On the Properties of the Rank-Two Null Space of Nonsparse and Canonical-Sparse Blind Deconvolution

Blind deconvolution is a ubiquitous nonlinear inverse problem in applications such as wireless communications and image processing. This problem is generally ill-posed, and there have been efforts to use sparse models for regularization to promote signal identifiability. Herein, the ambiguity in blind deconvolution is characterized by lifting to a rank-one matrix recovery problem and analyzing the rank-two null space of the resultant linear operator. A novel dimension-wise tight representation of this rank-two null space is stated and proved to show unidentifiability of blind deconvolution for almost every pair of unconstrained input signals. This representation is further used to establish, somewhat surprisingly, the ill-posedness of the canonical-sparse blind deconvolution problem by exemplifying the dimension of the unidentifiable signal sets. An important conclusion of this paper is that canonical sparsity occurring naturally in applications may be insufficient for signal identifiability in blind deconvolution, necessitating the use of coding.