Rank-Awareness Sparse Blind Deconvolution Using Modulated Input

This paper presents rank-awareness algorithms to solve sparse blind deconvolution using modulated input. We consider sparse blind deconvolution as a rank-one column-sparse matrix recovery problem, so the proposed algorithms can use both the rank-one property and the sparsity of the unknowns. Unknown input s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{s}$$\end{document} is first multiplied by a random sign sequence r\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{r}$$\end{document} and then convolved with an arbitrary filter h\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{h}$$\end{document} to obtain the measurements y\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{y}$$\end{document}. The unknown signal s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{s}$$\end{document} is assumed to have a sparse representation. Sparse blind deconvolution using modulated input has unique applications, such as the blind calibration of the random demodulation system. When the number of measurements has satisfied certain conditions, blind deconvolution can be solved without considering signal sparsity. This paper mainly studies how to use signal sparsity to reduce the number of measurements required for sparse blind deconvolution. We propose two methods to solve this problem. The first method uses the ℓ1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\ell _1$$\end{document}-norm regularization to promote the unknown signal to iterate in the direction of sparsity. The second method transforms the sparse blind deconvolution problem into a rank-one constrained block-sparse signal recovery problem, and we propose the rank-awareness sparse blind demodulation algorithm to solve it. Our proposed methods could effectively reduce the number of measurements required for sparse blind deconvolution. Under certain conditions, our proposed sparse blind deconvolution algorithms required 320 and 160 measurements, while 400 measurements were required when signal sparsity was not considered. The simulation results verify the effectiveness of the proposed algorithms.