Abstract
The detection performance of direct data domain (D3) space-time adaptive processing (STAP) will be extremely degraded when there are mismatches between the actual and the presumed signal steering vectors. In this paper, a robust D3 STAP method for multiple-input multiple-output (MIMO) radar is developed. The proposed method utilizes the worst-case performance optimization (WCPO) to prevent the target self-nulling effect. An upper bound for the norm of the signal steering vector error is given to ensure that the WCPO problem has an admissible solution. Meanwhile, to obtain better detection performance in the low signal-to-noise ratio (SNR) environment, the proposed method gives a modified objective function to minimize the array noise while mitigating the interferences. Simulation results demonstrate the validity of our proposed method.
1. Introduction
Space-time adaptive processing (STAP) is a key technology for airborne surveillance radar system to detect moving targets in a strong interference environment [1]. The working principles of STAP can be classified into two categories: statistical STAP and deterministic STAP (also known as direct data domain (D3) STAP) [2–7]. In statistical STAP, the covariance matrices of the interference and the clutter environment are usually estimated from the target-free training data, which is collected from range cells adjacent to the range gate under test [2, 3]. To approach the adequate detection performance, the required amount of the training data is at least twice the STAP filter dimension [8]. However, in the scenarios where the ranges of the interferences change fast or the clutters are nonhomogeneous in the target-free training data, the statistical characteristics of the interference environment are difficult to be precisely estimated. To cope with the aforementioned problems, some methods based on the D3 approach are given in [5–7]. The D3 approach can minimize the interference power while preserving unit gain for the target in a coherent processing interval (CPI). In the D3 approach, the target signal is firstly removed from the received signal to remain the contributions of the interferences. However, in the practical application, there are mismatches between the actual and the presumed signal steering vectors due to the direction of arrival (DOA) and Doppler frequency errors, imperfect array calibration, and distorted antenna shape [9]. Hence, in such situations, the processed signal also retains the contribution of the target, which may cause the target self-nulling effect.
In this paper, we propose a robust D3 STAP method for colocated multiple-input multiple-output (MIMO) radar, where superior capabilities arouse significant attentions among radar scholars [10–14]. In the presented robust method, the worst-case performance optimization (WCPO) [9] is used as the approach to avoid the target self-nulling effect. An upper bound for the norm of the target steering vector error is given to ensure the existence of a numerical solution for the WCPO problem. For the reason that the statistical characteristics of the array noise are changed in the D3 approach, the D3 approach based on the WCPO cannot minimize the array noise power effectively while mitigating the interferences. Hence, a modified D3 approach is given to obtain a better detection performance in the low signal-to-noise ratio (SNR) environment. In the modified method, the objective function of the D3 approach is replaced by a linear combination of the original objective function and the output of the array noise. Simulation results show that the proposed method can avoid the target self-nulling effect and outperforms the conventional robust D3 approach in the low SNR environment.
2. Signal Model
Consider a MIMO radar system equipped with a transmit uniform linear array (ULA) of M elements and a receive ULA of N elements. Each transmit array element emits a burst of L pulses in a CPI with a pulse repetition frequency (PRF) . The waveforms of M transmit array elements are mutually orthogonal and repeated pulse to pulse. Hence, the waveform of the ith transmit array element in a CPI can be written as [15] where is the pulse repetition interval (PRI) and is the ith orthogonal waveform which satisfies the following property: where denotes the conjugate operator.
Figure 1 depicts the structure of the MIMO radar receiver in a CPI. As is shown, received signals are collected over MN matched filters in L successive pulses. Assume that there are a point target located at angle with Doppler frequency and K interferences located at with Doppler frequency in the range cell under test. Then, the mth matched filter output from the nth receive array element at the lth pulse can be represented as [16] where is the interspace distance of the adjacent transmit array elements, is the interspace distance of the adjacent receive array elements, is the carrier wavelength, and is the noise with zero mean and variance .
By rearranging , the match filter outputs can be written in a vectorial form as where is the noise vector, denotes the Kronecker product, , , and are Doppler steering vector, receive spatial steering vector, and transmit steering vector, respectively, which are given by
3. Proposed Method
In a rapidly changing environment, the D3 STAP outperforms the statistical STAP for MIMO radar, due to the inaccurate interference covariance matrix estimation of the statistical STAP [7]. However, the detection performance of the D3 STAP will be extremely degraded, when there are steering vector errors [6]. Moreover, the statistical characteristics of the array noise are changed in the D3 approach, which will cause an adverse effect on the detection performance in the low signal-to-noise ratio (SNR) environment. To tackle the above problems, we introduce a modified robust D3 STAP for MIMO radar in the following subsections.
3.1. Conventional D3 STAP for MIMO Radar
In this subsection, we briefly introduce the D3 STAP for MIMO radar in absence of steering vector errors. The corresponding work can also be found in [7]. Define three complex scalar quantities as follows: which are determined by the DOA and the Doppler frequency of the target under test. Through removing the contributions of the target, we can obtain the following seven quantities which only contain the interference and the noise contributions:
By arranging the relationships in (9) as rows in a linear system matrix F, a weight vector w of dimension can be designed to minimize the interferences while preserving unit gain for the target [7]. The corresponding problem can be formulated as where is the conjugate transpose operator and denotes the Euclidean norm. is a truncated version of the target steering vector, while , , and are the , , and truncated versions of the steering vectors , , and , respectively. Note that is only minimized (not nulled) when the number of the rows in F is greater or equal to , and the degrees of freedom (DOFs) in the temporal and spatial domain are set to , , and . It is easy to see that problem (10) is equivalent to the following MVDR problem [17]:
The solution is given by
3.2. D3 STAP for MIMO Radar Based on the WCPO
3.2.1. Formulation of the Problem Based on the WCPO
When there are mismatches between the actual and the presumed target steering vectors, the linear system matrix F also contains the contribution of the target. Hence, the solution of problem (10) will cause the target self-nulling effect. In this subsection, a robust D3 STAP method based on the WCPO is presented to prevent the target self-nulling effect. Let e be a norm-bounded target steering vector error with . By employing the WCPO, problem (10) can be reformulated as
According to the derivation in [9], problem (13) is equivalent to the following problem: where denotes the imaginary part of the matrix in the bracket. The problem stated in (14) is convex, which can be easily solved by the convex optimization toolbox CVX [18]. In the next subsection, we will discuss the upper bound for the norm of the signal steering vector error.
3.2.2. Upper Bound for the Norm of the Signal Steering Vector Error
Let , , and be the error vectors of the transmit spatial steering vector, receive spatial steering vector, and the Doppler steering vector, respectively. Then, the actual signal steering vector can be written as
Inspecting on (15), the signal steering vector error is
Assume that the transmit spatial steering vector error , the receive spatial steering vector error , and the Doppler steering vector error are norm-bounded with , , and . Then, the upper bound for the norm of e is given by
Additionally, to ensure that problem (14) has an admissible solution, the norm of the signal steering vector error also has an upper bound as follows [6]:
Hence, the upper bound for the norm of the signal steering vector error e can be found as
3.3. Modified Robust D3 STAP for MIMO Radar
The output signal-to-interference-plus-noise ratio (SINR) of the D3 STAP for MIMO radar is defined by where is the interference-plus-noise covariance matrix, which is given by
We can clearly see from (20) and (21) that the average noise power of the system output is . However, the weight vector w is designed to minimize the interference and noise contributions in the linear system matrix F. And the statistical characteristics of the noise contributions in the linear system matrix F are determined by the relationships in (9). For example, one of the noise vectors in the linear system matrix F can be represented as
From (22), it is easy to see that the noise component is correlated with the other noise component . Therefore, the noise vector of any row of the linear system matrix F does not obey the distribution of zero mean and covariance matrix . In this situation, minimizing the objective function of problem (14) cannot ensure the minimum array noise power. To minimize array noise power while mitigating the interferences, we reformulate the robust D3 STAP problem as follows: where is a parameter which balances the weight between the performance of mitigating interferences and that of mitigating noises.
Inspecting on (23), the first and the second parts of the objective function are used to minimize the power of the interferences and array noise, respectively. When η is equal to one, problem (23) is equivalent to problem (14). And when η is equal to zero, the output signal power spectrum can only preserve a gain for the target without mitigating the interferences. Hence, it is necessary to select an appropriate η to solve problem (23). Generally, the value of η is proportional to the interference-to-interference-plus-noise ratio. The solution of problem (23) can also be solved by the convex optimization toolbox CVX.
4. Simulations
In this section, serval examples are given to investigate the performances of the proposed methods. Consider that the transmit and the receive ULAs are equipped with elements and elements. The interspace distance of the adjacent receive array elements is half the carrier wavelength , and the interspace distance of the adjacent transmit array elements is given as . The PRF is , and the number of pulses in a CPI is . The DOFs of the proposed methods are set to , , and . The upper bound for the norm of the signal steering vector error in (14) and (22) is . In the range cell under test, we suppose that the presumed target is located at with Doppler frequency . There are also two interferences located at with Doppler frequency and with Doppler frequency . The interference-to-noise ratio (INR) is assumed to be 20 dB for both interferences. And in the following examples, is chosen for the modified robust D3 STAP.
Example 1. In the first example, the SNR is assumed to be 10 dB and there are no mismatches between the presumed and the actual signal steering vectors. Figures 2(a) and 2(b) depict the two-dimensional adaptive weight patterns for the conventional D3 STAP and the modified robust D3 STAP, respectively. As shown in Figure 2, both the adaptive weight patterns can form a gain in the target position and place nulls in the interference positions. In addition, we can see that the sidelobe of the adaptive weight pattern obtained by the modified robust D3 STAP is lower than that of the other.
(a) The conventional D3 STAP
(b) The modified robust D3 STAP
Example 2. In this example, the signal steering vector error is considered. The SNR is assumed to be 10 dB, and the actual target is located at with Doppler frequency Hz. From Figures 3(a) and 3(b), both the adaptive weight patterns place nulls in the interference positions. However, the adaptive weight pattern obtained by the conventional D3 STAP places a null in the actual target position while the adaptive weight pattern obtained by the modified robust D3 STAP preserves a gain in the actual target position.
(a) The conventional D3 STAP
(b) The modified robust D3 STAP
Example 3. In this example, the output SINRs versus the input SNR are shown in Figures 4(a) and 4(b). The actual target position for Figure 4(b) is the same as that in Example 2. For a comparison, the loaded D3 approach is also given in the simulations, which is a robust method to design the weight vector of (11) like the loaded SMI (LSMI) [19]. The fixed diagonal loading parameter is chosen for the loaded D3 STAP. From Figures 4(a) and 4(b), we can see that the modified robust D3 STAP shows better performance than other methods. This improvement is especially remarkable when the SNR is less than 0 dB.
(a) Without signal steering vector error
(b) With signal steering vector error
5. Conclusion
In this paper, a robust D3 STAP method for MIMO radar has been presented. The proposed method uses the WCPO to avoid the target self-nulling effect. Moreover, to improve the output SINR, a modified objective function is proposed to minimize the array noise output while mitigating the interferences. Simulation results show that the proposed method can prevent the target self-nulling effect and provide higher output SINR than other methods in the low SNR environment.
Conflicts of Interest
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (Grant no. 61401204), Postdoctoral Science Foundation of Jiangsu Province (Grant no. 1501104C), and Technology Research and Development Program of Jiangsu Province (Grant no. BY2015004-03).