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 Hilbert transform. The analytic representation of a real-valued function is an analytic signal, comprising the original function and its Hilbert transform. This representation facilitates many mathematical manipulations. The basic idea is that the negative frequency components of the Fourier transform (or spectrum) of a real-valued function are superfluous, due to the Hermitian symmetry of such a spectrum. These negative frequency components can be discarded with no loss of information, provided one is willing to deal with a complex-valued function instead. That makes certain attributes of the function more accessible and facilitates the derivation of modulation and demodulation techniques, such as single-sideband. As long as the manipulated function has no negative frequency components (that is, it is still analytic), the conversion from complex back to real is just a matter of discarding the imaginary part. The analytic representation is a generalization of the phasor concept: while the phasor is restricted to time-invariant amplitude, phase, and frequency, the analytic signal allows for time-variable parameters. Definition If is a real-valued function with Fourier transform , then the transform has Hermitian symmetry about the axis: where is the complex conjugate of . The function: where is the Heaviside step function, is the sign function, contains only the non-negative frequency components of . And the operation is reversible, due to the Hermitian symmetry of : The analytic signal of is the inverse Fourier transform of : where is the Hilbert transform of ; is the binary convolution operator; is the imaginary unit. Noting that this can also be expressed as a filtering operation that directly removes negative frequency components: Negative frequency components Since , restoring the negative frequency components is a simple matter of discarding which may seem counter-intuitive. We can also note that the complex conjugate comprises only the negative frequency components. And therefore restores the suppressed positive frequency components. Another viewpoint is that the imaginary component in either case is a term that subtracts frequency components from The operator removes the subtraction, giving the appearance of adding new components. Examples Example 1 where Then: The last equality is Euler's formula, of which a corollary is In general, the analytic representation of a simple sinusoid is obtained by