Chebyshev filter

Chebyshev filters are continuous frequency filters that are designed for the sharpest possible kink in the frequency response at the cutoff frequency ω _g . Instead , the amplification in the pass band or in the stop band does not run monotonously, but has a ripple to be defined. Within an order, the greater the permitted ripple, the steeper the drop. They are named after Pafnuti Lwowitsch Chebyshev (formerly transcribed as Chebyshev).

A distinction is made between Chebyshev filters of type I and type II. Chebyshev filters of type I have an oscillating course of the transfer function in the pass band . Chebyshev filters of type II have the ripple of the transfer function in the blocking range and are also referred to in the specialist literature as inverse Chebyshev filters .

Transfer function

Transfer function of a 4th order Chebyshev filter of type I with frequency response related to the cutoff frequency

For the area that have Chebyshev polynomials the desired properties. For the Chebyshev polynomials grow monotonically. ${\displaystyle 0\leq x\leq 1}$ ${\displaystyle T_{n}}$ ${\displaystyle x\geq 1}$

In order to produce a low pass with the help of the Chebyshev polynomials, one sets

${\displaystyle \left|{\underline {A}}\right|^{2}={\frac {kA_{0}^{2}}{1+\varepsilon ^{2}T_{n}^{2}(P)}}}$

with is chosen such that x = 0 . is a measure of the ripple. ${\displaystyle k}$ ${\displaystyle \left|{\underline {A}}\right|^{2}=A_{0}^{2}}$ ${\displaystyle \varepsilon }$

Coefficients

The transfer function is brought into the form

${\displaystyle A[P]={\frac {A_{0}}{\prod _{i}(1+a_{i}P+b_{i}P^{2})}}}$

result for the coefficients and the following relationships: ${\displaystyle a_{i}}$ ${\displaystyle b_{i}}$

Order n of the filter even:

${\displaystyle a_{i}^{\prime }=2b_{i}^{\prime }\sinh \gamma \cos {\frac {(2i-1)\pi }{2n}}}$

${\displaystyle b_{i}^{\prime }={\frac {1}{\cosh ^{2}\gamma -\cos ^{2}{\frac {(2i-1)\pi }{2n}}}}}$

Order n of the filter is odd:

${\displaystyle a_{1}^{\prime }={\frac {1}{\sinh \gamma }}}$

${\displaystyle b_{1}^{\prime }=0}$

${\displaystyle a_{i}^{\prime }=2b_{i}^{\prime }\sinh \gamma \cos {\frac {(i-1)\pi }{n}}}$

${\displaystyle b_{i}^{\prime }={\frac {1}{\cosh ^{2}\gamma -\cos ^{2}{\frac {(i-1)\pi }{n}}}}}$

These coefficients are chosen so that the cutoff frequency is normalized to the last frequency at which the selected gain is last accepted. ${\displaystyle \omega _{\mathrm {g} }}$

properties

The Chebyshev filter has the following properties:

• wavy frequency course depending on the type in the pass band or in the stop band.
• very steep kink at the cutoff frequency , improves with the order and the ripple.
• considerable overshoot in step response , deteriorates with order and ripple.
• if the ripple is allowed to approach 0, the Chebyshev filter changes into a Butterworth filter .
• no constant group delay in the passband.

Digital realization

For a digital implementation of the Chebyshev filter, the individual biquads are first transformed using bilinear transformation and then cascaded with the corresponding coefficients and . In the following, this has been carried out for a low-pass filter with even order n. ${\displaystyle a_{i}}$ ${\displaystyle b_{i}}$

The Z-transform of a biquad generally looks like this:

${\displaystyle S(Z)={\frac {a(Z)}{b(Z)}}={\frac {\alpha _{0}+\alpha _{1}\cdot Z^{-1}+\alpha _{2}\cdot Z^{-2}}{1+\beta _{1}\cdot Z^{-1}+\beta _{2}\cdot Z^{-2}}}}$.

This equation transforms into the time domain as follows:

${\displaystyle y[n]=\alpha _{0}\cdot x[n]+\alpha _{1}\cdot x[n-1]+\alpha _{2}\cdot x[n-2]-\beta _{1}\cdot y[n-1]-\beta _{2}\cdot y[n-2]}$

The coefficients and are calculated from the coefficients and as follows: ${\displaystyle \alpha _{i}}$ ${\displaystyle \beta _{i}}$ ${\displaystyle a_{i}}$ ${\displaystyle b_{i}}$

${\displaystyle K=\tan \left(\pi {\frac {\text{Frequenz}}{\text{Abtastrate}}}\right)}$ (Prewarp the frequency)

${\displaystyle b_{i}={\frac {1}{\cosh(\gamma )^{2}-\cos ^{2}{\frac {(2i-1)\cdot \pi }{n}}}}}$

${\displaystyle a_{i}=K\cdot 2b_{i}\cdot \sinh(\gamma )\cdot \cos {\frac {(2i-1)\cdot \pi }{n}}}$

${\displaystyle \gamma }$ is a measure of the overshoot:

${\displaystyle \gamma ={\frac {\operatorname {arsinh} ({\text{Ripple in dB}})}{n}}}$

The coefficients are then calculated as follows:

${\displaystyle \alpha _{0}=K\cdot K}$

${\displaystyle \alpha _{1}=2\cdot K^{2}}$

${\displaystyle \alpha _{2}=K\cdot K}$

${\displaystyle \beta _{0}^{\prime }=(a_{0}+K^{2}+b_{0})}$

${\displaystyle \beta _{1}^{\prime }=2\cdot (b_{1}-K^{2})}$

${\displaystyle \beta _{2}^{\prime }=(a_{2}-K^{2}-b_{2})}$

${\displaystyle \beta _{1}=\beta _{1}^{\prime }/\beta _{0}^{\prime }}$

${\displaystyle \beta _{2}=\beta _{2}^{\prime }/\beta _{0}^{\prime }}$

In order to implement higher-order filters, you only need to cascade several biquad sections. The implementation of digital Chebyshev filters takes place in IIR filter structures (recursive filter structure).

See also

• Bessel filter
• Cauer filter
• Butterworth filter

literature

• Lutz v. Wangenheim: Active Filters and Oscillators . 1st edition. Springer Verlag, Bremen 2007, ISBN 978-3-540-71737-9 .

Web links

• Calculate Chebyshev low-pass filter