Abstract
In this paper we discuss recovering two signals from their convolution in 3 dimensions. One of the signals is assumed to lie in a known subspace and the other one is assumed to be sparse. Various applications such as super resolution, radar imaging, and direction of arrival estimation can be described in this framework. We introduce a method to estimate parameters of a signal in a low-dimensional subspace which is convolved with another signal comprised of some impulses in time domain. We transform the problem to a convex optimization in the form of a positive semi-definite program using lifting and the atomic norm. We demonstrate that unknown parameters can be recovered by lowpass observations. Numerical simulations show excellent performance of the proposed method.
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References
Ahmed, A., & Demanet, L. (2018). Leveraging diversity and sparsity in blind deconvolution. IEEE Transactions on Information Theory,64(6), 3975–4000.
Ahmed, A., Recht, B., & Romberg, J. (2014). Blind deconvolution using convex programming. IEEE Transactions on Information Theory,60(3), 1711–1732.
Candes, E. J., & Fernandez-Granda, C. (2014). Towards a mathematical theory of super-resolution. Communications on Pure and Applied Mathematics,67(6), 906–956.
Candes, E. J., & Plan, Y. (2011). A probabilistic and RIPless theory of compressed sensing. IEEE Transactions on Information Theory,57(11), 7235–7254.
Candes, E. J., Strohmer, T., & Voroninski, V. (2013). Phaselift: Exact and stable signal recovery from magnitude measurements via convex programming. Communications on Pure and Applied Mathematics,66(8), 1241–1274.
Chandrasekaran, V., Recht, B., Parrilo, P. A., & Willsky, A. S. (2012). The convex geometry of linear inverse problems. Foundations of Computational Mathematics,12(6), 805–849.
Chi, Y. (2016). Guaranteed blind sparse spikes deconvolution via lifting and convex optimization. IEEE Journal of Selected Topics in Signal Processing,10(4), 782–794.
Chi, Y., & Chen, Y. (2015). Compressive two-dimensional harmonic retrieval via atomic norm minimization. IEEE Transactions on Signal Processing,63(4), 1030–1042.
Dumitrescu, B. (2007). Positive trigonometric polynomials and signal processing applications (Vol. 103). Berlin: Springer.
Grant, M., & Boyd, S. (2014). CVX: Matlab software for disciplined convex programming, version 2.1.
Heckel, R. (2016). Super-resolution MIMO radar. In 2016 IEEE international symposium on information theory (ISIT) (pp. 1416–1420). IEEE.
Heckel, R., & Soltanolkotabi, M. (2018). Generalized line spectral estimation via convex optimization. IEEE Transactions on Information Theory,64(6), 4001–4023.
Kundur, D., & Hatzinakos, D. (1996). Blind image deconvolution. IEEE Signal Processing Magazine,13(3), 43–64.
Lee, K., Li, Y., Junge, M., & Bresler, Y. (2017). Blind recovery of sparse signals from subsampled convolution. IEEE Transactions on Information Theory,63(2), 802–821.
Li, Y., & Chi, Y. (2016). Off-the-grid line spectrum denoising and estimation with multiple measurement vectors. IEEE Transactions on Signal Processing,64(5), 1257–1269.
Li, Y., Lee, K., & Bresler, Y. (2016). Identifiability in blind deconvolution with subspace or sparsity constraints. IEEE Transactions on Information Theory,62(7), 4266–4275.
Li, X., Ling, S., Strohmer, T., & Wei, K. (2019). Rapid, robust, and reliable blind deconvolution via nonconvex optimization. Applied and Computational Harmonic Analysis,47(3), 893–934.
Ling, S., & Strohmer, T. (2017). Blind deconvolution meets blind demixing: Algorithms and performance bounds. IEEE Transactions on Information Theory,63(7), 4497–4520.
Litwin, L. R. (1999). Blind channel equalization. IEEE Potentials,18(4), 9–12.
Naylor, P. A., & Gaubitch, N. D. (Eds.). (2010). Speech dereverberation. Berlin: Springer.
Tang, G., Bhaskar, B. N., Shah, P., & Recht, B. (2013). Compressed sensing off the grid. IEEE Transactions on Information Theory,59(11), 7465–7490.
Tropp, J. A. (2012). User-friendly tail bounds for sums of random matrices. Foundations of computational mathematics,12(4), 389–434.
Valiulahi, I., Daei, S., Haddadi, F., & Parvaresh, F. (2017). Two-dimensional super-resolution via convex relaxation. arXiv preprint arXiv:1711.08239.
Weiss, A. J., & Friedlander, B. (1990). Eigenstructure methods for direction finding with sensor gain and phase uncertainties. Circuits, Systems and Signal Processing,9(3), 271–300.
Xu, W., Cai, J. F., Mishra, K. V., Cho, M., & Kruger, A. (2014). Precise semidefinite programming formulation of atomic norm minimization for recovering d-dimensional (d ≥ 2) off-the-grid frequencies. In 2014 information theory and applications workshop (ITA) (pp. 1–4). IEEE.
Yang, D., Tang, G., & Wakin, M. B. (2016). Super-resolution of complex exponentials from modulations with unknown waveforms. IEEE Transactions on Information Theory,62(10), 5809–5830.
Yilmaz, Ö. (2001). Seismic data analysis: Processing, inversion, and interpretation of seismic data. Society of Exploration Geophysicists.
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Appendices
Appendix A
1.1 Proof of the primary problem SDP
By defining one atom in the form of \( \varvec{A}\left( {\tau ,\varvec{b}} \right) = \varvec{c}\left( \tau \right)\varvec{b}^{\varvec{*}} \) with \( \tau_{i} \in \left[ {0,1} \right) \), \( \varvec{b} \in {\mathbb{R}}^{L} \), \( \left\| \varvec{b} \right\|_{2} = 1 \) and \( \varvec{Z} = \mathop \sum \limits_{k} a_{k} \varvec{A}\left( {\varvec{\tau}_{k} , \varvec{b}_{k} } \right), a_{k} \ge 0 \). We have:
Proof
We show the right hand side of the above equation with \( SDP\left( \varvec{Z} \right) \). assuming \( \varvec{Z} = \mathop \sum \limits_{k} a_{k} \varvec{c}\left( {\varvec{\tau}_{k} } \right)\varvec{b}_{k}^{\varvec{*}} , a_{k} \ge 0 \) and \( S\left( \varvec{T} \right) = \mathop \sum \limits_{k} a_{k} \varvec{c}\left( {\varvec{\tau}_{k} } \right)\varvec{c}\left( {\varvec{\tau}_{k} } \right)^{*} \) we have:
Therefore:
Which is: \( SDP\left( \varvec{Z} \right) \le \left\| \varvec{Z} \right\|_{{\mathcal{A}}} \).□
We assume \( \left[ {\begin{array}{*{20}c} {{\text{S}}\left( {\mathbf{T}} \right)} & \varvec{Z} \\ {\varvec{Z}^{\varvec{*}} } & \varvec{W} \\ \end{array} } \right]{ \succcurlyeq }0 \), while \( S\left( \varvec{T} \right) = \varvec{VDV}^{\varvec{*}} \) and \( \varvec{D} = {\text{diag}}\left( {d_{i} } \right), d_{i} > 0 \). we can see that \( \varvec{Z} \) is in the range space of \( \varvec{V} \). So \( \varvec{Z} = \varvec{VB} \) and recall that \( \varvec{b}_{k} \) is the kth column of \( \varvec{B}^{\varvec{T}} \). Also \( \varvec{W} \) is positive semi-definite, therefore \( \varvec{W} = \varvec{B}^{\varvec{*}} \varvec{EB} \) where \( \varvec{E} \) a is positive semi-definite matrix too. We have:
Thus:
Therefore, \( \left\| \varvec{Z} \right\|_{{\mathcal{A}}} \le SDP\left( \varvec{Z} \right) \) and we conclude that \( \left\| \varvec{Z} \right\|_{{\mathcal{A}}} = SDP\left( \varvec{Z} \right) \).
However, in contrary to the proposed SDP in Tang et al. (2013), since \( S\left( \varvec{T} \right) \) might not have a description as \( \varvec{VDV}^{\varvec{*}} \), we cannot guarantee that \( \left\| \varvec{Z} \right\|_{{\mathcal{A}}} = SDP\left( \varvec{Z} \right) \).
Appendix B
2.1 Proof of Proposition 1
Fist we show that each \( \varvec{q} \) satisfying (15) and (16) is dual feasible. We have:
Therefore, \( \left\langle {\chi^{*} \left( \varvec{q} \right) ,\varvec{Z}_{{\mathbf{0}}} } \right\rangle _{{\mathbb{R}}} = \left\| {\varvec{Z}_{{\mathbf{0}}} } \right\|_{{\mathcal{A}}} \) and based on strong duality \( \varvec{q} \) is the optimum dual for the problem which \( \varvec{Z}_{{\mathbf{0}}} \) is its primary optimum.
To show uniqueness of the solution, assume \( \tilde{\varvec{Z}} = \sum\nolimits_{k} {\tilde{a}_{k} \varvec{c}\left( {\tilde{\varvec{\tau }}_{k} } \right)\tilde{\varvec{h}}_{k}^{\text{T}} } \) is another solution of the problem and \( \varvec{Z}_{{\mathbf{0}}} \) and \( \tilde{\varvec{Z}} \) have the same support set T. Set of atoms in \( {\mathbf{\mathcal{T}}} \) are independent so they should be equal. Therefore we have:
Which contradicts the strong duality. Therefore, the solution of (8) is unique.
Appendix C
3.1 Proof of Lemma 2
Assume that \( \left( {2\pi n_{1}\upkappa} \right)^{2} \le 13 \) and \( B \) is defined as:
Also:
Now, Bernstein’s non-commutative inequality gives:
will give:
Therefore, if \( M^{3} \ge \frac{100\mu KL}{{\upvarepsilon_{1}^{2} }}{ \log }\left( {\frac{8KL}{{\delta_{1} }}} \right) \) then \( \left\| {{\varvec{\Gamma}} - {\bar{\boldsymbol{\varGamma }}}} \right\| \le\upvarepsilon_{1} \) with probability at least \( 1 - \delta_{1} \).
Appendix D
4.1 Proof of Lemma 6
We use the matrix Bernstein inequality to prove the Lemma. First define:
The following elementary bounds (Tang et al. 2013; Candes and Fernandez-Granda 2014) are useful.
Thus, we have
Also we can write
Applying matrix Bernstein inequality (Rectangular Case) (Tropp 2012; Yang et al. 2016) and using \( \left( {D.2} \right) \) and \( \left( {D.3} \right) \), we can see that for a fixed \( \left( {i_{1} ,i_{2} ,i_{3} } \right) \),
In order to make this failure probability less than \( \delta_{2} \), we require that:
Which results in the following bound on \( M \)
Using a union bound for \( i_{1} ,i_{2} ,i_{3} = 0,1,2,3 \), we have
Provided that \( M^{3} \ge \frac{{4000 \times 4^{{2\left( {i_{1} + i_{2} + i_{3} } \right)}} \mu LK}}{{3\upvarepsilon_{2}^{2} }}\log \left( {\frac{4KL + L}{{\delta_{2} }}} \right) \).
Appendix E
5.1 Proof of Lemma 7
According to Lemma 6, conditioned on the events \( \upvarepsilon_{{1,\upvarepsilon_{1} }} \) with \( {\upvarepsilon}_{1} \epsilon \left({0,\left. {\frac{1}{4} } \right]} \right. \) and
We have
Where the second inequality uses the fact that \( \varvec{L} \) is a submatrix of \( {\varvec{\Gamma}}^{ - 1} \), and the third and fourth inequalities follow from Lemmas 4 and 1, respectively.
Using a union bound for \( i_{1} ,i_{2} ,i_{3} = 0,1,2,3 \), we have
Appendix F
6.1 Proof of Lemma 9
There exist a numerical constant C such that
Where the inequality follows from proof of Lemma IV.9 in Tang et al. (2013). Conditioned on \( {\varvec{\upvarepsilon}}_{{1,\upvarepsilon_{1} }} \) with \( {\varvec{\upvarepsilon}}_{1} \epsilon \left({0,\frac{1}{4}} \right] \), if \( \left\{ {\varvec{\tau}_{k} } \right\} \) are well separated to the sequence \( \left\{ {K^{{\left( {i_{1} ,i_{2} ,i_{3} } \right)}} \left( {\varvec{\tau}-\varvec{\tau}_{k} } \right)} \right\} \) decreases, for some numerical constant C we have
Where the second inequality uses Lemmas 1 and 4 and the fact that \( \varvec{L} - \bar{\varvec{L}} \otimes \varvec{I}_{L} \) is a submatrix of \( {\varvec{\Gamma}}^{ - 1} - {\bar{\boldsymbol{\varGamma }}}^{ - 1} \).
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Shojaei, S., Haddadi, F. Blind three dimensional deconvolution via convex optimization. Multidim Syst Sign Process 31, 1029–1049 (2020). https://doi.org/10.1007/s11045-019-00696-x
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DOI: https://doi.org/10.1007/s11045-019-00696-x