- An ordinary circular convolution of vectors of real numbers can be written as a matrix-vector product:
$${\bf Av},\, \text{ where } {\bf A} \text{ is circulant : } {\bf A}_{i+1,j+1} = {\bf A}_{i,j}$$
- We also have from representation theory that elements of groups (and fields) can be represented by matrices in the sense that the operations (+) and ($\cdot$) of elements are mapped onto the operations of addition and multiplication of matrices having elements of some more primitive group (or field).
An example is the complex numbers $$z=a+bi : \begin{bmatrix}a&b\\-b&a\end{bmatrix}, a,b \in \mathbb{R}, z \in \mathbb{C}$$
- The property of matrix block-multiplication lets us implement matrices of such more advanced groups or fields. For example implementing convolution using block-matrices with the complex vector of just one element : $a+bi$ would be represented by the following matrix:
$$\begin{bmatrix}a&b&0&0&0&0\\-b&a&0&0&0&0\\0&0&\ddots&\ddots&0&0\\0&0&\ddots&\ddots&0&0\\0&0&0&0&a&b\\0&0&0&0&-b&a\\\end{bmatrix}$$
And the vector would consist of $2\times 2$ blocks, one new for each new complex scalar in the vector.
Consider the case $a=b=1$, then our $2\times 2$ blocks would be: $$\begin{bmatrix}1&1\\-1&1\end{bmatrix}$$
But this leads to exactly the dyadic Haar-filterbank ( asum & diff pair followed by downsampling a factor of 2), if we instead had let the matrix constructed as above operate on a normal ($1\times 1$ block) vector.
Now to the question: is there some theory as to when and how it can be fruitful for filter-design or is it just a coincidence that the Haar-bank is so readily constructible with this approach over $\mathbb{C}$?