1
An Efficient Coherent Integration Method for Maneuvering Target Detection Xiaolong Li, Guolong Cui, Wei Yi and Lingjiang Kong School of Electronic Engineering University of Electronic Science and Technology of China Chengdu, Sichuan, P.R. China; Email:
[email protected]
Abstract—This paper considers the coherent integration problem for detecting a high speed maneuvering target, involving range migration (RM) and Doppler frequency migration (DFM) within one coherent pulse interval. An efficient coherent integration method based on keystone transform (KT), fold factor phase compensation (FFPC) and fractional Fourier transform (FRFT), i.e., KT-FFPC-FRFT, is proposed. It can not only correct the RM effect by KT and FFPC, but also remove the DFM effect and achieve the coherent integration via selecting proper rotation angle. Compared with Radon-fractional Fourier transform (RFRFT) algorithm, the computational cost of the presented method is reduced significantly under the premise that the two methods have same coherent integration detection performance. Finally, several simulations are provided to demonstrate the effectiveness.
I. I NTRODUCTION With the development of science technology, especially the supersonic and stealth technology, there is a growing need for detection of maneuvering targets with high speed in radar society [1]–[4]. Unfortunately, it becomes more challenging for the radar target detection technology because these high speed maneuvering targets often have low radar cross section (RCS), which results in weak radar returns. It is known that the coherent integration can increase the signal-to-noise-ratio (SNR) and thus improve the radar detection performance via compensating the target’s phase fluctuation among multiple radar pulse samplings [5]–[7]. However, the complex motions of maneuvering targets, e.g., high speed, and acceleration, involves the range migration (RM) and the Doppler frequency migration (DFM) within one coherent pulse interval (CPI), which result in serious performance loss for the coherent integration processing. The keystone transform (KT), via rescaling the time axis for each frequency, has been employed to correct the RM and obtain the coherent integration for high speed moving targets [8]–[10]. Besides, a method named Radon Fourier transform (RFT) was proposed to realize the coherent integration for target with RM [11]–[13]. Nevertheless, the DFM induced by the target’s acceleration cannot be mitigated by the KT or RFT and thus it will suffer integration loss in case of the DFM. To remove the DFM and obtain the coherent integration for a maneuvering target with acceleration, the Radon transform with minimum entropy criterion was studied in [14]. However, the entropy criterion based method has a higher demand on the input SNR, so it is not suitable for weak target signal detection. The Radon-fractional Fourier transform (RFRFT)
was then presented to achieve the coherent integraiton for high speed and accelerated target under low SNR background [15]. Although RFRFT can effectively remove the effect of DFM and realize the long time coherent integration without RM effect, it is often computationally prohibitive, since it involves the solution of a multi-dimensional searching in the motion parameter space. In this paper, we consider the coherent integration problem for high speed maneuvering target detection, involving RM and DFM within one CPI. An efficient coherent integration method based on KT, fold factor phase compensation (FFPC) and fractional Fourier transform (FRFT), i.e, KT-FFPC-FRFT, is proposed. It can not only correct the RM effect by KT and FFPC, but also eliminate the DFM effect and achieve the coherent integration via selecting proper rotation angle. Compared with RFRFT algorithm [15], the computational cost of the presented method is reduced significantly meanwhile they have same coherent integration detection performance. The remainder of this paper is organized as follows. In Section II, we present the coherent integration problem for high speed maneuvering target detection. In Section III, the coherent integration algorithm based on KT-FFPC-FRFT is proposed. In Section IV, we evaluate the performance of the presented method via several numerical experiments. Finally, in Section V, we provide some concluding remarks and possible future research tracks. II. P ROBLEM F ORMULATION Suppose that the radar transmits a linear frequency modulated (LFM) signal, i.e., τ s(t, τ ) = rect exp jπγτ 2 exp(j2πfc t), (1) Tp where
1 rect (x) = 0
|x| ≤ 12 , |x| > 12 ,
Tp is the pulsewidth, γ is the frequency modulated rate, fc is the carrier frequency, t = nTr (n = 0, 1, · · ·, N − 1) is the slow time, Tr denotes the pulse repetition time, N is the number of coherent integrated pulses, and τ is the fast time. Under the assumption that |v|/c 1 (v is the target’s radial velocity and c denotes the light speed), the received baseband
2
signal can be stated as [16]–[18] 2 2r(t) τ − 2r(t)/c exp jπγ τ − sr (t, τ ) =A0 rect Tp c 4πr(t) 4πvτ × exp −j exp −j , λ λ (2) where A0 is the target reflectivity, r(t) is the slant range between target and radar at slow time t, and λ = c/fc denotes the wavelength. The received signal after pulse compression in the slow time-range frequency (t − f ) domain can be expressed as π 2 f + fd /2 exp −j fd Sr (t, f ) =A1 rect B − fd γ (3) 4π(f + fc + fd )r(t) , × exp −j c where B is the bandwidth of the transmitted signal and fd = 2v/λ denotes the Doppler frequency. Neglecting the high-order components, the instantaneous slant range r(t) satisfies r(t) = R0 + vt + at2 ,
(5)
where nk is the fold factor, va is the blind velocity, and v0 is the unambiguous velocity. If the radar pulse repetition frequency is fp , va is thus computed to be va = λfp /2. Substituting (4) and (5) into (3) yields 4π(f + fc + fd )R0 Sr (t, f ) =A2 exp −j c 4π(f + fc + fd ) × exp −j v0 t c (6) 4π(f + fc + fd ) 2 at × exp −j c × H1 (t, f ),
+fd /2 π 2 exp −j f where A2 = A1 rect fB−f γ d , and d H1 (t, f ) = exp [−j4π(f + fc + fd )nk va t/c] .
H1 (t, f ) = exp [−j4πnk va t(1 + ξ)/λ] = exp(−j2πnk fp t) exp(−j4πnk va tξ/λ) = exp(−j4πnk va tξ/λ).
(9)
It is worth pointing out that (9) is obtained by substituting exp(−j2πnk fp t) = 1 [14] into (9). From (8), it is shown that there are three exponential terms of slow time t, which are all coupled with the range frequency f . More specifically,
4πv t • exp −j λ 0 (1 + ξ) indicates the first-order phase term due to the unambiguous velocity. • H1 (t, f ) is the fold factor phase term because of undersampling. 4πa1 t2 • exp −j (1 + ξ) indicates the quadratic phase terλ m induced by the target’s radial acceleration. The first-order terms of t will result in RM while the quadratic term will result in DFM which would make the signal energy defocused. Both RM and DFM will bring difficulties to coherent accumulation of target energy. In the following, a coherent integration algorithm based on KT-FFPC-FRFT is proposed.
(4)
where R0 is the initial slant range from the radar to the target, and a denotes the target’s radial acceleration. Due to the high speed of target and the low pulse repetition frequency (PRF), undersampling would occur [1], [14]. Therefore, the velocity of the target can be expressed as v = nk va + v0 ,
where
(7)
Suppose that ξ = (f + fd )/fc and ignoring the effect of fd on ξ, ξ can be simplified into ξ ≈ f /fc. Hence, (6) and (7) can be rewritten as 4πR0 (1 + ξ) H1 (t, f ) Sr (t, f ) = A2 exp −j λ (8) 4πv0 t 4πat2 × exp −j (1 + ξ) exp −j (1 + ξ) , λ λ
III. C OHERENT I NTEGRATION V IA KT-FFPC-FRFT A. Keystone Transform fc First of all, the KT, which performs scaling t = f +f ta in c the t − f domain [8], can be used to correct the RM caused by the unambiguous velocity. Substituting the scaling formula into (8) yields 4πR0 SKT (ta , f ) = A2 exp −j (1 + ξ) H1 (ta , f ) λ (10) 4πat2 fc 4πv0 ta exp −j × exp −j , λ λ f + fc
where
4πnk va ta f . H1 (ta , f ) = exp −j λ f + fc
(11)
Under the narrow band environment f fc , we have f ≈ 1 and f +f ≈ ffc . Thus, (10) and (11) can be recast c
fc f +fc
as
4πR0 (1 + ξ) H2 (ta , f ) SKT (ta , f ) ≈A2 exp −j λ (12) 4πv0 ta 4πat2 exp −j , × exp −j λ λ
where
4πnk va ta f . H2 (ta , f ) ≈ exp −j λ fc
(13)
By (12), the coupling between v0 and f is removed and the RM caused by the unambiguous velocity v0 has been corrected. However, the residual RM induced by integral multiples of blind velocity, i.e, the fold factor phase term, is still present.
3
B. FFPC
C. Integration Via FRFT
Define the fold factor phase compensation function as follows 4πn va ta f Ha (ta , f ; n ) = exp j , (14) λ fc
As show in (19), after KT and FFPC, the selected signal becomes a chirp signal in the slow time dimension. Therefore, the FRFT, which has excellent performance in making the chirp signal accumulated as an impulse in the proper FRFT domain [19], [20], can be applied for realizing the coherent integration and estimating the motion parameters. The FRFT of ssel,R0 (ta ) with angle α is defined as
where n denotes the searching fold factor. Multiplying (12) with (14) yields 4πR0 (1 + ξ) SKT (ta , f ; n ) =A2 exp −j λ 4πat2 4πv0 ta exp −j × exp −j λ λ 4π(nk − n )va ta f × exp −j . λ fc (15) Applying inverse Fourier transform (IFT) on (15) with respect to f , we have 2(nk − n )va ta 2R0 − sKT (ta , τ ; n ) =A3 sinc τ − c c 4πv0 ta 4πR0 × exp −j exp −j λ λ 2 4πata . × exp −j λ (16) When n = nk , i.e, the searching fold factor equals to the fold factor of target, (16) can be written as 4πR0 2R0 exp −j sKT (ta , τ ) =A3 sinc τ − c λ (17) 4πv0 ta 4πat2a exp −j . × exp −j λ λ It can be seen from (17) that the RM of target is removed, i.e., the the target energy located in the same range cell (corresponding to the target’s initial slant range R0 ). With a certain searching range r, the selected signal ssel,r (ta ) can be expressed as 2r ssel,r (ta ) =sKT ta , c 4πR0 2r 2R0 exp −j =A3 sinc − (18) c c λ 4πat2a 4πv0 ta exp −j . × exp −j λ λ From (18), we can see that when the searching range equals to the target’s initial slant range (i.e., r = R0 ), the amplitude of the selected signal ssel,r (ta ) reaches its maximum value. Then, the selected signal can be stated as 4πv0 ta 4πR0 exp −j ssel,R0 (ta ) =A3 exp −j λ λ (19) 2 4πata × exp −j λ
Xα (u) = Fα [ssel,R0 (ta )] ∞ ssel,R0 (ta )Ka (ta , u)dta , =
(20)
−∞
where α = P π/2 is the rotation angle, P is searching transform order, Fα denotes the FRFT operator, and the transform kernel Ka (ta , u) is given by Ka (ta , u) = Aα exp[j(0.5t2a cot α − ut csc α + 0.5u2 cot α)] δ[u − (−1)n t]
α = nπ α = nπ (21)
where Aα = (1 − j cot α)/2π. The analysis above shows that the proposed method is a three-dimensional searching process. When the searching fold factor, searching range and searching transform order match respectively the target’s fold factor, initial slant range and radial acceleration, the FRFT output can reach its maximum value. Then, based on the peak location, the parameters estimation of target, i.e., fold factor n ˆ k , initial slant range ˆ0, μ R ˆ and transform order Pˆ can be obtained. After that, the unambiguous velocity and radial acceleration of target can be estimated by [15] λ u ˆ csc(Pˆ π/2), 2S λ a ˆ = − 2 cot(Pˆ π/2), S
vˆ0 =
(22) (23)
where S = T /fp and T denotes the total coherent integration time . With the estimated fold factor n ˆ k and unambiguous velocity vˆ0 , the radial velocity of target can be determined as follows ˆ k va . vˆ = vˆ0 + n
(24)
The flowchart of the presented algorithm based on KT, FFPC and FRFT is given in Fig. 1. 5DZ GDWD
,)7 RYHU IDVW WLPH
6HDUFKLQJ UDQJH
3XOVH FRPSUHVVLRQ
)ROG IDFWRU SKDVH FRPSHQVDWLRQ
([WUDFW GDWD DFFRUGLQJ WR UDQJH FHOOV
)7 RYHU IDVW WLPH DQG .7
6HDUFKLQJ IROG IDFWRU
)5)7
Fig. 1.
Flowchart of the proposed method.
4
D. Computational Cost
20
20
40
40 60 Pulse number
60 Pulse number
Denote the number of range cells, searching fold factor, searching transform order, searching velocity, searching acceleration by Mr , Mnk , MP , Mv and Ma , respectively. The computational complexities of the proposed method and RFRFT are listed in Table I. Suppose that Mr = Mnk = MP = Mv = Ma = N , then the computational complexities of the proposed method and RFRFT are respectively O(N 4 log2 N ) and O(N 5 log2 N ).
80 100 120
140
160
160
180
180
200 150
200
250
300 Range cell
400
200 150
450
200
250
300 Range cell
350
400
450
(b)
20 40 Pulse number
60 80 100 120 140 160 180 200 150
200
250
300 Range cell
350
400
450
(c)
(d)
(e)
(f)
0.15 GHz 2 MHz 10 MHz 200 Hz 50 us 201
20
B
80 100
C
120 140 160 180
100
200 300 Range cell
Target B 500 km 3000 m/s 20 m/s2
0.6 0.4 0.2 0 40
500
Searching fold factor
1
0
250
300
350
400
450
Searching range cell
(b) B
C
0.8 0.6 0.4 0.2 0 30 20 0
0
(c)
Target C 500 km 3600 m/s −10 m/s2
400
(a)
10
TABLE III Motion Parameters of Targets
B
0.8
20
200
Searching fold factor
B. Coherent Integration for Multiple Targets
A
1
A
60
Normalized amplitude
40 Pulse number
We first analyse the coherent integration performance of the proposed method for a single target in Fig. 2, where the motion parameters of a moving target are R0 = 499.625km, v = 1580m/s, a = 20m/s2 . Besides, the SNR of the target is 6dB after pulse compression. In Fig. 2(a), the result after pulse compression shows that serious RM occurs because of target’s high velocity. Fig. 2(b) shows the signal after KT and Fig. 2(c) shows the result after fold factor phase term compensation using the target’s fold factor. It can be seen from Fig. 2(c) that the RM is correct, i.e., the target energy located in the same range cell. Fig. 2(d) shows the FRFT outputs of the selected signal after RM correction. The energy of target is integrated as an obvious peak in FRFT domain. This is benefited from two points, one is the RM correction via KT and fold factor phase term compensation. The other is the transform angle, with which the target energy can be accumulated well in the corresponding optimal domain. In addition, Fig. 2(e) and Fig. 2(f) show respectively the integration results of MTD and RFT. It can be seen that the energy spread, which makes it hard to detect the target.
Fig. 2. Coherent integration for a single target. (a) Echoes after pulse compression. (b) Keystone transform of the compressed signal. (c) Result after fold factor phase term compensation. (d) Integration result of the proposed method. (e) Integration result of MTD. (f) Integration result of RFT.
Normalized amplitude
A. Coherent Integration for A Single Target
Target A 501.125 km 3600 m/s 20 m/s2
350
(a)
TABLE II Simulation Parameters of Radar
Motion parameters Initial slant range Radial velocity Radial acceleration
120
140
IV. N UMERICAL R ESULTS This section is devoted to evaluating the performance of the proposed method via computer simulations, where the radar parameters are listed in Table II. For the sake of comparison, we will also simulate the moving target detection (MTD) method, RFT [11] and RFRFT [15].
Carrier frequency fc Bandwidth B Sample frequency fs Pulse repetition frequency fp Pulse duration Tp Number of pulses N
80 100
0.5
1
1.5
2
Searching transform order
(d)
Fig. 3. Coherent integration for multiple targets under high SNR (8 dB). (a) Echoes after pulse compression. (b) Fold factor-range response slice (searching transform order P = 1.126). (c) Fold factor-transform order response slice (searching range cell r = 325). (d) Range-transform order response slice (searching fold factor n = 18).
5
TABLE I Computational Complexity of Proposed Method and RFRFT Methods Proposed method RFRFT
Additions Mr Mnk MP N log2 N +N Mr log2 Mr + N 2 Mr Mr Mv Ma MP N log2 N
20 Normalized amplitude
Pulse number
60 80 100 120 140 160
B
0.8 0.6 0.4 0.2 30
180
20 10
200
100
200 300 Range cell
400
500
Searching fold factor
(a)
0
250
300
350
400
450
Searching range cell
(b) B
C
1 Normalized amplitude
A
1
40
0.8 0.6 0.4 0.2 30 20 10
Searching fold factor
0
0
(c)
0.5
1
1.5
2
Searching transform order
(d)
Fig. 4. Coherent integration for multiple targets under low SNR (−4 dB). (a) Echoes after pulse compression. (b) Fold factor-range response slice (searching transform order P = 1.126). (c) Fold factor-transform order response slice (searching range cell r = 325). (d) Range-transform order response slice (searching fold factor n = 18).
Multiplications
1 Mr Mnk MP N log2 N 2 1 + 2 N Mr log2 Mr + N (N − 1)Mr 1 Mr Mv Ma MP N log2 N 2
and the four methods as corresponding detectors. The Gaussian noises are added to the target echoes and the false alarm ratio is set as Pf a = 10−4 . Fig. 5 shows the detection probability of the four detectors versus different SNR levels and in each case, 1000 times of Monte Carlo simulations are done. The simulation results show that the probability of the detector based on the proposed method is the same as the RFRFT detector. Besides, the detection performance of the presented method is superior to the MTD and RFT thanks to its ability to deal with the RM and DFM. Furthermore, Fig. 6 illustrates the computational complexities of RFRFT and the proposed method. The range cell number is 500 and vmax = 6000m/s. The maximum value of the target’s radial acceleration is amax = 40m/s2 . The computational cost curves for different pulse numbers are plotted. It is evident that the computational cost of RFRFT is much larger than the proposed method. For N = 201, the computational complexity of the presented method is about 4.1 × 10−3 of the RFRFT. 1
C. Detection Performance and Computational Complexity For simplicity, we only consider the target A in the scene. The detection performances of MTD, RFT, RFRFT and the proposed method are further investigated by Monte Carlo trials. We combine the constant false alarm (CFAR) detector
0.8
d
0.6 P
In Fig. 3 and Fig. 4, we evaluate the coherent integration performance of the presented method for multiple targets under different SNR background, where the motion parameters of three high speed maneuvering targets are listed in Table III. 1) High SNR background. The SNRs of the three targets are all 8 dB after pulse compression, as shown in Fig. 3(a). Fig. 3(b)−Fig. 3(d) are respectively the fold factor-range response slice, fold factor-transform order response slice, and rangetransform order response slice of the proposed method. We can see that the targets’ energy are all coherent integrated well in the corresponding optimal domain, which indicates the motion parameters estimation of the targets. 2) Low SNR background. The SNRs of the three targets are all − 4 dB after pulse compression, as shown in Fig. 4(a). It can be seen that the targets’ echoes are totally buried in the noise. Fig. 4(b)−Fig. 4(d) are respectively the fold factor-range response slice, fold factor-transform order response slice, and range-transform order response slice of the proposed method. With the SNR gain of coherent integration, the targets’ energy are all accumulated as obvious peaks in the corresponding domain of the presented method, which is helpful to the target detection.
0.4 MTD RFT RFRFT Proposed
0.2
0 −15
Fig. 5. method.
−10
−5
0
5 10 SNR (dB)
15
20
25
Detection probability of MTD, RFT, RFRFT and the proposed
V. C ONCLUSIONS In this paper, we have addressed the coherent integration problem for high speed maneuvering targets detection, where the velocity and acceleration result in respectively the RM and DFM within one CPI. Summarizing: • We proposed an efficient method based on KT, FFPC and FRFT to realize the coherent integration for the high speed maneuvering target. This method can effectively remove the two migrations (RM and DFM) and achieve the coherent accumulation of target energy. • We also analysed the computational complexity of the presented method. It shows that the proposed method
6
18
10
16
Computational cost
10
14
10
12
10
RFRFT Proposed
10
10
Fig. 6.
200
400 600 Number of pulses (N)
800
1000
Computational cost versus integration pulse number.
can reduce the complexity order from O(N 5 log2 N ) to O(N 4 log2 N ) in comparison with the RFRFT algorithm. • We have evaluated the performance of the presented method by several numerical simulations. Compared with RFRFT method, the results highlighted that, the proposed algorithm can obtain the same coherent integration detection ability with much lower computational cost. In addition, the presented method is superior to MTD and RFT in terms of detection performance for a high speed target with DFM. A possible future research work might concern the coherent integration for maneuvering targets detection if the range curvature should be considered [21], [22]. Finally, it might be of interest to develop coherent integration methods for the targets with motion-order higher than acceleration [23]. VI. ACKNOWLEDGMENTS This work is supported by the National Natural Science Foundation of China (61201276, 61178068, and 61301266), the Fundamental Research Funds of Central Universities (ZYGX2012Z001, ZYGX2013J012, ZYGX2014J013 and ZYGX2014Z005), the Chinese Postdoctoral Science Foundation (2014M550465), and by the Program for New Century Excellent Talents in University (A1098524023901001063). R EFERENCES [1] J. Tian, W. Cui, and S. Wu, “A novel method for parameter estimation of space moving targets,” IEEE Geoscience and Remote Sensing Letters, vol. 11, no. 2, pp. 389–393, February 2014. [2] S. Q. Zhu, G. S. Liao, D. Yang, and H. H. Tao, “A new method for radar high-speed maneuvering weak target detection and imaging,” IEEE Geoscience and Remote Sensing Letters, vol. 11, no. 7, pp. 1175–1179, July 2014. [3] X. L. Li, G. L. Cui, W. Yi, and L. J. Kong, “Coherent integration for maneuvering target detection based on Radon-Lv’s distribution,” IEEE Signal Processing Letters, vol. 22, no. 9, pp. 1467–1471, September 2015. [4] X. L. Li, G. L. Cui, L. J. Kong, W. Yi, X. B. Yang, and J. J. Wu, “High speed maneuvering target detection based on joint keystone transform and CP function,” in Proceedings of 2014 IEEE Radar Conference, May 2014, pp. 0436–0440. [5] M. A. Richards, “Coherent integration loss due to white gaussian phase noise,” IEEE Signal Processing Letters, vol. 10, no. 7, pp. 208–210, July 2003.
[6] J. Yu, J. Xu, and Y. N. Peng, “Upper bound of coherent integration loss for symmetrically distributed phase noise,” IEEE Signal Processing Letters, vol. 15, pp. 661–664, November 2008. [7] X. L. Li, G. L. Cui, W. Yi, and L. J. Kong, “A fast maneuvering target motion parameters estimation algorithm based on ACCF,” IEEE Signal Processing Letters, vol. 22, no. 3, pp. 270–274, March 2015. [8] R. P. Perry, R. C. Dipietro, and R. L. Fante, “SAR imaging of moving targets,” IEEE Transactions on Aerospace and Electronic Systems, vol. 35, no. 1, pp. 188–200, January 1999. [9] S. J. Yuan, T. Wu, M. Mao, G. J. Mei, and X. Wei, “Application research of keystone transform in weak high-speed target detection in lowPRF narrowband chirp radar,” in Proceedings of 2008 9th International Conference on Signal Processing, Beijing, China, October 2008, pp. 2452–2456. [10] S. S. Zhang, T. Zeng, T. Long, and H. P. Yuan, “Dim target detection based on keystone transform,” in Proceedings of 2005 IEEE International Radar Conference, May 2005, pp. 889–894. [11] J. Xu, J. Yu, Y. N. Peng, and X. G. Xia, “Radon-fourier transform (RFT) for radar target detection (I): generalized doppler filter bank processing,” IEEE Transactions on Aerospace and Electronic Systems, vol. 47, no. 2, pp. 1186–1202, April 2011. [12] ——, “Radon-fourier transform (RFT) for radar target detection (II): blind speed sidelobe suppression,” IEEE Transactions on Aerospace and Electronic Systems, vol. 47, no. 4, pp. 2473–2489, October 2011. [13] J. Yu, J. Xu, Y. N. Peng, and X. G. Xia, “Radon-fourier transform (RFT) for radar target detection (III): optimality and fast implementations,” IEEE Transactions on Aerospace and Electronic Systems, vol. 48, no. 2, pp. 991–1004, April 2012. [14] M. D. Xing, J. H. Su, G. Y. Wang, and Z. Bao, “New parameter estimation and detection algorithm for high speed small target,” IEEE Transactions on Aerospace and Electronic Systems, vol. 47, no. 1, pp. 214–224, January 2011. [15] X. L. Chen, J. Guan, N. B. Liu, and Y. He, “Maneuvering target detection via Radon-Fractional Fourier transform-based long-time coherent integration,” IEEE Transactions on Signal Processing, vol. 62, no. 4, pp. 939–953, February 2014. [16] M. R. Sharif and S. S. Abeysekera, “Efficient wideband signal parameter estimation using a radon-ambiguity transform slice,” IEEE Transactions on Aerospace and Electronic Systems, vol. 43, no. 2, pp. 673–688, April 2007. [17] ——, “Efficient wideband sonar parameter estimation using a single slice of radon-ambiguity transform,” in Proceedings of IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 5, March 2005, pp. 605–608. [18] E. J. Kelly and R. P. Wishner, “Matched-filter theory for high-velocity, accelerating targets,” IEEE Transactions on Military Electronics, vol. 9, no. 1, pp. 56–69, January 1965. [19] H. B. Sun, G. S. Liu, H. Gu, and W. M. Su, “Application of the Fractional Fourier transform to moving target detection in airborne SAR,” IEEE Transactions on Aerospace and Electronic Systems, vol. 38, no. 4, pp. 1416–1424, October 2002. [20] S. C. Pei and J. Ding, “Fractional Fourier transform, wigner distribution, and filter design for stationary and nonstationary random processes,” IEEE Transactions on Signal Processing, vol. 58, no. 8, pp. 4079–4092, August 2010. [21] L. J. Kong, X. L. Li, G. L. Cui, W. Yi, and Y. C. Yang, “Coherent integration algorithm for a maneuvering target with high-order range migration,” IEEE Transactions on Signal Processing, accept. [22] X. L. Li, G. L. Cui, W. Yi, L. J. Kong, and J. Y. Yang, “Range migration correction for maneuvering target based on generalized keystone transform,” in Proceedings of 2015 IEEE Radar Conference, May 2015, pp. 0095–0099. [23] X. L. Li, G. L. Cui, W. Yi, L. J. Kong, J. Y. Yang, and Y. L. Huang, “A fast non-searching method for maneuvering target detection based on ACCF,” in Proceedings of 2015 IEEE Radar Conference, May 2015, pp. 0011–0014.
