International Journal of Antennas and Propagation

Volume 2016 (2016), Article ID 4315616, 10 pages

http://dx.doi.org/10.1155/2016/4315616

## A Fast Algorithm of Generalized Radon-Fourier Transform for Weak Maneuvering Target Detection

Key Laboratory of Radar Imaging and Microwave Photonics, Ministry of Education, College of Electronic and Information Engineering, Nanjing University of Aeronautics and Astronautics, Nanjing 211100, China

Received 1 July 2016; Accepted 5 October 2016

Academic Editor: Ana Alejos

Copyright © 2016 Weijie Xia et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

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.

#### 1. Introduction

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 [1].

Concentrating on coherent integration, a lot of work has been carried out. The most commonly used method is moving target detection (MTD) [2], which achieves integration by using Doppler filter bank. However, MTD method can only deal with the target with uniform velocity and will become invalid if the range migration (RM) exceeds one range bin during the integration time [3]. It is of vital importance to eliminate RM since the high-speed target can easily exceed several range units even in a short time. To deal with RM, keystone transform (KT) [4, 5] was performed by rescaling the time axis for each frequency and is often performed before MTD. In actual detecting environment, for example, the velocity, acceleration, and jerk will result in first-order RM, second-order RM, and third-order RM, respectively. Unfortunately, conventional KT can only correct the first-order RM. Thus [6–8] studied second-order KT to correct the second-order RM and Kong et al. [9] proposed a coherent integration method via generalized KT and generalized dechirp process (GKTGDP) for maneuvering targets with arbitrary high-order RM. It is worth paying attention to that KT could be invalid without ambiguity correction if Doppler ambiguity happens. Algorithms for Doppler ambiguity correction are hardly independent of Doppler ambiguous integers searching; thus, the computational burden will greatly increase.

In recent years, a new method called Radon-Fourier transform (RFT) [10–12] has been proposed to realize long-time coherent integration via jointly searching along range and velocity directions. The detection performance of the high-speed and weak targets with constant velocity can be significantly improved by RFT if one of the searching pairs matches well with actual values. In consideration of maneuvering targets, generalized RFT (GRFT) [10, 13] was also defined for targets with arbitrary parameterized motion. Based on the idea of GRFT, a lot of work has been done recently [14–17]. Actually, the GRFT suffers from heavy computational burden and is impractical without fast implementations because of the multidimensional ergodic search. Fortunately, the realization of GRFT can be converted into an optimization problem in parameter space. Thus, intelligent optimization algorithms can be utilized to eliminate a large number of unnecessary searching paths. Another drawback of GRFT is the BSSL problem [10, 11] derived from discrete pulse sampling, finite range resolution, and limited integration time, which will lead to intelligent optimization algorithms converging to local optimum easily. Following consideration of the above issues, Qian et al. [18] have proposed BSSL learning-based particle swarm optimization (BPSO) to fast implement GRFT. By using the relation of BSSL and the main lobe, the local convergence can be avoided and the convergence speed can be accelerated simultaneously.

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) [19, 20] is a stochastic nature inspired, population based iterative heuristic global optimization method based on atmospheric motion. Compared with the traditional PSO, WDO employs additional terms in the velocity update equation, providing robustness and extra degrees of freedom for fine-tuning. However, it could be difficult in choosing optimum WDO coefficients for GRFT because the location of the optimum point depends on the motion parameters which have large dynamic ranges. In order to deal with the difficulty in choosing coefficients and further improve the global optimization ability of WDO in a noisy environment, we propose a modified WDO method which adjusts the control coefficients in WDO with random distributions, namely, MWDO. Detailed numerical experiments demonstrate that the proposed BMWDO method can improve the detection performance with a similar running time compared with BPSO.

#### 2. Signal Model and GRFT

##### 2.1. Signal Model

Suppose that radar transmits linear frequency modulated (LFM) signal, that is, where is the pulse width, is the frequency modulated rate, and is the carrier frequency. Let denote the slow time, where denotes the pulse repetition time, is the number of pulses, and is the fast time.

The received radar echo after carrier frequency demodulation can be denoted aswhere is the distance between radar and target at the radar line-of-sight and is the amplitude of the echo. The time delay of the echo is , where is the speed of light. After pulse compression via using the baseband transmitted signal as the reference signal, that is,the received signal in the time domain can be expressed as In the above equation, is the amplitude after pulse compression and denotes the system bandwidth. The range between radar and target in radial direction varies with the slow time and can usually be expressed as a polynomial function of , which can be expanded into Taylor series [16], that is,where is the motion order, is the coherent integration time, and is the motion parameter vector. It is obvious that the peak location of the sinc function varies with and the changes will exceed the range resolution easily if the integration time is long or the motion parameters are not very small, which means that the across range unit (ARU) effect will happen. The Doppler frequency can be calculated aswhere is the wavelength, is the central frequency of Doppler, and the parameter is the target’s velocity. If the target has acceleration or higher motion parameters, the Doppler frequency will be time-varying. If the changes of exceed the Doppler resolution , the Doppler frequency migration (DFM) will come across. In order to coherently accumulate the target’s energy, we need to correct both the ARU and the DFM.

##### 2.2. Definition and Analysis of GRFT

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 [10] is given as follows.

*Definition 1. *Suppose a 2D complex function is defined in the plane and a parameterized -dimensional function is used for searching a certain time-varied curve in the plane, where . Then GRFT can be defined aswhere is a known constant with respect to .

Let ; then (8) can be rewritten aswhere . Suppose that ; then GRFT degenerates into RFT which deals with the case of uniform velocity. From (9), we can easily know that when the searching values of motion parameters are exactly the target’s real motion values , the coherent integration could be achieved and the peak would be formed in the parameter space, that is,Then the target can be detected and the motion parameters can be easily obtained by the location of the peak in the parameter space. However, because of limited integration time, discrete pulse sampling, and finite range resolution, the BSSL [10, 11] will also be formed in parameter space, which influences target detection performance. The causes of BSSL and the relations between BSSL and the main lobe are discussed as follows.

Equation (9) can be rewritten aswhereWhen and , where is the blind speed and is the blind speed integer, we have in (11). It is obvious that the phase can be compensated even though , which results in the BSSL phenomenon. Slice of BSSL can be denoted asThe blind speed integer , where is the searching range of velocity.

By analyzing (11), (12), and (13), we can see that the properties of BSSL are irrelevant to . Thus, the case of constant velocity is taken as an example to analyze the relations between BSSL and the main lobe. The sketch map of BSSL is illustrated in Figure 1. Suppose that the target’s velocity is , the initial range is , and the one-time blind speed is . The shadow region is formed so that the searching lines with one-time blind speed intersect the range-walk line in the integration time . The searching range of the shadow region is ; thus the length of the supporting area isBecause of the finite range resolution, the overlapped pulse number in the intersection of the range-walk line and the searching line isThe pulses are also coherently integrated. Thus, the primary lobe to side lobe ratio (PSLR) can be denoted asIn general, in the case that the blind speed integer ,With the increase of , the supporting area of BSSL becomes longer and the amplitude of the side lobe decreases.