Digital Signal Processing Reference
In-Depth Information
7. Constraints for guaranteed BIBO stability
In order for the FRM digital filter consisting of CSD multiplier coefficients
M
FR M
to be BIBO stable,
it is both necessary and sufficient for the bilinear-LDI IIR interpolation digital subfilters
H
A
(
Z
)
and
H
B
(
Z
)
to be BIBO stable. Likewise, in order for the interpolation digital subfilters
H
A
(
Z
)
and
H
B
(
Z
)
to
be BIBO stable, it is both necessary and sufficient for the bilinear-LDI allpass digital networks
G
0
(
Z
)
and
G
1
(
Z
)
to be BIBO stable. In this way, it is required that the bilinear-LDI digital allpass networks
G
0
(
Z
)
and
G
1
(
Z
)
remain BIBO stable throughout the course of the PSO algorithm.
In the course of PSO algorithm, the infinite-precision multiplier coefficients
M
L
I
and
M
C
I
can only take
quantized values
M
L
I
and
M
C
I
that belong to
CSD
(
L
,
L
,
F
)
. In order for the bilinear-LDI digital allpass
networks
G
0
(
Z
)
and
G
1
(
Z
)
to remain BIBO stable, it is required that the values of the corresponding
quantized reactive elements
L
I
and
C
I
remain positive [
45
] in the course of optimization. This is due to
the properties of the bilinear frequency transformation from analog to digital domain. In order to find
the conditions for BIBO stability and in accordance with Eqns. (
35
) and (
36
), one has:
T
M
L
I
L
I
=
(37)
T
M
C
I
C
I
=
(38)
Moreover, in accordance with Eqns. (
31
-
34
), one has:
L
′
1
=
L
1
(39)
C
I
T
2
2
+
T
2
M
I
=2
C
′
1
=
C
1
+
T
4
L
I
C
I
+
T
2
+
(40)
4
L
1
4
L
I
2
4
C
I
+
T
2
3
5
2
4
L
I
L
I
=
L
I
(41)
C
I
C
I
C
I
+
T
2
C
I
=
(42)
4
L
I
where
L
1
=
∞
for odd-ordered allpass network
G
0
(
Z
)
.
By substituting Eqns. (
37
) and (
38
) into Eqns. (
39
-
42
), and by solving the resulting equations for the
reactive elements
L
I
and
C
I
, one can obtain
T
M
L
1
L
1
=
(43)
