The concept of signed frequency (negative and positive frequency) can indicate both the rate and sense of
rotation
Rotation, or spin, is the circular movement of an object around a '' central axis''. A two-dimensional rotating object has only one possible central axis and can rotate in either a clockwise or counterclockwise direction. A three-dimensional ...
; it can be as simple as a wheel rotating clockwise or counterclockwise.
The rate is expressed in units such as revolutions (a.k.a. ''cycles'') per second (
hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that o ...
) or
radian/second (where 1 cycle corresponds to 2''π''
radian
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 ...
s).
Sinusoids
Let be a nonnegative
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit ti ...
with units of radians/second. Then the function has slope , which is called a negative frequency. But when the function is used as the argument of a cosine operator, the result is indistinguishable from . Similarly, is indistinguishable from . Thus any
sinusoid
A sine wave, sinusoidal wave, or just sinusoid is a mathematical curve defined in terms of the ''sine'' trigonometric function, of which it is the graph. It is a type of continuous wave and also a smooth periodic function. It occurs often in ...
can be represented in terms of positive frequencies. The sign of the underlying phase slope is ambiguous.
The ambiguity is resolved when the cosine and sine operators can be observed simultaneously, because leads by 1/4 cycle (= ''π''/2 radians) when , and lags by 1/4 cycle when . Similarly, a vector, ,
rotates counter-clockwise if , and clockwise if .
The sign of is also preserved in the
complex-valued function:
since R(''t'') and I(''t'') can be separately observed and compared. Although
clearly contains more information than either of its components, a common interpretation is that it is a simpler function, because it simplifies multiplicative
trigonometric calculations, which leads to its formal description as the ''
analytic representation'' of
.
The sum of an analytic representation with its complex conjugate extracts the actual real-valued function they represent. For instance:
which gives rise to the interpretation that cos(''ωt'') comprises ''both'' positive and negative frequencies. But the sum is actually a cancellation that results in an ambiguity (the sign of ''ω''). Any measure that indicates both frequencies includes a false positive (or ''alias''), because ''ω'' can have only one sign. The
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
, for instance, merely tells us that cos(''ωt'') cross-correlates equally well with as with . Nevertheless, it is sometimes useful (and mathematically valid) to treat this as the combination of two different (positive and negative) frequencies.
Applications
Simplifying the Fourier transform
Perhaps the most well-known application of negative frequency is the formula:
:
which is a measure of the energy in function
at frequency
When evaluated for a continuum of argument
the result is called the
Fourier transform
A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed ...
.
For instance, consider the function:
:
And:
:
Infinite duration is a simplification that facilitates this discussion. Looking at the first term above, when
the negative frequency
cancels the positive frequency, leaving just the constant coefficient
(because
), which causes the infinite integral to diverge. At other values of
the residual oscillations cause the integral to converge to zero. This idealized ''Fourier transform'' is usually written as:
:
For realistic durations, the divergences and convergences are less extreme, and smaller non-zero convergences (
spectral leakage) appear at many other frequencies, but the concept of negative frequency still apples.
Fourier's original formulation (
the sine transform and the cosine transform) requires an integral for the cosine and another for the sine. And the resultant trigonometric expressions are often less tractable than complex exponential expressions. (see
Analytic signal
In mathematics and signal processing, an analytic signal is a complex-valued function that has no negative frequency components. The real and imaginary parts of an analytic signal are real-valued functions related to each other by the Hil ...
,
Euler's formula#Relationship to trigonometry, and
Phasor
In physics and engineering, a phasor (a portmanteau of phase vector) is a complex number representing a sinusoidal function whose amplitude (''A''), angular frequency (''ω''), and initial phase (''θ'') are time-invariant. It is related to ...
)
Sampling of positive and negative frequencies and aliasing
See also
*
Angle#Sign
Notes
Further reading
Positive and Negative Frequencies* Lyons, Richard G. (Nov 11, 2010). Chapt 8.4. ''Understanding Digital Signal Processing'' (3rd ed.). Prentice Hall. 944 pgs. .
{{DEFAULTSORT:Negative Frequency
Waves
Concepts in physics