# Blind Deconvolution Using Modulated Inputs

This paper considers the blind deconvolution of multiple modulated signals/filters, and an arbitrary filter/signal. Multiple inputs are modulated (pointwise multiplied) with random sign sequences , respectively, and the resultant inputs are convolved against an arbitrary input to yield the measurements , where and denote pointwise multiplication, and circular convolution. Given , we want to recover the unknowns and . We make a structural assumption that unknowns are members of a known K-dimensional (not necessarily random) subspace, and prove that the unknowns can be recovered from sufficiently many observations using a regularized gradient descent algorithm whenever the modulated inputs are long enough, i.e, (to within logarithmic factors, and signal dispersion/coherence parameters). Under the bilinear model, this is the first result on multichannel blind deconvolution with provable recovery guarantees under near optimal (in the case) sample complexity estimates, and comparatively lenient structural assumptions on the convolved inputs. A neat conclusion of this result is that modulation of a bandlimited signal protects it against an unknown convolutive distortion. We discuss the applications of this result in passive imaging, wireless communication in unknown environment, and image deblurring. A thorough numerical investigation of the theoretical results is also presented using phase transitions, image deblurring experiments, and noise stability plots.
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