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Simple filters[show]

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.

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[edit] Type I Chebyshev Filters

The frequency response of a fourth-order type I Chebyshev low-pass filter with ε = 1

These are the most common Chebyshev filters. The gain (or amplitude) response as a function of angular frequency ω of the nth order low pass filter is

$G_n(\omega) = \left | H_n(j \omega) \right | = \frac{1}{\sqrt{1+\epsilon^2 T_n^2\left(\frac{\omega}{\omega_0}\right)}}$

where ε is the ripple factor, ω₀ is the cutoff frequency and T_n() is a Chebyshev polynomial of the nth order.

The passband exhibits equiripple behavior, with the ripple determined by the ripple factor ε. In the passband, the Chebyshev polynomial alternates between 0 and 1 so the filter gain will alternate between maxima at G=1 and minima at $G=1/\sqrt{1+\epsilon^2}$ . At the cutoff frequency ω₀ the gain again has the value $1/\sqrt{1+\epsilon^2}$ but continues to drop into the stop band as the frequency increases. This behavior is shown in the diagram on the right. (note: the common definition of the cutoff frequency to −3 dB does not hold for Chebyshev filters!)

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.

The ripple is often given in dB:

Ripple in dB = $20 \log_{10}\frac{1}{\sqrt{1+\epsilon^2}}$

so that a ripple amplitude of 3 dB results from ε = 1.

An even steeper roll-off can be obtained if we allow for ripple in the stop band, by allowing zeroes on the jω-axis in the complex plane. This will however result in less suppression in the stop band. The result is called an elliptic filter, also known as Cauer filters.

[edit] 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 ω₀ = 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, assume that the cutoff frequency is equal to unity. The poles (ω_pm) of the gain of the Chebyshev filter will be the zeroes of the denominator of the gain. Using the complex frequency s:

$1+\epsilon^2T_n^2(-js)=0$

Defining − js = cos(θ) and using the trigonometric definition of the Chebyshev polynomials yields:

$1+\epsilon^2T_n^2(\cos(\theta))=1+\epsilon^2\cos^2(n\theta)=0$

solving for θ

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

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:

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

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

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

$s_{pm}^\pm=\pm \sinh\left(\frac{1}{n}\mathrm{arsinh}\left(\frac{1}{\epsilon}\right)\right)\sin(\theta_m)$

$+j \cosh\left(\frac{1}{n}\mathrm{arsinh}\left(\frac{1}{\epsilon}\right)\right)\cos(\theta_m)$

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

$\theta_m=\frac{\pi}{2}\,\frac{2m-1}{n}$

This may be viewed as an equation parametric in θ_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(arsinh(1 / ε) / n) and an imaginary semi-axis of length of cosh(arsinh(1 / ε) / n)

[edit] 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 will be 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

$H(s)= 2^{n-1}\epsilon\ \prod_{m=1}^{n} \frac{1}{(s-s_{pm}^-)}$

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

[edit] 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 stop band.

[edit] Type II Chebyshev Filters

The frequency response of a fifth-order type II Chebyshev low-pass filter with ε = 0.01

Also known as inverse Chebyshev, this 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:

$G_n(\omega,\omega_0) = \frac{1}{\sqrt{1+ \frac{1} {\epsilon^2 T_n ^2 \left ( \omega_0 / \omega \right )}}}.$

In the stop band, the Chebyshev polynomial will oscillate between 0 and 1 so that the gain will oscillate between zero and

$\frac{1}{\sqrt{1+ \frac{1}{\epsilon^2}}}$

and the smallest frequency at which this maximum is attained will be the cutoff frequency ω₀. The parameter ε is thus related to the stopband attenuation γ in decibels by:

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

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

$f_H = f_C \cosh \left(\frac{1}{n} \cosh^{-1}\frac{1}{\epsilon}\right).$