Formally, it is defined as the analytic signal corresponding to the real field.

This error can be measured by the analytic signal phase measurement method described earlier.

Since is an analytic signal, it is either zero or complex-valued.

The resulting function is the analytic signal of rather than of .

The complex conjugate of an analytic signal contains only negative frequency components.

The analytic signal can also be expressed in terms of complex polar coordinates, where:

The analytic signal can also be represented as:

For signals of two or more variables, an analytic signal can be defined in different ways, and two approaches are presented below.

The concept of analytic signal is well-defined for signals of a single variable which typically is time.

The inverse Fourier transform of X(f) is the analytic signal:

Another way to achieve a spectrum of negative frequencies is to frequency-shift the analytic signal sufficiently far in the negative direction.

According to the above result, however, it is possible to obtain an analytic signal by convolving the signal with a quadrature filter .

Obviously the real component of the complex conjugate is the same as the real component of the analytic signal.

The researchers claim that by using 2D analytic signal processing, the magnitude and phase of the field can be extracted for study and comparison to theoretical models.

Analytic signals are often shifted in frequency (down-converted) toward 0 Hz, which creates [non-symmetrical] negative frequency components.

A straightforward generalization of the analytic signal can be done for a multi-dimensional signal once it is established what is meant by negative frequencies for this case.

The real and imaginary parts of the analytic signal correspond to the two elements of the vector-valued monogenic signal, as it is defined for one-variable signals.

The Wigner distribution, or Wigner-Ville distribution (WVD) for analytic signals, also has applications in time frequency analysis.

The analytic signal is then produced by removing all negative frequencies and multiply the result by 2, in accordance to the procedure described for the case of one-variable signals.

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.

Usually one waveform is the Hilbert transform of the other waveform and the complex-valued function, is called an analytic signal, whose Fourier transform is zero for all negative values of frequency.

In the field of time-frequency signal processing, it was shown that the analytic signal was needed in the definition of the Wigner-Ville distribution so that the method can have the desirable properties needed for practical applications.