Optimal Conguration of Array Elements for Hybrid Distributed PA-MIMO Radar System Based on Target Detection

: How to utilize limited system resources budget to maximize the effectiveness potential of the entire system through optimal allocation has always been a hot issue in radar resource management. This paper establishes a hybrid distributed phased array multiple - input and multiple - output (PA - MIMO) radar system model. It combines coherent processing gain and spatial diversity gain to synergistically improve the target detection performance of the radar system. For the hybrid distributed PA - MIMO radar system, we derive a likelihood ratio test (LRT) detector based on the Neyman - Pearson (NP) criterion. The coherent processing gain and spatial diversity gain are jointly optimized by implementing subarray - level and array element - level optimal configuration at both transceiver and transmitter ends. Moreover, a quantum particle swarm optimization - based stochastic rounding (SR - QPSO) solution algorithm is proposed for the integer planning - based configuration model. And the optimal array element configuration strategy is guaranteed to be obtained with fewer iterations and realize the joint optimization between subarray and array levels. Finally, numerical simulations are carried out using three typical optimization problems to demonstrate the effectiveness of the optimal configuration of the hybrid distributed PA - MIMO radar system.


Introduction
The diversity of modern radar targets and the complexity of the battlefield environment have exposed the inadequacies of existing radar regimes and detection techniques. In order to cope with complex targets and environments [1] and seek breakthroughs in target detection theory and technology, regime modification and resource management for radar are being continuously and intensively carried out [2][3][4]. Maximizing the ability of radar sensor systems to obtain electromagnetic information [5], optimizing the utilization of existing radar resources [6,7], and improving the target detection capability of radar systems are fundamental topics and practical and urgent tasks faced in the field of radar information processing and optimal resource management [8].
Multiple-input multiple-output (MIMO) radar has recently received extensive attention as a novel radar system [9]. Generally, the MIMO radar can be divided into two categories based on the antennas' spatial configuration: One is the collocated MIMO radar [10], with array elements spaced at half-wavelength level, which mainly uses tuned detection signals to achieve superior waveform diversity. The other is distributed MIMO radar [11], which achieves joint signal processing through the spatially scattered configuration of array elements. And the spatial diversity gain of echo signals caused by angular extension can effectively overcome target scintillation to improve detection performance [12]. Compared to distributed MIMO radars, the T/R elements of conventional phased arrays are more closely distributed in space and have a strong correlation between channels, allowing for excellent spatial sampling capability and freedom of information processing [13].
Both coherent processing gain and spatial diversity gain can improve radar detection performance [14]. However, distributed MIMO radars transmitting orthogonal waveforms will lose spatial coherence gain while obtaining spatial diversity gain due to the different modes of radar operation. Whether the distributed MIMO radar using diversity gain or phased array radar with coherent processing gain is non-optimal with a certain number of array elements. It is far from sufficient to simply increase the total resources without considering the cooperation between individual terminals.
The proposal of phased-array multiple-input multiple-output (PA-MIMO) radar [15] opens a new avenue for developing MIMO radar. The hybrid distributed PA-MIMO radar is a combination of traditional phased array radar technology and MIMO radar technology. It utilizes the coherent processing gain and spatial diversity gain, which are obtained simultaneously from the coherence of array elements signal within the subarray and the orthogonality of the inter-subarray signal, respectively. So that the hybrid distributed PA-MIMO radar system can maintain the advantages of MIMO radar while having the benefits of coherent processing of phased-array radar, which is a compromise and effective implementation scheme [16].
Numerous scholars have conducted in-depth studies on the array elements configuration of the radar systems. The division of the transmitting array into multiple overlapable subarrays in the [17] improves the angular resolution and target capacity. Ref. [18] studies the optimal sparse array optimization configuration problem in the presence of multiple sources of interest (SOI). Ref. [19] proposed an algorithm for the joint arrangement of transmitter and receiver in distributed MIMO radar to improve the positioning accuracy. There is also a solid research foundation on improving system detection performance through array element configuration optimization. Ref. [20] optimizes the target detection capability by deploying the array elements in space through an exhaustive method and completing the power allocation through a waterfilling-type algorithm. The ref. [16] divides the transmit array into uniformly overlapping subarrays to obtain coherent processing gain and waveform diversity gain, and the theoretical derivation and simulation experiments demonstrate the superiority of phased array MIMO radar. In [21], the optimal allocation of two gains in a MIMO-MSRS system is proposed from the receiver side through the configuration of the spatial location of the array elements. Refs [22,23] investigated the optimal configuration of digital array radar arrays to study the effect of array space configuration optimization on radar system performance improvement from the receiver side. However, few references simultaneously consider the allocation and optimization of coherent processing gain and spatial diversity gain in a radar system from both the transmitter and receiver sides.
Aiming at solving the issues above, we establish the radar system signal model and array space configuration model based on a hybrid distributed PA-MIMO radar. Then, the likelihood ratio test (LRT) detector are derived under fixed noise and construct the array space configuration model based on Neyman-Pearson (NP) criterion. On this basis, three typical optimization problems are discussed, i.e., maximizing the detection probability, maximizing the effective radar range, and minimizing the radar system equipment volume for a given detection index. In this regard, the respective closed-form approximate solutions are constructed and solved by the proposed quantum particle swarm optimization-based stochastic rounding (SR-QPSO) algorithm to obtain the optimal strategy for the array element configuration. Finally, the solution realizes the optimal cooperation among the array elements to improve the radar detection performance based on the total amount of existing radar resources.
The rest of the paper is organized as follows: The hybrid distributed PA-MIMO system model along with the system configuration based on the diversity conditions are demonstrated in Section 2. Section 3 derives the LRT detector from the NP criterion based on the signal processing flow of the hybrid distributed PA-MIMO radar. In Section 4, three optimization scenarios are considered and propose a solution algorithm based on integer programming. Section 5 presents numerical results and analysis. Finally, Section 6 concludes the paper.

Configuration
The correlation between the elements of the target scattering coefficient matrix can be adjusted by changing the distance between each subarray of the radar system [12], thus changing the processing mode of the echo signal in the radar system. Without loss of generality, the correlation of the spatial signal is defined by the array element spacing d as where D is the tangential length of the target.
The spatial configuration of the hybrid distributed PA-MIMO radar and the allocation of the proportions of the two gains in the radar system essentially change the correlation of elements in the target scattering coefficient matrix H . If the array elements spacing does not satisfy the space diversity condition in (4), the subarrays are combined into a phased array radar; on the contrary, the sub-arrays follow the MIMO radar signal processing mechanism.
Therefore, changing the distance between the radar elements ensures that the corresponding target scattering coefficients are perfectly correlated or uncorrelated.
Once the target scattering coefficients lk  and nm  are completely uncorrelated, the radar system possesses an angular broadening of the space target, resulting in a spatial diversity gain. Accordingly, when lk  and nm  are entirely correlated, the radar system performs coherent processing on lk r and nm r to improve the signal-to-noise ratio (SNR) of the target echo signals. Herein, the target scattering coefficient matrix is reorganized and divided according to the correlation of each channel, and the target scattering coefficient matrix Ĥ will be reconstructed as (5) after the configuration. Simultaneously, each subarray also transmits and receives independent orthogonal signals as a MIMO radar for diversity processing. So that the radar system has coherent processing gain and space diversity gain at the same time. Spatial diversity processing can improve the detection performance by increasing the number of independent channels, and coherent processing improves the detection performance by increasing the detection SNR of each channel. By dividing the target scattering coefficient matrix into blocks, the proportion of space diversity gain and coherent processing gain in the radar system is coordinated and allocated to optimize the target detection performance of the radar system.

Hybrid Distributed
Therefore, the SNR corresponding to the sampling value of each subarray can be approximated as where s T is the duration of transmitting pulse. For simplicity, we define

LRT Detector of the Hybrid Distributed PA-MIMO Radar System
The spatial configuration of different subarrays satisfying the spatial diversity condition in equation (4) makes the outputs of each subarray independent and orthogonal to each other.
Assuming that the received noise level is known, then each echo signal is an independent identically distributed (IID) complex Gaussian random variable. Therefore, the square-law detection outputs of different stations of the radar system are given by where 2ˆˆn m nm Xr = , furthermore, the target detection problem can be expressed as follows Based on the NP criterion, this paper constructs a hybrid distributed PA-MIMO radar LRT detector, which must be expressed as the following structure We have where ( ) Further taking logarithms on both sides of (13), the log-likelihood ratio of the whole hybrid distributed phased-array MIMO radar is the sum of different field points.  (17) where 0  is a threshold preset according to the preset constant F P . Finally, the weighted sum of the ratio detector is used for LRT. Hence, the test statistic of hybrid distributed PA-MIMO radar system will be performed under the 0 and 1 assumptions, respectively.
where ˆnm R is a 2 DOF cardinality distribution variable, i.e., a standard exponential distribution variable [26]. For the test statistic ( ) Therefore, considering equations (17), (20) and (25) together, the LRT detector of the hybrid distributed PA-MIMO radar system is obtained as follows.

Overview of the Optimization Problem for Hybrid Distributed PA-MIMO Radar Systems
Optimizing the array element configuration of the hybrid distributed PA-MIMO radar system aims to improve detection performance. Generally speaking, the evaluation criteria for target detection performance are as follows: detection probability D P [28], detection range E max R [21], resolution and SNR [29], etc. However, different optimal configuration strategies may be used for different optimization purposes. D P is usually the most intuitive performance index used to describe the detection capability of the radar system; In addition, for a certain detection probability D P and false alarm probability FA P , the maximum operating distance E max R of the radar system is pursued; Thirdly, it is also of great practical significance to reduce the amount of equipment of the radar system with the given false alarm probability FA P and detection probability D P . Therefore, according to system design purposes, the optimal configuration of the hybrid distributed PA-MIMO radar can be divided into the following three optimization problems: based on the LRT detector of the radar system.

Detection Performance Analysis of Typical Hybrid Distributed PA-MIMO Radar System
In fact, the above three optimization problems consider improving the target detection capability from different perspectives, but the core problem is to optimize the vector β and γ . However, there is also a parameter coupling problem in the process of optimizing high-dimensional integer programming problems, which makes the analytical solution complex and impossible. To reduce the search time and solution complexity, array elements are divided into a certain number of non-overlapping subarrays, i.e.
According to the array element configuration scheme, the hybrid distributed PA-MIMO radar system can be decomposed into four typical structures: (1) Distributed MIMO radar with full diversity processing; (2) Phased array radar with full coherent processing; (3) MISO radar with full diversity processing at the transmitter side and full coherent processing at the receiver side; (4) SIMO radar with full diversity processing at the receiver side and coherent processing at the transmitter side.
Similarly, the distribution of test statistics of these typical radars is obtained as A more general implication of the equation (38) is that configuring the subarray should utilize as many receive array elements as possible to achieve the diversity number so that the transmitter can improve the coherent processing gain with a minimum division strategy [30].

Optimal Uniform Configuration for Hybrid Distributed PA-MIMO Radar System
Both the transmitter and the receiver sides are configured in a uniform non-overlapping manner, and in combination with (18) and (28) (

3) Model of optimization problem 3
While satisfying the intended target detection performance of the radar system, the volume of the required equipment is minimized through the optimal array elements configuration. Clearly, the most intuitive and logical way to reduce the amount of system equipment is to increase system integration by sharing antenna transceivers in a time-sharing manner. Accordingly, the total volume of radar system equipment is

QPSO-Based Stochastic Rounding Optimization Solution Algorithm
Considering that the optimization problem is integer programming and the objective function is complex and challenging to solve, although the optimal solution can be obtained by exhaustive search, the problem size is large and the computational effort is considerable.
Therefore, we propose a stochastic optimization rounding algorithm incorporating quantum-behaved particle swarm optimization (SR-QPSO) [31]. The particle swarm optimization algorithm with quantum behavior improves the algorithm to cover the whole search space during iteration by simulating the substantial uncertainty of state superposition in quantum systems. So it improves the global search weakness at the tail end of the classical PSO algorithm search and enhances the global optimization capability of the algorithm [32].
In this paper, a random rounding method is adopted in which the fractional part of the particle position parameter is used as the probability value for upward rounding the parameter decimal part. Although the rounded problem is no longer equivalent to the original problem, the solution set of the original problem is included in the feasible solutions of the rounded optimization problem, i.e., the maximum value of the latter is not smaller than the maximum value of the original optimization problem. The entire algorithm flow is shown in Algorithm

Parameter Settings
In order to verify the effectiveness of the hybrid distributed PA-MIMO radar array configuration on target detection capability enhancement, some numerical simulations based on (41)-(43) are presented in this section. In the following, the defaulted radar system configuration parameters are M=N=100,  Essentially, this is because the diversity at the transmitter side reduces the coherent processing gain, while the contribution of the spatial diversity gain is much smaller than the coherent processing gain. Combined with Fig. 5, it can be seen that increasing the diversity at the receiver side can improve the detection performance only if the gain at the transmitter side meets a certain level. . It is clear that increasing the number of transmitter diversity DOF does not improve the range of the radar system compared to the optimal solution, but rather reduces the effective range. Moreover, the corresponding optimal transmit array division DOF for the transmit array division scheme decreases as the number of transmitter sites increases. This is still essentially a decrease in channel SNR due to transmitter side diversity. Therefore, the subsequent analysis will be based on the 1 M = . , three curves of E max R versus N are plotted according to (42), respectively. Varying M , it was found that transmitter diversity also caused an attenuation of the effective range of the radar system. It can be observed that the optimal configuration strategy at the transmitter side for condition 1 M = is 5 N = , and for 2 M = is 3 N = . Therefore, 1 M = is not always the optimal transmitter-side diversity DOF for a certain N . And the maximum benefit can only be achieved in coordination with the receiver-side array division. . It can be obtained that as the detection probability increases, the effective action distance decreases accordingly. Secondly, the optimal receiver diversity DOF increases with the increase of the detection probability the lower the false alarm probability FA P , the shorter the effective range. In contrast with Fig.   11 and Fig. 12, FA P has less impact on the diversity DOF at the receiver side compared to D P , but the optimal strategies N all increase as the detection accuracy rises. In Fig. 13, the volume equipment of the radar system M curves versus system diversity M . The optimal radar system design is given according to (43). Also, the result with respect to D 0 6 0 7 0 8 0 9 0 99 P . , . , . , . , . = are all provided. Obviously, the higher the detection probability, the larger the amount of radar system equipment required. When the D P exceeds 0.8, the number of required T/R array elements first decreases and then increases with the number of division sites. Herein, the optimal division sites number is 2. Conversely, when D P is less than 0.8, the optimal array element configuration scheme is 1 M = . That is, the PA-MIMO is configured as a phased-array radar without diversity. In Fig. 14, the total volume of the radar system M curves versus system diversity M . The optimal radar system design is given according to (43). Also the result with respect to However, when the FA P is greater than 10, the optimal array configuration scheme is phased array radar. A. With the increase of diversity DOF for both the transmitter and receiver side, the radar detection performance will deteriorate when they exceed the optimal values. Normally， different optimization objectives have different optimal configuration schemes. And the radar detection probability and false alarm probability also affect the value of optimal diversity DOF M and N .
B. The essence of the hybrid distributed PA-MIMO radar possessing superior quality detection performance lies in its coherent processing improving the local SRN within each subarray, based on which the spatial diversity gain generated between each independent subarrays will further improve the target detection capability. In particular, for all optimization problems, only a small transmitter side diversity DOF is required, because the gain generated by transmitter-side diversity does not compensate for the lost coherent processing gain.

Conclusions
This paper investigates the optimal array elements configuration scheme for hybrid distributed PA-MIMO radar based on target detection. Its essence is to change the coherence between the array signals through the array elements configuration, coordinate the proportion of the coherence gain and the spatial diversity gain in the radar system, and ultimately improve the target detection performance of the radar system without increasing resources.
And a SR-QPSO method is proposed to solve the optimal array element configuration scheme.
From the analysis in the paper, it is clear that neither distributed MIMO radar nor phased-array radar merely using diversity gain or coherent processing gain is optimal. Therefore, the system SNR is improved by coherent processing at the transmitter side, and the target detection performance will be further optimized by diversity gain at the receiver side

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