Digital Signal Processing Reference
In-Depth Information
6.1 IMPULSE RESPONSE INVARIANT DESIGN
The impulse response invariant design method (or impulse invariant
transformation) is based on creating a digital filter with an impulse response that is
a sampled version of the impulse response of the analog filter. We first start with
an analog filter's transfer function
H
(
s
), and by using the inverse Laplace
transform, we determine the system's continuous impulse response
h
(
t
). We next
sample that response to determine the system's discrete-time impulse response
h
(
nT
). We then take the
z
-transform of this sampled impulse response to find the
discrete-time transfer function
H
(
z
). As an illustration, consider the following
example.
Example 6.1 Impulse Response Invariant Transformation
Problem:
Assume that we wish to convert the following continuous-time
transfer function to a discrete-time transfer function using the impulse invariant
transformation method:
12
H
(
s
)
=
(
s
+
2
⋅
(
s
+
5
Solution:
We first use basic partial fraction expansion techniques to write the
transfer function in a form suitable for inverse transformation:
4
4
H
(
s
)
=
−
(
s
+
2
(
s
+
5
Then recognizing the Laplace transform pair
⎧
A
⎫
−
1
−
at
L
H
(
s
)
=
=
A
⋅
e
⋅
u
(
t
)
=
h
(
t
)
(
s
+
a
)
we can easily find the impulse response as
−
2
t
−
5
t
h
(
t
)
=
(
4
⋅
e
−
4
⋅
e
)
⋅
u
(
t
)
If we then sample this impulse response at intervals of
T
, we will have the
discrete-time impulse response. Effectively, we simply replace every
t
with
nT
to
denote the
n
th sample at intervals of
T
:
−
2
nT
−
5
nT
h
(
nT
)
=
(
4
⋅
e
−
4
⋅
e
)
⋅
u
(
nT
)
