Digital Signal Processing Reference
In-Depth Information
2
4
3
5
0.7281 0.5229 0.4146
−
0.5351 0.7281
−
0.1282
0.1282
−
0.4146
B
opt
A
opt
=
.
(109)
d
opt
C
opt
0.0316
The controllability Gramian
K
(
opt
)
0
and the observability Gramian
W
(
opt
)
0
are given as follows:
0.5100
−
0.0870
−
0.0870
K
(
opt
)
0
=
(110)
0.4901
0.4901
−
0.0870
−
0.0870
W
(
opt
)
0
=
.
(111)
0.5100
We have to note that the controllability Gramian
K
(
opt
)
0
W
(
opt
)
0
and the observability Gramian
satisfy the sufficient condition of the absence of limit cycles given in Eq. (106) with
B
opt
=
diag
(
0.9803, 1.0201
)
.
(112)
Therefore,
(
A
opt
,
B
opt
,
C
opt
,
d
opt
)
is
the
minimum
L
2
-sensitivity
realization
without
limit
cycles.
We demonstrate the absence of limit cycles in the minimum
L
2
-sensitivity realization by
observing its zero-input response. We calculate the zero-input responses of the minimum
L
2
-sensitivity realization and the dierct form II, setting the initial state as
X
(
0
) = [
0.8
−
0.8
]
T
.
We let the dynamic range of signals to be
[−
1, 1
)
and adopt two's complement as the overflow
characteristic. The zero-input responses are shown in Fig. 8(a) and 8(b). We assume that each
filter coefficient and signal have 16[bits] fixed-point representation, of which lower 14[bits]
are fractional bits. In this numerical example, the overflow of the state variables occurs in
both cases. It is desirable that the effect of the overflow is decreasing since the digital filter
H(z)
in Eq. (108) is stable. For the minimum
L
2
-sensitivity realization synthesized by our
proposed method, the state variables
x
1
(n)
and
x
2
(n)
converge to zero after the overflow, as
shown in Fig. 8(a). Therefore, there are no limit cycles. On the other hand, for the direct form
II, a large-amplitude autonomous oscillation is observed as shown in Fig.
8(b).
Therefore,
the direct form II generates the limit cycles.
5.2.2. High-order digital filters
We can demonstrate the validity of the proposed method for also high-order digital filters.
Consider a fourth-order band-pass digital filter
H
(
z
)
given by
H
(
z
) =
0.0178
−
0.0252
z
−
1
+
0.0173
z
−
2
−
0.0252
z
−
3
+
0.0178
z
−
4
(113)
1
−
2.6977
z
−
1
+
3.5410
z
−
2
−
2.3340
z
−
3
+
0.7497
z
−
4
The frequency response of the digital filter
H
(
z
)
in Eq. (113) is shown in Fig. 9. We obtain
the limit cycle free minimum
L
2
-sensitivity realization
(
A
opt
,
B
opt
,
C
opt
,
d
opt
)
by successive
approximation method:
