In Signal_processing, the '''instantaneous phase''' (or local phase or simply phase) of a signal is a function defined as
:
where is the Analytic_signal of . In general, is a complex-valued function and the phase is the argument of that value seen as a function of .
== Phase representations ==
=== Unwrapped phase ===
In certain applications, the phase is treated as a continuous function of provided that also is a continuous function of . This implies that the argument function should not be restricted to an interval of length . This unrestricted phase is sometimes referred to as ''unwrapped phase''.
For example, in some application it is the absolute difference in phase between two points and which is the interesting parameter. If the phase value is restricted to a limited range before the difference is taken, any multiple of is then lost in the measurement.
Also, Instantaneous_frequency is often defined as the derivative of the phase with respect to , and in this case it is important that continuity of the phase is preserved and the unwrapped phase is used.
=== Wrapped phase ===
In practice, it is sometimes necessary to obtain the phase by means of a function which restricts the value to an interval of length , e.g., the interval . This is the ''wrapped phase''. The wrapped phase, however, introduces discontinuities at the interval boundaries which can make it tricky to use this phase value in further computations, e.g., differentiations.
Some applications work better with the wrapped phase. For example, if the phase is used to find local extreme points, it is better to have a phase value that is restricted to a limited range.
=== Complex representation ===
In some applications, it may be useful to represent the phase as the complex number
:
This representation is similar to the wrapped phase representation in that it does not distinguish between multiples of in the phase, but similar to the unwrapped phase representation since it is continuous.
== Applications ==
For a sinusoid signal
:
which has an analytic signal
:
it follows that
:
depending on if the phase value should be limited or not. Using the second version, it is the case that
* when assumes a local maximum value
* when assumes a local minimum value
* for all where changes with maximum rate from a minimum to maximum value, or vice versa.
Consequently, for signals which are approximately sinusoidal, the phase value indicates local properties of the signal in terms of extreme points and sudden transitions from a larger value to a smaller, or vice versa. This can be used, e.g., in Image_processing and Computer_vision to detect points which are close to edges or lines, and also to measure the position of these points with sub-pixel accuracy.
== References ==
* Leon Coen, Time-Frequency Analysis, Prentice Hall, 1995.
* Granlund and Knutsson, Signal Processing for Computer Vision, Kluwer Academic Publishers, 1995.
== See also ==
* Analytic_signal
* Instantaneous_frequency
Category:Signal_processing