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:
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.
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.
2m1π and m2π are the integer multiples of π necessary to add to unwrap the phase. At values of time, t, where there is no change to integer m2, the derivative of φ(t) is
For discrete-time functions, this can be written as a recursion:
Discontinuities can then be removed by adding 2π whenever Δφ[n] ≤ −π, and subtracting 2π whenever Δφ[n] > π. That allows φ[n] to accumulate without limit and produces an unwrapped instantaneous phase. An equivalent formulation that replaces the modulo 2π operation with a complex multiplication is:
where the asterisk denotes complex conjugate. The discrete-time instantaneous frequency (in units of radians per sample) is simply the advancement of phase for that sample
In some applications, such as averaging the values of phase at several moments of time, it may be useful to convert each value to a complex number, or vector representation:
This representation is similar to the wrapped phase representation in that it does not distinguish between multiples of 2π 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.