Buku Transmitting and Gaining Data by Rudolf AhlswedeAlexander Ahlswede Ingo Althöfer Christian Deppe & Ulrich Tamm
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Buku Transmitting and Gaining Data by Rudolf AhlswedeAlexander Ahlswede Ingo Althöfer Christian Deppe & Ulrich Tamm

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Transmitting and Gaining Data by Rudolf AhlswedeAlexander Ahlswede Ingo Althöfer Christian Deppe & Ulrich Tamm

Author:Rudolf AhlswedeAlexander Ahlswede, Ingo Althöfer, Christian Deppe & Ulrich Tamm

Language: eng

Format: epub

Publisher: Springer International Publishing, Cham

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Transmitting and Gaining Data by Rudolf AhlswedeAlexander Ahlswede Ingo Althöfer Christian Deppe & Ulrich Tamm

Since , it follows that for at least one , say , we must have . The interpretation of this inequality is straightforward; combined with m it permits us to “transmit” the sequence in any order we choose, and still receive each sequence correctly a fraction of at least of the times that it is transmitted, as this number goes to infinity, according to the law of large numbers (see also Exercise concerning Remark 2.12 on p. 104 of [10]). The only remaining point is that we still have sequences, but their length is now . However, since is fixed while may be taken arbitrarily large, the way out is clear.

We prove the theorem for some satisfying and take so large that . The number of sequences is then given by . These explanations are taken from [10]. “Wolfowitz [58, p.60], refers to it as an excellent description of the customary treatment”.

He calls the approach presented in the previous section as now conventional. We explained earlier that he shares this radically different view with Feinstein [11]. It is briefly described. In coding theory stochastic inputs serve only as a tool in proving coding theorems and are not of interest per se. Given is only the channel. Its operationally defined capacities , , (weak and strong capacities) are to be characterized and if possible in a computable form. It was shown for the DFMC that the weak capacity could be computed to any desired degree of accuracy by computing the rate of transmission for a suitable stochastic input of i.i.d. random sequences of length , the sequence distribution may be nonstationary.

 

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Transmitting and Gaining Data by Rudolf AhlswedeAlexander Ahlswede Ingo Althöfer Christian Deppe & Ulrich Tamm

Author:Rudolf AhlswedeAlexander Ahlswede, Ingo Althöfer, Christian Deppe & Ulrich Tamm , Date: June 27, 2019

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Author:Rudolf AhlswedeAlexander Ahlswede, Ingo Althöfer, Christian Deppe & Ulrich Tamm

Language: eng

Format: epub

Publisher: Springer International Publishing, Cham
Since , it follows that for at least one , say , we must have . The interpretation of this inequality is straightforward; combined with m it permits us to “transmit” the sequence in any order we choose, and still receive each sequence correctly a fraction of at least of the times that it is transmitted, as this number goes to infinity, according to the law of large numbers (see also Exercise concerning Remark 2.12 on p. 104 of [10]). The only remaining point is that we still have sequences, but their length is now . However, since is fixed while may be taken arbitrarily large, the way out is clear.

We prove the theorem for some satisfying and take so large that . The number of sequences is then given by . These explanations are taken from [10]. “Wolfowitz [58, p.60], refers to it as an excellent description of the customary treatment”.

He calls the approach presented in the previous section as now conventional. We explained earlier that he shares this radically different view with Feinstein [11]. It is briefly described. In coding theory stochastic inputs serve only as a tool in proving coding theorems and are not of interest per se. Given is only the channel. Its operationally defined capacities , , (weak and strong capacities) are to be characterized and if possible in a computable form. It was shown for the DFMC that the weak capacity could be computed to any desired degree of accuracy by computing the rate of transmission for a suitable stochastic input of i.i.d. random sequences of length , the sequence distribution may be nonstationary.

Whether the resulting is ergodic or whether are questions which do not arise in this approach, which does not use the AEP. But the present approach helps in answering the questions.

For the DFMC Tsaregradskii’s proof is complete and gives the positive answer by showing that for , which he introduces(!), and . Feinstein [11, p.42], remarks that he independently followed the same approach. Nedoma proved the equality for his DFMC, again by equating the two quantities with another capacity, his . Again for the DFMC Breiman [5] also proves the equation. Moreover, he shows that there is always at least one ergodic assuming the supremum: , a noticeable result. It was at the beginning of ergodic decomposition formulas by Parthasarathy and Jacobs (see Sect. 3.8).

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