Instantaneous phase and frequency

Instantaneous phase and frequency are important concepts in signal processing that occur in the context of the representation and analysis of time-varying functions.

When φ(t) is constrained to its principal value, either the interval (−π, π] or [0, 2π), it is called wrapped phase.

In the next example, we also see that the phase offset of a real-valued sinusoid is ambiguous unless a reference (sin or cos) is specified.

In both examples the local maxima of s(t) correspond to φ(t) = 2πN for integer values of N. This has applications in the field of computer vision.

A vector-average phase can be obtained as the arg of the sum of the complex numbers without concern about wrap-around.

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.