Chebyshev filter

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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

The frequency response of a fourth-order type I Chebyshev low-pass filter with

\varepsilon=1

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

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

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

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

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

\varepsilon ={\sqrt {10^{0.1\delta }-1}}.

At the cutoff frequency $\omega _{0}$ the gain again has the value $1/{\sqrt {1+\varepsilon ^{2}}}$ 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 ω_H is related to ω₀ by:

\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 zeroes on the $j\omega$ -axis in the complex plane. However, this results in less suppression in the stopband. The result is called an elliptic filter, also known as Cauer filter.

Poles and zeroes

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

\omega _{0}=1

. 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.

For simplicity, it is assumed that the cutoff frequency is equal to unity. The poles $(\omega _{{pm}})$ 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:

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

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

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

Solving for $\theta$

\theta ={\frac {1}{n}}\arccos \left({\frac {\pm j}{\varepsilon }}\right)+{\frac {m\pi }{n}}

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:

s_{{pm}}=j\cos(\theta )\,

=j\cos \left({\frac {1}{n}}\arccos \left({\frac {\pm j}{\varepsilon }}\right)+{\frac {m\pi }{n}}\right).

Using the properties of the trigonometric and hyperbolic functions, this may be written in explicitly complex form:

s_{{pm}}^{\pm }=\pm \sinh \left({\frac {1}{n}}{\mathrm {arsinh}}\left({\frac {1}{\varepsilon }}\right)\right)\sin(\theta _{m})

+j\cosh \left({\frac {1}{n}}{\mathrm {arsinh}}\left({\frac {1}{\varepsilon }}\right)\right)\cos(\theta _{m})

where m = 1, 2,..., n and

\theta _{m}={\frac {\pi }{2}}\,{\frac {2m-1}{n}}.

This may be viewed as an equation parametric in $\theta_n$ and it demonstrates that the poles lie on an ellipse in s-space centered at s = 0 with a real semi-axis of length $\sinh({\mathrm {arsinh}}(1/\varepsilon )/n)$ and an imaginary semi-axis of length of $\cosh({\mathrm {arsinh}}(1/\varepsilon )/n).$

The transfer function

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 are 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

H(s)={\frac {1}{2^{{n-1}}\varepsilon }}\ \prod _{{m=1}}^{{n}}{\frac {1}{(s-s_{{pm}}^{-})}}

where $s_{{pm}}^{-}$ are only those poles of the gain with a negative sign in front of the real term in the above equation for the poles.

The group delay

Gain and group delay of a fifth-order type I Chebyshev filter with ε = 0.5.

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.

\tau _{g}=-{\frac {d}{d\omega }}\arg(H(j\omega ))

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.

Type II Chebyshev filters

The frequency response of a fifth-order type II Chebyshev low-pass filter with

\varepsilon =0.01

Also known as inverse Chebyshev filters, the Type II Chebyshev filter 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:

G_{n}(\omega ,\omega _{0})={\frac {1}{{\sqrt {1+{\frac {1}{\varepsilon ^{2}T_{n}^{2}\left(\omega _{0}/\omega \right)}}}}}}.

In the stopband, the Chebyshev polynomial oscillates between -1 and 1 so that the gain will oscillate between zero and

{\frac {1}{{\sqrt {1+{\frac {1}{\varepsilon ^{2}}}}}}}

and the smallest frequency at which this maximum is attained is the cutoff frequency $\omega _{o}$ . The parameter ε is thus related to the stopband attenuation γ in decibels by:

\varepsilon ={\frac {1}{{\sqrt {10^{{0.1\gamma }}-1}}}}.

For a stopband attenuation of 5 dB, ε = 0.6801; for an attenuation of 10 dB, ε = 0.3333. The frequency f₀ = ω₀/2π is the cutoff frequency. The 3 dB frequency f_H is related to f₀ by:

f_{H}={\frac {f_{0}}{\cosh \left({\frac {1}{n}}\cosh ^{{-1}}{\frac {1}{\varepsilon }}\right)}}.