Digital Signal Processing Reference
In-Depth Information
6. Design and Simulation Results
The equations derived in the previous section have unknown parameters
H
i
(
z
)
,
G
i
(
z
)
, and
H
(
z
)
. The filter
H
(
z
)
in the direct scaling structure should be known, or designed, in ad‐
vance. Thus,
H
i
(
z
)
and
G
i
(
z
)
are obtained by solving the equations with a pre-designed de‐
sired
H
(
z
)
so that the filter bank structure will be exactly or approximately equivalent to the
direct scaling structure.
The concrete design procedure depends on applications the filter banks are utilized in. For
example, several constraints should be imposed on the analysis filters
H
i
(
z
)
for the filter
banks to have non DC leakage, high coding gain, etc., when the filter banks are exploited in
sub-band image coding.
The analysis filters should have high attenuation in the stop band, or excellent frequency de‐
composition performance, when the filter banks are employed for watermarking, since the
watermarks are often embedded only into one narrow sub-band that is robust to attacks.
These tell that the analysis filters typically suffer from severe constraints. In this section, we
survey the case where the analysis and synthesis filters are separately designed, i.e., only
G
i
(
z
)
are determined so that the equations are satisfied with not only a pre-designed
H
(
z
)
but also
H
i
(
z
)
, in order not to lay much burden on
H
i
(
z
)
. On the other hand, we do not
impose the symmetry constraints on
G
i
(
z
)
to satisfy the equations more precisely.
With this design policy, we here design filter banks. As an example, two-channel and three-
channel filter banks are designed and explored. In the two-channel case, the scaling factor is
3/2, and in the three channel one, it is 2/3. The length of the linear phase analysis filters is 6
in the two-channel case, and 8 in the three-channel one. The length of the obtained (non-lin‐
ear) synthesis filters is 9, and 6, respectively. The length of the linear phase filter in the direct
scaling structure is 3, and 2, respectively, and it has third and half band amplitude charac‐
teristics, respectively. This is summarized in Table1.
Please note that in the introduced filter bank structures, the direct scaling filters are not re‐
quired for scaling since the filter banks perform not only sub-band processing but also scal‐
ing. The direct scaling filters are used only to design the filter banks. The round brackets
“(· )” are used in this sense in Table 1. It is observed that the direct scaling filters have some
delays so that the conditions will be satisfied.
The number of
channels
The scaling
Factors
The length of the analysis, synthesis and direct scaling filters
2
3/2
6
9
(3)
3
2/3
8
6
(2)
Table 1.
The length of the designed filters in the introduced filter banks.
