Negative frequency

In mathematics, the concept of signed frequency (negative and positive frequency) can indicate both the rate and sense of rotation; it can be as simple as a wheel rotating clockwise or counterclockwise.

The rate is expressed in units such as revolutions (a.k.a.

has a positive frequency of +1 radian per unit of time and rotates counterclockwise around a unit circle, while the vector

has a negative frequency of -1 radian per unit of time, which rotates clockwise instead.

Let ω > 0 be an angular frequency with units of radians/second.

Then the function f(t) = −ωt + θ has slope −ω, which is called a negative frequency.

But when the function is used as the argument of a cosine operator, the result is indistinguishable from cos(ωt − θ).

Thus any sinusoid can be represented in terms of a positive frequency.

The sign of the underlying phase slope is ambiguous.

The ambiguity is resolved when the cosine and sine operators can be observed simultaneously, because cos(ωt + θ) leads sin(ωt + θ) by 1⁄4 cycle (i.e. π⁄2 radians) when ω > 0, and lags by 1⁄4 cycle when ω < 0.

Similarly, a vector, (cos ωt, sin ωt), rotates counter-clockwise if ω > 0, and clockwise if ω < 0.

In Eq.2 the second term looks like an addition, but it is actually a cancellation that reduces a 2-dimensional vector to just one dimension, resulting in the ambiguity.

Eq.2 also shows why the Fourier transform has responses at both

What the false response does is enable the inverse transform to distinguish between a real-valued function and a complex one.

Perhaps the best-known application of negative frequency is the formula: which is a measure of the energy in function

[A] For instance, consider the function: And: Note that although most functions do not comprise infinite duration sinusoids, that idealization is a common simplification to facilitate understanding.

cancels the positive frequency, leaving just the constant coefficient

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 applies.

Fourier's original formulation (the sine transform and the cosine transform) requires an integral for the cosine and another for the sine.

(see Analytic signal, Euler's formula § Relationship to trigonometry, and Phasor)

The counterclockwise-rotating vector (cos t , sin t ) has a positive frequency of +1 radian per unit of time . Not shown is a clockwise-rotating vector (cos (− t ), sin (− t )) which has a negative frequency of -1 radian per unit of time. Both go around a unit circle every 2 π units of time, but in opposite directions.
A negative frequency causes the sin function (violet) to lead the cos (red) by 1/4 cycle.
This figure depicts two complex sinusoids, colored gold and cyan, that fit the same sets of real and imaginary sample points. They are thus aliases of each other when sampled at the rate ( f s ) indicated by the grid lines. The gold-colored function depicts a positive frequency, because its real part (the cos function) leads its imaginary part by 1/4 of one cycle. The cyan function depicts a negative frequency, because its real part lags the imaginary part.