The circular Bedrosian identity and multidimensional periodic analytic signals

The analytic signal method via the Hilbert transform is a classical manner of defining without ambiguity the instantaneous amplitude and frequency of one-dimensional signals. In real applications such as image processing, signals are usually of finite duration and belong to higher dimensions. Extension of this method to the multidimensional space is needed. We shall justify that partial Hilbert transforms can be used to define multidimensional periodic analytic signals. The important associated circular Bedrosian identity for partial Hilbert transforms is then studied, where f and g are multidimensional periodic signals. Several characterizations and a necessity theorem are established. These characterizations provide us an efficient way of constructing a larger class of multivariate intrinsic mode functions with nonlinear phases, which are key ingredients for the celebrated empirical mode decomposition.