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Instantaneous phase
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In signal processing, the instantaneous phase (or "local phase" or simply "phase") of a complex-valued function is the real-valued function:
(see arg function)
And for a real-valued signal it is determined from the signal's analytic representation, :
When is constrained to an interval such as or it is called the wrapped phase. Otherwise it is called unwrapped, which is a continuous function of argument assuming is a continuous function of Unless otherwise indicated, the continuous form should be inferred.
- Example 1: where and are positive values.
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- Example 2:
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For both of these sinusoidal examples, the local maxima of s(t) correspond to:
for integer values of Similarly, the local minima correspond to:
and the maximum rates of change correspond to:
For signals that are approximately sinusoidal, these properties can be used, e.g., in image processing and computer vision, to detect points that are close to edges or lines, and also to measure the position of these points with sub-pixel accuracy.
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- and the instantaneous frequency (Hz) is:
- .
Conversely, the unwrapped phase can be represented in terms of an instantaneous frequency.
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Encyclopedia
In signal processing, the instantaneous phase (or "local phase" or simply "phase") of a complex-valued function is the real-valued function:
(see arg function)
And for a real-valued signal it is determined from the signal's analytic representation, :
When is constrained to an interval such as or it is called the wrapped phase. Otherwise it is called unwrapped, which is a continuous function of argument assuming is a continuous function of Unless otherwise indicated, the continuous form should be inferred.
- Example 1: where and are positive values.
-
- Example 2:
-
For both of these sinusoidal examples, the local maxima of s(t) correspond to:
for integer values of Similarly, the local minima correspond to:
and the maximum rates of change correspond to:
For signals that are approximately sinusoidal, these properties can be used, e.g., in image processing and computer vision, to detect points that are close to edges or lines, and also to measure the position of these points with sub-pixel accuracy.
Instantaneous frequency
In general, the instantaneous angular frequency is defined as:
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- and the instantaneous frequency (Hz) is:
- .
Conversely, the unwrapped phase can be represented in terms of an instantaneous frequency. When it is actually constructed/derived this way, this process is called phase unwrapping:
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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:
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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. A vector-average phase can be obtained as the arg of the sum of the complex numbers.
See also
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