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Chapter 7

Chapter 7. The Channel and Mutual Information. symbols can’t be swallowed. a 1 : : a q. b 1 : : b s. A. B. alphabet of symbols sent. alphabet of symbols received. P ( b j |a i ). or randomly generated. Information Through a Channel.

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Chapter 7

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  1. Chapter 7 The Channel and Mutual Information

  2. symbols can’t be swallowed a1:: aq b1:: bs A B alphabet of symbols sent alphabet of symbols received P(bj|ai) or randomly generated Information Through a Channel For example, in an error correcting code over a noisy channel, s≥ q. If two symbols sent are indistinguishable when received, s < q. Characterize a stationary channel by a matrix of conditional probabilities: received row column sent s Pi,j P = Pi,j = P(bj | ai) q 7.1, 7.2, 7.3

  3. For p(ai) = probability of source symbols, let p(bj) = probability of being received [p(a1) … p(aq)]P = [p(b1) … p(bs)] no noise: Pi,j = I; p(bj) = p(aj) all noise: Pi,j = 1/s; p(bj) = 1/s The probability that ai was sent and bj was received is: Baye’s Theorem P(ai, bj) = p(ai) ∙ P(bj | ai) = p(bj) ∙ P(ai | bj). [coincidental probability] So if p(bj) ≠ 0, the backwards conditional probabilities are: 7.1, 7.2, 7.3

  4. Binary symmetric Channel P0,0 a = 0 b = 0 p(a = 0) = p p(a = 1) = 1 − p P0,0 = P1,1 = P P0,1 = P1,0 = Q P0,1 P1,0 P1,1 a = 1 b = 1 a = 0 a = 1 p(b = 0) p(b = 1) ( pP + (1 − p)QpQ + (1 − p)P ) P Q Q P (p 1−p) = 7.4

  5. P = 1 Q = 0 P = Q = ½ If p = 1 − p = ½ (equiprobable) then: P(a = 1 | b = 0) = P(a = 0 | b = 1) = Q P(a = 0 | b = 0) = P(a = 1 | b = 1) = P P Q Q P 7.4

  6. H(A| B) H(B| A) H(A) Input entropy H(B) Output entropy System Entropies condition on bj average over all bj Similarly The information loss in the channel, called equivocation (or noise entropy). The average uncertainty about the symbol sent or received. 7.5

  7. H(A, B) Joint Entropy Intuition: taking snapshots A B Define : H(A| B) H(B| A) H(B | A) H(A) H(A | B) H(A, B) = H(B) 7.5

  8. H(A, B) a priori a posteriori joint H(A| B) H(B| A) P(ai | bj) p(ai) I(A; B) The amount of information they are sharing corresponds to Information gain upon receiving bj : I(ai) − I(ai | bj) . shared Mutual Information By symmetry: If ai and bj are independent (all noise), then P(ai, bj) = p(ai) ∙ p(bj) and hence P(ai | bj) = p(ai)  I(ai ; bj) = 0. No information gained in channel. 7.6

  9. from symmetry Similarly: Average over all ai: By Gibbs I(A; B) ≥ 0. Equality only if P(ai, bj) = p(ai)∙p(bj) [independence]. = H(A) + H(B) − H(A, B) ≥ 0  H(A, B)  H(A) + H(B) We know H(A, B) = H(A) + H(B | A) = H(B) + H(A | B).  I(A ; B) = H(A) − H(A | B) = H(B) − H(B | A) ≥ 0  H(A | B)  H(A) and H(B | A)  H(B). 7.6

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