Wikipedia / Chebyshev filter
Wikipedia / 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, but they achieve this with ripples in the frequency response. This type of filter is named after Pafnuty Chebyshev because its mathematical characteristics are derived from Chebyshev polynomials. Type I Chebyshev filters are usually referred to as "Chebyshev filters", while type II filters are usually called "inverse Chebyshev filters". Because of the passband ripple inherent in Chebyshev filters, filters with a smoother response in the passband but a more irregular response in the stopband are preferred for certain applications.
Type I Chebyshev filters (Chebyshev filters)
Type I Chebyshev filters are the most common types of Chebyshev filters. The gain (or amplitudeth-order low-pass filter is equal to the absolute value of the transfer function:
is the cutoff frequency is a Chebyshev polynomialth order.
.
The ripple factor ε is thus related to the passband ripple δ in decibels by:
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 −3dB 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 by:
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-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
, these occur when:
and using the trigonometric definition of the Chebyshev polynomials yields:
. The poles of the Chebyshev gain function are then:
Using the properties of the trigonometric and hyperbolic functions, this may be written in explicitly complex form:
and
and it demonstrates that the poles lie on an ellipse in s-space
The transfer function
. 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
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:
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 while maintaining the equi-ripple response of the pass band.
The needed modification involves mapping each pole of the Chebyshev transfer function in a manner that maps the lowest frequency reflection zero to zero and the remaining poles as needed to maintain the equi-ripple pass band. The lowest frequency reflection zero may be found from the Chebyshev Nodes. The complete Chebyshev pole mapping function is shown below.
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.
where:
are the pass band ripple frequency and maximum ripple attenuation in dB
are the stop band frequency and attenuation at that frequency in dB
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. The general definition of the Chebyshev function is required, which may be derived from the Chebyshev Polynomials, complex hyperbolic identities.
Using simple algebra on the above equations and references, the expression to scale each Chebyshev poles is:
Where:
is the relocated pole positioned to set the desired cutoff attenuation.
is a ripple cutoff pole that lies on the oval.
is the passband attenuation ripple in dB (.05 dB, 1 dB, etc.)).
is the desired passband attenuation at the cutoff frequency in dB (1 dB, 3 dB, 10 dB, etc.)
is the number of poles (the order of the filter).
A quick sanity check 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.
Where:
is the relocated pole positioned to set the desired cutoff attenuation.
is a ripple cutoff pole that has been modified for even order pass bands.
is the passband attenuation ripple in dB (.05 dB, 1 dB, etc.)).
is the desired passband attenuation at the cutoff frequency in dB (1 dB, 3 dB, 10 dB, etc.)
is the number of poles (the order of the filter).
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:
In the stopband, the Chebyshev polynomial oscillates between -1 and 1 so that the gain will oscillate between zero and
. The parameter ε is thus related to the stopband attenuation γ in decibels by:
For a stopband attenuation of 5dB, ε = 0.6801; for an attenuation of 10dB, ε = 0.3333. The frequency f0 = ω0/2π is the cutoff frequency. The 3dB frequency fH is related to f0 by:
Poles and zeroes
of the gain of the Chebyshev filter are the zeroes of the denominator of the gain:
The poles of gain of the type II Chebyshev filter are the inverse of the poles of the type I filter:
of the type II Chebyshev filter are the zeroes of the numerator of the gain:
The zeroes of the type II Chebyshev filter are therefore the inverse of the zeroes of the Chebyshev polynomial.
.
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. However, even order Chebyshev II filters may be modified by translating the highest frequency finite transmission zero to infinity, while maintaining the equi-ripple functions of the Chebyshev II stop band. To do this translation, an even order modified Chebyshev function is used in place of the standard Chebyshev function to define the Chebyshev II poles needed to create the even order modified Chebyshev II transfer function. Zeros are created using the roots of the even order modified Chebyshev polynomial, which are the even order modified Chebyshev nodes.
while maintaining an equi-ripple stop band frequency response.
Minimum order
To design an Inverse Chebyshev filter using the minimum required number of elements, the minimum order of the Inverse Chebyshev filter may be calculated as follows. 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.
where:
are the pass band frequency and attenuation at that frequency in dB
are the stop band frequency and minimum stop band attenuation in dB
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..
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.
Implementation
Cauer topology
A passive LC Chebyshev low-pass filter may be realized using a Cauer topology. The inductor or capacitorth-order Chebyshev prototype filter may be calculated from the following equations:
or
G1 to Gk are the capacitor or inductor element values,
G0 is the input termination impedance.
Gn+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
The coefficients A, γ, β, Ak, and Bk may be calculated from the following equations:
.
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 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,
The calculation of K(S) relies upon the following observed equality.
, imaginary conjugate pairs, quadruplet conjugate pairs, or real opposing signed pairs.
.
The design process for K(S) using the above expression is below.
solution for real and imaginary pairs.
, if needed.
The, "rational terms only" indicates to keep the rational part of the product, and to discard the irrational part. The rational term may be obtained by manually performing the polynomial arithmetic, or with the short cut below which is a solution derived from polynomial arithmetic and uses binomial coefficients. The algorithm is extremely efficient if the Binomial coefficients are implemented from a look-up table of pre-calculated values.
will be the standard Chebyshev equation. Even order finite transmission zero Chebyshev filters have the same limitation as the all-pole case in that they cannot be constructed using equally terminated passive networks. The same even order modification, to make equally terminated passive network implementations possible. However, the even order modification will also move the finite transmission zeros slightly. This movement may be significantly mitigated by propositioning the transmission zeros with the inverse of the even order modification using the lowest Chebyshev node.
Simple transmission zeros example
:
transfer function, do the following.
.
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, equation produce usable results. Real and complex quadruplet transmission zeros may also be created using this technique and are useful to modify the group delay, to create an asymmetric Chebyshev band pass filter is shown below.
, if needed.
Simple asymmetric example
, may be calculated as follows.
to 1 at s=j:
to scale the magnitude.
. Once accomplished, the final transfer function for the designed asymmetric Chebyshev filter is shown below.
, and Chebyshev style steepened slopes near the pass band edges.
Constricting the pass band ripple
Standard low pass Chebyshev filter design creates an equi-ripple pass band beginning from 0 rad/sec to a frequency normalized value of 1 rad/sec. However, some design requirements do not need an equi-ripple pass band at the low frequencies. A standard full-equi-ripple Chebyshev filter for this application would result in an over designed filter. Constricting the equi-ripple to a defined percentage of the pass band creates a more efficient design, reducing the size of the filter and potentially eliminating one or two components, which is useful in maximizing board space efficiency and minimizing production costs for mass produced items.
, which is typical of even order standard Chebyshev design, so for standard even order Chebyshev designs, the process is complete at this step. It will be necessary to insert a single reflection zero for even order modified designs. Added reflection zeros introduces a noticeable error in the pass band that is likely to be objectionable. This error may be removed quickly and accurately by repositioning the finite reflection zeros with the use of Newton's method for systems of equations.
Application of Newton's method
Positioning the reflection zeros with Newton's method requires three pieces of information:
The location of each pass band ripple minima that exists at frequencies higher than the constricted ripple frequency.
The Jacobian matrix of partial derivative for the constriction frequency and at each minima above the constriction frequency. with respect to each reflection zero.
, with the use of a root finding algorithm is obtainable from standard polynomial derivative definitions.
as a function of its conjugate root pairs, as shown below.
in the above expression.
, the expression above may be rewritten, as shown below.
designates a specific reflection zero conjugate pair.
may be easily computed following standard derivative rules terms that were moved to the front. The differentiable expression may be rewritten as follows.
and then simplifying. The result is below.
, the Jacobian matrix may be constructed as follows.
are the reflection zeros.
, so the solution vector entries are all 1, and the iterative equations to solve for Newton's method is
to keep their values inside the constricted ripple range during convergence.
Constricted pass band example
Design a 7 pole Chebyshev filter with a 1 dB equi-ripple pass band constricted to 55% of the pass band.
Step 1: = 0.45) .
Step 2: from step 1. (two reflection zero additions would be required for even order modified filters)
Step 3:.Step 4at each constricted and derivative zero point.Step 5: Create the B.Step 6:Step 7 using the B vector from step 5.Step 8: above from the past iteration of reflection zero positions.
is shown below.
is also 1dB. The summary of the computation below validates the example synthesis process.The final magnitude frequency response of the forward, is shown below.
Chebyshev II stop band ripple constricting
. However, some design requirements do not need an equi-ripple pass band at the high frequencies. A standard full-equi-ripple Inverse Chebyshev filter for this application would result in an over designed filter. Constricting the equi-ripple to a defined percentage of the stop band creates a more efficient design, reducing the size of the filter and potentially eliminating one or two components, which is useful in maximizing board space efficiency and minimizing production costs for mass produced items.
Inverse Chebyshev filters with constricted stop band ripple are synthesized in exactly the same process as standard a inverse Chebyshev. A constricted ripple Chebyshev is the stop band attenuation in dB, the poles and zeros of the designed constricted ripple Chebyshev filter are inverted, and the cut-off attenuation is set. Since standard Chebyshev equations will not work with constricted ripple design, the cut-off attenuation must be set using the process described in the Elliptic Hourglass design.
Below are the |S11| and |S12| scattering parameters for a 7 pole constricted ripple Inverse Chebyshev filter with 3dB cut-off attenuation.
Non-standard cut-off attenuation and transmission zeros
in step 2.
. These roots should not be included in the pass band minima used in the computations..
obtainable from the equations above.
for lossless networks), and pass band cut-off attenuation = 20dB.
, are below.
The validation consists of calculating scattering parameters 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. are shown below.
See also
References
External links