Chebyshev filter

Chebyshev filters are analog or digital filters that have a steeper roll-off than Butterworth filters, and have either passband ripple (type I) or stopband ripple (type II). Chebyshev filters have the property that they minimize the error between the idealized and the actual filter characteristic over the operating frequency range of the filter,

<math>P' = \left [ \sqrt{\left ( \frac{ P^2 + cos^2 \Bigl(\frac{\pi (n-1)}{ 2n } \Bigl)}

{1 - {cos^2 \Bigl(\frac{\pi (n-1)}{ 2n }\Bigl) \right )} \right ]_{\text{Left Half Plane } } </math>

Where:

n is the order of the filter (must be even)

P is a traditional Chebyshev transfer function pole

P' is the mapped pole for the modified even order transfer function.

"Left Half Plane" indicates to use the square root containing a negative real value.

When complete, a replacement equi-ripple transfer function is created with reflection zero scattering matrix values for S12 of one and S11 of zero when implemented with equally terminated passive networks. The illustration below shows an 8th order Chebyshev filter modified to support even order equally terminated passive networks by relocating the lowest frequency reflection zero from a finite frequency to 0 while maintaining an equi-ripple pass band frequency response.

alt=Even order modified Chebyshev illustration|center|thumb|660x660px|Even order modified Chebyshev illustration

The LC element value formulas in the Cauer topology are not applicable to the even order modified Chebyshev transfer function, and cannot be used. It is therefore necessary to calculate the LC values from traditional continued fractions of the impedance function, which may be derived from the reflection coefficient, which in turn may be derived from the transfer function.

Minimum order

To design a Chebyshev filter using the minimum required number of elements, the minimum order of the Chebyshev filter may be calculated as follows. The equations account for standard low pass Chebyshev filters, only. Even order modifications and finite stop band transmission zeros will introduce error that the equations do not account for.

<math>n = ceil \bigg[\frac{\cosh^{-1}{\sqrt{\frac{10^{\alpha_s/10}-1}{10^{\alpha_p/10}-1{\cosh^{-1}{(\omega_s /\omega_p)\bigg]</math>

where:

<math>\omega_p</math> and <math>\alpha_p</math> are the pass band ripple frequency and maximum ripple attenuation in dB

<math>\omega_s</math> and <math>\alpha_s</math> are the stop band frequency and attenuation at that frequency in dB

<math>n</math> is the minimum number of poles, the order of the filter.

ceil[] is a round up to next integer function.

Setting the cutoff attenuation

Pass band cutoff attenuation for Chebyshev filters is usually the same as the pass band ripple attenuation, set by the computation above. However, many applications such as diplexers and triplexers, The equations account for standard low pass Inverse Chebyshev filters, only. Even order modifications will introduce error that the equations do not account for. The equations is identical to that used for Chebyshev filter minimum order, with a slightly different variable definitions.

<math>n = ceil \bigg[\frac{\cosh^{-1}{\sqrt{\frac{10^{\alpha_s/10}-1}{10^{\alpha_p/10}-1{\cosh^{-1}{(\omega_s /\omega_p)\bigg]</math>

where:

<math>\omega_p</math> and <math>\alpha_p</math> are the pass band frequency and attenuation at that frequency in dB

<math>\omega_s</math> and <math>\alpha_s</math> are the stop band frequency and minimum stop band attenuation in dB

<math>n</math> is the minimum number of poles, the order of the filter.

ceil[] is a round up to next integer function.

Setting the cutoff attenuation

The standard cutoff attenuation as described is the same at the pass band ripple attenuation. However, just as in Chebyshev filters, it is useful to set the cutoff attenuation to a desired value, and for the same reasons. Setting the Chebyshev II cutoff attenuation is the same as for Chebyshev cutoff attenuation, except the arithmetic attenuation and ripple entries are inverted in the equation and the poles and zeros are multiplied by the result, as opposed to divided by in the Chebyshev case..

<math>\begin{align}

p_A & = p_1 * T_n^{-1}\Biggr(\sqrt{\frac{10^, n \Biggr)

\qquad & \text{For } 0 < \delta < \infty \text{ and } 0 \leq \alpha < \infty \\

& = p_1 *cosh \Biggr(\frac{1}{n}cosh^{-1}\Bigr(\sqrt{\frac{10^ - 1}{10^\Bigr)\Biggr)-cos^2(\frac{\pi(n-1)}{2n})}

{1-{cos^2(\frac{\pi(n-1)}{2n})

}

\text{ For } 0 < \delta < \infty \text { and } \delta \leq \alpha < \infty \\

\end{align} </math>

Implementation

Cauer topology

A passive LC Chebyshev low-pass filter may be realized using a Cauer topology. The inductor or capacitor values of an <math>n</math>th-order Chebyshev prototype filter may be calculated from the following equations:

G₁ to G_k are the capacitor or inductor element values,

G₀ is the input termination impedance.

G_n+1 is the output termination impedance when the last element is a shunt capacitor, and the output termination admittance when the last element is a series inductor. Note that for odd order Chebyshev the distinction for the last element is moot.

The 3&nbsp;dB frequency is calculated with: <math>f_H = f_0 \cosh \left(\frac{1}{n} \cosh^{-1}\frac{1}{\varepsilon}\right)</math>

The coefficients A, γ, β, A_k, and B_k may be calculated from the following equations:

:<math>\gamma = \sinh \left ( \frac{ \beta }{ 2n } \right )</math>

:<math>\beta = \ln\left [ \coth \left ( \frac{ \delta }{ 17.37 } \right ) \right ]</math>

:<math>A_k=\sin\frac{ (2k-1)\pi }{ 2n },\qquad k = 1,2,3,\dots, n </math>

:<math>B_k=\gamma^{2}+\sin^{2}\left ( \frac{ k \pi }{ n } \right ),\qquad k = 1,2,3,\dots,n </math>

where <math>\delta</math> is the passband ripple in decibels.

The number <math>17.37</math> is rounded from the exact value <math>40/\ln(10)</math>.

thumb|upright=1.6|Low-pass filter using Cauer topology

The calculated G_k values may then be converted into shunt capacitors and series inductors as shown on the right, or they may be converted into series capacitors and shunt inductors. For example,

• C_{1 shunt} = G₁, L_{2 series} = G₂, ...

or

• L_{1 series} = G₁, C_{2 shunt} = G₂, ...

Note that when G₁ is a shunt capacitor or series inductor, G₀ corresponds to the input resistance or conductance, respectively. The same relationship holds for G_n+1 and G_n. 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 is warped. Alternatively, the Matched Z-transform method may be used, which does not warp the response.

Comparison with other linear filters

The following illustration shows the Chebyshev filters next to other common filter types obtained with the same number of coefficients (fifth order):

center

Chebyshev filters are sharper than the Butterworth filter; they are not as sharp as the elliptic one, but they show fewer ripples over the bandwidth.

Advanced Topics in Chebyshev Filters

Chebyshev filter design flexibility may be augmented by more advanced design methods documented in this section. Transmission zeros may be inserted into the stop band to neutralize specific undesired frequencies or increase the cut-off attenuation, or may be inserted off-axis to obtain a more desirable group delay. Asymmetric Chebyshev band pass filters may be created that contain differing number of poles on each side of the pass band to meet frequency asymmetric design requirements more efficiently. The equi-ripple pass bands and that Chebyshev filters are known for may be restricted to a percentage of the pass band to meet design requirements more efficiently that only call for a portion of the pass band to be equi-ripple.

Chebyshev transmission zeros

Chebyshev filters may be designed with arbitrarily placed finite transmission zeros in the stop band while retaining an equi-ripple pass band. Stop band zeros along the <math>j\omega</math> axis are generally used to eliminate unwanted frequencies. Stop band zeros along the real axis or quadruplet stop band zeros in the complex plane may be used to modify the group delay to a more desirable shape. The transmission zeros design utilizes characteristic polynomials, K(S), to place the transmission and reflection zeros, which in turn are used to create the transfer function, <math>G(s)</math>,

<math>G(s) = \sqrt{\frac{1}{1+\varepsilon ^2K(s)K(-s)\bigg|_{\text{left half plane (LHP) poles</math>

The calculation of K(S) relies upon the following observed equality.

<math>\begin{align}

&\varepsilon^2 = 10^{1dB/10.} - 1. = .25892541 \\

&G(s) = \sqrt{G(s)G(-s)}\bigg|_{\text{LHP poles = \sqrt{\frac{1}{1+\varepsilon ^2K(s)K(-s)\bigg|_{\text{LHP poles = \sqrt{\frac{K(s)_{den}K(-s)_{den{K(s)_{den}K(-s)_{den}+\varepsilon ^2K(s)_{num}K(-s)_{num}\bigg|_{\text{LHP poles\\

&= \sqrt{\frac{\{0.25(s)^2 + 1\}\{\{0.25(s)^2 + 1\}\\bigg|_{\text{LHP poles \\

&=\frac{0.25(s)^2 + 1}{\sqrt{-3.1393872s^6 - 4.8638872s^4 -1.4326456s^2 + 1 }\bigr|_{\text{LHP poles} \\

\end{align}</math>

To obtain <math>G(s)</math> from the left half plane, factor the numerator and denominator to obtain the roots. Discard all roots from the right half plane of the denominator, half the repeated roots in the numerator, and rebuild <math>G(s)</math> with the remaining roots. Generally, normalize <math>|G(s)|</math> to 1 at <math>s=0</math>.

<math>\begin{align}

&G(s)= \frac{0.25s^2 + 1}{1.7718316s^3 + 1.7200107s^2 + 2.2074118s + 1} \\

\end{align}</math>

To confirm that the example <math>G(s)</math> is correct, the plot of <math>G(s)</math> along <math>j\omega</math> is shown below with a pass band ripple of 1 dB, a cut off frequency of 1 rad/sec, and a stop band zero at 2 rad/sec.

center|thumb|440x440px|Chebyshev transmission zero at 2 rad/sec

Asymmetric band pass filter

Chebyshev band pass filters may be designed with a geometrically asymmetric frequency response by placing the desired number of transmission zeros at zero and infinity with the use of the more generalized form of the Chebyshev transmission zeros equation above,), and pass band cut-off attenuation = 20dB.

The target value in step 5 is .01010101, and the <math>\varepsilon^2</math> to compute <math>G(s)</math> is 99. When complete, the characteristic polynomials ,<math>K(s)</math>, and forward transfer function,<math>G(s)</math>, are below.

<math>K(s) = \frac{2.3081085s^8+3.7315386s^6+1.8867298s^4+0.28974597s^2}{0.82644628s^2+1}</math>

<math>G(s) = \frac{K(s)_{den{\sqrt{K(s)_{den}K(-s)_{den}+\varepsilon^2K(s)_{num}K(-s)_{num|_{\text{LHP roots}</math>

<math>\text{Where }\varepsilon^2 = 10^{(20_{dB}/10)} - 1 = 99.0 </math>

<math>G(s) = \frac{0.82644628s^2+1}{22.96539s^8+39.774072s^7+71.570971s^6+73.962937s^5+65.358572s^4+40.848153s^3+19.393829S^2+6.0938301s+1}</math>

The validation consists of calculating scattering parameters <math>|S12| \text{ and } |S11|</math> (<math>|G(s)|</math> and <math>\sqrt{1-|G(s)|^2}</math> respectively) for the constriction frequency, the cutoff frequency, the remaining pass band minima frequencies in between, and the transmission zero frequency and as shown below.

{| class="wikitable"

|+8 pole Non-standard cut-off attenuation and transmission zeros validation summary

!<math>\omega_i</math>

!<math>|S12|=|G(j\omega_k)|</math>

!<math>|S12|=\sqrt{1-|G(j\omega_k)|^2}</math>

|-

|<math>\omega_1=0.45</math>

| -0.043648054 dB

| -20dB

|-

|<math>\omega_2=0.66133008</math>

| -0.043648054 dB

| -20dB

|-

|<math>\omega_3=0.82704812</math>

| -0.043648054 dB

| -20dB

|-

|<math>\omega_{cut}=1</math>

| -20 dB

| -0.043648054 dB

|-

|<math>\omega_{Tz_1}=1.1</math>

| -<math>\infty</math>

|0 dB

|}

The final magnitude frequency response of <math>|S_{12}| \text{ and } |S_{11}|</math> are shown below.

[[File:NewtCheb_8_Pole.png|alt=8 pole constricted ripple Chebyshev with finite transmission zero and non-standard cut-off attenuation|center|thumb|440x440px|

{|

!Step final:

|-

|8 pole 55% constricted ripple pass band for <math>|G(s)| = \sqrt{\frac{1}{1+\varepsilon^2|K(j\omega)|^2</math>

|-

|20dB S11 equi-ripple pass band

|-

|finite transmission zero at 1.1 rad/sec

|-

|non-standard S12 cut-off attenuation at 20dB

|-

|Geometric frequency scale

|}

]]

See also

• Bessel filter
• Butterworth filter
• Chebyshev nodes
• Chebyshev polynomial
• Comb filter
• Elliptic filter
• Filter design

References

</references>

External links