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Time-Limited Codewords over Band-Limited Channels: Data Rates and the Dimension of the W-T Space.

Youssef Jaffal1, Ibrahim Abou-Faycal1

  • 1Department of Electrical and Computer Engineering, American University of Beirut, Beirut 1107 2020, Lebanon.

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Summary

This study analyzes information rates and degrees of freedom in communication systems with finite block-length coding. Results provide bounds for finite WT values, showing they approach Shannon capacity as WT increases.

Keywords:
band-limiteddegrees of freedominformation ratesprolate spheroidal wave functionstime-limited

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Area of Science:

  • Information Theory
  • Digital Communications
  • Signal Processing

Background:

  • Shannon's capacity defines theoretical limits for communication over noisy channels.
  • In the asymptotic regime (WT→∞), achievable rates approach capacity with 2WT degrees of freedom.
  • Finite block-length coding introduces deviations from asymptotic capacity.

Purpose of the Study:

  • To investigate degrees of freedom and achievable information rates for finite values of WT.
  • To derive and numerically evaluate bounds on these parameters.
  • To understand the relationship between finite block-length coding and channel capacity.

Main Methods:

  • Utilizing prolate spheroidal wave functions for a discrete formulation.
  • Applying Polyanskiy's finite block-length coding theory.
  • Deriving upper and lower bounds on achievable rates and degrees of freedom.

Main Results:

  • Derived asymptotically tight upper and lower bounds for achievable rates and degrees of freedom at finite WT.
  • Numerically evaluated bounds for sample 2WT values, showing convergence as 2WT increases.
  • Established a logarithmic upper bound on the decrease in degrees of freedom from 2WT.

Conclusions:

  • The derived bounds are effective for finite block-length regimes.
  • The gap between bounds diminishes with increasing 2WT, confirming asymptotic behavior.
  • Finite block-length coding introduces a bounded loss in degrees of freedom compared to the ideal asymptotic case.