In signal processing, for a sinusoidal signal Signal processing is the processing, amplification and interpretation of signals. ... In mathematics, the trigonometric functions are functions of an angle, important when studying triangles and modeling periodic phenomena. ...

is called the angular frequency (usually radians/second) and is the frequency (usually in hertz or cycles/second). For a constant frequency is seen as the time-derivative of the argument of the sine or cosine function. So in general for a sinusoidal function of time with its argument expressed as a general angle that changes in time, Angular frequency is a measure of how fast an object is rotating In physics (specifically mechanics and electrical engineering), angular frequency ω (also called angular speed) is a scalar measure of rotation rate. ... See Radian (band) for the Austrian trio. ... Sine waves of various frequencies; the lower waves have higher frequencies than those above. ... The hertz (symbol Hz) is the SI unit of frequency. ... Jump to: navigation, search In mathematics, the derivative is one of the two central concepts of calculus. ...

,

the time-derivative of that unwrapped angle, , is the instantaneous frequency of that sinusoid at any given time . That is the instantaneous angular frequency is defined to be

and the instantaneous frequency (Hz) is

.

The angle is unwrapped if it is continuous everywhere except at places where the absolute value of the jump discontinuity is less than radians. When an angle is wrapped, it is always expressed as its principal value which has magnitude that is less than or equal to . The difference between the wrapped and unwrapped angle is always an integer multiple of radians. In mathematics, a continuous function is a function in which arbitrarily small changes in the input produce arbitrarily small changes in the output. ... In mathematics, the absolute value (or modulus1) of a real number is its numerical value without regard to its sign. ... Jump to: navigation, search Continuous functions are of utmost importance in mathematics and applications. ... See also Cauchy principal value for its use in describing improper integrals In considering complex multiple-valued functions in complex analysis, the principal values of a function are the values along one chosen branch of that function, so it is single-valued. ...

The angle will be unwrapped if it set to

and contains no dirac delta functions with strength as large as ¹/₂ in magnitude. Often a constant integer multiple of is added to so that , but that is not necessary to fully unwrap . The Dirac delta function, introduced by Paul Dirac, can be informally thought of as a function δ(x) that has the value of infinity for x = 0, the value zero elsewhere, and a total integral of one. ...

Explicitly, the sinusoid expressed in terms of its instantaneous frequency is

or

where

.

See also

• Frequency modulation