Low-complexity 2D parameter estimation of coherently distributed noncircular signals using modified propagator

Abstract

In this paper, a new algorithm is proposed for two-dimensional parameter estimation of coherently distributed noncircular signals. Based on the special sensor array geometry of L-shaped uniform linear arrays, 2D angular parameters are obtained independently by using the modified propagator method. Specifically, two selection matrices are employed to obtain rotation invariance matrices under considering signal noncircularity. By making use of the rotation invariance matrices, we then obtain the central elevation and azimuth direction-of-arrivals, respectively. After that, the pair-matching of them is accomplished by searching the minimums of a cost function of the estimated 2D angular parameters. Finally, numerical results to testify the effectiveness of the proposed algorithm are provided. It is also shown that the proposed algorithm performs well in a wide signal-to-noise ratio range.

Appendices

Appendix 1

In this appendix, we show the analytical expression of $\mathbf{c}\left( {\psi _k } \right)$ and $\bar{{\mathbf{c}}}\left( {\psi _k } \right)$. For simplicity, we neglect the subscript k in the following derivation.

In order to present the analytical expression of $\mathbf{c}\left( \psi \right)$, we consider $g(\vartheta ,\varphi ;\psi )$ is Cauchy distribution, and can be expressed as

$$\begin{aligned} g(\vartheta ,\varphi ;\psi )=\left\{ {\begin{array}{ll} \frac{1}{\pi ^{2}}\left( {\frac{\Delta _\theta \Delta _{\phi } }{\left( {(\vartheta -\theta )^{2}+\Delta _\theta ^2 } \right) \left( {(\varphi -\phi )^{2}+\Delta _{\phi } ^2 } \right) }} \right) ,&{}\quad \left| {\varphi -\phi } \right| >\varepsilon _\varphi ,\;\; \left| {\vartheta -\theta } \right| > \varepsilon _\theta \\ 0,&{}\quad \left| {\varphi -\phi } \right| <\varepsilon _\varphi ,\;\;\left| {\vartheta -\theta } \right| <\varepsilon _\theta \\ \end{array}} \right. \end{aligned}$$

(47)

where $\varepsilon _\theta$ and $\varepsilon _\varphi$ are small values associated with $\Delta _\theta$ and $\Delta _{\phi }$, respectively. For arbitrary $\theta$ and $\varphi$, we define $\vartheta =\theta +\tilde{\theta }$ and $\varphi =\phi +\tilde{\phi }$, where $\tilde{\theta }$ and $\tilde{\phi }$ are the small deviations. Assume that $g(\vartheta ;\psi )$ and $g(\varphi ;\psi )$ are independent identity distribution, $\int _0^\pi {g(\vartheta ;\psi )} d\vartheta =1$, and $\int _0^\pi {g(\varphi ;\psi )} d\varphi =1$.

Inserting (47) into (6), the mth element $\left[ {\mathbf{c}\left( \psi \right) } \right] _m$, $m=1,2,\ldots ,M$ of $\mathbf{c}\left( {\psi _k } \right)$ can be expressed as

$$\begin{aligned} \left[ \mathbf{c}\left( \psi \right) \right] _m= & {} \int _0^\pi \int _0^\pi e^{-j\pi (m-1)\cos (\tilde{\theta } +\theta )} \frac{1}{\pi ^{2}}\frac{\Delta _\theta }{\tilde{\theta }^{2}+\Delta _\theta ^2 } \frac{\Delta _{\phi } }{\tilde{\phi }^{2}+\Delta _{\phi } ^2 }d \tilde{\theta }d\tilde{\phi } \nonumber \\= & {} \frac{1}{\pi }\;e^{-j\pi (m-1)\cos (\tilde{\theta }+\theta )}\int _0^\pi e^{-j\pi (m-1)\tilde{\theta }\sin \theta }\frac{\Delta _\theta }{\tilde{\theta }^{2}+\Delta _\theta ^2 }d\tilde{\theta } \nonumber \\= & {} e^{-j\pi (m-1)\cos \theta }e^{-\pi (m-1)\left| {\Delta _\theta \sin \theta } \right| } \nonumber \\= & {} \left[ {\mathbf{a}(\theta )} \right] _m \left[ {\mathbf{w}(\psi )} \right] _m \end{aligned}$$

(48)

where $\left[ {\mathbf{w}(\psi )} \right] _m =e^{-\pi (m-1)\left| {\Delta _\theta \sin \theta } \right| }$, and $\mathbf{w}(\psi )$ is real-valued vector. It is important to note that $\mathbf{c}\left( \psi \right)$ and $\mathbf{a}(\theta )$ have the same phase.

In the similar way to obtain the analytical expression of $\bar{{\mathbf{c}}}\left( \psi \right)$, the mth element $\left[ {\bar{{\mathbf{c}}}\left( \psi \right) } \right] _m$ of $\bar{{\mathbf{c}}}\left( \psi \right)$ can be expressed as

$$\begin{aligned} \left[ {\bar{{\mathbf{c}}}\left( \psi \right) } \right] _m= & {} \int _0^\pi {\int _0^\pi {e^{-j\pi (m-1) \cos (\tilde{\phi }+\phi )}\frac{1}{\pi ^{2}} \frac{\Delta _{\phi } }{\tilde{\phi }^{2}+\Delta _{\phi } ^2 }\frac{\Delta _\theta }{\tilde{\theta }^{2}+ \Delta _\theta ^2 }} d\tilde{\theta }d\tilde{\phi }} \nonumber \\= & {} \frac{1}{\pi }e^{-j\pi (m-1)\cos \phi } \int _0^\pi {e^{-j\pi (m-1)\tilde{\phi }\sin \phi } \frac{\Delta _{\phi } }{\tilde{\phi }^{2}+\Delta _{\phi } ^2 }d\tilde{\phi }} \nonumber \\= & {} e^{-j\pi (m-1)\cos \phi }e^{-\pi (m-1) \left| {\Delta _{\phi } \sin \phi } \right| } \nonumber \\= & {} \left[ {\mathbf{b}(\phi )} \right] _m \left[ {\mathbf{v}(\psi )} \right] _m \end{aligned}$$

(49)

where $\left[ {\mathbf{v}(\psi )} \right] _m =e^{-\pi (m-1)\left| {\Delta _{\phi } \sin \phi } \right| }$, and $\mathbf{v}(\psi )$ is also a real-valued vector. Note that $\bar{{\mathbf{c}}}\left( \psi \right)$ and $\mathbf{b}(\phi )$ have the same phase.

Appendix 2

From (42), we partition $\mathbf{Q}$ into two submatrices as

$$\begin{aligned} \mathbf{Q}= \left[ \begin{array}{c} \mathbf{Q}_1 \\ \mathbf{Q}_2 \\ \end{array} \right] \begin{array}{c} {\left. \right\} } \\ {\left. \right\} } \\ \end{array} \begin{array}{c} {2M} \\ {2M} \\ \end{array} \end{aligned}$$

(50)

where $\mathbf{Q}_1$ and $\mathbf{Q}_2$ are both $2M\times (4M-K)$ matrices consisting of the first 2M and the last 2M rows of $\mathbf{Q}$.

Substituting (50) into (42), we have

$$\begin{aligned} f(\psi ,\mu )= & {} \mathbf{d}^{H}(\psi ,\mu )\left[ {\begin{array}{c@{\quad }c} \mathbf{Q}_1 \mathbf{Q}_1^H &{} \mathbf{Q}_1 \mathbf{Q}_2^H \\ \mathbf{Q}_2 \mathbf{Q}_1^H &{} \mathbf{Q}_2 \mathbf{Q}_2^H \\ \end{array}} \right] \mathbf{d}(\psi ,\mu ) \nonumber \\= & {} \left[ {\begin{array}{c} \mathbf{d}_X (\psi ,\mu ) \\ \mathbf{d}_Y (\psi ,\mu ) \\ \end{array}} \right] ^{H}\left[ {\begin{array}{c@{\quad }c} \mathbf{Q}_1 \mathbf{Q}_1^H &{}\mathbf{Q}_1 \mathbf{Q}_2^H \\ \mathbf{Q}_2 \mathbf{Q}_1^H &{}\mathbf{Q}_2 \mathbf{Q}_2^H \\ \end{array}} \right] \left[ {\begin{array}{c} \mathbf{d}_X (\psi ,\mu ) \\ \mathbf{d}_Y (\psi ,\mu ) \\ \end{array}} \right] \nonumber \\= & {} \mathbf{d}_X^H (\psi ,\mu )\mathbf{Q}_1 \mathbf{Q}_1^H \mathbf{d}_X (\psi ,\mu )+\mathbf{d}_X^H (\psi ,\mu )\mathbf{Q}_1 \mathbf{Q}_2^H \mathbf{d}_Y (\psi ,\mu ) \nonumber \\&+\,\mathbf{d}_Y^H (\psi ,\mu )\mathbf{Q}_2 \mathbf{Q}_1^H \mathbf{d}_X (\psi ,\mu )+\mathbf{d}_Y^H (\psi ,\mu )\mathbf{Q}_2 \mathbf{Q}_2^H \mathbf{d}_Y (\psi ,\mu ) \nonumber \\= & {} F_1 +F_2 +F_3 +F_4 \end{aligned}$$

(51)

Let $\mathbf{Q}_1 =[\mathbf{Q}_{11}^T ,\;\mathbf{Q}_{12}^T ]^{T}$ and $\mathbf{Q}_2 =[\mathbf{Q}_{21}^T ,\;\mathbf{Q}_{22}^T ]^{T}$, and each part of (51) can be separately expressed as

$$\begin{aligned} F_1= & {} \left[ {\begin{array}{c} \mathbf{c}(\psi ) \\ \mathbf{c}^{*} (\psi )e^{-j\mu } \\ \end{array}} \right] ^H \left[ {\begin{array}{c@{\quad }c} \mathbf{Q}_{11} \mathbf{Q}_{11}^H &{}\mathbf{Q}_{11} \mathbf{Q}_{12}^H \\ \mathbf{Q}_{12} \mathbf{Q}_{11}^H &{}\mathbf{Q}_{12} \mathbf{Q}_{12}^H \\ \end{array}} \right] \left[ {\begin{array}{c} \mathbf{c}(\psi ) \\ \mathbf{c}^{*} (\psi )e^{-j\mu } \\ \end{array}} \right] \nonumber \\= & {} \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] ^H \left[ {\begin{array}{c@{\quad }c} \mathbf{c}^H (\psi )\mathbf{Q}_{11} \mathbf{Q}_{11}^H \mathbf{c}(\psi )&{} \mathbf{c}^H (\psi )\mathbf{Q}_{11} \mathbf{Q}_{12}^H \mathbf{c}^{*} (\psi ) \\ \mathbf{c}^T (\psi )\mathbf{Q}_{12} \mathbf{Q}_{11}^H \mathbf{c}(\psi )&{} \mathbf{c}^T (\psi )\mathbf{Q}_{12} \mathbf{Q}_{12}^H \mathbf{c}^{*} (\psi ) \\ \end{array}} \right] \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] \nonumber \\= & {} \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] ^H \left[ {\begin{array}{cc} F_{11} &{} F_{12} \\ F_{13} &{} F_{14} \\ \end{array}} \right] \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] \end{aligned}$$

(52)

$$\begin{aligned} F_2= & {} \left[ {\begin{array}{c} \mathbf{c}(\psi ) \\ \mathbf{c}^{*} (\psi )e^{-j\mu } \\ \end{array}} \right] ^H \left[ {\begin{array}{c@{\quad }c} \mathbf{Q}_{11} \mathbf{Q}_{21}^H &{} \mathbf{Q}_{11} \mathbf{Q}_{22}^H \\ \mathbf{Q}_{12} \mathbf{Q}_{21}^H &{} \mathbf{Q}_{12} \mathbf{Q}_{22}^H \\ \end{array}} \right] \left[ {\begin{array}{c} \bar{{\mathbf{c}}}(\psi ) \\ \bar{{\mathbf{c}}}^{*} (\psi )e^{-j\mu } \\ \end{array}} \right] \nonumber \\= & {} \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] ^H \left[ {\begin{array}{c@{\quad }c} \mathbf{c}^H (\psi )\mathbf{Q}_{11} \mathbf{Q}_{21}^H \bar{{\mathbf{c}}}(\psi )&{} \mathbf{c}^H (\psi )\mathbf{Q}_{11} \mathbf{Q}_{22}^H \bar{{\mathbf{c}}}^{*} (\psi ) \\ \mathbf{c}^T (\psi )\mathbf{Q}_{12} \mathbf{Q}_{21}^H \bar{{\mathbf{c}}}(\psi )&{} \mathbf{c}^T (\psi )\mathbf{Q}_{12} \mathbf{Q}_{22}^H \bar{{\mathbf{c}}}^{*} (\psi ) \\ \end{array}} \right] \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] \nonumber \\= & {} \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] ^H \left[ {\begin{array}{cc} F_{21} &{} F_{22} \\ F_{23} &{} F_{24} \\ \end{array}} \right] \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] \end{aligned}$$

(53)

$$\begin{aligned} F_3= & {} \left[ {\begin{array}{c} \bar{{\mathbf{c}}}(\psi ) \\ \bar{{\mathbf{c}}}^{*} (\psi )e^{-j\mu } \\ \end{array}} \right] ^H \left[ {\begin{array}{c@{\quad }c} \mathbf{Q}_{21} \mathbf{Q}_{11}^H &{} \mathbf{Q}_{21} \mathbf{Q}_{12}^H \\ \mathbf{Q}_{22} \mathbf{Q}_{11}^H &{} \mathbf{Q}_{22} \mathbf{Q}_{12}^H \\ \end{array}} \right] \left[ {\begin{array}{c} \mathbf{c}(\psi ) \\ \mathbf{c}^{*} (\psi )e^{-j\mu } \\ \end{array}} \right] \nonumber \\= & {} \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] ^H \left[ {\begin{array}{c@{\quad }c} \bar{{\mathbf{c}}}^H (\psi )\mathbf{Q}_{21} \mathbf{Q}_{11}^H \mathbf{c}(\psi ) &{} \bar{{\mathbf{c}}}^H (\psi )\mathbf{Q}_{21} \mathbf{Q}_{12}^H \mathbf{c}^{*} (\psi ) \\ \bar{{\mathbf{c}}}^T (\psi )\mathbf{Q}_{22} \mathbf{Q}_{11}^H \mathbf{c}(\psi ) &{} \bar{{\mathbf{c}}}^T (\psi )\mathbf{Q}_{22} \mathbf{Q}_{12}^H \mathbf{c}^{*} (\psi ) \\ \end{array}} \right] \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] \nonumber \\= & {} \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] ^H \left[ {\begin{array}{c@{\quad }c} F_{31} &{} F_{32} \\ F_{33} &{} F_{34} \\ \end{array}} \right] \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] \end{aligned}$$

(54)

$$\begin{aligned} F_4= & {} \left[ {\begin{array}{c} \bar{{\mathbf{c}}}(\psi ) \\ \bar{{\mathbf{c}}}^{*} (\psi )e^{-j\mu } \\ \end{array}} \right] ^H \left[ {\begin{array}{c@{\quad }c} \mathbf{Q}_{21} \mathbf{Q}_{21}^H &{} \mathbf{Q}_{21} \mathbf{Q}_{22}^H \\ \mathbf{Q}_{22} \mathbf{Q}_{21}^H &{} \mathbf{Q}_{22} \mathbf{Q}_{22}^H \\ \end{array}} \right] \left[ {\begin{array}{c} \bar{{\mathbf{c}}}(\psi ) \\ \bar{{\mathbf{c}}}^{*} (\psi )e^{-j\mu } \\ \end{array}} \right] \nonumber \\= & {} \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] ^H \left[ {\begin{array}{c@{\quad }c} \bar{{\mathbf{c}}}^H (\psi )\mathbf{Q}_{21} \mathbf{Q}_{21}^H \bar{{\mathbf{c}}}(\bar{{\psi }}) &{} \bar{{\mathbf{c}}}^H (\psi )\mathbf{Q}_{21} \mathbf{Q}_{22}^H \bar{{\mathbf{c}}}^{*} (\psi ) \\ \bar{{\mathbf{c}}}^T (\psi )\mathbf{Q}_{22} \mathbf{Q}_{21}^H \bar{{\mathbf{c}}}(\bar{{\psi }}) &{} \bar{{\mathbf{c}}}^T (\psi )\mathbf{Q}_{22} \mathbf{Q}_{22}^H \bar{{\mathbf{c}}}^{*} (\psi ) \\ \end{array}} \right] \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] \nonumber \\= & {} \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] ^H \left[ {\begin{array}{c@{\quad }c} F_{41} &{} F_{42} \\ F_{43} &{} F_{44} \\ \end{array}} \right] \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] \end{aligned}$$

(55)

and then, (51) can be rewritten as

$$\begin{aligned} f(\psi ,\mu )=\left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] ^H \left[ {\begin{array}{cc} \tilde{F}_1 &{} \tilde{F}_2 \\ \tilde{F}_3 &{} \tilde{F}_4 \\ \end{array}} \right] \left[ {\begin{array}{c} 1 \\ e^{-j\mu } \\ \end{array}} \right] \end{aligned}$$

(56)

where $\tilde{F}_1 =F_{11} +F_{21} +F_{31} +F_{41}$, $\tilde{F}_2 =F_{12} +F_{22} +F_{32} +F_{42}$, $\tilde{F}_3 =F_{13} +F_{23} +F_{33} +F_{43}$, $\tilde{F}_4 =F_{14} +F_{24} +F_{34} +F_{44}$.

Through (56), we can see that $\mu$ is independent. For a given angular parameter $\psi$, we can find the minimum of $f(\psi ,\mu )$.

Let ${\partial f(\psi ,\mu )}/{\partial \mu }=0$, we obtain

$$\begin{aligned} e^{j\mu }=\pm \sqrt{{\tilde{F}_2 }/{\tilde{F}_3 }} \end{aligned}$$

(57)

The minimum of $f(\psi ,\mu )$ over $\mu$ is given when the right-hand side of (57) takes negative value. Using (57) into (56), the cost function can be written as

$$\begin{aligned} f(\psi )=\tilde{F}_1 +\tilde{F}_4 -2\rho \sqrt{\tilde{F}_2 \tilde{F}_3 } \end{aligned}$$

(58)