Inverse and Parseval identity for a periodic orthogonal compact-filter transform

Fix row types $a,b\in \{h,g\}$ and row indices $k,\mathrm{\ell }\in \{0,\dots ,N\mathrm{/}2-1\}$. A coordinate can contribute to both rows only when $2k+j\equiv 2\mathrm{\ell }+r(\mathrm{m}\mathrm{o}\mathrm{d}N)$ for taps $j,r\in \{0,\dots ,L-1\}$. Hence the row inner product is the sum of $a_j{b}_r$ over such pairs. The difference $j-r$ lies in $[-L+1,L-1]$, whose width is $2L-2<N$; so at most one representative of the congruence class modulo $N$ occurs. Since $N$ and $2(\mathrm{\ell }-k)$ are even, that representative, when present, is $2s$ for an integer $s$. The inner product is therefore the filter correlation ${\sum }_j{a}_j{b}_{j-2s}$, which equals $1$ exactly for the same filter and $s=0$, and equals $0$ otherwise by [F1]. In the same-filter case $s=0$ forces $k=\mathrm{\ell }$, because $k,\mathrm{\ell }$ lie in a complete residue system modulo $N\mathrm{/}2$. Thus the $N$ rows are orthonormal.

A square real matrix with orthonormal rows satisfies $W{W}^T=I_N$; hence it is invertible with inverse $W^T$, so also $W^{T}W=I_N$. It follows that $\mathrm{\parallel }Wx{\mathrm{\parallel }}_2^2=x^T{W}^{T}Wx=\mathrm{\parallel }x{\mathrm{\parallel }}_2^2$. A multilevel transform is a composition of these one-level orthogonal maps and coordinate permutations; composing their transposes in reverse order gives its inverse and preserves norm.