real-valued representation is no longer a simple matter of just extracting the real component. Up-conversion may be required, and if the signal has been sampled (discrete-time), interpolation (upsampling) might also be necessary to avoid aliasing. If is chosen larger than the highest frequency of then has no positive frequencies. In that case, extracting the real component restores them, but in reverse order; the low-frequency components are now high ones and vice versa. This can be used to demodulate a type of single-sideband signal called lower sideband or inverted sideband. Other choices of reference frequency are sometimes considered: Sometimes is chosen to minimize Alternatively, can be chosen to minimize the mean square error in linearly approximating the unwrapped instantaneous phase : or another alternative (for some optimum ): 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. Sometimes the phrase "complex envelope" is given the simpler meaning of the complex amplitude of a (constant-frequency) phasor; other times the complex envelope as defined above is interpreted as a time-dependent generalization of the complex amplitude. Their relationship is not unlike that in the real-valued case: varying envelope generalizing constant amplitude. Extensions of the analytic signal to signals of multiple variables The concept of analytic signal is well-defined for signals of a single variable which typically is time. For signals of two or more variables, an analytic signal can be defined in different ways, and two approaches are presented below. Multi-dimensional analytic signal based on an ad hoc direction 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. This can be done by introducing a unit vector in the Fourier domain and label any frequency vector as negative if . 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. However, there is no particular direction for which must be chosen unless there are some additional constraints. Therefore, the choice of is ad hoc, or application specific. The monogenic signal The real and imaginary parts of the analytic signal correspond to the two elements of the vector-valued monogenic signal, as