Copyright © 2016 by Wayne Stegall
Created December 31, 2016. See Document History at end for
details.
Chebychev Filter Synthesis
Introduction
It is often desirable to design filters of higher than first order to
some desirable specification such as flattest frequency response,
sharpest cutoff, or linear phase. the last article of this type
showed how Butterworth filters were synthesized. Next in order is
the Chebychev filter.
Figure
1:
Magnitude
plot
of
2nd
order
lowpass
Chebychev
filter. 
Figure
2:
Magnitude
plot
of
3rd
order
lowpass Chebychev
filter. 


Figure
3:
Magnitude
plot
of
4th order lowpass Chebychev filter. 
Figure
4:
Magnitude
plot
of
5th order lowpass Chebychev filter. 


Synthesis of poles
Like the Butterworth filter the mathematics for Chebychev synthesis
begins with designing a pole
plot. First Butterworth poles are created in the left half plane
of the sdomain on the unit circle, then altered to an ellipse the
multiplying by constants representing the filter specifications.
Figures 58
show Chebychev pole
plots from 2nd to 5th order.
Figure
5:
Sdomain
plot
of
2nd
order
lowpass
Chebychev
poles. 
Figure
6:
Sdomain
plot
of
3rd
order
lowpass
Chebychev
poles. 


Figure
7:
Sdomain
plot
of
4th
order
lowpass
Chebychev
poles. 
Figure
8:
Sdomain
plot
of
5th
order
lowpass
Chebychev
poles. 


First calculate Butterworth poles for the desired order of
filter. For N stages i to N the pole angles are calculated by:
(1)

θ_{polei} = 90°
+ 

180° × (2i – 1)
2N 

Here however the angles of the poles is not sufficient. Therefore
it is necessary to specify the poles in rectangular complex form.
(2)

α_{i} + jβ_{i}=
cos(θ_{polei}) + jsin(θ_{polei})









Perform the Chebychev conversion by multiplying the real part of each
pole by a constant.
(3)

α_{i}' + jβ_{i}'
= k_{cheb}⋅α_{i} + jβ_{i} 








The
Chebychev constant is
calculated as follows.
(4)

ε = 
10^{decibelripple/10}
– 1 







(5)

A = 
1
n 
sinh^{1}

1
ε 





Multiplication by the c alone specifies the design relative to its 3dB
point. To specify it by its ripple cutoff, the entire pole must
be multiplied by another constant as well.
(7)

α_{i}' + jβ_{i}'
= k_{ripple}(k_{cheb}α_{i}
+ jβ_{i}) 








Where the constant is calculated as
This is a equivalent to the following calculation.
(9)

α_{i}' + jβ_{i}'
= sinh(A)⋅α_{i} + jcosh(A)⋅β_{i}









Convert poles to polar form ω
_{i}∠
θ_{i}.
(10)

ω_{i} = α_{i}'
+ jβ_{i}'
= 
α_{i}'^{2}
+ β_{i}'^{2} 







(11)

θ_{i}
= tan^{1} 

β_{i}'
α_{i}' 

Highpass transformation
This point the filter is lowpass of frequency 1 rad/s. If a
highpass filter is required it is better to do the transform before
frequency scaling. For each pole ω
_{i}∠
θ_{i} invert the magnitude
around a frequency of 1rad/s as follows.
(12)

ω_{ihp}
= 
1
ω_{ilp} 
Deriving stage parameters from pole plot
It remains only to scale the magnitude to the desired frequency and
calculate Q from
θ_{i}.
(13)

f_{stagei} = 
ω_{i} × f_{3dB} 
(if designed relative to
3dB cutoff.)







(14)

f_{stagei} = 
ω_{i} × f_{ripplecutoff} 
(if designed relative to
ripple cutoff.)







It is acceptable to ignore the multiplying of mixed frequency units at
this point because the initial frequencies are only relative. It
is only when given a complete circuit specified to a certain frequency
that units have to be accounted.
(15)

Q_{stagei} =

1
2cos(θ_{i})

Example
Show synthesis of 5thorder highpass Chebychev at 20Hz.
First, calculate pole angles for 5th order lowpass Butterworth filter
according to
equation 1.
θ_{pole1} = 90°
+ 

180° × (2(1) – 1)
2(5) 

θ_{pole1} = 90°
+ 

180° × 1
10 

θ_{pole1} = 90°
+ 18° = 108° 
Continuing as above, each pole is 180/N° or 36° from the previous, thus
creating the following sequence of remaining angles.
θ
_{pole2} = 144°, θ
_{pole3} = 180°, θ
_{pole4}
= 216°, and θ
_{pole5} = 252°
Figure
9:
Polar
plot
of
5th
order
lowpass
Butterworth
poles 

Now plot those angles onto the unit circle in rectangular complex form
according to
equation 2.
The
first
pole
calculates:
α_{i} + jβ_{i} =
cos(108°) + jsin(108°)
=
–0.3090
+ 0.9511 
The remaining poles calculate to the values shown in the plot of
figure 10 below.
Figure
10:
Rectangular
plot
of
5th
order
lowpass
Butterworth
poles. 

Begin the Chebychev transformation by calculating the
Chebychev constant per equations
46.
(4)

ε = 
10^{1dB/10}
– 1 
=

10^{0.1}
– 1 
=

1.258925^{?}
– 1 
=

0.258925 
= 0.508847

(5)

A = 
1
n 
sinh^{1}

1
ε 
=

1
5 
sinh^{1} 
1
0.508847 
= 0.285595 
(6)

k_{cheb} = tanh(0.285595) =
0.278076 








Multiply the real part of each Butterworth pole by
k_{cheb}
to give the
lowpass Chebychev pole plot.
Figure
11:
Rectangular
plot
of
5th
order
lowpass Chebychev poles. 

Convert poles to polar form useful for highpass transformation and
calculation of stage parameters by
equations
10
and
11.
Figure
12:
Polar
plot
of
5th
order
lowpass
Chebychev
poles. 

Execute highpass transformation by taking the reciprocal of the
magnitude of each pole.
Figure
13:
Polar
plot
of
5th
order
highpass
Chebychev
poles. 

Calculate stage parameters from pole values. First scale
frequencies from magnitude for pole 1 (representing its conjugate as
well)..
(13)

f_{stage1} = ω_{1}
× f_{3dB} = 1.0472 × 20Hz = 20.944Hz 
Then calculate its Q from its angle.
(15)

Q_{stage1} =

1
2cos(θ_{1}) 
=

1
2cos(95.1628°) 
= 5.556422

Calculations from the remaining poles give a final result.
Stage


Frequency


Quality Factor

1


20.944Hz


5.556

2


31.778Hz


1.399

3


71.9229Hz


first order

Figure
14:
Frequency response of designed filter.


Hyperbolic functions
In case your calculator or programming library doe not have hyperbolic
functions these are how they are calculated.
(16)

sinh(x) =

e^{x} – e^{x}
2

(17)

cosh(x) =

e^{x} + e^{x}
2

(18)

tanh(x) =

e^{x} – e^{x}
e^{x} + e^{x}

(19)

sinh^{1}(x) = 
^{/}ln


x^{?}+

x^{2} + 1





^{1}The basis for most of the
calculations in this article are from notes taken long ago from (as
best I can remember):
Zverev, A. I., Handbook of Filter
Synthesis, John Wiley and Sons, New York, 1967.
Document History
December 31, 2016 Created.