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Spectral concentration problem

Index Spectral concentration problem

The spectral concentration problem in Fourier analysis refers to finding a time sequence of a given length whose discrete Fourier transform is maximally localized on a given frequency interval, as measured by the spectral concentration. [1]

Table of Contents

  1. 19 relations: Cosmology, Definite matrix, Discrete Fourier transform, Eigenvalues and eigenvectors, Fourier analysis, Fourier transform, Frequency, Geophysics, Matrix (mathematics), Multitaper, Orthogonality, Periodic function, Sampling (signal processing), Sequence, Sidelobes, Spectral band, Spectral density estimation, Spherical harmonics, Symmetry.

Cosmology

Cosmology is a branch of physics and metaphysics dealing with the nature of the universe, the cosmos.

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Definite matrix

In mathematics, a symmetric matrix \ M\ with real entries is positive-definite if the real number \ \mathbf^\top M \mathbf\ is positive for every nonzero real column vector \ \mathbf\, where \ \mathbf^\top\ is the row vector transpose of \ \mathbf ~. More generally, a Hermitian matrix (that is, a complex matrix equal to its conjugate transpose) is positive-definite if the real number \ \mathbf^* M \mathbf\ is positive for every nonzero complex column vector \ \mathbf\, where \ \mathbf^*\ denotes the conjugate transpose of \ \mathbf ~.

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Discrete Fourier transform

In mathematics, the discrete Fourier transform (DFT) converts a finite sequence of equally-spaced samples of a function into a same-length sequence of equally-spaced samples of the discrete-time Fourier transform (DTFT), which is a complex-valued function of frequency. Spectral concentration problem and discrete Fourier transform are Fourier analysis.

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Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector is a vector that has its direction unchanged by a given linear transformation.

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Fourier analysis

In mathematics, Fourier analysis is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions.

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Fourier transform

In physics, engineering and mathematics, the Fourier transform (FT) is an integral transform that takes a function as input and outputs another function that describes the extent to which various frequencies are present in the original function. Spectral concentration problem and Fourier transform are Fourier analysis.

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Frequency

Frequency (symbol f), most often measured in hertz (symbol: Hz), is the number of occurrences of a repeating event per unit of time.

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Geophysics

Geophysics is a subject of natural science concerned with the physical processes and physical properties of the Earth and its surrounding space environment, and the use of quantitative methods for their analysis.

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Matrix (mathematics)

In mathematics, a matrix (matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object.

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Multitaper

In signal processing, multitaper analysis is a spectral density estimation technique developed by David J. Thomson. Spectral concentration problem and multitaper are signal processing.

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Orthogonality

In mathematics, orthogonality is the generalization of the geometric notion of perpendicularity.

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Periodic function

A periodic function or cyclic function, also called a periodic waveform (or simply periodic wave), is a function that repeats its values at regular intervals or periods. Spectral concentration problem and periodic function are Fourier analysis.

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Sampling (signal processing)

In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. Spectral concentration problem and sampling (signal processing) are signal processing.

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Sequence

In mathematics, a sequence is an enumerated collection of objects in which repetitions are allowed and order matters.

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Sidelobes

In antenna engineering, sidelobes are the lobes (local maxima) of the far field radiation pattern of an antenna or other radiation source, that are not the main lobe.

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Spectral band

Spectral bands are regions of a given spectrum, having a specific range of wavelengths or frequencies.

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Spectral density estimation

In statistical signal processing, the goal of spectral density estimation (SDE) or simply spectral estimation is to estimate the spectral density (also known as the power spectral density) of a signal from a sequence of time samples of the signal.

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Spherical harmonics

In mathematics and physical science, spherical harmonics are special functions defined on the surface of a sphere. Spectral concentration problem and spherical harmonics are Fourier analysis.

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Symmetry

Symmetry in everyday life refers to a sense of harmonious and beautiful proportion and balance.

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References

[1] https://en.wikipedia.org/wiki/Spectral_concentration_problem