This module shows how to compute the scaling function. It also has a section with a proof for an assumption made for the computation.
Given coefficients
that satisfy the regularity conditions, we can
iteratively calculate samples of
on a fine grid of points
using the
cascade algorithm . Once we
have obtained
, the wavelet scaling equation can be used to construct
.
In this discussion we assume that
is causal with impulse response length
. Recall, from our
discussion of the
regularity conditions , that this implies
will have compact support on the interval
. The cascade algorithm is described below.
Consider the scaling function at integer times
:
Knowing that
for
, the previous equation can be written using an
x
matrix. In the case where
, we have
The matrix
is
structured as a
row-decimated convolution
matrix . From the matrix equation above (
[link] ), we see that
must be (some scaled version of) the eigenvector
of
corresponding to eigenvalue
. In general, the nonzero values of
,
i.e. ,
, can be calculated by appropriately scaling the eigenvector
of the
x
row-decimated convolution matrix
corresponding to the
eigenvalue
. It can be shown that this eigenvector must be
scaled so that
.
Given
, we can use the scaling equation to determine
:
This produces the
non-zero samples
.
Given
, the scaling equation can be used to find
:
where
denotes the impulse response of
,
i.e. , a 2-upsampled version of
, and where
. Note that
is the result of convolving
with
.
Given
, another convolution yields
:
where
is a 4-upsampled version of
and where
.
At the
stage,
is calculated by convolving the result of the
stage with a
-upsampled version of
:
For
, this gives a very good approximation of
. At this point, you could verify the key properties of
, such as orthonormality and the satisfaction of the
scaling equation.
In
[link] we show steps 1 through 5
of the cascade algorithm, as well as step 10, using Daubechies'db2 coefficients (for which
).