Image Processing Reference
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H optimization is thus minimization of the maximum value of a transfer function. This leads
to robustness against uncertainty in the frequency domain. Moreover, it is known that the H
norm of a transfer function G
2 -induced norm of
(
z
)
is equivalent to the
G
,thatis,
G
u
2
G
= G
:
=
sup
u
,
u
2
2
u
=
0
2 norm of u :
where
u
2 is the
n = 0 | u [ k ] |
2 1/2
u
2 :
=
.
The H optimization is minimization of the system gain when the worst case input is applied.
This fact implies that the H optimization leads to robustness against uncertainty in input
signals.
3. H Design problems of FIR digital filters
In this section, we consider two fundamental problems in signal processing: filter
approximation and inverse filtering. The problems are formulated as H optimization by
using the H norm defined in the previous section.
3.1 FIR approximation of IIR filters
The first problem we consider is approximation . In signal processing, there are a number
of design methods for IIR (infinite impulse response) filters, e.g., Butterworth, Chebyshev,
Elliptic, and so on (Oppenheim & Schafer, 2009). In general, to achieve a given characteristic,
IIR filters require fewer memory elements, i.e., z 1 , than FIR (finite impulse response) filters.
However, IIR filters may have a problem of instability since they have feedbacks in their
circuits, and hence, we prefer an FIR filter to an IIR one in implementation. For this reason,
we employ FIR approximation of a given IIR filter. This problem has been widely studied
(Oppenheim & Schafer, 2009). Many of them are formulated by H 2 optimization; they aim at
minimizing the average error between a given IIR filter and the FIR filter to be designed.
This optimal filter works well averagely , but in the worst case, the filter may lead a large
error. To guarantee the worst case performance, H optimization is applied to this problem
(Yamamoto et al., 2003). The problem is formulated as follows:
Problem 1 (FIR approximation of IIR filters) . Given an IIR filter P
(
z
)
, find an FIR (finite impulse
(
)
response) filter Q
z
which minimizes
P
W
e j ω )
e j ω )
e j ω )
(
P
Q
)
W
=
max
ω∈ [
(
Q
(
(
,
0,
π ]
where W is a given stable weighting function.
The procedure to solve this problem is shown in Section 4.
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