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Star Transform
In applied mathematics, the starred transform, or star transform, is a discrete-time variation of the Laplace transform, so-named because of the asterisk or "star" in the customary notation of the sampled signals. The transform is an operator of a continuous-time function x(t), which is transformed to a function in the following manner:Jury, Eliahu I. ''Analysis and Synthesis of Sampled-Data Control Systems''., Transactions of the American Institute of Electrical Engineers- Part I: Communication and Electronics, 73.4, 1954, p. 332-346. : \begin X^(s)=\mathcal (t)\cdot \delta_T(t)\mathcal ^(t) \end where is a Dirac comb function, with period of time T. The starred transform is a convenient mathematical abstraction that represents the Laplace transform of an ''impulse sampled'' function , which is the output of an ''ideal sampler'', whose input is a continuous function, x(t). The starred transform is similar to the Z transform In mathematics and signal processing, the Z-t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Applied Mathematics
Applied mathematics is the application of mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the professional specialty in which mathematicians work on practical problems by formulating and studying mathematical models. In the past, practical applications have motivated the development of mathematical theories, which then became the subject of study in pure mathematics where abstract concepts are studied for their own sake. The activity of applied mathematics is thus intimately connected with research in pure mathematics. History Historically, applied mathematics consisted principally of applied analysis, most notably differential equations; approximation theory (broadly construed, to include representations, asymptotic methods, variation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Laplace Transform
In mathematics, the Laplace transform, named after its discoverer Pierre-Simon Laplace (), is an integral transform that converts a function of a real variable (usually t, in the ''time domain'') to a function of a complex variable s (in the complex frequency domain, also known as ''s''-domain, or s-plane). The transform has many applications in science and engineering because it is a tool for solving differential equations. In particular, it transforms ordinary differential equations into algebraic equations and convolution into multiplication. For suitable functions ''f'', the Laplace transform is the integral \mathcal\(s) = \int_0^\infty f(t)e^ \, dt. History The Laplace transform is named after mathematician and astronomer Pierre-Simon, marquis de Laplace, who used a similar transform in his work on probability theory. Laplace wrote extensively about the use of generating functions in ''Essai philosophique sur les probabilités'' (1814), and the integral form of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Dirac Comb
In mathematics, a Dirac comb (also known as shah function, impulse train or sampling function) is a periodic function with the formula \operatorname_(t) \ := \sum_^ \delta(t - k T) for some given period T. Here ''t'' is a real variable and the sum extends over all integers ''k.'' The Dirac delta function \delta and the Dirac comb are tempered distributions. The graph of the function resembles a comb (with the \deltas as the comb's ''teeth''), hence its name and the use of the comb-like Cyrillic letter sha (Ш) to denote the function. The symbol \operatorname\,\,(t), where the period is omitted, represents a Dirac comb of unit period. This implies \operatorname_(t) \ = \frac\operatorname\ \!\!\!\left(\frac\right). Because the Dirac comb function is periodic, it can be represented as a Fourier series based on the Dirichlet kernel: \operatorname_(t) = \frac\sum_^ e^. The Dirac comb function allows one to represent both continuous and discrete phenomena, such as sampling ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Z Transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation. It can be considered as a discrete-time equivalent of the Laplace transform (s-domain). This similarity is explored in the theory of time-scale calculus. Whereas the continuous-time Fourier transform is evaluated on the Laplace s-domain's imaginary line, the discrete-time Fourier transform is evaluated over the unit circle of the z-domain. What is roughly the s-domain's left half-plane, is now the inside of the complex unit circle; what is the z-domain's outside of the unit circle, roughly corresponds to the right half-plane of the s-domain. One of the means of designing digital filters is to take analog designs, subject them to a bilinear transform which maps them from the s-domain to the z-domain, and then produce the digital filter by inspection, manipulation, or numerical ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Normalized Frequency (digital Signal Processing)
In digital signal processing (DSP), a normalized frequency () is a quantity that is equal to the ratio of a frequency and a characteristic frequency of a system. An example of a normalized frequency is the sampling frequency in a system in which a signal is sampled at periodically, in which it equals (with the unit ''cycle per sample''), where is a frequency and is the '' sampling rate''. For regularly spaced sampling, the continuous time variable, (with unit second), is replaced by a discrete ''sampling count'' variable, (with the unit sample), upon division by the sampling interval, (with the unit second per sample). The use of normalized frequency allows us to present concepts that are universal to all sample rates in a way that is independent of the sample rate. An example of such a concept is a digital filter design whose bandwidth is specified not in hertz, but as a percentage of the sample rate of the data passing through it. Formulas expressed in terms of or ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Z-transform
In mathematics and signal processing, the Z-transform converts a discrete-time signal, which is a sequence of real or complex numbers, into a complex frequency-domain (z-domain or z-plane) representation. It can be considered as a discrete-time equivalent of the Laplace transform (s-domain). This similarity is explored in the theory of time-scale calculus. Whereas the continuous-time Fourier transform is evaluated on the Laplace s-domain's imaginary line, the discrete-time Fourier transform is evaluated over the unit circle of the z-domain. What is roughly the s-domain's left half-plane, is now the inside of the complex unit circle; what is the z-domain's outside of the unit circle, roughly corresponds to the right half-plane of the s-domain. One of the means of designing digital filters is to take analog designs, subject them to a bilinear transform which maps them from the s-domain to the z-domain, and then produce the digital filter by inspection, manipulation, or nume ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Line Integral
In mathematics, a line integral is an integral where the function to be integrated is evaluated along a curve. The terms ''path integral'', ''curve integral'', and ''curvilinear integral'' are also used; '' contour integral'' is used as well, although that is typically reserved for line integrals in the complex plane. The function to be integrated may be a scalar field or a vector field. The value of the line integral is the sum of values of the field at all points on the curve, weighted by some scalar function on the curve (commonly arc length or, for a vector field, the scalar product of the vector field with a differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on intervals. Many simple formulae in physics, such as the definition of work as W=\mathbf\cdot\mathbf, have natural continuous analogues in terms of line integrals, in this case \textstyle W = \int_L \mathbf(\mathbf)\cdot d\mathbf, which computes the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Residue Theorem
In complex analysis, the residue theorem, sometimes called Cauchy's residue theorem, is a powerful tool to evaluate line integrals of analytic functions over closed curves; it can often be used to compute real integrals and infinite series as well. It generalizes the Cauchy integral theorem and Cauchy's integral formula. From a geometrical perspective, it can be seen as a special case of the generalized Stokes' theorem. Statement The statement is as follows: Let be a simply connected open subset of the complex plane containing a finite list of points , , and a function defined and holomorphic on . Let be a closed rectifiable curve in , and denote the winding number of around by . The line integral of around is equal to times the sum of residues of at the points, each counted as many times as winds around the point: \oint_\gamma f(z)\, dz = 2\pi i \sum_^n \operatorname(\gamma, a_k) \operatorname( f, a_k ). If is a positively oriented simple closed curve, if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |