The g-Invariant Wavelet Transform

Letbe the fundamental permutation (circulant) matrix [&make_named_href('', "node16.html#davis","[4]")],

Acting on vectors,, of length (period) n

Letbe the fundamental g-circulant matrix [4],

Acting on vectors,, of length (period) n,

We will need a few basic properties of circulant and g-circulant matrices.

1. 
2. 
3. For,

These imply that:

4.

where.

5.

For, or g=n-1,is equivalent to the anti-identity, or time-reversal transformation.

We letand define the set of transformations

The setis closed under multiplication (composition). Now, consider the set of transformations,, which is the direct sum

This set is closed under multiplication modulo multiplication from the left by even powers of. To see this, since, we need consider only terms fromandHowever, it is evident from the identity

that both sets belong to, modulo multiplication from the left by even powers of.

We define the g-invariant wavelet transform,. Consider a sequence,. Calculate the wavelet transforms,, of the sequences obtained from the transformations ofby the elements in. The g-invariant transformation is

where

for an additive information cost function, M. We note that any additive information cost function is g-invariant. To complete the transform when the minimum is not unique we proceed to distinguish the elements ofas described in the previous section.