This module derives 2-D filter bank from 1-D Haar transform.
Digital filter banks have been actively studied since the 1960s,
whereas Wavelet theory is a new subject area that was developedin the 1980s, principally by French and Belgian mathematicians,
notably Y. Meyer, I. Daubechies, and S. Mallat. The two topics arenow firmly linked and of great importance for signal analysis
and compression.
The 2-band filter bank
Recall the 1-D Haar transform from our
previous discussion .
where
We can write this in expanded form as:
More generally if
is a longer
sequence and the results are placed in two separate sequences
and
, we define the process as:
These can be expressed as 2 FIR filters with tap vectors
and
. Hence as z-transforms,
and
become:
where
where
.
(We shall later extend these filters to be more complicated.)
In practice, we only calculate
and
at alternate (say even) values of
so that the total number of
samples in
and
is the same as in
.
We may thus represent the Haar transform operation by a pair
of filters followed by downsampling by 2, as shown in
(a). This is known as a 2-band
analysis filter bank.
Two-band filter banks for analysis (a) and reconstruction
(b).
In
this
equation in our discussion of the Haar transform, to
reconstruct
from
we calculated
. For long sequences this may be written:
Since
and
are only calculated at even values of
, we may assume that they are
zero at odd values of
. We may
then combine
and
into a single expression for
, valid for all
:
or as z-transforms:
where
In
the signals
and
are not really the same as
and
in
and
because those in
and
have not had alternate samples set to
zero. Also, in
is the reconstructed output whereas in
and
it is the input signal.
To avoid confusion we shall use
,
and
for the signals in
so it becomes:
We may show this reconstruction operation as upsampling
followed by 2 filters, as in
(b).
If
and
are not the same as
and
, how do they relate to each other?
Now
Therefore
is a polynomial in
,
comprising
only the terms in even powers
of
from
. This may be written as:
Similarly
This is our general model for downsampling by 2, followed byupsampling by 2 as defined in
and
.
