  The analytic signal method via the Hilbert transform is a key tool in signal
analysis and processing, especially in the time-frquency analysis. Imaging and
other applications to multidimensional signals call for extension of the method
to higher dimensions. We justify the usage of partial Hilbert transforms to
define multidimensional analytic signals from both engineering and mathematical
perspectives. The important associated Bedrosian identity $T(fg)=fTg$ for
partial Hilbert transforms $T$ are then studied. Characterizations and several
necessity theorems are established. We also make use of the identity to
construct basis functions for the time-frequency analysis.
