Angle Estimation and Self-calibration Method for Bistatic MIMO Radar with Transmit and Receive Array Errors

Abstract

The mutual coupling, gain-phase error and sensor position error of the transmit and receive arrays would significantly degrade the performance of high-resolution direction of departure and direction of arrival estimation algorithms. By applying two well-calibrated instrumental sensors in both transmit and receive arrays, a novel angle estimation and self-calibration method, which considers the combined influences of the above three array errors, is proposed for bistatic MIMO radar. We show that the integrated array errors can be translated to angularly dependent gain-phase error. Then, a reduced dimensional method is used to estimate the angles and the angularly dependent gain-phase error coefficients. Based on the estimated angularly dependent gain-phase error coefficients, the nonlinear least squares optimization model for the three array errors are established and a quasi-Newton optimization method is applied to estimate the mutual coupling matrix, gain-phase error and sensors position errors of the transmit and receive arrays, respectively. Our simulation results corroborate our analysis.

Appendix

First of all, let

$$\begin{aligned} J_r = \hbox {trace}\left( {\varvec{P}_{\varvec{\varTheta }}^\bot \varvec{g}_r \varvec{g}_r^{\mathrm{H}} } \right) \end{aligned}$$

(43)

and

$$\begin{aligned} \varvec{V}_r \left( {{\varvec{\zeta }}_r^{\left( i \right) } } \right) =\frac{\partial J_r }{\partial {\varvec{\zeta }}_r^{\left( i \right) } }=\hbox {trace}\left( {\frac{\partial \varvec{P}_{\varvec{\varTheta }}^\bot }{\partial {\varvec{\zeta }}_r^{\left( i \right) } }\varvec{g}_r \varvec{g}_r^{\mathrm{H}} } \right) \end{aligned}$$

(44)

Since the derivative of the projection matrix $\varvec{P}_{\varvec{\varTheta }}^\bot$ with respect to ${\varvec{\zeta }}_r^{\left( k \right) }$ is [2]

$$\begin{aligned} \frac{\partial \varvec{P}_{\varvec{\varTheta }}^\bot }{\partial {\varvec{\zeta }}_r^{\left( k \right) } }=-\varvec{P}_{\varvec{\varTheta }}^\bot \frac{\partial {\varvec{\varTheta }}}{\partial {\varvec{\zeta }}_r^{\left( k \right) } }{\varvec{\varTheta }}^{\# }-\left( {\varvec{P}_{\varvec{\varTheta }}^\bot \frac{\partial {\varvec{\varTheta }}}{\partial {\varvec{\zeta }}_r^{\left( k \right) } }{\varvec{\varTheta }}^{\# }} \right) ^{\mathrm{H}} \end{aligned}$$

(45)

by substituting (36) into (35), we have

$$\begin{aligned} \varvec{V}_r \left( {{\varvec{\zeta }}_r^{\left( k \right) } } \right) =\frac{\partial J_r }{\partial {\varvec{\zeta }}_r^{\left( k \right) } }=-2 \hbox {Re}\left( {\varvec{g}_r^{\mathrm{H}} \varvec{P}_{\varvec{\varTheta }}^\bot \frac{\partial {\varvec{\varTheta }}}{\partial {\varvec{\zeta }}_r^{\left( k \right) } }{\varvec{\varTheta } \varvec{g}}_r } \right) \end{aligned}$$

(46)

For

(47)

(48)

(49)

(50)

where $\delta _{m,n} = {\left\{ {{\begin{array}{ll} 1, &{} m=n \\ 0, &{} m\ne n \\ \end{array} }}\right. }.$

By substituting (47)–(49) into (46), and after some manipulations, the gradient vector $\varvec{V}_r \left( {{\varvec{\zeta }}_r^{\left( k \right) } } \right)$ of (24) can be solved.