**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 the instantaneous phase.

And for a *real-valued* function *s*(*t*), it is determined from the function's analytic representation, *s*_{a}(*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, 2*π*), 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*) = 2π*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 phase**; otherwise, 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}π and *m*_{2}π 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 *m*_{2}, 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 Δ*φ*[*n*] ≤ −π, and subtracting 2π whenever Δ*φ*[*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

## Complex representation

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:^{[3]}

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.

## See also

## References

**^**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. doi:10.1109/TSP.2008.924856. ISSN 1053-587X.**^**Blackledge, Jonathan M. (2006).*Digital Signal Processing: Mathematical and Computational Methods, Software Development and Applications*(2 ed.). Woodhead Publishing. p. 134. ISBN 1904275265.**^**Wang, S. (2014). "An Improved Quality Guided Phase Unwrapping Method and Its Applications to MRI".*Progress in Electromagnetics Research*.**145**: 273–286. doi:10.2528/PIER14021005.

## Further reading

- Cohen, Leon (1995).
*Time-Frequency Analysis*. Prentice Hall. - Granlund; Knutsson (1995).
*Signal Processing for Computer Vision*. Kluwer Academic Publishers.