Multilevel compression of the plane wave transform using local directional sources and equivalence box

A sparse representation of discrete plane wave transformation (DPWT) matrix which relied on single-level organization of spatial degrees of freedom and multilevel organization of angular variables was presented in [1]. That scheme was to directly compress the DPWT of the incident fields and normal derivatives of fields on each child group. In this paper, a new algorithm to sparsely represent the DPWT matrix which relies on multilevel organization of spatial and angular variables is developed. The new algorithm starts from the DPWT of the incident fields and normal derivatives of field on a closed box which encloses the corresponding group of target at the finest level of spatial structures. The DPWT of the fields and normal derivatives of field on the target are related to those fields on the equivalence box by a transformation matrix. The DPWT matrixes of the equivalence boxes are compressed by applying a basis of spatially localized source modes that generate far-fields which are simultaneously bandlimited and localized with distinct regions of the far-field. Because the physical dimension and sample rate of the field for each equivalence box are selected to be the same, all DPWT matrices at a given level are identical. Therefore, we only need to deal with one group at each spatial level and automatically get the DPWT for other groups at same spatial level. This provides significant time and memory savings. Like the single-level algorithm of R. J. Adams, et al., the algorithm reported here provides an error-controlled, O(epsiv), representation of D.