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Instead of phase delay $td(f)$, use group delay $\tau(f)$: $$\tau(f) = -\frac{d}{d\omega}\phi(f),\tag{1}$$ calculated as the negative of the derivative of the phase $\phi(f)$ with respect to the angular frequency $\omega$, which is defined by: $$\omega = 2\pi f\quad\Leftrightarrow\quad f = \frac{\omega}{2\pi}.\tag{2}$$ Using your phase shift $\phi(f)$: $$\... 2 Because when a quantity can be complex, and even when it is just real, the absolute squared difference |f-g|^2 can be expressed in both domain (complex and real) as:$$|f-g|^2 = (f-g)^H(f-g) and of course this is correct as well for reals. This setting is often related to Hilbert spaces.