an analytic representation with its complex conjugate extracts the actual real-valued function they represent. For instance: which gives rise to the somewhat misleading interpretation that comprises both a positive and a negative frequency. But the "sum" involves a cancellation of all imaginary components . That cancellation merely results in an ambiguity about the sign of the frequency. Using either sign provides an equivalent representation of the same cosine wave. In any measure that indicates both frequencies, one of the two frequencies is a false positive or alias of the other, because can have only one sign. The Fourier transform, for instance, merely tells us that cross-correlates equally well with as with Nevertheless, treating a real sinusoid as the combination of a positive and a negative frequency is sometimes useful (and mathematically valid). 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. For instance, consider the function: And: Note that although most functions do not comprise infinite duration sinusoids, that idealization is a common simplification that facilitates understanding. Looking at the first term of this result, 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 applies. 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, , and Phasor) 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. . Waves Concepts in physics