With MM uniformly spaced
sub-bands, the sub-band width is
2πM
2
M
radians, implying that the sub-band signal can be
downsampled by a factor MM (but
not more than MM) without loss
of information. This is referred to as a "critically
sampled" filterbank. This maximal level of downsampling is
advantageous when storing or further processing the sub-band
signals. With critical sampling, the total number of
downsampled sub-band output samples equals the total number
of input samples. Assuming lossless sub-band processing,
the critically-sampled synthesis/analysis procedure is
illustrated in Figure 1:
Recall that one of our goals in filter design is to ensure
that
yn≃xn−d
y
n
x
n
d
for some integer delay
dd. From the block diagram above, one can see that
imperfect analysis filtering will contribute aliasing errors
to the sub-band signals. This aliasing distortion will
degrade
yn
y
n
if it is not cancelled by the synthesis filterbank. Though
ideal brick-wall filters
H
k
z
H
k
z
and
G
k
z
G
k
z
could easily provide perfect reconstruction
(i.e.,
yn=xn−d
y
n
x
n
d
), they would be unimplementable due to their
doubly-infinite impulse responses. Thus, we are interested
in the design of causal FIR filters that give near-perfect
reconstruction or, if possible, perfect reconstruction.
There are two principle approaches to the design of
filterbanks:
-
Classical: Approximate ideal brick wall filters to ensure
good sub-band isolation (i.e.,
frequency selectivity) and accept (a hopefully small
amount of) aliasing and thus reconstruction error.
-
Modern: Constrain the filters to give perfect (or
near-perfect) reconstruction and hope for good sub-band
isolation.
