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. 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 the instantaneous phase.
And for a "real-valued" function "s"("t"), it is determined from the function's analytic representation, "s"a("t"):
where formula_3 represents the Hilbert transform of "s"("t").
When "φ"("t") is constrained to its principal value, either the interval (−, ] or [0, 2), 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.
Examples.
Example 1.
where "ω" > 0.
In this simple sinusoidal example, the constant "θ" is also commonly referred to as "phase" or "phase offset". "φ"("t") is a function of time; "θ" is not. 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. "φ"("t") is unambiguously defined.
Example 2.
where "ω" > 0.
In both examples the local maxima of "s"("t") correspond to "φ"("t") = 2"N" for integer values of "N". This has applications in the field of computer vision.
Instantaneous frequency.
Instantaneous angular frequency is defined as:
and instantaneous (ordinary) frequency is defined as:
where "φ"("t") must be the "unwrapped" instantaneous phase angle. If "φ"("t") is wrapped, discontinuities in "φ"("t") will result in Dirac delta impulses in "f"("t").
The inverse operation, which always unwraps phase, is:
This instantaneous frequency, "ω"("t"), can be derived directly from the real and imaginary parts of "s"a("t"), instead of the complex arg without concern of phase unwrapping.
2"m"1 and "m"2 are the integer multiples of necessary to add to unwrap the phase. At values of time, "t", where there is no change to integer "m"2, the derivative of "φ"("t") is
This representation is similar to the wrapped phase representation in that it does not distinguish between multiples of 2 in the phase, but similar to the unwrapped phase representation since it is continuous. A vector-average phase can be obtained as the arg of the sum of the complex numbers without concern about wrap-around.