| SHEN SHIYI.[J].数学年刊A辑,1980,1(2):223~234 |
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| SOURCE CODING THEOREMS FOR GENERALSEQUENCE MODEL |
| Received:October 24, 1978 |
| DOI： |
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| 英文关键词: |
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| 英文摘要: |
| In this paper, we give a coding theorem for general source sequence. A source
sequence
\[{\mathcal{T}^{(n)}} = \{ [{X^{(n)}},{p^{(n)}}({X^{(n)}})],[{X^{(n)}} \otimes {Y^{(n)}},{\rho ^{(n)}}({X^{(n)}},{Y^{(n)}})]\} \]
is said to be \[({R^{(n)}},{d^{(n)}})\]-compress, if (i)\[{R^{(n)}}\], \[{d^{(n)}}\] are two sequences of real numbers
\[{R^{(n)}} \to \infty \]； (ii) there exist \[{\varepsilon ^{(n)}} > 0({\varepsilon ^{(n)}} \to 0)\] and set \[{B^{(n)}} \subset {Y^{(n)}}\] so that \[|{B^{(n)}}| \leqslant {2^{{R^{(n)}}(1 + {\varepsilon ^{(n)}})}}\]
and
\[{p^{(n)}}{\text{\{ }}({X^{(n)}}){\rho ^{(n)}}({X^{(n)}},{B^{(n)}}) \leqslant {d^{(n)}}\} \geqslant 1 - {\varepsilon ^{(n)}}\]
where \[{\rho ^{(n)}}({X^{(n)}},{B^{(n)}}) = \mathop {\min }\limits_{{y^{(n)}} \in {B^{(n}}} {\rho ^{(n)}}({X^{(n)}},{Y^{(n)}})\].
A \[{\mathcal{T}^{(n)}}\] is said to be \[({\mathcal{F}^{(n)}}{D^{(n)}})\]-information bounded, if (i)\[{\mathcal{F}^{(n)}} \to \infty \];(ii) there exist \[{\varepsilon ^{(n)}} > 0\] and conditional probability \[{Q^{(n)}}({Y^{(n)}}/{X^{(n)}})\] so that probability distribution
\[{p^{(n)}}({X^{(n)}}{Y^{(n)}}) = {p^{(n)}}({X^{(n)}}){Q^{(n)}}({Y^{(n)}}/{X^{(n)}})\] is satisfied by
\[\begin{gathered}
{p^{(n)}}\{ ({X^{(n)}},{Y^{(n)}}):i({X^{(n)}},{Y^{(n)}}) \leqslant {\mathcal{F}^{(n)}}(1 + {\varepsilon ^{(n)}}), \hfill \ {\rho ^{(n)}}({X^{(n)}},{Y^{(n)}}) \leqslant {\rho ^{(n)}}\} \geqslant 1 - {\varepsilon ^{(n)}} \hfill \\
\end{gathered} \]
where
\[i({X^{(n)}},{Y^{(n)}}) = \log \frac{{{p^{(n)}}({X^{(n)}},{Y^{(n)}})}}{{{p^{(n)}}({X^{(n)}}){p^{(n)}}({Y^{(n)}})}}\]
Theorem The necessary and sufficient conditions for a source sequence \[{\mathcal{F}^{(n)}}\] to be
\[({R^{(n)}},{a^{(n)}})\]-compress is that \[{\mathcal{F}^{(n)}}\] must be \[({R^{(n)}},{a^{(n)}})\]-information bounded.
From the theorem we obtain immediately the coding theorem and its converse for stationary and unstationary sources with memory. |
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