Leveraging Diversity and Sparsity in Blind Deconvolution

IEEE This paper considers recovering L-dimensional vectors w, and x1, x2,...,xN from their circular convolutions yn = w * x [subscript n]; n = 1, 2, 3,...,N. The vector w is assumed to be S-sparse in a known basis that is spread out in the Fourier domain, and each input x[subscript n] is a member of a known K-dimensional random subspace. We prove that whenever K + S log[superscript 2]S ≲ L/log[superscript 4](LN), the problem can be solved effectively by using only the nuclear-norm minimization as the convex relaxation, as long as the inputs are sufficiently diverse and obey N ≳ log[superscript 2](LN). By "diverse inputs", we mean that the x[subscript n]'s belong to different, generic subspaces. To our knowledge, this is the first theoretical result on blind deconvolution where the subspace to which w belongs is not fixed, but needs to be determined. We discuss the result in the context of multipath channel estimation in wireless communications. Both the fading coefficients, and the delays in the channel impulse response w are unknown. The encoder codes the K-dimensional message vectors randomly and then transmits coded messages x[subscript n]'s over a fixed channel one after the other. The decoder then discovers all of the messages and the channel response when the number of samples taken for each received message are roughly greater than (K + Slog[superscript 2]S) log[superscript 4](LN), and the number of messages is roughly at least log2(LN)

Leveraging Diversity and Sparsity in Blind Deconvolution

IEEE This paper considers recovering L-dimensional vectors w, and x1, x2,...,xN from their circular convolutions yn = w * x [subscript n]; n = 1, 2, 3,...,N. The vector w is assumed to be S-sparse in a known basis that is spread out in the Fourier domain, and each input x[subscript n] is a member of a known K-dimensional random subspace. We prove that whenever K + S log[superscript 2]S ≲ L/log[superscript 4](LN), the problem can be solved effectively by using only the nuclear-norm minimization as the convex relaxation, as long as the inputs are sufficiently diverse and obey N ≳ log[superscript 2](LN). By "diverse inputs", we mean that the x[subscript n]'s belong to different, generic subspaces. To our knowledge, this is the first theoretical result on blind deconvolution where the subspace to which w belongs is not fixed, but needs to be determined. We discuss the result in the context of multipath channel estimation in wireless communications. Both the fading coefficients, and the delays in the channel impulse response w are unknown. The encoder codes the K-dimensional message vectors randomly and then transmits coded messages x[subscript n]'s over a fixed channel one after the other. The decoder then discovers all of the messages and the channel response when the number of samples taken for each received message are roughly greater than (K + Slog[superscript 2]S) log[superscript 4](LN), and the number of messages is roughly at least log2(LN)

A convex approach to blind deconvolution with diverse inputs

This note considers the problem of blind identification of a linear, time-invariant (LTI) system when the input signals are unknown, but belong to sufficiently diverse, known subspaces. This problem can be recast as the recovery of a rank-1 matrix, and is effectively relaxed using a semidefinite program (SDP). We show that exact recovery of both the unknown impulse response, and the unknown inputs, occurs when the following conditions are met: (1) the impulse response function is spread in the Fourier domain, and (2) the N input vectors belong to generic, known subspaces of dimension K in ℝL. Recent results in the well-understood area of low-rank recovery from underdetermined linear measurements can be adapted to show that exact recovery occurs with high probablility (on the genericity of the subspaces) provided that K,L, and N obey the information-theoretic scalings, namely L ≳ K and N ≳ 1 up to log factors.Fonds national de la recherche scientifique (Belgium)MIT International Science and Technology InitiativesUnited States. Air Force. Office of Scientific ResearchUnited States. Office of Naval ResearchNational Science Foundation (U.S.)Total S