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 passband. 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 are the most common types of Chebyshev filters. The gain (or amplitude) response,
Gn(\omega)
\omega
n
Hn(s)
s=j\omega
Gn(\omega)=\left|Hn(j\omega)\right|=
1 | ||||||||||||
|
where
\varepsilon
\omega0
Tn
n
The passband exhibits equiripple behavior, with the ripple determined by the ripple factor
\varepsilon
G=1
G=1/\sqrt{1+\varepsilon2}
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
\omega0
1/\sqrt{1+\varepsilon2}
The 3 dB frequency
\omegaH
\omega0
\omegaH=\omega0\cosh\left(
1 | |
n |
\cosh-1
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
For simplicity, it is assumed that the cutoff frequency is equal to unity. The poles
(\omegapm)
s
2(-js)=0. | |
1+\varepsilon | |
n |
Defining
-js=\cos(\theta)
2(\cos(\theta))=1+\varepsilon | |
1+\varepsilon | |
n |
2\cos2(n\theta)=0.
Solving for
\theta
\theta= | 1 | \arccos\left( |
n |
\pmj | \right)+ | |
\varepsilon |
m\pi | |
n |
where the multiple values of the arc cosine function are made explicit using the integer index
m
spm=j\cos(\theta)
=j\cos\left( | 1 | \arccos\left( |
n |
\pmj | \right)+ | |
\varepsilon |
m\pi | |
n |
\right).
Using the properties of the trigonometric and hyperbolic functions, this may be written in explicitly complex form:
\pm=\pm | ||
s | \sinh\left( | |
pm |
1 | arsinh\left( | |
n |
1 | |
\varepsilon |
\right)\right)\sin(\thetam)
+j\cosh\left(
1 | arsinh\left( | |
n |
1 | |
\varepsilon |
\right)\right)\cos(\thetam)
where
m=1,2,...,n
\theta | ||||
|
2m-1 | |
n |
.
This may be viewed as an equation parametric in
\thetan
s
s=0
\sinh(arsinh(1/\varepsilon)/n)
\cosh(arsinh(1/\varepsilon)/n).
The above expression yields the poles of the gain
G
H(s)=
1 | |
2n-1\varepsilon |
n | |
\prod | |
m=1 |
1 | ||||||
|
where
- | |
s | |
pm |
The group delay is defined as the derivative of the phase with respect to angular frequency:
\tau | ||||
|
\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 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
\omega=0
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,
cosl(
\pi(n-1) | |
2n |
l)
P'=\left[\sqrt{\left(
| ||||||
|
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.
To design a Chebyshev filter using the minimum required number of elements, the minimum order of the Chebyshev filter may be calculated as follows.[2] 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[
| ||||||||||||||||||||||
where:
\omegap
\alphap
\omegas
\alphas
n
ceil[] is a round up to next integer function.
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,
Gn(\omega)
\varepsilon
Tn(\omega/\omega0)
Tn(\omega/
-1 | |
\omega | |
0)=cos(n\cos |
(\omega/\omega0))
-1 | |
T | |
n |
(\omega/
-1 | |
\omega | |
0)=cos(\cos |
(\omega/\omega0)/n)
\omega/\omega0\geq1
Tn(\omega/
-1 | |
\omega | |
0)=cosh(n\cosh |
(\omega/\omega0))
-1 | |
T | |
n |
(\omega/
-1 | |
\omega | |
0)=cosh(\cosh |
(\omega/\omega0)/n)
Using simple algebra on the above equations and references, the expression to scale each Chebyshev poles is:
\begin{align} pA&=p1/
-1 | ||
T | r(\sqrt{ | |
n |
10{/10 | |
- |
1}{10{\delta/10
Where:
pA
p1
\delta
\alpha
n
A quick sanity check on the above equation using passband ripple attenuation for the passband cutoff attenuation
(\alpha=\delta)
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)2becomes-\omega2)
\begin{align}pA=
p | |||||||||||||
|
Where:
pA
p1
\delta
\alpha
n
cos(
\pi(n-1) | |
2n |
)
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:
Gn(\omega)=
1 | |||||||||||||||
|
In the stopband, the Chebyshev polynomial oscillates between -1 and 1 so that the gain will oscillate between zero and
1 | |||||
|
and the smallest frequency at which this maximum is attained is the cutoff frequency
\omegao
\varepsilon=
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:
fH=
f0 | ||||||||
|
.
Assuming that the cutoff frequency is equal to unity, the poles
(\omegapm)
2(-1/js | |
1+\varepsilon | |
pm |
)=0.
The poles of gain of the type II Chebyshev filter are the inverse of the poles of the type I filter:
1 | ||||||
|
= \pm\sinh\left(
1 | arsinh\left( | |
n |
1 | |
\varepsilon |
\right)\right)\sin(\thetam)
+j\cosh\left(
1 | arsinh\left( | |
n |
1 | |
\varepsilon |
\right)\right)\cos(\thetam)
where
m=1,2,...n
(\omegazm)
2(-1/js | |
\varepsilon | |
zm |
)=0.
The zeroes of the type II Chebyshev filter are therefore the inverse of the zeroes of the Chebyshev polynomial.
1/szm=-j\cos\left(
\pi | |
2 |
2m-1 | |
n |
\right)
for
m=1,2,...n
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 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.
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.
The illustration below shows an 8th order Inverse Chebyshev filter modified to support even order equally terminated passive networks by relocating the highest frequency transmission zero from a finite frequency to
infty
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.[3] 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[
| ||||||||||||||||||||||
where:
\omegap
\alphap
\omegas
\alphas
n
ceil[] is a round up to next integer function.
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} pA&=p1*
-1 | ||
T | r(\sqrt{ | |
n |
10{/10 | |
- |
1}{10{\alpha/10
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, except the poles are multiplied by the result.
\begin{align}pA=
p | ||||||||||||||||
|
| ||||
-1}}r)r)-cos |
| ||||
)} {1-{cos |
)}} } For0<\delta<inftyand\delta\leq\alpha<infty\\ \end{align}
A passive LC Chebyshev low-pass filter may be realized using a Cauer topology. The inductor or capacitor values of an
n
G0=1
G1=
2A1 | |
\gamma |
Gk=
4Ak-1Ak | |
Bk-1Gk-1 |
, k=2,3,4,...,n
Gn+1=\begin{cases}1&ifnodd\\ \coth2\left(
\beta | |
4 |
\right)&ifneven\end{cases}
G1, Gk are the capacitor or inductor element values.fH, the 3 dB frequency is calculated with:
fH=f0\cosh\left(
1 | |
n |
\cosh-1
1 | |
\varepsilon |
\right)
The coefficients A, γ, β, Ak, and Bk may be calculated from the following equations:
\gamma=\sinh\left(
\beta | |
2n |
\right)
\beta=ln\left[\coth\left(
\delta | |
17.37 |
\right)\right]
A | ||||
|
, k=1,2,3,...,n
2 | |
B | |
k=\gamma |
+\sin2\left(
k\pi | |
n |
\right), k=1,2,3,...,n
\delta
17.37
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,
or
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.
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.
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.
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[4] .
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
G(s)
G(s)=\sqrt{
1 | |
1+\varepsilon2K(s)K(-s) |
The calculation of K(S) relies upon the following observed equality.
\begin{align} &\begin{array}{lcr}
| ||||||||||||||||
&|\prod | ||||||||||||||||
i=1 |
2}}{(1-j\omega/z | |
+\sqrt{1-\omega | |
i)}| |
=1.&&for0\leq\omega\leq1\end{array}\\ \end{align}
for all
zi=infty
Given the magnitude is always one in the pass bane (
0\leq\omega\leq1
K(s)
s=zi
The design process for K(S) using the above expression is below.
\begin{align} K(s)&=
| |||||||
)\} |
rationaltermonly
Use the positive
Mi
zi
Mi
zi
K(s)
|K(s)|=1ats=j
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.
\begin{align} &B=
N(M | |
\prod | |
is+1) |
\\ &K(s)num=
i\geq0,step=-2 | |
\sum | |
i=N |
[
j\geq0,step=-2 | |
\sum | |
j=i |
Bj\binom{(N-j)/2}{(N-i)/2}]si\\ &N=orderoftheChebyshevfilter\\ &B=apolynomialcreatedbytheproductofthespecifiedfactors\\ &Bj=thejthordercoefficentofpolynomialB\\ &\binom{n}{k}isthebinomialcoefficientfunction\\ \end{align}
When all M values are set to one, then
K(s)num
infty
K(s)
cos(\pi(N-1)/(2N))
\begin{align} &zi'=
2) | |
\sqrt{z | |
0 |
-
2} | |
C | |
0 |
2 | |
\\ &C | |
0 |
=
| ||||
cos |
)\\ &zi=desiredfinitetransmissionzero\\ &zi'=prepositionedfinitetransmissionzero\\ \end{align}
Design a 3 pole Chebyshev filter with a 1 dB pass band, a transmission zero at 2 rad/sec, and a transmission zero at
infty
\begin{align} &M1=M2=\sqrt{(j2)2+1}/j2=\sqrt{1-4}/j2=\sqrt3/2=0.866025,M3=\sqrt{infty2+1}/infty=1\\ &\\ &Fullpolynomialderivation:\\ &K(s)num=r(0.86602540s+\sqrt{s2+1}r)r(0.86602540s+\sqrt{s2+1}r)r(s+\sqrt{s2+1}r)\\ &K(s)num=3.4820508s3+2.7320508s+\sqrt{...}\\ &discardingtheirrational\sqrt{...}andkeepingonlytherationalpart:\\ &K(s)num=3.4820508s3+2.7320508s\\ &\\ &K(s)numshortcutderivation:\\ &B=(0.86602540s+1)(0.86602540s+1)(s+1)=.75s3+2.4820508s2+2.7320508s+1\\ &K(s)num=(0.75\binom{0}{0}+2.4820508\binom{1}{0})s3+(2.7320508\binom{1}{1})s\\ &K(s)num=3.4820508s3+2.7320508s\ &\\ &k(s)den=(
s | +1)( | |
j2 |
s | |
-j2 |
+1)=0.25s2+1\\ &Check|K(s)|ats=jtoinsureitisunity,andadjustwithaconstant,ifnecessary:\\ &|
K(s)num(s=j) | |
K(s)den(s=j) |
|=1Check!\\ &K(s)=
3.4820508s3+2.7320508s | |
0.25s2+1 |
\\ \end{align}
To find the
G(s)
\begin{align} &\varepsilon2=101dB/10.-1.=.25892541\\ &G(s)=\sqrt{G(s)G(-s)}|LHPpoles=\sqrt{
1 | |
1+\varepsilon2K(s)K(-s) |
To obtain
G(s)
G(s)
|G(s)|
s=0
\begin{align} &G(s)=
0.25s2+1 | |
1.7718316s3+1.7200107s2+2.2074118s+1 |
\\ \end{align}
To confirm that the example
G(s)
G(s)
j\omega
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 shown below. The
K(s)
\omega2
infty
\sqrt{\omega2}
K(s)
K(s)
\begin{align} K(s)&=
| |||||||
+\sqrt{s |
2+1})\} | |
rationaltermonly |
\end
K(s)
|K(s)|=1ats=j
Design an asymmetric Chebyshev filter with 1dB pass band ripple from 1 to 2 rad/sec, one transmission zero at
infty
K(s)
\begin{align} \omega2&=2\\ M1&=M2=M3=.5\\ M4&=1\\ K(s)&=
\{ (.5\sqrt{s2+22 | |
+\sqrt{s |
2+1}) (.5\sqrt{s2+22}+\sqrt{s2+1}) (.5\sqrt{s2+22}+\sqrt{s2+1}) (\sqrt{s2+22}+\sqrt{s
2+1}) \} | |
rationaltermonly |
Discarding the irrational part and normalizing
|K(s)|
\begin{align} K(s)&=
3s4+12.666667s2+10.666667 | |
s3 |
\\ \end{align}
Use the same process as in the low pass case to find
G(s)
K(s)
C
\begin{align} \varepsilon2&=101dB/10.-1.=.25892541\\ G(s)&=C\sqrt{
K(s)denK(-s)den | ||||||||||||
|
When reconstructing the denominator from the left half plane poles, it will be necessary to set the
G(s)
G(s)
|G(s)|
s=j
s=j2
\begin{align} G(s)&=
0.18424001s3 | |
0.28125000s4+0.34089984s3+1.3337548s2+0.54084155s+1 |
\\ \end{align}
Evaluating
|G(s)|
1<\omega<2
\omega=0
\omega=infty
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.
Constricted pass band ripple can be achieved by designing an asymmetric Chebyshev band pass filter using the techniques described above in this article with a 0 order asymmetric high pass side (no transmission zeros at 0) and an
\omega2
\omega=0
\omega=0
\omega=0
Positioning the reflection zeros with Newton's method requires three pieces of information:
|K(j\omega)|
|K(j)|=1
|K(j\omega)|K(j)|=1|
|K(s)|K(j)|=1|
|K(j\omega)|K(j)|=1|
Since the Chebyshev characteristic equations,
K(s)
j\omega
j\omega
j\omega
K(s)
(dK(s)/ds)num
(dK(s)/ds)num
(dK(s)/ds)num=K(s)den(d(K(s)num)/ds)-K(s)num(d(K(s)den)/ds)
The partial derivatives may be calculated digitally with
\partial|K(Rk,j\omega)K(j)=1|/\partialRk=|K(Rk,j\omega)|K(j)|=1|-|K(Rk+\vartriangleRk,j\omega)|K(j)|=1|)/\vartriangleRk
|K(s)|K(j)|=1|
K(s)
|K(j)|=1
K(s)
\begin{align} |K(s)|K(j)|=1|&=\begin{cases}Kfinite(s)&ifniseven\\ sKfinite(s)&ifnisodd\\ \end{cases}\\ &\\ K{finite}(s)&=
| ||||||||||||||||
|
| ||||||||||||||||
|
\ \end{align}
Where
K{finite}(s)
NRz
NTz
Rzi
Tzi
s
|K(s)|=1
s=j
Since only movement of the reflection zeros is needed to shape the Chebyshev pass band, the partial derivative expression only needs to be made on the
Rzi
Tzi
Rzi
|K(j\omega)|K(j)|=1|=
| |||||||
|
|K(j\omega)| | |||||||||
|
Where
2 | |
Rz | |
k |
This derivative of this expression with respect to
Rzk
2 | |
R | |
k |
|K(j\omega)|
2 | |
R | |
k |
|K(j\omega)|K(j)|=1|=
| |||||||
|
\{|K(j\omega)||
| |||||||
|
|\}constant
The partial derivative may then be determined by applying standard derivative procedures to
Rzk
\partial|K(j\omega)|K(j)|=1| | |
\partial|Rzk| |
=
| |||||||
|
|K(j\omega)|
Since the only frequencies of relevance are the frequencies at the constriction point and the
i=2toNRz
|(dK(s)/ds)num|
J(Rzk,\omegai)=\begin{bmatrix}
\partial{|K(Rz1,j\omega1)|K(j)|=1| | |
Where
\omega1
\omega(i>1)
|dK(s)/ds)num|
Rzk
Assuming that the filter cut-off attenuation is the same as the ripple magnitude, the value of
|K(j\omegai)|
\omegai
\begin{align}&[Bk]= \begin{bmatrix} |K(j\omega1)|K(j)|=1|-1\\ |K(j\omega2)|K(j)|=1|-1\\ \vdots
\\ |K(j\omega | |
NRz |
)|K(j)|=1|-1\\ \end{bmatrix}\\ &\\ &[J(Rk,\omegai)][\Deltak]=[Bk]\\ &\\ &[Rzk+1]=[Rzk]+[\Deltak]\\ \end{align}
Convergence is achieved when the sum of all
NRz | |
\sum | |
k=1 |
|\Deltak|<\delta
\delta
\Deltak
Rzk+1
Design a 7 pole Chebyshev filter with a 1 dB equi-ripple pass band constricted to 55% of the pass band.
Step 1: Design the
K(s)
infty
\omega2
K(s)=
63.089619s6+113.7979s4+60.897476s2+9.1891952 | |
1 |
Step 2: Insert a single reflection zero into the
K(s)
K(s)=
63.089619s7+113.7979s5+60.897476s3+9.1891952s | |
1 |
\omega(1toN)
|(d{K(s)}/ds)num|
\omega1
1 | 0.45 | 0.64670785 | 0.89924235 | |
2 | 0.45 | 0.68010003 | 0.9147864 | |
3 | 0.45 | 0.6710597 | 0.91089712 | |
4 | 0.45 | 0.66969972 | 0.91042253 | |
5 | 0.45 | 0.66967763 | 0.9104163 | |
6 | 0.45 | 0.66967762 | 0.9104163 |
Step 4: Determine the value of
|K(j\omegai)|
iterations | K(j\omega_1) | K(j\omega_2) | K(j\omega_3) | |||||||
1 | 0.45 | 0.64035786 | 0.89703503 | |||||||
2 | 1.3886545 | 1.1638033 | 1.0148793 | |||||||
3 | 1.045108 | 1.0133721 | 0.99991225 | |||||||
4 | 1.0007289 | 1.0001094 | 0.99998768 | |||||||
5 | 1.0000002 | 1 | 1 | |||||||
6 | 1 | 1 | 1 |
\omegak
|K(j)|=1
j
-1 | K(j\omega_2) | -1 | K(j\omega_3) | -1 | ||
1 | -0.55 | -0.35964214 | -0.10296497 | |||
2 | 0.38865445 | 0.1638033 | 0.014879269 | |||
3 | 0.045108043 | 0.013372137 | -8.7751135e-05 | |||
4 | 7.2893112e-04 | 1.0943442e-04 | -1.2324941e-05 | |||
5 | 1.7276985e-07 | 5.2176787e-09 | -2.6640391e-09 | |||
6 | 1.8873791e-14 | 1.5765167e-14 | -2.553513e-15 |
|K(j\omegai)|K(j)|=1|
\omega(1toN)
Rzk
\partial|K(j\omegai)|K(j)|=1| | |
\partial|Rzk| |
=
| |||||||
|
|K(j\omegai)K(j)=1|
|
|
| ||||||||||||||||||||||||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
|
|
|
[\Deltak]
[J(Rk,\omegai)][\Deltak]=[Bk]
1 | -0.033937389 | -0.040973291 | -0.0054977233 | .02680 | |
2 | 0.010159103 | 0.010436353 | 0.001099011 | .00723149 | |
3 | 0.0018170271 | 0.001314472 | 1.090765e-04 | .00108019 | |
4 | 3.4653892E-05 | 1.6843291E-05 | 1.2899974E-06 | 1.75957e-05 | |
5 | 9.0033707E-09 | 2.9081531E-09 | 2.3695501E-10 | 4.04949e-08 | |
6 | 0 | 0 | 0 | 0 |
[\Delta]
[Rzk]next=[Rzk]-[\Deltazk]
1 | 0.53982509 | 0.81637641 | 0.97841993 | |
2 | 0.52966599 | 0.80594006 | 0.97732092 | |
3 | 0.52784896 | 0.80462559 | 0.97721185 | |
4 | 0.52781431 | 0.80460874 | 0.97721056 | |
5 | 0.5278143 | 0.80460874 | 0.97721056 | |
6 | 0.5278143 | 0.80460874 | 0.97721056 |
NRz | |
\sum | |
k=1 |
|\Deltak|<\deltamin
K(s)
|K(j)|=1
K(s)
K(s)=
87.245248s7+164.10165s5+92.882626s3+15.026225s | |
1 |
G(s)=
K(s)den | ||||||||||||
|
|LHProots
Where\varepsilon2=10(1dB/10)-1=0.25892541
G(s)=
1 | |
44.394495s7+30.711417s6+94.125494s5+46.949428s4+58.490258s3+17.844618S2+9.7031614s+1 |
The synthesis process may be validated by doing a quick check of
|G(j\omegak)|
\omegak
\omega=1
\omega1=0.45 | -1 dB | |
\omega2=0.66967762 | -1 dB | |
\omega3=0.9104163 | -1 dB | |
\omegacut=1 | -1 dB |
|G(jw)|
Standard low pass Inverse Chebyshev filter design creates an equi-ripple stop band beginning from a normalized value of 1 rad/sec to
infty
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 designed with an inverted
\varepsilon
\varepsilon2=1/(10(\gamma/10)-1)
\gamma
Below are the |S11| and |S12| scattering parameters for a 7 pole constricted ripple Inverse Chebyshev filter with 3dB cut-off attenuation.
The constricted ripple example above is intentionally kept simple by keeping the cut-off attenuation equal to the pass band ripple attenuation, omitting optional transmission zeros, and using an odd order that does not potentially require even order modification. However, non-standard cutoff attenuations may be accommodated by calculating the target values in step 5 to be offset from the required 1 that exists at the cut-off frequency of
\omega=j
K(s)
K(s)
When including stop band transmission zeros, it is import to remember that the roots of
dK(s)/ds)num
\omega>1
Since
\varepsilon2
G(s)
K(s)
|K(j\omega)|
\begin{align} &|K(j\omega)|=\sqrt{
| |||||||
|
Consider a filter design of %constriction = 55, order = 8, single transmission zero at 1.1, pass band ripple attenuation = 0.043648054 (equivalent of S12 = 20dB attenuation based on the relation
|S11|2+|S12|2=1
The target value in step 5 is .01010101, and the
\varepsilon2
G(s)
K(s)
G(s)
K(s)=
2.3081085s8+3.7315386s6+1.8867298s4+0.28974597s2 | |
0.82644628s2+1 |
G(s)=
K(s)den | ||||||||||||
|
|LHProots
Where\varepsilon2=
(20dB/10) | |
10 |
-1=99.0
G(s)=
0.82644628s2+1 | |
22.96539s8+39.774072s7+71.570971s6+73.962937s5+65.358572s4+40.848153s3+19.393829S2+6.0938301s+1 |
|S12|and|S11|
|G(s)|
\sqrt{1-|G(s)|2}
= | G(j\omega_k) | S12 | =\sqrt | ||||
\omega1=0.45 | -0.043648054 dB | -20dB | |||||
\omega2=0.66133008 | -0.043648054 dB | -20dB | |||||
\omega3=0.82704812 | -0.043648054 dB | -20dB | |||||
\omegacut=1 | -20 dB | -0.043648054 dB | |||||
=1.1 | - infty | 0 dB |
The final magnitude frequency response of
|S12|and|S11|