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Bessel Filter
In electronics and signal processing, a Bessel filter is a type of analog linear filter with a maximally flat Group delay and phase delay, group/phase delay (maximally linear phase response), which preserves the wave shape of filtered signals in the passband. Bessel filters are often used in audio crossover systems. The filter's name is a reference to German mathematician Friedrich Bessel (1784–1846), who developed the mathematical theory on which the filter is based. The filters are also called Bessel–Thomson filters in recognition of W. E. Thomson, who worked out how to apply Bessel functions to filter design in 1949. The Bessel filter is very similar to the Gaussian filter, and tends towards the same shape as filter order increases. While the time-domain step response of the Gaussian filter has zero overshoot (signal), overshoot, the Bessel filter has a small amount of overshoot, but still much less than other common frequency-domain filters, such as Butterworth filters. I ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electronics
The field of electronics is a branch of physics and electrical engineering that deals with the emission, behaviour and effects of electrons using electronic devices. Electronics uses active devices to control electron flow by amplification and rectification, which distinguishes it from classical electrical engineering, which only uses passive effects such as resistance, capacitance and inductance to control electric current flow. Electronics has hugely influenced the development of modern society. The central driving force behind the entire electronics industry is the semiconductor industry sector, which has annual sales of over $481 billion as of 2018. The largest industry sector is e-commerce, which generated over $29 trillion in 2017. History and development Electronics has hugely influenced the development of modern society. The identification of the electron in 1897, along with the subsequent invention of the vacuum tube which could amplify and rectify small ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Polynomial
In mathematics, the Bessel polynomials are an orthogonal sequence of polynomials. There are a number of different but closely related definitions. The definition favored by mathematicians is given by the series :y_n(x)=\sum_^n\frac\,\left(\frac\right)^k. Another definition, favored by electrical engineers, is sometimes known as the reverse Bessel polynomials :\theta_n(x)=x^n\,y_n(1/x)=\sum_^n\frac\,\frac. The coefficients of the second definition are the same as the first but in reverse order. For example, the third-degree Bessel polynomial is :y_3(x)=15x^3+15x^2+6x+1 while the third-degree reverse Bessel polynomial is :\theta_3(x)=x^3+6x^2+15x+15. The reverse Bessel polynomial is used in the design of Bessel electronic filters. Properties Definition in terms of Bessel functions The Bessel polynomial may also be defined using Bessel functions from which the polynomial draws its name. :y_n(x)=\,x^\theta_n(1/x)\, :y_n(x)=\sqrt\,e^K_(1/x) :\theta_n(x)=\sqrt\,x^e^K_(x) ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Filters
Linear filters process time-varying input signals to produce output signals, subject to the constraint of linearity. In most cases these linear filters are also time invariant (or shift invariant) in which case they can be analyzed exactly using LTI ("linear time-invariant") system theory revealing their transfer functions in the frequency domain and their impulse responses in the time domain. Real-time implementations of such linear signal processing filters in the time domain are inevitably causal, an additional constraint on their transfer functions. An analog electronic circuit consisting only of linear components (resistors, capacitors, inductors, and linear amplifiers) will necessarily fall in this category, as will comparable mechanical systems or digital signal processing systems containing only linear elements. Since linear time-invariant filters can be completely characterized by their response to sinusoids of different frequencies (their frequency response), they are so ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel Function
Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary complex number \alpha, the ''order'' of the Bessel function. Although \alpha and -\alpha produce the same differential equation, it is conventional to define different Bessel functions for these two values in such a way that the Bessel functions are mostly smooth functions of \alpha. The most important cases are when \alpha is an integer or half-integer. Bessel functions for integer \alpha are also known as cylinder functions or the cylindrical harmonics because they appear in the solution to Laplace's equation in cylindrical coordinates. Spherical Bessel functions with half-integer \alpha are obtained when the Helmholtz equation is solved in spherical coordinates. Applications of Bessel functions The Bessel function is a generalizat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elliptic Filter
An elliptic filter (also known as a Cauer filter, named after Wilhelm Cauer, or as a Zolotarev filter, after Yegor Zolotarev) is a signal processing filter with equalized ripple (equiripple) behavior in both the passband and the stopband. The amount of ripple in each band is independently adjustable, and no other filter of equal order can have a faster transition in gain between the passband and the stopband, for the given values of ripple (whether the ripple is equalized or not). Alternatively, one may give up the ability to adjust independently the passband and stopband ripple, and instead design a filter which is maximally insensitive to component variations. As the ripple in the stopband approaches zero, the filter becomes a type I Chebyshev filter. As the ripple in the passband approaches zero, the filter becomes a type II Chebyshev filter and finally, as both ripple values approach zero, the filter becomes a Butterworth filter. The gain of a lowpass elliptic filter as a f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chebyshev Filter
Chebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple (type I) or stopband ripple (type II). Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter (See references eg. aniels utovac, but with ripples in the passband. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials. Type I Chebyshev filters are usually referred to as "Chebyshev filters", while type II filters are usually called "inverse Chebyshev filters". Because of the passband ripple inherent in Chebyshev filters, filters with a smoother response in the passband but a more irregular response in the stopband are preferred for certain applications. Type I Chebyshev filters (Chebyshev filters) Type I Chebyshev filters are the most common types of Chebyshev filter ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Comb Filter
In signal processing, a comb filter is a filter implemented by adding a delayed version of a signal to itself, causing constructive and destructive interference. The frequency response of a comb filter consists of a series of regularly spaced notches in between regularly spaced ''peaks'' (sometimes called ''teeth'') giving the appearance of a comb. Applications Comb filters are employed in a variety of signal processing applications, including: * Cascaded integrator–comb (CIC) filters, commonly used for anti-aliasing during interpolation and decimation operations that change the sample rate of a discrete-time system. * 2D and 3D comb filters implemented in hardware (and occasionally software) in PAL and NTSC analog television decoders, reduce artifacts such as dot crawl. * Audio signal processing, including delay, flanging, physical modelling synthesis and digital waveguide synthesis. If the delay is set to a few milliseconds, a comb filter can model the effect of acousti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Butterworth Filter
The Butterworth filter is a type of signal processing filter designed to have a frequency response that is as flat as possible in the passband. It is also referred to as a maximally flat magnitude filter. It was first described in 1930 by the British engineer and physicist Stephen Butterworth in his paper entitled "On the Theory of Filter Amplifiers". Original paper Butterworth had a reputation for solving "impossible" mathematical problems. At the time, filter design required a considerable amount of designer experience due to limitations of the theory then in use. The filter was not in common use for over 30 years after its publication. Butterworth stated that: Such an ideal filter cannot be achieved, but Butterworth showed that successively closer approximations were obtained with increasing numbers of filter elements of the right values. At the time, filters generated substantial ripple in the passband, and the choice of component values was highly interactive. Butterwo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bilinear Transform
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 to convert a transfer function H_a(s) of a linear, time-invariant ( LTI) filter in the continuous-time domain (often called an analog filter) to a transfer function H_d(z) of a linear, shift-invariant filter in the discrete-time domain (often called 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 to warp the frequency response of any discrete-time linear system (for example to approximate the non-linear frequency ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Series
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Complex Conjugate
In mathematics, the complex conjugate of a complex number is the number with an equal real part and an imaginary part equal in magnitude but opposite in sign. That is, (if a and b are real, then) the complex conjugate of a + bi is equal to a - bi. The complex conjugate of z is often denoted as \overline or z^*. In polar form, the conjugate of r e^ is r e^. This can be shown using Euler's formula. The product of a complex number and its conjugate is a real number: a^2 + b^2 (or r^2 in polar coordinates). If a root of a univariate polynomial with real coefficients is complex, then its complex conjugate is also a root. Notation The complex conjugate of a complex number z is written as \overline z or z^*. The first notation, a vinculum, avoids confusion with the notation for the conjugate transpose of a matrix, which can be thought of as a generalization of the complex conjugate. The second is preferred in physics, where dagger (†) is used for the conjugate tra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bessel 3rd-order Delay
Bessel may refer to: * Bessel beam * Bessel ellipsoid * Bessel function in mathematics * Bessel's inequality in mathematics * Bessel's correction in statistics. * Bessel filter, a linear filter often used in audio crossover systems * Bessel Fjord, NE Greenland * Bessel Fjord, NW Greenland * Bessel (crater), a small lunar crater * Bessel transform, also known as Fourier-Bessel transform or Hankel transform * Bessel window, in signal processing * Besselian date, see Epoch (astronomy)#Besselian years * , a German merchant ship in service 1928–45, latterly for the Kriegsmarine People * Friedrich Bessel Friedrich Wilhelm Bessel (; 22 July 1784 – 17 March 1846) was a German astronomer, mathematician, physicist, and geodesist. He was the first astronomer who determined reliable values for the distance from the sun to another star by the method ... (1784–1846), German mathematician, astronomer, and systematizer of the Bessel functions * Bessel Kok (born 1941), Dutch busines ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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