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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:
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, sa(t):[2]
where 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 sa(t) is a continuous function of t. Unless otherwise indicated, the continuous form should be inferred.
Instantaneous phase vs time. The function has two true discontinuities of 180° at times 21 and 59, indicative of amplitude zero-crossings. The 360° "discontinuities" at times 19, 37, and 91 are artifacts of phase wrapping.Instantaneous phase of a frequency-modulated waveform: MSK (minimum shift keying). A 360° "wrapped" plot is simply replicated vertically two more times, creating the illusion of an unwrapped plot, but using only 3x360° of the vertical axis.
^Sejdic, E.; Djurovic, I.; Stankovic, L. (August 2008). "Quantitative Performance Analysis of Scalogram as Instantaneous Frequency Estimator". IEEE Transactions on Signal Processing. 56 (8): 3837–3845. Bibcode:2008ITSP...56.3837S. doi:10.1109/TSP.2008.924856. ISSN 1053-587X. S2CID 16396084.
^Blackledge, Jonathan M. (2006). Digital Signal Processing: Mathematical and Computational Methods, Software Development and Applications (2 ed.). Woodhead Publishing. p. 134. ISBN 1904275265.
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Global Information
![{\displaystyle {\begin{aligned}\varphi (t)&=\arg\{s_{\mathrm {a} }(t)\}\\[4pt]&=\arg\{s(t)+j{\hat {s}}(t)\},\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1e6cc5d63de82e6f1472f9fbf5e297ebd5f3971d)
