Non-well-founded set theory

Fig 1: The vector (cos t, sin t) rotates counter-clockwise at 1 radian/sec, and completes a circle every 2π seconds. The vector (cos -t, sin -t) rotates in the other direction (not shown).

The concept of negative and positive frequency can be as simple as a wheel rotating one way or the other way. A signed value of frequency can indicate both the rate and direction of rotation. The rate is expressed in units such as revolutions (aka cycles) per second (hertz) or radian/second (where 1 cycle corresponds to 2π radians).

Fig.2: A negative frequency causes the sin function (violet) to lead the cos (red) by ¼ cycle.

Sinusoids

Similar to the rotating vector (Fig 1), the complex-valued function:

$e^{j\omega t}=\underbrace {\cos(\omega t)} _{R(t)}+j\cdot \underbrace {\sin(\omega t)} _{I(t)}$     

exhibits different properties for positive and negative values of the parameter ω. When ω > 0, R(t) appears to lead I(t) by ¼ cycle (= π/2 radians). When ω < 0, the roles are reversed. Fig 2 depicts a negative frequency. R(t) and I(t) are referred to as real and imaginary, respectively. The more-familiar real-valued sinusoid:

is an analytically more-complicated function than  $e^{j\omega t},$  because it comprises both a negative and positive frequency component. In fact,  $e^{j\omega t}$  is sometimes called the analytic representation of  $\cos(\omega t).$   When working with individual, real-valued sinusoids, a negative frequency has an equivalent positive-frequency form; i.e. $\scriptstyle \cos(-\omega t+\theta )=\cos(\omega t-\theta )=\sin(\omega t-\theta +\pi /2),$  so it is often sufficient to consider every frequency a positive one.

Applications

Perhaps the most well-known application of negative frequency is the calculation:

$X(\omega )=\int _{a}^{b}x(t)\cdot e^{-j\omega t}dt,$

which is a measure of the amount of frequency ω in the function x(t) over the interval (a,b). When evaluated as a continuous function of ω for the theoretical interval (-∞, ∞), it is known as the Fourier transform of x(t). A brief explanation is that the product of two complex sinusoids is also a complex sinusoid whose frequency is the sum of the original frequencies. So when ω is positive,  $e^{-j\omega t}$  causes all the frequencies of x(t) to be reduced by amount ω. Whatever part of x(t) that was at frequency ω is changed to frequency zero, which is just a constant whose amplitude level is a measure of the strength of the original ω content. And whatever part of x(t) that was at frequency zero is changed to a sinusoid at frequency -ω. Similarly, all other frequencies are changed to non-zero values. As the interval (a,b) increases, the contribution of the constant term grows in proportion. But the contributions of the sinusoidal terms only oscillate around zero. So X(ω) improves as a relative measure of the amount of frequency ω in the function x(t).

The Fourier transform of  $e^{j\omega t}$  produces a non-zero response only at frequency ω. The transform of  $\scriptstyle \cos(\omega t)$  has responses at both ω and -ω, as anticipated by Template:EquationNote.

Sampling of positive and negative frequencies and aliasing

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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

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References

• Positive and Negative Frequencies
• Lyons, Richard G. (Nov 11, 2010). Chapt 8.4. Understanding Digital Signal Processing (3rd ed.). Prentice Hall. 944 pgs. ISBN 0137027419.