Instantaneous frequency & phase

Once a narrowband signal is represented as an analytic signal (via Hilbert filtering or a complex wavelet), its instantaneous phase is the angle of the complex value, and its instantaneous frequency is the time derivative of that unwrapped phase divided by two pi. These quantities let one ask when in the cycle an event occurred, whether two rhythms are phase-locked, and how a rhythm's frequency drifts moment to moment, questions a static spectrum cannot answer.

Instantaneous frequency is fragile: differentiating a noisy phase amplifies noise, and any nonzero bandwidth means the estimate fluctuates. When the underlying signal is not a clean single tone, because of noise, overlapping rhythms, or non-sinusoidal (sawtooth-like) waveforms, the instantaneous frequency can even go negative or spike, which is a sign that the narrowband assumption has been violated rather than a physiological event.