Image Processing Reference
In-Depth Information
P
e
j
ω
Q
e
j
ω
−
γ
ω
ω
low
π
Fig. 2. Finite frequency approximation (
Problem 3
): the gain of the error
P
0
e
j
ω
)
−
e
j
ω
)
(
(
Q
is
Ω
low
=[
ω
low
]
minimized over the finite frequency range
0,
.
L
be a vector such
where
M
i
is a symmetric matrix and
x
i
is the
i
-th entry of
x
.Let
v
∈{
0, 1
}
that
v
x
2
. Our problem is then described by semidefinite programming as follows:
=
γ
minimize
v
x
subject to
M
(
x
)
≤
0.
By this, we can effectively approach the optimal parameters
α
N
by numerical
optimization softwares. For MATLAB codes of the semidefinite programming above, see
Section 7.
α
0
,
α
1
,...,
5. Finite frequency design of FIR digital filters
By the
H
∞
design discussed in the previous section, we can guarantee the maximum gain of
the frequency response of
T
=(
P
−
Q
)
W
(approximation) or
T
=(
QP
−
1
)
W
(inversion) over
the
whole frequency range
[
0,
π
]
. Some applications, however, do not need minimize the gain
over the whole range
[
0,
π
]
, but a finite frequency range
Ω
⊂
[
0,
π
]
. Design of noise shaping
ΔΣ
modulators is one example of such requirement (Nagahara & Yamamoto, 2009). In this
section, we consider such optimization, called
finite frequency optimization
.Wefirstconsider
the approximation problem over a finite frequency range.
Problem 3
(Finite frequency approximation)
.
Given a filter P
(
z
)
and a finite frequency range
Ω
⊂
[
0,
π
]
, find an FIR filter Q
(
z
)
which minimizes
e
j
ω
)
−
e
j
ω
)
Ω
(
−
)
=
(
(
V
P
Q
:
max
ω∈
Ω
P
Q
.
Figure 2 illustrates the above problem for a finite frequency range
Ω
=
Ω
low
=[
0,
ω
low
]
,
e
j
ω
)
−
where
ω
low
∈
(
0,
π
]
. We seek an FIR filter which minimizes the gain of the error
P
(
e
j
ω
)
Q
(
over the finite frequency range
Ω
, and do not care about the other range
[
0,
π
]
\
Ω
.We
can also formulate the inversion problem over a finite frequency range.
Problem4
(Finite frequency inversion)
.
Given a filter P
(
z
)
and a finite frequency range
Ω
⊂
[
0,
π
]
,
find an FIR filter Q
(
z
)
which minimizes
1
e
j
ω
)
e
j
ω
)
−
Ω
(
−
)
=
(
(
V
QP
1
:
max
ω∈
Ω
Q
P
.
These problems are also fundamental in digital signal processing. We will show in the
next section that these problems can be also described in semidefinite programming via
generalized KYP lemma.