Download Entropy and Information Theory by Robert M. Gray PDF

By Robert M. Gray

This e-book is an up-to-date model of the knowledge idea vintage, first released in 1990. approximately one-third of the ebook is dedicated to Shannon resource and channel coding theorems; the remaining addresses assets, channels, and codes and on details and distortion measures and their homes.

New during this edition:

  • Expanded therapy of desk bound or sliding-block codes and their relatives to standard block codes
  • Expanded dialogue of effects from ergodic concept appropriate to details theory
  • Expanded remedy of B-processes -- methods shaped via desk bound coding memoryless sources
  • New fabric on buying and selling off info and distortion, together with the Marton inequality
  • New fabric at the houses of optimum and asymptotically optimum resource codes
  • New fabric at the relationships of resource coding and rate-constrained simulation or modeling of random processes

Significant fabric now not lined in different info idea texts comprises stationary/sliding-block codes, a geometrical view of data concept supplied through procedure distance measures, and normal Shannon coding theorems for asymptotic suggest desk bound assets, that could be neither ergodic nor desk bound, and d-bar non-stop channels.

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Sample text

2: Suppose that P and M are two probability measures on a discrete space and that f is a random variable defined on that space, then D(Pf ||Mf ) ≤ D(P ||M ). The lemma, discussion, and corollaries can all be interpreted as saying that taking a measurement on a finite alphabet random variable lowers the entropy and the relative entropy of that random variable. By choosing U as (X, Y ) and f (X, Y ) = X or Y , the lemma yields the promised inequality of the previous lemma. Proof of Lemma: If HP ||M (R) = +∞, the result is immediate.

First observe that since PX (a) ≤ 1, all a, − ln PX (a) is positive and hence H(X) = − PX (a) ln PX (a) ≥ 0. 6) with M uniform as in the second interpretation of entropy above, if X is a random variable with alphabet AX , then H(X) ≤ ln ||AX ||. Since for any a ∈ AX and b ∈ AY we have that PX (a) ≥ PXY (a, b), it follows that H(X, Y ) = − PXY (a, b) ln PXY (a, b) a,b ≥− PXY (a, b) ln PX (a) = H(X). 1 we have that since PXY and PX PY are probability mass functions, H(X, Y ) − (H(X) + H(Y )) = PXY (a, b) ln a,b PX (a)PY (b) ≤ 0.

Measurements made on such processes, however, will always be assumed to be real. Suppose next we have a measurement f whose range space or alphabet f (Ω) ⊂ R of possible values is finite. Then f is called a discrete random variable or discrete measurement or digital measurement or, in the common mathematical terminology, a simple function. Given a discrete measurement f , suppose that its range space is f (Ω) = {bi , i = 1, · · · , N }, where the bi are distinct. Define the sets Fi = f −1 (bi ) = {x : f (x) = bi }, i = 1, · · · , N .

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