Digital Signal Processing (DSP) IIR Filter Lecture

1. Infinite Impulse Response (IIR) Filter
Supervised:
Prof .Dr. Bayan Mahdi Sabbar AL-Ibrahimy
Prepare by:
Fatimah Azeez
Al Nahrain University
College of Information Engineering
Department of Network Engineering

2. Contents
 Introduction to IIR Filter
 Structures for IIR Systems
 Direct-Form Structures
 Signal –Flow Graphs and Transposed Structures
 Cascade-Form Structures
 Parallel-Form Structures
 Lattice and Lattice-Ladder Structures for IIR Systems
 IIR Filter Design
 IIR Filter Design by Approximation of Derivatives
 IIR Filter Design by Impulse Invariance
 IIR Filter Design by the Bilinear Transformation
 Characteristics of Commonly Used Analog Filters
 Butterworth
 Chebyshev
 Elliptic filters
 Bessel filters
 Frequency Transformations
 Frequency Transformations in Analog Domain
 Frequency Transformations in Digital Domain
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3. Filter Design is Everywhere
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• Multimedia & Biomedical
Electronics
Noise Cancellation, Audio
Compression, Speech recognition,
Hearing Aids, EKG
• Communications
Cellular Phone , Cable, GPS,
Wireless Broadcast, WiFi
• Medical & Computer Vision
Process Control, Robotics,
Autonomous Navigation, Tracking
• Safety & Security Systems
Biometric access & control, Driver
awareness system
3

4. Recursive or Infinite Impulse Response
(IIR) Filters
A recursive filter has feedback from output to input, and in general its output is a
function of the previous output samples and the present and past input samples as
described by the following equation:
Thus are recursive filter is also known as an Infinite Duration Impulse
Response(IIR) filter.
Other names for an IIR filter include feedback filters, pole-zero filters and auto-
regressive-moving-average (ARMA) filter a term usually used in statistical signal
processing literature.
Where and are filter coefficients
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IIR Transfer Function:

5. Comparison FIR vs IIR
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vs
01
An FIR filter is less compact in
that it can usually achieve a
prescribed frequency response
with a smaller number of
coefficients
An IIR filter is more compact in
that it can usually achieve a
prescribed frequency response
with a smaller number of
coefficients
01
vs
02
FIR filters are less
efficient in memory
and computational
requirements
IIR filters are more
efficient in memory and
computational
requirements
02
vs
03
FIR filter is always
stable
An IIR filter unstable 03
vs
04
FIR can be designed
to be Linear Phase
filter or Non-Linear
Phase
IIR filter have Non-
Linear Phase
04

6. Structures
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7. Structure for IIR System
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1. Direct – Form Structures
2. Signal –Flow Graphs and Transposed Structures
3. Cascade – Form Structures
4. Parallel– Form Structures
5. Lattice and Lattice – Ladder Structures for IIR Systems

8. Direct – Form Structures
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• The System Function That characterizes an IIR System can
be viewed as form cascade , that is,
– Where 𝐻1(𝑧) consists Zeros of H(z)
Or 𝐻1(𝑧) is the transfer function of a feed-forward, all-zero, filter
given by
– H2(z): consists Poles of H(z)
Or H2(z): is the transfer function of a feed back, all-pole,
recursive filter given by

9. Direct – Form I Structures
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Requires M+N+1 memory locations,
M+N+1 multiplies, M+N Additions

10. Direct – Form II Structures
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Delayed versions of the sequence
{w(n)} Consequently, only single set
of memory locations is required for
storing the past values of {w(n)}
Requires
{M,N} memory locations
M+N+1 multiplies
M+N Additions

11. Signal –Flow Graphs and Transposed
Structures
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12. Signal –Flow Graphs and Transposed
Structures
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1& 3
summing
nodes
2, 4,& 5
branching
points

13. Transposed direct form II Structures
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1-reverse all
signal flow
directions
2-Change
nodes into
adders and
adders into
nodes
3-Interchange
i/p & o/p

14. Signal –Flow Graphs and Transposed
Structures
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The transposed direct form II realization that we
have obtained can be described by the set of
difference equations
Requires
Same No. of
memory locations
multiplies
Additions

15. Some Second-Order Modules for Discrete-Time
System
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16. Cascade – Form Structures
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(N>=M) The System can be factored into a cascade of second-
order subsystems, such that H(z) can be expressed as
Where K is the integer part of (N+1)/2.
Hk(z) has the general form
The coefficients and in the second-order
subsystems are real.

17. Cascade – Form Structures
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Quadratic factor any
two real valued zeros
can be paired
together
Quadratic factor in
numerator consist of
either a pair of real roots
or pair of complex –
conjugate roots

18. Parallel Filter Structure
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Obtained by performing a partial fraction expansion of H(z).
we obtain the result
Where {Pk} are the poles & {Ak} are the coefficients in the
partial-fraction expansion
the constant C is defined as C=bN/aN.
To avoid multiplications by complex
numbers, we can combine pairs of
complex-conjugate poles to form two-pole
subsystem

19. Parallel Filter Structure (Cont.)
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Where the coefficients {bki} and {aki}
are real-valued system parameters.
The overall function can now be
expressed as
Where K : integer part of (N+1)/2.
when N is odd, one of the Hk(z) is
really a single-pole system (i.e.,
bk1=ak2=0)

20. Example 3: Parallel Structure of IIR
Filter
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Example Determine the cascade and parallel realizations for
the system described by the system function
Solution: the cascade realization is easily obtained from this
form. One possible pairing of poles and zeros is

21. Example 3: Parallel Structure of IIR
Filter
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The cascade realization
To obtain the parallel-form, H(Z) must be expanded in partial functions

22. Example 3: Parallel Structure of IIR
Filter
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The parallel-form realization

23. Example 3: Parallel Structure of IIR
Filter
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A1=2.93,
A2=-17.68,
A3= 12.25-j14.57
A*3=12.25+j14.57

24. Lattice and Lattice – Ladder Structures
for IIR Systems
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25. Lattice and Lattice – Ladder Structures
for IIR Systems
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26. Lattice and Lattice – Ladder Structures
for IIR Systems
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27. Lattice and Lattice – Ladder Structures
for IIR Systems
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28. Lattice and Lattice – Ladder Structures
for IIR Systems
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29. Design of IIR filter
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30. Design IIR Filters from Analog Filter
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 Analog filter can be described by its systems functions
Where {αk} and {βk} are the filters coefficient,
 impulse response which is related to Ha(s) by the Laplace transform
 The analog filter having the relation system function H(s) can be
described by the linear constant-coefficient differential equation
Where x(t): input signal
y(x):output of filter signal
We begin design of digital filter in analog
domain (S-plane) and then convert the
design into digital domain (Z-plane)

31. Design IIR Filters from Analog Filter
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32. Design IIR Filters from Analog Filter
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Passband
edge
frequency
Stopband edge
frequency
Peak ripple
value
Peak ripple value
Transition band

33. Design IIR Filters from Analog Filter
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34. IIR Filter Design by Approximation of
Derivative
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1- j𝜴 𝒂𝒙𝒊𝒔 𝒊𝒏 𝒕𝒉𝒆 𝒔
− 𝒑𝒍𝒂𝒏𝒆 𝒔𝒉𝒐𝒖𝒍𝒅 𝒎𝒂𝒑 𝒊𝒏𝒕𝒐 𝒕𝒉𝒆 𝒖𝒏𝒊𝒕
𝒄𝒊𝒓𝒄𝒍𝒆 𝒊𝒏 𝒕𝒉𝒆 𝒛 − 𝒑𝒍𝒂𝒏𝒆
2-The left-half plane (LHP) of the S-
plane should map into the inside
of the unit circle in the Z-plane

35. IIR Filter Design by Approximation of
Derivative
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36. IIR Filter Design by Approximation of
Derivative
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37. Example of IIR Filter Design by
Approximation of Derivative
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38. IIR Filter Design by Impulse Invariance
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39. IIR Filter Design by Impulse Invariance
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40. IIR Filter Design by Impulse Invariance
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41. IIR Filter Design by Impulse Invariance
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42. Example: IIR Filter Design by Impulse
Invariance
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43. Example: IIR Filter Design by Impulse
Invariance
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44. IIR Filter Design by Bilinear Transform
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The mapping from the s-plane to the z-plane (thus avoiding
aliasing of frequency components ) is
Ω
This Method overcomes the limitation of
other two design methods (appropriate only
for lowpass filters and a limited class of
bandpass filters)

45. IIR Filter Design by Bilinear Transform
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46. Example- IIR Filter Design by Bilinear
Transform
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47. Example- IIR Filter Design by Bilinear
Transform
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48. IIR Filter Design by Bilinear Transform
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49. IIR Filter Design by Bilinear Transform
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50. Characteristic of commonly used
Analog Filters
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51. Characteristic of commonly used
Analog Filters- Butterworth Filters
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52. Characteristic of commonly used
Analog Filters- Butterworth Filters
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or

53. Butterworth Filters
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54. Butterworth Filters
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55. Butterworth Filters
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56. Chebyshev Filter
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57. Chebyshev Filter
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58. Chebyshev Filter
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59. Chebyshev Filter
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60. Chebyshev Filter
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61. Elliptic filter
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62. Elliptic filter
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63. Elliptic filter
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64. Bessel filter
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65. Bessel filter
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66. Frequency Transformations
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67. Frequency Transformations
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68. Frequency Transformations in Analog
Domain
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69. Frequency Transformations in Analog
Domain
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70. Frequency Transformations in Analog
Domain
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71. Frequency Transformations in Analog
Domain
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72. Example
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73. Frequency Transformations in Digital
Domain
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74. Frequency Transformations in Digital
Domain
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75. Example
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76. Example
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77. Frequency Transformations in Digital
Domain
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78. Bilinear Transformation Method
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Steps of
the design
procedure

79. IIR Filter Design by Bilinear Transform
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For LPF and HPF:
For BPF and BRF:
Frequency Warping
Prototype
Transformation
Obtained digital filter Transfer
Function:
From
analog LPF
to desired
analog filter

80. Bilinear Transformation Method
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81. Bilinear Transformation Method
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82. Example 1: Bilinear Transformation
Method
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82
Design a first-order high-pass digital Chebyshev filter with a
cut-off frequency of 3 kHz and 1 dB ripple on the pass-band
using a sampling frequency of 8,000 Hz.
Probl
em:
Solution:
Digital frequency
(rad/s):
Pre-warped analog
frequency :
First-order LP Chebyshev filter
prototype:
Applying transformation
LPF to HPF:
Dividing by 1.9625
Applying
BLT:

83. Example 2: Bilinear Transformation
Method
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Problem:
Solution:
Design a second-order digital band-pass Butterworth filter with the
following specifications:
Digital frequencies:
pre-warped analog
frequency:
A first-order LPF prototype will produce second-order BPF prototype.

84. IIR digital filter design using
MATLAB
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85. Design of Chebychev lowpass filter
using MATLAB
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Design of Chebychev lowpass filter
[z, p, k]=cheb1ap(N, Rp)
[N, wpo]=cheb1ord(wp, ws, Rp, As)
[N, wpo]=cheb1ord(wp, ws, Rp, As, ‘s’)
[B, A]=cheby(N, Rp, wpo, ‘ftype’)
[B, A]=cheby1(N, Rp, wpo, ‘ftype’, ‘s’)

86. Chebychev approximation
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Example
wp=2*pi*3000;ws=2*pi*12000;rp=0.1;as=60;
[N1,wp1]=cheb1ord(wp,ws,rp,as,’s’);
[B1,A1]=cheby1(N1,rp,wp1,’s’);
Subplot(221);
fk=0:12000/512:12000;wk=2*pi*fk;
Hk=frqs(B1,A1,wk);
Plot(fk/1000,20*log10(abs(Hk)));grid on

87. Chebychev approximation
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0 2 4 6 8 10 12
-70
-60
-50
-40
-30
-20
-10
0
Frequency（ Hz）
Magnitude(dB)
IIR Chebychev Lowpass Filter

88. Reference
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1) Digital Signal Processing, Principle , Algorithm and Application, Forth
Edition, John G. Proakis Dimitris G.Manolakis (Chapter 9 &10)
2) https://ocw.mit.edu/resources/res-6-008-digital-signal-processing-spring-
2011/video-lectures/lecture-14-design-of-iir-digital-filters-part-
1/MITRES_6_008S11_lec14.pdf