New methods of signal filtering based on atomic functions

Kravchenko, Vladislav V.; Басараб, М. А.

doi:10.1109/msmw.2001.946809

Cited by 4 publications

(6 citation statements)

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“…In [32,37,40,49], a new approach is set forth and substantiated to the use of atomic functions in problems of weighted processing of signals in constructing windows with required properties, i.e., the construction of weight windows for spectrum analysis by means of the discrete Fourier transform. Window functions constructed with the help of atomic functions offer some advantages in comparison with well-known window functions.…”

Section: Review Of the Results (1986-2006) On The Theory Of Atomic Fumentioning

confidence: 99%

Atomic functions: Generalization to the multivariable case and promising applications

Kolodyazhny

Rvachov

2007

Cybern Syst Anal

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517.95Main trends in the theory of atomic functions are outlined. These functions have been studied for more than 35 years and gave rise to new research areas in mathematical analysis, approximation theory, numerical methods, etc. Atomic functions are infinitely differentiable solutions with compact support to special functional-differential equations.Atomic functions whose investigation for the case of one independent variable has received significant attention until quite recently are infinitely differentiable compactly supported solutions to functional-differential equations of the formwhere L is a linear differential operator with constant coefficients. Atomic functions have been intensively investigated for more than 35 years (see [1] and reviews in [2-7]), and some new scientific directions arise in which typical properties inherent in atomic functions are necessarily required in theoretical and practical developments. Atomic functions and spaces generated by them are used in mathematical analysis [2,4,6], approximation theory [4, 5], in solving differential equations and boundary problems of mathematical physics [2, 3, 11, 15-21, 44, 48, 80], in numerical methods and other fields of applied mathematics [22, 46-47, 59-62], and also in technical realizations of bell-shaped functions [30, 31], digital signal processing [34-37, 41, 50-52], analysis and synthesis of antennas [49-52], etc. The history of atomic functions begins with [1], which was published by V. L. Rvachev and V. A. Rvachev in 1971 and in which, in particular, they used the method of contracting mappings to prove the existence and uniqueness of a compactly supported solution of the equation y x y x y x ¢ = + --( ) ( ) ( ) 2 2 1 2 2 1 under the conditions y ( ) 0 1 = and supp y x ( ) [ , ] = -1 1 . This problem was stated by V. L. Rvachev as long ago as in 1967 at a seminar on problems of applied mathematics, and its solution was connected with the advent of the most important atomic function up x ( ), up x itx t t dt k k k ( ) exp ( ) sin ( ) = -¥ ¥ = ¥ --ò Õ 1 2 2 2 1 p .The term "atomic function" was introduced in [2] in 1975. The paper considered the application of atomic functions in conjunction with the mathematical tools of the theory of R-functions in variational methods for solving problems of mathematical physics. In some works of foreign authors [9][10][11][12][13][14][15][16][17][18][19][20][21][22], the atomic function up x ( )also gained ground as "the Rvachev function." 893 1060-0396/07/4306-0893

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Section: Review Of the Results (1986-2006) On The Theory Of Atomic Fumentioning

confidence: 99%

Atomic functions: Generalization to the multivariable case and promising applications

Kolodyazhny

Rvachov

2007

Cybern Syst Anal

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“…On the basis of AF [1][2][3][4] Kravchenko-Kotelnikov generalised series are constructed. We write a signal s(t) with the limited band s(ω) in the form…”

Section: Generalized Kravchenko-kotelnikov Seriesmentioning

confidence: 99%

“…The parameter a (1 < a ≤ 2) is obtained by proceeding from frequency properties of signals [1]. Kravchenko-Kotelnikov (KK) weight function is defined as follows:…”

Section: The Kravchenko-kotelniko Filtermentioning

confidence: 99%

“…In this report the new designs of the digital filters [1][2][3][4][5] constructed on the basis of atomic functions (AF) [1][2][3][4][5][6][7][8][9][10][11] are suggested. They are defined through the Kravchenko-Kotelnikov (KK) generalized sampling Theorems [2].…”

Section: Introductionmentioning

confidence: 99%

“…Weight function (WF) in problems of digital signal processing (DSP) should satisfy some requirements [1][2][3][4] which provide a realizability of the concrete filter. Filter function should provide zero shift of a phase…”

Section: Introductionmentioning

confidence: 99%

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