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:
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.
^Blackledge, Jonathan M. (2006). Digital Signal Processing: Mathematical and Computational Methods, Software Development and Applications (2 ed.). Woodhead Publishing. p. 134. ISBN1904275265.