estimating the phase of a broadband signal

you want to extract instantaneous phase of a signal, the textbooks say to narrowband filter before computing the analytic signal. but why?

using a broadband signal, the resulting phase corresponds to the circular mean of the analytic signals of the constituents (here: 🔴=🟢+🔵). pic.twitter.com/uiK1t6q0kN

— Natalie Schaworonkow (@nschawor) September 30, 2022