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Chebyshev Filter Properties

Module by: C. Sidney Burrus. E-mail the author

Chebyshev Filter Properties

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 [4], [7].

The form for the magnitude squared of the frequency-response function for the Chebyshev filter is

| F ( j ω ) | 2 = 1 1 + ϵ 2 C N ( ω ) 2 | F ( j ω ) | 2 = 1 1 + ϵ 2 C N ( ω ) 2
(1)

where CN(ω)CN(ω) is an Nth-order Chebyshev polynomial and ϵϵ is a parameter that controls the ripple size. This polynomial in ωω has very special characteristics that result in the optimality of the response function Equation 1.

CHEBYSHEV POLYNOMIALS

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 by[4], [5], [1], [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 [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.

Figure 1: Chebyshev Polynomials for N = 0, 1, 2, 3, and 4
Figure one is a graph titled Nth order chebyshev polynomials. Its horizontal axis is labeled Frequency, ω, and ranges in value from -2 to 2 in increments of 0.5. The vertical axis is labeled C_N(ω) and ranges in value from -2 to 2 in increments of 0.5. There are four curves in this figure. The first is a diagonal line with constant positive slope that passes through the origin. The second is parabolic in shape with  its vertex as a minimum of the curve at (0, -1). The third starts from the bottom-left of the graph, increases to a peak at (-0.5, 1) and then decreases to a trough at (0.5, -1), where it finally increases to the top-right area of the graph. The fourth begins in the top-left as a decreasing function, and proceeds to make a trough, a peak, and a trough at (-0.5, -1), (0, 1) and (0.5, -1) respectively. The curve then increases and ends in the top-right area.

The filter frequency-response function for N=5N=5 is given in Figure 2 showing the passband ripple in terms of the parameter ϵϵ.

Figure 2: Fifth Order Chebyshev Filter Frequency Response
Figure two is a graph titled analog chebyshev filter. Its horizontal axis is labeled Frequency, ω, and ranges in value from 0 to 3 in increments of 0.5. Its vertical axis is labeled Magnitude Response and ranges in value from 0 to 1 in increments of 0.2. The curve in this graph begins at (0, 1) with a negative slope and moves in a wave-like motion of an amplitude of 0.05, completing two troughs of vertical value 0.9, and one two peaks. Two arrows point to the peaks and troughs, and are labeled, passband ripple. After the second peak, located at (1, 1), the curve then moves sharply downward with a strong negative slope, and begins decreasing at a decreasing rate until by (2, 0) it flattens along a horizontal asymptote at the vertical value 0.

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 [4], [5], [3], [2] and analog [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 circle[4], [5], [3], [6], [1], [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 Equation 5 from Butterworth Filter Properties.

Chebyshev Filter Design Procedures

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 filter[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 [4], [5], [3] and analog [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 1: 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 ) ( s + 0 . 25534 + j 0 . 97242 ) ( s + 0 . 25534 - j 0 . 97242 )
(38)
F ( s ) = 1 ( s + 0 . 51067 ) ( s 2 + 0 . 510675 s + 1 . 010789 ) F ( s ) = 1 ( s + 0 . 51067 ) ( s 2 + 0 . 510675 s + 1 . 010789 )
(39)
F ( s ) = 1 s 3 + 102135 s 2 + 1 . 271579 s + 0 . 516185 F ( s ) = 1 s 3 + 102135 s 2 + 1 . 271579 s + 0 . 516185
(40)

The frequency response is shown in Figure 3

Figure 3: Example Design of a Third Order Chebyshev Filter Frequency Response
Figure three is a graph titled, design of a chebyshev filter. The horizontal axis is labeled Frequency, ω, and ranges in value from 0 to 3 in increments of 0.5. The vertical axis is labeled Magnitude Response and ranges in value from 0 to 1 in increments of 0.2. There is one curve on this graph. It begins at (0, 1) and decreases in a wave-like motion to a wide trough at (0.5, 0.9), and then a narrow peak at (0.9, 1). Two arrows point to the peaks and troughs and label them, ripple d = 0.1. After the peak is a large downward-sloping portion, and after a while, the curve begins to become more shallow, terminating at (3, 0).

Inverse-Chebyshev Filter Properties

A second form of the mixture of a Chebyshev approximation and a Taylor's series approximation is called the Inverse Chebyshev filter or the Chebyshev II filter. This error measure uses a Taylor's series for the passband just as for the Butterworth filter and minimizes the maximum error over the total stopband. It reverses the types of approximation used in the preceding section. A fifth-order example is illustrated in Figure 1c from Design of Infinite Impulse Response (IIR) Filters by Frequency Transformations and Figure 4c.

Rather than developing the approximation directly, it is easier to modify the results from the regular Chebyshev filter. First, the frequency variable ωω in the regular Chebyshev filter, described in Equation 1, is replaced by 1/ω1/ω, which interchanges the characteristics at ωω equals zero and infinity and does not change the performance at ωω equals unity. This converts a Chebyshev lowpass filter into a Chebyshev highpass filter as illustrated in Figure 4 moving from the first to second frequency response.

Figure 4: Lowpass Chebyshev to Highpass Chebyshev to Lowpass Inverse Chebyshev
Figure four consists of three graphs. Each graph's horizontal axis is labeled Frequency ω, and range in value from 0 to 3 in increments of 1. Each graph's vertical axis is labeled Magnitude, and ranges in value from 0 to 1 in increments of 0.2. The first graph, titled chebyshev lowpass, begins at (0, 1), and begins decreasing first as part of a wavelike segment, with two troughs and two peaks. The amplitude of the waves is approximately 0.05. By (1, 1) the second peak is reached, and the curve moves sharply downward to an eventual horizontal asymptote along the horizontal axis by (2, 0). The second, titled chebyshev highpass, begins at the origin moving horizontally for 0.5 units, then sharply increases at an increasing rate to a peak at (1, 1). After the peak, the curve quickly decreases 0.1 units to a trough, and then slowly increases to a peak at (1.7, 1), after which it slowly decreases to a nearly horizontal segment where it terminates at (3, 0.9). The third, titled inverse cheby lowpass, begins horizontally for 0.5 units from (0, 1) to (0.5, 1), where it sharply decreases to (1, 0). This point is a kink in the graph, and from this point the curve increases to a small peak at (1.3, 0.1). After the peak, the graph decreases to another kink along the horizontal axis at (1.7, 0), and then the graph finishes by increasing to a horizontal portion that terminates at (3, 0.1).

This highpass characteristic is subtracted from unity to give the desired lowpass inverse-Chebyshev frequency response illustrated in Figure 4c. The resulting magnitude-squared frequency- response function is given by

F F ( j ω ) = ϵ 2 C N 2 ( 1 / ω ) 1 + ϵ 2 C N 2 ( 1 / ω ) F F ( j ω ) = ϵ 2 C N 2 ( 1 / ω ) 1 + ϵ 2 C N 2 ( 1 / ω )
(41)

Zero Locations

The zeros of the Chebyshev polynomial CN(ω)CN(ω) are easily found by

C N ( ω ) = 0 N cos - 1 ( ω ) = ( 2 k + 1 ) π / 2 C N ( ω ) = 0 N cos - 1 ( ω ) = ( 2 k + 1 ) π / 2
(42)

which requires

ω k = cos ( ( 2 k + 1 ) π / 2 N ) ω k = cos ( ( 2 k + 1 ) π / 2 N )
(43)

for k=0,1,N-1k=0,1,N-1, or

ω k = sin ( k π / 2 N ) ω k = sin ( k π / 2 N )
(44)

for k=0,±2,±4,...,±(N-1)k=0,±2,±4,...,±(N-1) : N odd

k=±1,±3,±5,...,±(N-1)k=±1,±3,±5,...,±(N-1) : N even

The zeros of the inverse-Chebyshev filter transfer function are derived from Equation 41 and Equation 43 to give

ω z k = 1 / ( cos ( ( 2 k + 1 ) π / 2 N ) ) ω z k = 1 / ( cos ( ( 2 k + 1 ) π / 2 N ) )
(45)

The zero locations are not a function of ϵϵ, i.e., they are independent of the stopband ripple.

Pole Locations

The pole locations are the reciprocal of those for the regular Chebyshev filter. If the poles for the inverse filter are denoted by

s k ' = σ k ' + j ω k ' s k ' = σ k ' + j ω k '
(46)

the locations are

σ k ' = σ k σ k 2 + ω k 2 σ k ' = σ k σ k 2 + ω k 2
(47)
ω k ' = ω k σ k 2 + ω k 2 ω k ' = ω k σ k 2 + ω k 2
(48)

Although this gives a straightforward formula for calculating the location of the poles and zeros of the inverse- Chebyshev filter, they do not lie on a simple geometric curve as did those for the Butterworth and Chebyshev filters. Note that the conditions for a Taylor's series approximation with preset zero locations are satisfied.

A partially factored form for the Butterworth filter and for the Chebyshev filter can be written for the inverse-Chebyshev filter using the zero locations from Equation 45 and the pole locations from the regular Chebyshev filter. For N even, this becomes

F ( s ) = k ( s 2 + ω z k 2 ) k ( s 2 - 2 ( σ k / ( σ k 2 + ω k 2 ) ) s + 1 / ( σ k 2 + ω k 2 ) F ( s ) = k ( s 2 + ω z k 2 ) k ( s 2 - 2 ( σ k / ( σ k 2 + ω k 2 ) ) s + 1 / ( σ k 2 + ω k 2 )
(49)

for k=1,3,5,,N-1k=1,3,5,,N-1. For N odd, F(s) has a single pole, and therefore, is of the form

F ( s ) = k ( s 2 + ω z k 2 ) ( s + 1 / sinh ( ν 0 ) ) k ( s 2 - 2 ( σ k / ( σ k 2 + ω k 2 ) ) s + 1 / ( σ k 2 + ω k 2 ) F ( s ) = k ( s 2 + ω z k 2 ) ( s + 1 / sinh ( ν 0 ) ) k ( s 2 - 2 ( σ k / ( σ k 2 + ω k 2 ) ) s + 1 / ( σ k 2 + ω k 2 )
(50)

for k=2,4,6,,N-1k=2,4,6,,N-1

Because of the relationships between the locations of the poles of the Butterworth, Chebyshev, and inverse-Chebyshev filters, it is easy to write a design program with many common calculations. That is illustrated in the program in the appendix.

Inverse-Chebyshev Filter Design Procedures

The natural form for the specifications of an inverse-Chebyshev filter is in terms of the flatness of the response at ωω to determine the passband, and a maximum allowable response in the stopband. The filter order and the stopband ripple are the parameters to be determined by the specifications. The rate of dropoff near the transition from pass to stopband is similar to the regular Chebyshev filter. Because practical specifications often allow more passband ripple than stopband ripple, the regular Chebyshev filter will usually have a sharper dropoff than the inverse-Chebyshev filter. Under those conditions, the inverse-Chebyshev filter will have a smoother phase response and less time-domain echo effects.

The stopband ripple d is simply defined as the maximum value that |F(jω)||F(jω)| assumes in the stopband, which is the set of frequencies 1<ω<1<ω<. An alternative specification is the minimum-allowed attenuation over stopband expressed in dB as b. The following formulas relate the stopband ripple δδ, the stopband attenuation b in positive dB, and the transfer function parameter ϵϵ in Equation 41

ϵ = δ 1 - δ 2 ϵ = δ 1 - δ 2
(51)
δ = ϵ 1 + ϵ 2 δ = ϵ 1 + ϵ 2
(52)
b = - 10 log ( ϵ 2 / ( 1 + ϵ 2 ) ) = - 20 log ( d ) b = - 10 log ( ϵ 2 / ( 1 + ϵ 2 ) ) = - 20 log ( d )
(53)

In some cases passband performance is not given in terms of degree of flatness at ω=0ω=0, but in terms of a minimum-allowed magnitude GG in the passband up to a certain frequency ωpωp, i.e., 1>|F|>G1>|F|>G for 0<ω<ωp<10<ω<ωp<1. For a given ϵϵ, this requirement will determine the order as the smallest positive integer satisfying

N > cosh - 1 ( G / ( ϵ 1 - G 2 ) ) cosh - 1 ( 1 / ω p ) N > cosh - 1 ( G / ( ϵ 1 - G 2 ) ) cosh - 1 ( 1 / ω p )
(54)

The design of an inverse-Chebyshev filter is summarized in the following steps:

  1. The maximum-allowed stopband response must be given in the form of δδ or b. From this, the parameter ϵϵ is calculated using Equation 51.
  2. The order N is determined from the desired flatness at ω=0ω=0, or from a minimum allowed response for frequencies up to ωpωp using Equation 54.
  3. ν0ν0 and sinh(ν0)sinh(ν0) and cosh(ν0)cosh(ν0) are calculated just as for the regular Chebyshev filter.
  4. The pole locations for the prototype Chebyshev filter are calculated from Equation 46 and Equation 48 and then "inverted" according to Equation 41 to give the inverse- Chebyshev filter pole locations.
  5. The pole locations are combined in Equation 41 to give the final filter transfer function denominator.
  6. The zero locations are calculated from Equation 45 and combined with the pole locations to give the total transfer function Equation 49 or Equation 50.

Example 2: Example Design of an Inverse-Chebyshev Filter

A third-order inverse-Chebyshev lowpass filter is desired with a maximum-allowed stopband ripple of d=0.1d=0.1 or b=20b=20 dB. This corresponds to an ϵϵ of 0.100504 and, together with N=3N=3, results in a ν0=0.99774ν0=0.99774. The scale factors are sinh=1.171717sinh=1.171717 and cosh=1.540429cosh=1.540429. The prototype Chebyshev filter transfer function is

F ( s ) = 1 ( s + 1 . 1717 ) ( s 2 + 1 . 1717 s + 2 . 0404 ) F ( s ) = 1 ( s + 1 . 1717 ) ( s 2 + 1 . 1717 s + 2 . 0404 )
(55)

The zeros are calculated from Equation 45, and the poles of the prototype are inverted to give, from Equation 50, the desired inverse- Chebyshev filter transfer function of

F ( s ) = s 2 + 4 / 3 ( s + 0 . 85345 ) ( s 2 + 0 . 57425 s + 0 . 490095 ) F ( s ) = s 2 + 4 / 3 ( s + 0 . 85345 ) ( s 2 + 0 . 57425 s + 0 . 490095 )
(56)

References

  1. Gold, B. and Rader, C. M. (1969). Digital Processing of Signals. New York: McGraw-Hill.
  2. Mitra, Sanjit K. (2006). Digital Signal Processing, A Computer-Based Approach. (Third). [First edition in 1998, second in 2001]. New York: McGraw-Hill.
  3. Oppenheim, A. V. and Schafer, R. W. (1999). Discrete-Time Signal Processing. (Second). [Earlier editions in 1975 and 1989]. Englewood Cliffs, NJ: Prentice-Hall.
  4. Parks, T. W. and Burrus, C. S. (1987). Digital Filter Design. New York: John Wiley & Sons.
  5. Rabiner, L. R. and Gold, B. (1975). Theory and Application of Digital Signal Processing. Englewood Cliffs, NJ: Prentice-Hall.
  6. Taylor, F. J. (1983). Digital Filter Design Handbook. New York: Marcel Dekker, Inc.
  7. Valkenburg, M.E. Van. (1982). Analog Filter Design. New York: Holt, Rinehart, and Winston.

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