In general, the instantaneous angular frequency is defined as
- and the instantaneous frequency (Hz) is:
-
- .
The inverse operation is:
For discrete-time functions, this can be written as a recursion:
Discontinuities can then be removed by adding 2π whenever and subtracing 2π whenever That allows to accumulate without limit and produces an unwrapped instantaneous phase. An equivalent formulation that replaces the modulo 2π operation with a complex multiplication is:
- where the asterisk denotes complex conjugate.
Read more about this topic: Instantaneous Phase
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