Pulsed-beam wavelets and their sources

Gerald Kaiser
Virginia Center for Signals and Waves

Acoustic and electromagnetic waves form a natural domain for multiscale analysis since their PDEs contain no a priori scale. This motivated the construction of acoustic and EM wavelets, families of localized solutions W_z that form frames in solution spaces. The parameters z are complex spacetime points generalizing the time and scale parameters in 1D wavelets. While the propagating wave W_z cannot have compact support, its source does. That gives z the following physical interpretation: x= Re z is the source center in space and time, and y=Im z specifies its extension about x. Specifically, the space components of y give the radius and orientation of a disk source launching W_z and its time component gives the duration of the excitation. Consequently, W_z is a pulsed beam whose direction and duration can be made arbitrarily wide or narrow by adjusting y (which must be restricted to the future cone in spacetime). In a sense to be made precise, W_z is the retarded solution due to a point source located at the complex point z and the imaginary displacement from x to x+iy causes a broadening of the source point in real spacetime. The associated wavelet analysis gives decompositions and reconstructions of general solutions in terms of pulsed beams with promising applications to such fields as radar and communications. We ask the following question: Can approximations to W_z be launched and detected by actual instruments? If so, then pulsed-beam wavelet analysis could become a practical tool as well as a mathematical method. Toward this end, we compute the source distributions needed to launch (and, by reciprocity, detect) the pulsed beams. This involves an extension of Newtonian potentials and their point sources from R^n to C^n called "complex-distance potential theory." As a byproduct, we obtain some interesting connections between fundamental solutions of elliptic and hyperbolic PDEs.


Preprint (PDF File) - Invited "Topical Review" for Journal of Physics A: Mathematical and General

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