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Topological Structures on DMC Spaces <sup>†</sup>.

Entropy (Basel, Switzerland)·2020
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This study demonstrates that key parameters and operations for discrete memoryless channels (DMC) remain continuous across different topological spaces. This continuity applies to measures like mutual information, channel capacity, and error probabilities.

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Area of Science:

  • Information Theory
  • Channel Coding
  • Mathematical Analysis

Background:

  • Discrete memoryless channels (DMC) are fundamental in information theory.
  • Understanding the topological properties of the space of DMCs is crucial for analyzing channel behavior.
  • Previous work has explored various aspects of DMCs, but continuity under topological spaces requires further investigation.

Purpose of the Study:

  • To investigate the continuity of essential channel parameters and operations within the space of equivalent discrete memoryless channels (DMC).
  • To establish a theoretical foundation for analyzing the stability and predictability of channel characteristics under varying topological conditions.
  • To provide insights into the robustness of information-theoretic measures and channel operations.

Main Methods:

  • Analysis of topological spaces applied to the set of equivalent discrete memoryless channels (DMC).
  • Mathematical proofs demonstrating the continuity of specific channel parameters.
  • Examination of the behavior of channel operations under different topological structures.

Main Results:

  • Mutual information, channel capacity, Bhattacharyya parameter, and probability of error (both fixed code and optimal) are shown to be continuous under various DMC topologies.
  • Channel operations including sums, products, interpolations, and Arıkan-style transformations are proven to be continuous.
  • The continuity of these fundamental properties ensures predictable behavior of DMCs in different topological settings.

Conclusions:

  • The continuity of key parameters and operations validates their reliable use in theoretical and practical applications of discrete memoryless channels (DMC).
  • This research provides a robust mathematical framework for understanding channel behavior under topological variations.
  • The findings contribute to the stability analysis of communication systems relying on discrete memoryless channels (DMC).