The Butterworth filter does not give a sufficiently good
approximation across the complete passband in many cases. The
Taylor's series approximation is often not suited to the way
specifications are given for filters. An alternate error measure is
the maximum of the absolute value of the difference between the
actual filter response and the ideal. This is considered over the
total passband. This is the Chebyshev error measure and was defined
and applied to the FIR filter design problem. For the IIR filter,
the Chebyshev error is minimized over the passband and a Taylor's
series approximation at ω=∞ω=∞ is used to determine the
stopband performance. This mixture of methods in the IIR case is
called the Chebyshev filter, and simple design formulas result, just
as for the Butterworth filter.
The design of Chebyshev filters is particularly interesting,
because the results of a very elegant theory insure that
constructing a frequency-response function with the proper form of
equal ripple in the error will result in a minimum Chebyshev error
without explicitly minimizing anything. This allows a
straightforward set of design formulas to be derived which can be
viewed as a generalization of the Butterworth formulas
Entry 4, Entry 7.
The form for the magnitude squared of the frequency-response
function for the Chebyshev filter is
The Chebyshev polynomial is a powerful function in
approximation theory. Although the function is a polynomial, it is
best defined and developed in terms of trigonometric functions
byEntry 4, Entry 5, Entry 1, Entry 7.
C
N
(
ω
)
=
cos
(
N
cos
-
1
(
ω
)
)
C
N
(
ω
)
=
cos
(
N
cos
-
1
(
ω
)
)
(2)
where CN(ω)CN(ω) is an Nth-order,
real-valued function of the real variable ωω. The development is made
clearer by introducing an intermediate complex variable φφ.
C
N
(
ω
)
=
cos
(
N
φ
)
C
N
(
ω
)
=
cos
(
N
φ
)
(3)
where
ω
=
cos
(
φ
)
ω
=
cos
(
φ
)
(4)
Although this definition of CN(ω)CN(ω) may not at first appear to
result in a polynomial, the following recursive relation derived
from (Equation 4) shows that it is a polynomial.
C
N
+
1
(
ω
)
=
2
ω
C
N
(
ω
)
-
C
N
-
1
(
ω
)
C
N
+
1
(
ω
)
=
2
ω
C
N
(
ω
)
-
C
N
-
1
(
ω
)
(5)
From (Equation 2), it is clear that C0=1C0=1 and C1=ωC1=ω,
and from (Equation 5), it follows that
C
2
=
2
ω
2
-
1
C
2
=
2
ω
2
-
1
(6)
C
3
=
4
ω
3
-
3
ω
C
3
=
4
ω
3
-
3
ω
(7)
C
4
=
8
ω
4
-
8
ω
2
+
1
C
4
=
8
ω
4
-
8
ω
2
+
1
(8)
etc.
Other relations useful for developing these polynomials are
C
N
2
(
ω
)
=
(
C
2
N
(
ω
)
+
1
)
/
2
C
N
2
(
ω
)
=
(
C
2
N
(
ω
)
+
1
)
/
2
(9)
C
M
N
(
ω
)
=
C
M
(
C
N
(
ω
)
)
C
M
N
(
ω
)
=
C
M
(
C
N
(
ω
)
)
(10)
where M and N are coprime.
These are remarkable functions Entry 7. They oscillate
between +1 and -1 for -1<ω<1-1<ω<1 and go monotonically to
+/- infinity outside that domain. All NN of their zeros are real
and fall in the domain of -1<ω<1-1<ω<1, i.e., CNCN is an equal ripple
approximation to zero over the range of ωω from -1 to +1. In
addition, the values for ωω where CNCN reaches its local maxima and
minima and is zero are easily calculated from (Equation 3) and (Equation 4). For
-1<ω<1-1<ω<1, a plot of CN(ω)CN(ω) can be made using the
concept of Lissajous figures. Example plots for C0C0, C1C1, C2C2, C3C3,
and C4C4 are shown in Figure 1.
The filter frequency-response function for N=5N=5 is given in
Figure 2 showing the passband ripple in terms of the parameter
ϵϵ.
The approximation parameters must be clearly understood. The
passband ripple is defined to be the difference between the maximum
and the minumum of |F||F| over the passband frequencies of 0<ω<10<ω<1. There can be confusion over this point as two
definitions appear in the literature. Most digital
Entry 4, Entry 5, Entry 3, Entry 2 and analog Entry 7 filter design books
use the definition just stated. Approximation literature, especially
concerning FIR filters, use one half this value which is a measure
of the maximum error, ||F|-|Fd||||F|-|Fd||, where |Fd||Fd| is the center
line in the passband of Figure 2, which |F||F| oscillates around.
The Chebyshev theory states that the maximum error over that
band is minimum and that this optimal approximation function has
equal ripple over the pass band. It is easy to see that e in
(Equation 1) determines the ripple in the passband and the order NN
determines the rate that the response goes to zero as ωω goes
to infinity.
Pole Locations
A method for finding the pole locations for the Chebyshev
filter transfer function is next developed. The details of this
section can be skipped and the results in (Equation 22,Equation 24) used
if desired.
From (Equation 1), it is seen that the poles of FF(s)FF(s) occur when
1
+
ϵ
2
C
N
2
(
s
/
j
)
=
0
1
+
ϵ
2
C
N
2
(
s
/
j
)
=
0
(11)
or
C
N
=
±
j
ϵ
C
N
=
±
j
ϵ
(12)
From (Equation 4), define φ=cos-1(ω)φ=cos-1(ω) with real and imaginary parts
given by
φ
=
cos
-
1
(
ω
)
=
u
+
j
v
φ
=
cos
-
1
(
ω
)
=
u
+
j
v
(13)
This gives,
C
N
=
cos
(
N
φ
)
=
cos
(
N
u
)
cosh
(
N
ν
)
-
j
sin
(
N
u
)
sinh
(
N
ν
)
=
±
j
ϵ
C
N
=
cos
(
N
φ
)
=
cos
(
N
u
)
cosh
(
N
ν
)
-
j
sin
(
N
u
)
sinh
(
N
ν
)
=
±
j
ϵ
(14)
which implies the real part of CNCN is
zero. This requires
cos
(
N
u
)
cosh
(
N
ν
)
=
0
cos
(
N
u
)
cosh
(
N
ν
)
=
0
(15)
which implies
cos
(
N
u
)
=
0
cos
(
N
u
)
=
0
(16)
which in turn implies that uu takes on
values of
u
=
u
k
=
(
2
k
+
1
)
π
/
2
N
,
k
=
0
,
1
,
.
.
.
N
-
1
u
=
u
k
=
(
2
k
+
1
)
π
/
2
N
,
k
=
0
,
1
,
.
.
.
N
-
1
(17)
For these values of uu, sin(nu)=±1sin(nu)=±1, we have
sinh
(
N
ν
)
=
1
/
ϵ
sinh
(
N
ν
)
=
1
/
ϵ
(18)
which requires νν to take on a value of
ν
=
ν
0
=
(
sinh
-
1
(
1
/
ϵ
)
)
/
N
ν
=
ν
0
=
(
sinh
-
1
(
1
/
ϵ
)
)
/
N
(19)
Using s=jωs=jω gives
s
=
j
ω
=
j
cos
(
φ
)
=
j
cos
(
u
+
j
ν
)
=
j
cos
(
(
2
k
+
1
)
π
/
2
N
+
j
ν
0
)
s
=
j
ω
=
j
cos
(
φ
)
=
j
cos
(
u
+
j
ν
)
=
j
cos
(
(
2
k
+
1
)
π
/
2
N
+
j
ν
0
)
(20)
which gives the location of the NN poles in the
ss plane as
s
k
=
σ
k
+
j
ω
k
s
k
=
σ
k
+
j
ω
k
(21)
where
σ
k
=
-
sinh
(
ν
0
)
cos
(
k
π
/
2
N
)
σ
k
=
-
sinh
(
ν
0
)
cos
(
k
π
/
2
N
)
(22)
ω
k
=
cosh
(
ν
0
)
sin
(
k
π
/
2
N
)
ω
k
=
cosh
(
ν
0
)
sin
(
k
π
/
2
N
)
(23)
for NN values of kk where
k
=
±
1
,
±
3
,
±
5
,
⋯
,
±
(
N
-
1
)
for N even
k
=
±
1
,
±
3
,
±
5
,
⋯
,
±
(
N
-
1
)
for N even
(24)
k
=
0
,
±
2
,
±
4
,
⋯
,
±
(
N
-
1
)
for N odd
k
=
0
,
±
2
,
±
4
,
⋯
,
±
(
N
-
1
)
for N odd
(25)
A partially factored form for F(s) can be derived using the same approach as for the Butterworth filter. For N
even, the form is
F
(
s
)
=
∏
k
1
s
2
-
2
σ
k
s
+
(
σ
k
2
+
ω
k
2
)
F
(
s
)
=
∏
k
1
s
2
-
2
σ
k
s
+
(
σ
k
2
+
ω
k
2
)
(26)
for k=1,3,5,⋯,N-1k=1,3,5,⋯,N-1. For NN odd, F(s)F(s) has a single real pole and, therefore,
the form
F
(
s
)
=
1
sinh
(
ν
0
)
F
(
s
)
=
∏
k
1
s
2
-
2
σ
k
s
+
(
σ
k
2
+
ω
k
2
)
F
(
s
)
=
1
sinh
(
ν
0
)
F
(
s
)
=
∏
k
1
s
2
-
2
σ
k
s
+
(
σ
k
2
+
ω
k
2
)
(27)
for k=2,4,6,...,N-1k=2,4,6,...,N-1 This is a
convenient form for the cascade and parallel realizations.
A single formula for both even and odd N is
σ
=
-
sinh
(
ν
0
)
sin
(
(
2
k
+
1
)
π
/
2
N
)
σ
=
-
sinh
(
ν
0
)
sin
(
(
2
k
+
1
)
π
/
2
N
)
(28)
ω
k
=
cosh
(
ν
0
)
cos
(
(
2
k
+
1
)
π
/
2
N
)
ω
k
=
cosh
(
ν
0
)
cos
(
(
2
k
+
1
)
π
/
2
N
)
(29)
for NN values of kk where k=0,1,2,⋯,N-1k=0,1,2,⋯,N-1
Note the similarity to the pole locations for the Butterworth
filter. Cross multiplying, squaring, and adding the terms in
(Equation 28,Equation 29) gives
(
σ
k
sinh
(
ν
0
)
)
2
+
(
ω
k
cosh
(
ν
0
)
)
2
=
1
(
σ
k
sinh
(
ν
0
)
)
2
+
(
ω
k
cosh
(
ν
0
)
)
2
=
1
(30)
This is the equation for an ellipse and
shows that the poles of a Chebyshev filter lie on an ellipse similar
to the way the poles of a Butterworth filter lie on a
circleEntry 4, Entry 5, Entry 3, Entry 6, Entry 1, Entry 7.
Summary
This section has developed the classical Chebyshev filter
approximation which minimizes the maximum error over the passband
and uses a Taylor's series approximation at infinity. This
results in the error being equal ripple in the passband. The
transfer function was developed in terms of the Chebyshev
polynomial and explicit formulas were derived for the location of
the transfer function poles. These can be expressed as a
modification of the pole locations for the Butterworth filter and
are implemented in the appendix.
It is possible to develop a theory for Chebyshev passband
approximation and arbitrary zero location similar to the Taylor's
series result in ((Reference)).
The Chebyshev filter has a passband optimized to minimize
the maximum error over the complete passband frequency range, and a
stopband controlled by the frequency response being maximally flat
at ω=∞ω=∞. The passband ripple and the filter order are
the two parameters to be determined by the specifications.
The form for the specifications that is most consistent with
the Chebyshev filter formulation is a maximum allowed error in the
passband and a desired degree of “flatness" at ω=∞ω=∞. The
slope of the response near the transition from pass to stopband at
ω=1ω=1 becomes steeper as both the order increases and the allowed
passband error ripple increases. The dropoff is more rapid than for the
Butterworth filterEntry 7.
As stated earlier, the design parameters must be clearly
understood to obtain a desired result. The passband ripple is defined to
be the difference between the maximum and the minimum of |F||F| over the
passband frequencies of 0<ω<10<ω<1. There can be confusion over
this point as two definitions appear in the literature. Most digital
Entry 4, Entry 5, Entry 3 and analog Entry 7 filter design books use the
definition just stated. Approximation literature, especially concerning
FIR filters, use half this value which is a measure of the maximum error,
||F|-|Fd||||F|-|Fd||, where |Fd||Fd| is the center line in the passband around
which |F||F| oscillates. The following formulas relate the passband ripple
δδ, the passband ripple aa in positive dB, and the transfer
function parameter ϵϵ.
a
=
10
log
(
1
+
ϵ
2
)
=
-
20
log
(
1
-
δ
)
,
a
=
10
log
(
1
+
ϵ
2
)
=
-
20
log
(
1
-
δ
)
,
(31)
ϵ
=
2
δ
-
δ
2
1
-
2
δ
+
δ
2
=
10
a
/
10
-
1
,
ϵ
=
2
δ
-
δ
2
1
-
2
δ
+
δ
2
=
10
a
/
10
-
1
,
(32)
δ
=
1
-
10
-
a
/
20
=
1
-
1
1
+
ϵ
2
δ
=
1
-
10
-
a
/
20
=
1
-
1
1
+
ϵ
2
(33)
In some cases, stopband performance is not given in terms of
degree of flatness at ω=∞ω=∞, but in terms of a maximum
allowed magnitude GG in the stopband above a certain frequency
ωsωs, i.e., G>|F|>0G>|F|>0 for 1<ωs<ω<∞1<ωs<ω<∞. For a given ϵϵ, this will determine the order as
the smallest positive integer satisfying
N
≥
cosh
-
1
(
1
-
G
2
ϵ
G
2
)
cosh
-
1
(
ω
s
)
N
≥
cosh
-
1
(
1
-
G
2
ϵ
G
2
)
cosh
-
1
(
ω
s
)
(34)
The design of a Chebyshev filter
involves the following steps:
- The maximum-allowed passband variation must be
in
the form of δδ or aa. From this, the parameter ϵϵ is
calculated using (Equation 32).
- The order NN is determined by the desired flatness at
ω=∞ω=∞ or a maximum-allowed response for frequencies
above ωsωs using (Equation 34).
- ν0ν0 is calculated from ϵϵ and nn using
(Equation 19), and the
scale factors sinh(ν0)sinh(ν0) and cosh(ν0)cosh(ν0) are then
determined.
- The pole locations are calculated from (Equation 22)
or (Equation 29). This can be done by scaling the poles
of a Butterworth prototype filter.
- These pole locations are combined in (Equation 27) and (Equation 28) to give
the final filter transfer function.
This process is easily programmed for computer aided design as
illustrated in Program 8 in the appendix.
If the design procedure uses (Equation 34) to determine the order
and the right-hand side of the equation is not exactly an integer,
it is possible to improve on the specifications. Direct use of the
order with ϵϵ from (Equation 32) gives a stopband gain at
ωsωs that is less than GG, or the same design can be viewed
as giving the maximum-allowed gain GG at a lower frequency than
ωsωs. An alternate approach is to solve (Equation 34) for a new
value of ϵϵ, then cause (Equation 34) to be an equation with
the specified ωsωs and GG. This gives a filter that exactly
meets the stopband specifications and gives a smaller passband
ripple than originally requested. A similar set of alternatives
exists for the elliptic-function filter.
Example 7-2. The Design of a Chebyshev Lowpass
Filter.
The design specifications require a maximum passband ripple
of δ=0.1δ=0.1 or a=0.91515a=0.91515 dB, and can allow no greater
response than G=0.2G=0.2 for frequencies above ωs=1.6ωs=1.6
radians per second.
Given δ=0.1δ=0.1 or a=0.91515a=0.91515, equation (Equation 32)
implies
ϵ
=
0
.
484322
ϵ
=
0
.
484322
(35)
Given G=0.2G=0.2 and ωs=1.6ωs=1.6, equation (Equation 34) implies an order of N=3N=3. From
ϵϵ and NN, ν0ν0 is 0.49074 from (Equation 19) and
sinh
(
ν
0
)
=
0
.
510675
sinh
(
ν
0
)
=
0
.
510675
(36)
cosh
(
ν
0
)
=
1
.
122849
cosh
(
ν
0
)
=
1
.
122849
(37)
These multipliers are used to scale the root
locations of the example third-order Butterworth filter to give
F
(
s
)
=
1
(
s
+
0
.
51067
)
(
s
+
0
.
25534
+
j
0
.
97242
)
(
s
+
0
.
25534
-
j
0
.
97242
)
F
(
s
)
=
1
(
s
+
0
.
51067
)
(
