2016-08-19

メモ2

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(N N N N N N N N N N N N 2 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1) 
(N N N N N N N N N N N N 2 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1) 
(N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N 2 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1) 
(N N N N N N N N N N N N 2 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1) 
(N N N N N N N N N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2) 
(N N N N N N N N N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2) 
(N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N 2 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1) 
(N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2) 
(N N N N N N N N N N N N 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2) 
(N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N 2 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1) 
(N N N N N N N N N N N N 2 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1) 
(N N N N N N N N N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2) 
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(N N N N N N N N N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2) 
(N N N N N N N N N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2) 
(N N N N N N N N N N N N 2 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1) 
(N N N N N N N N N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2) 
(N N N N N N N N N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2) 
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(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1) 
(N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1) 
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(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
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(N N N N N N N N N N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1) 
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(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
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(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
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(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
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(N N N N N N N N N N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1) 
(N N N N N N N N N N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1) 
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(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
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(N N N N N N N N N N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1) 
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(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2) 
(N N N N N N N N N N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1) 
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(N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N 2 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N 1 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N 1 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N 2 2 2 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
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(N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2) 
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(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2) 
(N N N N N N N N N N N N N N N 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N N 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2) 
(N N N N N N N N N N N N N N N 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1 2) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2) 
(N N N N N N N N N N N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 2 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1) 
(N N N N N N N N N N N N N N N 2 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 2 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1) 
(N N N N N N N N N N N N N N N 2 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1) 
(N N N N N N N N N N N N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2) 
(N N N N N N N N N N N N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 2 2 1 1 2 2 1 2 2 1 2 2 1 2 2 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
(N N N N N N N N N N N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1)

2016-08-19

メモ

k=1 (1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1) 
K=2 (N 1 1 2 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1 1) 
k=3 (N N N 1 1 1 2 1 1 2 1 1 2 1 1 2 1 1 2 1) 
k=4 (N N N 1 2 1 1 1 2 1 1 2 1 1 2 1 1 2 1 1) 
k=5 (N N N N 2 1 1 1 2 1 1 2 1 1 2 1 1 2 1 1) 
K=6 (N N N N 1 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2) 
k=7 (N N N N N 1 2 2 1 1 2 1 1 2 1 1 2 1 1 2) 
k=8 (N N N N N 2 1 1 1 2 1 1 2 1 1 2 1 1 2 1) 
k=9 (N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1) 
k=10(N N N N N 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1)

2016-06-19

ガウス雑音がある通信路の通信路容量

ガウス雑音がある通信路の通信路容量
$C=Wlog(\frac{S+N}{N})$
を証明します。以下は、甘利俊一著「情報理論」に書いてあることをまとめたものです。

証明のポイントは2つです。
加法的雑音があるとき、通信路容量の式が
$C=\frac{1}{T}\max_{p(\vec{x})}\left( H(Y)-H(Y|X) \right)=\frac{1}{T}\max_{p(\vec{x})}H(Y)-\frac{1}{T}H(N)$
となることと、
平均電力 $P$ が与えられたときの最大エントロピーが
$H(X)=2WTlog\sqrt{2\pi eP}$
となることです。

それでは、証明してみます。
標本化定理より、最大周波数 $W$ の送信信号 $x(t)$ は
$x(t)=\sum_i x_i a_i(t)=\sum_i x(\frac{i}{2W})\frac{sin \pi(2Wt-i)}{\pi(2Wt-i)}$
と表せます。
$a_i(t)=\frac{sin \pi(2Wt-i)}{\pi(2Wt-i)}$
は、標本化関数で直交完全系をなします。但し、ノルムの2乗は $\frac{1}{2W}$ です。
$\int_0^T a_i(t)^2 dt = \frac{1}{2W}$
$x(t)$ は $x_i$ を成分とするベクトル $\vec{x}$ で表せます。このベクトルの次元 $n$ は $T/(\frac{1}{2W})=2WT$ です。
受信信号 $y(t)$ も同様に $\vec{y}$ で表し、雑音信号 $n(t)$ も同様に $\vec{n}$ で表します。
$y(t) = x(t) + n(t)$
なので、
$\vec{y}=\vec{x}+\vec{n}$
となります。
通信路容量 $C$ は
$C=\frac{1}{T}\max_{p(\vec{x})}\left(H(Y)-H(Y|X)\right)$
です。ここで
$H(Y|X)=-\iint p(\vec{x})q(\vec{y}-\vec{x})log q(\vec{y}-\vec{x}) d\vec{x} d\vec{y}$
$=-\int p(\vec{x})d\vec{x}\int q(\vec{y}-\vec{x})log q(\vec{y}-\vec{x})d\vec{y}=H(N)$
と書けます。加法的雑音の場合 $p(\vec{y}|\vec{x})$ は雑音の確率分布 $q(\vec{n})=q(\vec{y}-\vec{x})$ に等しいことを使いました。
よって、
$C=\frac{1}{T}\max_{p(\vec{x})}H(Y)-\frac{1}{T}H(N)$
となります。
受信信号の電力は
$\frac{1}{T}\int_0^T y(t)^2 dt = \frac{1}{2WT}\vec{y}\cdot\vec{y}$
$=\frac{1}{2WT}(\vec{x}+\vec{n})\cdot(\vec{x}+\vec{n})$
$=\frac{1}{2WT}(\vec{x}\cdot\vec{x}+\vec{n}\cdot\vec{n})$
送信信号と雑音信号が独立であること $\vec{x}\cdot\vec{n}=0$ を使いました。
送信信号の平均電力を
$\frac{1}{2WT}\overline{\vec{x}\cdot\vec{x}}=\frac{1}{2WT}\int\vec{x}\cdot\vec{x}p(\vec{x})d\vec{x}=S$
とし、雑音信号の平均電力を
$\frac{1}{2WT}\overline{\vec{n}\cdot\vec{n}}=\frac{1}{2WT}\int\vec{n}\cdot\vec{n}q(\vec{n})d\vec{n}=N$
とすると、受信信号の平均電力は
$\frac{1}{2WT}\overline{\vec{y}\cdot\vec{y}}=S+N$
となります。
ここで、平均電力
$\frac{1}{2WT}\overline{\vec{x}\cdot\vec{x}}=\frac{1}{2WT}\int \vec{x}\cdot\vec{x}p(\vec{x})d\vec{x}=P$
が与えられているとき $H(X)$ を最大化する $p(\vec{x})$ を求めます。
ラグランジュの未定乗数 $\lambda$ 、 $\mu$ を導入した関数 $F$ を考えます。
$F=-\int p(\vec{x})log p(\vec{x})d\vec{x}-\lambda\int p(\vec{x})d\vec{x}-\mu\int \vec{x}\cdot\vec{x}p(\vec{x})d\vec{x}$
$p(\vec{x})\to p(\vec{x})+\delta p(\vec{x})$ としたとき $\delta F=0$ となる $p(\vec{x})$ は
$\delta F=-\int(log p(\vec{x}) + 1 + \lambda + \mu \vec{x}\cdot\vec{x})\delta p(\vec{x})d\vec{x}=0$
より、
$log p(\vec{x}) + 1 + \lambda + \mu \vec{x}\cdot\vec{x}=0$
すなわち
$p(\vec{x})=\frac{1}{(\sqrt{2\pi P})^{2WT}}e(-\frac{\vec{x}\cdot\vec{x}}{2P})$
となります。
このときエントロピーは
$H(X)=-\int p(\vec{x})log p(\vec{x})d\vec{x}=2WTlog\sqrt{2\pi e P}$
となります。
よって、通信路容量は
$C=\frac{2WT}{T}log\sqrt{2\pi e(S+N)}-\frac{1}{T}H(N)$
となります。雑音がガウス雑音のとき、雑音信号の確率分布は
$q(\vec{n})=\frac{1}{(\sqrt{2\pi N})^{2WT}}e(-\frac{\vec{x}\cdot\vec{x}}{2N})$
なので、
$H(N)=2WTlog\sqrt{2\pi e N}$
となり、
$C=\frac{2WT}{T}log\sqrt{2\pi e(S+N)}-\frac{2WT}{T}log\sqrt{2\pi e N}=Wlog\frac{S+N}{N}$
と示せました。

2016-06-12

初めての人のためのLISP 第6講

lisp

読者の宿題、atom-countを書いてみた。

(defun atom-count (x)
  (cond ((atom x) 1)
        (t (atom-count2 x))))

(defun atom-count2 (x)
  (cond ((null x) 0)
        ((atom (car x)) (1+ (atom-count2 (cdr x))))
        (t (+ (atom-count2 (car x)) (atom-count2 (cdr x))))))

2016-06-05

定常情報源のエントロピーレート

定常情報源のエントロピーレートには2つの表し方があります。
1つは
$lim_{n \to \infty} \frac{H(X_1, \cdots , X_n)}{n}$
で、もう1つは
$lim_{n \to \infty} H(X_n | X_1, \cdots , X_{n-1})$
です。
この2つの極限が等しいことを示します。
まず、以下が成り立ちます。これをチェインルールといいます。
$H(X_1, \cdots , X_n) = H(X_1) + H(X_2|X_1) + H(X_3|X_1X_2) + \cdots + H(X_n|X_1, \cdots , X_{n-1})$
すなわち
$\frac{H(X_1, \cdots , X_n)}{n} = \frac{1}{n}\sum_{i=1}^{n}H(X_i|X_1, \cdots , X_{i-1})$
です。
ここで、条件が多くなるとエントロピーが小さくなるため、以下の式が成り立ちます。
$H(X_{i+1}|X_1 , \cdots , X_i) \geq H(X_{i+1} | X_2 , \cdots , X_i)$
上式の右辺は定常情報源の性質から以下となります。
$H(X_{i+1}| X_2 , \cdots , X_i) = H(X_i | X_1 , \cdots , X_{i-1})$ 　
上の2つの式から、 $H(X_i|X_1 , \cdots , X_{i-1})$ はiについて、単調非増加であることを示しています。このことと、エントロピーが非負であることから、極限 $lim_{n \to \infty} H(X_n | X_1, \cdots , X_{n-1}) = H^'$ が存在することがわかります。

極限が存在することがわかったので、2つの表現が等しいことを示します。
以下のチェザロ平均aというものの性質を使います。
$lim_{n \to \infty}a_n = a,\,\,\, b_n = \frac{1}{n}\sum_{i=1}^n a_i \Rightarrow lim_{n \to \infty}b_n=a$
$a_i$ を $H(X_i|X_1, \cdots , X_{i-1})$ とみなし、 $b_n$ を $\frac{H(X_n|X_1, \cdots , X_{n-1})}{n}$ とみなすと、
$lim_{n \to \infty} H(X_n|X_1, \cdots , X_{n-1}) = H^'$
が成り立つことがわかります。

2016-04-12

プログラミングGauche メタオブジェクトプロトコル

Scheme

『プログラミングGauche』17章総称関数とオブジェクトの練習問題をやってみました。

総称関数subは、関数呼び出しの回数と、関数実行にかかった時間を保持するクラスと、引数と結果を出力するクラスを継承しています。

このままだと、クラスが保持する時間には、クラスによるログ出力にかかった時間が含まれてしまいます。

そこで、クラスにログのON/OFFをするためのスロットlog-on(初期値#t)を追加してみました。

(use gauche.time)

(define-class <logger-generic> (<generic>)
  ((log-on :init-value #t)))

(define-method apply-generic ((gf <logger-generic>) args)
  (if (ref gf 'log-on)
      (begin
        (format #t "args: ~s\n" args)
        (let ((return-values (next-method)))
          (format #t "result: ~s\n" return-values)
          return-values))
      (next-method)))

(define-class <profiler-generic> (<logger-generic>)
  ((counter :init-value 0)
   (time :init-form (make <real-time-counter>))))

(define-method apply-generic ((gf <profiler-generic>) args)
  (inc! (ref gf 'counter))
  (with-time-counter (ref gf 'time) (next-method)))

(define-method get-profiles ((gf <profiler-generic>))
  (format #t "~s: ~d times called and spent time ~d\n"
          (ref gf 'name) (ref gf 'counter)
          (time-counter-value (ref gf 'time))))

(define-method init-profile ((gf <profiler-generic>))
  (set! (ref gf 'counter) 0)
  (set! (ref gf 'time) (make <real-time-counter>)))

(define-generic sub :class <profiler-generic>)

(define-method sub ((num1 <number>) (num2 <number>))
  (- num1 num2))

以下のように使います。

gosh> (get-profiles sub)
sub: 0 times called and spent time 0
#<undef>
gosh> (define val (with-output-to-string
                      (lambda ()
                          (map sub (iota 10000 100 3) (iota 10000 300 5)))))
val
gosh> (get-profiles sub)
sub: 10000 times called and spent time 1.3829899999999549
#<undef>
gosh> (init-profile sub)
#<undef>
gosh> (get-profiles sub)
sub: 0 times called and spent time 0
#<undef>
gosh> (set! (ref sub 'log-on) #f)
#<undef>
gosh> (define val (with-output-to-string
                      (lambda ()
                          (map sub (iota 10000 100 3) (iota 10000 300 5)))))
val
gosh> (get-profiles sub)
sub: 10000 times called and spent time 0.182114999999972
#<undef>

2016-03-29

プログラミングGauche アドベンチャーゲーム

Scheme

アドベンチャーゲーム

;; -*- coding: utf-8 -*-

(use util.match)

(define *player*
  (make-player 'hp 320 'mp 66 'position (car *dungeon*)
               'inventory '(potion potion daggar cookie daggar)))

(define (reset!)
  (set! *player*
        (make-player 'hp 320 'mp 66 'position (car *dungeon*)
                     'inventory '(potion potion dagger cookie daggar))))

(define (make-player . args)
  (let loop ((lis args))
    (match lis
           [() '()]
           [(attr value . rest) (cons (cons attr value) (loop rest))]
           [_ (error "Number of arguments must be even:" args)])))

(define *dungeon*
  '(["あなたは森の北端にいる。道は南に続いている。"
     (s . 1)]
    ["あなたは鬱蒼とした森の中の道にいる。森は南北に伸びている。東に降りてゆく小径がある。"
     (n . 0)
     (s . 2)
     (e . 3)]
    ["足元がぬかるんでいる。道は直角に折れ、北と西に伸びている。西に続く道の先が明るくなっている。"
     (n . 1)
     (w . 4)]
    ["あなたは沼のほとりにいる。空気の動きが止まり、暑さを感じる。西に昇ってゆく小径がある。"
     (w . 1)]
    ["突然目の前が開けた。あなたは森の中の広場にいる。丈の短い、柔らかそうな草が一面に広場を覆っている。道が東に伸びている。"
     (e . 2)]))

(define *item-database*
  `((potion (drink . ,(cut add-hp! <> 50))
            (throw . ,(cut add-hp! <> -3)))
    (elixir (drink . ,(cut add-mp! <> 50))
            (throw . ,(cut add-mp! <> -3)))
    (pancake (eat . ,(cut add-hp! <> 30))
             (throw . ,(cut add-hp! <> -2)))
    (cookie (eat . ,(cut add-hp! <> 7))
            (throw . ,(cut add-hp! <> -1)))
    (dagger (throw . ,(lambda (_) #f)))))

(define (get-player-attr player attr)
  (cdr (assoc attr player)))

(define (update-player-attr! player attr updater)
  (let ((p (assoc attr player)))
    (set! (cdr p) (updater (cdr p)))))

(define (get-inventory player)
  (get-player-attr player 'inventory))

(define (has-item? player item)
  (member item (get-inventory player)))

(define (delete-item! player item)
  (update-player-attr! player 'inventory (cut delete-1 item <>)))

(define (delete-1 elt lis . options)
  (let-optionals* options ((cmp-fn equal?))
                  (let loop ((lis lis))
                    (cond [(null? lis) '()]
                          [(cmp-fn elt (car lis)) (cdr lis)]
                          [else (cons (car lis) (loop (cdr lis)))]))))

(define (add-item! player item)
  (update-player-attr! player 'inventory (cut cons item <>)))

(define (get-hp player)
  (get-player-attr player 'hp))

(define (add-hp! player n)
  (update-player-attr! player 'hp (cut + n <>)))

(define (get-mp player)
  (get-player-attr player 'mp))

(define (add-mp! player n)
  (update-player-attr! player 'mp (cut + n <>)))

(define (get-position player)
  (get-player-attr player 'position))

(define (set-position! player pos)
  (update-player-attr! player 'position (lambda (_) pos)))

(define (use-item! what item)
  (cond [(not (has-item? *player* item))
         (print item "を持っていません。")]
        [(itemproperty-get item what)
         => (lambda (action)
              (delete-item! *player* item)
              (action *player*))]
        [else (print item "を" what "することはできません。")])
  #t)

(define (item-properties item)
  (cond [(assoc item *item-database*) => cdr]
        [else '()]))

(define (itemproperty-get item what)
  (cond [(assoc what (item-properties item)) => cdr]
        [else #f]))

(define (describe)
  (print (car (get-position *player*)))
  #t)

(define (move! direction)
  (let ((position (get-position *player*)))
    (cond [(assoc direction (cdr position))
           => (lambda (p)
                (set-position! *player* (list-ref *dungeon* (cdr p))))]
          [else (print "そちらには移動できません。")]))
  #t)

(define (status)
  (print "hp :" (get-hp *player*))
  (print "mp :" (get-mp *player*))
  (print "inventory :" (get-inventory *player*))
  #t)