Contents
•
Digital IIR Filter Design Introduction
•
From s-plane to z-plane :
1.
Impulse Invariant Method
2.
Bilinear Transform.
•
Design case study: bilinear transform
applied to a notch filter.
•
Biquadratic filter structure.
•
Notch filter by direct pole/zero placement.
•
Summary

Digital IIR Filter Design Introduction
•
Usually, an IIR digital filter
H
(
z
) is designed from an
analogue filter
H
a
(
s
).
•
The basic idea behind the conversion of
H
a
(
s
) into
H
(
z
) is
to apply a mapping from the
s
-domain
to the
z
-domain
so
that essential
properties
of the analogue frequency
response are preserved.
•
Thus, mapping function should be such that:
1.
Imaginary
j
Ω
axis in the
s
-plane be mapped in to the
unit circle
of the
z
-plane.
2.
A
stable analogue
transfer function be mapped into a
stable
digital
transfer function.

Mapping from s-domain to z-domain
•
When the
Laplace transform
is performed on a
discrete-time signal (with each element of the
discrete-time sequence attached to a correspondingly
delayed unit impulse), the result is precisely the
z-
transform
of the discrete-time sequence with the
substitution of:
sT
e
z
snT
n
st
n
a
st
n
st
a
n
n
d
e
nT
h
dt
e
nT
t
nT
h
s
H
dt
e
nT
t
nT
h
dt
e
t
h
s
H
z
n
h
z
H
0
0
0
0
0
0
0
)
(
)
(
)
(
)
(
)
(
)
(
)
(
)
(
]
[
)
(

Analogue to Digital Methods of
Approximating
1)
Impulse invariant method
2)
Bilinear transform method
3) Matched
z-transform (pole-zero mapping)
Poles and zeros of the analogue filter are
mapped to the poles and zeros of the digital
filter using the
mapping.
sT
e
z

Impulse Invariant Method
1.
A suitable analogue transfer function
H(s)
is
designed.
2.
Its impulse response
h(n)
is obtained using
the inverse Laplace transform.
3.
The
h(n)
is suitably sampled to produce
h(nT)
4.
The desired
H(z)
is then obtained by
z-transforming
h(nT)
.

Impulse Invariant Method-Example I
p
is a pole in the s-domain which has now
been transformed to a pole
e
pT
in z-domain.
pt
Ce
p
s
C
L
s
H
L
t
h
p
s
C
s
H
1
1
)]
(
[
)
(
)
(
pnT
nT
t
Ce
t
h
nT
h
)
(
)
(
1
0
0
1
)
(
)
(
z
e
C
z
Ce
z
n
h
z
H
pT
n
n
pnT
n
n

Impulse Invariant Method-Example II
•
Using the impulse invariant method obtain the
transfer function
H(z)
of the digital filter which
approximates the following normalised analogue
transfer function:
•
This is the 2
nd
order Butterworth LPF filter.
•
The sampling frequency is 1.28 kHz and the 3 dB
cut off frequency should be 150 Hz.
1
2
1
)
(
2
s
s
s
H

Impulse Invariant Method-Example II-
Cont’d
•
Before applying the impulse invariant method,
we need to frequency scale the normalised
transfer function. This is achieved by replacing
s
by
s
/
α
, where
α
=2
π
×150=942.4778 [radian].

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