Blind Deconvolution using Modulated Inputs
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This paper considers the blind deconvolution of multiple modulated signals/filters, and an arbitrary filter/signal. Multiple inputs s_1, s_2, ..., s_N =: [s_n] are modulated (pointwise multiplied) with random sign sequences r_1, r_2, ..., r_N =: [r_n], respectively, and the resultant inputs (s_n r_n) ∈C^Q, n = [N] are convolved against an arbitrary input h∈C^M to yield the measurements y_n = (s_nr_n)h, n = [N] := 1,2,...,N, where , and denote pointwise multiplication, and circular convolution. Given [y_n], we want to recover the unknowns [s_n], and h. We make a structural assumption that unknown [s_n] are members of a known K-dimensional (not necessarily random) subspace, and prove that the unknowns can be recovered from sufficiently many observations using an alternating gradient descent algorithm whenever the modulated inputs s_n r_n are long enough, i.e, Q ≳ KN+M (to within log factors and signal dispersion/coherence parameters).
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