I Introduction
A MIMO radar array system employs multiple transmit antennas for emitting orthogonal waveforms and multiple receive antennas for receiving the echoes reflected by the targets [1, 2, 3]
and can exploit the waveform diversity to form a virtual array with increased degrees of freedom (DOFs) and a larger aperture compared to the traditional phasedarray radar. It has been shown that MIMO radar can provide enhanced spatial resolution, achieve better target detection performance, and significantly improve the system’s parameter identifiability
[3, 4, 5, 6].Many techniques have been proposed for angle estimation in MIMO radar using traditional direction of arrival (DOA) algorithms, such as MUSIC [7] and ESPRIT [8], by assuming perfect knowledge of the array manifold. However, these algorithms are sensitive to uncertainties in the array manifold, and their performance will degrade significantly in the presence of array model errors [9, 10, 11]. On the other hand, it is timeconsuming and expensive to calibrate the system in the case of large or timevarying arrays [10]. In addition, it is observed that in practice, even after initial calibration, antenna gain and phase errors still exist due to environmental changes and other factors [12]. To cope with the problem, in [13], a MUSICbased DOA estimation method in the presence of gain and phase errors was introduced. A subspacebased method for estimating the errors was proposed in [12]. Other methods were also proposed based on partially calibrated arrays [14, 11, 10]. Additionally, blind calibration is possible for nonGaussian signals by using higherorder statistics [15], although with a very high computational complexity.
In this work, we address the problem of joint DOA and array manifold estimation with a multiinput multioutput (MIMO) array configuration [2, 5], where only two transmit antennas are fully calibrated, while the receive antennas are uncalibrated [16, 17]. Since the two transmit antennas transmit orthogonal waveforms, we can extract the received data associated with each transmit antenna. With the two transmit antennas well calibrated, a rotational invariance property between the two sets of data can still be maintained without any knowledge of the array manifold of the uncalibrated side; then the ESPRIT algorithm can be used to find the initial DOAs of the targets. Starting with the initial DOA estimates, the antenna gains and phases can then be estimated through an appropriate modification of the MUSIC algorithm introduced in [13]. The estimated antenna gains and phases will be used in the more accurate estimation of DOAs via the MUSIC algorithm. This procedure will be repeated until some convergence criterion is met. The advantage of the scheme is that only two calibrated antennas are needed for high resolution DOA estimation and no specific requirement is imposed on the uncalibrated antennas. To our best knowledge, none of the existing DOA estimation methods for MIMO arrays has considered the joint DOA and array manifold estimation problem.
This paper is organized as follows. In Sec. II, the array model and a review of DOA estimation are provided, with the proposed method given in Sec. III. Simulation results are presented in Sec. IV and conclusions are drawn in Sec. V.
Ii Background
Consider a MIMO system with a uniform linear array (ULA) of
antennas used for both transmitting and receiving. For simplicity of notation and without loss of generality, we assume that the first two antennas are perfectly calibrated. The steering vector of the ULA is then given by
(1) 
where denotes the transpose operation, is the angle of the pointing direction, is the interelement spacing, is the signal wavelength, and and denote the gain and phase errors, respectively. Assume that targets are present. The output of the matched filters at the receiver is given by [5]
(2)  
where is the DOA of the th target, is the Kronecker product, , with being the complexvalued reflection coefficient of the th target and being the Doppler frequency, ,
(3) 
is the overall transmitreceive or virtual array manifold, and
is the white noise vector with a power
.Assume that all targetreflected signals and noise are uncorrelated. Then we have
(4)  
where and denote expectation and Hermitian transpose, respectively, , consists of the
principal eigenvalues of
,is the signal subspace, specified by the principal eigenvectors of
, and the remaining eigenvectors is the noise subspace. In practice, will be replaced by , where is the number of snapshots.The MUSIC algorithm for DOA estimation for MIMO radar can be constructed as [18, 19]
(5) 
The largest peaks of indicate the DOAs of the targets. It requires the spacing between two adjacent antennas to be within a half wavelength to avoid estimation ambiguity.
For ESPRIT estimator [20], it is based on the signal subspace . Let be the subset of , which relates to the first to the th transmit antennas, and be the subset of , which relates to the second to the th transmit antennas. We then have the following relationship
(6) 
where is an unknown nonsingular matrix and is a diagonal matrix, with its th main diagonal element being . Thus, the DOAs can be found from the eigenvalues of .
Iii Proposed Method
In this section, we first perform an initial DOA estimation using the two sets of received data associated with the first and the second transmit antennas by applying the ESPRIT algorithm, then the gain and phase errors can be estimated using the initial DOA results by applying a MUSICbased approach.
Iiia Estimating initial DOAs
Since the array manifold is unknown, we can not apply the traditional subspacebased methods directly. To solve the problem, define and as the first and the second rows of A, respectively, with
(7)  
(8)  
where Q is an diagonal matrix, with being its th main diagonal element.
Although there are model errors in both and , a rotational invariance property between and is still maintained, which enables the use of ESPRIT for DOA estimation. A and have a relationship determined by a unique nonsingular matrix T as
(9) 
Define and as the first and second rows of , respectively. We have
(10)  
(11) 
Then,
(12) 
Now using the traditional ESPRIT technique, the main diagonal elements of Q can be obtained via eigendecomposition of . Since the two transmit antennas have been well calibrated, can be obtained easily from Q.
Note that the rotational invariance property exploited here depends only on the two calibrated transmit antennas and is not related to the uncalibrated part. Thus, the initial DOAs can be estimated accurately without any knowledge of array model errors. Additionally, in this initial DOA estimation, the proposed ESPRITbased method imposes less constraints on the spacing of the uncalibrated part, which can be arranged to be much larger than a halfwavelength for a highresolution DOA estimation.
IiiB Estimating array manifold
From (5), with exactly known , the DOAs can also be found by solving the following equation [13]:
(13) 
The actual steering vector can also be expressed as
(14) 
where and . Therefore, the estimate of antenna gains and phases can be obtained using the initially estimated DOAs as follows:
(15) 
where is the gain and phase vector, with its elements being the diagonal elements of , , with being the initial DOA estimate of the th target, and . It should be noted that both the th and the th elements of should also be equal to 1; however, we find that the above two constraints are able to give a satisfactory result. The problem in (15) can be rewritten as
(16) 
where , , and . Its solution is given by
(17) 
Using the estimates (17), the DOAs can be estimated from the highest peaks of the following function:
(18) 
Since a set of initial DOA estimates has already been obtained, we can search for each DOA estimate over a small DOA region corresponding to each initial DOA estimate. Thus, the interelement spacing of the uncalibrated array does not have to be smaller than half wavelength to avoid estimation ambiguity. Actually, we can increase the interelement spacing of the uncalibrated array to improve the accuracy of estimation.
The proposed joint DOA and array manifold estimation scheme is summarized as follows:
1) Estimate the initial DOAs using the ESPRIT algorithm.
2) Estimate the array manifold using (17).
3) Use the results in Step 2 to find updated DOAs by local searching through (18).
4) Repeat Steps 2 and 3 until some convergence criterion is satisfied. One such a criterion could be the difference between the estimation results of the last round and the current one. When this difference is smaller than a preset threshold value, we can then stop the iteration.
Note that we have assumed implicitly that the antenna positions have been calibrated, and we consider the fixed uncalibrated gain and phase errors only. This is because the calibration of array position is more convenient than the calibration of gain and phase which may vary due to environmental changes. On the other hand, the position error can be transformed into phase errors. However, the phase errors caused by position errors are not fixed for the targets because the targets have different DOAs. In such a case, a simple way is to obtain the gain and phase errors corresponding to each target, i.e. we should estimate the gain and phase errors when obtaining one target’s DOA other than all the DOAs.
IiiC Complexity analysis
To estimate the sample covariance matrix, a computational complexity of is needed. The eigendecomposition operation needs a computational complexity of . The proposed ESPRIT requires a computational complexity of . In the estimation of array manifold, the computational complexity of is needed, where is the iteration number. Therefore, the proposed scheme has at least a complexity of .
IiiD CramérRao Bound for Uncalibrated Array
In this section, we derive the stochastic CRB for uncalibrated array by extending the results of [11, 21]. Define , i=3, , M, as the gain and phase error that corresponds to the th sensor and the vector containing the unknown parameters, where
(19)  
(20)  
(21) 
The snapshots are assumed to satisfy the stochastic model
(22) 
where
is the complex Gaussian distribution. The unknown parameters include the elements of
, the noise variance
, and the parameters of the source covariance matrix and .Iv Simulations
Simulations are carried out to investigate the performance of the proposed method compared with the traditional ESPRIT estimator in [20] and the MUSIC estimator. We consider a MIMO array with antennas and halfwavelength spacing. The first two antennas are perfectly calibrated. targets are located at , , and , respectively. Results from simulation runs are averaged to give the root mean square error (RMSE) of the estimates. For all simulations, the number of snapshots is used.
We first study the performance of the proposed ESPRITbased algorithm for initial DOA estimation. The antenna gain and phase errors are assumed to have a uniform distribution:
and . and change from run to run while remaining constant for all snapshots. Fig. 1 shows the RMSE results versus input SNR. We see that the gain and phase errors have significantly degraded the performance of the traditional MUSIC and ESPRIT algorithms. However, the proposed one is quite robust and has a much better performance. In this figure, we also showed the result of our proposed method with iterations, and a clear improvement can be observed compared to the initial estimation.In the second example, the effect of the iteration number on the performance of the proposed method is demonstrated. The input SNR is set to 20 dB and the antenna gain and phase errors are set as (the diagonal elements of )
(24) 
The RMSE for DOA estimation versus the iteration number is shown in Fig. 2 and the result for unknown parameters estimation is shown in Fig. 3. Clearly the first or two iterations have already led to an accurate enough result.
Now we study the effect of antenna spacing on the performance of the proposed method with iterations. The spacing between the two calibrated antennas is , while the spacing between the uncalibrated antennas is set to for the proposed method, and for the other methods. The other parameters remain the same as in Example 1. The results are shown in Fig. 4. We can see that the proposed ESPRITbased initial estimation has achieved a higher accuracy compared to Fig. 1, and the performance of the proposed method is much better than the corresponding result of Example 1 and significantly outperforms the other considered algorithms.
V Conclusions
A joint DOA and array manifold estimation scheme for a MIMO array system has been proposed, where only two antennas at the transmit side are initially calibrated, while the remaining part of the system is completely uncalibrated. By exploiting the rotational invariance property between two sets of received data associated with the two calibrated antennas, the ESPRIT algorithm is first employed to give a set of initial DOA estimation results, which is then used by the following MUSICbased algorithm for the joint estimation. Additionally, the proposed scheme does not require the adjacent antenna spacing in the uncalibrated part to be within a half wavelength, which provides further improvement to the estimation.
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