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An Integral Representation of the Logarithmic Function with Applications in Information Theory
1The Andrew and Erna Viterbi Faculty of Electrical Engineering, Israel Institute of Technology Technion City, Haifa 3200003, Israel.
Entropy (Basel, Switzerland)
|December 8, 2020
Summary
This study introduces an integral representation for logarithmic functions, simplifying calculations for expectations and moments. This method offers a rigorous alternative for information theory applications like data compression and channel capacity.
Area of Science:
- Information Theory
- Mathematical Physics
- Probability Theory
Background:
- Logarithmic functions are fundamental in various scientific domains.
- Existing methods for calculating moments and expectations of log-transformed variables can be complex.
- The replica method is popular but lacks mathematical rigor in certain applications.
Purpose of the Study:
- To present a novel integral representation of the logarithmic function.
- To demonstrate its utility in deriving exact formulas for expectations and moments.
- To offer a rigorous alternative to the replica method in specific contexts.
Main Methods:
- Utilizing a well-established integral representation of the logarithm.
- Applying this representation to derive formulas for statistical moments.
- Investigating applications in information theory and communication systems.
Main Results:
- Development of compact and computable exact formulas for log-variable expectations and moments.
- Successful application in universal lossless data compression.
- Accurate evaluation of entropy, differential entropy, and ergodic capacity for SIMO Gaussian channels.
Conclusions:
- The integral representation provides a powerful and rigorous tool for analyzing logarithmic functions.
- This approach simplifies complex calculations in information theory and related fields.
- It offers a mathematically sound alternative to less rigorous methods like the replica method.
Keywords:
SIMO channeldifferential entropyentropyergodic capacityintegral representationlogarithmic expectationmultivariate Cauchy distributionuniversal data compressionMore Related Videos
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