Editing Phase (waves) (section)

== Mathematical definition ==
Let the signal <math>F</math> be a periodic function of one real variable, and <math>T</math> be its period (that is, the smallest positive [[real number]] such that <math>F(t + T) = F(t)</math> for all <math>t</math>).  Then the ''phase of <math>F</math> at'' any argument <math>t</math> is 
<math display="block">\varphi(t) = 2\pi\left[\!\!\left[\frac{t - t_0}{T}\right]\!\!\right]</math>

Here <math>[\![\,\cdot\,]\!]\!\,</math><!-- The double square brackets benefit from the "hinted" latex renderer. --> denotes the fractional part of a real number, discarding its integer part; that is, <math>[\![ x ]\!]  = x - \left\lfloor x \right\rfloor\!\,</math>; and <math>t_0</math> is an arbitrary "origin" value of the argument, that one considers to be the beginning of a cycle.

This concept can be visualized by imagining a [[clock]] with a hand that turns at constant speed, making a full turn every <math>T</math> seconds, and is pointing straight up at time <math>t_0</math>.  The phase <math>\varphi(t)</math> is then the angle from the 12:00 position to the current position of the hand, at time <math>t</math>, measured [[clockwise]].

The phase concept is most useful when the origin <math>t_0</math> is chosen based on features of <math>F</math>.  For example, for a sinusoid, a convenient choice is any <math>t</math> where the function's value changes from zero to positive.

The formula above gives the phase as an angle in radians between 0 and <math>2\pi</math>.  To get the phase as an angle between <math>-\pi</math> and <math>+\pi</math>, one uses instead
<math display="block">\varphi(t) = 2\pi\left(\left[\!\!\left[\frac{t - t_0}{T} + \frac{1}{2}\right]\!\!\right] - \frac{1}{2}\right)</math>

The phase expressed in degrees (from 0° to 360°, or from −180° to +180°) is defined the same way, except with "360°" in place of "2π".

===Consequences===
With any of the above definitions, the phase <math>\varphi(t)</math> of a periodic signal is periodic too, with the same period <math>T</math>:
<math display="block">\varphi(t + T) = \varphi(t)\quad\quad \text{ for all } t.</math>

The phase is zero at the start of each period; that is
<math display="block">\varphi(t_0 + kT) = 0\quad\quad \text{ for any integer } k.</math>

Moreover, for any given choice of the origin <math>t_0</math>, the value of the signal <math>F</math> for any argument <math>t</math> depends only on its phase at <math>t</math>. Namely, one can write <math>F(t) = f(\varphi(t))</math>, where <math>f</math> is a function of an angle, defined only for a single full turn, that describes the variation of <math>F</math> as <math>t</math> ranges over a single period.

In fact, every periodic signal <math>F</math> with a specific [[waveform]] can be expressed as 
<math display="block">F(t) = A\,w(\varphi(t))</math>
where <math>w</math> is a "canonical" function of a phase angle in 0 to 2π, that describes just one cycle of that waveform; and <math>A</math> is a scaling factor for the amplitude. (This claim assumes that the starting time <math>t_0</math> chosen to compute the phase of <math>F</math> corresponds to argument 0 of <math>w</math>.)