How to translate “phase” into “delay”

Suppose that a system induces a phase of -90 degree, how do I practically measure the delay in the signal coming out of the system? I think time lag would be a more useful measure than phase.

Clearly these have different units, what is the conversion factor that tells me how much time lag is induced due to the system?

• Well, frequency will tell you how many times the oscillator spins a full rotation (360 degrees or 2pi radians) per second and you are trying to figure out how long it would take to spin some amount. If I'm understanding your question correctly, you should be able to figure that out pretty easily! – Alan Wolfe Apr 12 '15 at 3:28
• phase delay: $$\tau_\phi = -\frac{\phi}{\omega}$$ group delay: $$\tau_\text{g} = -\frac{d\phi}{d\omega}$$ gotta know what $\omega$ is. the practical way i used to measure delay was with a dual-channel scope and i would show the sinusoid applied to the input of the system along with the sinusoid coming out. i would measure (usually from zero crossings) the (phase) delay directly and calculate the phase shift from that. anyway, that's how i did it as an undergrad 4 decades ago. – robert bristow-johnson Apr 12 '15 at 3:37
• A pitfall with phase delay and group delay is that if the system's impulse response consists of two identical impulses far apart then the reported delay will be half way between the peaks. In real world you would experience that the signal is delayed by the position of the first peak and then repeats after the second peak. Looking at the complex envelope of the impulse response would tell you this more practical info. – Olli Niemitalo Apr 12 '15 at 6:11
• @OlliNiemitalo, great to see you here. (we gotta load you up with reputation... Dilip or Peter can you do something about that? Olli deserves at least 100 initial rep like i got when i showed up here.)  certainly there are special cases like $$h(t) = \delta(t-t_1) + \delta(t-t_2)$$ which detract from the generalization, but group delay is often what is "the reported delay". not with a single sinusoid (then it's phase delay), but with a "blap" of something localized in frequency. – robert bristow-johnson Apr 13 '15 at 1:52