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UNIT II INFINITE IMPULSE RESPONSE FILTERS

INTRODUCTION

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To remove or to reduce strength of unwanted signal like noise and to improve the

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quality of required signal filtering process is used. To use the channel full bandwidth
we mix up two or more signals on transmission side and on receiver side we would
like to separate it out in efficient way.

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Hence filters are used. Thus the digital filters are mostly used in

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1. Removal of undesirable noise from the desired signals

2.
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Equalization of communication channels

3. Signal detection in radar, sonar and communication

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4. Performing spectral analysis of signals.

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Analog and digital filters

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In signal processing, the function of a filter is to remove unwanted parts of the signal,
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such as random noise, or to extract useful parts of the signal, such as the components
lying within a certain frequency range.

The following block diagram illustrates the basic idea.

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There are two main kinds of filter, analog and digital. They are quite different in
their physical makeup and in how they work.

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An analog filter uses analog electronic circuits made up from components such as

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resistors, capacitors and op amps to produce the required filtering effect. Such filter
circuits are widely used in such applications as noise reduction, video signal
enhancement, graphic equalizers in hi-fi systems, and many other areas.
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In analog filters the signal being filtered is an electrical voltage or current which is
the direct analogue of the physical quantity (e.g. a sound or video signal or
transducer output) involved.
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A digital filter uses a digital processor to perform numerical calculations on sampled
values of the signal. The processor may be a general-purpose computer such as a
PC, or a specialized DSP (Digital Signal Processor) chip.
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The analog input signal must first be sampled and digitized using an ADC (analog
to digital converter). The resulting binary numbers, representing successive sampled
values of the input signal, are transferred to the processor, which carries out
numerical calculations on them. These calculations typically involve multiplying the
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input values by constants and adding the products together. If necessary, the results
of these calculations, which now represent sampled values of the filtered signal, are
output through a DAC (digital to analog converter) to convert the signal back to
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analog form.
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In a digital filter, the signal is represented by a sequence of numbers, rather than a

voltage or current.

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The following diagram shows the basic setup of such a system.

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BASIC BLOCK DIAGRAM OF DIGITAL FILTERS
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a. Samplers are used for converting continuous time signal into a discrete time
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signal by taking samples of the continuous time signal at discrete time instants.
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b. The Quantizer are used for converting a discrete time continuous amplitude
signal into a digital signal by expressing each sample value as a finite number of
digits.
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c. In the encoding operation, the quantization sample value is converted to the

binary equivalent of that quantization level.
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d. The digital filters are the discrete time systems used for filtering of sequences.

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e. These digital filters performs the frequency related operations such as low pass,

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high pass, band pass and band reject etc. These digital Filters are designed with

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digital hardware and software and are represented by difference equation.

DIFFERENCE BETWEEN ANALOG FILTER AND DIGITAL FILTER

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Analog Filter
1 Analog filters are used for filtering analog signals.
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2 Analog filters are designed with various components like resistor, inductor and
capacitor
3 Analog filters less accurate & because of component tolerance of active
components & more sensitive to environmental changes.

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4 Less flexible
5 Filter representation is in terms of system components.
6 An analog filter can only be changed by redesigning the filter circuit.

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Digital Filter
1 Digital filters are used for filtering digital sequences.
2 Digital Filters are designed with digital hardware like FF, counters shift registers,
ALU and software s like C or assembly language.

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3 Digital filters are less sensitive to the environmental changes, noise and

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disturbances. Thus periodic calibration can be avoided. Also they are extremely
stable.
4 These are most flexible as software programs & control programs can be easily
modified. Several input signals can be filtered by one digital filter.
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5 Digital filters are represented by the difference equation.
6 A digital filter is programmable, i.e. its operation is determined by a program
stored in the processor's memory. This means the digital filter can easily be changed
without affecting the circuitry (hardware).
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FILTER TYPES AND IDEAL FILTER CHARACTERISTIC

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Filters are usually classified according to their frequency-domain characteristic as

lowpass, highpass, bandpass and bandstop filters.
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Lowpass Filter
A lowpass filter is made up of a passband and a stopband, where the lower
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frequencies Of the input signal are passed through while the higher frequencies are
attenuated.
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Highpass Filter

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A highpass filter is made up of a stopband and a passband where the lower
frequencies of the input signal are attenuated while the higher frequencies are
passed. ee
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Bandpass Filter
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A bandpass filter is made up of two stopbands and one passband so that the lower
and higher frequencies of the input signal are attenuated while the intervening
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frequencies are passed.

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Bandstop Filter
A bandstop filter is made up of two passbands and one stopband so that the lower
and higher frequencies of the input signal are passed while the intervening
frequencies are attenuated. An idealized bandstop filter frequency response has the
following ee
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Multipass Filter
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A multipass filter begins with a stopband followed by more than one passband. By
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default, a multipass filter in Digital Filter Designer consists of three passbands and
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four stopbands. The frequencies of the input signal at the stopbands are attenuated
while those at the passbands are passed.

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Multistop Filter
A multistop filter begins with a passband followed by more than one stopband. By
default, a multistop filter in Digital Filter Designer consists of three passbands and
two stopbands.

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All Pass Filter

An all pass filter is defined as a system that has a constant magnitude response for

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all frequencies.

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|H(ω)| = 1 for 0 ≤ ω < ∏
The simplest example of an all pass filter is a pure delay system with system function
H(z) = Z-k. This is a low pass filter that has a linear phase characteristic.
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All Pass filters find application as phase equalizers. When placed in cascade with a
system that has an undesired phase response, a phase equalizers is designed to

compensate for the poor phase characteristic of the system and therefore to produce
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an overall linear phase response.

IDEAL FILTER CHARACTERISTIC

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1. Ideal filters have a constant gain (usually taken as unity gain) passband
characteristic and zero gain in their stop band.
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2. Ideal filters have a linear phase characteristic within their passband.

3. Ideal filters also have constant magnitude characteristic.
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4. Ideal filters are physically unrealizable.

TYPES OF DIGITAL FILTER
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Digital filters are of two types. Finite Impulse Response Digital Filter & Infinite
Impulse Response Digital Filter

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DIFFERENCE BETWEEN FIR FILTER AND IIR FILTER

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FIR Digital Filter

1. FIR system has finite duration unit sample response. i.e h(n) = 0 for n<0 and n ≥
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M Thus the unit sample response exists for the duration from 0 to M-1.
2. FIR systems are non recursive. Thus output of FIR filter depends upon present
and past inputs.

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3. Difference equation of the LSI system for FIR filters becomes

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4. FIR systems has limited or finite memory requirements.

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5. FIR filters are always stable
6. FIR filters can have an exactly linear phase response so that no phase distortion is
introduced in the signal by the filter.

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7. The effect of using finite word length to implement filter, noise and quantization

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errors are less severe in FIR than in IIR.
8. All zero filters
9. FIR filters are generally used if no phasedistortion is desired.
Example:
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System described by
Y(n) = 0.5 x(n) + 0.5 x(n-1) is FIR filter.
h(n)={0.5,0.5}
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IIR Digital Filter

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1. IIR system has infinite duration unit sample response. i. e h(n) = 0 for n<0 Thus
the unit sample response exists for the duration from 0 to ∞.
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2. IIR systems are recursive. Thus they use feedback. Thus output of IIR filter
depends upon present and past inputs as well as past outputs
3. Difference equation of the LSI system for IIR filters becomes
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4. IIR system requires infinite memory.

5. Stability cannot be always guaranteed.

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6. IIR filter is usually more efficient design in terms of computation time and
memory requirements. IIR systems usually requires less processing time and storage
as compared with FIR.
7. Analogue filters can be easily and readily transformed into equivalent IIR digital
filter. But same is not possible in FIR because that have no analogue counterpart.

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8. Poles as well as zeros are present.
9. IIR filters are generally used if sharp cutoff and high throughput is required.
Example:
System described by

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Y(n) = y(n-1) + x(n) is IIR filter.
h(n)=an u(n) for n≥0

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STRUCTURES FOR FIR SYSTEMS

FIR Systems are represented in four different ways

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1. Direct Form Structures

2. Cascade Form Structure
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3. Frequency-Sampling Structures
4. Lattice structures.
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DIRECT FORM STRUCTURE OF FIR SYSTEM

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The convolution of h(n) and x(n) for FIR systems can be written
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The above equation can be expanded as,

Y(n)= h(0) x(n) + h(1) x(n-1) + h(2) x(n-2) + …………… + h(M-1) x(n-M+1) (2)

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Implementation of direct form structure of FIR filter is based upon the above
equation.

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1. There are M-1 unit delay blocks. One unit delay block requires one memory
location. ee
Hence direct form structure requires M-1 memory locations.
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2. The multiplication of h(k) and x(n-k) is performed for 0 to M-1 terms. Hence M
multiplications and M-1 additions are required.

3. Direct form structure is often called as transversal or tapped delay line filter.
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CASCADE FORM STRUCTURE OF FIR SYSTEM

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In cascade form, stages are cascaded (connected) in series. The output of one system
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is input to another. Thus total K number of stages are cascaded. The total system
function 'H' is given by
H= H1(z) . H2(z)……………………. Hk(z) (1)
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H= Y1(z)/X1(z). Y2(z)/X2(z). ……………Yk(z)/Xk(z) (2)

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Each H1(z), H2(z)… etc is a second order section and it is realized by the direct form

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as shown in below figure.
System function for FIR systems

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Expanding the above terms we have
H(z)= H1(z) . H2(z)……………………. Hk(z)
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STRUCTURES FOR IIR SYSTEMS

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IIR Systems are represented in four different ways

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1. Direct Form Structures Form I and Form II

2. Cascade Form Structure
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3. Parallel Form Structure

4. Lattice and Lattice-Ladder structure.
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1.
DIRECT FORM STRUCTURE FOR IIR SYSTEMS

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IIR systems can be described by a generalized equations as

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Overall IIR system can be realized as cascade of two function H1(z) and H2(z). Here
H1(z) represents zeros of H(z) and H2(z) represents all poles of H(z).
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FIG - DIRECT FORM I REALIZATION OF IIR SYSTEM

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1. Direct form I realization of H(z) can be obtained by cascading the

realization of H1(z) which is all zero system first and then H2(z) which is
all pole system.

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2. There are M+N-1 unit delay blocks. One unit delay block requires one

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memory location. Hence direct form structure requires M+N-1 memory
locations.

3. Direct Form I realization requires M+N+1 number of multiplications and

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M+N number of additions and M+N+1 number of memory locations.

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DIRECT FORM - II

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1. Direct form realization of H(z) can be obtained by cascading the realization of
H1(z) which is all pole system and H2(z) which is all zero system.

2. Two delay elements of all pole and all zero system can be merged into single
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delay element.

3. Direct Form II structure has reduced memory requirement compared to Direct

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form I structure. Hence it is called canonic form.

4. The direct form II requires same number of multiplications(M+N+1) and

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additions (M+N) as that of direct form I.

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CASCADE FORM STRUCTURE FOR IIR SYSTEMS

In cascade form, stages are cascaded (connected) in series. The output of one system
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is input to another. Thus total K number of stages are cascaded. The total system
function 'H' is given by
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H= H1(z) . H2(z)……………………. Hk(z) (1)

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H= Y1(z)/X1(z). Y2(z)/X2(z). ……………Yk(z)/Xk(z) (2)

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Each H1(z), H2(z)… etc is a second order section and it is realized by the direct form

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as shown in below figure.
System function for IIR systems
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Thus Direct form of second order IIR system is shown as

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PARALLEL FORM STRUCTURE FOR IIR SYSTEMS

System function for IIR systems is given as

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The above system function can be expanded in partial fraction as follows

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H(z) = C + H1(z) + H2(z)…………………….+ Hk(z) (3)

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Where C is constant and Hk(z) is given as

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IIR FILTER DESIGN

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IMPULSE INVARIANCE

BILINEAR TRANSFORMATION

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BUTTERWORTH APPROXIMATION

IIR FILTER DESIGN - IMPULSE INVARIANCE METHOD

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Impulse Invariance Method is simplest method used for designing IIR Filters.
Important Features of this Method are
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1. In impulse variance method, Analog filters are converted into digital filter just by
replacing unit sample response of the digital filter by the sampled version of impulse
response of analog filter. Sampled signal is obtained by putting t=nT hence
h(n) = ha(nT) n=0,1,2. ………….
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where h(n) is the unit sample response of digital filter and T is sampling interval.

2. But the main disadvantage of this method is that it does not correspond to simple
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algebraic mapping of S plane to the Z plane. Thus the mapping from

analog frequency to digital frequency is many to one. The segments
(2k-1)∏/T ≤ Ω ≤ (2k+1) ∏/T of j Ω axis are all mapped on the unit circle ∏≤ω≤∏.
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This takes place because of sampling.

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3. Frequency aliasing is second disadvantage in this method. Because of frequency

aliasing, the frequency response of the resulting digital filter will not be identical to
the original analog frequency response.
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4. Because of these factors, its application is limited to design low frequency filters
like LPF or a limited class of band pass filters.

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RELATIONSHIP BETWEEN Z PLANE AND S PLANE

Z is represented as rejω in polar form and relationship between Z plane and S plane
is given as Z=eST where s= σ + j Ω.

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Here we have three condition
1. If σ = 0 then r=1

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2. If σ < 0 then 0 < r < 1
3. If σ > 0 then r> 1

Thus
1.
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Left side of s-plane is mapped inside the unit circle.
2. Right side of s-plane is mapped outside the unit circle.
3. jΩ axis is in s-plane is mapped on the unit circle.
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CONVERSION OF ANALOG FILTER INTO DIGITAL FILTER

Let the system function of analog filter is n

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Ha(s)= ΣK+1 Ck / s-pk (1) k=1

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where pk are the poles of the analog filter and ck are the coefficients of partial
fraction expansion. The impulse response of the analog filter ha(t) is obtained by
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inverse Laplace transform and given as

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The unit sample response of the digital filter is obtained by uniform sampling of
ha(t). h(n) = ha(nT)
n=0,1,2. ………….

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System function of digital filter H(z) is obtained by Z transform of h(n).

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Using the standard relation and comparing equation (1) and (4) system function of

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digital filter is given as

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STANDARD RELATIONS IN IIR DESIGN
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EXAMPLES - IMPULSE INVARIANCE METHOD

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IIR FILTER DESIGN - BILINEAR TRANSFORMATION METHOD (BZT)

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The method of filter design by impulse invariance suffers from aliasing. Hence in
order to overcome this drawback Bilinear transformation method is designed. In
analogue domain frequency axis is an infinitely long straight line while sampled data
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z plane it is unit circle radius. The bilinear transformation is the method of squashing
the infinite straight analog frequency axis so that it becomes finite.

Important Features of Bilinear Transform Method are

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1. Bilinear transformation method (BZT) is a mapping from analog S plane to digital

Z plane. This conversion maps analog poles to digital poles and analog zeros to
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digital zeros. Thus all poles and zeros are mapped.

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2. This transformation is basically based on a numerical integration techniques used

to simulate an integrator of analog filter.
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3. There is one to one correspondence between continuous time and discrete time
frequency points. Entire range in Ω is mapped only once into the range -∏≤ω≤∏.
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4. Frequency relationship is non-linear. Frequency warping or

frequency compression is due to non-linearity. Frequency warping means amplitude

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response of digital filter is expanded at the lower frequencies and compressed at the
higher frequencies in comparison of the analog filter.

5. But the main disadvantage of frequency warping is that it does change the shape

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of the desired filter frequency response. In particular, it changes the shape of the

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transition bands.

CONVERSION OF ANALOG FILTER INTO DIGITAL FILTER

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Z is represented as rejω in polar form and relationship between Z plane and S plane

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in BZT method is given as

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Comparing the above equation with S= σ + j Ω. We have

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Here we have three condition

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1. If σ < 0 then 0 < r < 1

2. If σ > 0 then r > 1
3. If σ = 0 then r=1

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When r =1

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The above equations shows that in BZT frequency relationship is non-linear. The

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frequency relationship is plotted as

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FIG - MAPPING BETWEEN FREQUENCY VARIABLE ω AND Ω IN BZT

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METHOD.
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DIFFERENCE - IMPULSE INVARIANCE Vs BILINEAR

TRANSFORMATION
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Impulse Invariance
1. In this method IIR filters are designed having a unit sample response h(n) that is
sampled version of the impulse response of the analog filter.
2. In this method small value of T is selected to minimize the effect of aliasing.
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3. They are generally used for low frequencies like design of IIR LPF and a limited
class of bandpass filter
4. Frequency relationship is linear.
5. All poles are mapped from the s plane to the z plane by the relationship Zk= epkT.
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But the zeros in two domain does not satisfy the same relationship.
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Bilinear Transformation
1. This method of IIR filters design is based on the trapezoidal formula for numerical
integration.
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2. The bilinear transformation is a conformal mapping that transforms the j Ω axis

into the unit circle in the z plane only once, thus avoiding aliasing of frequency
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components.
3. For designing of LPF, HPF and almost all types of Band pass and band stop filters
this method is used.

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4. Frequency relationship is non-linear. Frequency warping or frequency

compression is due to non-linearity.
5. All poles and zeros are mapped.

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LPF AND HPF ANALOG BUTTERWORTH FILTER TRANSFER
FUNCTION

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METHOD FOR DESIGNING DIGITAL FILTERS USING BZT
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step 1. Find out the value of ωc*,.

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step 2. Find out the value of frequency scaled analog transfer function
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Normalized analog transfer function is frequency scaled by replacing s by s/ωp *.

step 3. Convert into digital filter

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Apply BZT. i.e Replace s by the ((z-1)/(z+1)). And find out the desired transfer
function of digital function.

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Example:

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Q) Design first order high pass butterworth filter whose cutoff frequency is 1 kHz at
sampling frequency of 104 sps. Use BZT Method

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Step 1. To find out the cutoff frequency

ωc = 2∏f
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= 2000 rad/sec

Step 2. To find the prewarp frequency

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ωc* = tan (ωc Ts/2) = tan(∏/10)

Step 3. Scaling of the transfer function

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For First order HPF transfer function H(s) = s/(s+1) Scaled transfer function H*(s)
= H(s) |s=s/ωc*
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H*(s)= s/(s + 0.325)

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Step 4. Find out the digital filter transfer function. Replace s by (z-1)/(z+1)
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Q) Design second order low pass butterworth filter whose cutoff frequency is 1 kHz
at sampling frequency of 104 sps.

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Q) First order low pass butterworth filter whose bandwidth is known to be 1 rad/sec
. Use BZT method to design digital filter of 20 Hz bandwidth at sampling frequency

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60 sps.

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Q) Second order low pass butterworth filter whose bandwidth is known to be 1
rad/sec . Use BZT method to obtain transfer function H(z) of digital filter of 3 DB
cutoff frequency of 150 Hz and sampling frequency 1.28 kHz.
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Q) The transfer function is given as s2+1 / s 2+s+1 The function is for Notch filter
with frequency 1 rad/sec. Design digital Notch filter with the following specification
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(1) Notch Frequency= 60 Hz

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(2) Sampling frequency = 960 sps.

BUTTERWORTH FILTER APPROXIMATION
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The filter passes all frequencies below Ωc. This is called passband of the filter. Also
the filter blocks all the frequencies above Ωc. This is called stopband of the filter.
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Ωc is called cutoff frequency or critical frequency.

No Practical filters can provide the ideal characteristic. Hence approximation of the
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ideal characteristic are used. Such approximations are standard and used for filter
design.

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Such three approximations are regularly used.

a) Butterworth Filter Approximation
b) Chebyshev Filter Approximation
c) Elliptic Filter Approximation

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Butterworth filters are defined by the property that the magnitude response is
maximally flat in the passband.

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This equation gives the pole position of H(s) and H(-s).

FREQUENCY RESPONSE CHARACTERISTIC

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The frequency response characteristic of |Ha(Ω)|2 is as shown. As the order of the

filter N increases, the butterworth filter characteristic is more close to the ideal
characteristic. Thus at higher orders like N=16 the butterworth filter characteristic
closely approximate ideal filter characteristic. Thus an infinite order filter (N ∞) is

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required to get ideal characteristic.

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Ap= attenuation in passband.
As= attenuation in stopband.
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Ωp = passband edge frequency
Ωs = stopband edge frequency
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Specification for the filter is

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To determine the poles and order of analog filter consider equalities.

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Q) Design a digital filter using a butterworth approximation by using impulse

invariance.
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Filter Type - Low Pass Filter
Ap - 0.89125
As - 0.17783
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Ωp - 0.2∏
Ωs - 0.3∏
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Step 1) To convert specification to equivalent analog filter.

(In impulse invariance method frequency relationship is given as ω= Ω T while in

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Bilinear transformation method frequency relationship is given as Ω= (2/T) tan (ω/2)

If Ts is not specified consider as 1)
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|Ha(Ω)| ≥ 0.89125 for Ω ≤ 0.2∏/T and |Ha(Ω)| ≤ 0.17783 for Ω ≥ 0.3∏/T.

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Step 2) To determine the order of the filter.

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N= 5.88

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1. Order of the filter should be integer.

2. Always go to nearest highest integer vale of N.

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Hence N=6

Step 3) To find out the cutoff frequency (-3DB frequency)

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cutoff frequency Ωc = 0.7032
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Step 4) To find out the poles of analog filter system function.
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For stable filter all poles lying on the left side of s plane is selected. Hence

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S1 = -0.182 + j 0.679 S1* = -0.182 - j 0.679

S2 = -0.497 + j 0.497 S2* = -0.497 - j 0.497
S3 = -0.679 + j 0.182 S3* = -0.679 - j 0.182

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Step 5) To determine the system function (Analog Filter)

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Step 6) To determine the system function (Digital Filter)

(In Bilinear transformation replace s by the term ((z-1)/(z+1)) and find out the
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transfer function of digital function)

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Step 7) Represent system function in cascade form or parallel form if asked.

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Q) Given for low pass butterworth filter

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Ap= -1 db at 0.2∏
As= -15 db at 0.3∏

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a) Calculate N and Pole location

b) Design digital filter using BZT method.

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Q) Obtain transfer function of a lowpass digital filter meeting specifications
Cutoff 0-60Hz
Stopband > 85Hz
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Stopband attenuation > 15 db
Sampling frequency= 256 Hz . use butterworth characteristic.

Q) Design second order low pass butterworth filter whose cutoff frequency is 1 kHz
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at sampling frequency of 104 sps. Use BZT and Butterworth approximation.
FREQUENCY TRANSFORMATION
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When the cutoff frequency Ωc of the low pass filter is equal to 1 then it is called
normalized filter. Frequency transformation techniques are used to generate High
pass filter, Bandpass and bandstop filter from the lowpass filter system function.
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FREQUENCY TRANSFORMATION (ANALOG FILTER)

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FREQUENCY TRANSFORMATION ((DIGITAL FILTER)
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Example:

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Q) Design high pass butterworth filter whose cutoff frequency is 30 Hz at sampling

frequency of 150 Hz. Use BZT and Frequency transformation.
Step 1. To find the prewarp cutoff frequency
ωc* = tan (ωcTs/2) = 0.7265

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Step 2. LPF to HPF transformation

For First order LPF transfer function H(s) = 1/(s+1) Scaled

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transfer function H*(s) = H(s) |s=ωc*/s H*(s)= s/(s + 0.7265)

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Step 3. Find out the digital filter transfer function. Replace s by (z-1)/(z+1)
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Q) Design second order band pass butterworth filter whose passband of 200 Hz and
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300 Hz and sampling frequency is 2000 Hz. Use BZT and Frequency transformation.
Q) Design second order band pass butterworth filter which meet following
specification Lower cutoff frequency = 210 Hz
Upper cutoff frequency = 330 Hz Sampling Frequency = 960 sps
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Use BZT and Frequency transformation.

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