1. Lecture: IIR Filter Design
Assistant Prof. Yangmo Yoo Dept. of Electronic Engineering SOGANG UNIVERSITY

2. EEE4176 Application of Digital Signal Processing
IIR Filters
FIR vs. IIR
FIR : Ideal response (linear phase), stable
IIR :
Better magnitude response (sharper transition and/or lower stopband attenuation than FIR with the same number of parameters: HW efficient)  
Established filter types and design methods.
IIR filter design procedure
1) Set up digital filter spec
2) Determine the corresponding analog filter spec
(frequency translation involved)
3) Design the analog filter
4) Transform the analog filter to digital filter using various transformation methods
Impulse invariant method
Bilinear transformation
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EEE4176 Application of Digital Signal Processing

3. EEE4176 Application of Digital Signal Processing
IIR Filters
Important parameters
Passband ripple :
Stopband attenuation :
Discrimination factor :
Selectivity factor :
(-3dB) cutoff frequency :
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4. EEE4176 Application of Digital Signal Processing
IIR Filters
Frequency response
Transfer function : Rational
Asymptotic attenuation at high frequency
Attenuation function: (rational or polynomial function)
If is monotone, so is
If is oscillatory, exhibits ripple.
: Square magnitude frequency response
: reference frequency
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5. EEE4176 Application of Digital Signal Processing
IIR Filters
Frequency response
For real rational transfer function
Stability requirement
must include all poles of
on the left half of the s plane and only those.
Analog filter types
Butterworth
Chebyshev
Elliptic
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6. EEE4176 Application of Digital Signal Processing
Butterworth filters
Magnitude Squared Response
Properties of a LP Butterworth filter
Magnitude response : monotonically decreasing
Maximum gain : 0 at
Asymptotic attenuation at high frequency :
Maximally flat at DC (maximally flat filter)
: -3 dB point
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7. EEE4176 Application of Digital Signal Processing
Butterworth filters
Transfer function
2N poles:
: N poles are on the left side
of the complex plane
All pole filter
Normalized transfer function : Nth-order LP Butterworth filter
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EEE4176 Application of Digital Signal Processing

8. EEE4176 Application of Digital Signal Processing
Butterworth filters
LP Butterworth filter design procedure
Set up filter spec :
Compute N, using
Choose using
Compute the poles , using
Compute , using
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9. EEE4176 Application of Digital Signal Processing
Chebyshev filters
Chebyshev polynomial of degree
Monotone only in one band
Chebyshev Type I : equiripple in the passband
Chebyshev Type II : equiripple in the stopband
Sharper than Butterworth due to the ripples ! Why ?
Sharpest if equiripple in both bands, pass- and stop-bands.
Phase response : Better for maximally flat or monotonic mag response filters
Recursive formula:
If N is even(odd), so is
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EEE4176 Application of Digital Signal Processing

10. Chebyshev-I : Chebyshev Filter of the first kind
Properties
All-pole filter
For
Monotonically decreasing because
asymptotic attenuation :
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EEE4176 Application of Digital Signal Processing

11. Chebyshev-I : Chebyshev Filter of the first kind
Poles of a Nth-order LP Chebyshev-I filter
Transfer function
(N=3) case
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EEE4176 Application of Digital Signal Processing

12. Chebyshev-II : Chebyshev Filter of the second kind
Inverse Chebyshev filter or Chebyshev-II
Properties
Passband : monotonic Stopband : equi-ripple
Contains both the poles and zeros
for all
: monotonically decreasing
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13. EEE4176 Application of Digital Signal Processing
Elliptic Filters
Overview
Equiripple in both the passband and the stop band
Minimum possible order for a given spec : Sharpest (optimum)
Magnitude Squared Response: LP elliptic filter
: Jacobian elliptic function of degree N
Even(odd) function of for even(odd)
For , oscillates between -1 and +1
For
oscillates between 1 and for
oscillates between and for
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14. EEE4176 Application of Digital Signal Processing
Elliptic Filters
Example
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15. Frequency transformation
Analog filter design
1. Design a LPF (Butterworth, Chebyshev, elliptic)
2. Frequency transformation to obtain HPF, BPF, BRF
Definitions
: rational function ( , )
Transfer function of a LP filter :
Transformed filter
: rational function of
Class and stability of the filter is preserved after transformation.
Design domain :
Target domain :
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16. Frequency transformation
LP to LP transformation
LP to HP transformation
LP to BP transformation
LP to BS transformation
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17. Digital IIR filter design
1. Digital filter spec -> analog filter spec
2. Design analog filter
3. Transformation : Analog filter to digital filter
Transformation Goal
Requirements for
Real, causal, stable, rational
The order of should not be greater than that of if possible.
should be close to where
•transform should be simple, convenient to implement and applicable to all analog filter types and classes
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18. Digital IIR filter design
Impulse Invariant Transformation
Definition
Procedure 1. 2. 3.
High-pass filter cannot be
transformed !!
Filter orders are not changed
After transformation
Example)
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19. Digital IIR filter design
Bilinear Transform
Definition and Properties
For
(Approximation of continuous-time
integration by discrete-time
trapezoidal integration)
1) # of poles are preserved. => Preserve the filter order
2) # of zeros increase from q to p if p > q (p-1 zeros at z=-1)
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20. Digital IIR filter design
Bilinear Transform
Definition and Properties If
=> Preserve the stability
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21. Digital IIR filter design
Bilinear Transform
Definition and Properties
For
Frequency warping : One-to-one mapping,
3) Can be used for all filter types
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EEE4176 Application of Digital Signal Processing

22. Digital IIR filter design
Bilinear Transform
Prewarping
Prewarp the analog frequencies -> Bilinear transform -> desired digital frequencies
For convenience, set => Prewarping -> BLT gives the same result.
IIR filter Design procedure using BLT
1. Convert each specified band-edge frequency of the digital filter to a corresponding band-edge freq of an analog filter, using (A)
- Leave the ripple values unchanged.
2. Design
using BLT
(A)
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23. Digital IIR filter design
IIR filter Examples
Design the IIR filters specified in Ex) using MATLAB.  
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