Exp.No:
Design of IIR filters
Date:
AIM:
To design analog and digital Butterworth and Chebyshev Type I Low pass and High pass IIR filter
using Bilinear Transformation.
SOFTWARE USED:
Python
THEORY:
A recursive IIR digital filter is an LTI system based on a difference equation of the form:
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The impulse response h(n) is infinite in length.
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A system described by this type of difference equation is called an IIR (Infinite Impulse Response) filter,
a recursive filter, or an autoregressive moving-average (ARMA) filter.
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IIR filters have poles as well as zeros. They will be unstable if not all the poles are inside |z| = 1.
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IIR filters cannot have linear-phase.
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Methods for IIR filter design are either more complicated or less flexible than FIR design methods.
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The implementation of IIR filters is more sensitive to finite precision effects than FIR filters are.
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The advantage of IIR filters over FIR ones is that for the same filter complexity (number of filter
parameters) the magnitude response of an IIR filter can be significantly better than that of an FIR one.
The output y(n) of the filter can be written as
As the impulse response is infinite, the convolutional sum is an infinite sum. The transfer function of the system
H(z) can be written in term of the impulse response as
A digital filter, H(ejω) with infinite impulse response (IIR), can be designed by first transforming it into a
prototype analog filter H c (ejω) and then design this analog filter using a standard procedure. Once the analog
filter is properly designed, it is then mapped back to the discrete-time domain to obtain a digital filter that meets
the specifications. The commonly used analog filters are Butterworth filters (no ripples) and Chebychev filters
(ripples in the passband or in the stopband).
BUTTERWORTH FILTER
The magnitude square function of the Butterworth low pass filter is given by
The order N of the filter is calculated using
The Poles located in the Left Half Plane of the s-plane is calculated by
CHEBYSHEV FILTER
The order N of the filter is calculated using
Steps to design a digital filter using Impulse Invariance method:
● For the given specifications, find Ha (s), the transfer function of an analog filter.
● Select the sampling rate of the digital filter, T seconds per sample.
● Express the analog filter transfer function as the sum of single-pole filters.
●
Compute the z-transform of the digital filter by using the formula
● For high sampling rates use
o
Steps to design a digital filter using Bilinear Transform technique:
● From the given specifications, compute the prewarping analog frequencies using the formula
● Using the analog frequencies, compute H(s) of the analog filter.
● Select the sampling rate of the digital filter, T seconds per sample.
● Substitute
into the transfer function H(s)
Exercise Problems
1. Apply Bilinear transformation to design a Butterworth Low pass filter for the following specification:
Assume T=1sec
2. Apply Bilinear transformation to design a Chebyshev Type I digital Low pass filter with following
specifications:
αp = 0.2 db, ωp =60π rad/sec
αs =0.89 db, ωs= 150π rad/sec
Assume T=1sec
Result:
The analog and digital Butterworth Low pass IIR filter is designed using
Transformation.
Bilinear