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.


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