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.
 
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