A novel spatially spread electromagnetic vector sensor for high-accuracy 2-D DOA estimation

Abstract

In this paper, a new spatially spread electromagnetic vector sensor (SS-EMVS) is proposed by a two-step design. In addition, a novel DOA estimator with coarse-fine estimate combination is presented for the proposed array. The first step aims to make the configurations of SS-EMVS satisfy the “vector cross-product” estimator, leading to a coarse estimation of three direction-cosines. The second step focuses on extending the two dimensional (2-D) array apertures of SS-EMVS, resulting in two fine but ambiguous estimations on the direction-cosines by extracting inter-sensor phase-delay. Combination the coarse and fine estimations, the high-accuracy 2-D DOA estimation can be obtained by using the coarse estimation to disambiguate the fine estimation. The three- dipoles and loops of the proposed configuration are located separately, which are found to reduce mutual coupling as compared with collocated EMVS. Moreover, the new configuration is able to extend 2-D array aperture to improve the accuracy of 2-D direction-finding. Numerical Simulations are conducted to demonstrate the effectiveness of the proposed algorithm.

Appendix A: Derivation of the CRB for a single arbitrary SS-EMVS

The output signal is modeled as: ${\varvec{x}}(t)=\sum \limits _{k=1}^{K}{\tilde{{\varvec{a}}}_{k}s_{k}(t)} +{\varvec{n}}(t)$, where $\tilde{{\varvec{a}}}$ is defined in (3), $s_{k}(t)=\sigma _{s}\exp (j2\pi f_{k}t+j\varepsilon )$ denotes uncorrelated pure tones with different frequencies, which is the same as given in Wong and Zoltowski (1997); Monte et al. (2007); Luo and Yuan (2012) and Au Yeung and Wong (2009). ${\varvec{n}}(t)$ denotes Gaussian random process with zero-mean and covariance $\sigma _{n}^{2}$. The analysis method of Au Yeung and Wong (2009) can be used to derivate the CRBs for a single arbitrary SS-EMVS, as follows.

The data sampling instants occur at $t=l/f_{s}$, where $f_{s}$ denotes the sampling frequency. We collect all $L$ snapshots into a $6L\times 1$ data-set: ${\varvec{y}}=[{\varvec{x}}(1/{f_{s}})^{T},{\varvec{x}} (2/{f_{s}})^{T},\cdots {\varvec{x}}(L/{f_{s}})^{T}]^{T}$. The mean of ${\varvec{y}}$ equals ${\varvec{\mu }}=\sum \limits _{k=1}^{K}{{\varvec{s}}_{k}\otimes \tilde{{\varvec{a}}}_{k}}$, where ${\varvec{s}}_{k}=\sigma _{s}\exp (j\varepsilon )[\exp (j2\pi f_{k}^{\prime }),\exp (j4\pi f_{k}^{\prime }),\cdots ,\exp (j2L\pi f_{k}^{\prime })]^{T}$, $f_{k}^{\prime } ={f_{k}}/{f_{s}}$ denotes digital frequency. Note that the source parameters $\left\{ {\sigma _{s},\varepsilon ,f_{k}^{\prime }} \right\}$ will affect the CRB. The covariance of${\varvec{y}}$ equals ${\varvec{R}}=\sigma _{n}^{2}{\varvec{I}}_{6L}$. Therefore, ${\varvec{y}}\sim {\mathcal {N}}_{c}({\varvec{\mu }},{\varvec{R}})$. Then the $(i,j)$th entry of the Fisher Information Matrix (FIM) is (Trees 2002, Eq. (8.34)):

$$\begin{aligned} \left[ \hbox {FIM}\left( {\varvec{\psi }}\right) \right] _{i,j} =2\hbox {Re}\left[ \left( \frac{\partial {\varvec{\mu }}}{\partial \psi _{i}}\right) ^{H}{\varvec{R}}^{-1}\left( \frac{\partial {\varvec{\mu }}}{\partial \psi _{j}}\right) \right] , \end{aligned}$$

(31)

where ${\varvec{\psi }}=[\theta ,\phi ,\gamma ,\eta ]^{T}$ is the vector of parameters to be estimated. Note that $\frac{\partial {\varvec{\mu }}}{\partial \psi _{i}}=\frac{\partial \sum \limits _{k=1}^{K}{{\varvec{s}}_{k}\otimes \tilde{{\varvec{a}}}_{k}}}{\partial \psi _{i}}={\varvec{s}}_{i}\otimes \frac{\partial \tilde{{\varvec{a}}}_{i}}{\partial \psi _{i}}$. Then the FIM can be simplified to:

$$\begin{aligned} \left[ \hbox {FIM}\left( {\varvec{\psi }}\right) \right] _{i,j}= & {} \frac{2}{\sigma _{n}^{2}}\cdot \hbox {Re}\left\{ \left[ {\varvec{s}}_{i}\otimes \frac{\partial \tilde{{\varvec{a}}}_{i}}{\partial \psi _{i}}\right] ^{H}\left[ {\varvec{s}}_{j}\otimes \frac{\partial \tilde{{\varvec{a}}}_{i}}{\partial \psi _{j}}\right] \right\} \nonumber \\= & {} \frac{2}{\sigma _{n}^{2}}\cdot \hbox {Re}\left\{ \left( {\varvec{s}}_{i}^{H}{\varvec{s}}_{j}\right) \left[ \left( \frac{\partial \tilde{{\varvec{a}}}_{i}}{\partial \psi _{i}}\right) ^{H}\frac{\partial \tilde{{\varvec{a}}}_{j}}{\partial \psi _{j}}\right] \right\} \nonumber \\= & {} \frac{2\sigma _{s}^{2}}{\sigma _{n}^{2}}\cdot \hbox {Re}\left[ c_{ij} \left( \frac{\partial \tilde{{\varvec{a}}}_{i}}{\partial \psi _{i}}\right) ^{H}\frac{\partial \tilde{{\varvec{a}}}_{j}}{\partial \psi _{j}}\right] , \end{aligned}$$

(32)

where $c_{ij} =\exp [j2\pi (f_{j}^{\prime } -f_{i}^{\prime })]+\exp [j4\pi (f_{j}^{\prime } -f_{i}^{\prime })]+\cdots +\exp [j2L\pi (f_{j}^{\prime } -f_{i}^{\prime })]$. In the following, we list the relevant derivatives of the array manifold $\tilde{{\varvec{a}}}$ in (32). When $\psi$ denotes the elevation angle:

$$\begin{aligned} \frac{\partial \tilde{{\varvec{a}}}}{\partial \theta }= & {} \frac{\partial {\varvec{d}}\left( \theta ,\phi \right) \odot \left[ {\varvec{\varPhi }} \left( \theta ,\phi \right) {\varvec{g}}\left( \gamma ,\eta \right) \right] }{\partial \theta } \nonumber \\= & {} \frac{\partial {\varvec{d}}\left( \theta ,\phi \right) }{\partial \theta }\odot \left[ {{\varvec{\varPhi }} }\left( \theta ,\phi \right) {\varvec{g}}\left( \gamma ,\eta \right) \right] +{\varvec{d}}\left( \theta ,\phi \right) \odot \left[ \frac{\partial {\varvec{\varPhi }}\left( \theta ,\phi \right) }{\partial \theta }{\varvec{g}}\left( \gamma ,\eta \right) \right] \nonumber \\= & {} {\varvec{c}}_{1}{\varvec{d}}\left( \theta ,\phi \right) \odot \left[ {{\varvec{\varPhi }} }\left( \theta ,\phi \right) {\varvec{g}}\left( \gamma ,\eta \right) \right] +{\varvec{d}}\left( \theta ,\phi \right) \odot \left[ {\varvec{C}}_{2}{\varvec{g}}\left( \gamma ,\eta \right) \right] , \end{aligned}$$

(33)

where $\{{\varvec{\varPhi }} (\theta ,\phi ),{\varvec{g}}(\gamma ,\eta )\}$ and ${\varvec{d}}(\theta ,\phi )$ are defined in (1) and (3), respectively,

$$\begin{aligned} {\varvec{c}}_{1}=\left[ { \begin{array}{l} -j\frac{2\pi }{\lambda }\left( x_{ex} \cos \theta \cos \phi +y_{ex} \cos \theta \sin \phi -z_{ex} \sin \theta \right) \\ -j\frac{2\pi }{\lambda }\left( x_{ey} \cos \theta \cos \phi +y_{ey} \cos \theta \sin \phi -z_{ey} \sin \theta \right) \\ -j\frac{2\pi }{\lambda }\left( x_{ez} \cos \theta \cos \phi +y_{ez} \cos \theta \sin \phi -z_{ez} \sin \theta \right) \\ -j\frac{2\pi }{\lambda }\left( x_{hx} \cos \theta \cos \phi +y_{hx} \cos \theta \sin \phi -z_{hx} \sin \theta \right) \\ -j\frac{2\pi }{\lambda }\left( x_{hy} \cos \theta \cos \phi +y_{hy} \cos \theta \sin \phi -z_{hy} \sin \theta \right) \\ -j\frac{2\pi }{\lambda }\left( x_{hz} \cos \theta \cos \phi +y_{hz} \cos \theta \sin \phi -z_{hz} \sin \theta \right) \\ \end{array}}\right] , \end{aligned}$$

(34)

and

$$\begin{aligned} {\varvec{C}}_{2}=\left[ { \begin{array}{c} -u,-v,-w, 0, 0, 0 \\ 0, 0, 0, u, v, w \\ \end{array}}\right] ^{T}. \end{aligned}$$

(35)

When $\psi$ denotes the azimuth angle:

$$\begin{aligned} \frac{\partial \tilde{{\varvec{a}}}}{\partial \phi }= & {} \frac{\partial {\varvec{d}}\left( \theta ,\phi \right) }{\partial \phi }\odot \left[ {{\varvec{\varPhi }} }\left( \theta ,\phi \right) {\varvec{g}}\left( \gamma ,\eta \right) \right] +{\varvec{d}}\left( \theta ,\phi \right) \odot \left[ \frac{\partial {{\varvec{\varPhi }} }\left( \theta ,\phi \right) }{\partial \phi }{\varvec{g}}\left( \gamma ,\eta \right) \right] \nonumber \\= & {} {\varvec{c}}_{3}{\varvec{d}}\left( \theta ,\phi \right) \odot \left[ {{\varvec{\varPhi }} }\left( \theta ,\phi \right) {\varvec{g}}\left( \gamma ,\eta \right) \right] +{\varvec{d}}\left( \theta ,\phi \right) \odot \left[ {\varvec{C}}_{4}{\varvec{g}}\left( \gamma ,\eta \right) \right] , \end{aligned}$$

(36)

where

$$\begin{aligned} {\varvec{c}}_{3}=\left[ { \begin{array}{l} -j\frac{2\pi }{\lambda }\left( -x_{ex} \sin \theta \sin \phi +y_{ex} \sin \theta \cos \phi \right) \\ -j\frac{2\pi }{\lambda }\left( -x_{ey} \sin \theta \sin \phi +y_{ey} \sin \theta \cos \phi \right) \\ -j\frac{2\pi }{\lambda }\left( -x_{ez} \sin \theta \sin \phi +y_{ez} \sin \theta \cos \phi \right) \\ -j\frac{2\pi }{\lambda }\left( -x_{hx} \sin \theta \sin \phi +y_{hx} \sin \theta \cos \phi \right) \\ -j\frac{2\pi }{\lambda }\left( -x_{hy} \sin \theta \sin \phi +y_{hy} \sin \theta \cos \phi \right) \\ -j\frac{2\pi }{\lambda }\left( -x_{hz} \sin \theta \sin \phi +y_{hz} \sin \theta \cos \phi \right) \\ \end{array}}\right] , \end{aligned}$$

(37)

and

$$\begin{aligned} {\varvec{C}}_{4}=\left[ {\begin{array}{l} -\cos \theta \sin \phi ,\cos \theta \cos \phi ,0, -\cos \phi , -\sin \phi , 0 \\ -\cos \phi , -\sin \phi , 0, \cos \theta \sin \phi ,-\cos \theta \cos \phi ,0 \\ \end{array}}\right] ^{T}. \end{aligned}$$

(38)

When $\psi$ denotes the auxiliary polarization angle:

$$\begin{aligned} \frac{\partial \tilde{{\varvec{a}}}}{\partial \gamma }={\varvec{d}}(\theta ,\phi )\odot \left[ {{\varvec{\varPhi }} }(\theta ,\phi )\frac{\partial {\varvec{g}}(\gamma ,\eta )}{\partial \gamma }\right] ={\varvec{d}}(\theta ,\phi )\odot \left[ {{\varvec{\varPhi }} }(\theta ,\phi ){\varvec{c}}_{5}\right] , \end{aligned}$$

(39)

where ${\varvec{c}}_{5}=\left[ {\cos \gamma e^{j\eta },-\sin \gamma }\right] ^{T}$. When $\psi$ denotes the polarization phase difference:

$$\begin{aligned} \frac{\partial \tilde{{\varvec{a}}}}{\partial \eta }={\varvec{d}}(\theta ,\phi )\odot \left[ {{\varvec{\varPhi }} }(\theta ,\phi )\frac{\partial {\varvec{g}}(\gamma ,\eta )}{\partial \eta }\right] ={\varvec{d}}(\theta ,\phi )\odot \left[ {{\varvec{\varPhi }} }(\theta ,\phi ){\varvec{c}}_{6}\right] , \end{aligned}$$

(40)

where ${\varvec{c}}_{6}=\left[ {j\sin \gamma e^{j\eta },0}\right] ^{T}$.