3.6 Examples
This chapter discusses various IIR filter design methods. The four standard types of filters are used here:
The design method used here is known as bilinear transformation.
The IIR filter design process can be split into several steps described in Chapter 3.2.5
Designing IIR filters by bilinear transformation. These are:

 Defining filter specifications;

 Specifying the type of analog prototype filter;

 Computing the filter order according to the filter specifications and specified analog prototype filter;

 Computing the transfer function of reference analog prototype filter;

 Transformation into analog filter by range scaling;

 Transformation into digital filter by bilinear transformation; and

 If the resulting filter doesn’t satisfy the given specifications or if it is possible to decrease the filter order, then it is necessary to do it. The specified filter order is increased or decreased according to needs, and steps 4, 5 and 6 are repeated after that as many times as needed.
Some steps are skipped in some cases. If the filter order is known, step 3 is skipped. If the type of reference analog prototype filter is predetermined, step 2 is skipped.
In every given example, the IIR filter design process will be described through these steps in order to make it easier for you to observe similarities and differencies between various design methodes, analog prototype filters and design of various types of filters as well.
Figure 361 illustrates the design steps along with input and output data for each of them.
Figure 361. Steps in designing digital IIR filter
The first block refers to design of reference analog prototype filter of appropriate order. The output data is a reference analog prototype filter transfer function H
a(s). Regardless of the type of reference analog prototype filter in use, the transfer function is given by:
where:
H
0 is a constant;
z
k is the kth zero of the reference analog prototype filter transfer function;
M is a number of zeros of the reference analog prototype filter transfer function;
p
k is the kth pole of the reference analog prototype filter transfer function; and
N is a number of poles of the reference analog prototype filter transfer function and filter order as well.
Reference analog prototype filter is always a lowpass filter. The next step is the transformation into an analog filter of appropriate type. The expression used to transform the reference analog prototype filter transfer function depends on the type of filter that needs to be obtained. The final result is the transfer function H(s) given by:
where:
H
0 is a constant;
z
k is the kth zero of the reference analog prototype filter transfer function;
M is a number of zeros of the reference analog prototype filter transfer function;
p
k is the kth pole of the reference analog prototype filter transfer function; and
N is a number of poles of the reference analog prototype filter transfer function and filter order as well.
As seen, the transfer functions of reference analog prototype filter and analog filter are very similar. They differ only in the value of constant H
0, the values of the transfer function poles and zeros z
k and p
k as well as in the number of transfer function zeros M. The filter order is the same if the analog filter is a lowpass or highpass filter, whereas it is different if the analog filter is a bandpass or bandstop filter. In the later case, the analog filter order (N) is twice that of the reference analog prototype filter.
The next step is the transformation into appropriate digital IIR filter using bilinear transformation given by expression:
Filter Designer Tool is used for testing and analysing the resulting IIR filters in this chapter. All data are calculated with the accuracy of 4 decimal digits, which is sufficient for most examples.
3.6.1 Filter design using Butterworth filter
3.6.1.1 Example 1
Step 1:
Type of filter – lowpass filter
Filter specifications:

 Sampling frequency – fs=20KHz;

 Passband cutoff frequency – fc=2.5KHz; and

 Minimum stopband attenuation – ap=40dB.
Step 2:
Method filter design using Butterworth reference analog prototype filter.
Step 3:
Filter order is predetermined, N=2.
Step 4:
The Butterworth reference prototype filter transfer function has no zeros, only poles. These can be computed via expression:
As N = 2, the values of poles are:
The reference analog prototype filter transfer function is:
Step 5:
First it is necessary to compute the analog prototype filter cutoff frequency Ω
c.
The analog filter transfer function is obtained from the reference analog prototype filter transfer function using expression:
As the Butterworth reference prototype filter has no zeros, the expression for transfer function is simpler:
Step 6:
The transformation into a digital filter through bilinear transformation:
Generally, substituting the complex variable s into the expression for analog filter transfer function, the following is obtained:
where:
z
k are the zeros of analog filter transfer function; and
p
k are the poles of analog filter transfer function.
This general expression can be written in a simpler way in this example:
A more condensed form of the previous expression is:
The result is the IIR filter transfer function.
Step 7:
The filter order is predetermined.
There is no need to additionally change it.
Filter realization:
Figure 362 illustrates the direct realization of designed IIR filter, whereas Figure 363 illustrates the frequency response of the filter obtained using
Filter Designer Tool.
Figure 362. Digital IIR filter direct realization in this example
Figure 363. Digital IIR filter frequency characteristic in this example
3.6.1.2 Example 2
Step 1:
Type of filter – highpass filter
Filter specifications:

 Sampling frequency – fs=20KHz;

 Passband cutoff frequency – fc=5KHz; and

 Minimum stopband attenuation – ap=40dB.
Step 2:
Method filter design using Butterworth reference analog prototype filter
Step 3:
Filter order is predetermined, N = 3.
Step 4:
The Butterworth reference prototype filter transfer function has no zeros, only poles. These can be computed via expression:
As N = 3, the values of poles are:
The reference analog prototype filter transfer function is:
Step 5:
First it is necessary to compute the analog prototype filter cutoff frequency Ω
c.
The analog filter transfer function is obtained from the reference analog prototype filter transfer function using expression:
As the Butterworth reference prototype filter has no zeros, the expression for transfer function is simpler:
Step 6:
The transformation into a digital filter through bilinear transformation:
Generally, substituting the complex variable s into the expression for analog filter transfer function, the following is obtained:
where:
z
k are the zeros of analog filter transfer function; and
p
k are the poles of analog filter transfer function.
This general expression can be written in a simpler way in this example:
A more condensed form of the previous expression is:
The result is the IIR filter transfer function.
Step 7:
The filter order is predetermined.
There is no need to additionally change it.
Filter realization:
Figure 364 illustrates the direct realization of designed IIR filter.
Figure 364. Digital IIR filter direct realization in this example
Figure 365. Digital IIR filter frequency characteristic in this example
3.6.1.3 Example 3
Step 1:
Type of filter – bandpass filter
Filter specifications:

 Sampling frequency – fs=20KHz; and

 Passband cutoff frequency – fc1=4KHz, fc2=6KHz.
Step 2:
Method  filter design using Butterworth reference analog prototype filter
Step 3:
Filter order is predetermined, N = 4.
Step 4:
The Butterworth reference prototype filter transfer function has no zeros, only poles. These can be computed via expression:
When designing an IIR bandpass filter, the reference prototype filter order is half that of the required IIR filter order. In this example for N=4, the order of reference prototype filter is 2, so the values of poles are:
The reference analog prototype filter transfer function is:
Step 5:
First it is necessary to compute the analog prototype filter cutoff frequency Ω
c.
The analog filter transfer function is obtained from the reference analog prototype filter transfer function using expression:
As the Butterworth reference prototype filter has no zeros, the expression for transfer function is simpler:
Step 6:
The transformation into a digital filter through bilinear transformation:
Generally, by substituting the complex variable s into the expression for analog filter transfer function, the following is obtained:
where:
z
k are the zeros of analog filter transfer function; and
p
k are the poles of analog filter transfer function.
In this example, the general expression can be written in a simpler way:
A more condensed form of the previous expression is:
The result is the IIR filter transfer function.
Step 7:
Filter order is predetermined.
There is no need to additionally change it.
Filter realization:
Figure 366 illustrates the direct realization of designed IIR filter.
Figure 366. Digital IIR filter direct realization in this example
Figure 367. Digital IIR filter frequency characteristic in this example
3.6.1.4 Example 4
Step 1:
Type of filter – bandstop filter
Filter specifications:

 Sampling frequency – fs=20KHz; and

 Passband cutoff frequency – fc1=3000Hz, fc2=3300Hz.
Step 2:
Method  filter design using Butterworth reference analog prototype filter.
Step 3:
Filter order is predetermined, N = 4.
Step 4:
The Butterworth reference prototype filter transfer function has no zeros, only poles. These can be computed via expression:
When designing an IIR bandstop filter, the reference prototype filter order is half that of the required IIR filter order. In this example for N = 4, the order of reference prototype filter is 2, so the values of poles are:
The reference analog prototype filter transfer function is:
Step 5:
First it is necessary to compute the analog prototype filter cutoff frequency Ω
c.
The analog filter transfer function is obtained from the reference analog prototype filter transfer function using expression:
As the Butterworth reference prototype filter has no zeros, the expression for transfer function is simpler:
Step 6:
The transformation into a digital filter through bilinear transformation:
Generally, by substituting the complex variable s into the expression for analog filter transfer function, the following is obtained:
where:
z
k are the zeros of analog filter transfer function; and
p
k are the poles of analog filter transfer function.
In this example, the general expression can be written in a simpler way:
A more condensed form of the previous expression is:
The result is the IIR filter transfer function.
Step 7:
Filter order is predetermined.
There is no need to additionally change it.
Filter realization:
Figure 368 illustrates the direct realization of designed IIR filter.
Figure 368. Digital IIR filter direct realization in this example
Figure 369. Digital IIR filter frequency characteristic in this example
3.6.2 Filter design using Chebyshev filter
3.6.2.1 Example 1
Step 1:
Type of filter – lowpass filter
Filter specifications:

 Sampling frequency – fs=44100Hz;

 Passband cutoff frequency– fc1=15KHz;

 Stopband cutoff frequency – fc2=18KHz;

 Maximum passband attenuation – ap=1dB; and

 Minimum stopband attenuation – as=40dB.
Step 2:
Method  filter design using Chebyshev reference analog prototype filter
Step 3:
Filter order is not predetermined, so it is necessary to choose an initial solution from which iterative method starts. The solution is redefined progressively until some predetermined requirements are satisfied. Let’s assume that the initial filter order is 4.
Step 4:
The Chebyshev reference prototype filter transfer function has no zeros, only poles. These can be computed via expression:
As N = 4, the values of poles are:
The Chebyshev filter transfer function is expressed as:
In this case, the value of constant A
0 is:
A0 = 0.2457
so that the Chebyshev reference analog prototype filter transfer function is:
Step 5:
First it is necessary to compute the analog prototype filter cutoff frequency Ω
c.
The analog filter transfer function is obtained from the reference analog prototype filter transfer function using expression:
As the Chebyshev reference prototype filter has no zeros, the expression for transfer function is simpler:
Step 6:
The transformation into a digital filter through bilinear transformation:
Generally, by substituting the complex variable s into the expression for analog filter transfer function, the following is obtained:
where:
z
k are the zeros of analog filter transfer function; and
p
k are the poles of analog filter transfer function.
This general expression can be written in a simpler way in this example:
A more condensed form of the previous expression is:
The result is the IIR filter transfer function.
Step 7:
By analyzing the resulting filter using Filter Designer Tool, it is obvious that the attenuation amounting to 31.2dB approximately at the frequency of 18KHZ is not sufficient. The frequency characteristic of the resulting digital filter is illustrated in Figure 3610 below.
Figure 3610. Frequency characteristic of the resulting IIR filter
It is necessary to additionally redefine the filter order until the predefined requirements are satisfied. The filter order is incremented by 1 and is 5 therefore. All steps starting with step 3 are iterated.
Step3:
The filter order is incremented in the second iteration. A new filter order is 5.
Step 4:
The Chebyshev reference prototype filter transfer function has no zeros, only poles. These can be computed via expression:
As N = 5, the values of poles are:
The Chebyhsev filter transfer function is expressed as:
In this case, the value of constant A
0 is:
A0 =  0.1228
so the Chebyshev reference analog prototype filter transfer function is:
Step 5:
First it is necessary to compute the analog prototype filter cutoff frequency Ω
c.
The analog filter transfer function is obtained from the reference analog prototype filter transfer function using expression:
As the Chebyshev reference prototype filter has no zeros, the expression for transfer function is simpler:
Step 6:
The transformation into a digital filter through bilinear transformation:
Generally, substituting the complex variable s into the expression for analog filter transfer function, the following is obtained:
where:
z
k are the zeros of analog filter transfer function; and
p
k are the poles of analog filter transfer function.
This general expression can be written in a simpler way in this example:
A more condensed form of the previous expression is:
The result is the IIR filter transfer function.
Step 7:
By analyzing the resulting filter using
Filter Designer Tool, it is obvious that the attenuation amounting to 41.6dB approximately at the frequency of 18KHZ is not sufficient. The frequency characteristic of the resulting digital filter is illustrated in Figure 3611 below.
Figure 3611. Frequency characteristic of the resulting IIR filter
It is not necessary to further increase the filter order as this one is appropriate. Sometimes more iterations are needed to determine the filter order. The whole procedure is the same, only it takes more time.
Filter realization:
Figure 3612 illustrates the direct realization of designed IIR filter, whereas Figure 3613 illustrates the frequency characteristic of the filter obtained using Filter Designer Tool.
Figure 3612. Digital IIR filter direct realization
Figure 3613. Frequency characteristic of digital IIR filter
3.6.1.2 Example 2
Step 1:
Type of filter – highpass filter
Filter specifications:

 Sampling frequency – fs = 20KHz;

 Passband cutoff frequency – fc = 5KHz; and

 Maximum passband attenuation – ap = 1dB.
Step 2:
Method  filter design using Chebyshev reference analog prototype filter
Step 3:
Filter order is predetermined, N = 3.
Step 4:
The Chebyshev reference prototype filter transfer function has no zeros, only poles. These can be computed via expression:
As N = 3, the values of poles are:
The Chebyshev filter transfer function is expressed as:
In this case, the value of constant A
0 is:
A0 =  0.4913
The reference analog prototype filter transfer function is:
Step 5:
First it is necessary to compute the analog prototype filter cutoff frequency Ω
c.
The analog filter transfer function is obtained from the reference analog prototype filter transfer function using expression:
As the Chebyshev reference prototype filter has no zeros, the expression for transfer function is simpler:
After substitution of poles and Ω
c into expression:
Step 6:
The transformation into a digital filter through bilinear transformation:
Generally, substituting the complex variable s into the expression for analog filter transfer function, the following is obtained:
where:
z
k are the zeros of analog filter transfer function; and
p
k are the poles of analog filter transfer function.
This general expression can be written in a simpler way in this example:
A more condensed form of the previous expression is:
The result is the IIR filter transfer function.
Step 7:
The filter order is predetermined.
There is no need to additionally change it.
Filter realization:
Figure 3614 illustrates the direct realization of designed IIR filter.
Figure 3614. Direct realization of digital IIR filter in this example
Figure 3615. Frequency characteristic of digital IIR filter in this example
3.6.1.3 Example 3
Step 1:
Type of filter – bandpass filter
Filter specifications:

 Sampling frequency – fs = 20KHz;

 Passband cutoff frequencies – fc1 = 4KHz, fc2 = 6KHz; and

 Maximum passband attenuation – ap = 1dB.
Step 2:
Method filter design using Chebyshev reference analog prototype filter.
Step 3:
Filter order is predetermined, N = 4.
Step 4:
The Chebyshev reference prototype filter transfer function has no zeros, only poles. These can be computed via expression:
When designing an IIR bandpass filter, the reference prototype filter order is half that of the required IIR filter order. In this example for N = 4, the order of reference prototype filter is 2, so the values of poles are:
The Chebyshev filter transfer function is expressed as:
In this example, the value of constant A
0 is:
A0 = 0.9826
The reference analog prototype filter transfer function is:
Step 5:
First it is necessary to compute the analog prototype filter cutoff frequency Ω
c.
The analog filter transfer function is obtained from the reference analog prototype filter transfer function using expression:
As the Chebyshev reference prototype filter has no zeros, the expression for transfer function is simpler:
Step 6:
The transformation into a digital filter through bilinear transformation:
Generally, substituting the complex variable s into the expression for analog filter transfer function, the following is obtained:
where:
z
k are the zeros of analog filter transfer function; and
p
k are the poles of analog filter transfer function.
In this expression, the general expression can be written in a simpler way:
A more condensed form of the previous expression is:
The result is the IIR filter transfer function.
Step 7:
Filter order is predetermined.
There is no need to additionally change it.
Filter realization:
Figure 3616 illustrates direct realization of designed IIR filter.
Filter 3616. Direct realization of digital IIR filter in this example
Figure 3617. Frequency characteristic of digital IIR filter in this example
3.6.1.4 Example 4
Step 1:
Type of filter – bandstop filter
Filter specifications:

 Sampling frequency – fs = 20KHz;

 Passband cutoff frequencies – fc1 = 3000Hz, fc2 = 3300Hz; and

 Maximum passband attenuation – ap = 1dB.
Step 2:
Method  filter design using Chebyshev reference analog prototype filter.
Step 3:
Filter order is predetermined, N = 4.
Step 4:
The Chebyshev reference prototype filter transfer function has no zeros, only poles. These can be computed via expression:
When designing an IIR bandpass filter, the reference prototype filter order is half that of the required IIR filter order. In this example for N = 4, the order of reference prototype filter is 2, so the values of poles are:
The Chebyshev filter transfer function is expressed as:
In this example, the value of constant A
0 is:
A0 = 0.9826
The reference analog prototype filter transfer function is:
Step 5:
First it is necessary to compute the analog prototype filter cutoff frequency Ω
c.
The analog filter transfer function is obtained from the reference analog prototype filter transfer function using expression:
As the Chebyshev reference prototype filter has no zeros, the expression for transfer function is simpler:
Step 6:
The transformation into a digital filter through bilinear transformation:
Generally, substituting the complex variable s into the expression for analog filter transfer function, the following is obtained:
where:
z
k are the zeros of analog filter transfer function; and
p
k are the poles of analog filter transfer function.
In this expression, the general expression can be written in a simpler way:
A more condensed form of the previous expression is:
The result is the IIR filter transfer function.
Step 7:
Filter order is predetermined.
There is no need to additionally change it.
Filter realization
Figure 3618 illustrates the direct realization of designed IIR filter.
Figure 3618. Direct realization of digital IIR filter in this example
Figure 3619. Frequency characteristic of digital IIR filter in this example
3.6.3 Filter design using inverse Chebyshev filter
3.6.3.1 Example 1
Step 1:
Type of filter –lowpass filter
Filter specifications:

 Sampling frequency – fs = 44100Hz;

 Passband cutoff frequency – fc1 = 15KHz;

 Stopband cutoff frequency – fc2 = 18KHz;

 Maximum passband attenuation – ap = 1dB; and

 Minimum stopband attenuation – as = 40dB.
Step 2:
Method  filter design using inverse Chebyshev reference analog prototype filter.
Step 3:
Filter order is not predetermined, so it is necessary to choose an initial solution from which iterative method starts. The solution is redefined progressively until some predetermined requirements are satisfied. Let’s assume that the initial filter order is 4.
Step 4:
The Chebyshev reference prototype filter transfer function has no zeros, only poles. These can be computed via expression:
As N = 4, the values of poles are:
The inverse Chebyhsev filter transfer function is expressed as:
In this example, the value of constant H
0 is:
H0 = 0.01
so that the Chebyshev reference analog prototype filter transfer function is:
Step 5:
First it is necessary to compute the analog prototype filter cutoff frequency Ω
c.
The analog filter transfer function is obtained from the reference analog prototype filter transfer function using expression:
In this example, the analog filter transfer function becomes:
Step 6:
The transformation into a digital filter through bilinear transformation:
Generally, substituting the complex variable s into the expression for analog filter transfer function, the following is obtained:
where:
z
k are the zeros of analog filter transfer function; and
p
k are the poles of analog filter transfer function.
This general expression can be written in a simpler way in this example:
A more condensed form of the previous expression is:
The result is the IIR filter transfer function.
Step 7:
By analyzing the resulting filter using
Filter Designer Tool, it is obvious that the attenuation amounting to 5.2dB approximately at the frequency of 15KHZ is not sufficient. The frequency characteristic of the resulting digital filter is illustrated in Figure 3620 below.
Figure 3620. Frequency charactersitic of designed IIR filter
The attenuation at 15KHz is higher than 1dB and the resulting filter doesn’t satisfy the specifications therefore. It is necessary to additionally redefine the filter order until the predefined requirements are satisfied. The filter order is incremented by 1, and becomes 5 therefore. All steps starting with step 3 are iterated.
Step 3:
The filter order is incremented in the second iteration. A new filter order is 5.
Step 4:
The inverse Chebyshev reference prototype filter transfer function has no zeros, only poles.
These can be computed via expression:
As N = 5, the values of poles are:
The Chebyhsev filter transfer function is expressed as:
In this case, the value of constant H
0 is:
H0 =  0.05
so the inverse Chebyshev reference analog prototype filter transfer function is:
Step 5:
First it is necessary to compute the analog prototype filter cutoff frequency Ω
c.
The analog filter transfer function is obtained from the reference analog prototype filter transfer function using expression:
In this example, the analog filter transfer function becomes:
Step 6:
The transformation into a digital filter through bilinear transformation:
Generally, substituting the complex variable s into the expression for analog filter transfer function, the following is obtained:
where:
z
k are the zeros of analog filter transfer function; and
p
k are the poles of analog filter transfer function.
This general expression can be written in a simpler way in this example:
A more condensed form of the previous expression is:
The result is the IIR filter transfer function.
Step 7:
By analysing the resulting filter using
Filter Designer Tool, it is obvious that the attenuation at 18KHZ is less than 1dB (approximately 0.8dB). The frequency characteristic of the resulting digital filter is illustrated in Figure 3621 below.
Figure 3621. Frequency characteristic of IIR filter
It is not necessary to further increase the filter order as this one is approriate. Sometimes more iterations are needed to determine the filter order. The whole procedure is the same, only it takes more time.
Filter realization:
Figure 3622 illustrates the direct realization of designed IIR filter, whereas Figure 3623 illustrates the frequency characteristic of the filter obtained using Filter Designer Tool.
Figure 3622. Direct realization of digital IIR filter in this example
Figure 3623. Frequency characteristic of digital IIR filter in this example
3.6.1.2 Example 2
Step 1:
Type of filter – highpass filter
Filter specifications:

 Sampling frequency – fs = 20KHz;

 Passband cutoff frequency– fc = 5KHz; and

 Minimum stopband attenuation – 40dB.
Step 2:
Method  filter design using inverse Chebyshev reference analog prototype filter.
Step 3:
Filter order is predetermined, N=3.
Step 4:
The Chebyshev reference prototype filter transfer function has no zeros, only poles. These can be computed via expression:
As N = 3, the values of poles are:
The inverse Chebyshev filter transfer function is expressed as:
In this case, the value of constant H
0 is:
H0 = 0.03
The reference analog prototype filter transfer function is:
Step 5:
First it is necessary to compute the analog prototype filter cutoff frequency Ω
c.
The analog filter transfer function is obtained from the reference analog prototype filter transfer function using expression:
In this example, the transfer function is:
After substituting the poles and Ω
c into expression:
Step 6:
The transformation into a digital filter through bilinear transformation:
Generally, by substituting the complex variable s into the expression for analog filter transfer function, the following is obtained:
where:
z
k are the zeros of analog filter transfer function; and
p
k are the poles of analog filter transfer function.
This general expression can be written in a simpler way in this example:
A more condensed form of the previous expression is:
The result is the IIR filter transfer function.
Step 7:
The filter order is predetermined.
There is no need to additionally change it.
Filter realization:
Figure 3624 illustrates the direct realization of designed IIR filter.
Figure 3624. Direct realization of digital IIR filter in this example
Figure 3625. Frequency characteristic of digital IIR filter in this example
3.6.1.3 Example 3
Step 1:
Type of filter – bandpass filter
Filter specification:

 Sampling frequency – fs = 20KHz;

 Passband cutoff frequencies – fc1 = 4KHz, fc2 = 6KHz; and

 Minimum stopband attenuation – 40dB.
Step 2:
Method  filter design using inverse Chebyshev reference analog prototype filter.
Step 3:
Filter order is predetermined, N = 4.
Step 4:
The Chebyshev reference prototype filter transfer function has no zeros, only poles. These can be computed via expression:
When designing an IIR bandpass filter, the reference prototype filter order is half that of the required IIR filter order. In this example for N=4, the order of reference prototype filter is 2, so the values of poles are:
The Chebyshev filter transfer function is expressed as:
In this example, the value of constant H
0 is:
H0 = 0.01
The reference analog prototype filter transfer function is:
Step 5:
First it is necessary to compute the analog prototype filter cutoff frequency Ω
c.
The analog filter transfer function is obtained from the reference analog prototype filter transfer function using expression:
For this example, the analog filter transfer function is:
Step 6:
The transformation into a digital filter through bilinear transformation:
Generally, substituting the complex variable s into the expression for analog filter transfer function, the following is obtained:
where:
z
k are the zeros of analog filter transfer function; and
p
k are the poles of analog filter transfer function.
In this example, the general expression can be written in a simpler way:
A more condensed form of the previous expression is:
The result is the IIR filter transfer function.
Step 7:
Filter order is predetermined.
There is no need to additionally change it.
Filter realization:
Figure 3626 illustrates direct realization of designed IIR filter.
Figure 3626. Direct realization of digital IIR filter in this example
Figure 3627. Frequency characteristic of digital IIR filter in this example
3.6.1.4 Example 4
Step 1:
Type of filter – bandstop filter
Filter specification:

 Sampling frequency – fs=20KHz;

 Stopband cutoff frequencies – fc1=3000Hz, fc2=3300Hz; and

 Minimum stopband attenuation – 40dB.
Step 2:
Method  filter design using inverse Chebyshev reference analog prototype filter.
Step 3:
Filter order is predetermined, N = 4.
Step 4:
The inverse Chebyshev reference prototype filter transfer function has no zeros, only poles. These can be computed via expression:
When designing an IIR bandstop filter, the reference prototype filter order is half that of the required IIR filter order. In this example for N = 4, the order of reference prototype filter is 2, so the values of poles are:
The inverse Chebyshev filter transfer function is expressed as:
In this example, the value of constant H
0 is:
H0 = 0.01
The reference analog prototype filter transfer function is:
Step 5:
First it is necessary to compute the analog prototype filter cutoff frequency Ω
c.
The analog filter transfer function is obtained from the reference analog prototype filter transfer function using expression:
In this example, the analog filter transfer function is:
Step 6:
The transformation into a digital filter through bilinear transformation:
Generally, substituting the complex variable s into the expression for analog filter transfer function, the following is obtained:
where:
z
k are the zeros of analog filter transfer function; and
p
k are the poles of analog filter transfer function.
In this expression, the general expression can be written in a simpler way:
A more condensed form of the previous expression is:
The result is the IIR filter transfer function.
Step 7:
Filter order is predetermined.
There is no need to additionally change it.
Filter realization:
Figure 3628 illustrates the direct realization of designed IIR filter.
Figure 3628. Direct realization of digital IIR filter in this example
Figure 3629. Frequency characteristic of digital IIR filter in this example