Editing Analytic signal (section)

==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]] <math>\boldsymbol \hat{u}</math> in the Fourier domain and label any frequency vector <math>\boldsymbol \xi</math> as negative if <math>\boldsymbol \xi \cdot \boldsymbol \hat{u} < 0</math>. 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 <math>\boldsymbol \hat{u}</math> which must be chosen unless there are some additional constraints. Therefore, the choice of <math>\boldsymbol \hat{u}</math> 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 it is defined for one-variable signals. However, the monogenic signal can be extended to arbitrary number of variables in a straightforward manner, producing an {{nowrap|(''n'' + 1)}}-dimensional vector-valued function for the case of ''n''-variable signals.