Periodic finite orthogonal compact-filter wavelet transform

Let $L\ge 1$, let $N$ be even with $N\ge 2L$, and let $h=(h_0,\dots ,h_{L-1})\in {\mathbb{R}}^L$. Set $g_j=(-1{)}^j{h}_{L-1-j}$ for $0\le j<L$, and set $h_j=g_j=0$ for all other integers $j$. The pair $(h,g)$ is an admissible orthogonal filter pair when ${\sum }_j{h}_j=\sqrt{2},{\sum }_j{g}_j=0,{\sum }_j{h}_j{h}_{j-2r}={\delta }_{r0},{\sum }_j{g}_j{g}_{j-2r}={\delta }_{r0},{\sum }_j{h}_j{g}_{j-2r}=0(r\in \mathbb{Z})\mathrm{.}$ Here ${\delta }_{r0}=1$ for $r=0$ and $0$ for $r\mathrm{\ne }0$. For $x\in {\mathbb{R}}^N$, define $a_k={\sum }_{j=0}^{L-1}{h}_j{x}_{(2k+j)<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>N},d_k={\sum }_{j=0}^{L-1}{g}_j{x}_{(2k+j)<mrow><mi\; mathvariant="normal">m</mi><mi\; mathvariant="normal">o</mi><mi\; mathvariant="normal">d</mi></mrow>N}(0\le k<N\mathrm{/}2)\mathrm{.}$ This is the one-level periodic analysis map. It has no padding: the indices are reduced modulo $N$. Its matrix $W$ has the $N\mathrm{/}2$ rows for $a_k$ followed by the $N\mathrm{/}2$ rows for $d_k$. Synthesis is, by definition, multiplication by $W^T$. A multilevel transform repeats this one-level rule on the approximation vector while its current length $M$ is even and $M\ge 2L$, then stops; the boundary convention at every level is periodic modulo that level's current length.