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In-Depth Information
A
B
C
Low-pass
filter
Low-pass
filter
High-pas
s
filter
Low-pa
ss
filter
Low-pass
filter
High-pass
filter
High-pas
s
filter
Low-pass
filter
Low-pas
s
filter
High-pass
filter
High-pass
filter
High-pass
filter
Low-pas
s
filter
High-pass
filter
F I GU R E 14 . 19
Decomposition of an input sequence into multiple bands by recur-
sively using a two-band split.
filters used for the two-band split are quite good, when we employ them in the tree structure
showninFigure
14.19
, the spectral characteristicsmay not be very good. For example, consider
the four-tap filter with filter coefficients shown in Table
14.6
. In Figure
14.20
we show what
happens to the spectral characteristics when we look at the two-band split (at point
A
in Figure
14.19
), the four-band split (at point
B
in Figure
14.19
), and the eight-band split (at point
C
in
Figure
14.19
). For a two-band split the magnitude characteristic is flat, with some aliasing.
When we employ these same filters to obtain a four-band split from the two-band split, there
is an increase in the aliasing. When we go one step further to obtain an eight-band split, the
magnitude characteristic deteriorates substantially, as evidenced by Figure
14.20
. The various
bands are no longer clearly distinct. There is significant overlap between the bands, and hence
there will be a significant amount of aliasing in each band.
In order to see why there is an increase in distortion, let us follow the top branch of the tree.
The path followed by the signal is shown in Figure
14.21
(a). As we will show later (Section
14.8
), the three filters and downsamplers can be replaced by a single filter and downsampler
as shown in Figure
14.21
(b), where
z
2
z
4
A
(
z
)
=
H
L
(
z
)
H
L
(
)
H
L
(
)
(92)
