A new direction-of-arrival estimation method exploiting signal structure information

Abstract

A new method is proposed to estimate the direction-of-arrival (DOA) based on uniform linear array sampling and named as sparsity and temporal correlation exploiting (SaTC-E). By exploiting the structure information of source signals, including spatial sparsity and temporal correlation of sources, SaTC-E accomplishes DOA estimation with superior performance via block sparse bayesian learning methodology and grid refined strategy. SaTC-E is applicable into time-varying manifold scenario, such as wideband sources, time-varying array, provided that the array manifold matrix is determinable. It has improved performance with some other merits, including superior resolution, requirement for a few snapshots, no knowledge of source number, and applicability to spatially and temporally corrected sources. Real data tests and numerical simulations are carried out to demonstrate the advantages of SaTC-E.

Appendices

Appendix 1

The detail for the derivatives (20) is presented as follows.

Let $\mathbf {W}= {{\varvec{\Sigma }}_{-k}}+{{\tilde{\gamma }}_{k}}{\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \otimes \mathbf {B},$ and the derivatives are expressed as

$$\begin{aligned} \frac{\partial L\left( {{{\tilde{\gamma }}}_{k}},{{{\tilde{\vartheta }}}_{k}} \right) }{\partial {{{\tilde{\gamma }}}_{k}}}= & {} \frac{\partial \ln \left| \mathbf {W} \right| }{\partial {{{\tilde{\gamma }}}_{k}}}+\frac{\partial {{{\mathbf {z}}}^{H}}{{\mathbf {W}}^{-1}}{\mathbf {z}}}{\partial {{{\tilde{\gamma }}}_{k}}}\\= & {} \left\langle \frac{\partial \ln \left| \mathbf {W} \right| }{\partial \mathbf {W}},\frac{\partial \mathbf {W}}{\partial {{{\tilde{\gamma }}}_{k}}} \right\rangle +\left\langle \frac{\partial {{{\mathbf {z}}}^{H}}{{\mathbf {W}}^{-1}}{\mathbf {z}}}{\partial \mathbf {W}},\frac{\partial \mathbf {W}}{\partial {{{\tilde{\gamma }}}_{k}}} \right\rangle ,\\ \frac{\partial L\left( {{{\tilde{\gamma }}}_{k}},{{{\tilde{\vartheta }}}_{k}} \right) }{\partial {{{\tilde{\vartheta }}}_{k}}}= & {} \frac{\partial \ln \left| \mathbf {W} \right| }{\partial {{{\tilde{\vartheta }}}_{k}}}+\frac{\partial {{{\mathbf {z}}}^{H}}{{\mathbf {W}}^{-1}}{\mathbf {z}}}{\partial {{{\tilde{\vartheta }}}_{k}}} \\= & {} \left\langle \frac{\partial \ln \left| \mathbf {W} \right| }{\partial \mathbf {W}},\frac{\partial \mathbf {W}}{\partial {{{\tilde{\vartheta }}}_{k}}} \right\rangle +\left\langle \frac{\partial {{{\mathbf {z}}}^{H}}{{\mathbf {W}}^{-1}}{\mathbf {z}}}{\partial \mathbf {W}},\frac{\partial \mathbf {W}}{\partial {{{\tilde{\vartheta }}}_{k}}} \right\rangle . \end{aligned}$$

Employing the matrix differential formulas that

$$\begin{aligned} \frac{\partial \ln \left| \mathbf {W} \right| }{\partial \mathbf {W}}= & {} {{\mathbf {W}}^{-1}}, \\ \frac{\partial \mathbf {W}}{\partial {{{\tilde{\gamma }}}_{k}}}= & {} {\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \otimes \mathbf {B}, \\ \frac{\partial \mathbf {W}}{\partial {{{\tilde{\vartheta }}}_{k}}}= & {} {{\tilde{\gamma }}_{k}}\left( {\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{\mathbf {d}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) +\mathbf {d}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \right) \otimes \mathbf {B}, \\ \frac{\partial {{{\mathbf {z}}}^{H}}{{\mathbf {W}}^{-1}}{\mathbf {z}}}{\partial \mathbf {W}}= & {} -{{\mathbf {W}}^{-1}}{\mathbf {z}}{{{\mathbf {z}}}^{H}} {{\mathbf {W}}^{-1}}, \end{aligned}$$

where $\mathbf {W}$ is a conjugated symmetrical matrix, and $\mathbf {d}\left( {{{\tilde{\vartheta }}}_{k}}\right) ={\partial {\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}}\right) }/{\partial {{{\tilde{\vartheta }}}_{k}}}\;$, and let $\mathbf {V}={{\mathbf {W}}^{-1}}-{{\mathbf {W}}^{-1}}{\mathbf {z}} {{{\mathbf {z}}}^{H}}{{\mathbf {W}}^{-1}}$, we have

$$\begin{aligned} \frac{\partial L\left( {{{\tilde{\gamma }}}_{k}},{{{\tilde{\vartheta }}}_{k}} \right) }{\partial {{{\tilde{\gamma }}}_{k}}}= & {} \left\langle \mathbf {V},{\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \otimes \mathbf {B} \right\rangle \\= & {} {\text {tr}}\left\{ \left( \mathbf {V} \right) \left[ {\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \otimes \mathbf {B} \right] \right\} , \\ \frac{\partial L\left( {{{\tilde{\gamma }}}_{k}},{{{\tilde{\vartheta }}}_{k}} \right) }{\partial {{{\tilde{\vartheta }}}_{k}}}= & {} {{{\tilde{\gamma }}}_{k}}\left\langle \mathbf {V},\left( {\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{\mathbf {d}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) +\mathbf {d}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \right) \otimes \mathbf {B} \right\rangle \\= & {} {{{\tilde{\gamma }}}_{k}}{\text {tr}}\left\{ \mathbf {V}\left[ \left( {\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{\mathbf {d}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \right) \otimes \mathbf {B} \right] \right\} \\&+ \, {{{\tilde{\gamma }}}_{k}}{\text {tr}}\left\{ \mathbf {V}\left[ \left( \mathbf {d}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \right) \otimes \mathbf {B} \right] \right\} . \end{aligned}$$

In terms of the singular value decomposition (SVD), $\mathbf {B}$ can be rewritten as

$$\begin{aligned} \mathbf {B}=\sum \limits _{i=1}^{{{r}_{\mathbf {B}}}} {\mathbf {u}_{i}^{\mathbf {B}}{{\left( \mathbf {u}_{i}^{\mathbf {B}} \right) }^{H}}} \end{aligned}$$

with ${r}_{\mathbf {B}}={\text {rank}}\left( \mathbf {B} \right)$ and $\mathbf {u}_{i}^\mathbf {B}$ being the product of the ith singular value and the associated eigenvector of $\mathbf {B}$, So that

$$\begin{aligned} \quad \frac{\partial L\left( {{{\tilde{\gamma }}}_{k}},{{{\tilde{\vartheta }}}_{k}} \right) }{\partial {{{\tilde{\gamma }}}_{k}}}= & {} {\text {tr}}\left\{ \mathbf {V}\left[ {\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \otimes \sum \limits _{i=1}^{{{r}_{\mathbf {B}}}} {\mathbf {u}_{i}^{\mathbf {B}}{{\left( \mathbf {u}_{i}^{\mathbf {B}} \right) }^{H}}} \right] \right\} \\= & {} \sum \limits _{i=1}^{{{r}_{\mathbf {B}}}}{{\text {tr}}\left\{ \mathbf {V}\left[ \left( {\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \otimes \mathbf {u}_{i}^{\mathbf {B}} \right) \left( {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \otimes {{\left( \mathbf {u}_{i}^{\mathbf {B}} \right) }^{H}} \right) \right] \right\} }\\= & {} \sum \limits _{i=1}^{{{r}_{\mathbf {B}}}}{{{\left[ {\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \otimes \mathbf {u}_{i}^{\mathbf {B}} \right] }^{H}}\mathbf {V}\left[ {\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \otimes \mathbf {u}_{i}^{\mathbf {B}} \right] }. \end{aligned}$$

The derivation is accomplished.

Appendix 2

Proof of Proposition 2

when $\mathbf {B}={{{\mathbf {I}}}_{L}}$, we get ${{\varvec{\Sigma }}_{-k}}={{{\breve{{\varvec{\Sigma }} }}}_{-k}}\otimes {{{\mathbf {I}}}_{L}}$, and then $L\left( {{{\tilde{\gamma }}}_{k}},{{{\tilde{\vartheta }}}_{k}} \right)$ can be rewritten as

$$\begin{aligned} L\left( {{{\tilde{\gamma }}}_{k}},{{{\tilde{\vartheta }}}_{k}} \right)= & {} \log \left| \left( {{{{\breve{{\varvec{\Sigma }} }}}}_{-k}}+{{{\tilde{\gamma }}}_{k}}{\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \right) \otimes {{{\mathbf {I}}}_{L}} \right| \\&+\,{{{\mathbf {z}}}^{H}}{{\left( \left( {{{{\breve{{\varvec{\Sigma }} }}}}_{-k}}+{{{\tilde{\gamma }}}_{k}}{\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \right) \otimes {{{\mathbf {I}}}_{L}} \right) }^{-1}}{\mathbf {z}}, \end{aligned}$$

and the first term of $L\left( {{{\tilde{\gamma }}}_{k}},{{{\tilde{\vartheta }}}_{k}} \right)$ equals to the first term of $\breve{L}\left( {{{\tilde{\gamma }}}_{k}},{{{\tilde{\vartheta }}}_{k}} \right)$, because

$$\begin{aligned}&\log \left| \left( {{{{\breve{{\varvec{\Sigma }} }}}}_{-k}}+{{{\tilde{\gamma }}}_{k}}{\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \right) \otimes {{{\mathbf {I}}}_{L}} \right| \\&\quad = \log \left( {{\left| {{{{\breve{{\varvec{\Sigma }} }}}}_{-k}}+{{{\tilde{\gamma }}}_{k}}{\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \right| }^{L}}{{\left| {{{\mathbf {I}}}_{L}} \right| }^{M}} \right) \\&\quad =\log {{\left| {{{{\breve{{\varvec{\Sigma }} }}}}_{-k}}+{{{\tilde{\gamma }}}_{k}}{\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \right| }^{L}} \\&\quad = L\log \left| {{{{\breve{{\varvec{\Sigma }} }}}}_{-k}}+{{{\tilde{\gamma }}}_{k}}{\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \right| . \end{aligned}$$

Let $\mathbf {Q}= {{\left( {{{{\breve{{\varvec{\Sigma }} }}}}_{-k}}+{{{\tilde{\gamma }}}_{k}}{\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \right) }^{-1}}$, and write the array output matrix ${\mathbf {Y}}$ as the form of

$$\begin{aligned} {\mathbf {Y}}=\left[ \begin{array}{c@{~}l} {{{\mathbf {y}}}_{1}} \\ {{{\mathbf {y}}}_{2}} \\ \vdots \\ {{{\mathbf {y}}}_{M}} \\ \end{array} \right] \in {{{\mathbb {C}}}^{M\times L}},\, {{{\mathbf {y}}}_{i}}=\left[ {{y}_{i1}} \, {{y}_{i2}} \, \ldots \, {{y}_{iL}} \right] \in {{{\mathbb {C}}}^{1\times L}}, \end{aligned}$$

so that ${{{\mathbf {z}}}^{T}}=\left[ {{{\mathbf {y}}}_{1}} \, {{{\mathbf {y}}}_{2}} \, \ldots \, {{{\mathbf {y}}}_{M}} \right] \in {{{\mathbb {C}}}^{1\times ML}}$. The second terms of $L\left( {{{\tilde{\gamma }}}_{k}},{{{\tilde{\vartheta }}}_{k}} \right)$ and $\breve{L}\left( {{{\tilde{\gamma }}}_{k}},{{{\tilde{\vartheta }}}_{k}} \right)$ can be calculated, respectively, as

$$\begin{aligned}&{{{\mathbf {z}}}^{H}}{{\left( \left( {{{{\breve{{\varvec{\Sigma }} }}}}_{-k}}+{{{\tilde{\gamma }}}_{k}}{\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \right) \otimes {{{\mathbf {I}}}_{L}} \right) }^{-1}}{\mathbf {z}}\\&\quad ={{{\mathbf {z}}}^{H}}\left( \mathbf {Q}\otimes {{{\mathbf {I}}}_{L}} \right) {\mathbf {z}}\\&\quad =\sum \limits _{i=1}^{M}{\sum \limits _{j=1}^{M}{{{q}_{ij}} {{{\bar{\mathbf{y}}}}_{i}}{\mathbf {y}}_{j}^{T}}},\\&{\text {tr}}\left( {{{\mathbf {Y}}}^{H}}{{\left( {{{{\breve{{\varvec{\Sigma }} }}}}_{-k}}+{{{\tilde{\gamma }}}_{k}}{\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \right) }^{-1}}{\mathbf {Y}} \right) \\&\quad ={\text {tr}}\left( \mathbf {QY}{{{\mathbf {Y}}}^{H}} \right) \\&\quad =\sum \limits _{i=1}^{M}{\sum \limits _{j=1}^{M}{{{q}_{ij}} {{{\mathbf {y}}}_{j}}{\mathbf {y}}_{i}^{H}}}. \end{aligned}$$

Because of ${{{\mathbf {y}}}_{j}}{\mathbf {y}}_{i}^{H}= \sum \limits _{k=1}^{L}{{{y}_{jk}}{{{\bar{y}}}_{ik}}} ={{\bar{\mathbf {y}}}_{i}}{\mathbf {y}}_{j}^{T}$, we have

$$\begin{aligned}&{{{\mathbf {z}}}^{H}}{{\left( \left( {{{{\breve{{\varvec{\Sigma }} }}}}_{-k}}+{{{\tilde{\gamma }}}_{k}}{\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \right) \otimes {{{\mathbf {I}}}_{L}} \right) }^{-1}}{\mathbf {z}}\\&\quad = {\text {tr}}\left( {{{\mathbf {Y}}}^{H}}{{\left( {{{{\breve{{\varvec{\Sigma }} }}}}_{-k}}+{{{\tilde{\gamma }}}_{k}}{\mathbf {a}}\left( {{{\tilde{\vartheta }}}_{k}} \right) {{{\mathbf {a}}}^{H}}\left( {{{\tilde{\vartheta }}}_{k}} \right) \right) }^{-1}}{\mathbf {Y}} \right) , \end{aligned}$$

and thus Proposition 2 is proved. $\square$