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Hartley Transform
In mathematics, the Hartley transform (HT) is an integral transform closely related to the Fourier transform (FT), but which transforms real-valued functions to real-valued functions. It was proposed as an alternative to the Fourier transform by Ralph V. L. Hartley in 1942, and is one of many known Fourier-related transforms. Compared to the Fourier transform, the Hartley transform has the advantages of transforming real functions to real functions (as opposed to requiring complex numbers) and of being its own inverse. The discrete version of the transform, the discrete Hartley transform (DHT), was introduced by Ronald N. Bracewell in 1983. The two-dimensional Hartley transform can be computed by an analog optical process similar to an optical Fourier transform (OFT), with the proposed advantage that only its amplitude and sign need to be determined rather than its complex phase. However, optical Hartley transforms do not seem to have seen widespread use. Definition The Hartl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Symmetric Matrix
In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose. Formally, Because equal matrices have equal dimensions, only square matrices can be symmetric. The entries of a symmetric matrix are symmetric with respect to the main diagonal. So if a_ denotes the entry in the ith row and jth column then for all indices i and j. Every square diagonal matrix is symmetric, since all off-diagonal elements are zero. Similarly in characteristic different from 2, each diagonal element of a skew-symmetric matrix must be zero, since each is its own negative. In linear algebra, a real symmetric matrix represents a self-adjoint operator represented in an orthonormal basis over a real inner product space. The corresponding object for a complex inner product space is a Hermitian matrix with complex-valued entries, which is equal to its conjugate transpose. Therefore, in linear algebra over the complex numbers, it is often assumed that a symmetric m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oxford University Press, Inc
Oxford () is a city in England. It is the county town and only city of Oxfordshire. In 2020, its population was estimated at 151,584. It is north-west of London, south-east of Birmingham and north-east of Bristol. The city is home to the University of Oxford, the oldest university in the English-speaking world; it has buildings in every style of English architecture since late Anglo-Saxon. Oxford's industries include motor manufacturing, education, publishing, information technology and science. History The history of Oxford in England dates back to its original settlement in the Saxon period. Originally of strategic significance due to its controlling location on the upper reaches of the River Thames at its junction with the River Cherwell, the town grew in national importance during the early Norman period, and in the late 12th century became home to the fledgling University of Oxford. The city was besieged during The Anarchy in 1142. The university rose to domi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
McGraw-Hill
McGraw Hill is an American educational publishing company and one of the "big three" educational publishers that publishes educational content, software, and services for pre-K through postgraduate education. The company also publishes reference and trade publications for the medical, business, and engineering professions. McGraw Hill operates in 28 countries, has about 4,000 employees globally, and offers products and services to about 140 countries in about 60 languages. Formerly a division of The McGraw Hill Companies (later renamed McGraw Hill Financial, now S&P Global), McGraw Hill Education was divested and acquired by Apollo Global Management in March 2013 for $2.4 billion in cash. McGraw Hill was sold in 2021 to Platinum Equity for $4.5 billion. Corporate History McGraw Hill was founded in 1888 when James H. McGraw, co-founder of the company, purchased the ''American Journal of Railway Appliances''. He continued to add further publications, eventually establishin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The IEEE
The ''Proceedings of the IEEE'' is a monthly peer-reviewed scientific journal published by the Institute of Electrical and Electronics Engineers (IEEE). The journal focuses on electrical engineering and computer science. According to the ''Journal Citation Reports'', the journal has a 2017 impact factor of 9.107, ranking it sixth in the category "Engineering, Electrical & Electronic." In 2018, it became fifth with an enhanced impact factor of 10.694. History of the Proceedings The journal was established in 1909, known as the ''Proceedings of the Wireless Institute''. Six issues were published under this banner by Greenleaf Pickard and Alfred Goldsmith. Then in 1911, a merger between the Wireless Institute (New York) and the Society of Wireless Telegraph Engineers (Boston) resulted in a society named the Institute of Radio Engineers (IRE). In January 1913 newly formed IRE published the first issue of the ''Proceedings of the IRE''. Later, a 1000-page special issue commemor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of The Optical Society Of America
The ''Journal of the Optical Society of America'' is a peer-reviewed scientific journal of optics, published by Optica. It was established in 1917 and in 1984 was split into two parts, A and B. ''Journal of the Optical Society of America A'' Part A covers various topics in optics, vision, and image science. The editor-in-chief is Olga Korotkova (University of Miami, USA). ''Journal of the Optical Society of America B'' Part B covers various topics in the field of optical physics, such as guided waves, laser spectroscopy, nonlinear optics, quantum optics, lasers, organic and polymer materials for optics, and ultrafast phenomena In optics, an ultrashort pulse, also known as an ultrafast event, is an electromagnetic pulse whose time duration is of the order of a picosecond (10−12 second) or less. Such pulses have a broadband optical spectrum, and can be created by m .... The editor-in-chief is Kurt Busch ( Humboldt University of Berlin, Germany). References {{refli ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Proceedings Of The IRE
The ''Proceedings of the IEEE'' is a monthly peer-reviewed scientific journal published by the Institute of Electrical and Electronics Engineers (IEEE). The journal focuses on electrical engineering and computer science. According to the ''Journal Citation Reports'', the journal has a 2017 impact factor of 9.107, ranking it sixth in the category "Engineering, Electrical & Electronic." In 2018, it became fifth with an enhanced impact factor of 10.694. History of the Proceedings The journal was established in 1909, known as the ''Proceedings of the Wireless Institute''. Six issues were published under this banner by Greenleaf Pickard and Alfred Goldsmith. Then in 1911, a merger between the Wireless Institute (New York) and the Society of Wireless Telegraph Engineers (Boston) resulted in a society named the Institute of Radio Engineers (IRE). In January 1913 newly formed IRE published the first issue of the ''Proceedings of the IRE''. Later, a 1000-page special issue commemorated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fractional Fourier Transform
In mathematics, in the area of harmonic analysis, the fractional Fourier transform (FRFT) is a family of linear transformations generalizing the Fourier transform. It can be thought of as the Fourier transform to the ''n''-th power, where ''n'' need not be an integer — thus, it can transform a function to any ''intermediate'' domain between time and frequency. Its applications range from filter design and signal analysis to phase retrieval and pattern recognition. The FRFT can be used to define fractional convolution, correlation, and other operations, and can also be further generalized into the linear canonical transformation (LCT). An early definition of the FRFT was introduced by Edward Condon, Condon, by solving for the Green's function for phase-space rotations, and also by Namias, generalizing work of Norbert Wiener, Wiener on Hermite polynomials. However, it was not widely recognized in signal processing until it was independently reintroduced around 1993 by severa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cis (mathematics)
is a mathematical notation defined by , where is the cosine function, is the imaginary unit and is the sine function. The notation is less commonly used in mathematics than Euler's formula, which offers an even shorter notation for but cis(x) is widely used as a name for this function in software libraries. Overview The notation is a shorthand for the combination of functions on the right-hand side of Euler's formula: :e^ = \cos x + i\sin x, where . So, :\operatorname x = \cos x + i\sin x, i.e. "" is an acronym for "". The notation was first coined by William Rowan Hamilton in ''Elements of Quaternions'' (1866) and subsequently used by Irving Stringham in works such as ''Uniplanar Algebra'' (1893), or by James Harkness and Frank Morley in their ''Introduction to the Theory of Analytic Functions'' (1898). It connects trigonometric functions with exponential functions in the complex plane via Euler's formula. It is mostly used as a convenient shorthand notation to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trigonometry
Trigonometry () is a branch of mathematics that studies relationships between side lengths and angles of triangles. The field emerged in the Hellenistic world during the 3rd century BC from applications of geometry to astronomical studies. The Greeks focused on the calculation of chords, while mathematicians in India created the earliest-known tables of values for trigonometric ratios (also called trigonometric functions) such as sine. Throughout history, trigonometry has been applied in areas such as geodesy, surveying, celestial mechanics, and navigation. Trigonometry is known for its many identities. These trigonometric identities are commonly used for rewriting trigonometrical expressions with the aim to simplify an expression, to find a more useful form of an expression, or to solve an equation. History Sumerian astronomers studied angle measure, using a division of circles into 360 degrees. They, and later the Babylonians, studied the ratios of the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convolution
In mathematics (in particular, functional analysis), convolution is a mathematical operation on two functions ( and ) that produces a third function (f*g) that expresses how the shape of one is modified by the other. The term ''convolution'' refers to both the result function and to the process of computing it. It is defined as the integral of the product of the two functions after one is reflected about the y-axis and shifted. The choice of which function is reflected and shifted before the integral does not change the integral result (see commutativity). The integral is evaluated for all values of shift, producing the convolution function. Some features of convolution are similar to cross-correlation: for real-valued functions, of a continuous or discrete variable, convolution (f*g) differs from cross-correlation (f \star g) only in that either or is reflected about the y-axis in convolution; thus it is a cross-correlation of and , or and . For complex-valued fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Convolution Theorem
In mathematics, the convolution theorem states that under suitable conditions the Fourier transform of a convolution of two functions (or signals) is the pointwise product of their Fourier transforms. More generally, convolution in one domain (e.g., time domain) equals point-wise multiplication in the other domain (e.g., frequency domain). Other versions of the convolution theorem are applicable to various Fourier-related transforms. Functions of a continuous variable Consider two functions g(x) and h(x) with Fourier transforms G and H: \begin G(f) &\triangleq \mathcal\(f) = \int_^g(x) e^ \, dx, \quad f \in \mathbb\\ H(f) &\triangleq \mathcal\(f) = \int_^h(x) e^ \, dx, \quad f \in \mathbb \end where \mathcal denotes the Fourier transform operator. The transform may be normalized in other ways, in which case constant scaling factors (typically 2\pi or \sqrt) will appear in the convolution theorem below. The convolution of g and h is defined by: r(x) = \(x) \triangleq \int_^ g( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
