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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 numeri ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
John R
John R. (born John Richbourg, August 20, 1910 - February 15, 1986) was an American radio disc jockey who attained fame in the 1950s and 1960s for playing rhythm and blues music on Nashville radio station WLAC. He was also a notable record producer and artist manager. Richbourg was arguably the most popular and charismatic of the four announcers at WLAC who showcased popular African-American music in nightly programs from the late 1940s to the early 1970s. (The other three were Gene Nobles, Herman Grizzard, and Bill "Hoss" Allen.) Later rock music disc jockeys, such as Alan Freed and Wolfman Jack, mimicked Richbourg's practice of using speech that simulated African-American street language of the mid-twentieth century. Richbourg's highly stylized approach to on-air presentation of both music and advertising earned him popularity, but it also created identity confusion. Because Richbourg and fellow disc jockey Allen used African-American speech patterns, many listeners thought that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Causal System
In control theory, a causal system (also known as a physical or nonanticipative system) is a system where the output depends on past and current inputs but not future inputs—i.e., the output y(t_) depends only on the input x(t) for values of t \le t_. The idea that the output of a function at any time depends only on past and present values of input is defined by the property commonly referred to as causality. A system that has ''some'' dependence on input values from the future (in addition to possible dependence on past or current input values) is termed a non-causal or acausal system, and a system that depends ''solely'' on future input values is an anticausal system. Note that some authors have defined an anticausal system as one that depends solely on future ''and present'' input values or, more simply, as a system that does not depend on past input values. Classically, nature or physical reality has been considered to be a causal system. Physics involving special relat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Finite Impulse Response
In signal processing, a finite impulse response (FIR) filter is a filter whose impulse response (or response to any finite length input) is of ''finite'' duration, because it settles to zero in finite time. This is in contrast to infinite impulse response (IIR) filters, which may have internal feedback and may continue to respond indefinitely (usually decaying). The impulse response (that is, the output in response to a Kronecker delta input) of an Nth-order discrete-time FIR filter lasts exactly N+1 samples (from first nonzero element through last nonzero element) before it then settles to zero. FIR filters can be discrete-time or continuous-time, and digital or analog. Definition For a causal discrete-time FIR filter of order ''N'', each value of the output sequence is a weighted sum of the most recent input values: :\begin y &= b_0 x + b_1 x -1+ \cdots + b_N x -N\\ &= \sum_^N b_i\cdot x -i \end where: * x /math> is the input signal, * y /math> is the output signa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that category was abolished in 1995). The radian is defined in the SI as being a dimensionless unit, with 1 rad = 1. Its symbol is accordingly often omitted, especially in mathematical writing. Definition One radian is defined as the angle subtended from the center of a circle which intercepts an arc equal in length to the radius of the circle. More generally, the magnitude in radians of a subtended angle is equal to the ratio of the arc length to the radius of the circle; that is, \theta = \frac, where is the subtended angle in radians, is arc length, and is radius. A right angle is exactly \frac radians. The rotation angle (360°) corresponding to one complete revolution is the length of the circumference divided by the radius, which i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Argument
In mathematics (particularly in complex analysis), the argument of a complex number ''z'', denoted arg(''z''), is the angle between the positive real axis and the line joining the origin and ''z'', represented as a point in the complex plane, shown as \varphi in Figure 1. It is a multi-valued function operating on the nonzero complex numbers. To define a single-valued function, the principal value of the argument (sometimes denoted Arg ''z'') is used. It is often chosen to be the unique value of the argument that lies within the interval . Definition An argument of the complex number , denoted , is defined in two equivalent ways: #Geometrically, in the complex plane, as the 2D polar angle \varphi from the positive real axis to the vector representing . The numeric value is given by the angle in radians, and is positive if measured counterclockwise. #Algebraically, as any real quantity \varphi such that z = r (\cos \varphi + i \sin \varphi) = r e^ for some positive ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Imaginary Unit
The imaginary unit or unit imaginary number () is a solution to the quadratic equation x^2+1=0. Although there is no real number with this property, can be used to extend the real numbers to what are called complex numbers, using addition and multiplication. A simple example of the use of in a complex number is 2+3i. Imaginary numbers are an important mathematical concept; they extend the real number system \mathbb to the complex number system \mathbb, in which at least one root for every nonconstant polynomial exists (see Algebraic closure and Fundamental theorem of algebra). Here, the term "imaginary" is used because there is no real number having a negative square. There are two complex square roots of −1: and -i, just as there are two complex square roots of every real number other than zero (which has one double square root). In contexts in which use of the letter is ambiguous or problematic, the letter or the Greek \iota is sometimes used instead. For example, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Formal Power Series
In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial sums, etc.). A formal power series is a special kind of formal series, whose terms are of the form a x^n where x^n is the nth power of a variable x (n is a non-negative integer), and a is called the coefficient. Hence, power series can be viewed as a generalization of polynomials, where the number of terms is allowed to be infinite, with no requirements of convergence. Thus, the series may no longer represent a function of its variable, merely a formal sequence of coefficients, in contrast to a power series, which defines a function by taking numerical values for the variable within a radius of convergence. In a formal power series, the x^n are used only as position-holders for the coefficients, so that the coefficient of x^5 is the fifth ter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Two-sided Laplace Transform
In mathematics, the two-sided Laplace transform or bilateral Laplace transform is an integral transform equivalent to probability's moment generating function. Two-sided Laplace transforms are closely related to the Fourier transform, the Mellin transform, the Z-transform and the ordinary or one-sided Laplace transform. If ''f''(''t'') is a real- or complex-valued function of the real variable ''t'' defined for all real numbers, then the two-sided Laplace transform is defined by the integral :\mathcal\(s) = F(s) = \int_^\infty e^ f(t)\, dt. The integral is most commonly understood as an improper integral, which converges if and only if both integrals :\int_0^\infty e^ f(t) \, dt,\quad \int_^0 e^ f(t)\, dt exist. There seems to be no generally accepted notation for the two-sided transform; the \mathcal used here recalls "bilateral". The two-sided transform used by some authors is :\mathcal\(s) = s\mathcal\(s) = sF(s) = s \int_^\infty e^ f(t)\, dt. In pure mathematics the argum ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laurent Series
In mathematics, the Laurent series of a complex function f(z) is a representation of that function as a power series which includes terms of negative degree. It may be used to express complex functions in cases where a Taylor series expansion cannot be applied. The Laurent series was named after and first published by Pierre Alphonse Laurent in 1843. Karl Weierstrass may have discovered it first in a paper written in 1841, but it was not published until after his death.. Definition The Laurent series for a complex function f(z) about a point c is given by f(z) = \sum_^\infty a_n(z-c)^n, where a_n and c are constants, with a_n defined by a line integral that generalizes Cauchy's integral formula: a_n =\frac\oint_\gamma \frac \, dz. The path of integration \gamma is counterclockwise around a Jordan curve enclosing c and lying in an annulus A in which f(z) is holomorphic (analytic). The expansion for f(z) will then be valid anywhere inside the annulus. The annulus is shown in red ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abraham De Moivre
Abraham de Moivre FRS (; 26 May 166727 November 1754) was a French mathematician known for de Moivre's formula, a formula that links complex numbers and trigonometry, and for his work on the normal distribution and probability theory. He moved to England at a young age due to the religious persecution of Huguenots in France which reached a climax in 1685 with the Edict of Fontainebleau. He was a friend of Isaac Newton, Edmond Halley, and James Stirling. Among his fellow Huguenot exiles in England, he was a colleague of the editor and translator Pierre des Maizeaux. De Moivre wrote a book on probability theory, ''The Doctrine of Chances'', said to have been prized by gamblers. De Moivre first discovered Binet's formula, the closed-form expression for Fibonacci numbers linking the ''n''th power of the golden ratio ''φ'' to the ''n''th Fibonacci number. He also was the first to postulate the central limit theorem, a cornerstone of probability theory. Life Early years Abraham ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Generating Function
In mathematics, a generating function is a way of encoding an infinite sequence of numbers () by treating them as the coefficients of a formal power series. This series is called the generating function of the sequence. Unlike an ordinary series, the ''formal'' power series is not required to converge: in fact, the generating function is not actually regarded as a function, and the "variable" remains an indeterminate. Generating functions were first introduced by Abraham de Moivre in 1730, in order to solve the general linear recurrence problem. One can generalize to formal power series in more than one indeterminate, to encode information about infinite multi-dimensional arrays of numbers. There are various types of generating functions, including ordinary generating functions, exponential generating functions, Lambert series, Bell series, and Dirichlet series; definitions and examples are given below. Every sequence in principle has a generating function of each type (except ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
