Chebyshev filter

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(\omega/\omega_0)}} where \varepsilon is the ripple factor, \omega_0 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 \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^{\delta/10}-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 \omega_H is related to \omega_0 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 zeros on the \omega-axis in the complex plane. While this produces near-infinite suppression at and near these zeros (limited by the quality factor of the components, parasitics, and related factors), overall suppression in the stopband is reduced. The result is called an elliptic filter, also known as a Cauer filter. Poles and zeroes (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^2T_n^2(-js)=0.\, Defining -js=\cos(\theta) and using the trigonometric definition of the Chebyshev polynomials yields: :1+\varepsilon^2T_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, obtained from the above equation. The group delay The group delay is defined as the derivative of the phase with respect to angular frequency: :\tau_g=-\frac{d}{d\omega}\arg(H(j\omega)) The gain and the group delay for a 5th-order type I Chebyshev filter with ε=0.5 are plotted in the graph on the left. Its stop band has no ripples. But the ripples of group delay in its passband indicate that different frequency components have different delay, which along with the ripples of gain in its passband results in distortion of the waveform's shape. Even order modifications Even order Chebyshev filters implemented with passive elements, typically inductors, capacitors, and transmission lines, with terminations of equal value on each side cannot be implemented with the traditional Chebyshev transfer function without the use of coupled coils, which may not be desirable or feasible, particularly at the higher frequencies. This is due to the physical inability to accommodate the even order Chebyshev reflection zeros that result in a scattering matrix S12 values that exceed the S12 value at \omega=0. If it is not feasible to design the filter with one of the terminations increased or decreased to accommodate the pass band S12, then the Chebyshev transfer function must be modified so as to move the lowest even order reflection zero to \omega=0 while maintaining the equi-ripple response of the pass band. 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 } } 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. 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. 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] where: \omega_p and \alpha_p are the pass band ripple frequency and maximum ripple attenuation in dB \omega_s and \alpha_s are the stop band frequency and attenuation at that frequency in dB n 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, require a cutoff attenuation of -3.0103 dB in order to obtain the needed reflections. Other specialized applications may require other specific values for cutoff attenuation for various reasons. It is therefore useful to have a means available of setting the Chebyshev pass band cutoff attenuation independently of the pass band ripple attenuation, such as -1 dB, -10 dB, etc. The cutoff attenuation may be set by frequency scaling the poles of the transfer function. The scaling factor may be determined by direct algebraic manipulation of the defining Chebyshev filter function, G_n(\omega), including \varepsilon and T_n( \omega / \omega_0). The general definition of the Chebyshev function, T_n( \omega / \omega_0)=cos(n\cos^{-1}( \omega / \omega_0)) is required, which may be derived from the Chebyshev Polynomials equations, and the inverse Chebyshev function, T_n^{-1}( \omega / \omega_0)=cos(\cos^{-1}( \omega / \omega_0)/n). To keep the numbers real for values of \omega / \omega_0\geq 1, complex hyperbolic identities may be used to rewrite the equations as, T_n( \omega / \omega_0)=cosh(n\cosh^{-1 }( \omega / \omega_0)) and T_n^{-1}( \omega / \omega_0)=cosh(\cosh^{-1 }( \omega / \omega_0)/n). Using simple algebra on the above equations and references, the expression to scale each Chebyshev poles is: \begin{align} p_A & = p_1 / T_n^{-1}\Biggr(\sqrt{\frac{10^{{ \alpha} / 10 } - 1}{10^-1}}, n \Biggr) \qquad & \text{For } 0 Where: p_A is the relocated pole positioned to set the desired cutoff attenuation. p_1 is a ripple cutoff pole that lies on the oval. \delta is the passband attenuation ripple in dB (.05 dB, 1 dB, etc.)). \alpha is the desired passband attenuation at the cutoff frequency in dB (1 dB, 3 dB, 10 dB, etc.) n is the number of poles (the order of the filter). A quick sanity check on the above equation using passband ripple attenuation for the passband cutoff attenuation (\alpha = \delta) reveals that the pole adjustment will be 1.0 for this case, which is what is expected. Even order modified cutoff attenuation adjustment For Chebyshev filters being designed with modified for even order pass band ripple for passive equally terminated filters, the attenuation frequency computation needs to include the even order adjustment by performing the even order adjustment operation on the computed attenuation frequency. This makes the even order adjustment arithmetic slightly simpler, since frequency can be treated as a real variable, in this case ((J\omega)^2 \text { becomes }-\omega^2). \begin{align} p_A = p_1\sqrt{ \frac {1-{cos^2(\frac{\pi(n-1)}{2n})}} {cosh^2\Biggr(\frac{1}{n}cosh^{-1}\Bigr(\sqrt{\frac{10^-1}{10^-1}}\Bigr)\Biggr)-cos^2(\frac{\pi(n-1)}{2n})} } \text{ For } 0 Where: p_A is the relocated pole positioned to set the desired cutoff attenuation. p_1 is a ripple cutoff pole that has been modified for even order pass bands. \delta is the passband attenuation ripple in dB (.05 dB, 1 dB, etc.)). \alpha is the desired passband attenuation at the cutoff frequency in dB (1 dB, 3 dB, 10 dB, etc.) n is the number of poles (the order of the filter). cos (\frac{\pi (n-1)}{ 2n }) is the smallest even order Chebyshev Node == Type II Chebyshev filters (inverse Chebyshev filters) ==

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) = \frac{1}{\sqrt{1+\frac{1}{\varepsilon^2 T_n^2(\omega_0/\omega)}}} = \sqrt{\frac{\varepsilon^2 T_n^2(\omega_0/\omega)}{1+\varepsilon^2 T_n^2(\omega_0/\omega)}}. 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^{\gamma/10}-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 fH is related to f0 by: :f_H = \frac{f_0}{\cosh \left(\frac{1}{n} \cosh^{-1}\frac{1}{\varepsilon}\right)}. Poles and zeroes Assuming that the cutoff frequency is equal to unity, the poles (\omega_{pm}) of the gain of the Chebyshev filter are the zeroes of the denominator of the gain: :1+\varepsilon^2T_n^2(-1/js_{pm})=0. The poles of gain of the type II Chebyshev filter are the inverse of the poles of the type I filter: :\frac{1}{s_{pm}^\pm}= \pm \sinh\left(\frac{1}{n}\mathrm{arsinh}\left(\frac{1}{\varepsilon}\right)\right)\sin(\theta_m) :\qquad+j \cosh\left(\frac{1}{n}\mathrm{arsinh}\left(\frac{1}{\varepsilon}\right)\right)\cos(\theta_m) where m = 1, 2, ..., n. The zeroes (\omega_{zm}) of the type II Chebyshev filter are the zeroes of the numerator of the gain: :\varepsilon^2T_n^2(-1/js_{zm})=0.\, The zeroes of the type II Chebyshev filter are therefore the inverse of the zeroes of the Chebyshev polynomial. :1/s_{zm} = -j\cos\left(\frac{\pi}{2}\,\frac{2m-1}{n}\right) for m = 1, 2, ..., n. The transfer function The transfer function is given by the poles in the left half plane of the gain function, and has the same zeroes but these zeroes are single rather than double zeroes. The group delay The gain and the group delay for a fifth-order type II Chebyshev filter with ε=0.1 are plotted in the graph on the left. It can be seen that there are ripples in the gain in the stopband but not in the pass band. Even order modifications Just like Chebyshev filter even order filters, the standard Chebyshev II even order filter cannot be implemented with equally terminated passive elements without the use of coupled coils, which may not be desirable or feasible. In the Chebyshev II case, this is due to finite attenuation of S12 in the stop band. 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. 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] where: \omega_p and \alpha_p are the pass band frequency and attenuation at that frequency in dB \omega_s and \alpha_s are the stop band frequency and minimum stop band attenuation in dB n 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.. \begin{align} p_A & = p_1 * T_n^{-1}\Biggr(\sqrt{\frac{10^{{ \delta} / 10 } - 1}{10^-1}}, n \Biggr) \qquad & \text{For } 0 Even order modified cutoff attenuation adjustment The same even order adjustment to the poles and zeros that was used for the Chebyshev even order modified cutoff attenuation may also be used for the Chebyshev II case, both the poles and zeros are multiplied by the result. \begin{align} p_A = p_1\sqrt{ \frac {cosh^2\Biggr(\frac{1}{n}cosh^{-1}\Bigr(\sqrt{\frac{10^-1}{10^-1}}\Bigr)\Biggr)-cos^2(\frac{\pi(n-1)}{2n})} {1-{cos^2(\frac{\pi(n-1)}{2n})}} } \text{ For } 0 == Implementation ==

Cauer topology A passive LC Chebyshev low-pass filter may be realized using a Cauer topology. The inductor or capacitor values of an nth-order Chebyshev prototype filter may be calculated from the following equations: :G_{0} = 1 :G_{1} =\frac{ 2 A_{1} }{ \gamma } :G_{k} =\frac{ 4 A_{k-1} A_{k} }{ B_{k-1}G_{k-1} },\qquad k = 2,3,4,\dots,n :G_{n+1} =\begin{cases} 1 & \text{if } n \text{ odd} \\ \coth^{2} \left ( \frac{ \beta }{ 4 } \right ) & \text{if } n \text{ even and }G_{n}\text{ element shunt} \\ \tanh^{2} \left ( \frac{ \beta }{ 4 } \right ) & \text{if } n \text{ even and }G_{n}\text{ element series} \end{cases} G1, Gk are the capacitor or inductor element values. fH, the 3 dB frequency is calculated with: f_H = f_0 \cosh \left(\frac{1}{n} \cosh^{-1}\frac{1}{\varepsilon}\right) The coefficients A, γ, β, Ak, and Bk may be calculated from the following equations: :\gamma = \sinh \left ( \frac{ \beta }{ 2n } \right ) :\beta = \ln\left [ \coth \left ( \frac{ \delta }{ 17.37 } \right ) \right ] :A_k=\sin\frac{ (2k-1)\pi }{ 2n },\qquad k = 1,2,3,\dots, n :B_k=\gamma^{2}+\sin^{2}\left ( \frac{ k \pi }{ n } \right ),\qquad k = 1,2,3,\dots,n where \delta is the passband ripple in decibels. The number 17.37 is rounded from the exact value 40/\ln(10). The calculated Gk 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, • C1 shunt = G1, L2 series = G2, ... or • L1 series = G1, C2 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 Gn+1 and Gn. 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): 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 j\omega 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, G(s), G(s) = \sqrt{\frac{1}{1+\varepsilon ^2K(s)K(-s)}}\bigg|_{\text{left half plane (LHP) poles}} The calculation of K(S) relies upon the following observed equality. \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}\}}{\{0.25(s)^2 + 1\}\{{0.25(-s)^2 + 1}\}+.25892541\{3.4820508(s)^3 + 2.7320508(s)\}\{3.4820508(-s)^3 + 2.7320508(-s)\} }}\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} To obtain G(s) 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 G(s) with the remaining roots. Generally, normalize |G(s)| to 1 at s=0. \begin{align} &G(s)= \frac{0.25s^2 + 1}{1.7718316s^3 + 1.7200107s^2 + 2.2074118s + 1} \\ \end{align} To confirm that the example G(s) is correct, the plot of G(s) along j\omega 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. 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 \varepsilon^2 to compute G(s) is 99. When complete, the characteristic polynomials ,K(s), and forward transfer function,G(s), are below. K(s) = \frac{2.3081085s^8+3.7315386s^6+1.8867298s^4+0.28974597s^2}{0.82644628s^2+1} G(s) = \frac{K(s)_{den}}{\sqrt{K(s)_{den}K(-s)_{den}+\varepsilon^2K(s)_{num}K(-s)_{num}}|_{\text{LHP roots}}} \text{Where }\varepsilon^2 = 10^{(20_{dB}/10)} - 1 = 99.0 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} The validation consists of calculating scattering parameters |S12| \text{ and } |S11| (|G(s)| and \sqrt{1-|G(s)|^2} 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. The final magnitude frequency response of |S_{12}| \text{ and } |S_{11}| are shown below. ==See also==