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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, with a simple change of variables, where the starre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Applied Mathematics
Applied mathematics is the application of mathematics, mathematical methods by different fields such as physics, engineering, medicine, biology, finance, business, computer science, and Industrial sector, industry. Thus, applied mathematics is a combination of mathematical science and specialized knowledge. The term "applied mathematics" also describes the profession, 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 Mathematical analysis, applied analysis, most notably differential equations; approximation theory (broadly construed, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Laplace Transform
In mathematics, the Laplace transform, named after Pierre-Simon Laplace (), is an integral transform that converts a Function (mathematics), function of a Real number, real Variable (mathematics), variable (usually t, in the ''time domain'') to a function of a Complex number, complex variable s (in the complex-valued frequency domain, also known as ''s''-domain, or ''s''-plane). The transform is useful for converting derivative, differentiation and integral, integration in the time domain into much easier multiplication and Division (mathematics), division in the Laplace domain (analogous to how logarithms are useful for simplifying multiplication and division into addition and subtraction). This gives the transform many applications in science and engineering, mostly as a tool for solving linear differential equations and dynamical systems by simplifying ordinary differential equations and integral equations into algebraic equation, algebraic polynomial equations, and by simplifyin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Dirac Comb
In mathematics, a Dirac comb (also known as sha function, impulse train or sampling function) is a periodic function, periodic Function (mathematics), 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 Distribution_(mathematics)#Tempered_distributions_and_Fourier_transform, 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 script, Cyrillic letter Sha (Cyrillic), sha (ะจ) 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: \o ... [...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 valued frequency-domain (the z-domain or z-plane) representation. It can be considered a discrete-time equivalent of the Laplace transform (the ''s-domain'' or ''s-plane''). This similarity is explored in the theory of time-scale calculus. While the continuous-time Fourier transform is evaluated on the s-domain's vertical axis (the imaginary axis), the discrete-time Fourier transform is evaluated along the z-domain's unit circle. The s-domain's left half-plane maps to the area inside the z-domain's unit circle, while the s-domain's right half-plane maps to the area outside of the z-domain's unit circle. In signal processing, 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 inspec ... [...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 ratio of a variable frequency (f) and a constant frequency associated with a system (such as a '' sampling rate'', f_s). Some software applications require normalized inputs and produce normalized outputs, which can be re-scaled to physical units when necessary. Mathematical derivations are usually done in normalized units, relevant to a wide range of applications. Examples of normalization A typical choice of characteristic frequency is the '' sampling rate'' (f_s) that is used to create the digital signal from a continuous one. The normalized quantity, f' = \tfrac, has the unit ''cycle per sample'' regardless of whether the original signal is a function of time or distance. For example, when f is expressed in Hz (''cycles per second''), f_s is expressed in ''samples per second''. Some programs (such as MATLAB toolboxes) that design filters with real-valued coefficients prefer the Nyquist frequency (f_s/2) as ... [...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 valued frequency-domain (the z-domain or z-plane) representation. It can be considered a discrete-time equivalent of the Laplace transform (the ''s-domain'' or ''s-plane''). This similarity is explored in the theory of time-scale calculus. While the continuous-time Fourier transform is evaluated on the s-domain's vertical axis (the imaginary axis), the discrete-time Fourier transform is evaluated along the z-domain's unit circle. The s-domain's left half-plane maps to the area inside the z-domain's unit circle, while the s-domain's right half-plane maps to the area outside of the z-domain's unit circle. In signal processing, 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 in ... [...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 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 u(x) and v(x) with Fourier transforms U and V: :\begin U(f) &\triangleq \mathcal\(f) = \int_^u(x) e^ \, dx, \quad f \in \mathbb\\ V(f) &\triangleq \mathcal\(f) = \int_^v(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 u and v is defined by: :r(x) = \(x) \triangleq \int_^ u(\tau) ... [...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 (mathematics), 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 #Complex line integral, 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 Dot product, scalar product of the vector field with a Differential (infinitesimal), differential vector in the curve). This weighting distinguishes the line integral from simpler integrals defined on interval (mathematics), intervals. Many simple formulae in physics, such as the definition of Work (physics), work as have natural continuous analogues in terms of l ... [...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. The residue theorem should not be confused with special cases of the generalized Stokes' theorem; however, the latter can be used as an ingredient of its proof. Statement of Cauchy's residue theorem The statement is as follows: Residue theorem: Let U be a simply connected open subset of the complex plane containing a finite list of points a_1, \ldots, a_n, U_0 = U \smallsetminus \, and a function f holomorphic function, holomorphic on U_0. Letting \gamma be a closed rectifiable curve in U_0, and denoting the residue (complex analysis), residue of f at each point a_k by \operatorname(f, a_k) and the winding number of \gamma around a_k by \operatorname(\gamma, a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |