Chebychev filter (Type II)

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The Chebychev filter of type II is given by the normalized transfer function

Normalized transfer function for the Chebychev filter of type II

where n is the order of the filter, ε defines the size of the ripples of the filter's magnitude response in the stop band, the roots of the numerator rk are

Roots of the numerator of the transfer function of the type II Chebychev filter

and the roots of the denominator sk are

Roots of the denominator of the transfer function of the type II Chebychev filter

Here sinh and cosh are the hyperbolic sine and cosine functions and asinh and acosh are the inverse hyperbolic sine and cosine functions.

The transfer function above is shown irrespective of the cutoff frequency (i.e., it is normalized for the cutoff frequency 1) and the forms of the Chebychev type II filter require substitutions. Use S = s / ωc to get the Chebychev type II low pass filter, where s = jω, ωc is the cutoff frequency of the filter, and ω is the angular frequency spanning the frequency spectrum between 0 and π. Use S = ωc / s to get the Chebychev type II high pass filter. Use S = (s2 + ωc2) / (B s) to get the Chebychev type II band pass filter, where ωc is the midpoint of the pass band and B is the width of the band. Use S = B s / (s2 + ωc2) to get the Chebychev type II band stop filter.

This filter is also called the "inverse Chebychev filter". Its poles (the roots of the denominator) are the inverse of the Chebychev filter of type I and as its gain function (shown below) "inverts" the gain function of the Chebychev filter of type I. While the Chebychev filter of type I controls the size of ripples in the pass band, this filter exhibits a flat pass band and controls the size of the ripples in the stop band.

Example: Fourth order low pass Chebychev type II filer

Set n = 4 and use the substitution S = s / ωc. The transfer function of the fourth order low pass Chebychev type II filter is

Transfer function of the low pass Chebychev type II filter of order four

Note that r1 and r4 are complex conjugates, r2 and r3 are complex conjugates, s1 and s4 are complex conjugates, and s2 and s3 are complex conjugates. Moreover, the roots of the numerator rk have real parts equal to zero, and hence r1 + r4 = 0, and r3 + r2 = 0. To make our computations easier, we can rewrite the transfer function as follows.

Revised transfer function of the low pass Chebychev type II filter of order four

or

Final transfer function of the low pass Chebychev type II filter of order four

After the bilinear transformation s = 2 (z – 1) / (z + 1) the transfer function becomes a Z transform transfer function.

Transfer function of the type II Chebychev low pass filter or order four after the bilinear transformation

Say that the cutoff frequency of the filter is ωc = 0.6 and that ε = 0.1 (technically, the cutoff frequency of the filter will be ωc = 2 arctan(0.6/2) ≈ 0.583, due to the warping of the frequency domain by the bilinear transformation). The transfer function of this example Chebychev type 1 low pass filter, after scaling to obtain b0 = 1 is

Transfer function of an example low pass Chebychev type II filter of order four

and the filter itself is

y(k) = 0.090729 x(k) – 0.190160 x(k – 1) + 0.251568 x(k – 2)
– 0.190160 x(k – 3) + 0.090729 x(k – 4) + 2.662409 y(k – 1)
– 2.828952 y(k -2) + 1.378102 y(k – 3) – 0.264003 y(k – 4)

Suppose that the sampling frequency is 2000 Hz. The cutoff frequency then is ωc = (0.6 * 2000) / (2 π) = 191 Hz. The magnitude response of the filter is shown in the graph below.

Magnitude response of the example low pass Chebychev type II filter of order four

Size of the ripples in the stop band

The magnitude function of the Chebychev type II low pass filer can also be written as follows.

Magnitude of the Chebychev type II filter

where Tn are the Chebychev polynomials

Chebychev polynomials

In the stop band, where ωc / ω < 1, Tnc / ω) = cos(n acos(ωc / ω)) and 0 ≤ Tn2c / ω) ≤ 1. In the stop band

Stop band limits of the Chebychev type II filter gain function

Thus, smaller ε implies smaller ripples in the stop band and larger ε implies larger ripples. It should also be expected that |H(jω)| approaches 0 at ωc / ω approaches cos(π / 2n), since then Tnc / ω) = 0.

Using the same parameters as above, namely n = 2 and ωc = 0.6, we can plot the | H(jω)| for three different values of ε (0.1, 0.2, and 0.3).

Example behavior of the Chebychev type II filter gain function in the stop band

The magnitude response of the three corresponding filters is:

Example magnitude responses for low pass Chebychev type II filters with different values for epsilon

A larger ε creates a filter with a smaller transition band, but with larger ripples in the stop band and worse stop band attenuation. A smaller ε creates a filter with smaller ripples in the stop band and better stop band attenuation, but with larger transition band.

Example: Second order band stop Chebychev type II filter

Set n = 2 and S = B s / (s2 + ωc2). The transfer function of the second order band stop Chebychev type II filter is

Transfer function for the band stop Chebychev type II filter of order two

s1, s2, r1 and r2 are computed with the same formulae as before. After the bilinear transformation s = 2 (z – 1) / (z + 1), setting, for example, ε = 0.1, B = 0.5 and ωc = 0.8, and scaling to obtain 1 at the beginning of the denominator, we have the following band stop transfer function

Transfer function for the second order Chebychev type II filter after the bilinear transformation

If we suppose that the sampling frequency is 2000 Hz, then ωc = (0.8 * 2000) / (2 π) = 255 Hz, B = (0.5 * 2000) / (2 π) = 159 Hz, and the magnitude response of the filter is as follows.

Magnitude response of an example band stop Chebychev type II filter of order two

Higher order Chebychev type II filters

Higher order Chebychev type II filters can be designed either using the methods above, or by stacking lower order filters. For example, the transfer function for the fourth order low pass Chebychev type II filter could have also been written as

Transfer function for the fourth order low pass Chebychev type II filter

This means that the fourth order low pass Chebychev type II filter can be implemented by stacking two filters of order two (these are not Chebychev type II filters, as the roots of each filter of order two differ from the Chebychev roots). Multiplying two transfer functions means that the impulse response of the total filter is the convolution of the two impulse responses of the lower order filters and the output of the first filter is the input to the second filter. We combine the roots in the manner above, as s1 and s4, s2 and s3, r1 and r4, and r2 and r3 form complex conjugate pairs. It is not important whether ε is used in the first filter, the second filter, or split between the two.

The following is a comparison of the magnitude responses of the fourth order and second order Chebychef type II filters with ε = 0.1, sampling frequency of 2000 Hz, and ωc = 0.6.

Magnitude response of a fourth order and second order low pass Chebychev type II filters

Odd order Chebychev type II filters

When the Chebychev type II filter is of odd order n, one of the roots rk of the numerator cannot be computed, as the computation produces a division by zero. At k = (n – 1) / 2 we have rk = 1 / (-j cos(π / 2)). Thus, when n is odd, the middle root of the numerator is omitted from the transfer function, and the numerator becomes a polynomial of s of a lower degree than the denominator. Take, for example, the low pass Chebychev type II filter of order n = 3. Its transfer function, after omitting the middle root r2, is as follows.

Transfer function of the Chebychev type II filter of order 3 after omitting the middle root

Since r1 and r3 are complex conjugates, s1 and s3 are complex conjugates, and s2 is a real number, we can rewrite the transfer function above as a function with real coefficients in the following way.

Transfer function of the Chebychev type II filter of order 3 with real valued coefficients

After the bilinear transformation this filter has the transfer function

Z transform transfer function of the Chebychev type II filter of order 3

For example, when ωc = 0.6 and after scaling to obtain b0 = 1 in the denominator, the filter would be

An example Chebychev type II filter of order 3

This filter, however, produces a magnitude response in the pass band of about -5 dB. To ensure that the filter does not attenuate the pass band, we add some gain by scaling the coefficients in the numerator, so that H(s), evaluated at zero, is equal to 1. The gain is

Gain adjustment for the Chebychev type II filter of order 3

In the example above the gain is 1.8 and the filter itself changes to

An example Chebychev type II filter of order 3 after the gain adjustment

The magnitude response of this third order filter is shown below with the magnitude responses of two Chebychev type II filter so order 2 and 4.

Example Chebychev type II filters of orders 2, 3 and 4



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mic
We added an example of an odd order Chebychev type II computations. When the middle root of the numerator is dropped, the gain of the filter should be adjusted as described above.
Posted At 17-12-2011 03:38:08

mic
Yes. When the order of the filter is odd, one of the zeroes of the numerator becomes "infinite" (results in a division by zero). From what I have seen, this zero is simply removed from the numerator. This means that the (s + j * infinity) that you mentioned is not included in the numerator. The numerator becomes a polynomial of lower order than the polynomial in the denominator.
Posted At 03-12-2011 08:19:20

Werner Van Belle
Very nice summary. I found it remarkably hard to find the correct transfer function for the type 2 filter. So, everything is working out quite well now, except for the odd orders. What does one do with the middle zero ? This gives an expression (s+infinity*j)
Posted At 01-12-2011 09:06:00



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