Editing Analytic signal (section)

===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.