Robustness of the coherently distributed MUSIC algorithm to the imperfect knowledge of the spatial distribution of the sources

Abstract

The MUltiple SIgnal Classification (MUSIC) estimator has been widely studied for a long time for its high resolution capabilities in the domain of the directional of arrival (DOA) estimation, with the sources assumed to be point. However, when the actual sources are spatially distributed with angular dispersion, the performance of the conventional MUSIC is degraded. This paper deals with the sensitivity of MUSIC to modeling error due to coherently distributed (CD) sources. A performance analysis of an extended MUSIC taking into account a generalized steering vector based on a CD source model (CD-MUSIC) is first studied. We establish closed-form expressions of the DOA estimation bias and mean square error due to both the model error and the effects of a finite number of snapshots. The aim of this paper is also to determine when the point source assumption is acceptable for standard MUSIC. The analytical results are validated by numerical simulations and discussed in different configurations.

Appendix A

Based on (9), we study the case that the number of snapshots N is big enough, so that ${\varDelta }\hat{\mathbf{R }}, B_2, C_2$ is small, we can make the approximation :

$$\begin{aligned}&\sqrt{B^2 - 4AC} \nonumber \\ {}&\quad \approx \sqrt{B_1^2 - 4AC_1}\left( 1 + \frac{1}{2}\frac{2B_1B_2 +B_2^2 - 4AC_2}{B_1^2 - 4AC_1}\right) . \end{aligned}$$

(23)

Introducing (23) in (9), and keeping only first-order terms in ${\varDelta }\hat{\mathbf{R }}$, the expression of DOA estimation error can be given by:

$$\begin{aligned} {\varDelta }\theta _i \approx \frac{-(B_1+ B_2) + \sqrt{B_1^2-4AC_1}\left( 1+\frac{1}{2}\frac{2B_1B_2 - 4AC_2}{B_1^2 - 4AC_1} \right) }{2A}.\nonumber \\ \end{aligned}$$

(24)

${\varDelta }\hat{\mathbf{R }}$ is a Wishart distribution [24] matrix with the property $E[{\varDelta }\hat{\mathbf{R }}] = 0$, so that $E[B_2] = E[C_2] = 0$. It follows that the DOA estimation bias can be derived as (10). Similarly, the DOA estimation MSE can be given by:

$$\begin{aligned} E[{\varDelta }\theta _i^2]&= \frac{B_1^2 - 2AC_1 - B_1\sqrt{B_1^2 - 4AC_1}}{2A^2} \nonumber \\&\quad + \left( 1/2A^2 - \frac{B_1}{2A^2\sqrt{B_1^2 - 4AC_1}}\right) E[B_2^2]\nonumber \\&\quad +\frac{1}{A\sqrt{B_1^2 - 4AC_1}}E[B_2C_2]. \end{aligned}$$

(25)

Using the same method in [21], we obtain $E[B_2^2] = \frac{\sigma ^2_b}{2N}\varphi$ and $E[B_2C_2] = \frac{\sigma ^2_b}{2N}\chi$, where:

$$\begin{aligned} \varphi \triangleq&{\mathcal {R}}{\text {e}} \left\{ 4\dot{\mathbf{c }}_h(\theta _i)^H {\varPi }\dot{\mathbf{c }}_h(\theta _i) \dot{\mathbf{c }}_h(\theta _i)^H\mathbf{Q }\mathbf{R }\mathbf{Q }\dot{\mathbf{c }}_h(\theta _i) \right. \\&+ 4\ddot{\mathbf{c }}_h(\theta _i)^H {\varPi }\dot{\mathbf{c }}_h(\theta _i)\mathbf{c }_h(\theta _i)\mathbf{Q }\mathbf{R }\mathbf{Q }\mathbf{c }_h(\theta _i) \\&+ \left. \ddot{\mathbf{c }}_h(\theta _i)^H {\varPi }\ddot{\mathbf{c }}_h(\theta _i)\mathbf{c }_h(\theta _i)\mathbf{Q }\mathbf{R }\mathbf{Q }\mathbf{c }_h(\theta _i) \right\} , \\ \chi \triangleq&{\mathcal {R}}{\text {e}} \left\{ 2\dot{\mathbf{c }}_h(\theta _i)^H {\varPi }\dot{\mathbf{c }}_h(\theta _i)\dot{\mathbf{c }}_h(\theta _i)\mathbf{Q }\mathbf{R }\mathbf{Q }\mathbf{c }_h(\theta _i) \right. \\&\left. +\, \ddot{\mathbf{c }}_h(\theta _i)^H {\varPi }\dot{\mathbf{c }}_h(\theta _i)\mathbf{c }_h(\theta _i)\mathbf{Q }\mathbf{R }\mathbf{Q }\mathbf{c }_h(\theta _i) \right\} . \end{aligned}$$