Download E-books The Theory of Information and Coding (Encyclopedia of Mathematics and its Applications No. 86) PDF

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.

Show description

Read or Download The Theory of Information and Coding (Encyclopedia of Mathematics and its Applications No. 86) PDF

Similar Mathematics books

Real and Complex Analysis (Higher Mathematics Series)

This is often a complicated textual content for the single- or two-semester direction in research taught basically to math, technological know-how, desktop technology, and electric engineering majors on the junior, senior or graduate point. the fundamental recommendations and theorems of research are awarded in any such method that the intimate connections among its a number of branches are strongly emphasised.

Principles of Mathematical Analysis (International Series in Pure and Applied Mathematics) (International Series in Pure & Applied Mathematics)

The 3rd version of this renowned textual content keeps to supply a great origin in mathematical research for undergraduate and first-year graduate scholars. The textual content starts off with a dialogue of the genuine quantity process as an entire ordered box. (Dedekind's development is now taken care of in an appendix to bankruptcy I.

Numbers: A Very Short Introduction

Numbers are crucial to our daily lives and issue into virtually every thing we do. during this Very brief creation, Peter M. Higgins, a well known popular-science author, unravels the realm of numbers, demonstrating its richness and supplying an outline of the entire quantity forms that characteristic in glossy technology and arithmetic.

The Number Mysteries: A Mathematical Odyssey through Everyday Life (MacSci)

At any time when we obtain track, take a flight around the Atlantic or speak on our mobile phones, we're hoping on nice mathematical innovations. within the quantity Mysteries, one in every of our generation's optimal mathematicians Marcus du Sautoy bargains a playful and obtainable exam of numbers and the way, regardless of efforts of the best minds, the main basic puzzles of nature stay unsolved.

Additional resources for The Theory of Information and Coding (Encyclopedia of Mathematics and its Applications No. 86)

Show sample text content

1. 27 allow X and Z be self sustaining random variables with non-stop density features, and enable Y ˆ X ‡ Z. If h(Y ) and h( Z) exist, express that I(X ; Y ) ˆ 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 ˆ X ‡ Z. convey that Y has a continuing density, and that if h(Y ) and h( Z) exist, then I(X ; Y ) ˆ h(Y ) À h( Z). express that an identical outcome holds if X and Z are random vectors. 1. 29 If X ˆ (X 1 , . . . , X n ) are self sufficient Gaussian random variables with capability ì i and variances ó 2i and X9 ˆ (X 19 , . . . , X 9n ) are ditto with potential ì9i and variances ó 9i 2 , convey that X ‡ X9 ˆ (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 ) ˆ hello , express that 48 Entropy and mutual info h(X 1 ‡ Á Á Á ‡ X n ) > 2 n ˆ 1 2 log three e 2 hello , iˆ1 with equality iff the X i are Gaussian with variances ó 2i ˆ e 2 hello =2ðe. 1. 31 If X has a continual density, convey that sup H([X ]) ˆ ‡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 ) ˆ 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)] ˆ h(X ) ‡ p(x) logjJ j dx, the place J ˆ J (x1 , . . . , xn ) is the Jacobian of the transformation f . 1. 34 enable X ˆ (X 1 , . . . , X n ) be a random variable with density, and think E[(X i À ì i )(X j À ì j )] ˆ 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 [4], 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) ˆ zero if x P I p(x) f (x) dx ˆ A, the place inf( f ) , A , sup( f ). (For instance, if I ˆ (ÀI, I) and f (x) ˆ (x À ì)2, we'll arrive via a „new course on the onedimensional case of Theorem 1. eleven. ) De®ne G(s) ˆ I e Àsf (x) dx. exhibit that there exists s0 with G9(s0 )=G(s0 ) ˆ ÀA, and de®ne q(x) ˆ e Às0 f (x) =G(s0 ) for x P I, q(x) ˆ zero for x P = I. exhibit that h(X ) < log G(s0 ) ‡ s0 A with equality iff X 's density ˆ q(x) nearly all over the place. [ trace: See Prob. 1. eight. ] follow this basic strategy to 3 instances: (a) f (x) ˆ log x, I ˆ (1, I). (b) f (x) ˆ x, I ˆ (0, I). (c) f (x) ˆ jxj, I ˆ (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 [4], Vol.

Rated 4.45 of 5 – based on 32 votes