Instantaneous phase and frequency

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Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions.^[1] The instantaneous phase (also known as local phase or simply phase) of a complex-valued function s(t), is the real-valued function:

${\displaystyle \varphi (t)=\arg\{s(t)\},}$

where arg is the complex argument function. The instantaneous frequency is the temporal rate of change of the instantaneous phase.

And for a real-valued function s(t), it is determined from the function's analytic representation, s_a(t):^[2]

${\displaystyle {\begin{aligned}\varphi (t)&=\arg\{s_{\mathrm {a} }(t)\}\\[4pt]&=\arg\{s(t)+j{\hat {s}}(t)\},\end{aligned}}}$

where ${\displaystyle {\hat {s}}(t)}$ represents the Hilbert transform of s(t).

When φ(t) is constrained to its principal value, either the interval (−π, π] or [0, 2π), it is called wrapped phase. Otherwise it is called unwrapped phase, which is a continuous function of argument t, assuming s_a(t) is a continuous function of t. Unless otherwise indicated, the continuous form should be inferred.