EE264: Lecture 6
IIR Filters cont.
Differential equation to Difference equation Transformation
When a system has a transfer function
H
(
s
)
of rational form, the system relates its output to its input through
a Linear Constant Coefficient Differential Equation. Approximating derivatives with (discretetime) differ
ences allows us to translate the differential equation to a difference equation that defines a Discretetime
system which will approximate the differential equation if the sampling period is small enough.
Example:
Let
x
(
t
)
and
y
(
t
)
be the input and output of a continuoustime system with the following transfer function:
H
c
(
s
) =
s

1
(
s
+ 1)(
s
+ 2)
=
s

1
s
2
+ 3
s
+ 2
then
Y
(
s
)
X
(
s
)
=
s

1
s
2
+ 3
s
+ 2
Y
(
s
)(
s
2
+ 3
s
+ 2) =
X
(
s
)(
s

1)
d
2
y
(
t
)
dt
2
+ 3
dy
(
t
)
dt
+ 2
y
(
t
) =
dx
(
t
)
dt

x
(
t
)
Let
x
[
n
] =
x
(
nT
)
, we can approximate derivatives at times
nT
as follows:
dx
(
t
)
dt
vextendsingle
vextendsingle
vextendsingle
vextendsingle
t
=
nT
≈
x
[
n
+ 1]

x
[
n
]
T
forward difference
dx
(
t
)
dt
vextendsingle
vextendsingle
vextendsingle
vextendsingle
t
=
nT
≈
x
[
n
]

x
[
n

1]
T
backward difference
dx
(
t
)
dt
vextendsingle
vextendsingle
vextendsingle
vextendsingle
t
=
nT
≈
x
[
n
+ 1]

x
[
n

1]
2
T
symmetrical difference
d
2
x
(
t
)
dt
2
vextendsingle
vextendsingle
vextendsingle
vextendsingle
vextendsingle
t
=
nT
≈
x
[
n
+1]

x
[
n
]
T

x
[
n
]

x
[
n

1]
T
T
=
x
[
n
+ 1]

2
x
[
n
] +
x
[
n

1]
T
2
Using the forward difference approximation we obtain:
y
[
n
+ 1]

2
y
[
n
] +
y
[
n

1]
T
2
+ 3
y
[
n
+ 1]

y
[
n
]
T
+ 2
y
[
n
] =
x
[
n
+ 1]

x
[
n
]
T

x
[
n
]
Taking the ztransform on both sides of the difference equation we get:
H
(
z
) =
Y
(
z
)
X
(
z
)
=
1
T

(
1
T
+ 1)
z

1
(
1
T
2
+
3
T
)

(
2
T
2
+
3
T

2)
z

1
+
1
T
2
z

2
Copyright c
circlecopyrt
1995–1996 by G. Plett. Copyright c
circlecopyrt
1998–2004 by M. Kamenetsky. Copyright c
circlecopyrt
2005 by A. Flores
65
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66
IIR Filters cont.
Lect. 6
Bilinear
(but not linear)
Transformation
To avoid aliasing, we need a onetoone mapping from the splane to the zplane. Consider
s
=
2
T
parenleftBigg
1

z

1
1 +
z

1
parenrightBigg
⇒
H
(
z
) =
H
c
parenleftBigg
2(1

z

1
)
T
(1 +
z

1
)
parenrightBigg
which defines a bilinear transformation
z
=
1 +
T
2
s
1

T
2
s
⇔
s
=
2(1

z

1
)
T
(1 +
z

1
)
Q: Is this a legal transformation?
A:
Let
s
=
j
Ω
. Then
z
=
1 +
j
Ω
T
2
1

j
Ω
T
2
=
radicalBig
1 + Ω
2
T
2
4
negationslash
tan

1
parenleftBig
Ω
T
2
parenrightBig
radicalBig
1 + Ω
2
T
2
4
negationslash
tan

1
parenleftBig

Ω
T
2
parenrightBig
= 1
negationslash
2 tan

1
parenleftbigg
Ω
T
2
parenrightbigg
So,

z

= 1
. Also, if
σ <
0
,

z

<
1
, so OK.
Bilinear transformation is a nonlinear mapping, which maps
Ω
in
∞
<
Ω
<
∞
to the unit circle
e
jω
,

π
≤
ω < π
. The relationship between
Ω
and
ω
can be expressed as
j
Ω =
2
T
parenleftBigg
1

e
jω
1 +
e
jω
parenrightBigg
=
j
2
T
parenleftbigg
sin
ω/
2
cos
ω/
2
parenrightbigg
⇒
Ω =
2
T
tan(
ω/
2)
(the frequency warp).
π

π
Ω
ω
Procedures of bilinear transformation:
1.
Perform frequency prewarp to obtain the corresponding analog filter specs (pick any
T
if the specs
are given in the discretetime domain.)
2.
Design the analog filter
H
c
(
s
)
using any one of the analog filter prototypes.
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 Winter '09
 Eltanany
 Digital Signal Processing, Lowpass filter, G. Plett, A.65 Flores, M. Kamenetsky, IIR Filters cont

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