In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
and
mathematics, the phase of a
periodic function
A periodic function is a function that repeats its values at regular intervals. For example, the trigonometric functions, which repeat at intervals of 2\pi radians, are periodic functions. Periodic functions are used throughout science to des ...
of some
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
variable
(such as time) is an
angle
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the '' vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
-like quantity representing the fraction of the cycle covered up to
. It is denoted
and expressed in such a
scale that it varies by one full
turn as the variable
goes through each
period
Period may refer to:
Common uses
* Era, a length or span of time
* Full stop (or period), a punctuation mark
Arts, entertainment, and media
* Period (music), a concept in musical composition
* Periodic sentence (or rhetorical period), a concept ...
(and
goes through each complete cycle). It may be
measured in any
angular unit
In Euclidean geometry, an angle is the figure formed by two rays, called the '' sides'' of the angle, sharing a common endpoint, called the ''vertex'' of the angle.
Angles formed by two rays lie in the plane that contains the rays. Angles a ...
such as
degrees or
radians
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before that ...
, thus increasing by 360° or
as the variable
completes a full period.
This convention is especially appropriate for a
sinusoidal function, since its value at any argument
then can be expressed as
, the
sine of the phase, multiplied by some factor (the
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of am ...
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
is also a periodic function, with the same period as
, that repeatedly scans the same range of angles as
goes through each period. Then,
is said to be "at the same phase" at two argument values
and
(that is,
) if the difference between them is a whole number of periods.
The numeric value of the phase
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
with a shifted version
of it. If the shift in
is expressed as a fraction of the period, and then scaled to an angle
spanning a whole turn, one gets the ''phase shift'', ''phase offset'', or ''phase difference'' of
relative to
. If
is a "canonical" function for a class of signals, like
is for all sinusoidal signals, then
is called the ''initial phase'' of
.
Mathematical definition
Let
be a periodic signal (that is, a function of one real variable), and
be its period (that is, the smallest positive
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every ...
such that
for all
). Then the ''phase of ''
''at'' any argument
is
: