Factorization of filter coefficient matrices

Recall from Section 4, that the wavelet matrix A can be partitioned into m X m submatrices as follows A = (AqAi. .. Aq). Provided that the orthogonality condition ) satisfied, the wavelet matrix can 

The symbol 0 denotes the polynomial product which is defined by 

If m = 3 and q = 2 then A = (Ao A Ai) with each Aj having dimension 4x4 thus, A has size mxm(q-l-l)= 3x9. Assuming the orthogonality condition is satisfied then 

Our aim is to construct Q and each projection matrix Ri (for i = 1. q). We first consider the representation of Q. The regularity condition Ik = /m places a constraint on the first row of Q. This is equivalent to setting the first row of Q to (l/yinjlm where Im denotes an m x 1 column vector of ones. 

The remaining m - 1 rows are constructed ensuring the orthogonality of Q is maintained. If the last m - 1 rows are calculated by 

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