Blind Deconvolution Using Modulated Inputs

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  • February 2, 2022
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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 \odot r_n ) \in \mathbb{C}^Q , n \in [N]?are convolved against an arbitrary input?h \in \mathbb{C}^M?to yield the measurements?y_n = (s_n \odot r_n ) \circledast h, n \in [N] := 1, 2, ... ,N, where?\odot?and?\circledast?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 unknowns?[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 a regularized gradient descent algorithm whenever the modulated inputs?s_n \odot r_n?are long enough, i.e,?Q \geq KN + M?(to within logarithmic factors, and signal dispersion/coherence parameters). Under the bilinear model, this is the first result on multichannel?(N \geq 1)?blind deconvolution with provable recovery guarantees under near optimal (in the?N = 1?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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Categories: Research