Blind linear system identification (or recovery) arises in several applications in engineering (e.g. channel equalization, super-resolution, MRI and SAR image formation). This is a special case of a bi-linear inverse problem, and is sometimes equivalent to range-based operator recovery.
The aim of this research is to study the structure of solutions for range-based identification, which is typically an affine or projective variety, and is usually ambiguous (containing more than one element - not identifiable).
Algebraic geometry was utilized to derive a generic range-space based identification algorithm and identifiability test. The properties of irreducible complex varieties were used to derive a numerical identifiability guarantee for complex parametric families. In addition, an alternative approach (of so-called preserving pre-compositions) examined the ambiguity from a non-parametric viewpoint, searching for operations that preserve both the structure of a system as well as its range space.
The established framework and results were then used to determine cases wherein the recovery of sampled multichannel finite impulse response (FIR) configurations, particularly blind sampled deconvolution, is ambiguous.
The last chapter of this work offers some insights about the spatial structure of data eigen-patches, that were used in previous chapters in the process of system identification. Empirical results indicate that those eigen-patches tend to exhibit wave-like shapes, and the sample covariance operator is approximately Toeplitz. A heuristic explanation for those two phenomena is offered with some statistical analysis, which could be further developed later into a complete and rigorous explanation of the observations
