Digital Signal Processing Reference
In-Depth Information
The lattice structure can be quite useful for applications in adaptive filtering and
speech processing. Although this structure is not as computationally efficient as the
direct or cascade forms, requiring more multiplication operations, it is less sensitive
to quantization effects [6-8].
5.3 BILINEAR TRANSFORMATION
The BLT is the most commonly used technique for transforming an analog filter
into a discrete filter. It provides one-to-one mapping from the analog
s
-plane to the
digital
z
-plane, using
sK
z
z
-
+
1
1
=
(5.29)
The constant
K
in (5.29) is commonly chosen as
K
2/
T
, where
T
represents a
sampling variable. Other values for
K
can be selected, since it has no consequence
in the design procedure. We choose
T
=
=
2 or
K
=
1 for convenience to illustrate the
BLT procedure. Solving for
z
in (5.29) gives us
1
1
+
-
s
s
z
=
(5.30)
This transformation allows the following:
1.
The left region in the
s
-plane, corresponding to
s<
0, maps
inside
the unit
circle in the
z
-plane.
2.
The right region in the
s
-plane, corresponding to
s>
0, maps
outside
the unit
circle in the
z
-plane.
3.
The imaginary
j
w
axis in the
s
-plane maps
on
the unit circle in the
z
-plane.
Let
w
A
and
w
D
represent the analog and digital frequencies, respectively. With
s
=
j
w
A
and
z
=
e
j
w
D
T
, (5.29) becomes
(
)
e
e
j
w
T
-
+
1
1
e
j
w
T
2
e
j
w
T
2
-
e
-
j
w
T
2
D
D
D
D
(5.31)
j
w
=
=
A
(
)
j
w
T
j
w
T
2
j
w
T
2
-
j
w
T
2
e
e
+
e
D
D
D
D
Using Euler's expressions for sine and cosine in terms of complex exponential func-
tions,
w
A
from (5.31) becomes
w
D
T
w
=
tan
(5.32)
A
2
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