Hartley function

Abstract: Shannon’s Formula and Hartley’s Rule: A Mathematical Coincidence? Olivier Rioul∗ and Jose Carlos Magossi† ∗ Telecom ParisTech - Institut Mines-Telecom - CNRS LTCI, Paris, France † School of Technology (FT) - Unicamp, Campinas, Brazil Abstract. Shannon’s formula C = 1 2 log(1+P/N) is the emblematic expression for the information capacity of a communication channel. Hartley’s name is often associated to it, owing to Hartley’s rule: counting the highest possible number of distinguishable values for a given amplitude A and precision ±∆ yields a similar expression C0 = log(1+A/∆). 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 transmis- sion rate, bandwidth, and signal-to-noise ratio came out unexpected in 1948; (3) Hartley’s rule is an imprecise relation while Shannon’s formula is exact; (4) Hartley’s expression is not an appropriate formula for the capacity of a communication channel. We show that all these four statements are questionable, if not wrong. Keywords: Shannon

Abstract: The invention discloses a Hartley output method of rigid body space motion states, which enables three velocity components and a ternary figure of a body axis to form linear simultaneous differential equations by defining the ternary figure, and adopts a Hartley function to conduct approximate approaching description of roll rate P, pitching rate Q and yaw rate R. Users can solve a state-transition matrix of a system in accordance with arbitrary order keeper modes, and then obtain expression of rigid motion discrete state equations, thereby avoiding gesture equation strange problems and obtaining main motion states of a rigid body. The Hartley output method enables the state-transition matrix to be a blocking upper triangular form by introducing the ternary figure, can reduce order to solve the state-transition matrix, greatly simplifies computational complexity and is convenient to use in construction.

Abstract: The Hartley transform is an integral transformation that maps a real valued function into a real valued frequency function via the Hartley kernel, thereby avoiding complex arithmetic as opposed to the Fourier transform. Approximation of the Hartley integral by the trapezoidal quadrature results in the discrete Hartley transform, which has proven a contender to the discrete Fourier transform because of its involutory nature. In this paper, a discrete transform is proposed as a real transform with a convolution property and is an alternative to the discrete Hartley transform. Copyright © 2015 John Wiley & Sons, Ltd.

Abstract: Although measures of the various types of uncertainty-based information (Table 3.5) are not sufficient in human communication [Cherry, 1957], they are highly effective tools for dealing with systems problems of virtually any kind [Klir, 1985]. For the classical information measures (Hartley function and Shannon entropy), which were originally conceived solely as tools for analyzing and designing telecommunication systems, this broad utility is best demonstrated by Ashby [1958, 1965, 1969, 1972] and Conant [1969, 1974, 1976, 1981].