On Shannon's formula and Hartley's rule: Beyond the mathematical coincidence

Abstract :

In the information theory community, the following “historical” statements are generally well accepted: (1) Hartley did put forth his rule twenty years before Shannon; (2) Shannon’s formula as a fundamental tradeoff between transmission rate, bandwidth, and signal-to-noise ratio came out unexpected in 1948; (3) Hartley’s rule is inexact while Shannon’s formula is characteristic of the additive white Gaussian noise channel; (4) Hartley’s rule is an imprecise relation that is not an appropriate formula for the capacity of a communication channel. We show that all these four statements are somewhat wrong. A careful calculation shows that “Hartley’s rule” in fact coincides with Shannon’s formula. We explain this mathematical coincidence by deriving necessary and sufficient conditions on an additive noise channel such that its capacity is given by Shannon’s formula and construct a sequence of such channels that makes the link between the uniform (Hartley) and Gaussian (Shannon) channels.

Document type :
Journal articles
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https://hal.telecom-paristech.fr/hal-02286945
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Submitted on : Friday, September 13, 2019 - 4:23:15 PM
Last modification on : Thursday, October 17, 2019 - 12:36:59 PM

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  • HAL Id : hal-02286945, version 1

Citation

Olivier Rioul, José Carlos Magossi. On Shannon's formula and Hartley's rule: Beyond the mathematical coincidence. Entropy, Special Issue on Information, Entropy and their Geometric Structures, 2014, 16 (9), pp.4892-4910. ⟨hal-02286945⟩

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