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The quantity you might be looking for is signal-front delay, which is the delay of the beginning of a signal passing through a linear system. It is simply the largest value $\tau_{sf}$ (in samples) for which $$h[n]=0,\qquad n<\tau_{sf}\tag{1}$$ is satisfied, where $h[n]$ is the system's impulse response. If an input, or change in the input, starts at $n_0$...

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