Chebyshev Filters

Chebyshev filters come in two flavors: the more popular Type-I and the Type-II Chebyshev filter.

The Chebyshev Type-I Low-Pass Filter

Chebyshev Type-I filters are also all-pole filters, with poles of the normalized ($\omega _{0}=1\,\mathrm {\tfrac {rad}{s}}$) filter located on an ellipse inside the unit circle, as shown in Fig. 1. This arrangement of poles results in (1) a shorter transition band, and therefore a steeper roll-off compared to the Butterworth low-pass filter of the same order, shown in Fig. 2, (2) poorer group delay characteristics as depicted in Figs. 3 and 4, and (3) ripples in the pass-band also seen in Fig. 2. By allowing larger ripples in the pass-band, we get: (1) a narrower transition band, and hence sharper cut-off, (2) higher Q poles, and (3) more degradation in the phase response.

Figure 1: The low-pass Chebyshev Type-I pole-zero with $N=5$ and a $3\,{\text{dB}}$ ripple.	Figure 2: The low-pass Chebyshev Type-I low-pass filter magnitude response with $N=5$ and a $3\,{\text{dB}}$ ripple.
Figure 3: The low-pass Chebyshev Type-I low-pass filter phase response with $N=5$ and a $3\,{\text{dB}}$ ripple.	Figure 4: The low-pass Chebyshev Type-I low-pass filter group delay with $N=5$ and a $3\,{\text{dB}}$ ripple.

The Chebyshev Type-I magnitude response is given by:

$\left|H\left(j\omega \right)\right|={\frac {1}{\sqrt {1+\varepsilon ^{2}\cdot T_{N}^{2}\left({\frac {\omega }{\omega _{0}}}\right)}}}$

(1)

Where $N$ is the filter order, $\omega _{0}$ is the cut-off frequency, $\varepsilon ={\sqrt {10^{\frac {\delta }{10}}-1}}$ is the ripple factor with ripple $\delta$ in dB, and $T_{N}\left(x\right)$ are solutions to the Chebyshev differential equation:

$\left(1-x^{2}\right){\frac {\partial ^{2}y}{\partial x^{2}}}-x{\frac {\partial y}{\partial x}}+N^{2}y=0$

(2)

For a normalized filter, or $\omega _{0}=1\,\mathrm {\tfrac {rad}{s}}$, the poles of the Chebyshev Type-I low-pass filter are the roots of $1+\varepsilon ^{2}\cdot T_{N}^{2}\left(-js\right)=0$. In practice, instead of solving this equation directly, we can easily get the roots (and hence the poles) of the filter transfer function using tools such as Python or Matlab.

The Chebyshev Type-II Low-Pass Filter

The Chebyshev Type-II filter of order $N$ has $N$ poles and $N$ zeros for even $N$, and $N-1$ zeros for odd $N$. It has poles inside and outside the unit circuit, and complex conjugate zeros on the $j\omega$-axis, creating nulls in the stop-band. It is also less common than the Type-I filter since the roll-off is less steep, and due to its zeros, it requires more components.

The Chebyshev Type-II magnitude response is given by:

$\left|H\left(j\omega \right)\right|={\frac {1}{\sqrt {1+{\frac {1}{\varepsilon ^{2}\cdot T_{N}^{2}\left({\frac {\omega }{\omega _{0}}}\right)}}}}}$

(3)

Where $N$ is the filter order, $\omega _{0}$ is the cut-off frequency, and $\varepsilon ={\sqrt {10^{\frac {\gamma }{10}}-1}}$ is the attenuation factor with stop-band attenuation $\gamma$ in dB. The poles are the roots of:

$1+\varepsilon ^{2}T_{N}^{2}\left(-{\frac {1}{js}}\right)=0$

(4)

And the zeros are given by:

${\frac {1}{z_{m}}}=-j\cos \left({\frac {\pi }{2}}\cdot {\frac {2m-1}{N}}\right)$

(5)

For $m=1,2,\ldots ,N$. The characteristics of the Chebyshev Type-II filter are shown in Figs. 1-4.

Figure 1: The low-pass Chebyshev Type-II pole-zero with $N=5$ and a <math40\,\text{dB}</math> stop-band attenuation.	Figure 2: The low-pass Chebyshev Type-II magnitude response with $N=5$ and a <math40\,\text{dB}</math> stop-band attenuation.
Figure 3: The low-pass Chebyshev Type-II phase response with $N=5$ and a <math40\,\text{dB}</math> stop-band attenuation.	Figure 4: The low-pass Chebyshev Type-II group delay with $N=5$ and a <math40\,\text{dB}</math> stop-band attenuation.