Joint DOD and DOA estimation for bistatic MIMO radar

doi:10.1016/j.sigpro.2008.08.003

Signal Processing

Volume 89, Issue 2, February 2009, Pages 244-251

https://doi.org/10.1016/j.sigpro.2008.08.003 Get rights and content

Abstract

A joint direction of arrivals (DOAs) and direction of departures (DODs) estimation algorithm for bistatic multiple-input multiple-output (MIMO) radar via ESPRIT by means of the rotational factor produced by multi-transmitter is presented. The DOAs and DODs of targets can be solved in closed form and paired automatically. Furthermore, the spatial colored noise can be cancelled in the case of three-transmitters configuration by using this method. Simulation results confirm the performance of the algorithm.

Introduction

A multiple-input multiple-output (MIMO) [1], [2], [3], [4], [5], [6], [7], [8] radar has a number of potential advantages over conventional phased-array radar. By exploiting the spatial diversity, statistical MIMO radar [1], [3], [5], whose transmit (or both transmit and receive) antennas are spaced far away from each other, can overcome performance degradations caused by target scintillations. Unlike statistical MIMO radar, co-located MIMO radar [2], [4], [7], [8] (to simplify, we call it MIMO radar later), whose elements in transmit and receive arrays are closely spaced, can achieve coherent processing gain. In [2], it was shown that MIMO radar allows one to obtain virtual aperture which is larger than real aperture and this results in narrower beamwidth and lower sidelobes. Parameter identifiability was also discussed in [8], which proved that the maximum number of targets that can be unambiguously identified by the MIMO radar is increasing greatly. MIMO radar ground-moving target detection was treated in [6]. In [7], adaptive techniques were applied to MIMO radar for estimating the radar cross-section (RCS) of targets. In [9], the capon-based estimation of direction of arrivals (DOAs) and direction of departures (DODs) of targets for bistatic MIMO radar was presented. However, to obtain DOAs and DODs of the targets, it is assumed that the reflection coefficient is random process in [9] and two-dimension (2-D) angle search is necessitated.

ESPRIT is a high-resolution parameter estimation technique. In [10], DOA matrix method was proposed to estimate azimuth and elevation. Several methods have also been proposed to estimate DOAs and DODs for MIMO Communication Systems. The multiple signal classification (MUSIC) method was introduced in [14] to estimate DOAs and DODs. But it needs multi-dimension search. Miao et al. [11] proposed 2-D Unitary ESPRIT to estimate DOAs and DODs. High estimation performance could be achieved under the condition of high signal-to-noise ratio (SNR). But the precondition is that the channel matrix should be estimated accurately. Furthermore, the number of DOAs and DODs that the method can estimate is limited by not only the number of transmitters but also the number of receivers.

In this paper, we present an ESPRIT-based method for bistatic MIMO radar DODs and DOAs estimation. The DOAs and DODs of targets can be solved in closed form and paired automatically. The number of identifiable targets is more than that of the algorithm proposed in [11]. In the case of three-transmitters configuration, the spatial colored noise can be cancelled by using this algorithm.

This paper is organized as follows. The bistatic MIMO radar signal model is presented in Section 2. In Section 3, ESPRIT method is applied to bistatic MIMO radar. Both two-transmitters configuration and three-transmitters configuration systems are considered. The closed-form solution of angles is derived. Cramer–Rao bounds (CRB) for target angle are given in Section 4. Section 5 compares the estimation performance of the two systems. Finally, Section 6 concludes the paper.

Section snippets

Bistatic MIMO radar signal model

Consider a narrowband bistatic MIMO radar system with M closely spaced transmit antennas and N closely spaced receive antennas, shown in Fig. 1. The transmit antennas transmit M orthogonal coded signals. Assume that the aperiodic autocorrelation and cross-correlation sidelobes of the signals are very low even when Doppler shift exists. The transmitted baseband coded signals are denoted by s_m∈C^1×K, where m denotes the mth transmitter and s_ms_m^H=K. In this paper, a class of binary sequences with

Esprit method for bistatic MIMO radar

In the case of two elements at the transmit side, from (4) we can obtain $Y_{1} = A_{r} D_{1} Φ + N_{1},$ $Y_{2} = A_{r} D_{2} Φ + N_{2},$ where Y₁ and Y₂ denote the received data from the first and the second transmitters. The covariance matrix of the noises is as follows: $E [N_{i} N_{j}^{H}] = E [(\frac{1}{\sqrt{K}} {Zs}_{i}^{H}) {(\frac{1}{\sqrt{K}} {Zs}_{j}^{H})}^{H}] = \frac{1}{K} E {[s_{i}^{*} \otimes I_{M}] vec (Z) {vec}^{H} (Z) [s_{j}^{T} \otimes I_{M}]} = \frac{1}{K} [s_{i}^{*} \otimes I_{M}] [I_{L} \otimes Q] [s_{j}^{T} \otimes I_{M}] = {\begin{matrix} Q, i = j \\ 0, i \neq j \end{matrix}$ where (·)*, (·)^T, ⊗ and vec(·) denote the complex conjugate, the transpose, the Kronecker matrix product and the vectorization operator, respectively. Eq. (10) shows

Cramer–Rao bound

In the case of three transmit antennas configuration, the Cramer–Rao bound (CRB) of DOAs and DODs are considered here. Rewrite the received data as $Y = [\begin{matrix} Y_{1} \\ Y_{2} \\ Y_{3} \end{matrix}] = [\begin{matrix} A_{r} D_{1} Φ \\ A_{r} D_{2} Φ \\ A_{r} D_{3} Φ \end{matrix}] + [\begin{matrix} N_{1} \\ N_{2} \\ N_{3} \end{matrix}] = [a_{t} (ϕ_{1}) \otimes a_{r} (θ_{1}), \dots, a_{t} (ϕ_{P}) \otimes a_{r} (θ_{P})] Φ + N = K (θ, φ) Φ + N .$

The Fisher information matrix (FIM) with respect to θ=[θ₁,…,θ_P] and ϕ=[ϕ₁,…,ϕ_P] can be written as $F = [\begin{matrix} F_{11} F_{12} \\ F_{21} F_{22} \end{matrix}] .$

Note that the (i, j)th element of F₁₁ is [12] given by $F (θ_{i}, θ_{j}) = 2 Re tr [{(\frac{\partial K (θ, φ) Φ}{\partial θ_{i}})}^{H} ζ^{- 1} \frac{\partial K (θ, φ) Φ}{\partial θ_{j}}] = 2 Re tr [({\dot{K}}_{θ} e_{i} e_{i}^{T} Φ)^{H} ζ^{- 1} ({\dot{K}}_{θ} e_{j} e_{j}^{T} Φ)] = 2 Re [(e_{i}^{T} {\dot{K}}_{θ}^{H} ζ^{- 1} {\dot{K}}_{θ} e_{j}) (e$

Simulation results

First of all, the transmitted waveforms are constructed. The start vectors X^m and Y^m in [13] are given by $X^{m} = [+ - - + - + - - - + + - - - - - - + + -]$ $Y^{m} = [- + + - - - - - - + - + + + - + - + + -]$

According to the method proposed in [13], a waveform set with family size 4, sequence length 160 and zero correlation zone 20 are designed. The first, third and fourth sequences are chosen as transmitted signals. The aperiodic normalized autocorrelation and cross-correlation of the transmitted waveforms within ZCZ are shown in Fig. 2. Fig. 2

Conclusions

In this paper, ESPRIT method is applied to bistatic MIMO radar to estimate target angles by exploiting the rotational factor produced by multi-transmitter. The closed-form solution of DODs and DOAs is derived. The effect of sidelobes on estimate performance is very little and this is validated by computer simulation. Both two-transmitters and three-transmitters systems are considered. But theoretical analysis together with simulation results has shown that the MIMO system with three-transmit

Acknowledgments

This research is supported by the Key Project of Ministry of Education of PR China under Contract no. 107102. The authors are grateful to four anonymous reviewers for providing them with a large number of detailed suggestions for improving the submitted manuscript.

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