Chebyshev filter

Chebyshev filters are analog or digital filters having a steeper roll-off and more passband ripple (type I) or stopband ripple (type II) than Butterworth filters. 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. ,), but with ripples in the passband.This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials.The type I Chebyshev filters are called usually as just 'Chebyshev filters', the type II ones are usually called 'inverse Chebyshev filters'. Chebyshev filters are analog or digital filters having a steeper roll-off and more passband ripple (type I) or stopband ripple (type II) than Butterworth filters. 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. ,), but with ripples in the passband.This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials.The type I Chebyshev filters are called usually as just 'Chebyshev filters', the type II ones are usually called 'inverse Chebyshev filters'. 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. Type I Chebyshev filters are the most common types of Chebyshev filters. The gain (or amplitude) response, G n ( ω ) {displaystyle G_{n}(omega )} , as a function of angular frequency ω {displaystyle omega } of the nth-order low-pass filter is equal to the absolute value of the transfer function H n ( s ) {displaystyle H_{n}(s)} evaluated at s = j ω {displaystyle s=jomega } : where ε {displaystyle varepsilon } is the ripple factor, ω 0 {displaystyle omega _{0}} is the cutoff frequency and T n {displaystyle T_{n}} is a Chebyshev polynomial of the n {displaystyle n} th order. The passband exhibits equiripple behavior, with the ripple determined by the ripple factor ε {displaystyle varepsilon } . In the passband, the Chebyshev polynomial alternates between -1 and 1 so the filter gain alternate between maxima at G = 1 and minima at G = 1 / 1 + ε 2 {displaystyle G=1/{sqrt {1+varepsilon ^{2}}}} . The ripple factor ε is thus related to the passband ripple δ in decibels by: At the cutoff frequency ω 0 {displaystyle omega _{0}} the gain again has the value 1 / 1 + ε 2 {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 ωH is related to ω0 by: 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.