Chebyshev filter Totally Explained

Everything about Chebyshev Filter totally explained

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 filter characteristic and the actual over the range of the filter, but with ripples in the passband. This type of filter is named in honor of Pafnuty Chebyshev because their mathematical characteristics are derived from Chebyshev polynomials.
Because of the passband ripple inherent in Chebyshev filters, filters which have a smoother response in the passband but a more irregular response in the stopband are preferred for some applications.

Type I Chebyshev Filters

These are the most common Chebyshev filters. The gain (or amplitude) response as a function of angular frequency

omega

of the nth order low pass filter is ť

G_n(omega) = left | H_n(j omega) 
ight | = frac
ight)

, k = 1,2,3,...n

:where R_dB is the passband ripple in decibels.
The calculated G_k values may then be converted into shunt capacitors and top inductors as shown on the right, or they may be converted into top capacitors and shunt inductors.

For example, C_{1 shunt}=G₁, L_{2 top}=G₂, ...

or L_{1 shunt} = G₁, C_{1 top}=G₂, ... 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, band-pass, and band-stop filters of any desired cutoff frequency or bandwidth.

Digital

As with most analog filters, the Chebyshev may be converted to a digital (discrete-time) recursive form via the bilinear transform. However, as digital filters have a finite bandwidth, the response shape of the transformed Chebyshev will be warped. Alternatively, the Matched Z-transform may be used, which doesn't warp the response.

Comparison with other linear filters

Here is an image showing the Chebyshev filters next to other common kind of filters obtained with the same number of coefficients:
As is clear from the image, Chebyshev filters are sharper than the Butterworth filter; they're not as sharp as the elliptic one, but they show fewer ripples over the bandwidth.

Further Information

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