Coupled Canonical Polyadic Decompositions and Multiple Shift Invariance in Array Processing

The canonical polyadic decomposition (CPD) plays an important role for signal separation in array processing. The CPD model requires arrays composed of several displaced but identical subarrays. Consequently, it is less appropriate for more complex array geometries. In this paper, we explain that coupled CPD allows a much more flexible modeling that can handle multiple shift-invariance structures, i.e., arrays that can be decomposed into multiple but not identical displaced subarrays. Both deterministic and generic identifiability conditions are presented. We also point out that, under mild conditions, the signal separation problem can in the exact case be solved by means of an eigenvalue decomposition. This is similar to ESPRIT, although the working conditions are much more relaxed. Borrowing tools from algebraic geometry, we derive generic uniqueness bounds for L-shaped, frame-shaped, and triangular-shaped arrays that come close to bounds that are necessary for uniqueness. Recognizing multiple shift invariance can be a bit of an art by itself. We explain that any centrosymmetric array processing problem can be interpreted in terms of a coupled CPD. In addition, we demonstrate that the coupled CPD model allows us to significantly relax the far-field assumption commonly used in CPD-based array processing.