Joint-block-sparsity for efficient 2-D DOA estimation with multiple separable observations

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 order to cope with this issue, 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 enforces the sparsity in multiple dimensions, namely azimuth, 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 solves the 2-D DOA estimation problem with significantly low complexity and sufficient accuracy.

Appendices

Appendices

1.1 Derivation of (18)

Using $\textit{factored representation}$ as in Cotter et al. (2005) and Rao and Kreutz-Delgado (1999), the gradient of the cost function (18), $J(\{\mathbf{P}_k\}_{k=1}^K)$, is given as follows

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

(22)

where $\bar{\mathbf{H}}_{\phi } = [\mathbf{Q}_1^H\mathbf{A}_{\phi },\ldots ,\mathbf{Q}_K^H\mathbf{A}_{\phi }]$. The gradient of the first term in (22) is $\nabla _{{P}} \{ || \tilde{\mathbf{P}}||_{2,2,1}\} = {\varPi }(\mathbf{P})\mathbf{P}$ (Cotter et al. 2005) where ${\varPi }(\mathbf{P}) \in \mathbb {C}^{N_{\theta } \times N_{\theta }}$ can be computed as

$$\begin{aligned} \varPi (\mathbf{P}) = l \cdot \text {diag}(\tilde{\mathbf{p}}^{l-2} ) \end{aligned}$$

(23)

where $l\le 1$ (Cotter et al. 2005; Rao and Kreutz-Delgado 1999), $\tilde{\mathbf{p}} \in \mathbb {C}^{N_{\theta }}$ and

$$\begin{aligned} \tilde{p}_{n_{\theta }}= \left( \sum _{k = 1}^{K}\sum _{n = 1}^{N} |[\mathbf{P}]_{n_{\theta },N(k-1)+n }|^2\right) ^{1/2}. \end{aligned}$$

(24)

By equating the gradient in (22) to zero we get

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

(25)

Then we have the solution for $\mathbf{P}$ as

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

(26)

1.2 Derivation of (19)

In a similar way as in Appendix 7.1 (Cotter et al. 2005; Rao and Kreutz-Delgado 1999), we have the gradient of the cost function in (19), $J(\{\mathbf{Q}_k\}_{k=1}^K)$, as

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

where $\tilde{\mathbf{H}}_{\phi } = [\mathbf{A}_{\phi }{} \mathbf{P}_1^T,\ldots ,\mathbf{A}_{\phi }{} \mathbf{P}_K^T]\in \mathbb {C}^{N_{\phi }\times N_{\theta }K }$, ${\varPi }(\mathbf{Q}) = l \cdot \text {diag}(\tilde{\mathbf{q}}^{l-2} )$ for $\tilde{\mathbf{q}} \in \mathbb {C}^{N_{\phi }}$ and

$$\begin{aligned} \tilde{q}_{n_{\phi }}= \left( \sum _{k = 1}^{K}\sum _{n = 1}^{N} |[\mathbf{Q}]_{n_{\phi },N(k-1)+n }|^2\right) ^{1/2}. \end{aligned}$$

(27)

By equating the gradient to zero we get

$$\begin{aligned}&\lambda {\varPi }(\mathbf{Q})\mathbf{Q} + \gamma _{2} \big ( {\mathbf{A}}_{\phi }{\mathbf{A}}_{\phi }^H \mathbf{Q} - \tilde{\mathbf{H}}_{\phi } \big ) = 0 \nonumber \\&\big (\lambda {\varPi }(\mathbf{Q}) + \gamma _{2} {\mathbf{A}}_{\phi }{\mathbf{A}}_{\phi }^H \big )\mathbf{Q} = \tilde{\mathbf{H}}_{\phi }. \end{aligned}$$

(28)

Then we have the solution for $\mathbf{Q}$ as

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

(29)