Phase reversal

In physics and mathematics, the phase of a periodic function $F$ of some real variable $t$ (such as time) is an angle-like quantity representing the fraction of the cycle covered up to $t$. It is denoted $\phi(t)$ and expressed in such a scale that it varies by one full turn as the variable $t$ goes through each period (and $F(t)$ goes through each complete cycle). It may be measured in any angular unit such as degrees or radians, thus increasing by 360° or $2\pi$ as the variable $t$ completes a full period.

This convention is especially appropriate for a sinusoidal function, since its value at any argument $t$ then can be expressed as $\phi(t)$, the sine of the phase, multiplied by some factor (the amplitude of the sinusoid). (The cosine may be used instead of sine, depending on where one considers each period to start.)

Usually, whole turns are ignored when expressing the phase; so that $\phi(t)$ is also a periodic function, with the same period as $F$, that repeatedly scans the same range of angles as $t$ goes through each period. Then, $F$ is said to be "at the same phase" at two argument values $t_1$ and $t_2$ (that is, $\phi(t_1) = \phi(t_2)$) if the difference between them is a whole number of periods.

The numeric value of the phase $\phi(t)$ depends on the arbitrary choice of the start of each period, and on the interval of angles that each period is to be mapped to.

The term "phase" is also used when comparing a periodic function $F$ with a shifted version $G$ of it. If the shift in $t$ is expressed as a fraction of the period, and then scaled to an angle $\varphi$ spanning a whole turn, one gets the ''phase shift'', ''phase offset'', or ''phase difference'' of $G$ relative to $F$. If $F$ is a "canonical" function for a class of signals, like $\sin(t)$ is for all sinusoidal signals, then $\varphi$ is called the ''initial phase'' of $G$.

Mathematical definition

Let $F$ be a periodic signal (that is, a function of one real variable), and $T$ be its period (that is, the smallest positive real number such that $F(t + T) = F(t)$ for all $t$). Then the ''phase of ''$F$ ''at'' any argument $t$ is
:\phi(t) = 2\pi\left !\!\left[\frac\right!\!\right">frac\right.html" ;"title="!\!\left[\frac\right">!\!\left[\frac\right!\!\right/math>

Here $[\![\,\cdot\,]\!]\!\,$ denotes the fractional part of a real number, discarding its integer part; that is, $[\![ x ]\!] = x - \left\lfloor x \right\rfloor\!\,$; and $t_0$ 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 $T$ seconds, and is pointing straight up at time $t_0$. The phase $\phi(t)$ is then the angle from the 12:00 position to the current position of the hand, at time $t$, measured clockwise.

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

The formula above gives the phase as an angle in radians between 0 and $2\pi$. To get the phase as an angle between $-\pi$ and $+\pi$, one uses instead
:\phi(t) = 2\pi\left(\left !\!\left[\frac + \frac\right!\!\right">frac_+_\frac\right.html" ;"title="!\!\left[\frac + \frac\right">!\!\left[\frac + \frac\right!\!\right- \frac\right)

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 $\phi(t)$ of a periodic signal is periodic too, with the same period $T$:
:$\phi(t + T) = \phi(t)\quad\quad$ for all $t$.

The phase is zero at the start of each period; that is
:$\phi(t_0 + kT) = 0\quad\quad$ for any integer $k$.

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

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

Adding and comparing phases

Since phases are angles, any whole full turns should usually be ignored when performing arithmetic operations on them. That is, the sum and difference of two phases (in degrees) should be computed by the formulas
:360\,\left !\!\left[\frac\right!\!\right">frac\right.html" ;"title="!\!\left[\frac\right">!\!\left[\frac\right!\!\rightquad\quad and \quad\quad 360\,\left !\!\left[\frac\right!\!\right">frac\right.html" ;"title="!\!\left[\frac\right">!\!\left[\frac\right!\!\right/math>

respectively. Thus, for example, the sum of phase angles is 30° (, minus one full turn), and subtracting 50° from 30° gives a phase of 340° (, plus one full turn).

Similar formulas hold for radians, with $2\pi$ instead of 360.

Phase shift

The difference $\varphi(t) = \phi_G(t) - \phi_F(t)$ between the phases of two periodic signals $F$ and $G$ is called the ''phase difference'' or ''phase shift'' of $G$ relative to $F$. At values of $t$ when the difference is zero, the two signals are said to be ''in phase'', otherwise they are ''out of phase'' with each other.

In the clock analogy, each signal is represented by a hand (or pointer) of the same clock, both turning at constant but possibly different speeds. The phase difference is then the angle between the two hands, measured clockwise.

The phase difference is particularly important when two signals are added together by a physical process, such as two periodic sound waves emitted by two sources and recorded together by a microphone. This is usually the case in linear algebra">linear