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, but with ripples in the passband. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials.

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, ${\displaystyle G_{n}(\omega )}$, as a function of angular frequency ${\displaystyle \omega }$ of the nth-order low-pass filter is equal to the absolute value of the transfer function ${\displaystyle H_{n}(s)}$ evaluated at ${\displaystyle s=j\omega }$:

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