Quadrature mirror filter
= Quadrature mirror filter
{title2=$|m(\xi)|^2+|m(\xi+\pi)|^2=1$}
For an orthonormal <wavelet> refinement filter, $\sum_kh_k\overline{h_{k-2n}}=\delta_{n0}$ is equivalent to the displayed identity. A high-pass filter can be chosen with coefficients $g_k=(-1)^k\overline{h_{1-k}}$. In frequency, its symbol is $e^{-i\xi}\overline{m(\xi+\pi)}$, up to a constant phase. The two channels split an approximation space into a coarser approximation and its <orthogonal complement>.