIA Ecole Polytechnique France

  2. How to measure complexity ? Deterministic entropy: A system Z with N states : N very large base 2 log for the minimum number of bits to describe the system It is the information quantity contained in the system Tribit (base 3) log. For alphabet, base 26 log. We will omit the log base (logons). Information Theory

  3. If If f(Z) is a function of Z Properties of entropy

  4. If Z is a Cartesian product : If Z is a binary sequence of length n If Z is the set of texts of length n in alphabet A Properties of entropy

  5. Compression limit The information in a system (a text, a code, etc) cannot be compressed within less symbols on alphabet A. More compression → Information loss Information Theory

  6. Z is the set of binary sequences with n1 « 1 » and n0 « 0 »: Exercise: show that Application: entropy of statistic mixtures taux d’entropie

  7. Mutual Information or exchanged information • A systeme with two components X and Y • Since • Mutual Information

  8. Conditionnal entropy N(Y|X) average number of different Y for a fixed X H(Y|X)=h(X,Y)-h(Y)

  9. X: n bits sequence θcontains n1 « 1 » et n0 « 0 » Application: seven error game

  10. Probabilistic entropy distribution over two elements : Uniform distribution in general Quantity exp(h(X)) is the number of probabilistic distinct X

  11. Entropy over two elements p1

  12. If X and Y are independent, then Otherwise (by convexity) Random n bits sequence Probabilistic entropy

  13. Transmission of information • System (emitter, receiver): Z=(X,Y) • Transmitted information: I(X,Y) • I(X,Y)=received entropy - channel entropy. • Received entropy: h(Y) • Channel entropy: h(Y|X) Emitted Code X Received Code :Y

  14. Transmission of information • System (emitter, receiver): Z=(X,Y) • Transmitted information: I(X,Y) • Encoding function: • Decoding function: • We want to retrieve information: i=j. Emitted Code X Received Code :Y

  15. Noisy channel • X is a number between 0 and N-1 • Y=X+ß, ß is a number between 0 and B-1 X X +ß

  16. Noisy channel • i is a number between 0 and • Decoding Y=X+ß, ß is a number between 0 and B-1 X X +ß

  17. Encoding and decoding • Let a piece of information I: • Encoding • Send X, receive Y. • Decoding • We want with high probability

  18. Shannon Theorem • Let • There exists an encoding and decoding functions which sends information nR • Error rate arbitrarily low when n→∞ • Let • Every encoding and decoding function to send Rn fails. • Error rate arbitrarily close to 100% when n→∞

  19. Decoding-encoding X Y J I Y X ambiguity

  20. Encoding Capacity in general average number of probilistic Distinct Y X1 X2 average number of Y for a fixed X N(X) number of encodable codes without ambiguity X3 N(Y) Encoding capacity (number of distinct encodable codes) :

  21. Capacity of communication channel • Exemples :

  22. wireless channel • N is signal energy •  discretization • B is noise level (gaussian) • W is bandwidth (number of symbols per sec) X X +ß

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