Almost Sure Convergence of Relative Frequency of Occurrence of Burst Errors on Channels with Memory

Mitsuru HAMADA

Almost Sure Convergence of Relative Frequency of Occurrence of Burst Errors on Channels with Memory

Mitsuru HAMADA

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Motivated by intention to evaluate asymptotically multiple-burst-error-correcting codes on channels with memory, we will derive the following fact. Let {Z_i } be a hidden Markov process, i. e. , a functional of a Markov chain with a finite state space, and W_b(Z₁Z₂Z_n) denote the number of burst errors that appear in Z₁Z₂Z_n, where the number of burst errors is counted using Gabidulin's burst metric , 1971. As the main result, we will prove the almost sure convergence of relative burst weight W_b(Z₁Z₂Z_n)/n, i. e. , the relative frequency of occurrence of burst errors, for a broad class of functionals { Z_i } of finite Markov chains. Functionals of Markov chains are often adopted as models of the noises on channels, especially on burst-noise channels, the most famous model of which is probably the Gilbert channel proposed in 1960. Those channel models described with Markov chains are called channels with memory (including channels with zero-memory, i. e. , memoryless ones). This work's achievement enables us to extend Gilbert's code performance evaluation in 1952, a landmark that offered the well-known Gilbert bound, discussed its relationship to the (memoryless) binary symmetric channel, and has been serving as a guide for the-Hamming-metric-based design of error-correcting codes, to the case of the-burst-metric-based codes (burst-error-correcting codes) and discrete channels with or without memory.

Publication: IEICE TRANSACTIONS on Fundamentals Vol.E82-A No.10 pp.2022-2033

Publication Date: 1999/10/25

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Type of Manuscript: Special Section PAPER (Special Section on Information Theory and Its Applications)

Category: Coding Theory

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Mitsuru HAMADA, "Almost Sure Convergence of Relative Frequency of Occurrence of Burst Errors on Channels with Memory" in IEICE TRANSACTIONS on Fundamentals, vol. E82-A, no. 10, pp. 2022-2033, October 1999, doi: .
Abstract: Motivated by intention to evaluate asymptotically multiple-burst-error-correcting codes on channels with memory, we will derive the following fact. Let {Z_i } be a hidden Markov process, i. e. , a functional of a Markov chain with a finite state space, and W_b(Z₁Z₂Z_n) denote the number of burst errors that appear in Z₁Z₂Z_n, where the number of burst errors is counted using Gabidulin's burst metric , 1971. As the main result, we will prove the almost sure convergence of relative burst weight W_b(Z₁Z₂Z_n)/n, i. e. , the relative frequency of occurrence of burst errors, for a broad class of functionals { Z_i } of finite Markov chains. Functionals of Markov chains are often adopted as models of the noises on channels, especially on burst-noise channels, the most famous model of which is probably the Gilbert channel proposed in 1960. Those channel models described with Markov chains are called channels with memory (including channels with zero-memory, i. e. , memoryless ones). This work's achievement enables us to extend Gilbert's code performance evaluation in 1952, a landmark that offered the well-known Gilbert bound, discussed its relationship to the (memoryless) binary symmetric channel, and has been serving as a guide for the-Hamming-metric-based design of error-correcting codes, to the case of the-burst-metric-based codes (burst-error-correcting codes) and discrete channels with or without memory.
URL: https://global.ieice.org/en_transactions/fundamentals/10.1587/e82-a_10_2022/_p

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@ARTICLE{e82-a_10_2022,
author={Mitsuru HAMADA, },
journal={IEICE TRANSACTIONS on Fundamentals},
title={Almost Sure Convergence of Relative Frequency of Occurrence of Burst Errors on Channels with Memory},
year={1999},
volume={E82-A},
number={10},
pages={2022-2033},
abstract={Motivated by intention to evaluate asymptotically multiple-burst-error-correcting codes on channels with memory, we will derive the following fact. Let {Z_i } be a hidden Markov process, i. e. , a functional of a Markov chain with a finite state space, and W_b(Z₁Z₂Z_n) denote the number of burst errors that appear in Z₁Z₂Z_n, where the number of burst errors is counted using Gabidulin's burst metric , 1971. As the main result, we will prove the almost sure convergence of relative burst weight W_b(Z₁Z₂Z_n)/n, i. e. , the relative frequency of occurrence of burst errors, for a broad class of functionals { Z_i } of finite Markov chains. Functionals of Markov chains are often adopted as models of the noises on channels, especially on burst-noise channels, the most famous model of which is probably the Gilbert channel proposed in 1960. Those channel models described with Markov chains are called channels with memory (including channels with zero-memory, i. e. , memoryless ones). This work's achievement enables us to extend Gilbert's code performance evaluation in 1952, a landmark that offered the well-known Gilbert bound, discussed its relationship to the (memoryless) binary symmetric channel, and has been serving as a guide for the-Hamming-metric-based design of error-correcting codes, to the case of the-burst-metric-based codes (burst-error-correcting codes) and discrete channels with or without memory.},
keywords={},
doi={},
ISSN={},
month={October},}

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TY - JOUR
TI - Almost Sure Convergence of Relative Frequency of Occurrence of Burst Errors on Channels with Memory
T2 - IEICE TRANSACTIONS on Fundamentals
SP - 2022
EP - 2033
AU - Mitsuru HAMADA
PY - 1999
DO -
JO - IEICE TRANSACTIONS on Fundamentals
SN -
VL - E82-A
IS - 10
JA - IEICE TRANSACTIONS on Fundamentals
Y1 - October 1999
AB - Motivated by intention to evaluate asymptotically multiple-burst-error-correcting codes on channels with memory, we will derive the following fact. Let {Z_i } be a hidden Markov process, i. e. , a functional of a Markov chain with a finite state space, and W_b(Z₁Z₂Z_n) denote the number of burst errors that appear in Z₁Z₂Z_n, where the number of burst errors is counted using Gabidulin's burst metric , 1971. As the main result, we will prove the almost sure convergence of relative burst weight W_b(Z₁Z₂Z_n)/n, i. e. , the relative frequency of occurrence of burst errors, for a broad class of functionals { Z_i } of finite Markov chains. Functionals of Markov chains are often adopted as models of the noises on channels, especially on burst-noise channels, the most famous model of which is probably the Gilbert channel proposed in 1960. Those channel models described with Markov chains are called channels with memory (including channels with zero-memory, i. e. , memoryless ones). This work's achievement enables us to extend Gilbert's code performance evaluation in 1952, a landmark that offered the well-known Gilbert bound, discussed its relationship to the (memoryless) binary symmetric channel, and has been serving as a guide for the-Hamming-metric-based design of error-correcting codes, to the case of the-burst-metric-based codes (burst-error-correcting codes) and discrete channels with or without memory.
ER -