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Sensor array calibration with joint-block-sparsity in the presence of multiple separable observations

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Abstract

In sparsity-based optimization problems, one of the major issue is computational complexity, especially when the unknown signal is represented in multi-dimensions such as in the problem of 2-D (azimuth and elevation) direction-of-arrival (DOA) estimation. In this paper, a low-complexity sparsity-based method is proposed for DOA estimation in the presence of array imperfections such as mutual coupling. In order to reduce the complexity of the optimization problem, this paper introduces a new sparsity structure that can be used to model the optimization problem in case of multiple data snapshots and multiple separable observations where the dictionary can be decomposed into two parts: azimuth and elevation dictionaries. The proposed sparsity structure is called joint-block-sparsity which exploits the sparsity in both multiple dimensions, namely azimuth and elevation, and data snapshots. In order to model the joint-block-sparsity in the optimization problem, triple mixed norms are used. In the simulations, the proposed method is compared with both sparsity-based techniques and subspace-based methods as well as the Cramer–Rao lower bound. It is shown that the proposed method effectively calibrates the sensor array with significantly low complexity and sufficient accuracy.

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Notes

  1. If the source signals are coherent where there is only a single significant singular value, then the dimension reduction procedure can still be done as in [19].

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Correspondence to Ahmet M. Elbir.

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Appendix

Appendix

Using \(\textit{factored representation}\) as in [31, 32], the gradient of the cost function (22), \(J(\{\mathbf{P }_k\}_{k=1}^K)\), is given as follows

$$\begin{aligned}&\nabla _{P}\{J\left( \left\{ \mathbf{P }_k\}_{k=1}^K\right) \right\} = \nabla _{{P}} \{ || \tilde{\mathbf{P }}||_{2,2,1} \} \nonumber \\&\quad +\,\frac{\gamma _{1}}{2} \left( 2{\mathbf{A }}_{\theta }^H{\mathbf{A }}_{\theta }\mathbf{P } -2{\mathbf{A }}_{\theta }^H \tilde{\mathbf{Y }}^{\text {SV}} \right) +\frac{\gamma _{2}}{2} \left( 2 \mathbf{P } - 2 \tilde{\mathbf{H }}_{\phi }^Q \right) , \end{aligned}$$
(24)

where \({\mathbf{P }} = [\mathbf{P }_1,\dots ,\mathbf{P }_K]\), \(\tilde{\mathbf{H }}_{\phi }^Q = [\mathbf{Q }_1^H\mathbf{A }_{\phi },\dots ,\mathbf{Q }_K^H\mathbf{A }_{\phi }]_{N\times KN_{\theta }}\) and the gradient of the first term is \(\nabla _{{P}} \{ || \tilde{\mathbf{P }}||_{2,2,1}\} = {\varPi }(\mathbf{P })\mathbf{P }\) [31]. \({\varPi }(\mathbf{P }) \in {\mathbb {C}}^{N_{\theta } \times N_{\theta }}\) can be computed as \( \varPi (\mathbf{P }) = l \cdot \text {diag}(\tilde{\mathbf{p }}^{l-2} )\) for \(l\le 1\) [31, 32], \(\tilde{\mathbf{p }} \in {\mathbb {C}}^{N_{\theta }}\) and \({\tilde{p}}_{n_{\theta }}= (\sum _{k = 1}^{K}\sum _{n = 1}^{N} |[\mathbf{P }]_{n_{\theta },N(k-1)+n }|^2)^{1/2}. \) By equating the gradient in (24) to zero, we get

$$\begin{aligned}&{\varPi }(\mathbf{P }) \mathbf{P } + \gamma _{1} \left( {\mathbf{A }}_{\theta }^H{\mathbf{A }}_{\theta }\mathbf{P } - \mathbf{A }_{\theta }^H \tilde{\mathbf{Y }}^{\text {SV}}\right) + \gamma _{2}\left( \mathbf{P } - \tilde{\mathbf{H }}_{\phi }^Q\right) = 0 \nonumber \\&\big ( {\varPi }(\mathbf{P })+\gamma _{1} {\mathbf{A }}_{\theta }^H{\mathbf{A }}_{\theta } + \gamma _{2} \mathbf{I }\big ) \mathbf{P } = {\mathbf{A }}_{\theta }^H \tilde{\mathbf{Y }}^{\text {SV}} + \gamma _{2}\tilde{\mathbf{H }}_{\phi }^Q. \end{aligned}$$
(25)

Then, we have the solution for \(\mathbf{P }\) as

$$\begin{aligned} \mathbf{P } = \big ( {\varPi }(\mathbf{P })+\gamma _{1} {\mathbf{A }}_{\theta }^H{\mathbf{A }}_{\theta } + \gamma _{2} \mathbf{I }\big )^{\dagger } \big ( {\mathbf{A }}_{\theta }^H \tilde{\mathbf{Y }}^{\text {SV}} + \gamma _{2}\tilde{\mathbf{H }}_{\phi }^Q\big ). \end{aligned}$$
(26)

Similar to \(J(\{\mathbf{P }_k\}_{k=1}^K)\), we have the gradient of the cost function in (23), \(J(\{\mathbf{Q }_k\}_{k=1}^K)\), as

$$\begin{aligned} \nabla _{Q}\left\{ J\left( \{\mathbf{Q }_k\}_{k=1}^K\right) \right\} = \lambda {\varPi }(\mathbf{Q })\mathbf{Q } +\frac{\gamma _{2}}{2} \left( 2 {\mathbf{A }}_{\phi }{\mathbf{A }}_{\phi }^H \mathbf{Q } - 2 \tilde{\mathbf{H }}_{\phi }^P \right) , \end{aligned}$$

where \({\mathbf{Q }} = [\mathbf{Q }_1,\dots ,\mathbf{Q }_K]\), \(\tilde{\mathbf{H }}_{\phi }^P = [\mathbf{A }_{\phi }\mathbf{P }_1^T,\dots ,\mathbf{A }_{\phi }\mathbf{P }_K^T]_{N_{\phi }\times N_{\theta }K }\), \({\varPi }(\mathbf{Q }) = l \cdot \text {diag}(\tilde{\mathbf{q }}^{l-2} ) \) for \(\tilde{\mathbf{q }} \in {\mathbb {C}}^{N_{\phi }}\) and \({\tilde{q}}_{n_{\phi }}= (\sum _{k = 1}^{K}\sum _{n = 1}^{N} |[\mathbf{Q }]_{n_{\phi },N(k-1)+n }|^2)^{1/2}\). Equating the gradient to zero, we get \(\lambda {\varPi }(\mathbf{Q })\mathbf{Q } + \gamma _{2} \big ( {\mathbf{A }}_{\phi }{\mathbf{A }}_{\phi }^H \mathbf{Q } - \tilde{\mathbf{H }}_{\phi }^P \big ) = 0 \) and

$$\begin{aligned} \big (\lambda {\varPi }(\mathbf{Q }) + \gamma _{2} {\mathbf{A }}_{\phi }{\mathbf{A }}_{\phi }^H \big )\mathbf{Q } = \tilde{\mathbf{H }}_{\phi }^P . \end{aligned}$$
(27)

Then, we have the solution for \(\mathbf{Q }\) as

$$\begin{aligned} \mathbf{Q } = \big (\lambda {\varPi }(\mathbf{Q }) + \gamma _{2} {\mathbf{A }}_{\phi }{\mathbf{A }}_{\phi }^H \big )^{\dagger }\tilde{\mathbf{H }}_{\phi }^P . \end{aligned}$$
(28)

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Elbir, A.M. Sensor array calibration with joint-block-sparsity in the presence of multiple separable observations. SIViP 13, 905–913 (2019). https://doi.org/10.1007/s11760-019-01427-2

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