If a signal consists of a single component and has the form $f(t) = A(t) e^{2πiφ(t)}$, then $A(t) = |f(t)|^2$$A(t) = |f(t)|$ is its amplitude, $φ(t) = \arg(f(t))/2π$ is its phase, and its instantaneous frequency is the time derivative of the phase $ν_f(t) = φ'(t)$. As such, it satisfies the identity $$|f(t)|^2 ν_f(t) = \overline{f(t)} \left({1\over{4πi}}\overleftrightarrow{d\over dt}\right) f(t),$$ where $\overleftrightarrow{d/dt}$ denotes $d/dt$ applied to the right, minus $d/dt$ applied to the left - a standard notation in the physics literature. Alternatively, and equivalently, it's the imaginary part of $(1/2π) d/dt \ln f(t)$.
The spectrum for the signal will be a line which resides at frequency $ν_f(t)$ at time $t$, with an intensity proportional to $A(t)$.
If a signal consists of multiple components, then they each need to be separated out first. An example can be seen here
https://www.youtube.com/watch?v=nd2J4xTrSHQ
Notice, in particular, what happens when the voice and tone overrun each other - there's a collision of the two at around 17 seconds in the video. A sufficient (but not necessary) condition to separate out the components is that their corresponding instantaneous spectra not be colliding like that. If they're not colliding, then they can be each be separately filtered out, and the single-component definition applied to each one. If they are colliding, then you need to find another way to separate them first (a kind of intelligent "unmixing" algorithm that unmixes even the places where they overlap). An example vividly displaying that kind of separation is here
https://www.youtube.com/shorts/Sl1SwkiIo30
(The scalogram, however, continues to depict the mixed sound. I don't show the separation there, it's only shown in the spectrogram.)
