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Nonuniform Sampling
Nonuniform sampling is a branch of sampling theory involving results related to the Nyquist–Shannon sampling theorem. Nonuniform sampling is based on Lagrange interpolation and the relationship between itself and the (uniform) sampling theorem. Nonuniform sampling is a generalisation of the Whittaker–Shannon–Kotelnikov (WSK) sampling theorem. The sampling theory of Shannon can be generalized for the case of nonuniform samples, that is, samples not taken equally spaced in time. The Shannon sampling theory for non-uniform sampling states that a band-limited signal can be perfectly reconstructed from its samples if the average sampling rate satisfies the Nyquist condition. Therefore, although uniformly spaced samples may result in easier reconstruction algorithms, it is not a necessary condition for perfect reconstruction. The general theory for non-baseband and nonuniform samples was developed in 1967 by Henry Landau. He proved that the average sampling rate (uniform or other ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nyquist–Shannon Sampling Theorem
The Nyquist–Shannon sampling theorem is a theorem in the field of signal processing which serves as a fundamental bridge between continuous-time signals and discrete-time signals. It establishes a sufficient condition for a sample rate that permits a discrete sequence of ''samples'' to capture all the information from a continuous-time signal of finite bandwidth. Strictly speaking, the theorem only applies to a class of mathematical functions having a Fourier transform that is zero outside of a finite region of frequencies. Intuitively we expect that when one reduces a continuous function to a discrete sequence and interpolates back to a continuous function, the fidelity of the result depends on the density (or sample rate) of the original samples. The sampling theorem introduces the concept of a sample rate that is sufficient for perfect fidelity for the class of functions that are band-limited to a given bandwidth, such that no actual information is lost in the sampling proc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lagrange Interpolation
In numerical analysis, the Lagrange interpolating polynomial is the unique polynomial of lowest degree that interpolates a given set of data. Given a data set of coordinate pairs (x_j, y_j) with 0 \leq j \leq k, the x_j are called ''nodes'' and the y_j are called ''values''. The Lagrange polynomial L(x) has degree \leq k and assumes each value at the corresponding node, L(x_j) = y_j. Although named after Joseph-Louis Lagrange, who published it in 1795, the method was first discovered in 1779 by Edward Waring. It is also an easy consequence of a formula published in 1783 by Leonhard Euler. Uses of Lagrange polynomials include the Newton–Cotes method of numerical integration and Shamir's secret sharing scheme in cryptography. For equispaced nodes, Lagrange interpolation is susceptible to Runge's phenomenon of large oscillation. Definition Given a set of k + 1 nodes \, which must all be distinct, x_j \neq x_m for indices j \neq m, the Lagrange basis for polynomials of degr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Henry Landau
Henry Jacob Landau is an American mathematician known for his contributions to information theory, including the theory of bandlimited functions and on moment issues. Landau attended the Bronx High School of Science. He received an A.B. (1953), A.M. (1955) and Ph.D. (1957) from Harvard University. His thesis ''On Canonical Conformal Maps of Multiply Connected Regions'' was advised by Lars Ahlfors and Joseph Leonard Walsh. Landau later became Distinguished Member of Technical Staff at Bell Laboratories and a twice visiting member at the Institute for Advanced Study in Princeton. He has also served as an adjunct professor at City University of New York, the Chinese University of Hong Kong, and Columbia University. Publications The following is a list of publications: on his homepage (as of 1995). # On Uniform App ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compressed Sensing
Compressed sensing (also known as compressive sensing, compressive sampling, or sparse sampling) is a signal processing technique for efficiently acquiring and reconstructing a Signal (electronics), signal, by finding solutions to Underdetermined system, underdetermined linear systems. This is based on the principle that, through optimization, the sparsity of a signal can be exploited to recover it from far fewer samples than required by the Nyquist–Shannon sampling theorem. There are two conditions under which recovery is possible. The first one is sparsity, which requires the signal to be sparse in some domain. The second one is incoherence, which is applied through the isometric property, which is sufficient for sparse signals. Overview A common goal of the engineering field of signal processing is to reconstruct a signal from a series of sampling measurements. In general, this task is impossible because there is no way to reconstruct a signal during the times that the signa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spectrum
A spectrum (plural ''spectra'' or ''spectrums'') is a condition that is not limited to a specific set of values but can vary, without gaps, across a continuum. The word was first used scientifically in optics to describe the rainbow of colors in visible light after passing through a prism. As scientific understanding of light advanced, it came to apply to the entire electromagnetic spectrum. It thereby became a mapping of a range of magnitudes (wavelengths) to a range of qualities, which are the perceived "colors of the rainbow" and other properties which correspond to wavelengths that lie outside of the visible light spectrum. Spectrum has since been applied by analogy to topics outside optics. Thus, one might talk about the " spectrum of political opinion", or the "spectrum of activity" of a drug, or the "autism spectrum". In these uses, values within a spectrum may not be associated with precisely quantifiable numbers or definitions. Such uses imply a broad range of condition ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Stability
In the mathematical subfield of numerical analysis, numerical stability is a generally desirable property of numerical algorithms. The precise definition of stability depends on the context. One is numerical linear algebra and the other is algorithms for solving ordinary and partial differential equations by discrete approximation. In numerical linear algebra, the principal concern is instabilities caused by proximity to singularities of various kinds, such as very small or nearly colliding eigenvalues. On the other hand, in numerical algorithms for differential equations the concern is the growth of round-off errors and/or small fluctuations in initial data which might cause a large deviation of final answer from the exact solution. Some numerical algorithms may damp out the small fluctuations (errors) in the input data; others might magnify such errors. Calculations that can be proven not to magnify approximation errors are called ''numerically stable''. One of the common task ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Entire Functions
In complex analysis, an entire function, also called an integral function, is a complex-valued Function (mathematics), function that is holomorphic function, holomorphic on the whole complex plane. Typical examples of entire functions are polynomials and the exponential function, and any finite sums, products and compositions of these, such as the trigonometric functions sine and cosine and their hyperbolic function, hyperbolic counterparts hyperbolic sine, sinh and hyperbolic cosine, cosh, as well as derivatives and integrals of entire functions such as the error function. If an entire function has a root of a function, root at , then , taking the limit value at , is an entire function. On the other hand, the natural logarithm, the reciprocal function, and the square root are all not entire functions, nor can they be analytic continuation, continued analytically to an entire function. A Transcendental function, transcendental entire function is an entire function that is not a pol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernstein Space
Bernstein is a common surname in the German language, meaning "amber" (literally "burn stone"). The name is used by both Germans and Jews, although it is most common among people of Ashkenazi Jewish heritage. The German pronunciation is , but in English it is pronounced either as or . Notable people sharing the surname "Bernstein" A–H * Aaron Bernstein (1812–1884), German short story writer and historian * Abe Bernstein (1892–1968), American mobster * Abraham Bernstein (other), several people * Adam Bernstein (born 1960), American film, music video, television director, and screenwriter * Al Bernstein (born 1950), American sportscaster, writer, stage performer, recording artist, and speaker * Al Bernstein (born 1949), Austrian contemporary artist * Al Bernstein (born 1950), American boxing commentator * Alan Bernstein (born 1947), Canadian medical researcher * Alexander Bernstein, Baron Bernstein of Craigweil (1936–2010), British television executive and poli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
