Blind Deconvolution Using Modulated Inputs
This paper considers the blind deconvolution of multiple modulated signals/filters, and an arbitrary filter/signal. Multiple inputs <inline-formula><tex-math notation="LaTeX">$\boldsymbol{s}_1, \boldsymbol{s}_2, \ldots, \boldsymbol{s}_N =: [\boldsymbol{s}_n]$</tex-math></inline-formula> are modulated (pointwise multiplied) with random sign sequences <inline-formula><tex-math notation="LaTeX">$\boldsymbol{r}_1, \boldsymbol{r}_2, \ldots, \boldsymbol{r}_N =: [\boldsymbol{r}_n]$</tex-math></inline-formula>, respectively, and the resultant inputs <inline-formula><tex-math notation="LaTeX">$(\boldsymbol{s}_n \odot \boldsymbol{r}_n) \in \mathbb {C}^Q, \ n \in [N]$</tex-math></inline-formula> are convolved against an arbitrary input <inline-formula><tex-math notation="LaTeX">$\boldsymbol{h} \in \mathbb {C}^M$</tex-math></inline-formula> to yield the measurements <inline-formula><tex-math notation="LaTeX">$\boldsymbol{y}_n = (\boldsymbol{s}_n\odot \boldsymbol{r}_n)\circledast \boldsymbol{h}, \ n \in [N] := 1,2,\ldots,N,$</tex-math></inline-formula> where <inline-formula><tex-math notation="LaTeX">$\odot$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\circledast$</tex-math></inline-formula> denote pointwise multiplication, and circular convolution. Given <inline-formula><tex-math notation="LaTeX">$[\boldsymbol{y}_n]$</tex-math></inline-formula>, we want to recover the unknowns <inline-formula><tex-math notation="LaTeX">$[\boldsymbol{s}_n]$</tex-math></inline-formula> and <inline-formula><tex-math notation="LaTeX">$\boldsymbol{h}$</tex-math></inline-formula>. We make a structural assumption that unknowns <inline-formula><tex-math notation="LaTeX">$[\boldsymbol{s}_n]$</tex-math></inline-formula> are members of a known <inline-formula><tex-math notation="LaTeX">$K$</tex-math></inline-formula>-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 <inline-formula><tex-math notation="LaTeX">$\boldsymbol{s}_n \odot \boldsymbol{r}_n$</tex-math></inline-formula> are long enough, i.e, <inline-formula><tex-math notation="LaTeX">$Q \gtrsim KN+M$</tex-math></inline-formula> (to within logarithmic factors, and signal dispersion/coherence parameters). Under the bilinear model, this is the first result on multichannel (<inline-formula><tex-math notation="LaTeX">$N\geq 1$</tex-math></inline-formula>) blind deconvolution with provable recovery guarantees under near optimal (in the <inline-formula><tex-math notation="LaTeX">$N=1$</tex-math></inline-formula> 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.