In signal processing, the instantaneous phase (or "local phase" or simply "phase") of a complex-valued function <math>x(t)\,</math> is the real-valued function:
- <math>
\phi(t) = \arg(x(t)).\,</math> (see arg function) And for a real-valued signal <math>s(t)\,</math> it is determined from the signal's analytic representation, <math>s_\mathrm{a}(t)\,</math>:
- <math>
\phi(t) = \mathrm{arg}( s_\mathrm{a}(t) ) \,</math> When <math>\phi(t)\,</math> is constrained to an interval such as <math>(-\pi, \pi]\,</math> or <math>[0, 2\pi),\,</math> it is called the wrapped phase. Otherwise it is called unwrapped, which is a continuous function of argument <math>t,\,</math> assuming <math>s_\mathrm{a}\,</math> is a continuous function of <math>t.\,</math> Unless otherwise indicated, one should generally infer the continuous form.
- Example 1: <math>s(t) = A\cdot \cos(\omega t + \theta),\,</math> where <math>A\,</math> and <math>\omega\,</math> are positive values.
- <math>
s_\mathrm{a}(t) = A\cdot e^{i (\omega t +\theta)} \,</math>
-
- <math>\phi(t) = \omega t + \theta\,</math>
- Example 2: <math>s(t) = A\cdot \sin(\omega t) = A\cdot \cos\left(\omega t -\begin{matrix} \frac{\pi}{2}\end{matrix}\right)\,</math>
- <math>
s_\mathrm{a}(t) = A\cdot e^{i \left(\omega t -\begin{matrix} \frac{\pi}{2}\end{matrix}\right)} \,</math>
-
- <math>\phi(t) = \omega t -\begin{matrix} \frac{\pi }{2}\end{matrix}\,</math>
For both of these sinusoidal examples, the local maxima of s(t) correspond to:
- <math>\phi(t) = N\cdot 2\pi,\,</math>
for integer values of <math>N.\,</math> Similarly, the local minima correspond to:
- <math>\phi(t) = \pi + N\cdot 2\pi,\,</math>
and the maximum rates of change correspond to:
- <math>\phi(t)= \begin{matrix} \frac{\pi}{2}\end{matrix} + N\cdot \pi,\,</math>
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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Instantaneous frequency
In general, the instantaneous angular frequency is defined as:
-
- <math>\omega(t) = \phi^\prime(t) = {d \over dt} \phi(t)\,</math>
- and the instantaneous frequency (Hz) is:
- <math> f(t) = \frac{1}{2 \pi} \phi^\prime(t) \ </math>.
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:
-
<math>\phi(t) = 2 \pi \int_{-\infty}^{t} f(\tau)\, d \tau \ </math> <math>= 2 \pi \int_{0}^{t} f(\tau)\, d \tau + 2 \pi \int_{-\infty}^{0} f(t)\, dt </math> <math>= 2 \pi \int_{0}^{t} f(\tau)\, d \tau + \phi(0) </math>
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:
-
<math>e^{i \phi(t)}\,</math> s_\mathrm{a}(t)|}\,</math> <math>= \cos(\phi(t)) + i\cdot \sin(\phi(t))\,</math> (Euler's formula)
This representation is similar to the wrapped phase representation in that it does not distinguish between multiples of <math>2 \pi</math> 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.
References
- Leon Coen, Time-Frequency Analysis, Prentice Hall, 1995.
- Granlund and Knutsson, Signal Processing for Computer Vision, Kluwer Academic Publishers, 1995.
