For an a real signal we define the analytic signal corresponding to it by:
where denotes the Hilbert transform of :
Then the instantaneous (angular) frequency of is:
To a handwaving level of exposition we are finding a complex signal of the form:
where is a real amplitude, and is a real phase such that:
Then the instantaneous frequency is:
Often the analytic signal coresponding to a real signal can be written down without going via the Hilbert Transform.