The generalized Radon-Fourier transform (GRFT) has been proposed to detect radar weak maneuvering targets by realizing coherent integration via jointly searching in motion parameter space. Two main drawbacks of GRFT are the heavy computational burden and the blind speed side lobes (BSSL) which will cause serious false alarms. The BSSL learning-based particle swarm optimization (BPSO) has been proposed before to reduce the computational burden of GRFT and solve the BSSL problem simultaneously. However, the BPSO suffers from an apparent loss in detection performance compared with GRFT. In this paper, a fast implementation algorithm of GRFT using the BSSL learning-based modified wind-driven optimization (BMWDO) is proposed. In the BMWDO, the BSSL learning procedure is also used to deal with the BSSL phenomenon. Besides, the MWDO adjusts the coefficients in WDO with Levy distribution and uniform distribution, and it outperforms PSO in a noisy environment. Compared with BPSO, the proposed method can achieve better detection performance with a similar computational cost. Several numerical experiments are also provided to demonstrate the effectiveness of the proposed method.

With the development of aircraft stealth technology, there is a growing need for radar to detect weak maneuvering targets in a noisy background. It is a known fact that pulse integration especially coherent integration can improve the signal-to-noise ratio (SNR) and ultimately improve the detection performance [

Concentrating on coherent integration, a lot of work has been carried out. The most commonly used method is moving target detection (MTD) [

In recent years, a new method called Radon-Fourier transform (RFT) [

Although BPSO-based GRFT is efficient, it suffers from apparent detection performance loss compared with GRFT. To improve the detection performance, this paper proposes the BSSL learning-based modified wind-driven optimization (BMWDO). The wind-driven optimization (WDO) [

Suppose that radar transmits linear frequency modulated (LFM) signal, that is,

The received radar echo after carrier frequency demodulation can be denoted as

GRFT is a coherent integration algorithm via jointly searching in multidimensional parameter space. By using GRFT, the trace of the target can be extracted and the DFM can be compensated at the same time. The definition of GRFT in [

Suppose a 2D complex function

Let

Equation (

By analyzing (

Sketch map of the BSSL phenomenon.

The wind-driven optimization (WDO) algorithm [

WDO is very similar to PSO [

Before performing WDO, the four coefficients

Through analyzing the characteristics of Levy distribution, we can find that the random number

When applying MWDO in GRFT, a great number of unnecessary searching paths can be eliminated, which means the GRFT can be calculated efficiently. However, big values of BSSL in GRFT may cause local convergence to side lobes. To settle this matter, we propose a BSSL learning-based MWDO to find the main lobe by using the relations between side lobes and the main lobe. The detailed description of the proposed method is given as follows and the whole target detection procedure based on BMWDO is shown in Figure

Flow chart of the target detection method via BMWDO.

Generate the values of coefficients of MWDO via (

Further update

Repeat Step

It should be pointed out that when condition

In this section, several numerical experiments are provided to demonstrate the effectiveness of the proposed fast implementation method of GRFT. The BSSL phenomenon is firstly verified and then BMWDO and BPSO are compared in convergence performance. The detection performances of the two fast implementation methods as well as the traditional GRFT, RFT, and moving target detection (MTD) are compared. The running time of the traditional GRFT, the BPSO-based GRFT, and the BMWDO-based GRFT is also provided.

In this part, we suppose that the radar pulse repetition interval

Range and velocity slice of GRFT.

From Figure

In the following simulations, the radar parameters listed in Table

Simulation parameters of radar.

Carrier frequency | 1 GHz |

Bandwidth | 15 MHz |

Sample frequency | 60 MHz |

Pulse duration | 25 |

Pulse repetition frequency | 100 Hz |

Pulse number | 100 |

Average convergence graphs of BPSO and BMWDO with 20 runs under different SNR (before pulse compression). (a) SNR = −10 dB. (b) SNR = −28 dB.

Figure

The detection performances of traditional GRFT, BPSO-based GRFT, BMWDO-based GRFT, RFT, and MTD are investigated via Monte Carlo trials. The false alarm probability is set as

Detection performances of five different detectors with different motion orders. (a) Motion order is 1 (

Figure

It is not difficult to discover that the decline of the detection probability is not obvious when motion order changes from 2 to 3. This is because the high-order motion parameters have much lower influence on ARU and DFM compared with low-order parameters. When applying BMWDO or BPSO, the low-order motion parameters are dominant in deciding the movement trend of air parcels. Thus, based on the radar parameters adopted in this paper, it is possible to neglect the higher motion parameters and set

Detection probabilities of BMWDO-based GRFT with motion orders no less than 3.

In this simulation, the computational costs of the traditional GRFT, the BPSO-based GRFT, and the BMWDO-based GRFT are investigated. The searching ranges of parameters in the traditional GRFT are the same as that in BMWDO and the searching interval of each parameter can be determined according to [

Computational cost of GRFT, BPSO-based GRFT, and the proposed BMWDO-based GRFT.

Figure

In this paper, we propose a fast implementation method for GRFT to reduce the computational burden, namely, BMWDO. By applying BMWDO, a large number of unnecessary searching paths can be eliminated and the local convergence to BSSL can be avoided. Several numerical experiments are provided to analyze the performance of BMWDO in detail, including the convergence performance, the detection performance, and the computational burden. Compared with the traditional ergodic-search GRFT, the proposed method can realize the weak maneuvering target detection in a much more efficient way. Compared with BPSO, the BMWDO has better antinoise performance, which indicates that BMWDO has greater chance to converge to the target’s main lobe in a relatively low SNR. The simulation results show that BMWDO has better detection performance and slightly longer running time compared with BPSO, which verify the effectiveness of the proposed method. At last, we should notice that although the BMWDO obviously improves the detection performance compared with BPSO, it still suffers from detection performance loss compared with the traditional GRFT. The reason is that the BMWDO is a stochastic optimization method and it cannot jump out of the convergence to noise peaks each time. Thus, our future work may further study the WDO method and combine it with other algorithms to obtain stronger antinoise performance.

The authors declare that there are no competing interests regarding the publication of this paper.

The work was supported by the National Natural Science Foundation of China (Grant no. 61201366), the Fundamental Research Funds for the Central Universities (Grant no. NS2016040), the Fundamental Research Funds for the Central Universities (Grant no. NJ20150020), and the Priority Academic Program Development of Jiangsu Higher Education Institutions.