1. IIR Filters FIR vs. IIR IIR filter design procedure
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
IIR Filters
DSP

2. IIR Filters Important parameters Passband ripple :
Stopband attenuation :
Discrimination factor :
Selectivity factor :
(-3dB) cutoff frequency :
IIR Filters
DSP

3. 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
IIR Filters
DSP

4. IIR Filters Frequency response Analog filter types
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
IIR Filters
DSP

5. 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
IIR Filters
DSP

6. 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
IIR Filters
DSP

7. Butterworth filters LP Butterworth filter design procedure
Set up filter spec :
Compute N, using
Choose using
Compute the poles , using
Compute , using
IIR Filters
DSP

8. 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
IIR Filters
DSP

9. Chebyshev-I : Chebyshev Filter of the first kind
Properties
All-pole filter
For
Monotonically decreasing because
asymptotic attenuation :
IIR Filters
DSP

10. Chebyshev-I : Chebyshev Filter of the first kind
Poles of a Nth-order LP Chebyshev-I filter
Transfer function
(N=3) case
IIR Filters
DSP

11. 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
IIR Filters
DSP

12. 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
IIR Filters
DSP

13. Elliptic Filters
Example
IIR Filters
DSP

14. 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 :
IIR Filters
DSP

15. Frequency transformation
LP to LP transformation
LP to HP transformation
LP to BP transformation
LP to BS transformation
IIR Filters
DSP

16. 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
IIR Filters
DSP

17. 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)
IIR Filters
DSP

18. 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)
IIR Filters
DSP

19. Digital IIR filter design
Bilinear Transform
Definition and Properties If
=> Preserve the stability
IIR Filters
DSP

20. Digital IIR filter design
Bilinear Transform
Definition and Properties
For
Frequency warping : One-to-one mapping,
3) Can be used for all filter types
IIR Filters
DSP

21. 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)
IIR Filters
DSP