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Frequency Warping
The bilinear transform (also known as Tustin's method, after Arnold Tustin) is used in digital signal processing and discrete-time control theory to transform continuous-time system representations to discrete-time and vice versa. The bilinear transform is a special case of a conformal mapping (namely, a Möbius transformation), often used for converting a transfer function H_a(s) of a linear, time-invariant ( LTI) filter in the continuous-time domain (often named an analog filter) to a transfer function H_d(z) of a linear, shift-invariant filter in the discrete-time domain (often named a digital filter although there are analog filters constructed with switched capacitors that are discrete-time filters). It maps positions on the j \omega axis, \mathrm 0 , in the s-plane to the unit circle, , z, = 1 , in the z-plane. Other bilinear transforms can be used for warping the frequency response of any discrete-time linear system (for example to approximate the non-linear freque ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arnold Tustin
Arnold Tustin, (16 July 1899 – 9 January 1994), was a British engineer and Professor of Engineering at the University of Birmingham and at Imperial College London who made important contributions to the development of control engineering and its application to electrical machines. Biography Tustin started working in 1914 at the age of 16 as an apprentice to the C. A. Parsons and Company, of Newcastle upon Tyne. He entered Armstrong College, later part of Newcastle University, in 1916, served in the Royal Engineers in World War I, and eventually received his master's degree in science in 1922. In 1922 he joined Metropolitan-Vickers (Metro-Vick) as a graduate trainee. In the early 1930s he worked for Metro-Vick in Russia for two years, advising and selling equipment to the government companies. Here, he wrote his first book on the design of electric motors, which was also translated into Russian. In the late 1930s and during World War II Tustin was working on the Metadyne co ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frequency Response
In signal processing and electronics, the frequency response of a system is the quantitative measure of the magnitude and Phase (waves), phase of the output as a function of input frequency. The frequency response is widely used in the design and analysis of systems, such as audio system, audio and control systems, where they simplify mathematical analysis by converting governing differential equations into algebraic equations. In an audio system, it may be used to minimize audible distortion by designing components (such as microphones, Audio power amplifier, amplifiers and loudspeakers) so that the overall response is as flat (uniform) as possible across the system's Bandwidth (signal processing), bandwidth. In control systems, such as a vehicle's cruise control, it may be used to assess system Stability theory, stability, often through the use of Bode plots. Systems with a specific frequency response can be designed using analog filter, analog and digital filters. The frequency ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Number
In mathematics, a complex number is an element of a number system that extends the real numbers with a specific element denoted , called the imaginary unit and satisfying the equation i^= -1; every complex number can be expressed in the form a + bi, where and are real numbers. Because no real number satisfies the above equation, was called an imaginary number by René Descartes. For the complex number is called the , and is called the . The set of complex numbers is denoted by either of the symbols \mathbb C or . Despite the historical nomenclature, "imaginary" complex numbers have a mathematical existence as firm as that of the real numbers, and they are fundamental tools in the scientific description of the natural world. Complex numbers allow solutions to all polynomial equations, even those that have no solutions in real numbers. More precisely, the fundamental theorem of algebra asserts that every non-constant polynomial equation with real or complex coefficie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pole (complex Analysis)
In complex analysis (a branch of mathematics), a pole is a certain type of singularity (mathematics), singularity of a complex-valued function of a complex number, complex variable. It is the simplest type of non-removable singularity of such a function (see essential singularity). Technically, a point is a pole of a function if it is a zero of a function, zero of the function and is holomorphic function, holomorphic (i.e. complex differentiable) in some neighbourhood (mathematics), neighbourhood of . A function is meromorphic function, meromorphic in an open set if for every point of there is a neighborhood of in which at least one of and is holomorphic. If is meromorphic in , then a zero of is a pole of , and a pole of is a zero of . This induces a duality between ''zeros'' and ''poles'', that is fundamental for the study of meromorphic functions. For example, if a function is meromorphic on the whole complex plane plus the point at infinity, then the sum of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Logarithm
In mathematics, the logarithm of a number is the exponent by which another fixed value, the base, must be raised to produce that number. For example, the logarithm of to base is , because is to the rd power: . More generally, if , then is the logarithm of to base , written , so . As a single-variable function, the logarithm to base is the inverse of exponentiation with base . The logarithm base is called the ''decimal'' or ''common'' logarithm and is commonly used in science and engineering. The ''natural'' logarithm has the number as its base; its use is widespread in mathematics and physics because of its very simple derivative. The ''binary'' logarithm uses base and is widely used in computer science, information theory, music theory, and photography. When the base is unambiguous from the context or irrelevant it is often omitted, and the logarithm is written . Logarithms were introduced by John Napier in 1614 as a means of simplifying calculation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trapezoidal Rule
In calculus, the trapezoidal rule (or trapezium rule in British English) is a technique for numerical integration, i.e., approximating the definite integral: \int_a^b f(x) \, dx. The trapezoidal rule works by approximating the region under the graph of the function f(x) as a trapezoid and calculating its area. It follows that \int_^ f(x) \, dx \approx (b-a) \cdot \tfrac(f(a)+f(b)). The integral can be even better approximated by Partition of an interval, partitioning the integration interval, applying the trapezoidal rule to each subinterval, and summing the results. In practice, this "chained" (or "composite") trapezoidal rule is usually what is meant by "integrating with the trapezoidal rule". Let \ be a partition of [a,b] such that a=x_0 < x_1 < \cdots < x_ < x_N = b and be the length of the -th subinterval (that is, ), then |
Numerical Integration
In analysis, numerical integration comprises a broad family of algorithms for calculating the numerical value of a definite integral. The term numerical quadrature (often abbreviated to quadrature) is more or less a synonym for "numerical integration", especially as applied to one-dimensional integrals. Some authors refer to numerical integration over more than one dimension as cubature; others take "quadrature" to include higher-dimensional integration. The basic problem in numerical integration is to compute an approximate solution to a definite integral :\int_a^b f(x) \, dx to a given degree of accuracy. If is a smooth function integrated over a small number of dimensions, and the domain of integration is bounded, there are many methods for approximating the integral to the desired precision. Numerical integration has roots in the geometrical problem of finding a square with the same area as a given plane figure ('' quadrature'' or ''squaring''), as in the quadrature of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Dirac Delta Function
In mathematical analysis, the Dirac delta function (or distribution), also known as the unit impulse, is a generalized function on the real numbers, whose value is zero everywhere except at zero, and whose integral over the entire real line is equal to one. Thus it can be Heuristic, represented heuristically as \delta (x) = \begin 0, & x \neq 0 \\ , & x = 0 \end such that \int_^ \delta(x) dx=1. Since there is no function having this property, modelling the delta "function" rigorously involves the use of limit (mathematics), limits or, as is common in mathematics, measure theory and the theory of distribution (mathematics), distributions. The delta function was introduced by physicist Paul Dirac, and has since been applied routinely in physics and engineering to model point masses and instantaneous impulses. It is called the delta function because it is a continuous analogue of the Kronecker delta function, which is usually defined on a discrete domain and takes values ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Padé Approximant
In mathematics, a Padé approximant is the "best" approximation of a function near a specific point by a rational function of given order. Under this technique, the approximant's power series agrees with the power series of the function it is approximating. The technique was developed around 1890 by Henri Padé, but goes back to Georg Frobenius, who introduced the idea and investigated the features of rational approximations of power series. The Padé approximant often gives better approximation of the function than truncating its Taylor series, and it may still work where the Taylor series does not converge. For these reasons Padé approximants are used extensively in computer calculations. They have also been used as auxiliary functions in Diophantine approximation and transcendental number theory, though for sharp results ad hoc methods—in some sense inspired by the Padé theory—typically replace them. Since a Padé approximant is a rational function, an artificial sin ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nyquist Frequency
In signal processing, the Nyquist frequency (or folding frequency), named after Harry Nyquist, is a characteristic of a Sampling (signal processing), sampler, which converts a continuous function or signal into a discrete sequence. For a given Sampling (signal processing), sampling rate (''samples per second''), the Nyquist frequency ''(cycles per second'') is the frequency whose cycle-length (or period) is twice the interval between samples, thus ''0.5 cycle/sample''. For example, audio compact disc, CDs have a sampling rate of 44100 ''samples/second''. At ''0.5 cycle/sample'', the corresponding Nyquist frequency is 22050 ''cycles/second'' (hertz, Hz). Conversely, the Nyquist rate for sampling a 22050 Hz signal is 44100 ''samples/second''. When the highest frequency (Bandwidth (signal processing), bandwidth) of a signal is less than the Nyquist frequency of the sampler, the resulting discrete-time sequence is said to be free of the distortion known as aliasing, and the corre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
