Infinite Impulse Response (IIR) Filters

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Infinite Impulse Response (IIR) Filters
Recursive Filters
with constant coefficients.
Advantages very selective filters with a few
parameters Disadvantages a) in general
nonlinear phase, b) can be unstable.
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Design Techniques discretization of analog
filters
Problem we need to map the derivative
operator s into the time shift operator
z, and make sure that the resulting system is
still stable.
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• Two major techniques
• Euler Approximation (easiest),
• Bilinear Transformation (best).

Euler Approximation of the differential operator
approximation of s
take the z-Transform of both sides
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Example take the analog filter with transfer
function and
discretize it with a sampling frequency
. By Eulers approximation
The filter is implemented by the difference
equation
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Problem with Euler Approximation it maps the
whole stable region of the s-plane into a subset
of the stable region in the z-plane
s-plane
z-plane
since
if Reslt0.
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Bilinear Transformation. It is based on the
relationship
Take the z-Transform of both sides
which yields the bilinear transformation
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Main Property of the Bilinear Transformation it
preserves the stability regions.
s-plane
z-plane
since
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Mapping of Frequency with the Bilinear
Transformation.
Magnitude
Phase
where
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See the meaning of this
it is a frequency mapping between analog
frequency and digital freqiency.
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Example we want to design a digital low pass
filter with a bandwith and a
sampling frequency .
Use the Bilinear Transformation.
Solution

• Step 1 specs in the digital freq. domain
• Step 2 specs of the analog filter to be
  digitized

or equivalently

• Step 3 design an analog low pass filter (more
  later) with a bandwith
  
• Step 4 apply Bilinear Transformation to obtain
  desired digital filter.
  

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Design of Analog Filters
Specifications
pass band
transition band
stop band
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Two Major Techniques Butterworth, Chebychev
Butterworth
Specify from passband, determine
N from stopband
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Poles of Butterworth Filter
which yields the poles as solutions
N2
and choose the N poles in the stable region.


poles


s-plane
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Example design a low pass filter, Butterworth,
with 3dB bandwith of 500Hz and 40dB attenuation
at 1000Hz. Solution
solve for N from the expression
poles at
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Chebychev Filters. Based on Chebychev Polynomials
Property of Chenychev Polynomials within the
interval Chebychev
polynomials have least maximum deviation from 0
compared to polynomials of the same degree and
same highest order coefficient
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Why? Suppose there exists
with smaller deviation then
But
has
degree 2 and it cannot have
three roots!!!
So you cannot find a P(x) which does better (in
terms of deviation from 0) then the Chebychev
polynomial.
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Chebychev Filter
Since (easy to show from the
definition), then
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Design of Chebychev Filters Formulas are tedious
to derive. Just give the results
Given the passband, and
which determines the ripple in the
passband, compute the poles from the formulae
s-plane
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• Example design a Chebychev low pass filter with
  the following specs
• passband with a 1dB
  ripple,
• stopband with
  attenuation of at least 40dB.

Step 1 determine . The passband
frequency For 1dB ripple,
Step 2 determine the order N. Use the formula
with , to obtain
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Frequency Transformations We can design high
pass, bandpass, bandstop filters from
transformations of low pass filters.
Low Pass to High Pass
same value at
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Low Pass to Band Pass
The tranformation
maps
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Low Pass to Band Stop
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How to make the transformation Consider the
transfer function
then with we obtain
with zeros and poles solutions of
also n-m extra zeros at s where
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IIR filter design using Matlab In Matlab there
are three functions for each class of filters
(Butterworth, Chebytchev1, Chebytchev2) BUTTAP C
HEB1AP CHEB2AP Poles and Zeros of Analog
Prototype Filter BUTTER CHEBY1 CHEBY2 Numerator
and Denominator from N and BUTTORD CHEBY1ORD CH
EBY2ORD N and from specifications
Example. We want to design an IIR Digital Filter
with the following specifications Pass Band 0
to 4kHz, with 1dB ripple Stop Band gt 8kHz with
at least 40 dB attenuation Sampling frequency
40kHz Type of Filter Butterworth. Using Matlab
gtgt N, fcbutterord(fp, fs, Rp, Rs) fp,
fspassband and stopband freq relative to Fs/2 gtgt
B, Abutter(N, fc) B, A vectors of
numerator and denominator coefficients. In our
case N, fcbutterord(4/20, 8/20, 1, 40),
would yield N7, fc0.2291 B, Abutter(7,
0.2291), would yield the transfer function
B(z)/A(z).
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Lets verify these numbers Step 1
specifications in the digital frequency domain
Step 2 specifications for analog filter from the
transformation
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Step 3 choose (say) Butterworh Filter
with
and from the ripple
specification
Step 4 determine order N from attenuation of
40dB
with yields N7
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Step 5 finally the cutoff frequency, from the
equation
Which yields
, corresponding to a digital
frequency
Step 6 the desired Filter is obtained by the
function num, den butter( 7 , 0.6889/?)
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Magnitude and Phase Plots