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| {{Linear analog electronic filter|filter2=hide|filter3=hide}}
| | Ellos son algunas de las situaciones en las que transcribir de audio puede ser el mejor factor de lograr. Transcribir audio es definitivamente un elementos cruciales para los músicos de jazz y por lo que pensar en ello una sustancial sugieren de escolaridad. Esto realmente no es sorprendente, teniendo en cuenta que el jazz está en función principalmente de la improvisación. Fuera de las ciudades usted podría conocimiento correcto elegancia natural de Italia. |
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| '''Chebyshev filters''' are [[analog filter|analog]] or [[digital filter|digital]] filters having a steeper [[roll-off]] and more [[passband]] [[ripple (filters)|ripple]] (type I) or [[stopband]] ripple (type II) than [[Butterworth filter]]s. Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the range of the filter,{{fact|date=December 2013}} but with ripples in the passband.
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| This type of filter is named after [[Pafnuty Chebyshev]] because its mathematical characteristics are derived from [[Chebyshev polynomials]].
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| Because of the passband ripple inherent in Chebyshev filters, the ones that have a smoother response in the passband but a more irregular response in the stopband are preferred for some applications.
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| == Type I Chebyshev filters ==
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| [[File:Chebyshev Type I Filter Response (4th Order).svg|thumb|350px|The frequency response of a fourth-order type I Chebyshev low-pass filter with <math>\varepsilon=1</math>]]
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| These are the most common Chebyshev filters. The gain (or [[amplitude]]) response as a function of angular frequency <math>\omega</math> of the ''n''th-order low-pass filter is equal to the absolute value of the transfer function <math>H_n(j \omega)</math>:
| | Hierba normalmente debe ser de dos a tres pulgadas de alto. Sólo cortar una tercera con la parte superior de una sola vez para evitar que la hierba entre en shock. Su imperativo que la cuchilla del cortacésped es siempre fuerte para detener el daño hierba. tire suavemente de los hilos separados con los dedos. . Después se separa cada sección, mantenga esa sección del cabello junto con una banda elástica cubierta de tela o torcer la sección y mantenga de forma segura con un clip. |
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| :<math>G_n(\omega) = \left | H_n(j \omega) \right | = \frac{1}{\sqrt{1+\varepsilon^2 T_n^2\left(\frac{\omega}{\omega_0}\right)}}</math>
| | amazines.comSportsarticle_detail.cfm? articleid = Cómo identificar si su plancha de pelo GHD es auténtica Están llenos de una gran cantidad de altas tecnologías características y mecanismos de seguridad. por lo que recurren a eBay. La compra de una plancha de pelo ghd fuera de ebay le puede ahorrar dinero, pero también puede venir con riesgo. Usted puede terminar pagando la mitad del costo de su plancha de pelo ghd, rizos así como giros en su cabello. |
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| where <math>\varepsilon</math> is the ripple factor, <math>\omega_0</math> is the [[cutoff frequency]] and <math>T_n</math> is a [[Chebyshev polynomial]] of the <math>n</math>th order.
| | La característica más atractiva de este increíble producto es que tiene una tensión universal. ue a la presencia de esta función se puede llevar a cualquier parte del mundo que va a viajar a sin tener que llevar un convertidor.<br><br>If you have any concerns with regards to the place and how to use [http://tinyurl.com/ntsklkt ghd plancha], you can make contact with us at our own page. |
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| The passband exhibits equiripple behavior, with the ripple determined by the ripple factor <math>\varepsilon</math>. In the passband, the Chebyshev polynomial alternates between -1 and 1 so the filter gain will alternate between maxima at ''G'' = 1 and minima at <math>G=1/\sqrt{1+\varepsilon^2}</math>. At the cutoff frequency <math>\omega_0</math> the gain again has the value <math>1/\sqrt{1+\varepsilon^2}</math> but continues to drop into the stop band as the frequency increases. This behavior is shown in the diagram on the right. The common practice of defining the cutoff frequency at −3 [[decibel|dB]] is usually not applied to Chebyshev filters; instead the cutoff is taken as the point at which the gain falls to the value of the ripple for the final time.
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| The order of a Chebyshev filter is equal to the number of [[Reactance (electronics)|reactive]] components (for example, [[inductor]]s) needed to realize the filter using [[analog electronics]].
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| The ripple is often given in [[decibel|dB]]:
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| :Ripple in dB = <math>20 \log_{10}\sqrt{1+\varepsilon^2}</math>
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| so that a ripple amplitude of 3 dB results from <math>\varepsilon = 1.</math>
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| An even steeper [[roll-off]] can be obtained if we allow for ripple in the stop band, by allowing zeroes on the <math>j\omega</math>-axis in the complex plane. This will however result in less suppression in the stop band. The result is called an [[elliptic filter]], also known as Cauer filter.
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| <br style="clear:both;" />
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| === Poles and zeroes ===
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| [[File:Chebyshev Type I Filter s-Plane Response (8th Order).svg|right|thumb|300px|Log of the absolute value of the gain of an 8th order Chebyshev type I filter in [[complex frequency space]] (''s'' = ''σ'' + ''jω'') with ε = 0.1 and <math>\omega_0=1</math>. The white spots are poles and are arranged on an ellipse with a semi-axis of 0.3836... in σ and 1.071... in ω. The transfer function poles are those poles in the left half plane. Black corresponds to a gain of 0.05 or less, white corresponds to a gain of 20 or more.]]
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| For simplicity, assume that the cutoff frequency is equal to unity. The poles <math>(\omega_{pm})</math> of the gain function of the Chebyshev filter are the zeroes of the denominator of the gain function. Using the complex frequency ''s'', these occur when:
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| :<math>1+\varepsilon^2T_n^2(-js)=0.\,</math>
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| Defining <math>-js=\cos(\theta)</math> and using the trigonometric definition of the Chebyshev polynomials yields:
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| :<math>1+\varepsilon^2T_n^2(\cos(\theta))=1+\varepsilon^2\cos^2(n\theta)=0.\,</math>
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| Solving for <math>\theta</math>
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| :<math>\theta=\frac{1}{n}\arccos\left(\frac{\pm j}{\varepsilon}\right)+\frac{m\pi}{n}</math>
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| where the multiple values of the arc cosine function are made explicit using the integer index ''m''. The poles of the Chebyshev gain function are then:
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| :<math>s_{pm}=j\cos(\theta)\,</math>
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| ::::<math>=j\cos\left(\frac{1}{n}\arccos\left(\frac{\pm j}{\varepsilon}\right)+\frac{m\pi}{n}\right).</math>
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| Using the properties of the trigonometric and hyperbolic functions, this may be written in explicitly complex form:
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| :<math>s_{pm}^\pm=\pm \sinh\left(\frac{1}{n}\mathrm{arsinh}\left(\frac{1}{\varepsilon}\right)\right)\sin(\theta_m)</math>
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| ::::<math>+j \cosh\left(\frac{1}{n}\mathrm{arsinh}\left(\frac{1}{\varepsilon}\right)\right)\cos(\theta_m)
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| </math>
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| where ''m'' = 1, 2,..., ''n'' and
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| :<math>\theta_m=\frac{\pi}{2}\,\frac{2m-1}{n}.</math>
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| This may be viewed as an equation parametric in <math>\theta_n</math> and it demonstrates that the poles lie on an ellipse in [[Complex frequency space|''s''-space]] centered at ''s'' = 0 with a real semi-axis of length <math>\sinh(\mathrm{arsinh}(1/\varepsilon)/n)</math> and an imaginary semi-axis of length of <math>\cosh(\mathrm{arsinh}(1/\varepsilon)/n).</math>
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| === The transfer function ===
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| The above expression yields the poles of the gain ''G''. For each complex pole, there is another which is the complex conjugate, and for each conjugate pair there are two more that are the negatives of the pair. The [[transfer function]] must be stable, so that its poles will be those of the gain that have negative real parts and therefore lie in the left half plane of complex frequency space. The transfer function is then given by
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| :<math>H(s)= \frac{1}{2^{n-1}\varepsilon}\ \prod_{m=1}^{n} \frac{1}{(s-s_{pm}^-)}</math>
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| where <math>s_{pm}^-</math> are only those poles with a negative sign in front of the real term in the above equation for the poles.
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| === The group delay ===
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| [[Image:Chebyshev5 GainDelay.png|300px|left|thumb|Gain and group delay of a fifth-order type I Chebyshev filter with ε = 0.5.]]
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| The [[group delay]] is defined as the derivative of the phase with respect to angular frequency and is a measure of the distortion in the signal introduced by phase differences for different frequencies.
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| :<math>\tau_g=-\frac{d}{d\omega}\arg(H(j\omega))</math>
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| The gain and the group delay for a fifth-order type I Chebyshev filter with ε=0.5 are plotted in the graph on the left. It can be seen that there are ripples in the gain and the group delay in the passband but not in the stopband.
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| <br style="clear:both;" />
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| ==Type II Chebyshev filters==
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| [[Image:ChebyshevII response.png|thumb|350px|The frequency response of a fifth-order type II Chebyshev low-pass filter with <math>\varepsilon=0.01</math>]]
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| Also known as inverse Chebyshev, this type is less common because it does not roll off as fast as type I, and requires more components. It has no ripple in the passband, but does have equiripple in the stopband. The gain is:
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| :<math>G_n(\omega,\omega_0) = \frac{1}{\sqrt{1+ \frac{1} {\varepsilon^2 T_n ^2 \left ( \omega_0 / \omega \right )}}}.</math>
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| In the stopband, the Chebyshev polynomial will oscillate between -1 and 1 so that the gain will oscillate between zero and
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| :<math>\frac{1}{\sqrt{1+ \frac{1}{\varepsilon^2}}}</math>
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| and the smallest frequency at which this maximum is attained will be the cutoff frequency <math>\omega_o</math>. The parameter ε is thus related to the [[stopband]] [[attenuation]] γ in [[decibel]]s by:
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| :<math>\varepsilon = \frac{1}{\sqrt{10^{0.1\gamma}-1}}.</math>
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| For a stopband attenuation of 5dB, ε = 0.6801; for an attenuation of 10dB, ε = 0.3333. The frequency ''f''<sub>0</sub> = ''ω''<sub>0</sub>/2''π'' is the cutoff frequency. The 3dB frequency ''f''<sub>H</sub> is related to ''f''<sub>0</sub> by:
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| :<math>f_H = \frac{f_0}{\cosh \left(\frac{1}{n} \cosh^{-1}\frac{1}{\varepsilon}\right)}.</math>
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| === Poles and zeroes ===
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| [[File:Chebyshev Type II Filter s-Plane Response (8th Order).svg|right|thumb|300px|Log of the absolute value of the gain of an 8th order Chebyshev type II filter in complex frequency space (s=σ+jω) with ε = 0.1 and <math>\omega_0=1</math>. The white spots are poles and the black spots are zeroes. All 16 poles are shown. Each zero has multiplicity of two, and 12 zeroes are shown and four are located outside the picture, two on the positive ω axis, and two on the negative. The poles of the transfer function will be poles on the left half plane and the zeroes of the transfer function will be the zeroes, but with multiplicity 1. Black corresponds to a gain of 0.05 or less, white corresponds to a gain of 20 or more.]]
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| Again, assuming that the cutoff frequency is equal to unity, the poles <math>(\omega_{pm})</math> of the gain of the Chebyshev filter will be the zeroes of the denominator of the gain:
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| :<math>1+\varepsilon^2T_n^2(-1/js_{pm})=0.</math>
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| The poles of gain of the type II Chebyshev filter will be the inverse of the poles of the type I filter:
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| :<math>\frac{1}{s_{pm}^\pm}=
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| \pm \sinh\left(\frac{1}{n}\mathrm{arsinh}\left(\frac{1}{\varepsilon}\right)\right)\sin(\theta_m)</math>
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| :<math>\qquad+j \cosh\left(\frac{1}{n}\mathrm{arsinh}\left(\frac{1}{\varepsilon}\right)\right)\cos(\theta_m)
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| </math>
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| where ''m'' = 1, 2, ..., ''n'' . The zeroes <math>(\omega_{zm})</math> of the type II Chebyshev filter will be the zeroes of the numerator of the gain:
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| :<math>\varepsilon^2T_n^2(-1/js_{zm})=0.\,</math>
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| The zeroes of the type II Chebyshev filter will thus be the inverse of the zeroes of the Chebyshev polynomial.
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| :<math>1/s_{zm} = -j\cos\left(\frac{\pi}{2}\,\frac{2m-1}{n}\right)</math>
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| for ''m'' = 1, 2, ..., ''n''.
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| === The transfer function ===
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| The transfer function will be given by the poles in the left half plane of the gain function, and will have the same zeroes but these zeroes will be single rather than double zeroes.
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| === The group delay ===
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| [[Image:ChebyshevII5 GainDelay.png|300px|left|thumb|Gain and group delay of a fifth-order type II Chebyshev filter with ε = 0.1.]]
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| The gain and the group delay for a fifth-order type II Chebyshev filter with ε=0.1 are plotted in the graph on the left. It can be seen that there are ripples in the gain in the stop band but not in the pass band.
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| <br style="clear:both;" />
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| == Implementation ==
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| === Cauer topology ===
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| A passive LC Chebyshev [[low-pass filter]] may be realized using a [[Cauer topology (electronics)|Cauer topology]]. The inductor or capacitor values of a nth-order Chebyshev [[prototype filter]] may be calculated from the following equations:<ref>Matthaei et. al 1980, p.99</ref>
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| :<math>G_{0} = 1</math>
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| :<math>G_{1} =\frac{ 2 A_{1} }{ \gamma }</math>
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| :<math>G_{k} =\frac{ 4 A_{k-1} A_{k} }{ B_{k-1}G_{k-1} },\qquad k = 2,3,4,\dots,n </math>
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| :<math>G_{n+1} =\begin{cases} 1 & \text{if } n \text{ odd} \\
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| \coth^{2} \left ( \frac{ \beta }{ 4 } \right ) & \text{if } n \text{ even}
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| \end{cases}</math>
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| G<sub>1</sub>, G<sub>k</sub> are the capacitor or inductor element values.
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| f<sub>H</sub>, the 3 dB frequency is calculated with: <math>f_H = f_0 \cosh \left(\frac{1}{n} \cosh^{-1}\frac{1}{\varepsilon}\right)</math>
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| The coefficients ''A'', ''γ'', ''β'', ''A''<sub>''k''</sub>, and ''B''<sub>''k''</sub> may be calculated from the following equations:
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| :<math>\gamma = \sinh \left ( \frac{ \beta }{ 2n } \right )</math>
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| :<math>\beta = \ln\left [ \coth \left ( \frac{ R_{db} }{ 17.37 } \right ) \right ]</math>
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| :<math>A_k=\sin\frac{ (2k-1)\pi }{ 2n },\qquad k = 1,2,3,\dots, n </math>
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| :<math>B_k=\gamma^{2}+\sin^{2}\left ( \frac{ k \pi }{ n } \right ),\qquad k = 1,2,3,\dots,n </math>
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| where ''R''<sub>dB</sub> is the passband ripple in decibels.
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| [[File:Cauer Topology Lowpass Filter.svg|thumb|right|450px|Low-pass filter using Cauer topology]]
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| The calculated ''G''<sub>''k''</sub> values may then be converted into [[shunt (electrical)|shunt]] capacitors and [[series (circuit)|series]] inductors as shown on the right, or they may be converted into series capacitors and shunt inductors. For example,
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| *''C''<sub>1 shunt</sub> = G<sub>1</sub>, ''L''<sub>2 series</sub> = ''G''<sub>2</sub>, ...
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| or
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| *''L''<sub>1 shunt</sub> = ''G''<sub>1</sub>, ''C''<sub>1 series</sub> = ''G''<sub>2</sub>, ...
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| Note that when G<sub>1</sub> is a shunt capacitor or series inductor, G<sub>0</sub> corresponds to the input resistance or conductance, respectively. The same relationship holds for G<sub>n+1</sub> and G<sub>n</sub>. The resulting circuit is a normalized low-pass filter. Using [[frequency transformations]] and [[impedance scaling]], the normalized low-pass filter may be transformed into [[high-pass filter|high-pass]], [[band-pass filter|band-pass]], and [[band-stop filter|band-stop]] filters of any desired [[cutoff frequency]] or [[Bandwidth (signal processing)|bandwidth]].
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| === Digital ===
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| As with most analog filters, the Chebyshev may be converted to a digital (discrete-time) [[recursive filter|recursive]] form via the [[bilinear transform]]. However, as [[digital filter]]s have a finite bandwidth, the response shape of the transformed Chebyshev will be [[bilinear transform#Frequency warping|warped]]. Alternatively, the [[Matched Z-transform method]] may be used, which does not warp the response.
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| ==Comparison with other linear filters==
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| Here is an image showing the Chebyshev filters next to other common kind of filters obtained with the same number of coefficients (all filters are fifth order):
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| [[Image:Electronic linear filters.svg|500px|center]]
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| As is clear from the image, Chebyshev filters are sharper than the [[Butterworth filter]]; they are not as sharp as the [[Elliptic filter|elliptic one]], but they show fewer ripples over the bandwidth.
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| ==See also==
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| :''[[Filter design]]''
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| * [[Bessel filter]]
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| * [[Comb filter]]
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| * [[Elliptic filter]]
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| :''[[Mathematics]]''
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| * [[Chebyshev nodes]]
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| * [[Chebyshev polynomial]]
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| ==Notes==
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| {{reflist}}
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| == References ==
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| *{{cite book | last=Daniels | first=Richard W. |authorlink= | coauthors= | title=Approximation Methods for Electronic Filter Design |year=1974 |publisher=McGraw-Hill |location=New York |isbn=0-07-015308-6 }}
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| * {{Cite book | last1 = Williams | first1= Arthur B. |last2=Taylors | first2=Fred J. | title =Electronic Filter Design Handbook | location= New York | publisher = McGraw-Hill | year = 1988 | isbn = 0-07-070434-1 | postscript = <!--None-->}}
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| * {{cite book | last1 = Matthaei | first1 = George L. | last2 = Young | first2 = Leo | last3 = Jones | first3 = E. M. T. | title = Microwave Filters, Impedance-Matching Networks, and Coupling Structures | location = Norwood, MA | publisher = Artech House | year = 1980 | isbn = 0-89-006099-1}}
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| [[Category:Linear filters]]
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| [[Category:Network synthesis filters]]
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| [[Category:Electronic design]]
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Ellos son algunas de las situaciones en las que transcribir de audio puede ser el mejor factor de lograr. Transcribir audio es definitivamente un elementos cruciales para los músicos de jazz y por lo que pensar en ello una sustancial sugieren de escolaridad. Esto realmente no es sorprendente, teniendo en cuenta que el jazz está en función principalmente de la improvisación. Fuera de las ciudades usted podría conocimiento correcto elegancia natural de Italia.
Inland esto realmente está dentro de la clase de picos de piedra caliza sin hueso por profundos barrancos y cascada ho en las piscinas de piedra. Cerca de la costa, encontrará excelentes paseos junto al mar y una gran cantidad de estaciones de costa incluyendo Fano en el interior del norte y Pedaso y San Benedetto del Tronto en el interior del sur. http:www.articlebiz.comarticle--ghd-vs-ghd-which-one-is-better-for-usGhd Vs. Chi comparar las características ghd y chi son dos fabricantes más conocidos de las mejores herramientas de peinado del cabello de venta en el mercado.
articleid = ghd planchas para el pelo Lo mejor para tu cabello Las mujeres siempre han estado buscando mejores formas de embellecer a sí mismos. Y el pelo siempre tiene un poco de atención especial en este proceso de embellecimiento de las mujeres. Ciencia también ha hecho una gran cantidad de contribución para el mejor estilo de las mujeres. Hay un montón de cosas que son resultado de la ciencia pura y son utilizados con fines de embellecimiento y peinado pelo de las mujeres.
puede ser una mujer común y corriente por un tiempo, pero no toda la vida. Así se dice que no hay chicas feas únicos pero perezosos en el mundo. Sin embargo, iamond ghd puede resolver todos los problemas de su cabello desordenado. Otras grandes características de alisadores de pelo CHI incluyen un interruptor de apagado automático del sensor de movimiento. Un año garantía del fabricante, que le proporcionará la tranquilidad de saber acerca de su compra. CHI también tienen un ghdp regulador de calor auto, Algunos paja es valioso para su césped por la razón que anima a la descomposición de restos de césped y con sujeción natural.
Hierba normalmente debe ser de dos a tres pulgadas de alto. Sólo cortar una tercera con la parte superior de una sola vez para evitar que la hierba entre en shock. Su imperativo que la cuchilla del cortacésped es siempre fuerte para detener el daño hierba. tire suavemente de los hilos separados con los dedos. . Después se separa cada sección, mantenga esa sección del cabello junto con una banda elástica cubierta de tela o torcer la sección y mantenga de forma segura con un clip.
amazines.comSportsarticle_detail.cfm? articleid = Cómo identificar si su plancha de pelo GHD es auténtica Están llenos de una gran cantidad de altas tecnologías características y mecanismos de seguridad. por lo que recurren a eBay. La compra de una plancha de pelo ghd fuera de ebay le puede ahorrar dinero, pero también puede venir con riesgo. Usted puede terminar pagando la mitad del costo de su plancha de pelo ghd, rizos así como giros en su cabello.
La característica más atractiva de este increíble producto es que tiene una tensión universal. ue a la presencia de esta función se puede llevar a cualquier parte del mundo que va a viajar a sin tener que llevar un convertidor.
If you have any concerns with regards to the place and how to use ghd plancha, you can make contact with us at our own page.