PPT - IIR Filter Design | Signals and Systems - Electronics and Communication Engineering (ECE) PDF Download

 Page 1


ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete
 Objectives:
Filters and Difference Equations
Signal Flow Graphs
FIR and IIR Filters
Bilinear Transform
Digital Conversion of Filters
Design of Analog Filters
 Resources:
ISIP: Filter Transformations
Wiki: Digital Filter Design
JOS: Digital Filters
Wiki: Bilinear Transform
CNX: IIR Design
DESIGN OF IIR FILTERS
URL:
Page 2


ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete
 Objectives:
Filters and Difference Equations
Signal Flow Graphs
FIR and IIR Filters
Bilinear Transform
Digital Conversion of Filters
Design of Analog Filters
 Resources:
ISIP: Filter Transformations
Wiki: Digital Filter Design
JOS: Digital Filters
Wiki: Bilinear Transform
CNX: IIR Design
DESIGN OF IIR FILTERS
URL:
EE 3512: Lecture 36, Slide 1
 Recall our expression for a linear, constant-coefficient difference equation:
 This equation can be written succinctly using summations:
 We can draw a signal flow graph implementation of this equation:
 This is known as the Direct Form I implementation of the above difference
equation. Can we implement this more efficiently?
Converting Difference Equations To Signal Flow Graphs
] [ ... ] 1 [ ] [ ] [ ... ] 2 [ ] 1 [ ] [
1 0 2 1
M n x b n x b n x b N n y a n y a n y a n y
M N
- + + - + = - + + - + - +
å å
= =
- + - - =
M
l
l
N
k
k
l n x b k n y a n y
0 1
] [ ] [ ] [
. . .
. . .
+
1 -
z
1 -
z
0
b
1
b
2
b
1 -
z
1 -
z
1
a -
2
a -
] [n x ] [n y
+
+
+
+
+
Page 3


ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete
 Objectives:
Filters and Difference Equations
Signal Flow Graphs
FIR and IIR Filters
Bilinear Transform
Digital Conversion of Filters
Design of Analog Filters
 Resources:
ISIP: Filter Transformations
Wiki: Digital Filter Design
JOS: Digital Filters
Wiki: Bilinear Transform
CNX: IIR Design
DESIGN OF IIR FILTERS
URL:
EE 3512: Lecture 36, Slide 1
 Recall our expression for a linear, constant-coefficient difference equation:
 This equation can be written succinctly using summations:
 We can draw a signal flow graph implementation of this equation:
 This is known as the Direct Form I implementation of the above difference
equation. Can we implement this more efficiently?
Converting Difference Equations To Signal Flow Graphs
] [ ... ] 1 [ ] [ ] [ ... ] 2 [ ] 1 [ ] [
1 0 2 1
M n x b n x b n x b N n y a n y a n y a n y
M N
- + + - + = - + + - + - +
å å
= =
- + - - =
M
l
l
N
k
k
l n x b k n y a n y
0 1
] [ ] [ ] [
. . .
. . .
+
1 -
z
1 -
z
0
b
1
b
2
b
1 -
z
1 -
z
1
a -
2
a -
] [n x ] [n y
+
+
+
+
+
EE 3512: Lecture 36, Slide 2
 One of the more elementary aspects of the field of digital signal processing is
to develop more efficient implementations of digital filters, as well as improve
their ability to produce accurate results with less numerical precision.
 A more efficient implementation of our filter is a Direct Form II:
 This filter has the same transfer function, but shares the delay element
between the feedforward (moving average/finite impulse response) and
feedback (autoregressive/infinite impulse response) portions of the filter.
 Analog differential equations can be represented by similar signal flow graphs,
but their implementation involves physical components (e.g., RLCs, op amps).
Direct Form II: Sharing Delay Elements (Memory)
. . .
+
1 -
z
1 -
z
0
b
1
b
2
b
1
a -
2
a -
] [n x ] [n y
+
+
+
+
+
Page 4


ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete
 Objectives:
Filters and Difference Equations
Signal Flow Graphs
FIR and IIR Filters
Bilinear Transform
Digital Conversion of Filters
Design of Analog Filters
 Resources:
ISIP: Filter Transformations
Wiki: Digital Filter Design
JOS: Digital Filters
Wiki: Bilinear Transform
CNX: IIR Design
DESIGN OF IIR FILTERS
URL:
EE 3512: Lecture 36, Slide 1
 Recall our expression for a linear, constant-coefficient difference equation:
 This equation can be written succinctly using summations:
 We can draw a signal flow graph implementation of this equation:
 This is known as the Direct Form I implementation of the above difference
equation. Can we implement this more efficiently?
Converting Difference Equations To Signal Flow Graphs
] [ ... ] 1 [ ] [ ] [ ... ] 2 [ ] 1 [ ] [
1 0 2 1
M n x b n x b n x b N n y a n y a n y a n y
M N
- + + - + = - + + - + - +
å å
= =
- + - - =
M
l
l
N
k
k
l n x b k n y a n y
0 1
] [ ] [ ] [
. . .
. . .
+
1 -
z
1 -
z
0
b
1
b
2
b
1 -
z
1 -
z
1
a -
2
a -
] [n x ] [n y
+
+
+
+
+
EE 3512: Lecture 36, Slide 2
 One of the more elementary aspects of the field of digital signal processing is
to develop more efficient implementations of digital filters, as well as improve
their ability to produce accurate results with less numerical precision.
 A more efficient implementation of our filter is a Direct Form II:
 This filter has the same transfer function, but shares the delay element
between the feedforward (moving average/finite impulse response) and
feedback (autoregressive/infinite impulse response) portions of the filter.
 Analog differential equations can be represented by similar signal flow graphs,
but their implementation involves physical components (e.g., RLCs, op amps).
Direct Form II: Sharing Delay Elements (Memory)
. . .
+
1 -
z
1 -
z
0
b
1
b
2
b
1
a -
2
a -
] [n x ] [n y
+
+
+
+
+
EE 3512: Lecture 36, Slide 3
More About Types of Filters
 Consider a filter with only
feedforward components:
 The transfer function is:
 Since the impulse response of this filter, h[n], has a finite number of  nonzero
terms, this filter is referred to as a finite impulse response (FIR) filter. Observe
that this filter only has zeroes.
 Next, consider a filter with only
feedback components:
 The transfer function is:
 This is an all-pole filter with an infinite impulse response (IIR). Why?
å
å
=
-
=
= =
- =
M
l
l
l
M
l
l
z b
z X
z Y
z H
l n x b n y
0
1
) (
) (
) (
] [ ] [
å
å
=
-
=
+
=
+ - - =
N
k
k
k
N
k
k
z a
b
z H
n x b k n y a n y
1
0
0
1
1
) (
] [ ] [ ] [
Page 5


ECE 8443 – Pattern Recognition EE 3512 – Signals: Continuous and Discrete
 Objectives:
Filters and Difference Equations
Signal Flow Graphs
FIR and IIR Filters
Bilinear Transform
Digital Conversion of Filters
Design of Analog Filters
 Resources:
ISIP: Filter Transformations
Wiki: Digital Filter Design
JOS: Digital Filters
Wiki: Bilinear Transform
CNX: IIR Design
DESIGN OF IIR FILTERS
URL:
EE 3512: Lecture 36, Slide 1
 Recall our expression for a linear, constant-coefficient difference equation:
 This equation can be written succinctly using summations:
 We can draw a signal flow graph implementation of this equation:
 This is known as the Direct Form I implementation of the above difference
equation. Can we implement this more efficiently?
Converting Difference Equations To Signal Flow Graphs
] [ ... ] 1 [ ] [ ] [ ... ] 2 [ ] 1 [ ] [
1 0 2 1
M n x b n x b n x b N n y a n y a n y a n y
M N
- + + - + = - + + - + - +
å å
= =
- + - - =
M
l
l
N
k
k
l n x b k n y a n y
0 1
] [ ] [ ] [
. . .
. . .
+
1 -
z
1 -
z
0
b
1
b
2
b
1 -
z
1 -
z
1
a -
2
a -
] [n x ] [n y
+
+
+
+
+
EE 3512: Lecture 36, Slide 2
 One of the more elementary aspects of the field of digital signal processing is
to develop more efficient implementations of digital filters, as well as improve
their ability to produce accurate results with less numerical precision.
 A more efficient implementation of our filter is a Direct Form II:
 This filter has the same transfer function, but shares the delay element
between the feedforward (moving average/finite impulse response) and
feedback (autoregressive/infinite impulse response) portions of the filter.
 Analog differential equations can be represented by similar signal flow graphs,
but their implementation involves physical components (e.g., RLCs, op amps).
Direct Form II: Sharing Delay Elements (Memory)
. . .
+
1 -
z
1 -
z
0
b
1
b
2
b
1
a -
2
a -
] [n x ] [n y
+
+
+
+
+
EE 3512: Lecture 36, Slide 3
More About Types of Filters
 Consider a filter with only
feedforward components:
 The transfer function is:
 Since the impulse response of this filter, h[n], has a finite number of  nonzero
terms, this filter is referred to as a finite impulse response (FIR) filter. Observe
that this filter only has zeroes.
 Next, consider a filter with only
feedback components:
 The transfer function is:
 This is an all-pole filter with an infinite impulse response (IIR). Why?
å
å
=
-
=
= =
- =
M
l
l
l
M
l
l
z b
z X
z Y
z H
l n x b n y
0
1
) (
) (
) (
] [ ] [
å
å
=
-
=
+
=
+ - - =
N
k
k
k
N
k
k
z a
b
z H
n x b k n y a n y
1
0
0
1
1
) (
] [ ] [ ] [
EE 3512: Lecture 36, Slide 4
Design of Digital Filters Using Analog Prototypes
 Analog filter design theory was developed in the mid-1900’s.
 As digital signal processing developed, it seemed reasonable to leverage
existing knowledge in analog filter design.
 Our strategy will be to design the filter in the analog domain, and then
transform the filter to the digital domain.
 We can derive this transformation by recalling the relationship between the
Laplace transform and the z-transform:
 We can approximate the logarithm using a Taylor series:
 This transformation is known as the bilinear transform. It maps the left-half
s-plane to the interior of the unit circle in the z-plane.
 Unfortunately, it also “warps” the frequency axis, so the analog filter design
must be prewarped so that it lands at the proper frequency in the z-plane.
Let s = s + j W and z = re
j w
:
) ln(
1
z
T
s e z
sT
= Û =
÷
÷
ø
ö
ç
ç
è
æ
+
-
=
÷
ø
ö
ç
è
æ
+
-
» =
-
-
1
1
1
1 2
1
1 2
) ln(
1
z
z
T z
z
T
z
T
s
÷
÷
ø
ö
ç
ç
è
æ
+
-
= W +
-
w
w
s
j
j
re
re
T
j
1
1 2