By Rudolf Ahlswede (auth.), Alexander Ahlswede, Ingo Althöfer, Christian Deppe, Ulrich Tamm (eds.)
ISBN-10: 3319054783
ISBN-13: 9783319054780
ISBN-10: 3319054791
ISBN-13: 9783319054797
The quantity “Storing and Transmitting info” relies on Rudolf Ahlswede's introductory path on "Information thought I" and provides an creation to Shannon conception. Readers, regularly occurring or surprising with the technical intricacies of knowledge thought, will profit significantly from operating throughout the e-book; specifically bankruptcy VI with its energetic reviews and uncensored insider perspectives from the area of technology and learn deals informative and revealing insights. this is often the 1st of a number of volumes that might function a gathered learn documentation of Rudolf Ahlswede’s lectures on info thought. every one quantity comprises reviews from an invited famous specialist. Holger Boche contributed his insights within the complement of the current volume.
Classical details processing matters the most projects of gaining wisdom, garage, transmitting and hiding info. the 1st job is the best aim of records. For the 2 subsequent, Shannon provided a powerful mathematical thought referred to as info idea, which he in response to probabilistic types. the idea principally consists of the idea that of codes with small errors chances regardless of noise within the transmission, that's modeled by means of channels. The lectures awarded during this paintings are compatible for graduate scholars in arithmetic, and in addition in Theoretical computing device technology, Physics, and electric Engineering with history in simple arithmetic. The lectures can be utilized because the foundation for classes or to complement classes in lots of methods. Ph.D. scholars also will locate learn difficulties, frequently with conjectures, that supply capability topics for a thesis. extra complex researchers may possibly locate the root of complete study programs.
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Additional info for Storing and Transmitting Data: Rudolf Ahlswede’s Lectures on Information Theory 1
Example text
1, 0) . P(n, {0, 1}) = (0, 1), ( , n n n n n n So the empirical classes are just the levels in the poset P({1, . . , n}) and the typical sequences to P = ( ni , n−i n ) are those sequences with exactly i 0’s and (n − i) 1’s. 26 2 Data Compression Observe that the set of entropy-typical sequences E(n, P, δ) is the union of ED classes, since permutations of the xi ’s in x n = (x1 , . . , xn ) do not change the probn ability Pn (x n ) = x⊂X P(x)(x |x) . The following four lemmas will later be used in the second proof of the Coding Theorem for the DMS.
Pn ), for which the three axioms n pi · log pi . of the theorem hold, must be of the form H ( p1 , . . , pn ) = −K i=1 n On the other hand, it is easily verified that −K i=1 pi · log pi is continuous in pi , i = 1, . . , n and monotonely increasing in n for P = ( n1 , . . , n1 ). The grouping axiom holds, since for P = ( p1 , . . , pn ) and Q = (q1 , . . , qs ), with n q j = n jj−1 +1 pi , j = 1, . . , qj nj s q j · log q j + j=1 pn j qj j=1 i=n j−1 +1 ⎞ pi pi ⎠ · log qj qj 42 3 The Entropy as a Measure of Uncertainty ⎛ = −K · ⎝ s ⎛ nj ⎝q j · log q j + pi ⎠⎠ pi · log pi − log q j · i=n j−1 +1 j=1 ⎞⎞ nj i=n j−1 +1 n = −K · pi · log pi = H (P).
But Qn (TQn ) → (n + 1)−|X | , since by the Lemmas 3 and 6 (n + 1)|X | · Qn (TQn ) → Qn (TQn ) → P⊂P (n,X ) Qn (TPn ) P⊂P (n,X ) Qn (x n ) = = P⊂P (n,X ) x n ⊂TPn Qn (x n ) = 1 x n ⊂X n and hence |TQn | → (n + 1)−|X | exp{n · H(Q)}. Summarizing the results of the preceding lemmas the entropy can be combinatorially interpreted as a measure for the cardinality of empirical classes. More exactly, with Lemma 7 |TQn | is asymptotically exp{n · H(Q)}, since (n + 1)|X | is polynomial in n. The relative entropy can be regarded as a measure for errors.
Storing and Transmitting Data: Rudolf Ahlswede’s Lectures on Information Theory 1 by Rudolf Ahlswede (auth.), Alexander Ahlswede, Ingo Althöfer, Christian Deppe, Ulrich Tamm (eds.)
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