IIR Filter Design: Basic Approaches

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IIR Filter Design Basic Approaches

• Most common approach to IIR filter design
• Convert specifications for the digital filter
  into equivalent specifications for an
  analog prototype lowpass filter
• Determine the analog lowpass filter transfer
  function
• (3) Transform into the desired
  digital transfer function

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Digital Filter Design Basic Approaches

• An analog transfer function to be denoted as
• where the subscript a specifically indicates
  the analog domain
• A digital transfer function derived from will
  be denoted as

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Digital Filter Design Basic Approaches

• Basic idea behind the conversion of
  into is to apply a mapping from the
  s-domain to the z-domain so that essential
  properties of the analog frequency response are
  preserved
• Thus mapping function should be such that
• Imaginary axis in the s-plane be mapped
  onto the unit circle of the z-plane
• A stable analog transfer function be mapped into
  a stable digital transfer function

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S plane to Z plane mapping
Preserve stability Pole in the right half plan
should map inside the circle in the z plan.
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Euler Approximation
Is the sampling interval
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IIR Filter Design by Bilinear Transformation
(1) Design Concept
- s-plane to z-plane conversion

• any mapping than maps stable region is s-plane
  (left half plane)
• to stable region in z-plane (inside u.c) ?

or
bilinear transform!
Td inserted for convention may put to any
convenient value for practical use.
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(2) Properties
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IIR Digital Filter Design Bilinear
Transformation Method

• Bilinear transformation
• Above transformation maps a single point in the
  s-plane to a unique point in the z-plane
  and vice-versa
• Relation between and is
  then given by

T inserted for convention may put to any
convenient value for practical use.
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Bilinear Transformation

• Digital filter design consists of 4 steps
• (1) Develop the specifications of HD(z)
• (2) Develop the specifications of
• (3) Design
• (4) Determine HD(z) by applying bilinear
  transformation to

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IIR Filter Design Procedure
Given specification in digital domain Convert it
into analog filter specification Design analog
filter (Butterworth, Chebyshov,
elliptic)H(s) Apply bilinear transform to get
H(z) out of H(s)
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• Design a digital filter equivalent of a 2nd order
  Butterworth low-pass filter with a cut-off
  frequency fc 100 Hz and a sampling frequency fs
  1000 samples/sec.
• The normalised cut-off frequency of the digital
  filter is given by the following equation
• the equivalent analogue filter cut-off frequency
  ?ac, The value of K is immaterial so let K 1.

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• H(s) for a Butterworth filter is
• Hence the transfer function of the Butterworth
  filter becomes

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• Next, convert the analogue filter into an
  equivalent digital filter by applying the
  bilinear z-transform. This is achieved by making
  a substitution for s in the transfer function.
• The finite difference equation of the filter is
  found by inverting the transfer function

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Direct form 2nd order
http//ccrma.stanford.edu/jos/filters/Direct_Form
_II.html
http//ccrma.stanford.edu/jos/filters/Direct_Form
_II.html
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Direct realisation for a 2nd order Butterworth
equivalent filter.
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Matlab Bilinear

• a1
• b1, 1.141, 1
• c, dbilinear(a, b, 1000)