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A Fast Maneuvering Target Motion Parameters Estimation Algorithm Based on ACCF Xiaolong Li, Guolong Cui, Wei Yi, and Lingjiang Kong
Abstract—This letter considers the motion parameters estimation problem for a maneuvering target with arbitrary parameterized motion. The slant range of the target is modeled as a polynomial function in terms of its multiple motion parameters and a fast estimation method based on adjacent cross correlation function (ACCF) is proposed, where the iterative adjacent cross correlation operation is employed to remove the range migration and reduce the order of Doppler frequency migration. Then the motion parameters are estimated via Fourier transform. Compared with the generalized Radon Fourier transform (GRFT), the proposed method can estimate the parameters without searching procedure and acquire close estimation performance at high signal-to-noise ratio (SNR) with a much lower computational cost. Finally, simulations are provided to demonstrate the effectiveness. Index Terms—Adjacent cross correlation function, maneuvering target, motion parameters estimation, range migration.
I. INTRODUCTION
W
ITH the development of science technology, especially the highly maneuvering target and high-resolution imaging technique, there is a growing need for high precision motion estimation of maneuvering targets with complex motions in radar society [1]–[4]. The accurate motion parameters estimation of moving targets affects the detection and imaging of targets significantly [5]–[7]. It is known that the coherent integration can increase the signal-to-noise-ratio (SNR) and thus improve the parameters estimation accuracy of targets [8]–[10]. Unfortunately, the complex motions of maneuvering targets, e.g., high velocity, acceleration and jerk, involves the range migration (RM) and the Doppler frequency migration (DFM) within one coherent pulse interval, which result in serious performance loss for the coherent integration processing. Some typical estimation algorithms, such as keystone transform (KT) [11], [12] and Radon Fourier transform (RFT) [13],
Manuscript received August 04, 2014; revised September 07, 2014; accepted September 10, 2014. Date of publication September 16, 2014; date of current version September 23, 2014. This work was supported by the National Natural Science Foundation of China under Grants 61201276, 61178068, and 61301266, the Fundamental Research Funds of Central Universities under Grants ZYGX2012Z001 and ZYGX2013J012, the Chinese Postdoctoral Science Foundation under Grant 2014M550465, and by the Program for New Century Excellent Talents in University under Grant A1098524023901001063. The associate editor coordinating the review of this manuscript and approving it for publication was Prof. X. Li, G. Cui, W. Yi, and L. Kong are with the School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu, China (e-mail:
[email protected];
[email protected];
[email protected];
[email protected]). Color versions of one or more of the figures in this paper are available online at http://ieeexplore.ieee.org. Digital Object Identifier 10.1109/LSP.2014.2358230
[14], were presented for the maneuvering target with RM. Nevertheless, the DFM induced by the target’s acceleration cannot be mitigated by the KT or RFT and thus it will suffer performance loss in case of the DFM. To remove the DFM and obtain the motion parameters estimation of a target with acceleration, the Radon transform with minimum entropy criterion was studied [15]. In addition, the Radon-fractional Fourier transform was proposed to eliminate the DFM and achieve the motion parameters estimation [16]. However, the radial jerk of target is not taken into consideration. For the maneuvering targets with complex motions, it is highly possible that the radial jerk exists and then the algorithms aforementioned will not be appropriate. Also, many literatures have discussed motion estimation of a maneuvering target based on polynomial phase signal model, such as cubic phase function [17], [18] and discrete polynomial-phase transform [19]. Unfortunately, the RM effect of the maneuvering target is not considered and thus they would become invalid in the case of RM. The generalized Radon Fourier transform (GRFT) was then presented for motion estimation of a maneuvering target with arbitrary parameterized motion [13], [20]. Although GRFT can obtain parameters estimation via searching in motion parameter space, it has two drawbacks. First, the blind speed sidelobe (BSSL) will appear in the GRFT, which brings about estimation performance loss [21]. Second, it is often computationally prohibitive because of the multi-dimensional searching. An improved particle swarm optimization (PSO) was introduced to reduce the computational burden of GRFT [22]. However, the PSO is sensitive to the initial parameters set and easily gets trapped in the local optimization. This letter considers the motion parameters estimation problem for a maneuvering target with arbitrary parameterized motion. First, the slant range of the maneuvering target is model as a polynomial function in terms of its multiple motion parameters. Then a fast non-searching motion estimation method based on adjacent cross correlation function (ACCF) is proposed, where the iterative adjacent cross correlation operation is applied to eliminate the RM effect and reduce the order of DFM. After that, the motion parameters are estimated via Fourier transform (FT). The rest is organized as follows. In Section II the signal model is presented. In Section III the motion parameters estimation algorithm is proposed. In Section IV several numerical experiments are provided. In Section V conclusions are given. II. SIGNAL MODEL Suppose that the radar transmits a linear frequency modulated (LFM) signal, i.e.,
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(1)
LI et al.: FAST MANEUVERING TARGET MOTION PARAMETERS ESTIMATION ALGORITHM BASED ON ACCF
where
,
is the pulsewidth,
de-
notes the carrier frequency, is the FM rate, is the fast time, is the number of radar denotes pulses, is the pulse repetition interval, and the slow time. The received baseband signal of a moving target can be expressed as [23]
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Substituting (7) and (8) into (6), we have
(9)
where (10)
(2)
Second, the ACCF of
denotes the target reflectivity, is the slant range where between target and radar at slow time and is the light speed. is modeled as a polynoThe instantaneous slant range mial function in terms of the target’s motion parameters [20], i.e.
can be expressed as
(11)
(3) is the initial slant range from the radar to the target, denote the target’s motion parameters with different orders as velocity, acceleration, , respectively. Assume . that Substituting (3) into (2) yields where
where Similarly, after the ACCF of
. times adjacent cross correlation operation, can be written as
(4) III. PARAMETERS ESTIMATION ALGORITHM VIA ACCF where
A. Property of ACCF
(12)
First, the ACCF of signal (4) can be stated as follows [23] (5) Note that the correlation of time domain signals is equal to the inverse Fourier transform (IFT) of the conjugate multiplication of their corresponding frequency domain signals, i.e., (6) where
(7)
(13) • From (12), it can be seen that both the envelope of the sinc function and the phase of the exponential term are order functions of slow time . With the increase of , the order of the envelope migration and exponential term’s phase reduces. , and • Although the changes of are different with respect to the slow time , the peak of can be considered to be located in the same range cell as long as the shift of the target between the adjacent time is less than a range cell, i.e., the following condition should be satisfied: (14)
(8) and over
,
denote respectively the FT over is the bandwidth of the transmitted signal.
and IFT
and denote respectively the radar pulse repewhere tition frequency and the sampling rate. Generally speaking the condition shown in (14) can be satisfied. (We only consider the situation that (14) is satisfied in this letter.) • From (13), we can see that , and are directly determined by , and . If we obtain the estimations of
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of vice-versa.
, and , and
, then the estimations can be achieved, and
B. Algorithm Description Note that after tion, the ACCF
times adjacent cross correlation operais
Applying FT on (15) with respect to
yields
(15) Fig. 1. RM correction. (a) Raw data. (b) Result after compression. (c) Result of adjacent cross correlation. (d) FT output.
via (13). Then construct the phase compensation function as follows
(16) . The parameter can where . Then, be estimated via the peak location of can be achieved by (13). Using the the estimation of to construct the phase compensation function estimated . with yields Multiplying
(17) Applying FT on (17) with respect to
, we have
(19)
Step 4) Multiplying
with (19), we have
(20) Step 5) Applying FT on (20) with respect to
yields
(18)
(21)
. By (18), can be where . Based on estimated via the peak location of and , we can obtain the estimations the estimated , and via ACCF iteratively, the implementation of of which is described in detail as follows. Step 1) Obtain , and , via adjacent cross correlation operation iteratively. Step 2) Applying FT on with respect to and obtain the estimation of based on the peak location of , . , Step 3) With the estimated we can achieve the estimations of
where . via the peak loStep 6) Achieve the estimation of cation of . Hence, the estimations , and are all obof tained. . Step 7) Repeat the step 3 to step 6 if . Then we can , and . achieve the estimations of Using the estimated , and to construct the phase compensation function, i.e., (22)
LI et al.: FAST MANEUVERING TARGET MOTION PARAMETERS ESTIMATION ALGORITHM BASED ON ACCF
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Fig. 2. Estimation MSEs versus SNR. (a) Target jerk estimation MSE versus SNR. (b) Target acceleration estimation MSE versus SNR. (c) Target velocity estimation MSE versus SNR.
Step 8) Perform FT on to , we have
shown in (9) with respect
TABLE I SIMULATION PARAMETERS OF RADAR AND THE MOVING TARGET
(23) Multiplying
with
yields (24)
Step 9) Taking IFT to (24) with respect to respect to , respectively, we have
and FT with
(25) can be estimated by the peak The parameter . As a result, the estimations location of , and are all achieved. of Based on the relationships shown in (10), the motion parameters of the maneuvering target, i.e., , and , can be obtained according to (26) where
, ,
.. .
.. .
..
.
.. .
(27)
C. Computational Cost Denote the number of range cells, searching motion paby , , respectively. rameter Then the computational cost of the presented method is . On the other hand, the computational cost will be
for
GRFT [20]. Suppose that , then the computational cost of the proposed method is , which is linear to the target’s motion order; whereas the com, which is exponential putational burden of GRFT is
to the target’s motion order. Therefore, the proposed method requires a much lower computational load than GRFT for the . target with motion-order higher than acceleration, i.e., IV. NUMERICAL RESULTS This section is devoted to evaluating the performance of the proposed method via computer simulations, where the parameters of radar and a moving target are shown in Table I. With these parameters, the condition shown in (14) is satisfied. We first evaluate the RM correction performance of the adjacent cross correlation operation in Fig. 1. Fig. 1(a) shows the received echoes of the maneuvering target, which are blurry in dB. The pulse compression result is the noise with given in Fig. 1(b). Due to 20 dB SNR gain of the pulse compression, the RM is clearly shown in this figure. Fig. 1(c) shows the envelope of the ACCF of the received data, where the RM is corrected. After two adjacent cross correlation operations, the FT result is given in Fig. 1(d). Based on the peak location, the jerk of target can be obtained. We also compare the motion parameters estimation performance of the proposed method and GRFT method via Monte Carlo trials in Fig. 2, where the SNR for the signal was varied in [ dB, dB]. In order to facilitate fair comparison, the SNR is defined before pulse compression for the proposed method while it is defined after pulse compression for GRFT. For each SNR value, 500 simulations were performed, and the measured mean square errors (MSEs) for the parameters estimation are shown in Fig. 2. The MSEs of the presented method are in general close to those of GRFT method at high SNR. V. CONCLUSIONS This letter has presented a motion parameters estimation algorithm based on ACCF for a maneuvering target with arbitrary parameterized motion. The characteristics of the proposed method include the following: 1) It is a nonsearching method; 2) adjacent cross correlation operation removes the RM and reduces the order of DFM; 3) the computational cost is linear to the target’s motion order; 4) it can acquire close estimation performance at high SNR with a much lower computational cost in comparison with the GRFT. Simulations demonstrated the effectiveness of the presented method. An important element of future work is the extension of the proposed method to the scenario with multiple targets [24].
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