Digital Signal Processing Reference
In-Depth Information
1
−
z
−
N
1
−
α
z
−
N
.
H
MN
(
z
) =
1
+
α
2
(82)
This filter has N notches at the frequency 2
π
k/N[rad] for k
=
1,
· · ·
, N. One can easily show
that Eq. (82) can be rewritten as the form of Eq. (77) where
α
−
z
−
1
1
−
α
z
−
1
.
θ
=
1
2
,
ρ
=
1
H
AP
(
z
) =
2
,
(83)
4.4. Numerical examples
This subsection gives numerical examples of synthesis of the minimum L
2
-sensitivity
realizations for various types of digital filters with all second-order modes equal.
4.4.1. First-order FIR digital filters
Consider a first-order FIR digital filter H
FIR
(
z
)
given by
H
FIR
(
z
) =
0.5
+
0.5z
−
1
(84)
of which frequency magnitude and phase responses are shown in Fig. 5 (a). The second-order
mode of the digital filter H
FIR
(
z
)
is
θ
=
0.5. The balanced realization
(
A
b
,
B
b
,
C
b
, d
b
)
, which
is equal to the minimum L
2
-sensitivity realization, of H
FIR
(
z
)
is derived as
A
b
B
b
C
b
0
0.7071
=
d
b
0.7071
0.5
(85)
K
(
b
)
0
and observability Gramians
W
(
b
)
0
and controllability Gramian
are calculated as
K
(
b
0
=
W
(
b
0
=
0.5.
(86)
4.4.2. First-order IIR digital filters
Consider a first-order IIR digital filter H
IIR
(
z
)
given by
H
IIR
(
z
) =
0.25
+
0.25z
−
1
1
−
0.5z
−
1
(87)
of
which
frequency
magnitude
and
phase
responses
are
shown
in
Fig.
5
(b).
The
second-order
mode
of
the
digital
filter
H
IIR
(
z
)
is
θ
=
0.5.
The
balanced
realization
