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 operating frequency range of the filter,^[1][2] but they achieve this 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".^[3] 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.^[4]

Type I Chebyshev filters (Chebyshev filters)

Type I Chebyshev filters are the most common types of Chebyshev filters. The gain (or amplitude) response, ${\displaystyle G_{n}(\omega )}$, as a function of angular frequency ${\displaystyle \omega }$ of the ${\displaystyle n}$th-order low-pass filter is equal to the absolute value of the transfer function ${\displaystyle H_{n}(s)}$ evaluated at ${\displaystyle s=j\omega }$:

${\displaystyle G_{n}(\omega )=\left|H_{n}(j\omega )\right|={\frac {1}{\sqrt {1+\varepsilon ^{2}T_{n}^{2}(\omega /\omega _{0})}}}}$

where ${\displaystyle \varepsilon }$ is the ripple factor, ${\displaystyle \omega _{0}}$ is the cutoff frequency and ${\displaystyle T_{n}}$ is a Chebyshev polynomial of the ${\displaystyle n}$th order.

The passband exhibits equiripple behavior, with the ripple determined by the ripple factor ${\displaystyle \varepsilon }$. In the passband, the Chebyshev polynomial alternates between -1 and 1 so the filter gain alternate between maxima at ${\displaystyle G=1}$ and minima at ${\displaystyle G=1/{\sqrt {1+\varepsilon ^{2}}}}$.

The ripple factor ε is thus related to the passband ripple δ in decibels by:

${\displaystyle \varepsilon ={\sqrt {10^{\delta /10}-1}}.}$

At the cutoff frequency ${\displaystyle \omega _{0}}$ the gain again has the value ${\displaystyle 1/{\sqrt {1+\varepsilon ^{2}}}}$ but continues to drop into the stopband as the frequency increases. This behavior is shown in the diagram on the right. The common practice of defining the cutoff frequency at −3 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.

The 3 dB frequency ${\displaystyle \omega _{H}}$ is related to ${\displaystyle \omega _{0}}$ by:

${\displaystyle \omega _{H}=\omega _{0}\cosh \left({\frac {1}{n}}\cosh ^{-1}{\frac {1}{\varepsilon }}\right).}$

The order of a Chebyshev filter is equal to the number of reactive components (for example, inductors) needed to realize the filter using analog electronics.

An even steeper roll-off can be obtained if ripple is allowed in the stopband, by allowing zeros on the ${\displaystyle \omega }$-axis in the complex plane. While this produces near-infinite suppression at and near these zeros (limited by the quality factor of the components, parasitics, and related factors), overall suppression in the stopband is reduced. The result is called an elliptic filter, also known as a Cauer filter.

Poles and zeroes

For simplicity, it is assumed that the cutoff frequency is equal to unity. The poles ${\displaystyle (\omega _{pm})}$ of the gain function of the Chebyshev filter are the zeroes of the denominator of the gain function. Using the complex frequency ${\displaystyle s}$, these occur when:

${\displaystyle 1+\varepsilon ^{2}T_{n}^{2}(-js)=0.\,}$

Defining ${\displaystyle -js=\cos(\theta )}$ and using the trigonometric definition of the Chebyshev polynomials yields:

${\displaystyle 1+\varepsilon ^{2}T_{n}^{2}(\cos(\theta ))=1+\varepsilon ^{2}\cos ^{2}(n\theta )=0.\,}$

Solving for ${\displaystyle \theta }$

${\displaystyle \mathrm{\theta <!--\; \theta \; -->}}$Zdroj:https://en.wikipedia.org?pojem=Chebyshev_filter
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