Modeling and blind deconvolution via sparse representations
Abstract:
The impact of sparsity-inducing techniques in signal analysis has been recognized for several years now and has been the key to a growing literature on the subject —commonly referred to as compressive sensing. In this talk we will discuss the potential such sparsity-inducing techniques in the context of system identification, spectral analysis, and blind deconvolution. In particular we will consider the problem of separating sinusoids in colored noise and identifying the dynamics that generate this wide-bandwidth noise-component at the same time. The formalism relies on modeling the data as a superposition of a few unknown sinusoidal signals together with the output of an auto-regressive filter driven by white noise. Naturally, since neither the underlying dynamics nor possible sinusoids present are known, the problem is ill-posed. We seek a sparse selection of sinusoids which together with the auto-regressive component can account for the data-set. To this end, suitable modification of sparsity-inducing functionals (a la LASSO/Basis pursuit/etc.) which can generate admissible solutions appear quite effective. We will also discuss the relevance of sparse/jointly-sparse representations in spectral coherence analysis, delay estimation, etc. The talk is based on joint work with Lipeng Ning and Allen Tannenbaum.