Abstract
A method of direction of arrival (DOA) and direction of departure (DOD) angle estimation based on polynomial rooting for bistatic multiple-input multiple-output (MIMO) radar with uniform circular array (UCA) configuration is proposed in this paper. The steering vector of the UCA is firstly transformed into a steering vector with a Vandermonde structure by using the Jacobi-Anger expansion. Then the null-spectrum function of the MIMO radar can be written as an expression in which the transmit and receive steering vectors are decoupled. Finally, a two-step polynomial rooting is used to estimate DOA and DOD of targets instead of two-dimensional multiple signal classification (MUSIC) search method for bistatic UCA MIMO radar. The angle estimation performance of the proposed method is similar to that of the MUSIC spectral search method, but the computation burden of the proposed polynomial rooting algorithm is much lower than that of the conventional MUSIC method. The simulation results of the proposed algorithm are presented and the performances are investigated and analyzed.
1. Introduction
Research on multiple-input multiple-output (MIMO) radar has been growing as evidenced by an increasing body of literature [1–10]. MIMO radar is characterized by using multiple antennas to simultaneously transmit orthogonal waveforms and multiple antennas to receive the reflected signals. MIMO radar has been shown to provide a number of potential benefits as compared with conventional radar, such as enhancing angle resolution, improving parameter identifiability, and increasing flexibility for transmit beam pattern design. In particular, the problem of multitarget localization in bistatic MIMO radar has received great research interests. Many methods in bistatic MIMO radar are proposed to identify and locate multiple targets [3–9] in which both the transmit array and the receive array are uniform linear arrays (ULAs). In order to avoid angle search, estimation of signal parameters via rotational invariance techniques (ESPRIT) algorithm is applied to bistatic MIMO radar [3–7] by exploiting the invariance property of the transmit and receive arrays. In [8, 9], several algorithms based on polynomial root finding procedure are proposed to estimate DOA and DOD of targets.
Unfortunately, both the ESPRIT and the polynomial rooting method are designed for ULAs. The steering vector of the ULA is dependent on (where is the interelement spacing, is DOA of the source, and is the signal wavelength), and, hence, DOA estimation with ULA becomes ambiguous [10] beyond the range of 180° (from −90° to +90°). The following properties of uniform circular arrays (UCAs) [11–14] make them attractive in the context of DOA estimation. UCAs can provide 360° azimuthal coverage. In addition, direction patterns synthesized with UCAs can be electronically rotated in the plane of the array without significant change of beam shape. ULAs, in contrast, provide only 180° coverage, and beams formed with ULAs broaden as the array is steered away from boresight. Especially, UCAs configuration is very suitable for the MIMO radar, which often transmits orthogonal waveforms in each of the transmit antennas in order to detect the whole 360° in the azimuth angle simultaneously.
In order to come up with computationally efficient high-resolution DOA estimators for UCAs, the so-called beamspace transform [11–13] may be applied. It rebuilds desired Vandermonde structure for the steering vectors. Consequently, methods like root-MUSIC [15–17] and ESPRIT may be applied to find DOAs. In [11], Tewfik and Hong have shown that it is possible to extend the Root-MUSIC to UCA using the phase mode excitation concept. In [12], Mathews and Zoltowski proposed real beamspace MUSIC to UCA that yields reduced computational complexity and better resolution. The beamspace transform works properly only under certain conditions on the array configuration that may be difficult to satisfy in some applications [13]. These algorithms require a sufficiently large number of antenna elements to avoid spatial aliasing and mapping errors that may cause error floor and excess variance [13]. It also shows that there is a significant difference in the performance of the UCA root-MUSIC technique depending on whether an even or odd number of elements is used [14]. Manifold separation technique [18, 19] shows an alternative method to map the array steering vectors to a Vandermonde structured virtual array with a significantly smaller fitting error than for beamspace transform.
In this paper, direction finding for bistatic MIMO radar with UCA configuration employing polynomial rooting is presented. Transmit and receive steering vectors are firstly decomposed using the Jacobi-Anger expansion [20]. Then two-dimensional direction finding in bistatic MIMO radar is transformed into double one-dimensional direction finding procedure [9]. At last, an algorithm based on polynomial root finding to estimate DOA and DOD of targets in bistatic MIMO radar with UCA configure is proposed. The computational complexity of the proposed method is low without the requirement of costly space searching procedure.
The remainder of this paper is organized as follows. In Section 2, we describe our bistatic UCA MIMO radar scheme and the associated data model. In Section 3, the proposed bistatic UCA MIMO root finding algorithm is described. Moreover, the simulation results of the proposed algorithm are presented and the performances are investigated in Section 4. Finally, Section 5 concludes the paper.
2. Signal Model
Consider a narrowband bistatic MIMO radar system with -element transmit antennas and -element receive antennas, both of which are UCAs with radii and , respectively. At the transmit site, different narrow-band pulse waveforms are emitted simultaneously, which have identical bandwidth and center frequency, but are temporally orthogonal. In each receiver, the echoes are processed for all of the transmitted pulse waveforms. It is assumed that the Doppler frequencies have almost no effect on the orthogonality of the signals. Therefore, the variety of the phase within pulses caused by Doppler frequency can be ignored. There are uncorrelated targets located at the same range cell. The directions of the th target with respect to the normal direction of transmit array and receive array denoted by transmit angle and receive angle , respectively. The output of the entire matched filters at the receivers can be expressed as [10] where is a matrix composed of the steering vectors.
Cosider that is the Kronecker product of the transmit and the receive steering vectors for the th target. , with and being the reflection coefficient depending on the target radar cross-section (RCS) and the Doppler frequency of the th target, respectively. is the pulse repeat frequency, denotes the transpose, denotes a noise vector assumed to be independent and identically distributed, zero-mean complex Gaussian distribution and spatially white with covariance matrix , where is noise variance and is the identity matrix.
The array covariance matrix can be written as where denotes Hermitian transpose and denotes the statistical expectation.
The signals and the noises are assumed to be stationary, uncorrelated random processes; substituting from (1) into (3), we have where is the source covariance matrix and is the noise power at array sensors.
In practical situations, the exact array covariance matrix is unavailable and its sample estimate is used, where is the number of snapshots.
The eigenvalue decomposition of yields [10] where are the eigenvalues of and () are the corresponding eigenvectors. The matrices are composed of the signal and the noise subspace eigenvectors of the array covariance matrix, respectively.
The MUSIC null-spectrum function is defined as [18] where denotes the vector 2-norm. The spectral MUSIC technique estimates the signal DOAs from the minima of this function by means of a two-dimensional search over and .
3. Direction Finding for Bistatic MIMO Radar with UCA Configuration
The transmit and receive steering vectors of the UCAs can be denoted as [11–14]where , is wavelength and and are the azimuth angles of the th element of transmit and receive arrays, respectively.
By using the Jacobi-Anger expansion, we can mathematically express the th element of the transmit steering vector as [10, 20] where is the th element of the sampling matrix and is the Bessel function of the first kind of order . Consequently, we can express the transmit steering vector by The th component of is Truncating and from to (suppose is odd), the transmit steering vector can be approximated as where is an matrix and We can see that is a Vandermonde vector which depends on the steering angle and the parameter . The accuracy of the approximation equation (14) increases with increasing the value of .
The receive steering vector can be treated in the same way as the transmit steering vector; therefore, we obtain where is an matrix and We can also see that is a Vandermonde vector which depends on the steering angle and the parameter .
Using the notations and , (15) and (17) can be written as Using (2) and (16), the null-spectrum function equation (8) can be written as where Therefore, to solve the set in expression (19), we can first find satisfying [9] Suppose . Obviously matrix B is an matrix that can be written as where is matrix. Then we get The roots inside and closest to the unitary circle of the polynomial allow estimating the DOA angles given by . We can see that the computational complexity is burdensome because of the high degree of the polynomial. Here we use a fast root-MUSIC method [17] to reduce the computational complexity. The algorithm, no matter how large the degree of the polynomial is, only needs to calculate roots (the number of targets).
By substituting the obtained roots and using (14) in the expression (19), we constitute the following equation: where . We can again use the polynomial root technique to get the roots with the polynomial (24) equal to zero. The correspondent DOD angle can be given by . Note that the pairing is automatically obtained between the DOA and DOD angles, which avoid the traditional bistatic radar problem of synchronization.
4. Simulation Results
Here, we present simulation results showing the statistical performance of the proposed algorithm when using UCA configuration. Consider a narrowband bistatic MIMO radar system with 4 transmit antennas and 3 receive antennas; both are UCAs with radius equal to . For the simulations, the following settings have been used: two uncorrelated sources impinging the arrays with equal powers from the angles and , 256 snapshots, and 100 independent Monte Carlo trials. Figure 1 shows the angle estimation result of two targets for bistatic MIMO radar under the conditions SNR = 10 dB and . We can observe that the target directions are well localized and DOA and DOD are automatically paired.
With the same configuration of the simulation as before, Figure 2 shows the DOA and the DOD estimation versus signal noise ratio (SNR) by using the proposed polynomial root finding algorithm. We observe that the proposed algorithm has almost the same performance, no matter if the direction of the target is near or far away from the boresight. It is the advantage of the UCA configuration.
(a) DOA
(b) DOD
Finally, we compare angle estimation performance of MUSIC spectrum with the searching step 0.01° and polynomial rooting method with . The root mean squared error (RMSE) of the th target direction estimation is defined as RMSE = . Figure 3 demonstrates that the proposed polynomial rooting algorithm provides a similar angle estimation performance to the MUSIC spectral search method with the search interval 0.01 degree, but the MUSIC spectral search method is computationally expensive for such a fine grid search over the whole angle range.
(a) The first target
(b) The second target
5. Conclusion
In this paper, we have proposed a new technique to transform the steering vector of the UCA configuration into a steering vector with a Vandermonde structure in bistatic MIMO radar by using the Jacobi-Anger expansion. The two-step polynomial root finding algorithm has then been used to estimate DOA and DOD of the targets. The simulation results show that the proposed algorithm provides good performances in angle estimation. In addition, the proposed polynomial rooting angle estimation method avoids spectral search and reduces the computational complexity for bistatic MIMO radar with UCA configure.
Acknowledgments
This work was supported in part by the National Natural Science Foundation of China (61372136, 61172137, and 61271290) and in part by the Fundamental Research Funds for the Central Universities (K5051202005, and K5051302089).