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Fourier Series
In mathematics, a Fourier series
Fourier series
(English: /ˈfʊəriˌeɪ/)[1] is a way to represent a function as the sum of simple sine waves. More formally, it decomposes any periodic function or periodic signal into the sum of a (possibly infinite) set of simple oscillating functions, namely sines and cosines (or, equivalently, complex exponentials). The discrete-time Fourier transform
Fourier transform
is a periodic function, often defined in terms of a Fourier series. The Z-transform, another example of application, reduces to a Fourier series
Fourier series
for the important case z=1. Fourier series
Fourier series
are also central to the original proof of the Nyquist–Shannon sampling theorem
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Eigenvalue, Eigenvector And Eigenspace
In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that only changes by a scalar factor when that linear transformation is applied to it. More formally, if T is a linear transformation from a vector space V over a field F into itself and v is a vector in V that is not the zero vector, then v is an eigenvector of T if T(v) is a scalar multiple of v
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Signal Processing
Signal
Signal
processing concerns the analysis, synthesis, and modification of signals, which are broadly defined as functions conveying "information about the behavior or attributes of some phenomenon",[1] such as sound, images, and biological measurements.[2] For example, signal processing techniques are used to improve signal transmission fidelity, storage efficiency, and subjective quality, and to emphasize or detect components of interest in a measured signal.[3]Contents1 History 2 Application fields 3 Typical devices 4 Mathematical methods applied 5 Categories5.1 Analog signal processing 5.2 Continuous-time signal processing 5.3 Discrete-time signal
Discrete-time signal
processing 5.4 Digital signal processing 5.5 Nonlinear signal processing6 See also 7 Notes and references 8 External linksHistory[edit] According to Alan V. Oppenheim and Ronald W
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Linear Combination
In mathematics, a linear combination is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of x and y would be any expression of the form ax + by, where a and b are constants).[1][2][3] The concept of linear combinations is central to linear algebra and related fields of mathematics. Most of this article deals with linear combinations in the context of a vector space over a field, with some generalizations given at the end of the article.Contents1 Definition 2 Examples and counterexamples2.1 Euclidean vectors 2.2 Functions 2.3 Polynomials3 The linear span 4 Linear independence 5 Affine, conical, and convex combinations 6 Operad theory 7 Generalizations 8 Application 9 References 10 External linksDefinition[edit] Suppose that K is a field (for example, the real numbers) and V is a vector space over K. As usual, we call elements of V vectors and call elements of K scalars
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Function (mathematics)
In mathematics, a function[1] is a relation between a set of inputs and a set of permissible outputs with the property that each input is related to exactly one output. An example is the function that relates each real number x to its square x2. The output of a function f corresponding to an input x is denoted by f(x) (read "f of x"). In this example, if the input is −3, then the output is 9, and we may write f(−3) = 9. Likewise, if the input is 3, then the output is also 9, and we may write f(3) = 9. (The same output may be produced by more than one input, but each input gives only one output.) The input variable(s) are sometimes referred to as the argument(s) of the function. Functions of various kinds are "the central objects of investigation"[2] in most fields of modern mathematics. There are many ways to describe or represent a function. Some functions may be defined by a formula or algorithm that tells how to compute the output for a given input
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Integral
In mathematics, an integral assigns numbers to functions in a way that can describe displacement, area, volume, and other concepts that arise by combining infinitesimal data. Integration is one of the two main operations of calculus, with its inverse, differentiation, being the other. Given a function f of a real variable x and an interval [a, b] of the real line, the definite integral ∫ a b f ( x ) d x displaystyle int _ a ^ b !f(x),dx is defined informally as the signed area of the region in the xy-plane that is bounded by the graph of f, the x-axis and the vertical lines x = a and x = b. The area above the x-axis adds to the total and that below the x-axis subtracts from the total. Roughly speaking, the operation of integration is the reverse of differentiation
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Académie Française
The Académie française
Académie française
(French pronunciation: ​[akademi fʁɑ̃sɛz]) is the pre-eminent French council for matters pertaining to the French language. The Académie was officially established in 1635 by Cardinal Richelieu, the chief minister to King Louis XIII.[1] Suppressed in 1793 during the French Revolution, it was restored as a division of the Institut de France
France
in 1803 by Napoleon
Napoleon
Bonaparte.[1] It is the oldest of the five académies of the institute. The Académie consists of forty members, known informally as les immortels (the immortals).[2] New members are elected by the members of the Académie itself. Academicians hold office for life, but they may resign or be dismissed for misconduct
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Electrical Engineering
Electrical engineering
Electrical engineering
is a professional engineering discipline that generally deals with the study and application of electricity, electronics, and electromagnetism. This field first became an identifiable occupation in the later half of the 19th century after commercialization of the electric telegraph, the telephone, and electric power distribution and use. Subsequently, broadcasting and recording media made electronics part of daily life. The invention of the transistor, and later the integrated circuit, brought down the cost of electronics to the point they can be used in almost any household object. Electrical engineering
Electrical engineering
has now subdivided into a wide range of subfields including electronics, digital computers, computer engineering, power engineering, telecommunications, control systems, robotics, radio-frequency engineering, signal processing, instrumentation, and microelectronics
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Oscillation
Oscillation
Oscillation
is the repetitive variation, typically in time, of some measure about a central value (often a point of equilibrium) or between two or more different states. The term vibration is precisely used to describe mechanical oscillation
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Daniel Bernoulli
Daniel Bernoulli
Daniel Bernoulli
FRS (German pronunciation: [bɛʁˈnʊli];[1] 8 February 1700 – 17 March 1782) was a Swiss mathematician and physicist and was one of the many prominent mathematicians in the Bernoulli family
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Jean Le Rond D'Alembert
Jean-Baptiste le Rond d'Alembert (/ˌdæləmˈbɛər/;[1] French: [ʒɑ̃ batist lə ʁɔ̃ dalɑ̃bɛːʁ]; 16 November 1717 – 29 October 1783) was a French mathematician, mechanician, physicist, philosopher, and music theorist. Until 1759 he was also co-editor with Denis Diderot
Denis Diderot
of the Encyclopédie. D'Alembert's formula for obtaining solutions to the wave equation is named after him.[2][3][4] The wave equation is sometimes referred to as d'Alembert's equation.Contents1 Early years 2 Studies and adult life 3 Career 4 Music theories 5 Personal life 6 Death 7 Legacy 8 Fictional portrayal 9 List of works 10 See also 11 Notes 12 References 13 External linksEarly years[edit] Born in Paris, d'Alembert was the natural son of the writer Claudine Guérin de Tencin and the chevalier Louis-Camus Destouches, an artillery officer. Destouches was abroad at the time of d'Alembert's birth
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Optics
Optics
Optics
is the branch of physics which involves the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it.[1] Optics
Optics
usually describes the behaviour of visible, ultraviolet, and infrared light. Because light is an electromagnetic wave, other forms of electromagnetic radiation such as X-rays, microwaves, and radio waves exhibit similar properties.[1] Most optical phenomena can be accounted for using the classical electromagnetic description of light. Complete electromagnetic descriptions of light are, however, often difficult to apply in practice. Practical optics is usually done using simplified models. The most common of these, geometric optics, treats light as a collection of rays that travel in straight lines and bend when they pass through or reflect from surfaces
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Acoustics
Acoustics
Acoustics
is the branch of physics that deals with the study of all mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician while someone working in the field of acoustics technology may be called an acoustical engineer. The application of acoustics is present in almost all aspects of modern society with the most obvious being the audio and noise control industries. Hearing
Hearing
is one of the most crucial means of survival in the animal world, and speech is one of the most distinctive characteristics of human development and culture. Accordingly, the science of acoustics spreads across many facets of human society—music, medicine, architecture, industrial production, warfare and more
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Image Processing
Digital image
Digital image
processing is the use of computer algorithms to perform image processing on digital images. As a subcategory or field of digital signal processing, digital image processing has many advantages over analog image processing
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Complex Exponential
Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the fundamental relationship between the trigonometric functions and the complex exponential function. Euler's formula
Euler's formula
states that for any real number x e i x = cos ⁡ x + i sin ⁡ x , displaystyle e^ ix =cos x+isin x, where e is the base of the natural logarithm, i is the imaginary unit, and cos and sin are the trigonometric functions cosine and sine respectively, with the argument x given in radians. This complex exponential function is sometimes denoted cis x ("cosine plus i sine")
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Quantum Mechanics
Quantum mechanics (QM; also known as quantum physics or quantum theory), including quantum field theory, is a fundamental theory in physics which describes nature at the smallest scales of energy levels of atoms and subatomic particles.[2] Classical physics
Classical physics
(the physics existing before quantum mechanics) is a set of fundamental theories which describes nature at ordinary (macroscopic) scale
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