|
Walsh Function
In mathematics, more specifically in harmonic analysis, Walsh functions form a complete orthogonal set of functions that can be used to represent any discrete function—just like trigonometric functions can be used to represent any continuous function in Fourier analysis. They can thus be viewed as a discrete, digital counterpart of the continuous, analog system of trigonometric functions on the unit interval. But unlike the sine and cosine functions, which are continuous, Walsh functions are piecewise constant. They take the values −1 and +1 only, on sub-intervals defined by dyadic fractions. The system of Walsh functions is known as the Walsh system. It is an extension of the Rademacher system of orthogonal functions. Walsh functions, the Walsh system, the Walsh series, and the fast Walsh–Hadamard transform are all named after the American mathematician Joseph L. Walsh. They find various applications in physics and engineering when analyzing digital signals. Histori ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural And Sequency Ordered Walsh 16
Nature is an inherent character or constitution, particularly of the ecosphere or the universe as a whole. In this general sense nature refers to the laws, elements and phenomena of the physical world, including life. Although humans are part of nature, human activity or humans as a whole are often described as at times at odds, or outright separate and even superior to nature. During the advent of modern scientific method in the last several centuries, nature became the passive reality, organized and moved by divine laws. With the Industrial Revolution, nature increasingly became seen as the part of reality deprived from intentional intervention: it was hence considered as sacred by some traditions (Rousseau, American transcendentalism) or a mere decorum for divine providence or human history (Hegel, Marx). However, a vitalist vision of nature, closer to the pre-Socratic one, got reborn at the same time, especially after Charles Darwin. Within the various uses of the word t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Digital Signal Processing
Digital signal processing (DSP) is the use of digital processing, such as by computers or more specialized digital signal processors, to perform a wide variety of signal processing operations. The digital signals processed in this manner are a sequence of numbers that represent Sampling (signal processing), samples of a continuous variable in a domain such as time, space, or frequency. In digital electronics, a digital signal is represented as a pulse train, which is typically generated by the switching of a transistor. Digital signal processing and analog signal processing are subfields of signal processing. DSP applications include Audio signal processing, audio and speech processing, sonar, radar and other sensor array processing, spectral density estimation, statistical signal processing, digital image processing, data compression, video coding, audio coding, image compression, signal processing for telecommunications, control systems, biomedical engineering, and seismology ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Transform
In mathematics, the Fourier transform (FT) is an integral transform that takes a function as input then outputs another function that describes the extent to which various frequencies are present in the original function. The output of the transform is a complex-valued function of frequency. The term ''Fourier transform'' refers to both this complex-valued function and the mathematical operation. When a distinction needs to be made, the output of the operation is sometimes called the frequency domain representation of the original function. The Fourier transform is analogous to decomposing the sound of a musical chord into the intensities of its constituent pitches. Functions that are localized in the time domain have Fourier transforms that are spread out across the frequency domain and vice versa, a phenomenon known as the uncertainty principle. The critical case for this principle is the Gaussian function, of substantial importance in probability theory and statist ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fourier Series
A Fourier series () is an Series expansion, expansion of a periodic function into a sum of trigonometric functions. The Fourier series is an example of a trigonometric series. By expressing a function as a sum of sines and cosines, many problems involving the function become easier to analyze because trigonometric functions are well understood. For example, Fourier series were first used by Joseph Fourier to find solutions to the heat equation. This application is possible because the derivatives of trigonometric functions fall into simple patterns. Fourier series cannot be used to approximate arbitrary functions, because most functions have infinitely many terms in their Fourier series, and the series do not always Convergent series, converge. Well-behaved functions, for example Smoothness, smooth functions, have Fourier series that converge to the original function. The coefficients of the Fourier series are determined by integrals of the function multiplied by trigonometric func ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Line
A number line is a graphical representation of a straight line that serves as spatial representation of numbers, usually graduated like a ruler with a particular origin (geometry), origin point representing the number zero and evenly spaced marks in either direction representing integers, imagined to extend infinitely. The association between numbers and point (geometry), points on the line links elementary arithmetic, arithmetical operations on numbers to geometry, geometric relations between points, and provides a conceptual framework for learning mathematics. In elementary mathematics, the number line is initially used to teach addition and subtraction of integers, especially involving negative numbers. As students progress, more kinds of numbers can be placed on the line, including fractions, decimal fractions, square roots, and transcendental numbers such as the pi, circle constant : Every point of the number line corresponds to a unique real number, and every real number to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Haar Wavelet
In mathematics, the Haar wavelet is a sequence of rescaled "square-shaped" functions which together form a wavelet family or basis. Wavelet analysis is similar to Fourier analysis in that it allows a target function over an interval to be represented in terms of an orthonormal basis. The Haar sequence is now recognised as the first known wavelet basis and is extensively used as a teaching example. The Haar sequence was proposed in 1909 by Alfréd Haar. Haar used these functions to give an example of an orthonormal system for the space of square-integrable functions on the unit interval , 1 The study of wavelets, and even the term "wavelet", did not come until much later. As a special case of the Daubechies wavelet, the Haar wavelet is also known as Db1. The Haar wavelet is also the simplest possible wavelet. The technical disadvantage of the Haar wavelet is that it is not continuous, and therefore not differentiable. This property can, however, be an advantage ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bounded Function
In mathematics, a function f defined on some set X with real or complex values is called bounded if the set of its values (its image) is bounded. In other words, there exists a real number M such that :, f(x), \le M for all x in X. A function that is ''not'' bounded is said to be unbounded. If f is real-valued and f(x) \leq A for all x in X, then the function is said to be bounded (from) above by A. If f(x) \geq B for all x in X, then the function is said to be bounded (from) below by B. A real-valued function is bounded if and only if it is bounded from above and below. An important special case is a bounded sequence, where ''X'' is taken to be the set \mathbb N of natural numbers. Thus a sequence f = (a_0, a_1, a_2, \ldots) is bounded if there exists a real number M such that :, a_n, \le M for every natural number n. The set of all bounded sequences forms the sequence space l^\infty. The definition of boundedness can be generalized to functions f: X \rightarrow Y taking ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square-integrable Function
In mathematics, a square-integrable function, also called a quadratically integrable function or L^2 function or square-summable function, is a real- or complex-valued measurable function for which the integral of the square of the absolute value is finite. Thus, square-integrability on the real line (-\infty, +\infty) is defined as follows. One may also speak of quadratic integrability over bounded intervals such as ,b/math> for a \leq b. An equivalent definition is to say that the square of the function itself (rather than of its absolute value) is Lebesgue integrable. For this to be true, the integrals of the positive and negative portions of the real part must both be finite, as well as those for the imaginary part. The vector space of (equivalence classes of) square integrable functions (with respect to Lebesgue measure) forms the L^p space with p = 2. Among the L^p spaces, the class of square integrable functions is unique in being compatible with an inner product, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hilbert Space
In mathematics, a Hilbert space is a real number, real or complex number, complex inner product space that is also a complete metric space with respect to the metric induced by the inner product. It generalizes the notion of Euclidean space. The inner product allows lengths and angles to be defined. Furthermore, Complete metric space, completeness means that there are enough limit (mathematics), limits in the space to allow the techniques of calculus to be used. A Hilbert space is a special case of a Banach space. Hilbert spaces were studied beginning in the first decade of the 20th century by David Hilbert, Erhard Schmidt, and Frigyes Riesz. They are indispensable tools in the theories of partial differential equations, mathematical formulation of quantum mechanics, quantum mechanics, Fourier analysis (which includes applications to signal processing and heat transfer), and ergodic theory (which forms the mathematical underpinning of thermodynamics). John von Neumann coined the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthonormal Basis
In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite Dimension (linear algebra), dimension is a Basis (linear algebra), basis for V whose vectors are orthonormal, that is, they are all unit vectors and Orthogonality_(mathematics), orthogonal to each other. For example, the standard basis for a Euclidean space \R^n is an orthonormal basis, where the relevant inner product is the dot product of vectors. The Image (mathematics), image of the standard basis under a Rotation (mathematics), rotation or Reflection (mathematics), reflection (or any orthogonal transformation) is also orthonormal, and every orthonormal basis for \R^n arises in this fashion. An orthonormal basis can be derived from an orthogonal basis via Normalize (linear algebra), normalization. The choice of an origin (mathematics), origin and an orthonormal basis forms a coordinate frame known as an ''orthonormal frame''. For a general inner product space V, an orthono ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthonormality
In linear algebra, two vector space, vectors in an inner product space are orthonormal if they are orthogonality, orthogonal unit vectors. A unit vector means that the vector has a length of 1, which is also known as normalized. Orthogonal means that the vectors are all perpendicular to each other. A set of vectors form an orthonormal set if all vectors in the set are mutually orthogonal and all of unit length. An orthonormal set which forms a basis (linear algebra), basis is called an ''orthonormal basis''. Intuitive overview The construction of orthogonality of vectors is motivated by a desire to extend the intuitive notion of perpendicular vectors to higher-dimensional spaces. In the Cartesian coordinate system#Cartesian coordinates in two dimensions, Cartesian plane, two Vector (geometry), vectors are said to be ''perpendicular'' if the angle between them is 90° (i.e. if they form a right angle). This definition can be formalized in Cartesian space by defining the dot produc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Binary Representation
A binary number is a number expressed in the base-2 numeral system or binary numeral system, a method for representing numbers that uses only two symbols for the natural numbers: typically "0" (zero) and "1" (one). A ''binary number'' may also refer to a rational number that has a finite representation in the binary numeral system, that is, the quotient of an integer by a power of two. The base-2 numeral system is a positional notation with a radix of 2. Each digit is referred to as a bit, or binary digit. Because of its straightforward implementation in digital electronic circuitry using logic gates, the binary system is used by almost all modern computers and computer-based devices, as a preferred system of use, over various other human techniques of communication, because of the simplicity of the language and the noise immunity in physical implementation. History The modern binary number system was studied in Europe in the 16th and 17th centuries by Thomas Harriot, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
