Digital Signal Processing Reference
In-Depth Information
y[n] = (2aa + 2bb)x[n1]:
The Haar transform is a very simple wavelet transform. If we carefully choose
our coecients a and b above, we will have the Haar transform. If 2aa + 2bb = 1,
then y[n] = x[n1], meaning that the output is the same as the input, only delayed
by 1. We can handle the coecients a and b in one of two ways. Either we can
make them both 1, which makes the lters easier to implement, and then remember
to divide all y[n] values by (21
2
+ 21
2
) = 4, or we can choose our a and b
values to make sure that (aa + bb) = 1. If we force aa = 1=2, and bb = 1=2, then
a = b = 1=
p
2. These values correspond to the Haar transform. The reason we
want (aa + bb) = 1 and not 2(aa + bb) = 1 has to do with the down/up-samplers,
discussed later in this chapter.
9.2
Quadrature Mirror Filters and Conjugate
Quadrature Filters
Sometimes two-channel lter banks are referred to as subband coders. Here, we
will use the terms interchangeably, though subband coders may have more than two
channels. You may also see a two-channel lter bank called a quadrature mirror lter
(QMF), or a conjugate quadrature lter (CQF), though \two-channel lter bank"
is the most general of these three terms. The QMF and CQF both put conditions
on the lter coecients to cancel aliasing terms and get perfect reconstruction. In
other words, a quadrature mirror lter and a conjugate quadrature lter are both
types of two-channel lter banks, with the same structure. The only dierences
between a QMF, CQF, or another two-channel lter bank are the lter coecients.
We specify the coecients in terms of one of the analysis lters, and we will label
these coecients h
0
. The other lter coecients (h
1
;g
0
;g
1
) are typically obtained
from h
0
. After all, we cannot just use any values we want for the lter coecients
or the end result will not match our original signal.
Essentially, a quadrature mirror lter species that h
1
uses the same lter co-
ecients as h
0
, but negates every other value. The reconstruction lter g
0
is the
same as h
0
, and the lter coecients g
1
=h
1
, meaning that we copy h
1
to g
1
and
negate all of g
1
's values. See Figure 9.4 for an example using two coecients. An
interesting note about the QMF is that this does not work in general. Practically
speaking, the QMF has only two coecients (unless one uses IIR lters).
The conjugate quadrature lter species h
1
as a reversed version of h
0
, with
