By Robert J. McEliece
This revised version of McEliece's vintage is a self-contained creation to all easy ends up in the speculation of data and coding. This conception was once built to house the basic challenge of conversation, that of reproducing at one element, both precisely or nearly, a message chosen at one other element. there's a brief and trouble-free evaluate introducing the reader to the idea that of coding. Following the most effects, the channel and resource coding theorems is 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 direction at college. It comprises dozens of labored examples and several other hundred difficulties for resolution.
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1. 27 allow X and Z be self sustaining random variables with non-stop density features, and enable Y X Z. If h(Y ) and h( Z) exist, express that I(X ; Y ) h(Y ) À h( Z). exhibit that a similar formulation holds if X and Z are random vectors. 1. 28 (Continuation). allow X and Z be self reliant random variables, X discrete and Z with non-stop density, and allow Y X Z. convey that Y has a continuing density, and that if h(Y ) and h( Z) exist, then I(X ; Y ) h(Y ) À h( Z). express that an identical outcome holds if X and Z are random vectors. 1. 29 If X (X 1 , . . . , X n ) are self sufficient Gaussian random variables with capability ì i and variances ó 2i and X9 (X 19 , . . . , X 9n ) are ditto with potential ì9i and variances ó 9i 2 , convey that X X9 (X 1 X 19 , . . . , X n X 9n ) are ditto with ability ì i ì9i and variances ó 2i ó 9i 2 . 1. 30 If X 1 , X 2 , . . . , X n are self sufficient random variable with differential entropies h(X i ) hello , express that 48 Entropy and mutual info h(X 1 Á Á Á X n ) > 2 n 1 2 log three e 2 hello , i1 with equality iff the X i are Gaussian with variances ó 2i e 2 hello =2ðe. 1. 31 If X has a continual density, convey that sup H([X ]) I, the place the supremum is taken over all discrete quantizations of X . 1. 32 build a random variable whose density p(x) is continuing for all actual x such that h(X ) ÀI. 1. 33 allow X be a random n-dimensional vector with non-stop n-dimensional density p(x) and differential entropy h(X). enable f be a continually differentiable one-to-one mapping of Euclidean n-space En onto itself. express that h[ f (X)] h(X ) p(x) logjJ j dx, the place J J (x1 , . . . , xn ) is the Jacobian of the transformation f . 1. 34 enable X (X 1 , . . . , X n ) be a random variable with density, and think E[(X i À ì i )(X j À ì j )] rij for all i, j. turn out that h(X) < n log 2ðe(ó 2 )1= n , 2 the place ó 2 is the determinant of the matrix ( rij ). in addition, express that equality holds iff X is an n-dimensional general random variable whose covariance matrix is ( rij ). (See Feller , Vol. 2, part III. 6. ) 1. 35 permit f (x) be a continual real-valued functionality de®ned on an period I. the item of this challenge is to ®nd out how huge the differential entropy h(X ) can = I, and be if X has a density functionality that satis®es p(x) zero if x P I p(x) f (x) dx A, the place inf( f ) , A , sup( f ). (For instance, if I (ÀI, I) and f (x) (x À ì)2, we'll arrive via a new course on the onedimensional case of Theorem 1. eleven. ) De®ne G(s) I e Àsf (x) dx. exhibit that there exists s0 with G9(s0 )=G(s0 ) ÀA, and de®ne q(x) e Às0 f (x) =G(s0 ) for x P I, q(x) zero for x P = I. exhibit that h(X ) < log G(s0 ) s0 A with equality iff X 's density q(x) nearly all over the place. [ trace: See Prob. 1. eight. ] follow this basic strategy to 3 instances: (a) f (x) log x, I (1, I). (b) f (x) x, I (0, I). (c) f (x) jxj, I (ÀI, I). 1. 36 If X and Y are based random variables, convey that there exist discrete quantizations [X ] and [Y ] that are additionally established. the following 3 difficulties are for readers with a few wisdom of basic Markov chains; see, for instance, Feller , Vol.