chapter 3_ iir filters - digital filter design

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9/22/13 Chapter 3: IIR filters - Digital Filter Design - mikroElektronika

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TOC Chapter 1 Chapter 2 Chapter 3

Digital Filter Design

Chapter 3: Infinite Impulse Response (IIR) Filters

3.1. Introduction3.2. IIR filter design

3.3. Reference analog prototype filter

3.4. Analog prototype filter to analog filter transformation

3.5. Bilinear transformation

3.6. Examples

3.1 Introduction

IIR filters are digital filters with infinite impuls e response. Unlike FIR filters, they have the feedback (a recursive part of a filter) and are

known as recursive digital filters therefore.

Figure 3-1-1. Block d iagrams of FIR and IIR filters

For this reason IIR filters have much be tter frequency response than FIR filters of the sam e order. Unlike FIR filters, their phase

characteristic is not linear which can cause a problem to the systems which need phase linearity. For this reas on, it is not preferable to

use IIR filters in digital signal processing when the phase is of the essence.

Otherwise, when the linear phase characteristic is not important, the use of IIR filters is an excellent solution.

There is one problem known as a potential instability that is typical of IIR filters only. FIR filters do not have such a p roblem as they do not

have the feedback. For this reason, it is always necessary to check after the design process whether the resulting IIR filter is stable or not.

IIR filters can be designed using different methods. One of the most comm only used is via the reference analog prototype filter. This

method is the best for designing all standard types of filters such as low-pass , high-pass, band-pass and band-stop filters.

This book des cribes the des ign method us ing reference analog prototype filter. Figure 3-1-2 illustrates the block diagram of this m ethod.

Figure 3-1-2. Block diagram of design method using reference analog prototype filter

FIR filters can have linear phase characteristic, which is not typical of IIR filters. When it is necess ary to have linear phas e characteristic,

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FIR filters are the only available solution. In other cases when linear phase characteristic is not necess ary, such as speech s ignal

process ing, FIR filters are not good sol ution. IIR filters s hould be us ed instead. The resulting filter order is cons iderably lower for the sam e

frequency respons e.

The filter order determines the number of filter delay lines, i.e. number of input and output samples that should be s aved in order that the

next output sample can be computed. For instance, if the filter order is 10, it means that it is neces sary to save 10 input samples plus 10

output samples preceeding the current sample. All these 21 sam ples will affect the next output sample.

The IIR filter transfer function is a ratio of two po lynomials of com plex variable z-1. The numerator defines location of zeros, whereas the

denominator defines location of poles of the resulting IIR filter transfer function.

Figure 3-1-3. illustrates input and output signals of the system with non-linear phase characteristic.

Figure 3-1-3. The effect of non-linear phase characteristic

The system introduces phas e shift of 0 radians at frequency of , and radians at three times hi gher frequency. Input signal consis ts of

nature frequency and harmonics with the same am plitude at three times higher frequency. Figure on the left illustrates an input s ignal,

whereas Figure on the right illustrates an output signal. As seen, these two signals have different waveforms. Neither the power of the

signal nor am plitudes of particular harmonics have been changed, but the phase of the second harm onic.

Assum e that an input represents a speech s ignal where the phas e is not important. In this case s uch phas e dis torsion would be

negligable as the system satisfies the s tated requirements. Otherwise, if the phase is important, such a huge distorsion m ustnt be

allowed.

3.2 Infinite impulse response (IIR) filter design

The most comm only used IIR filter design method uses reference analog prototype filter. It is the best method to use when des igning

standard filters such as low-pass, high-pass, bandpass and band-stop filters.

The filter design process starts with s pecifications and requirem ents of the desirable IIR filter. A type of reference analog prototype filter to

be used is specified according to the s pecifications and after that everything is ready for analog p rototype filter desi gn.

The next step in the des ign process is s caling of the frequency range of analog prototype filter into desirable frequency range. This is how

an analog prototype filter is converted into an analog filter.

After the analog filter is designed, it is time to go through the last step in the dig ital IIR filter des ign process . It is conversion from ana log to

digital filter. The most popular and m ost comm only used converting method is bilinear transformation method. The resulting filter, obtained

in this way, is always stable. However, instability of the resulting filter, when bilinear transformation is used, may be caused only by the

finite word-length side-effect.

3.2.1 Basic concepts and IIR filter specification

First of all, it is neces say to learn the basic concepts that will be us ed further in this book. You should be aware that without being familiar

with these concepts, it is not poss ible to understand analyses and s ynthesis of digital filters.

Figure 3-2-1 illustrates a low-pass digital filter specification. The word specification refers to the frequency response specification.


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Figure 3-2-1a. Low-pass digital filter specification

Figure 3-2-1b. Low-pass digital filter specification

p normalized passband cut-off frequency;

s normalized stopband cut-off frequency;

1 maximum passband ripples;

2 minimum stopband attentuation;

passband attenuation parameter;

A stopband attenuation parameter;

ap maximum passband ripples [dB]; and

as minimum stopband attenuation [dB].

Frequency normalization can be expressed as follows:

where:

fs is the sampling frequency;

f is the frequency to normalize; and

is the normalized frequency.

Specifications for high-pass , band-pass and band-stop filters are defined almost the same way as those for low-pass filters. Figure 3-2-2

illustrates a high-pass filter specification, whereas Figure 3-2-3 illustrates a band-pass filter specification.


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Figure 3-2-2a. High-pass digital filter specification

Figure 3-2-2b. High-pass d igital filter sp ecification

Comparing these two Figures 3-2-1 and 3-2-2, it is obvious that low-pass and high-pas s filters have similar specifications. The sam e

parameters are defined in both cases with the difference that in the later case the pas sband is s ubstituded by the stopband and vice versa.

Figure 3-2-3 illustrates a band -pass specification.

Filter 3-2-3a. Band-pass digital filter specification

Figure 3-2-3b. Band-pass digital filter specification

Figure 3-2-4 illustrates band-s top digital filter specification


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Figure 3-2-4a. Band-stop d igital filter s pecification

Figure 3-2-4b. Band-stop digital filter specification

3.2.2 Z-transform

The Z-transform is performed upon dis crete-time s ignals. It converts a discrete timedomain s ignal into a complex frequency-domin

representation. It is very suitable for analyzing dis crete time-domain s ignals and systems as well. The z-transform is derived from the

Fourier discrete time-domain transformation and is considered the bas ic operation in digital filter design process .

The Z-transform is defined as :

where z is the complex number.

Example:

Assum e that samples of a discrete-tim e signal x(n) are known. It is necess ary to transform this s ignal wi th the z-transform and Fourie r

fransform.

x(n)={1,2,3,4,5,4,3,2,1} ; 0 n 8

z-transform is defined via express ion:

It becomes :

The last expression is the z-transform of the given si gnal.

The Fourier transformation can be found by rewriting the previous expression in terms of z as z=e^j. It becomes:


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Figure 3-2-5 illustrates the (frequency) spectrum of the given signal.

Figure 3-2-5. Frequency spectrum of the given signal

Comparing Z and Fourier transforms, it is easy to notice som e sim ilarities between them:

In polar coordinates, the complex number z may be expressed as follows:

The two last expressions lead us to the conclusion that Fourier transform is just a s pecial form of the z-transform forr=1.

In the z plane, the Fourier transform is represented as a unit circle, which can be seen in Figure 3-2-6 below.


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Figure 3-2-6. Fourier transform in the z plane

The z-transform of the transfer function is of great importance for IIR filters. The location of poles in the z plane is used for testing stability of

designed IIR filter. The poles of the IIR filter transfer function mus t be located within the unit circle in o rder that filter is stable.

Figure 3-2-7a illustrates zeros and poles of the transfer function of a s table IIR filter in the z plane.

Figure 3-2-7a Stable IIR filter

Transfer function zeros are denoted by small circles , whereas its poles are denoted by sm all crosses.


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Re Real axis

Im Imaginary axis

As seen in Figure 3-2-7, one trans fer function zero is located outs ide the unit circle. It doesn t cause any problem as the location of poles is

the only thing that matters. All four poles of transfer function are located within the unit circle, which guarantees tha stability of IIR filters.

According to the location of poles i n the z plane, it is easy to determine whether it refers to FIR or IIR filter. The poles of the FIR filter transfer

function are located at the origin. It is obviously not the case in Figure 3-2-7, which means that it refers to IIR, not FIR filter.

Also, Figure 3-2-7a clearly indicates interrelation between zeros and poles in the z plane . If a zero or a pole is located on the real axe in the

z plane, i.e. the imagi nary part is zero, then it is single. If either of them is not located on the real axis i n the z plane, then it has the

corresponding pair having the sam e real value and the sam e imagina ry value with the opposite s ign. In the z plane, it is illustrated as a

pair of zeros or poles whi ch are symmetric around the real axis. Such a pair is also called a complex-conjugated pair of zeros or poles.

Figure 3-2-7b illustrates the zeros and poles of the transfer function of an instable IIR filter in the z plane.

Figure 3-2-7b. Instable IIR filter

As seen from Figure 3-2-7b, two poles loca ted outside the unit circle make this IIR filter ins table. If bilinear transformation is used in the

filter des ign, the resulting filter is s table before the coefficient quantization starts. This quantization changes the location of zeros and poles

of the resulting IIR filter, which can cause one pole o r one pair of poles to be located outside the unit circle. The result of such a

quantization is a filter that is not stable.

3.2.3 Transfer function of discrete-time systems

The Z-transform is primarily used for finding the transfer function of linear discrete-time s ystems. When the transfer function is found, it is

necess ary to cons ider the zeros and po les of the transfer function in the z plane. The transfer function of discrete-time s ystems is defined

to be:

where:

bi are the feedforward filter coefficients (non-recursive part);

aj are the feedback filter coefficients (recursive part);

H0 is a constant;

qi are the zeros of the transfer function;


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pj are the poles of the transfer function;

B(z) is the transfer function of non-recursive part of the system; and

A(z) the transfer function of recursive part of the system (feedback).

The recursive part of the transfer function is actually a discrete-time system feedback. Unlike the FIR filters, the IIR filters have feedback

which enable them to have greater selectivity as well as nonlinearity of phase characteristic than FIR filters.

Figure 3-2-8. illustrates block diag ram of discrete-time system with feedback.

Figure 3-2-8 Discrete-time system with feedback

In the time dom ain, the discrete-time s ystem shown in Figure 3-2-8 can be expressed as follows:

OR

The later expression is more convenient for software IIR filters realization.

In the frequency domain, the discrete-time s ystem shown in Figure 3-2-8 can be expressed as the mu ltiplication of Z-transform input

signal X(z) and the transform function H(z):

The first way of representing discrete-time systems is s uitable for both software and hardware IIR filter implementation, whereas the

representation in the z domain is suitable for analyzes of des igned filters and synthesis itself (design proces s).

Example:

The transfer function of a 3th order IIR filter, designed us ing Chebyshev function is :

The following express ion describes the filtering process :

This process is als o known as convolution. Another express ion for convolusion that is m ore useful in practical applications is:


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After making s ubs titutions of impulse response coefficients , it becom es:

Using expression:

it is pos sible to find function for particular normalized frequency.

For example, when = 0.2:

The numerator is com puted first:

Then denominator:


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Figure 2-2-8 illus trates a hardware realization of this IIR filter.

Figure 3-2-9. Realization of IIR filter in th is examp le

The software realization would require two buffers each of minimum length 3. One buffer would be used for input samples and another

one for output sam ples. These are usually circular buffers whose length can be expressed as 2^n, which in this case m eans that the

circular buffer is 4 = 2^2 in length.

By complexity, the given IIR filter corresponds to a 6th order FIR filter. Selectivity and attenuation of this filter are much higher than those of

any 6th order FIR filter. The result of the feedback, which provides so high selectivity and attenuation, is a non-linear phas e characteristic.

3.2.4 Effects of the poles and zeros of the transfer function

The location of poles and zeros of the transfer function is very important for discrete-time system analyses and synthesis . According to their

location it is pos sible to test stability of a discrete-time s ystem, detect round-off errors made due to so ftware implem entation of a filter as

well as coefficient errors encountered during hardware implementation of a filter.

In order that a discrete-time s ystem is s table, all poles of the discrete-time system transfer function must be located within the unit circle,

as s hown in Figure 3-2-6. If this requirem ent is not satisfied, the system becom es uns table, which is very dangerous. The location of

zeroes doesnt affect the stabilty of discrete-time systems . Recalling that FIR flters do not have a feedback, which m akes them stable.

However, this doesnt apply on IIR filters. Therefore, it is preferable to use bilinear transformation becaus e it always m akes filter stable. In

this case, filter stability is ques tioned only due to coefficient quantization which is performed at the end of the des ign process.

It always happens due to software and hardware implem entation that an error in coefficients represen tation is produced. In software

implem entation, an error is triggered by the finite word-length effect, whereas in hardware implem entation, it ocurrs due to impossibility of

representing the coefficients w ith apsolute accuracy. The resu lt in both cases is that the actual value of coefficients differs from their value

obtained in design process . A direct result of such errors is deviation of the frequency of designed dis crete-time s ystem.

Deviation of frequency depends on the spacing between the zeros and poles of the FIR filter transfer function and the origin in the z plane.

The FIR filter coefficient error affects more the frequency respons e as the spacing between the zero and pole of the transfer function and

the origin narrows. This property is particularly typical of high-order filters because their zeros are very close each other. Besides , the pole

quantization, by rule, affects more frequency characteristic. Slight errors in coefficient representation m ay cause large frequency deviations.

Figure 3-2-9 illus trates the required and obtained frequency characteristic of an IIR filter. The finite word-length effect on the transform


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function of an IIR filter is clearly marked in this figure.

Figure 3-2-10. Deviation from required frequency characteristic

The frequency deviation shown in Figure 3-2-10 is basically s light deviation, even though it is very large at certain frequencies. The

minimum attenuation and the width of transition region o f the resulting IIR filter remain unchanged, so that such deviation is acceptable.

3.2.5 IIR filters design using bil inear transformation

The IIR filter design us ing bilinear transformation can be s plit into several steps:

The final objective of defining IIR filter specifications is to find the des irable norm alized cutoff frequencies (c, c1, c2), transition width,

maximum pas sband attenuation and minim um s topband attenuation. The type of analog prototype filter as well as the filter order will be

specified according to these parameters.

Now, it is time to specify the type of reference analog prototype filter. Be aware that every type has its good and bad sides. It is only

important that its characteristics can satisfy the given specifications. However, it is preferable to s pecify such a type of analog prototype

filter that can produce the lowest order IIR filter.

After this step , that is, when the type of analog proptotype filter is known, it is necess ary to specify or compute the filter order required for a

given set of specifications. The initial value of the filter order is roughly estimated and is changed after that depending on the obtained

characteristics and requirements.

When both type and o rder of analog prototype filter are known, it is poss ible to find its transfer function.

The transfer function of analog prototype filter depends on frequencies which are not s caled into the des irable range. For this reas on, it is

necess ary to perform s caling of the transfer function so that cut-off frequencies go into the des irable range. This operation is actually

conversion of reference analog prototype filter into analog filter with des irable characteristic.

Finally, the transfer function of the specified type of reference analog prototype filter is obtained by converting analog filter into digital one.

This book represents the most commonly used conversion known as bilinear transformation.

If the resulting filter doesnt satisfy the given specifications, or if the filter order can be less than actual one, it should be changed. IIR filters

have much greater selectivity and attenuation than FIR filters o f the same order. For this reason, it is preferable to increase or decrease the

filter order by 1. After changing the filter order, the entire IIR filter design process, i.e. computing of the transfer function of reference analog

prototype filter, scaling and obtaining analogue filters and conversion into digital filter, is repeated.

3.2.6 IIR filter realization

FIR filter transfer function can be expressed as :

where:

N is the filter order;

bk the coefficient of non-recursive part of IIR filter; and

ak the coefficient of recursive part (feedback) of IIR filter.

The coefficients bk and ak are of interest for IIR filter realization (both hardware and software). Figure 3-2-11 illus trates the b lock diagram of

1. Defining filter specification;

2. Specifying analog prototype filter;

3. Computing the filter order required for a given set of specifications and speci fied analog prototype filter;

4. Computing the transfer function of reference analog prototype filter;

5. Conversion into analog filter via scaling;

6. Conversion into digital filter via bilinear transformation; and

7. If the obtained filter doesnt sat isfy the given specifications or if it is possible to decrease the filter order, then it is

necessary to do it. The filter order can be increased or decreased according to needs and after that steps 4, 5 and

6 are repeated as many times as needed.


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IIR filter.

Figure 3-2-11. Block diagram of IIR filter

There are several types of IIR filter realization. This chapter covers direct, direct transpos e, direct canonic, direct transpose canonic and

cascade realizations. All of them a re very convenient and mos t commonly us ed for both hardware and s oftware IIR filter realization. Each of

them will be described in detail along with their advantages and dis advantages.

3.2.6.1 Direct realization

Direct realization of IIR filters s tarts with this express ion:

The first part of the expression refers to non-recursive part and the other refers to recursive part of IIR filter. In IIR filter direct realization,

these two parts are s eparately considered and realized.

The realization of non-recursive part of IIR filter is identical to the direct realization of FIR filter. Figure 3-2-12. illustrates the block diagram of

direct realization of non-recursive part of IIR filter.

Figure 3-2-12. Direct re alization of non-recurs ive part of IIR filter


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As seen from Figure 3-2-12 above, multiplication coefficients are identical to those of the trans fer function.

Realization of non-recursive part of IIR filter is similar to that of recursive part. Figure 3-2-13. illustrates the direct realization of the filter

recursive part.

Figure 3-2-13. Direct re alization of non-recurs ive part of IIR filter

As non-recurs ive and recurs ive part of IIR filter are separately realized, it doesn t matter which of them will be used firs t in filtering process.

Figures 3-2-14a and 3-2-14b illustrate block diagram s o f IIR filter realization when non-recursive part is used before and after recursive

part of IIR filter, respectively.


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Figure 3-2-14a. IIR filter direct realization, non-recursive part is used first

Figure 3-2-14b. IIR filter direct realization, recursive part is used first

This structure is als o known as a direct form I structure. As s een from Figures 3-2-14a and 3-2-14b, direct realization requires in total of 2N

delay lines, (2N+1) multiplications and 2N additions.

Direct realization is very convenient for software implementation and this is whe re it is m ost comm only used. Some of disadvantages of


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this realization are the greatest sensitivity to accuracy of realized coefficients (i.e. the largest finite word-length effect), and the greatest

complexity due to implem entation (i.e. needs m ost resources ).

On IIR filter software implementation with direct structure, it is necess ary to have two buffers with at least N+1 s amples , where N is the IIR

filter order. For their simplicity and effectiveness, mos t commonly used are the so called circular buffers the length of which can be

expressed as 2^k. The value of constant k is defined as a minim um value for which N 2^k is valid. Accordingly:

where the operator represents rounding down to a less value.

Figure 3-2-15. Circular buffe r of length 16 = 2^4

Since the buffer length is 16, location addressing in the circular buffer is pe rformed via module 16 operations:

Example:

A 6th order FIR filter is used in this example. It is necessary to design this filter us ing direct structure with circular bu ffer. The length of the

buffer needs to be 2^k.

The length of circular buffer is obtained from the following express ion:

It means that the minimum length of circular buffer is 2^3 = 8.

The contents of the buffer after receiving the first 10 s amples is s hown in the table 3-2-1. Input samples are denoted by x[n] and each

shaded cell repres ents changed location in buffer.

STEPADDR.

7

ADDR.

6

ADDR.

5

ADDR.

4

ADDR.

3

ADDR.

2

ADDR.

1

ADDR.

0

0

1 x[0]

2 x[1] x[0]

3 x[2] x[1] x[0]

4 x[3] x[2] x[1] x[0]

5 x[4] x[3] x[2] x[1] x[0]

6 x[5] x[4] x[3] x[2] x[1] x[0]

7 x[6] x[5] x[4] x[3] x[2] x[1] x[0]

8 x[7] x[6] x[5] x[4] x[3] x[2] x[1] x[0]

9 x[7] x[6] x[5] x[4] x[3] x[2] x[1] x[8]

10 x[7] x[6] x[5] x[4] x[3] x[2] x[9] x[8]

Table 2-2-2. Input circular buffer after receiving 10 samples

3.2.6.2 Direct transpose re alization

Direct transpose realization is sim ilar to direct realization. The only difference is in the pos ition of delay lines, i.e. buffer if it is about

software implementation. Here, it is also necessary to have two buffers of minim um length N+1, where N is the filter order.

Figures 3-2-16 and 3-2-17 illus trate the block diagram describing IIR filter direct transpos e realization structure of IIR filter.


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Figure 3-2-16. IIR filter direct transpose realization, non-recursive part is used first

Figure 3-2-17. IIR filter direct transpose realization, recursive part is used first

There are no s ignificant differences be tween direct and direct transpose realizations. Both structures have the sam e multiplication

coefficients. The only difference is in the position of delay lines. Simila r to direct realization s tructure, the direct transpose realization

structure uses 2N delay lines, (2N+1) multiplications and 2N additions.


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3.2.6.3 Direct Canonical Realization

Direct canonical realization structure has reduced numbe r of delay lines to the minimum , that is, N delay lines. This way, one of the main

disadvantages of direct and direct transpos e realization structures is eliminated. Recursive and non-recurs ive parts of IIR filter are not

considered separately, which causes implem entation to be more com plex than for direct realization s tructure. A good thing is that the

coefficients are the s ame as for direct realization.

Figure 3-2-18 illustrates the block diagram des cribing direct canonic realization structure of IIR filter.

Figure 3-2-18. Direct canonic realization structur e block diagram

Similarities between direct canonic s tructure block diagram and direct realization s tructure shown in Figure 3-2-14b are obvious. The

difference between realization structures shown in Figures 3-2-14b and 3-2-18 is that non-recursive and recurs ive part for direct canonic

realization s tructure cannot be treated s eparately, although it is easy to differentiate between them.

Direct canonic structure uses N delay elements, (2N+1) m ultipilications and 2N additions. Sensitivity to the accuracy of coefficients is the

sam e as for all previously described s tructures, which is the main disadvantage of this realization structure.

3.2.6.4 Direct transpose canonical realization

Direct transpos e canonical realization structure has reduced number of delay lines to the minimum o f N delay lines as well as reduced

number of adders to N+1. Recursive and nonrecursi ve parts of IIR filter are not considered s eparately, which causes implem entation to be

more com plex than for direct realization structure, but similar to direct canonical s tructure. A good thing is that the coefficients are the sam e

as for direct realization.

Figure 3-2-19 illustrates the block diagram des cribing direct transpose canonical realization structure of IIR filter.


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Figure 3-2-19. Direct trans pose canonic realization structur e block diagram

Similarities between direct transpose canonical structure block diagram and direct transpose realization structure shown in Figure 3-2-16

are obvious. The difference between realization structures shown in Figures 3-2-16 and 3-2-19 is that non-recursive and recurs ive part for

direct transpose canonical realization structure cannot be treated separately, although it is easy to differentiate between them.

Direct transpose canonic structure uses N delay elements, (2N+1) m ultipilication elements and N+1 adders. Sens itivity to the accuracy of

coefficients is the same as for all previously described s tructures, which is the main dis advantage of this realization structure.

3.2.6.5 Cascade Realization

Cascade realization s tructure is the mos t difficult to obtain from the transfer function (comparing to other realization s tructures given in this

book). It is very convenient for its m odular s tructure and less sens itivity to the accuracy of non-recursive and recurs ive coefficients

realization. On cascade IIR filter realization, a filter is divided into several, mutually independent sections of the first or s econd order.

Individual sections are m ostly realized in di rect canonical or direct transpose canonical s tructure.

Since the sections are mutually independent after design process , the finite word-length effect on the accuracy of coefficients, m odulation

of frequency respons e and IIR filter stability are sepa rately examined for each section. The analyse is simp lified this way.

The IIR filter transfer function is expressed as :

where:

bi are the coefficients of transfer function numerator (non-recursive part);

aj are the coefficients of transfer function denominator (recursive part);

H0 is a constant;

qi are the zeros of the transfer function;

pj are the poles of the transfer function;

B(z) is the transfer function of non-recursive part;

A(z) is the transfer function of recursive part (feedback); and

M is the number of sections in cascade realization structure.

Cascade realization requires the given expression to be factorized so that the transfer function is expressed as follows:


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where:

a[i, k] are the coefficients of recursive part of the i-th IIR filter section;

b[i, k] are the coefficients of non-recursive part of the i-th IIR filter section.

Individual sections are of the first or s econd order. Direct transpose canonical s tructure is mos t frequently used in realization. Figure 3-2-

20 illus trates a first-order section.

Figure 3-2-20. First-order section

Figure 3-2-21 illustrates a second-order section.

Figure 3-2-21. Second-order section

The use of direct transpose realization structure reduces necess ary number of delay lines and adders as well . Filter dividing in

independent sections reduces the sens itivity to the accuracy of quantization coefficients and simp lifies analysing the stability of the

resulting filter. Besides, the poss ibility that IIR filter becomes instable after quantization is drastically reduced as the coefficients

quantization is performed after dividing filter in sections, so the changes of poles locations are sm aller, therefore.

Software realization requires M buffer of length 2 or 1 . Each section mus t have its own buffer for saving sam ples of intermediate signals .

Such complexity and needed factorization are two m ain dis advantages of this realization structure.

Figure 3-2-21 illustrates the block diagram des cribing cascade IIR filter structure.

Figure 3-2-22. Cascade IIR filter structur e

3.3 Reference Analog Prototype Filter

IIR filter design process starts with reference analog prototype filter. This book explains Butterworth, Chebyshev (Chebyshev I) and inverse

Chebyshev filter (Chebyshev II).

The transform function of analog filter Hsa(s) is expressed as:


21/45



where:


s is the complex frequency (s = + j); and

M N.

In order that a system described via expression above is stable, it is necessary that all poles (the square roots of polinomial Aa(s)) are

located in the left half of S plane. Figure 3-3-1 illustrates S plane.

Figure 3-3-1. S plane and re gion of stability

A low-pass filter is used for analog filter des ign. The conversion into the appropriate type of filter (high-pas s, band-pass or band-s top) is

performed by converting into analog filter, i.e. frequency axis scaling.

3.3.1 Butterworth analog filter

Low-pass Butterworth analog filters are filters whose frequency response is a monotonious descending function. They are also known as

maximally flat magnitude filters at the frequency of = 0, as the first 2N - 1 derivatives of the transfer function when = 0 are equal to

zero.

Butterworth filter is characterized by 3dB attenuation at the frequency of =1, no matter the filter order is. Figure 3-3-2 illustrates frequency

respons es for a few various parameters N (filter order).

Figure 3-3-2. Frequency response of Butterworth filter

Butterworth filter is defined via expression:

where:

is the frequency; and

N is the filter order.


22/45



Figure 3-3-3. IIR filter s pecification

Figure 3-3-3 illus trates IIR filter specification with parame ters of m ost interest for Butterworth filter.

To design Butterworth reference analog prototype filter, it is necessary to know the filter order. All poles of the resulting filter mus t be

located in the left half of the S plane, i.e. to the left of the imaginary axis.

Note:

In the z plane, filter is stable if all poles are located within the unit circle. In the s plane, filter is stable if all poles are located in the left half of

the s plane. Z-transform is used for digital filters, whereas Laplase transform (s plane) is used for analog filters. Even though these two types

of transformations are similar to some extent, they should not b e mixed up concerning the filter stability analyse.

When the filter order is known , it is easy to find its poles using express ion:

Butterworth poles are equally allocated (equidistantly) on the unit circle within the left half of the s plane. The location of poles for N=5 and

N=6 is shown in Figure 3-3-4.

Figure 3-3-4. Position of Butterw orth filter poles fo r N=5 and N=6

The transfer function of the Butterworth reference analog prototype filter is expressed as follows:

where:

Sk is the k-th pole of the Butterworth filter transfer function

For N=5, the transfer function is:

3.3.2 Chebyshev Analog Filter

Chebyshev analog low-pass filter of the first kind is a type of analog filter that has the least oscillation in frequency respons e in the entire

pass band. Therefore it is characterized by equal ripple in the passband and the stopband frequency respons e is m onotoniously

descending function.

Figure 3-3-5 illustrates frequency response for a 4th order band-s top Chebyshev reference analog filter.

Figure 3-3-5. Frequency response of Chebyshev analog filter

To design Chebyshev reference analog prototype filter, it is necessary to know the filter order.

Chebyshev analog filter is defined via expression:

where:

is the frequency;


is a parameter used to define maximum os cillations in the pass band frequency respons e; and

TN is the Chebyshev polynomial.

The Chebyshev polynomial TN() can be obtained via recursive relations:

If the filter order is known in advance, neither recursive relations nor express ion for the square of frequency respons e are neces sary. The

design process starts from the values of poles of a 1s t order Chebyshev reference analog filter.

The values of poles are expressed as :

where:

si is the i-th transfer function pole of analog prototype filter (complex value);

i is the pole; and

i is the imaginary pole.

where:

N is the filter order; and

i=1, 2, ..., N.

The value of parameter is obtained via expression:

Transfer function is expressed as :


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The value of A0 is found via express ion:


Figure 3-3-6. Location of the poles o f Chebyshev filter for N=5

3.3.3 Inverse Chebyshev Analog Filter

Inverse Chebyshev analog filter is also known as Chebyshev analog filter of the second kind. The frequency response o f this filter

monotoniously falls i n the passband and transition region. Similar to Butterworth filter, the frequency response is extremely flat function at

the frequency of = 0, as the first 2N - 1 derivatives of the transfer function for = 0 are equal to zero. In the stopband, inverse Chebyshev

filter has the least oscillation in the frequency response.

Figure 3-3-7 illustrates the frequency response for an inverse Chebyshev reference analog band-stop filter of the 4th order.

Figure 3-3-7. Freque ncy characteristic of inver se Che byshev analog filter

To design inverse Chebyshev reference analog pototype filter, it is necessary to know the filter order.

Inverse Chebyshev analog filter is defined via expression:

where: is the frequency;


is the parameter of maximum os cillation in the passband frequency response; and

TN is the Chebyshev polynomial.

The Chebyshev polynomial TN() can be obtained from recurs ive relation:

If the filter order is familiar in advance, neither these recursive relations nor expression for the square of frequency response are

necess ary. The design process s tarts from the values of poles of a 1st order inverse Chebyshev reference analog filter.

The poles of the transfer function of inverse Chebyshev analog filter are considered reciprocal poles of the transfer function of a 1s t order

Chebyshev analog filter.

The poles of a 1s t order Chebyshev analog filter are expressed as :

where:

si is the i-th pole of the transfer function of analog prototype filter (complex value);

i is the real pole; and

i is the imaginary pole.

where:

N is the filter order; and

i=1, 2, ..., N.

The poles of the transfer function of inverse Chebyshev analog filter are found via expression:

where:

si i s the pole of the transfer function of a 1s t order Chebyshev analog filter; and

s2i is the po le of the transfer function of inverse Chebyshev analog filter.

Transfer function is expressed as :

The coefficient k in numerator can be only an odd number. Table 3-3-1 provides a few examples of values ofk.

N M I N M A X V A L UES

5 1 5 1, 3, 5

6 1 5 1, 3, 5

7 1 7 1, 3, 5, 7

8 1 7 1, 3, 5, 7

Table 3-3-1. coefficient k in the transfer function numerator

The values k are found via express ion:


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The value H0 is found via express ion:


Figure 3-3-8. Location of poles and zeros of inverse Chebyshev filter for N=5

As seen from Figure 3-3-8 and express ion for k, the zeros of the transfer function are always com plex-conjugated values, which is not the

case wi th the poles o f the transfer function.

3.4 Analog prototype filter to analog filter transformation

All analog prototype filters, regardless of their type, have scaled frequency range so that the passband cut-off frequency amounts to = 1.

For this reason, it is neces sary to scale filter during the design process, so that passband and stopband cut-off frequencies have the

appropriate, predetermined values.

Reference analog prototype filter is als o a low-pas s filter so it requires to be transformed into the appropriate type of filter, i.e. high-pass ,

band-pass or band-stop filter, if needed.

Transformation from an analog prototype filter to appropriate analog filter is performed before transforming it in digital filter.

Frequency scaling depends on the type of analog filter being desi gned. Scaling is explained for low-pass , high-pass, band-pass and

band-stop filters.

All the results ob tained in this chapter are tes ted in the Filter designer tool program.

3.4.1 Low-pass filter

The transformation from a reference analog prototype filter to a low-pass analog filter is the simp lest type of transformation. Analog

prototype filter is a low-pass filter with the cut-off frequency of p=1. In this case, the transformation comes to a sim ple frequency scaling.

In the transform function, s\\c is used instead ofs, where c is a desirable cut-off frequency in the passband.

Generally, the transform function of the reference analog prototype filter can be expressed as follows:

where

H0 is a cos tant;

zk is the k-th zero of the transfer function of the reference analog prototype filter;

M is a number of zeros of the transfer function of the reference analog prototype filter;

pk is the k-th pole of the transfer function of the reference analog prototype filter; and

N is a number of poles of the transfer function of the reference analog prototype filter and the filter order as wel l.

By performing the transformation:

each expression within brackets in the transfer function numerator is transformed into:

and the whole numerator is transformed into:

Each bracket in denominator is transformed s imilarly:

and the entire denominator is transformed into:

By replacing numerator and denom inator by their transformed express ions, the transform function of the analog filter is obtained:

Example:

The transform function of the Butterworth reference analog prototype filter of the 3rd order is expressed as follows:

The transformation in a band-pass analog filter with the cut-off frequency c = 0.2929 in the passband is obtained via express ion:

Example:

The transform function of inverse Chebyshev reference analog prototype filter of the 3rd order is expressed as follows:


25/45



The transformation to a band-pass analog filter with the cut-off frequency c=0.3719 in the pass band is ob tained via expression:

3.4.2 High-pass filter

Analog prototype filter is a low-pas s filter with the cut-off frequency p = 1. Its transformation to a high-pass ana log filter can be split into

two steps. The first step refers to the transformation to a high-pas s analog filter, whereas the second one refers to frequency scaling. The

final objective is that passband cut-off frequency of the resulting high-pass analog filter amounts to c.

The transformation to a high-pas s ana log filter:

Scaling of frequency axis is performed by transformation:

These two transformations can be represented by one transformation:

Generally, the transform function of the reference analog prototype filter can be expressed as follows:

where:

H0 is a constant;





By performing the following transformation:

each bracket in the numerator of the transfer function is transformed into

and the entire numerator is transformed into:


and the entire denominator is transformed into:

Substituting these transformed expressions for numerator and denom inator, the transform function of the analog filter is obtained:

Example:

The transfer function of the Chebyshev reference analog prototype filter of the 3rd order is expressed as follows:

The transformation in a band-pas s analog filter with the cut-off frequency c = 0.3721 in the passband is expressed as:

The result is the transfer function of analog filter:

Example:

The transfer function of inverse Chebyshev reference analog prototype filter of the 2nd order is expressed as follows:

The transformation in a band-pass analog filter with the cut-off frequency c=0.1434 in the passband is expressed as:

The result is the transform function of analog filter:

3.4.3 Band-pass filter

The transformation in a band-pass analog filter is m ore complex than the transformation in a low-pass and high-pass analog filters. The

filter order is doubled by this transformation. This is why it is not poss ible to design an odd o rder band-pass filter.


26/45


27/45




3.4.4 Band-stop filter

The transformation in a band-stop analog filter is s imilar to the transformation in a bandpas s ana log filter. Similarly, the filter order is

doubled and the order of a band-s top digital filter cannot be an odd number.

When designing, the required filter order is divided by two. The resulting filter order is used for des igning a low-pas s reference analog

prototype filter. The transformation into a bandstop analog filter causes the filter order to double. This is how the required filter order is

obtained.

Example:

It is necessary to des ign a band-s top digital filter of the 10th order.

A low-pass reference ana log filter of the 5th order is des igned firs t.

The reference analog filter is trans formed in a band-s top analog filter. This transformation causes the filter order to double. The result is a

10th order filter.

The transformation in a band-s top analog filter:

The value of the constant 0 can be found via express ion:

where:

s1 is a lower cut-off frequency in the stopband (refer to figure 3-4-2); and

s2 is a higher cut-off frequency in the stopband (refer to figure 3-4-2).

Figure 3-4-2. Band-stop filter specification

The value of c, which is neces sary for normalization, is found via expression:

The transfer function of the reference analog prototype filter is transformed first into a band-stop analog filter, and normalized after that with

c.

Generally, the transfer function of the reference analog prototype filter can be expressed as follows:

where:H0 is a cons tant;





The first step refers to normalization with the frequency c:

The transformation is performed in the second step:

and each bracket in the transfer function numerator is transformed in:

and the entire numerator is transformed in:


and the entire denominator is transformed in:

Substituting the transformed express ions for numerator and denom inator, the transform function of analog filter is obtained:

Example:

The transfer function of inverse Chebyshev reference analog prototype filter of the 2nd order is expressed as :


28/45



The transformation into a band-stop analog filter with the cut-off frequency c=0.252.5727 in the pas sband is express ed as :


3.5 Bilinear transformation

Digital IIR filters are des igned us ing analog filters. After the frequency scaling and transformation into a des irable type of filter have been

performed, it is neces sary to transform the resulting analog filter into a digi tal one. It is done by transforming the analog filter transfer

function into a di gital one.

The transformation is supposed to:

This transformation also transforms s plane into z plane.

Analog filter is stab le if the poles o f the transfer function are located in the left half of s plane, whereas digital filter is s table if the poles are

located within the unit circle. For this reason, the transformation mus t provide that the left half of s plane coincides with the area within the

unit circle of z plane, as s hown in Figure 3-5-1.

Figure 3-5-1. Transformation of s p lane into z plane

One of most comm only used method of transforming analog filters into appropriate IIR filters is known as bilinear transformation. It is

defined via express ion:

Using the previous express ion, the transformation of the analog filter transform function into a digital one can be expressed as:

As seen, the transformation is performed by a simpl e change of variable s in the express ion for the transfer function of the resul ting analog

filter.

The analog filter transfer function can be express ed as:

where:

H0 is a constant. If s=0 then H(s)=H0 ;

zk is the zero of the analog filter transfer function; and

pk is the pole of the analog filter transfer function.

After transformation, the analog filter trans fer function is further transformed in to:

where:

Hoz is a constant of the digital IIR filter transfer function

Example:

The transfer function of a s econd-order high-pass analog filter (inverse Chebyshev, fc=2KHz, fs=44100Hz, 60dB) is expressed as :

It is neces sary to transform the given analog filter into the appropriate digital filter by bilinear transformation.

Using express ion for linear transformation:

we obtain:

where:

N=2

M=2

z1=j0.1014

z2=-j0.1014

p1=-2.267+j2.2692

p2=-2.267-j2.2692

1. faithfully approximate the frequency response of analog filter; and

2. provide that the resulting digital filter is guaranteed to be stable.


29/45



The result is the transfer function of a digital high-pass IIR filter. Realization structure is illus trated in Figure 3-5-2 below.

Figure 3-5-2. IIR filter realization

Digital filters designed via bilinear trasnformation are guaranteed to be s table. However, the accurate values of coefficients are obtained

immedi ately after the implementation of bilinear transformation. On filter realization, it is imposs ible to represent coefficients without an

error. In software digital filter realization (implem entation), the resulting coefficients are quantized, which also generates a certain error. Any

error made during the quantization of coefficients affects more or less the frequency respons e, which may further cause the s topband

attenuation to decrease.

The quantization effect on digital filter stability is much m ore dangerous . Special care is required when quantizing feedback coefficients asit causes the location of the digital IIR filter transfer function poles to change their location. It is very important to prevent the poles from

being located outside the unite circle. However, if it happens, the resulting IIR filter is not stable and is useles s therefore.

A disadvantage of the bilinear transform ation is a non-linear transform ation of the analog filter frequency axis into a digi tal one. When

designing, the cut-off frequencies are defined on the bas is of the given specifications and type of a filter. When transforming, these

frequencies have appropriate locations, which is not the case with the res t of the frequency axis. Such a non-linear transformation of

analog filter frequencies caus es the phase characteristic distorsion, so that it is not linear.

3.6 Examples

This chapter discusses various IIR filter design methods . The four standard types of filters are us ed here:

low-pass filter;

high-pass filter;

band-pass filter; and

band-stop filter.The design method used here is known as bilinear transformation.

The IIR filter design process can be split into several steps des cribed in Chapter 3.2.5 Designing IIR filters by b ilinear transformation.

These are:

Some s teps are skipped in some cas es. 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 des cribed through these steps in order to make it eas ier for you to observe

sim ilarities and differencies between various des ign methodes , analog prototype filters and desi gn of various types of filters as well.

Figure 3-6-1 illustrates the design steps along with input and output data for each of them.

Figure 3-6-1. Steps in de signing 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 Ha(s). Regardles s of the type of reference analog prototype filter in use, the transfer function is given by:

where:

H0 is a cons tant;

zk is the k-th zero of the reference analog prototype filter transfer function;

M is a number of zeros of the reference analog prototype filter transfer function;

pk is the k-th 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 low-pass filter. The next step is the transformation into an analog filter of appropriate type. The

expression us ed to transform the reference analog p rototype 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:

H0 is a cons tant;

zk is the k-th zero of the reference analog prototype filter transfer function;

1. Defining filter specifications;

2. Specifying the type of analog prototype filter;

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

4. Computing the transfer function of reference analog prototype filter;

5. Transformation into analog filter by range scaling;

6. Transformation into digital filter by bilinear transformation; and

7. If the resulting filter doesnt satis fy 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.


30/45



M is a number of zeros of the reference analog prototype filter transfer function;

pk is the k-th 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 pro totype filter and analog filter are very simila r. They differ only in the value of constant

H0, the values of the transfer function poles and zeros zk and pk as well as in the numbe r of transfer function zeros M. The filter order is the

sam e if the analog filter is a low-pass or high-pass filter, whereas it is different if the analog filter is a band-pass or band-stop 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 res ulting IIR filters in this chapter. All data are calculated with the accuracy of 4

decimal digits, which is s ufficient for mos t examples .

3.6.1 Filter design using Butterworth filter

3.6.1.1 Example 1

Step 1:

Type of filter low-pass filter

Filter specifications:

Filter order N=2;

Sampling frequency fs=20KHz;

Passband cut-off 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 com puted via expression:

As N = 2, the values of poles a re:

The reference analog prototype filter transfer function is :

Step 5:

First it is necessary to compute the analog prototype filter cut-off frequency 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 express ion for transfer function is simpler:

Step 6:

The transformation into a digital filter through bilinear transformation:

Generally, subs tituting the com plex variable s into the expression for analog filter transfer function, the following is obtained:

where:

zk are the zeros of analog filter transfer function; and

pk are the poles of analog filter transfer function.

This general expression can be w ritten in a sim pler way in this example:

A more condens ed form of the previous expression is :


31/45



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 3-6-2 illustrates the direct realization of designed IIR filter, whereas Figure 3-6-3 illustrates the frequency response of the filter

obtained using Filter Designer Tool.

Figure 3-6-2. Digital IIR filter dir ect re alization in this e xample

Figure 3-6-3. Digital IIR filter fr eque ncy characteris tic in this e xample

3.6.1.2 Example 2

Step 1:

Type of filter high-pass filter


Filter order N=3;

Sampling frequency fs=20KHz;

Passband cut-off 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:




Step 5:




Step 6:



where:





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Step 7:



Filter realization:

Figure 3-6-4 illus trates the direct realization of des igned IIR filter.



3.6.1.3 Example 3

Step 1:

Type of filter band-pass filter


Filter order N=4;

Sampling frequency fs=20KHz; andPassband cut-off frequency fc1=4KHz, fc2=6KHz.

Step 2:

Method - filter des ign us ing Butterworth reference analog prototype filter

Step 3:


Step 4:


When designing an IIR band-pass 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:


Step 5:




Step 6:


Generally, by subs tituting the complex variable s into the expression for analog filter transfer function, the following is obtained:

where:




33/45



In this example, the general express ion can be written in a s impler way:



Step 7:

Filter order is predetermined.


Filter realization:




3.6.1.4 Example 4

Step 1:

Type of filter band-stop filter


Filter order N=4;

Sampling frequency fs=20KHz; and

Passband cut-off frequency fc1=3000Hz, fc2=3300Hz.

Step 2:

Method - filter design using Butterworth reference analog prototype filter.

Step 3:


Step 4:


When designing an IIR band-stop 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:


Step 5:




Step 6:



where:




34/45



In this example, the general express ion can be written in a s impler way:



Step 7:



Filter realization:




3.6.2 Filter design using Chebyshev filter

3.6.2.1 Example 1

Step 1:

Type of filter low-pass filterFilter specifications:

Sampling frequency fs=44100Hz;

Passband cut-off frequency fc1=15KHz;

Stopband cut-off 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 pre-determined, so it is neces sary to choose an initial solution from which iterative method s tarts. The solution is

redefined progressively until som e pre-determined requirements are s atisfied. Lets as sum e that the initial filter order is 4.

Step 4:

The Chebyshev reference prototype filter transfer function has no zeros, only poles. These can be com puted via expression:


The Chebyshev filter transfer function is expressed as:

In this case, the value of cons tant A0 is :

A0 = 0.2457

so that the Chebyshev reference analog prototype filter transfer function is :

Step 5:



As the Chebyshev reference prototype filter has no zeros , the express ion for transfer function is simpler:

Step 6:


35/45





where:






Step 7:

By analyzing the resulting filter using Filter Designer Tool, it is obvious that the attenuation am ounting to 31.2dB approximately at the

frequency of 18KHZ is not sufficient. The frequency characteristic of the resulting digital filter is illustrated in Figure 3-6-10 below.

Figure 3-6-10. Freque ncy characteris tic of the r esulting IIR filter

It is necess ary to additionally redefine the filter order until the predefined requiremen ts are s atisfied. The filter order is incremented by 1

and is 5 therefore. All s teps s tarting with s tep 3 are iterated.

Step3:

The filter order is incremented in the s econd iteration. A new filter order is 5.

Step 4:



The Chebyhsev filter transfer function is expressed as:


A0 = - 0.1228

so the Chebyshev reference analog prototype filter transfer function is:

Step 5:




Step 6:



where:




36/45






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 3-6-11 below.

Figure 3-6-11. Freque ncy characteris tic of the r esulting IIR filter

It is not necessa ry 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 s ame, only it takes m ore time.

Filter realization:

Figure 3-6-12 illus trates the direct realization of des igned IIR filter, whereas Figure 3-6-13 illustrates the frequency characteristic of the filter

obtained using Filter Designer Tool.

Figure 3-6-12. Digital IIR filter dire ct re alization

Figure 3-6-13. Frequency characteristic of digital IIR filter

3.6.1.2 Example 2

Step 1:

Type of filter high-pass filter


Filter order N = 3;

Sampling frequency fs = 20KHz;

Passband cut-off frequency fc = 5KHz; and

Maximum passband attenuation ap = 1dB.

Step 2:

Method - filter design using Chebyshev reference analog prototype filter

Step 3:


Step 4:





A0 = - 0.4913


Step 5:





37/45



After subs titution of poles and c into express ion:

Step 6:



where:






Step 7:



Filter realization:


Figure 3-6-14. Direct r ealization of digital IIR filter in this example

Figure 3-6-15. Freque ncy characteris tic of digital IIR filter in this example

3.6.1.3 Example 3

Step 1:

Type of filter band-pass filter


Filter order N = 4;


Passband cut-off frequencies fc1 = 4KHz, fc2 = 6KHz; and


Step 2:

Method filter design using Chebyshev reference analog p rototype filter.

Step 3:


Step 4:


When designing an IIR band-pass filter, the reference prototype filter order is half that of the required IIR filter order. In this example for N =



In this example, the value of constant A0 is:

A0 = 0.9826



38/45



Step 5:




Step 6:



where:



In this expression, the general expression can be written in a sim pler way:



Step 7:



Filter realization:

Figure 3-6-16 illus trates direct realization of des igned IIR filter.

Filter 3-6-16. Direct r ealization of digital IIR filter in this example


3.6.1.4 Example 4

Step 1:

Type of filter band-stop filter


Filter order N = 4;


Passband cut-off frequencies fc1 = 3000Hz, fc2 = 3300Hz; and


Step 2:

Method - filter design using Chebyshev reference analog prototype filter.

Step 3:


Step 4:


When designing an IIR band-pass filter, the reference prototype filter order is half that of the required IIR filter order. In this example for N =




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In this example, the value of constant A0 is:

A0 = 0.9826


Step 5:




Step 6:



where:



In this expression, the general expression can be written in a sim pler way:



Step 7:



Filter realization


Figure 3-6-18. Direct r ealization 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 low-pass filter


Sampling frequency fs = 44100Hz;

Passband cut-off frequency fc1 = 15KHz;

Stopband cut-off 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 pre-determined, so it is neces sary to choose an initial solution from which iterative method s tarts. The solution is


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redefined progressively until som e pre-determined requirements are s atisfied. Lets as sum e that the initial filter order is 4.

Step 4:



The inverse Chebyhsev filter transfer function is expressed as :

In this example, the value of constant H0 is:

H0 = 0.01

so that the Chebyshev reference analog prototype filter transfer function is :

Step 5:


The analog

chapter 3_ iir filters - digital filter design

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