1. Introduction
With the rapid advancement of modern electronic technology, electronic warfare represented by electronic jamming has brought severe challenges and threats to the detection performance of radar systems [
1,
2]. Deception jammers generate false target information on the radar signal receiving system through delay, modulation, and forwarding to cover true targets, which prevents radar station from effectively completing target detection or accurately estimating target parameters [
3]. Therefore, a strong jamming countermeasure capability is an important guarantee for the survival of radar system under complex electromagnetic interference conditions.
For deception jamming, monostatic radar can identify false targets utilizing transmission signal optimization [
4,
5], polarization information [
6,
7], inverse tracking [
8], time-frequency analysis [
9], DRFM quantization error [
10], and clustering discrimination [
11]. However, the target detection process of monostatic radar has only one perspective, and it is impossible to obtain rich environmental information. At the same time, jamming countermeasures highly depend on the radar hardware to a large extent. It is only suitable for some special jamming scenarios. Hence, in the case of high-fidelity deceptive false targets, the anti-jamming effect of monostatic radar is not ideal, and it is difficult to deal with the existing complex electronic jamming.
For the limitations of monostatic radar, multi-radar coordinated operations have become a trend [
12,
13,
14]. In the future battlefield, our various radars will form a multistatic radar system with designated configurations and formations. The system must be equipped with several radars of the same type or different types to construct a large number of radar groups. By connecting radars in different spatial distributions, the multistatic radar system forms a networked detection system that occupies multiple observation angles, multiple frequencies, and multiple working modes to obtain a high-density signal space [
15,
16,
17,
18]. Multi-platforms and multi-sensors can effectively improve the target detection performance and parameter estimation accuracy, in contrast to monostatic radar system. [
19,
20,
21,
22].
In the fusion center of a multistatic radar system, the information collection and processing require the signal sharing and fusion of each radar, which can effectively distinguish interference and greatly improve the anti-jamming capability of the system. Meanwhile, by managing and scheduling the working mode of each radar, the technical complexity of unified jamming for the whole system is greatly increased.
Most of the existing anti-jamming methods detect targets separately in the multistatic radar system; then, the preprocessed echo information is uniformly sent to the information fusion center. According to the detection results from each radar station, the fusion center obtains the final system detection results for the target based on certain distributed detection criteria [
23,
24,
25]. However, in non-ideal environments, such as a low signal to noise ratio (SNR) or partial radar tracking loss, it is difficult for each radar to detect targets independently. The fusion center cannot jointly process the detection results [
26,
27,
28].
To deal with this problem, multistatic radar system needs to adopt the target joint detection mode to ensure the overall detection probability. At this point, the radar system will work as a whole. Each radar directly transmits the original target echo data to the system information fusion center. In the fusion center, joint detection or parameter estimation is carried out for the target, so as to achieve the optimal effect of jamming countermeasures.
The false target discrimination method based on parameter joint estimation is an effective discrimination algorithm [
29,
30]. It can be proved that the number of radars is an important parameter of performance, and increasing the number of transmitters or receivers can effectively improve false target discrimination capability. However, as the number of radars increases, communication requirements and computational complexity increase, resulting in unnecessary consumption of equipment resources [
31]. Therefore, how to strike a balance between radar resources and identification performance becomes a problem of resource scheduling. For multiple radar detection systems, resource scheduling can be divided into radar layout and parameter selection. For layout optimization, Hana proposed a performance-driven resource allocation scheme [
32], and minimized the number of transmitting and receiving radars employed in the estimation process. Hadi clustered the sensor yield to the accuracy threshold and successfully applied a minimum number of utilized sensors [
33]. For transmitted parameter selection, Zheng selected an optimal subset of sensors with the predetermined size and implemented the power allocation and bandwidth strategies among them, which achieved better performance within the same resource constraints [
34]. Nil extended optimization to multi-target cases, and put forward a joint power and bandwidth allocation strategy [
35]. The posterior Cramer Rao Lower Bound was suggested to minimize subarray utilization to a predetermined tracking accuracy and minimized the total utilized power [
36]. However, for anti-jamming techniques, resource scheduling has not been further explored.
To solve the above problems, this paper proposes subset selection strategies of gradual shrinkage for false target discrimination. For an existing multistatic radar system, on the premise of satisfied preset false target discrimination performance or limited device resources, multiple iteration screening can choose some transmitters or receivers with better spatial distribution or discrimination ability to form the radar subset, so as to achieve the approximate discrimination capability of the original system. The required equipment is reduced. The amount of data that needs to be processed in the fusion center and the required communication links are reduced. So, this strategy effectively reduces the operating costs and optimizes the radar configuration.
The remainder of the paper is organized as follows.
Section 2 introduces the signal model for multistatic radar system with deception jamming. In
Section 3, the shrinkage model for different situations is proposed, and the reduced time complexity is deduced.
Section 4 presents the simulation results, and conclusions are drawn in
Section 5.
3. Shrinkage Model
3.1. Construction of Shrinkage Model
Based on the deceptive false target discrimination method for joint parameter estimation, the number of radars in the system is an important parameter that affects the discrimination probability. The more radars, the stronger the discrimination capability. However, a large number of radars can easily cause equipment redundancy. Due to the low utilization efficiency of certain sites, a waste of resources is unavoidable.
To settle this problem, the selection vectors and can be introduced to construct a partial radar subset in which some transmitters/receivers are selected to replace the whole system, where , and . When the system selects the l-th transmitter or the k-th receiver, and are set as 1, otherwise, they are set as 0, so as to construct the extended design of .
For the final selected radar subset
,
and
respectively contain all transmitting stations
or receiving stations
that can be selected. So, CRLB of
can be defined as
. In this paper, only the estimation accuracy of jamming distance is applied, and
’s MSE of radar subset
is shown as follows:
When the l-th transmitter and the k-th receiver are selected, the corresponding and in the selection vectors and are set to 1. Based on the partial radars selected in the formula, the deceptive distance estimation accuracy can be calculated, and the discrimination probability of the false target can be further improved.
In a given multistatic radar system, some transmitters and receivers contribute to better jamming discrimination than others because of the different relative positions and parameters for the target and each sensor.
The existing algorithm needs to search all combinations to find the best performing subset. Such an exhaustive search method requires multiple iterations with high complexity. Therefore, this paper proposes some subset selection methods to effectively reduce the number of radar stations in the multistatic radar system. These algorithms reduce station numbers and save communication resources on the premise that the deceptive target discrimination performance meets the system requirements or the best discrimination performance balances the limited equipment resources. In the meantime, compared with the exhaustive search, the computational complexity is effectively reduced.
3.2. Rapid Shrinkage
For multistatic radar system, it is an asymptotic optimization problem to obtain the preset discrimination index with the smallest number of stations. The target function is set to the minimum number of radars, and the constraint condition is that the false target discrimination performance reaches the preset value
, as is shown in the formula:
This section proposes a rapid shrinkage strategy for subset selection to obtain the minimum number of devices under a given threshold of discrimination performance. To reduce the time complexity of the screening process, we can select the radar combination with the shortest target distance as the initial subset. Then, the radar station with the best discrimination performance is selected in each round to form the radar subset. After many iterations, the number of selected radars gradually increases. When the discrimination probability
reaches the preset threshold or all radars are selected, the iterative process needs to be stopped, and the corresponding selection vectors
and
are obtained. The operation is shown below, and the specific selection steps are shown in Algorithm 1.
Algorithm 1: Algorithm of Selection Strategy for Rapid Shrinkage |
|
Where represents the union of sets, \ represents the matrix deletion.
On the subset selection strategy of the rapid shrinkage method, for a multistatic radar system, the time complexity of finding the initial radar combination is . After that, in each round, one radar station is iteratively selected from the remaining radar stations to join the radar subset, and the time complexity of each round is . Then, the total time complexity of the strategy is , where J is the total number of radars in the final subset. The time complexity of exhaustive search is . A conclusion can be drawn that the time complexity is reduced from the exponential order of exhaustive search to the linear order by the rapid shrinkage method.
3.3. Global Shrinkage
In the process of subset construction, the subsequent selection results can be directly affected by the selection of the initial transmitters and receivers. The selection strategy of rapid shrinkage method is proposed in
Section 3.2 for the shrinkage model that can achieve the lowest time complexity. However, the number of stations is only the local minimum, which can not achieve the global optimum. Therefore, there may still be equipment redundancy. This section considers the selection strategy of global shrinkage. Based on
Section 3.2, each branch can be scanned from
MN channel pairs, and all local minimum values need to be compared to obtain the subset with the best discrimination performance. Although the complexity is increased, the local optimal problem is effectively alleviated. Operations are shown as follows, and specific selection steps are shown in Algorithm 2.
Algorithm 2: Algorithm of Selection Strategy for Global Shrinkage |
|
Where represents a belonging relationship between sets.
Through the steps above, the global shrinkage subset selection strategy is obtained. The exploratory shrinkage method reduces the time complexity to
, where
J is the number of selected radars. According to
MN different initial choices, all of the unknown radar combinations in
Section 3.2 can be obtained, so as to achieve the effect of global optimal contraction.
3.4. Predetermined Size
In order to use limited resources reasonably, a balance needs to be achieved between discrimination performance and equipment resources. Making full use of existing radars is particularly meaningful for resource optimization, especially when utilization of the system infrastructure is restricted. Motivated by this, an operational policy is proposed in this section. The goal is to select a subset of a predetermined size
X that has the greatest distinguishing ability. This knapsack problem can be written as follows:
The selection of the initial transmitter and receiver has a considerable impact on the final result. As mentioned above, there are two ways to find the best pairs as the initial subset. In this section, we choose the initial subset selection method for global shrinkage. For MN preselected initial subset, all pairs are compared to find the best choice as the selected subset.
The selection steps are the same as in
Section 3.3 on the shrinkage model, but the main difference lies in the ending conditions. In this section, when the size of the preselected matrix is
X, the selection must stop. Then, with the comparison of preselected matrixes, the best subset is singled out. At the same time, the vectors
and
are updated to suit the choice of the selected transmitters or receivers. The detailed steps can be seen in Algorithm 3.
Algorithm 3: Algorithm of Selection Strategy for Predetermined Size |
|
Through the steps above, the predetermined size subset selection strategy is obtained. The time complexity of this algorithm is determined through the initial subset selection, when the initial subset is selected with the shortest target distance. It is similar to the calculation method of Algorithm 1 in
Section 3.1. The total time complexity of the strategy is
, only replacing the total number of radars
J in the final subset by preset amount
X. When the initial subset is scanned from
MN channel pairs, the predetermined size method reaches the time complexity of
. The calculation is similar to Algorithm 2 in
Section 3.2.