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Continuity of Channel Parameters and Operations under Various DMC Topologies.

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Topological Structures on DMC Spaces †.

Rajai Nasser1

  • 1École Polytechnique Fédérale de Lausanne, Route Cantonale, 1015 Lausanne, Switzerland.

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|December 3, 2020
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Summary

This study defines natural topologies for equivalent channels, proving they are σ-compact, separable, and path-connected. A new noisiness topology is introduced, offering a natural metric for channel comparison.

Keywords:
Blackwell measurediscrete memoryless channelstopologytotal-variation distance

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Area of Science:

  • Information Theory
  • Topology
  • Channel Equivalence

Background:

  • Equivalent channels are fundamental in information theory, representing systems degradable from each other.
  • The space of equivalent channels requires a topological structure for analysis.
  • Existing topological structures may not adequately capture channel properties.

Purpose of the Study:

  • To define and characterize natural topologies on the space of equivalent channels.
  • To investigate the properties of the finest natural topology (strong topology).
  • To introduce and analyze a new metric-based topology (noisiness topology) for comparing channel noise levels.

Main Methods:

  • Utilizing quotient topology to define natural topologies on equivalent channel spaces.
  • Analyzing topological properties such as compactness, separability, and path-connectedness.
  • Introducing a metric distance to define the noisiness topology and studying its properties.

Main Results:

  • All natural topologies are proven to be σ-compact, separable, and path-connected.
  • The strong topology is compactly generated, sequential, and T4, but not first-countable or metrizable.
  • The newly introduced noisiness topology is shown to be natural.

Conclusions:

  • Natural topologies provide a robust framework for studying equivalent channels.
  • The strong topology offers advanced topological properties but lacks countability.
  • The noisiness topology offers a metrizable approach to comparing channel noise levels.