High Speed Maneuvering Target Detection Based on Joint Keystone Transform and CP Function Xiaolong Li, Guolong Cui, Lingjiang Kong, Wei Yi, Xiaobo Yang, Junjie Wu 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 detection problem for high speed maneuvering target with a constant jerk, and proposes a new coherent integration detection method involves Keystone transform (KT) and cubic phase function (CPF). Specifically, it employs KT to correct the effect of the range migration. Then, the estimates of the acceleration and jerk with CPF are obtained to compensate Doppler frequency migration. Finally, Fourier Transform (FT) is conducted to integrate coherently the target returns. Compared with the third-order RFT algorithm [8, 10], the proposed method can avoid the problem of blind speed lobe (BSSL) and has a much lower computational complexity under the premise that the two methods have similar coherent integration performance. We evaluate the performance via several numerical experiments.
I.
INTRODUCTION
With the increasing exploitation of near space resources, near space target detection is receiving a growing attention and significant research efforts in the field of radar [1-3]. It becomes more challenging for the radar target detection technology because these high speed targets have low radar cross section (RCS) and complex motions, e.g., acceleration and jerk. It is known coherent integration is an effective method to improve radar detection performance [8-10]. With the increase in coherent integration time, the signal-to-noise ratio (SNR), as well as ultimate target detection performance, can be remarkably improved. Unfortunately, for the high speed maneuvering target, the long coherent integration time would result in the occurrence of the range migration (RM) and Doppler frequency migration (DFM), which limit the performance of the traditional FFT coherent integration method. Therefore, it is necessary to deal with the RM effect and DFM effect of moving targets in order to achieve long time coherent integration for high speed maneuvering targets. Perry et al. [4] have proposed a method named Keystone transform (KT) to correct RM over the integration time in synthetic aperture radar (SAR) imaging. Yuan et al. [5] introduced an algorithm based on KT for high-speed target detection. Xing et al. [6] proposed a method based on Radon
transform and entropy criterion for high speed small target detection. Tao et al. [7] proposed a method based on the scaling processing and the fractional Fourier transform to remove RM and DFM. However, the radial jerk of the target is not taken into consideration in [4-7], and these algorithms may not be appropriate for the target with a jerk. Xu et al. [8-10] proposed a coherent detection method-Radon Fourier transform (RFT), which realizes the echo spatial temporal decoupling via searching in the motion parameters space. But for the high maneuvering target with a jerk, a huge calculation load is need for four-dimension joint searching of range, velocity, acceleration and jerk. Besides, due to the limited integration time and limited range resolution, the blind speed side lobes (BSSL) with high peak values may exist in the RFT output [8]. In this paper, a new method is proposed for coherent integration detection of high speed maneuvering target with a jerk. In this method, KT is used to correct the RM. Then the quadratic phase term and CP term are estimated and compensated by the CPF [11-12]. Thus, the DFM is corrected. Finally, coherent integration result is obtained via FT. Compared with the third-order RFT algorithm [8, 10], the computational burden of the proposed method is reduced under the condition that they have similar coherent integration performance. Besides, the blind speed side lobe (BSSL) phenomenon would not appear in the proposed method. The remainder of this paper is organized as follows. In Section II the signal model is established. In Section III coherent integration method is proposed. In Section IV several numerical experiments are provided. In Section V conclusions are given. II.
PROBLEM FORMULATION AND SIGNAL MODEL
Consider the radar transmitting a narrowband linear modulation frequency signal with bandwidth B and carrier frequency f c , i.e., tˆ s ( tˆ, tn ) = rect T p
2 exp ( jπµ tˆ ) exp ( j 2π f c t )
This work was supported the National Natural Science Foundation of China (61301266, 61178068 and 61201276), Fundamental Research Funds of Central Universities (ZYGX2012Z001, ZYGX2013J012) and Program for New Century Excellent Talents in University (A1098524023901001063).
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(1)
where Tp , T and µ = B Tp are respectively denotes the pulse duration, the pulse repetition time and the frequency modulated rate. tˆ = t − nT is the fast time, tn = nT represents the slow time. Assume that the slant range between the radar and a high speed maneuvering target with a constant jerk is as follows [8-10] R ( tn ) = R0 + a1tn + a2 tn2 + a3tn3
(2)
where R0 is the initial slant range, a1 , a2 , a3 are the velocity, acceleration and jerk of the target along slant range at tn = 0 , respectively.
Under the assumption that a1 c 1 , c is the speed of light, then the received baseband signal can be expressed as [6, 13, 14] 2 R ( tn ) tˆ − c exp − j 4π f c R ( tn ) sr ( tˆ, tn ) = A0 rect Tp c 2 2 R ( tn ) 4π f c a1tˆ × exp jπµ tˆ − exp − j c c
(3)
Then after pulse compression, the compressed signal can be written as the following [6]: (4)
Equation (4) shows that the target envelope changes with the slow time after pulse compression. In order to remove the effect of target motion and realize coherent integration of target energy, a Fourier transform (FT) to the variable tˆ is employed to transform the compressed signal in (4) into the range frequency domain, which can be written as 4π ( f + f d + f c ) R ( tn ) exp − j c
(5)
exp ( jφ0 )
(6)
where φ0 = − 4π f c (1 + γ ) R0 c , φ1 = − 4π f c (1 + γ ) a1tn c , ,
φ3 = − 4π f c (1 + γ ) a3tn3 c
γ = ( fr + fd ) fc .
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Substitute (7) into (6) and combine λ = c f c , then (6) can be expressed as 2R f S ( f , tn ) = A3 exp − j 2π d + 0 f exp ( − j 2π (1 + γ ) f d tn ) c µ (8) 2 4π (1 + γ ) a2 tn 4π (1 + γ ) a3tn3 × exp − j exp − j λ λ
For the high speed movement and low radar pulse repetition frequency, there would be much severe Doppler ambiguity. In this situation, the target’s true Doppler frequency can be expressed as [5-6] (9)
where f a is the ambiguous Doppler frequency, nk is the fold factor and f p denotes the radar pulse repetition frequency. Substituting (9) into (8), we can obtain 2R f S ( f , tn ) = A3 exp − j 2π d + 0 f exp ( − j 2π (1 + γ ) f a tn ) c µ × exp ( − j 2π nk f p tn ) exp ( − j 2πγ nk f p tn ) 4π (1 + γ ) a2 tn2 × exp − j λ
(10)
4π (1 + γ ) a3tn3 exp − j λ
2R f S ( f , tn ) = A3 exp − j 2π d + 0 f exp ( − j 2π (1 + γ ) f a tn ) (11) c µ 2 3 4π (1 + γ ) a2tn 4π (1 + γ ) a3tn × exp ( − j 2πγ nk f p tn ) exp − j exp − j λ λ
× exp ( jφ1 ) exp ( jφ2 ) exp ( jφ3 )
φ2 = − 4π f c (1 + γ ) a2 tn2 c
(7)
It is worth pointing out that in the third exponential term is a multiple of 2π . This term in (10) becomes exp ( − j 2π nk f p tn ) = 1 . So S ( f , tn ) can be written as
Substitute (2) into (5), then S ( f , tn ) can be rewritten as 2π ff d f S ( f , tn ) = A2 rect exp − j B µ
f fc
f d = f a + nk f p
where sin c ( x ) = sin (π x ) π x is the sinc function. f d = 2a1 λ is the true Doppler frequency of the target, and λ denotes the radar wavelength.
2π ff d f S ( f , tn ) = A2 rect exp − j µ B
γ=
4π f c R0 f where A3 = A2 rect exp − j . c B
where A0 denotes the target reflectivity.
2 R ( tn ) f d 4π ( f c + f d ) R ( tn ) sc ( tˆ, tn ) = A1 sin c B tˆ − − exp − j µ c c
The definition of γ in (6) shows that γ is a function of Doppler frequency, radar carrier frequency, and range frequency. The contribution of f d to γ can be neglected if the radar carrier frequency is very high, i.e., the ratio of the true Doppler frequency to the carrier frequency and the ratio of the range frequency to the radar carrier frequency are far less than 1. Under this assumption, γ can be simplified as
,
There are four exponential terms which vary with slow time and couple with the range frequency f in (11). The first one and the second one indicate the Doppler term because of the target’s radial velocity. The third one is the frequency modulation term induced by the target’s radial acceleration and the last term
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results from the radial jerk of the target. The CP term and quadratic phase term of tn will result in DFM which make the signal energy defocusing. And the first-order term of tn will result in RM. The RM and RFM will create difficulties during coherent integration of target energy. Therefore, it is necessary to correct the RM and DFM. III.
PROPOSED COHERENT INTEGRATION ALGORITHM
In this section, a new method based on joint KT and CP function is proposed to remove the RM and DFM of the target echo and then achieve the coherent integration of target energy.
A. Range Migration Correction To correct the range migration, the Keystone transform is applied to the received signal in (11) and then we can obtain new spectrum function as follows fc S KT ( f , tn ) = S f , tn f + fc
2R 4π a2 tn2 f = A3 exp − j 2π d + 0 f exp ( − j 2π f a tn ) exp − j c µ λ (1 + γ ) 4π a3tn3 × exp − j λ (1 + γ )2
(12)
(13)
According to (18), if the selected fold factor equals it true value, the peak time echoes fix at 2 R0 c + f d µ . In this way, the signal energy will concentrate in a range cell parallel to the slow time axis. Therefore, the signal energy that is accumulated with the integration along the slow axis as shown in (19) may be used to estimate the fold factor N
2
(19)
corresponding n′ is the estimation of the fold factor, i.e., nk = arg max E ( n′ )
(20)
n′
B. Doppler Frequency Migration Correction
(14)
Define a phase ambiguity function as follows: (15)
where n′ is the searching fold factor. After multiplying (15) with (14), and we can obtain 4π a2 tn2 2R f S KT ( f , tn ) = A3 exp − j 2π d + 0 f exp ( − j 2π f a tn ) exp − j λ (16) c µ 4π a3tn3 f × exp − j exp − j 2π ( nk − n′ ) f p tn λ fc
Then an inverse Fourier transform (IFT) is taken to the signal with variable f , we can obtain the time-domain signal as follows:
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(18)
When the energy E ( n′ ) reaches its maximum value, the
For the same reason, the γ in (12) also does not need to be considered [6], and then (12) can be simply written as
f H a ( n′ ) = exp j 2π n′f p tn fc
2R f sKT ( tˆ, tn ) = A4 sin c B tˆ − 0 − d exp ( − j 2π f a tn ) µ c 4π a3 3 4π a2 2 × exp − j tn exp − j tn λ λ
n =1
In the narrowband environment f f c , and we can have
2R f S KT ( f , tn ) = A3 exp − j 2π d + 0 f exp ( − j 2π f a tn ) c µ 4π a2 tn2 4π a3tn3 f × exp − j exp − j exp − j 2π nk f p tn fc λ λ
When the searching fold factor equals it true value, then (17) can be written as
E ( n′ ) = ∑ sKT ( tˆ, tn ; n′ )
f exp − j 2π nk f p tn f f + c
f f ≈ f + fc fc
2R ( nk − n′ ) f p f sKT ( tˆ, tn , n′ ) = A4 sin c B tˆ − 0 − d − tn µ c fc (17) 4π a3 3 4π a2 2 tn exp − j tn × exp ( − j 2π f a tn ) exp − j λ λ
In the preceding subsection we have remove the RM by Keystone transform and the peak of the target is located in a certain range cell as shown in (18). However, the DFM is still presented. The following we estimate the quadratic phase term and CP term of the signal in (18) and then compensate them. Thus, the DFM could be corrected. For simplicity only one range cell signal in (18) is considered, and the sinc term in (18) is a constant in one range cell. Therefore, we neglect the effect of the sinc term and then the selected signal can be expressed as 4π a3 3 4π a2 2 sKT ( tn ) = A4 exp ( − j 2π f a tn ) exp − j tn exp − j tn λ λ N −1 N −1 4π 2 3 = A4 exp − j ( b1tn + a2tn + a3tn ) , − 2 ≤ n ≤ 2 λ
(21)
where b1 = f a λ 2 . N is assumed to be odd. We rewrite sKT ( tn ) as sKT ( n ) , where tn = nT , and then sKT ( n ) could be expressed as 2 3 4π sKT ( n ) = A4 exp − j b1 ( nT ) + a2 ( nT ) + ( nT ) λ
(
)
(22)
The discrete-time CP function [11-12] for signal sKT ( n ) is given by
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( N −1)
CP ( n, Ω ) =
∑
m=0
2
4π sKT ( n + m ) sKT ( n − m ) exp j Ωm 2T 2 (23) λ
Substituting (22) into (23) yields 2 3 8π CP ( n, Ω ) = A42 exp − j b1nT + a2 ( nT ) + a3 ( nT ) λ
(
( N −1)
×
∑
2
m=0
(24)
4π exp − j ( 2 ( a2 + 3a3 nT ) − Ω ) m2T 2 λ
other hand, NMN a1 N a2 N a3 Mc and
Ω denotes the instantaneous frequency rate (IFR).
It is not hard not see that CP ( n, Ω ) peaks along the curve (25)
ˆ and Ω ˆ , at times n and n First, estimate Ω 1 2 1 2 ˆ = arg max CP ( n , Ω ) Ω 1 1
(26)
ˆ = arg max CP ( n , Ω ) Ω 2 2
(27)
Ω
(Suggested defaults for n1 and n2 are 0 and round ( 0.11N ) , respectively [12]). T ˆ , X = 2 6n1T . Ω 2 2 6n T 2 Then compute aˆ (i.e., the vector of a2 and a3 estimates) according to
Second, let aˆ = [ aˆ2
T ˆ = Ω ˆ aˆ3 ] , R 1
ˆ aˆ = X −1R
(28)
With the estimated motion parameter vector aˆ , the quadratic phase term and CP term of the signal in (18) can be compensated. Thus, we could achieve the signal energy coherent accumulation via Fourier transform (FT). C. Computational Complexity
In what follows, the computational complexity of major steps in the proposed method will be analyzed in terms of the number of operations, i.e., complex multiplications (Mc) and additions (Ac). Suppose that the coherent integration pulse number is N . Denote the number of range cell, searching velocity, searching acceleration, searching jerk, searching fold factor, and searching IFR M , N a1 , N a2 , N a3 N F , N Ω , respectively. For the RM correct, NM log 2 M + N 2 M + N F ( M log 2 M + 2 MN ) Mc and NM log 2 M 2 + N ( N − 1) M + N F ( M log 2 M 2 + 2 M ( N − 1) )
Ac are needed. As to acceleration and jerk estimation via CPF,
2
N a3 are
SIMULATION VERIFICATION
In this section, detailed numerical experiments are carried out to validate the effectiveness of the proposed method. The radar parameters are given as follows. Transmitting LFM pulse duration Tp = 100us , the pulse repetition time T = 5ms , bandwidth B = 1MHz , radar carrier frequency f c = 1.5GHz , sampling frequency f s = 1MHz . Assume that there is a point target moves toward to the radar. The motion parameters of the fast approaching target are 2 3 R0 = 600km , a1 = 3600m s , a2 = 20m s , a3 = 10m s . The pulse number N = 201 . The signal-to-noise ratio (SNR) of the target’s raw data is -8 dB. Fig.1 (a) shows the result after pulse compression and it can be seen serious range migration occurs due to the target’s high speed. Fig.1 (b) shows the result of fold factor searching, where the peak value indicates the estimated fold factor. Fig.1 (c) is the results after Keystone transform. It is clearly shown that the envelope of the target is concentrated in a certain range cell. Fig.1 (d) shows CP function results. From fig.1 (d), it is can be 0 ˆ = [39.9900 46.8900]T . Besides, X = 2 see that R 2 0.690 , T −1 ˆ therefore, aˆ = X R = [19.9950 10.0000] . The FT outputs after Doppler frequency compensation with the estimated parameters is given in Fig.1 (e) and it shows that the target energy is coherent integrated well with the proposed method. For comparison, the coherent integration result via third-order RFT method is also given in Fig.1 (f). It is can be seen that the BSSL phenomenon appears in the searching range-velocity plane. This simulation demonstrates that the proposed method is valid and does not have the problem of BSSL. 20 50
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A. Analysis of the Proposed Method
From the above equation, it is clear that if the Ω is known at two different time position, then a2 and a3 can be determined. The specification of the algorithm could be described as follows.
Ω
IV.
Fold factor
Ω = 2 ( a2 + 3a3 nT )
( N − 1) MN a N a
needed for the third-order RFT algorithm [8, 10]. It is worth to notice that only the computational complexity after pulse compression is considered for both of the cases. As is evident in the above analysis, the computational burden of the proposed method is sharply reduced compared with the third-order RFT algorithm.
Pulse number
where
)
3 N Ω M ( N + 1) Mc and N Ω M ( N + 1) Ac are needed. On the
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The proposed algorithm firstly corrects the RM of the target echo using the KT. Quadratic phase term and CP term of the target echo are then estimated and compensated by the CP function. As a result, the DFM is corrected. Finally, FT is used for energy integration and target detection. The proposed method has similar coherent integration performance to the third-order RFT algorithm, but does not have the problem of BSSL and has a lower computational burden. Numerical experiments also validate the effectiveness of the proposed method. REFERENCES [1]
[2]
(e)
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[3]
Fig.1. Target detection result of the proposed method (a) result after range compression (b) result of fold factor searching (c) result after Keystone transform (d) CP function output (e) coherent integration result of proposed method (f) coherent integration result of third-order RFT method
[4]
B. Coherent Integration Performance of the Proposed Method
[5]
One thousand times of Monte Carlo simulations are implemented for the proposed method and the third-order RFT algorithm with different SNRs (defined before range compression). The radar parameters and moving parameters of target are the same as those in Section IV.A.
[6]
[7] 1 0
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[9]
-4 The proposed mehtod Third-order RFT method
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[10]
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Fig.2. coherent integration performance curves
[12]
Fig.2 shows the coherent integration performance curves of the two methods. It is can be seen that the proposed method has the similar coherent integration performance as the third-order RFT method.
[13]
[14]
V.
CONCLUSION
This paper proposes a new coherent integration detection method for high speed maneuvering target with a constant jerk.
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