Negative frequency

The concept of signed frequency (negative and positive frequency) can indicate both the rate and sense of rotation; in 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) or radian/second (where 1 cycle corresponds to 2π radians).

Sinusoids

Let $ω$ be a nonnegative 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 $cos(ωt - θ)$ . Similarly, $sin(- ωt + θ)$ is indistinguishable from $sin(ωt - θ + π)$ . Thus any sinusoid can be represented in terms of positive frequencies. The sign of the underlying phase slope is ambiguous.

A negative frequency causes the sin function (violet) to lead the cos (red) by 1/4 cycle.

The vector

(cos t, sin t)

rotates counter-clockwise at 1 radian/second, and completes a circle every

2 π

seconds. The vector

(cos - t, sin - t)

rotates in the other direction (not shown).

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 (= 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 $ω < 0$ .

The sign of $ω$ is also preserved in the complex-valued function:

$e^{i\omega t}=\underbrace {\cos(\omega t)} _{R(t)}+i\cdot \underbrace {\sin(\omega t)} _{I(t)},$ ^[A]

(Eq.1)

since R(t) and I(t) can be separately observed and compared. Although $e^{i\omega t}$ clearly contains more information than either of its components, a common interpretation is that it is a simpler function, because it simplifies many important trigonometric calculations, which leads to its formal description as the analytic representation of $\cos(\omega t)$ .^[B]

The sum of an analytic representation with its complex conjugate extracts the actual real-valued function they represent. For instance:

$\cos(\omega t)={\begin{matrix}{\frac {1}{2}}\end{matrix}}\left(e^{i\omega t}+e^{-i\omega t}\right),$

(Eq.2)

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.^[C] The Fourier transform, for instance, merely tells us that cos(ωt) cross-correlates equally well with cos(ωt) + i sin(ωt) as with cos(ωt) − i sin(ωt).^[D] 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:

{\hat {f}}(\omega )=\int _{-\infty }^{\infty }f(t)e^{-i\omega t}dt,

which is a measure of the energy in function $f(t)$ at frequency $\omega .$ When evaluated for a continuum of argument $\omega ,$ the result is called the Fourier transform.^[E]

For instance, consider the function:

f(t)=A_{1}e^{i\omega _{1}t}+A_{2}e^{i\omega _{2}t},\ \forall \ t\in \mathbb {R} ,\ \omega _{1}>0,\ \omega _{2}>0.

And:

{\begin{aligned}{\hat {f}}(\omega )&=\int _{-\infty }^{\infty }[A_{1}e^{i\omega _{1}t}+A_{2}e^{i\omega _{2}t}]e^{-i\omega t}dt\\&=\int _{-\infty }^{\infty }A_{1}e^{i\omega _{1}t}e^{-i\omega t}dt+\int _{-\infty }^{\infty }A_{2}e^{i\omega _{2}t}e^{-i\omega t}dt\\&=\int _{-\infty }^{\infty }A_{1}e^{i(\omega _{1}-\omega )t}dt+\int _{-\infty }^{\infty }A_{2}e^{i(\omega _{2}-\omega )t}dt\end{aligned}}

Infinite duration is a simplification that facilitates this discussion. Looking at the first term above, when $\omega =\omega _{1},$ the negative frequency $-\omega _{1}$ cancels the positive frequency, leaving just the constant coefficient $A_{1}$ (because $e^{i0t}=e^{0}=1$ ), which causes the infinite integral to diverge. At other values of $\omega$ the residual oscillations cause the integral to converge to zero. This idealized Fourier transform is usually written as:

{\hat {f}}(\omega )=2\pi A_{1}\delta (\omega -\omega _{1})+2\pi A_{2}\delta (\omega -\omega _{2}).

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, Euler's formula#Relationship to trigonometry, and Phasor)

Sampling of positive and negative frequencies and aliasing

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.

Notes

^ The equivalence is called Euler's formula
^ See Euler's formula § Relationship to trigonometry and Phasor § Addition for examples of calculations simplified by the complex representation.
^ Conversely, any measure that indicates only one frequency has made an assumption, perhaps based on collateral information.
^ cos(ωt) and sin(ωt) are orthogonal functions, so the imaginary parts of both correlations are zero.
^ There are several forms of the Fourier transform. This is the non-unitary form in angular frequency of time

Negative frequency

Contents

Sinusoids

Applications

Simplifying the Fourier transform

Sampling of positive and negative frequencies and aliasing

See also

Notes

Further reading

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