By Robert McEliece
This can be a revised edition of McEliece's 1977 vintage, released with scholars in brain. it's a self-contained advent to all simple leads to the idea of knowledge and coding. This conception used to be built to accommodate the elemental challenge of communique, that of reproducing at one aspect, both precisely or nearly, a message chosen at one other aspect.
There is a quick and undemanding evaluate introducing the reader to the concept that of coding. Then, following the most effects, the channel and resource coding theorems, there's a learn of particular coding schemes which are used for channel and resource coding.
This quantity can be utilized both for self-study, or for a graduate/undergraduate point path at collage. It comprises dozens of labored examples and a number of other hundred difficulties for resolution. The exposition should be simply understandable to readers with a few previous wisdom of chance and linear algebra.
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Extra info for The Theory of Information and Coding (2nd Edition) (Encyclopedia of Mathematics and its Applications, Volume 86)
Sample text
2), we obtain several important alternative forms for I(X ; Y ): p(xj y) I(X ; Y ) , (1:5) p(x, y) log p(x) x, y p(x, y) log p(x, y) , p(x) p( y) (1:6) p(x, y) log p( yjx) : p( y) (1:7) x, y x, y (The details are left as Prob. ) log 34); however, we shall now prove the important fact that I(X ; Y ) cannot be negative. This is surely reasonable, given our heuristics: we do not expect to be misled (on the average) by observing the output of the channel. 3 For any discrete random variables X and Y, I(X ; Y ) > 0.
24)) that H([X ]) is the sum of two terms; one is an approximating sum for h(X ), but the other, p(i) log(Äxi )À1 , is a kind of measure of the mesh size and clearly approaches I as the Äxi become smaller and smaller. Indeed, it is easy to show that H([X ]) itself always approaches I (Prob. 31) even if the integral h(X ) converges to ÀI! Hence, although h(X ) is a useful quantity to know, it is not in any sense a measure of the randomness of X . Indeed, h(X ) is not even invariant under a coordinate transformation (see Prob.
Show that the same formula holds if X and Z are random vectors. 28 (Continuation). Let X and Z be independent random variables, X discrete and Z with continuous density, and let Y X Z. Show that Y has a continuous density, and that if h(Y ) and h( Z) exist, then I(X ; Y ) h(Y ) À h( Z). Show that the same result holds if X and Z are random vectors. 29 If X (X 1 , . . , X n ) are independent Gaussian random variables with means ì i and variances ó 2i and X9 (X 19 , . . , X 9n ) are ditto with means ì9i and variances ó 9i 2 , show that X X9 (X 1 X 19 , .