4.4. Equiripple filters and the Chebyshev approximation – Digital Filters Design for Signal and Image Processing

4.4. Equiripple filters and the Chebyshev approximation

4.4.1. Characteristics of the Chebyshev approximation

Butterworth filters are widely used, but their use has the drawback of an elevated degree of polynomials with standard applications. To get around this problem, an alternative solution consists in using equiripple filters and, more specifically, Chebyshev filters.

The type I Chebyshev approximation (or of type II or inverse) distributes the approximation error throughout the entire passband (or throughout the attenuated band). Unlike Butterworth filters, the frequency response curve then presents, with the Chebyshev approximation, an oscillation in this frequency band. This is an equal amplitude oscillation.

We should bear in mind that the maximum value of admissible error in relation to the reference level is minimized. Moreover, we can demonstrate that the amplitude in the attenuated band decreases in monotonically and much more quickly, for filters of an order above 1, than is the case with Butterworth filters.

4.4.2. Type I Chebyshev filters

4.4.2.1. The Chebyshev polynomial

We represent Cn(x), the Chebyshev function, sometimes called the Chebyshev polynomial, of order n as follows:

In Table 4.3, we give the Chebyshev functions of 0 to 4 for n. We see that these are even functions if n is even and odd functions if n is odd.

Figure 4.13. Variation curves of the first Chebyshev functions

Degree Chebyshev polynomials
0 C0(x) = 1
1 C1(x) = x
2 C2(x) = 2x2 − 1
3 C3(x) = 4x3 −3x
4 C4(x) = 8x4 − 8x2 + 1
n Cn(x) = 2xCn-1(X) − Cn−2(X) n≥2

Table 4.3. Chebyshev polynomial values

4.4.2.2. Type I Chebyshev filters

For a type I Chebyshev filter, the square of the normalized amplitude is in the form:

where is a parameter that regulates the ripple value in the passband. We should remember that here we introduce the normalized frequency in relation to the frequency limit of the passband fp, and not in relation to the cut-off frequency fc to −3dB.

Figure 4.14. Squared amplitude of type I Chebyshev filters for different orders = 1/3

According to Figure 4.14, we see that the extrema numbers present In the passband is equal to the filter order.

Moreover, the attenuation satisfies the relation:

Figure 4.15. Example of attenuation in type I Chebyshev filters for different orders = 1/3

4.4.2.3. Pole determination

The poles pk are expressed by the denominator roots of |H(s)|2:

So:

We then introduce the quantities sk, uk, and νk so that:

From here, equation (4.34) becomes:

By identifying the real and imaginary parts of the two portions of equation (4.7), the poles of the transfer function H(p) are determined from the system following the two equations with two unknown factors:

Since cosh nνk ≠ 0, the relation in equation (4.37) is reduced to:

So we have:

From there, since sin nuk = ±1, equation (4.37) is equivalent to:

We can easily demonstrate that the poles are situated on an ellipse. So equation (4.35) can be written as follows:

By identifying the real and imaginary part of equation (4.41), we get:

We can then write:

4.4.2.4. Determining the cut-off frequency at −3 dB and the filter order

Figure 4.16. Low-pass filter specification characterized by angular frequencies at the end and the passband and the beginning and the attenuated band

Let Ap be the maximum attenuation that we wish for the angular frequency fp, and Aa the maximum attenuation that we wish for angular frequency fa. Determining the minimal order satisfying these two conditions is done as follows below.

Given equation (4.31) and the constraints linked to the specifications, we have, on the one hand, since .

and on the other hand:

From there, taking into account equations (4.44) and (4.45), we obtain:

or:

Instead of using the function arg ch(), we can exploit an alternative relation. In addition, the hyperbolic cosines and the hyperbolic sinus verify the following relation:

Using equation (4.48), we can then write ch(x) + sh(x) as follows:

With the equality in equation (4.49), the function x = arg ch(y) can be written as follows using the Neperian logarithm:

From equations (4.47) and (4.50), we then get:

Using equation (4.45), we easily obtain the value of the coefficient :

We can easily express the cut-off frequency at −3 dB; it satisfies the following relation:

or:

4.4.2.5. Application

We want to synthesize a Chebyshev filter with an attenuation of 40 dB to 4,000 Hz and of 0.5 dB to 3,200 Hz.

Figure 4.17. Synthesis of a continuous, type I Chebyshev filter

By applying the above formulae, we find that the order of the filter is 10 instead of 26 for the Butterworth filter described earlier.

4.4.2.6. Realization of a Chebyshev filter

In order to realize this type of filter, the following two constraints must be satisfied:

– we must respect the ripple value in the reference band;

– we must obtain the minimum attenuation value in the attenuated band.

The transfer function is written in the following form:

where n, the polynomial degree, is determined by the required attenuation in the attenuated band.

There are tables showing the denominator values of the transfer function of the filter for different values of n and of the ripple in the reference band, expressed in dB (also see Tables 4.4 and 4.5).

Degree Polynomial for 0.5 dB of ripple
1 s + 2.863
2 s2 + 1.425s + 1.516
3 s3 + 1.253s2 + 1.535s + 0.716 = (s + 0.626)(s2 + 0.626s + 1.142)
4 s4 + 1.197s3 + 1.717s2 + 1.025s + 0.379 = (s2 + 0.351s + 1.064)(s2 + 0.845s + 0.356)
5 s5 + 1.172s4 + 1.937s3 + 1.309s2 + 0.753s + 0.179 = (s + 0.362)(s2 + 0.224s + 1.036)(s2 + 0.586s + 0.477)

Table 4.4. Denominator of H(s) for 0.5 dB of ripple

Degree Polynomial for 1 dB of ripple
1 s + 1.965
2 s2 + 1.098s + 1.103
3 s3 + 0.988s2 + 1.238s + 0.491 = (s + 0.494)(s2 + 0.494s + 0.994)
4 s4 + 0.953s3 + 1.454s2 + 0.743s + 0.276 = (s2 + 0.279s + 0.987)(s2 + 0.674s + 0.279)
5 s5 + 0.937s4 + 1.689s3 + 0.974s2 + 0.581s + 0.123 = (s + 0.289)(s2 + 0.179s + 0.988)(s2 + 0.468s + 0.429)

Table 4.5. Denominator of H(s) for 1 dB of ripple

4.4.2.7. Asymptotic behavior

Here we look at asymptotic behavior in the gain curve of a Chebyshev filter of order n. We can demonstrate that:

So:

Now, for a Butterworth filter of order n, there is a drop of 20n dB per decade, so:

To the same degree, we observe that a Chebyshev filter presents more attenuation than a Butterworth filter.

4.4.3. Type II Chebyshev filter

For a type II Chebyshev filter, the square of the normalized amplitude possess both poles and zeros:

4.4.3.1. Determining the filter order and the cut-off frequency

We determine the minimal order that will satisfy these two conditions as described below.

Using equation (4.59) and the constraints linked to the specifications, we have, on the one hand:

and on the other hand:

The constraints in equations (4.60) and (4.61) lead to the same condition of order as with the type I Chebyshev filter, so we have:

or:

We also have:

4.4.3.2. Application

Suppose we wish to synthesize a type II Chebyshev filter with an attenuation of 40 dB to 4,000 and of 0.5 at 3,200 Hz.

Figure 4.18. Synthesis of a continuous-type II Chebyshev filter. Frequential representation: ripple in the attenuated band