# Invertibility conditions of signal processed by filter banks

Professor Strang, crediting Professor Daubechies, concludes that the conditions for invertibility of a filtered signal are summarized in

\begin{align} F_0(\omega)H_0(\omega) - H_0(\omega+\pi)F_0(\omega+\pi)&=2\exp(-i\omega l)\\[2ex] P_0(\omega) - P_0(\omega + \pi) &=2\exp(-i\omega l) \end{align}

with $$H_0$$ and $$F_0$$ being the low-pass filter (analysis filter) and its inverse (synthesis filter) in a filter bank, respectively.

The factor $$e^{-i\omega l}$$ is attributed to a certain latency $$l$$ resulting from the causal nature of the filters (lower triangular matrix), yet the specific formula is not derived. I see intuitive the latency concept, but I want to ask for a concrete derivation of the exponential factor.