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In this paper, we first propose a novel multiaircraft formation. (MAF) signal model for narrowband coherent radar in a long coherent integration time (LCIT).
Multiaircraft Formation Identification for Narrowband Coherent Radar in a Long Coherent Integration Time

respectively. Subsequently, we propose to apply third-order polynomial Fourier transform (PFT), to accurately estimate the MAF parameters, which include aircraft number and different order motion parameters for each aircraft. The performance of third-order PFT is compared with the existing third-order polynomial generalized Radon transform (GRT) in time-frequency domain. It is shown that the third-order PFT has higher parameter estimation resolution than the third-order polynomial GRT, as well as higher estimation accuracies with some additional computational complexity. Finally, some numerical experiments are provided to demonstrate the effectiveness of the proposed MAF model and the parameter estimation method by using simulation and real measured data.

I. INTRODUCTION SHI-BAO PENG Tsinghua University Beijing, China JIA XU, Senior Member, IEEE Beijing Institute of Technology Beijing, China XIANG-GEN XIA, Fellow, IEEE University of Delaware Newark, DE, USA FENG LIU TENG LONG, Senior Member, IEEE Beijing Institute of Technology Beijing, China JIAN YANG, Senior Member, IEEE YING-NING PENG, Senior Member, IEEE Tsinghua University Beijing, China

In this paper, we first propose a novel multiaircraft formation (MAF) signal model for narrowband coherent radar in a long coherent integration time (LCIT). We then give a quantitative analysis on the phase order approximation of the MAF echoes according to the LCIT duration with a given set of target motion and radar system parameters. It is shown that these echoes in a certain LCIT can be well approximated by multiple chirp signals or multiple cubic phase signals for MAFs with rigid or nonrigid structures, Manuscript received June 7, 2014; revised November 28, 2014; released for publication February 28, 2015. DOI. No. 10.1109/TAES.2015.140427. Refereeing of this contribution was handled by R. Adve. This work was supported in part by the National Natural Science Foundation of China under Grant 6127139, by the China Ministry Research Foundation under Grants 9140A21050014HT05338 and 9140A07021012JW0101, and by the Postdoctor Foundation of China under Grant 2013M530626. Authors’ addresses: S.-B. Peng, J. Yang, Y.-N. Peng, Department of Electronic Engineering, Tsinghua University, Beijing 100084, China; J. Xu, F. Liu, T. Long, School of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China. E-mail: ([email protected]). X.-G. Xia, Department of Electrical and Computer Engineering, University of Delaware, Newark, DE 19716. C 2015 IEEE 0018-9251/15/$26.00 

As an all-day and all-weather sensor, radar has been widely used in many fields since the 1940s. Apart from conventional aircraft detection, a modern radar needs to obtain an aircraft’s information from echoes as much as possible, which includes range, velocity, orientation, shape, etc. [1–5]. In practice, multiple aircrafts may appear in a formation for real applications, which is called multiaircraft formation (MAF) in this paper. The unknown MAF parameters to be retrieved for radar include aircraft number and motion parameters for each aircraft. It is known that aircraft number can be retrieved from the peaks of high range resolution profile (HRRP) [2] for a modern wideband radar, where several Giga-Hertz (GHz) bandwidth can be used to obtain high range resolution of several centimeters. Nevertheless, the large bandwidth will increase the analog-to-digital (A/D) requirement, data storage, and algorithm implementation on the radar signal processor, as well as more demanded wideband properties on the radar T/R channels. Moreover, it is known that high range resolution will also bring about difficulties for target detection because the target’s backscattering energy will be distributed among several range units in a single pulse. Besides, the effect of across-range unit (ARU) [6, 16–19] with the increase of integration time may also cause the serious pulse-by-pulse integration difficulties for moving targets. If the above problems can be well solved, the target detection performance with a high range resolution may possibly approach and/or outperform that with a low range resolution with the consideration of target radar cross section (RCS) changes and high detection probability. Unfortunately, solutions of the above high resolution detection problems are still in development and have difficulties in real applications. Also, most existing air search radars are narrowband radar with only several Mega-Hertz (MHz) bandwidth with respect to tens of meters range resolution, and the aircraft number of MAF cannot be easily obtained from these low-resolution range profiles. With these reasons, in his paper we focus on narrowband radars for MAF. For most existing air search narrowband radars, pulse-by-pulse transmitting waveforms are coherent and the Doppler information of an aircraft can be retrieved from the echoes. Besides, the above ARU effect can also be remarkably alleviated due to the reduced range

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resolution. It is known that the Doppler filter bank, i.e., moving target detection (MTD), has been widely used for suppressing clutter and increasing signal-to-noise ratio (SNR) for coherent radars. However, the MTD target detection is normally accomplished in a single Doppler unit and an aircraft’s Doppler frequency should appear in the output of a single Doppler filter. This condition also limits the possible integration time for an arbitrary moving target via MTD processing, and effective methods in the long coherent integration time (LCIT) [7–20] have been widely investigated in recent years to overcome the above limitations. Specifically, LCIT in this paper refers to the integration time during which a target demonstrates a time-varied Doppler modulation and its echoes have obvious across Doppler unit (ADU) effect. A typical example is the over-the-horizon radar (OTHR) with extremely low range resolution on tens of kilometers and the phase-only information, i.e., the Doppler frequency, is used for target detection and motion estimation. MTD and estimation in LCIT always attract great attention and many papers have discussed MTD, estimation, and tracking for a narrowband coherent radar. For example, Barnum [7] proposed autoregressive spectral analysis techniques to improve Doppler resolution for ship detection in sky wave OTHR. Fante et al. [8] proposed the fluctuating and nonfluctuating target signal model for OTHR MTD. Howard et al. [9] demonstrated the effectiveness by using Wigner-Ville distribution (WVD) to detect an accelerating target in the OTHR and Chen et al. [10] proposed fractional Fourier transform for target detection under sea clutter in LCIT. Tao et al. [11] proposed some effective methods for range and Doppler compensation in LCIT for linear frequency modulation pulse compression radar. Wang et al. [12] used adaptive chirplet transform to adaptively reject clutter and realize the OTHR maneuvering target detection. Zhang et al. [13] proposed a high resolution time-frequency distribution method and Astola et al. [14] applied the bispectrum estimation of time-frequency analysis in OTHR and surveillance pulse-Doppler radar for multiple maneuvering targets detection, respectively. Thayaparan et al. proposed joint time-frequency analysis (JTFA) [15–17], local polynomial Fourier transform (LPFT) [18, 37–39], and S-method [19] to detect maneuvering air targets in sea clutter for a high-frequency radar and to remove the micro-Doppler effect for a target with micromotion [20–22]. Also, the keystone transform [23] and modified keystone transform method [24] have been introduced from synthetic aperture radar (SAR) ground moving target imaging into target detection in LCIT. Recently, we proposed the Radon-Fourier transform and generalized Radon-Fourier transform (RFT/GRFT) for detection of radar moving targets [25–29]. It was shown that the RFT is an effective method to overcome a target’s ARU, ADU, and even for across beam unit (ABU) effect [28] in LCIT. Furthermore, the maximum likelihood estimators (MLEs) were derived in [29] for motion estimation of a maneuvering target based on joint envelope 2122

and phase measurement, phase-only measurement, and envelope-only measurement, respectively. It is shown that the proposed MLEs in an additive white Gaussian noise (AWGN) environment are to search the maximums of the outputs of the GRFT, generalized Radon transform (GRT) and generalized Fourier transform (GFT), respectively. Most of the existing LCIT-based methods mainly focus on the single target detection, estimation, and tracking, and only a few are on MAF identification. In [30], Chen et al. proposed an MAF discrimination by JTFA. Nevertheless, the existing MAF signal models include many different types for identification, e.g., multiple component chirp signals (mcCS) or multiple component polynomial phase signals (mcPPS). Normally, the mcPPS model may be a generalized choice because an arbitrary time-varying Doppler frequency can be well approximated via a polynomial phase signal (PPS) as long as the polynomial order is high enough. Much of the literature [31–36] has discussed motion estimation of a maneuvering target based on the PPS model. The LPFT [18, 37–39] and product high-order ambiguity function (PHAF) [40, 41], as a high-order generalization of the short-time Fourier transform (STFT), has been developed and used for many different applications in recent years. However, they may be too complicated to be applied for PPS in real applications with too high orders of the phases. Therefore, it is important to establish a suitable signal model in a possible integration time to demonstrate the relationship among approximation order with parameters of a radar system and targets’ motions. In this paper, for a narrowband air search radar, the condition is given for the integration time during which an aircraft in the MAF can be modeled as a point target at first. Then, during the integration time, the MAF signal model is derived and the condition is further given for phase order approximation. In this paper, the possible MAFs are categorized into four types with rigid formation structure and nonrigid formation structure, respectively. The first is MAF with rigid structure and each target moves with the same uniform velocity. The second is MAF with rigid structure and each target moves with the same velocity and acceleration. The third is the MAF with nonrigid structure and each target moves with different uniform velocities. The fourth is the MAF with nonrigid structure and each target moves with different velocities and accelerations. Interestingly, it is shown that echoes of all four MAF types can be expressed as mcPPS with order lower than fourth order. The second part of this paper is to introduce an effective method for the MAF identification. It is known that the existing methods normally accomplish MAF identification via JTFA. PPS estimation resolution and robustness should be balanced for the conventional JTFA-based methods with the window width, and the cross-term interference of JTFA will cause false detections. Although some JTFA-based methods, e.g., those in [16, 17], LPFT [18], S-method [19], PHAF [40], local polynomial Wigner distribution [41], and

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compressed sensing methods [42], can significantly remove/reduce the cross-term interference and improve the computational efficiency, these methods will further be explored and applied in future studies. Also, the MAF information including aircraft number and motion parameters cannot be obtained easily in the time-frequency plane [16–19]. In [29], we have proven that GFT is an MLE, i.e., statistically optimal estimator, for maneuvering targets’ motion estimation with a given aircraft number. Nevertheless, the aircraft number is also unknown for an arbitrary MAF and it should be retrieved from echoes. Therefore, the resolution in certain parameter space may be more important for MAF identification among different signal components. Obviously, effective methods are needed for MAF identification with higher resolution and more automatic implementations. In this paper, we apply an effective method called third-order polynomial Fourier transform (PFT) to MAF identification based on the above signal model. For comparison of the identification performance, a method called third-order polynomial generalized Radon transform (GRT) is used via STFT. Meanwhile, the MAF aircraft number can be obtained as well as motion parameters of each target from the local maximums of third-order PFT or third-order polynomial GRT via an automatic watershed method. Numerical experiments of simulated and real measured data are provided to show that the third-order PFT has higher parameter resolution than third-order polynomial GRT with some additional computational burden. Generally speaking, it is shown that third-order PFT is not only an optimal estimator for motion parameters but also a better choice for MAF identification due to its higher resolution. The remainder of this paper is organized as follows. In Section II, MAF signal model and structure are derived and discussed for narrowband coherent radar in LCIT. In Section III, the methods of third-order PFT as well as third-order polynomial GRT are provided for comparison. In Section IV, some experimental results are provided based on simulated data and real data to demonstrate the effectiveness of the proposed MAF model and identification method. In Section V, some conclusions are drawn. II. MAF SIGNAL MODEL FOR NARROWBAND COHERENT RADAR IN LCIT A. MAF Scenario of Narrowband Coherent Radar

A two-dimensional (2D) geometry of an MAF scenario is shown in Fig. 1, where only two aircraft are used to depict the MAF for simplicity. It is assumed that there are unknown K aircraft in range unit R0 of a narrowband radar. In order to describe the targets in MAF clearly, three coordinates are used in this paper. The first is radar coordinate UOV and the radar is located at the coordinate origin O. The second is reference coordinate XoY, where |Oo| = R0 and | · | is the absolute value operator. The origin of reference coordinate is located at (0, R0 ) in UOV and the axis Y coincides with the axis V in UOV. The

Fig. 1. MAF geometry for narrowband radar.

third is the targets’ coordinate {x(k) o(k) y(k) , k = 1, · · · , K}, where o(k) is the k-th aircraft geometry center, K is the aircraft number of the MAF and the origin of the k-th targets’ coordination is located at (X0(k) , Y0(k) ) in XoY. For each target’s coordination, we have    (k)  X0  ≤ (R0 + R) θ/2

0 ≤ Y0(k) ≤ R

(1)

where θ is the azimuth beamwidth, R is the range resolution of the narrowband radar. The radar line of sight (LOS) angle is θ0(k) , corresponding to the angle ∠UOo(k) . The radial direction is denoted as the axis of Y and the tangential direction is the axis of X. In real applications, aircraft in an MAF cannot maneuver rapidly in order to keep MAF configuration and to avoid collisions among each other. Furthermore, the LCIT will not be longer than several seconds normally for MAF identification. Therefore, only 2D velocity and acceleration motion, i.e., the motions lower than third order, are considered in this paper for the targets in MAF. Furthermore, a target’s velocity is decomposed into two components, i.e., vx(k) and vy(k) , along radial direction and tangential direction, respectively. Also, a target’s acceleration is decomposed into two parts, i.e., ax(k) and ay(k) , along radial direction and tangential directions, respectively. Thus, an arbitrary scatterer P of the k-th target is located at (xp(k) , yp(k) )T in the k-th target coordinates, which corresponds to the location (X0(k) + xp(k) , Y0(k) + yp(k) )T in radar coordinates and (·)T is the transpose operator. Thus, the range versus t from the scatterer P to the radar can be expressed as   rp(k) (t) = |OP (t)| = Oo(k) (t) + o(k) P (t)

PENG ET AL.: MAF IDENTIFICATION FOR NARROWBAND COHERENT RADAR IN LCIT

(2) 2123

Due to |Oo(k) (t)|  |o(k) P(t)|, (2) can be approximated as T    rp(k) (t) ≈ Oo(k) (t) + i(k) · o(k) P (t) los (t) (k) (t) = rt(k) (t) + rp,r

where

(4)

(9)

During several seconds of LCIT, the range caused by a target’s translation can be expressed as rt(k) (t) 

2 2   R0 + Y0(k) + vy(k) (t) dt + X0(k) + vx(k) (t) dt = ≈ R0 +

(t) dt +

2

 +

vx(k)

(t) dt

/2R0

(10) The above Fennel approximation is satisfied under the condition of R0  Y0(k) . Because the vector o(k) P is (xp(k) , yp(k) )T , the range caused by a target’s rotation range can be expressed as   (k) (t) = yp(k) sin θ0(k) − θ (k) (t) rp,r   (11) + xp(k) cos θ0(k) − θ (k) (t) Equation (11) can be further simplified as (k) (t) = y˜p(k) cos θ (k) (t) − x˜p(k) sin θ (k) (t) rp,r

2124

cos θ0(k)



sin θ0(k)

xp(k) yp(k)

 (13)

Due to θ (k) → 0 in LCIT, we have sin θ (k) (t) ≈ θ (k) (t) = t, cos θ (k) (t) ≈ 1. Then, (12) can be further expressed as (14)



(6)

θ (k) (t) ≈ t

X0(k)

=

− sin θ0(k) cos θ0(k)

Substituting (14) and (10) into (3), the scatterer range in LCIT can be expressed as

(k) (t) is the equivalent rotation velocity generated by the target’s tangential motion. Furthermore, the rotation velocity (k) (t) of each target can be assumed to be identical and denoted by  in this paper. Thus, the MAF rotation velocity can be expressed as

vy(k)

(5)

rt(k) (t) is the time-varied range caused by the translation (k) motion of the k-th aircraft, and rp,r (t) is the time-varied range caused by the rotation motion of the scatterer P. i(k) los (t) is the LOS, which can be further expressed as     T (k) (k) (t) , sin θ0(k) − θ (k) (t) i(k) los (t) = cos θ0 − θ (7) where θ (k) (t) is the target rotation angle, which can be expressed as  (8) θ (k) (t) = vx(k) (t) dt/R0 = (k) (t) t





(k) (t) = y˜p(k) − x˜p(k) t rp,r

  (t) = Oo(k) (t)

 T (k) (t) = i(k) (t) · o(k) P (t) rp,r los

Y0(k) +

x˜p(k) y˜p(k)

(3)

 (k)  (k)   i(k) los (t) = Oo (t) / Oo (t) rt(k)

where

(12)

rp(k)

(t) =

vy(k) (t) dt 2  + X0(k) + vx(k) (t) dt /2R0 − x˜p(k) t (15)

Note that the constant component of the time-varied range has been omitted in (15) for simplicity. B. Signal Modeling for MAF of Narrowband Coherent Radar

It is assumed that a narrowband radar transmits an LFM coherent pulse sequence signal with carrier frequency fo , bandwidth B, and pulse duration Tp . Normally, the bandwidth B is only several MHz for the narrowband radar and the signal after the matched filtering can be expressed as P K        σp(k) sinc B tˆ − τp(k) (t) s tˆ, t = (k)

k=1 p=1

  × exp −j 2πf0 τp(k) (t)

(16)

where tˆ and t represent fast-time and slow-time, i.e., the sampling times in the pulse and across the pulses, respectively, P (k) is the scatterer number of the k-th aircraft, σp(k) and τp(k) (t) are the intensity and time delay of the p-th scatterer in the k-th target, which are module and phase of the scatterer reflected coefficient, respectively. The time delay can be expressed as τp(k) (t) =

2rp(k) (t) c

(k) (t) = τt(k) (t) + τp,r

(17)

where c is light speed, τt(k) (t) = 2rt(k) (t)/c and (k) (k) τp,r (t) = 2rp,r (t)/c are the time delays generated by the target’s translation motion and rotation motion, respectively. It is assumed that each aircraft in the MAF locates at the same range unit during LCIT due to the narrow bandwidth. Thus, τp(k) (t) of different scatterers can be represented with an identical τ (t) and (16) can be

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further simplified as      s tˆ, t = sinc B tˆ − τ (t) P K   (k)

×

  σp(k) exp −j 2πf0 τp(k) (t) (18)

k=1 p=1

As discussed above, the motions of the target in the MAF are modeled in this paper with their velocity and acceleration, and the k-th target’s velocity versus t can be expressed as (k) + ax(k) t vx(k) (t) = vx0 (k) vx0

(k) vy(k) (t) = vy0 + ay(k) t

(19)

(k) vy0

and are target’s initial tangential velocity where and radial velocity, respectively. Substituting (19) into (10), the time-varied translation range can be expressed as    (k) (k) (k) vy0 + ay(k) t dt rt (t) = R0 + Y0 +    2 (k) (k) (k) vx0 + ax t dt /2R0 (20) + X0 + Omitting the constant component, (20) can be further expanded as   (k) 2 ) (vx0 v (k) a (k) 1 (k) (k) (k) rt (t) = vy0 t + t 2 + x0 x t 3 ay + 2 R0 2R0  (k) 2 a X(k) v (k) X(k) a (k) + x t 4 + 0 x0 t + 0 x t 2 (21) 8R0 R0 2R0 Thus, the scatterers’ time-varied phase can be obtained as  4π  (k) (k) (t) φp(k) (t) = rt (t) + rpr  l = φ1(k) (t) + φ2(k) (t) + φ3(k) (t) + φ4(k) (t) (k) (k) (t) + φ1,c (t) + φ1,p

φ2(k) (t) =

2π l





(k) vy,0 t

2

(v (k) ) X(k) a (k) ay(k) + x0 + 0 x R0 R0

φ3(k) (t) =

φ4(k)

(23)  t2

(k) 2π vx0 ax(k) 3 t l R0

 2 π ax(k) 4 (t) = t 2l R0

(k) (t) = − φ1,p

(k) φ1,c

l

4π l

l

According to the frequency resolution theory [2], as (k) (k) long as phase difference |φ1,p (T ) − φ1,p (T )| ≥ 2π, the 1 2 p1 th scatterer and p2 th scatterer will contribute two resolvable peaks in frequency domain via Fourier transform. Therefore, the minimum integration time Tp,min for scatterer identification can be obtained as Tp,min =

xp(k) t

(k) (k) 4π X0 vx,0 4π (k) (t) = t≈ X0 t l R0 l

(26)

(27)

l

(k2 ) − φ1,c (T )| ≥ 2π, As long as phase difference the k1 th scatterer and k2 th scatterer will contribute two resolvable peaks in frequency domain via Fourier transform. Therefore, the needed minimum integration time Tc,min for target identification can be obtained as

Tc,min =

l

(32)

2X Normally, due to xp  X, we have Tp,min  Tc,min

(33)

Because rotation speed  is normally in the degree of milliradian per second for conventional aircrafts, the Tc,min is a value of approximate subsecond while the Tp,min may be tens of seconds. In most scenarios, the real LCIT with several seconds meets (34)

Tc,min < T < Tp,min

(k) (k) In this case, due to |φ1,p (T ) − φ1,p (T )|  2π for any 1 2 p1 th and p2 th scatterer in a same target, (22) may be approximated as   (k) (t) + φ1(k) (t) + φ2(k) (t) + φ3(k) (t) + φ4(k) (t) φp(k) (t) ≈ φ1,c (35) It means that each target in MAF contributes to one component and the component number is the aircraft number K. Then, the MAF signal model (18) can be rewritten as K         (k) (t) s tˆ, t = sinc B tˆ − τ (t) σ (k) (t) exp j φ1,c k=1

(28)

(30)

(k1 ) (T ) |φ1,c

(24)

(25)

l

2xp 

Also, it is assumed that the cross-range interval between two targets is X, and the phase difference in LCIT caused by two targets is   4π  (k)  XT (31) φ1,c (T ) =

(22)

where φ1(k) (t) =

where φ1(k) (t), φ2(k) (t), φ3(k) (t), φ4(k) (t) are the phase terms generated by the target translational motion modelled as (k) (k) (19), and φ1,p (t) and φ1,c (t) are the phase differences among different scatters and different components, respectively, generated by the target rotation velocity . It is assumed that the maximum cross-range among scatterers in the aircraft is xp , and the phase difference in LCIT caused by two scatterers is  4π    (k) xp T (29) φ1,p (T ) =

+ φ1(k)

PENG ET AL.: MAF IDENTIFICATION FOR NARROWBAND COHERENT RADAR IN LCIT

(t) +

φ2(k)

(t) + φ3(k) (t) + φ4(k) (t)

 (36) 2125

Assume σ (k) (t) ≈ σ (k) , (36) can be further rewritten as K         (k) (t) s tˆ, t = sinc B tˆ − τ (t) σ (k) exp j φ1,c k=1

+ φ1(k) (t) + φ2(k) (t) + φ3(k) (t) + φ4(k) (t)

f3(k) = 0

 (37)

The amplitude of (37) can be expressed as      sa tˆ, t tˆ=τ (t) = s tˆ, t tˆ=τ (t)  K     (k) (t) + φ1(k) (t) + φ2(k) (t) = σ (k) exp j φ1,c  k=1    (k) (k) (38) +φ3 (t) + φ4 (t)   It is seen from (38) that the MAF signal amplitude is that of a summation of K moving scatterers with constant backscattering coefficients and periodically changed phases for different targets. Thus, due to the phase interference among different components, the amplitude of MAF echoes will display “interference fringes,” i.e., fringes with periodically strong and weak distributions, as shown in Section V. It is shown from (37) that the MAF echo is a superposition of K-component PPS signals. Also, let’s rewrite the MAF signal versus t in the range unit R0 as s (t) =

K 

  (k) (t) + φ1(k) (t) + φ2(k) (t) σ (k) exp j φ1,c

k=1

+ φ3(k)

(t) +

φ4(k)

(t)

(39)

   σ (k) exp j 2π f1(k) t + f2(k) t 2 + f3(k) t 3 + f4(k) t 4

k=1

(40) where fl(k) , l = 1, 2, 3, 4, is the l-th order phase coefficient, which can be expressed as   (k) /l f1(k) = 2 X0(k)  + vy,0 ⎛ ⎜ f2(k) = ⎝ay(k) +

 2 (k) vx,0 R0

f3(k) =

f4(k) 2126

f4(k) = 0

(45)

s (t) =

K 

   σ (k) exp j 2π f1(k) t + f2(k) t 2

(46)

k=1

It is shown from (45) that the echo is a K-component chirp signals. Otherwise the MAF signal is a superposition of K-component fourth-order PPS when ax(k) = 0. Because when fourth-order phase term in LCIT satisfies |φ4(k) (T ) − φ4(k) (0)| ≤ π/2 it can be omitted for simplicity [43, 44]. Accordingly, the minimum integration time T4,min during which the fourth-order phase can be omitted can be obtained by 

 ! 4  (k) T4,min  π φ  ≤ ⇒ T4,min = !  (47) 4  "  4   (k)  2 2 f4  Normally, the LCIT for MAF identification satisfies the condition T < T4,min

(48)

That is, the fourth-order phase can be omitted for MAF echoes in a limited LCIT and echoes can be expressed as K 

   σ (k) exp j 2π f1(k) t + f2(k) t 2 + f3(k) t 3

k=1



s (t) K 

and

The MAF signal in (40) can be expressed as

s (t) =

and (39) can be denoted as

=

when each target in the MAF moves with uniform velocity and ax(k) = 0, we have

(41)

⎞ +

(k) (k) ax vx,0

lR0

 (k) 2 a = x 4lR0

X0(k) ax(k) ⎟ R0

⎠ /l

(42)

(43)

(44)

(49) This implies that an MAF signal can be modeled by a third-order multiple component PPS, i.e., the multiple component cubic phase signal (mcCPS), and the number of components is the number of aircrafts. Furthermore, the mcPPS can be degenerated into the mcCS when ax(k) = 0. C. MAF Classification Based on Different Formation Structures

In the real scenarios, the MAF will appear with different formation structures. Take the relative position among aircraft for consideration, MAF can be categorized into two types. The first is the formation with rigid structure, where all aircrafts strictly keep the constant relative position among each other. The second is the formation with nonrigid structure, where the relative positions among different aircrafts may be changed over the time. It is obvious that the rigid formation structure requires that all aircraft move identically, while the aircraft in nonrigid formation structure can have motions that are different from each other. MAF can be further divided into four types for narrowband coherent radar in LCIT as shown in Table I. Type I and II are the MAFs with rigid structure. For type I, different aircraft in MAF fly with the same uniform velocity without acceleration, and the signal can be modeled as mcCS with the same chirp rate. For type II, all

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TABLE I MAF Types with Different Formation Structures Structure

Rigid Structure

Nonrigid Structure

Type

I

III

II

IV

Motions

The same uniform velocity without acceleration

The same velocity and the same acceleration

Different velocities without acceleration

Different velocities and different accelerations

Models

mcCS with the same chirp rate

mcCPS with same chirp rate and cubic rate

mcCS with different chirp rates

mcCPS with different chirp rates and cubic rates Fig. 2. Parameter space of MAF via third-order PFT/GRT.

the aircraft in MAF fly with the same uniform velocity and the same acceleration, and the signal can be modeled as mcCPS with the same chirp rate and cubic rate. Notably, there may be a special case for MAF with rigid formation structure where the cross-range difference X = 0 holds for two targets, for which these two targets cannot be discriminated from each other because the mcCS have the same frequency history. Types III and IV are MAFs with the nonrigid structure. For type III, different aircraft in the formation fly with different uniform velocities without acceleration, and the signal can be represented as mcCS with different chirp rates. For type IV, different aircraft in the formation fly with different uniform velocities and different accelerations, and the echoes can be modeled as mcCPS with different chirp rates and cubic rates. III. MAF IDENTIFICATION USING THIRD-ORDER PFT

For the MAF identification, the most essential parameter is aircraft number K. Thus, the problem in this paper is to estimate the aircraft number based on the signal model as (46) and (47). For the aircraft number information retrieved, the key is to discriminate each target in the parameter domain. Thus, a high-resolution method is needed to obtain the aircraft number as well as motion parameters for each target. In this paper, an effective method, the third-order PFT, is applied to realize the MAF identification. It should be pointed out that although the MAF signal model can be divided into four types, they are usually PPS with order lower than fourth order. Therefore, we can use the same third-order PFT to accomplish the identification of the above four types of MAFs. It is true that the time-frequency methods such as LPFT [18], S-method [19], and PHAF [40, 41] can significantly improve the time-frequency distributions in Fig. 5 and Fig. 6. Normally, these mentioned methods are nonlinear time-frequency analysis methods, and they have made great improvements on the cross-term suppressing and time-frequency resolution based on S-methods, reassigned smooth, etc. They are proposed for the single-component PPS signal at first and generalized into the multicomponent PPS case via clean method [47] or other recursive methods. Although this is the case, it is

impossible for a quadratic distribution to completely avoid the cross terms. Nevertheless, the focus of this paper is on the MAF parameter estimation, i.e., the component number as well as the multiple order parameter estimation for multicomponent PPS via linear methods. It is critical to avoid any cross terms. This is the reason why we use linear methods and STFT + third-order GRT [45] is chosen on the JTFA plane for comparison with the proposed third-order PFT. A. Third-Order PFT Method

The third-order PFT method can be expressed as H (f1 , f2 , f3 )        =  s (t) exp −j 2π f1 t + f2 t 2 + f3 t 3 dt 

(50)

It is obvious that the echoes’ energy of the k-th target will be coherently integrated [45] and focused at the local maximum at (f1(k) , f2(k) , f3(k) ) in the three-dimensional parameter space as shown in Fig. 2. Thus, the aircraft number and target’s motion parameters can be determined from the number of local maximums as well as their locations. The third PFT has straightforward relationships with some existing methods. It is a special case of the GFT [29] or the LPFT [37] with certain phase function or window function, respectively. Also it is an order extended chirp-Fourier transform (CFT) [31, 32]. Subsequently, the flowchart of the third-order PFT method is given step-by-step as follows. Step 1: Give a parameter combination (f1 , f2 , f3 ) and generate a compensation function h(t) as    (51) h (t) = exp j 2π f1 t + f2 t 2 + f3 t 3 Step 2: Obtain the compensated signal as s (t) = s (t) h (t)

(52)

Step 3: Perform integration of the signal along t and extract the amplitude of the result as      (53) H (f1 , f2 , f3 ) =  s (t)dt 

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Step 4 : Repeat step 1 to step 3 to obtain all the H (f1 , f2 , f3 ) for f1 min < f1 < f1 max , f2 min < f2 < f2 max , and f3 min < f3 < f3 max , where fi min and fi max , i = 1, 2, 3 are the search bounds for the ith-order phase coefficient. Step 5: Search the number and locations of the local maximums in the four-dimensional space H (f1 , f2 , f3 ) to obtain the aircraft number as well as motion parameters for each aircraft. For comparison, the existing third-order polynomial GRT method can be expressed as G (f1 , f2 , f3 )    = |W (t, f )| δ f − f1 − 2f2 t − 3f3 t 2 dt

(54)

where W (t, f ) is a time-frequency distribution of s(t) like WVD, STFT, and so on. For the consideration of linear processing, such as PFT, the STFT is used in this paper as  (55) W (t, f ) = s (t)hw (t) exp(−j 2πf t)dt where hw (t) is a local time window with length Tw < T . The similar flowchart of GRT on the 2D time-frequency distribution as (53)–(57) can be found in [45] to obtain the aircraft number as well as motion parameters for each aircraft. B. Parameter Identification Resolutions of Third-Order PFT

In real applications, the aircraft number identification accuracy is closely related to the resolutions of each phase coefficient fi(k) , i = 1, 2, 3. For the third-order PFT method, parameter identification resolution can be obtained from the third-order ambiguity function as  χ (τ, f1 , f2 , f3 ) =

T /2

−T /2

s (t)s ∗ (t + τ )

   × exp −j 2π f1 t + f2 t 2 +f3 t 3 dt (56) The ambiguity function in terms of f1 can be expressed as  χ (0, f1 , 0, 0) = = =

T /2

−T /2  T /2 −T /2

s (t)s ∗ (t) exp {−j 2πf1 t} dt exp {−j 2πf1 t}dt

sin πf1 T = T sinc (πf1 T ) πf1

(57)

The resolution δf1 for f1 is the 3 dB width of the |χ(0, f1 , 0, 0)| and it can be expressed as δf1 = 2128

1 T

(58)

Fig. 3. |χ (0, 0, 0, f3 )| with T = 1.28s.

The ambiguity function about f2 can also be expressed as  T /2   χ (0, 0, f2 , 0) = s (t)s ∗ (t) exp −j 2πf2 t 2 dt −T /2 T /2

 =

−T /2

  exp −j 2πf2 t 2 dt

(59)

The exact expression cannot be obtained from (68) but the bounds can be expressed as #   0.25πT 2 sin c 0.25πT 2 f < |χ (0, 0, f2 , 0)| #   (60) < 0.5πT 2 sin c 0.5πT 2 f The proof of (60) can be found in the Appendix. Due to the integration time T > 1 in this paper, (60) can be refined as 2 4 < δf2 < 2 (61) 2 T T Also, the upper bound is selected for the f2 resolution in this paper, i.e., 4 (62) T2 The ambiguity function about f3 can be expressed as  T /2   χ (0, 0, 0, f3 ) = s (t)s ∗ (t) exp −j 2πf3 t 3 dt δf2 =

−T /2 T /2

 =

−T /2

  exp −j 2πf3 t 3 dt

(63)

The resolution δf3 for f3 is the 3 dB width of the |χ(0, 0, 0, f3 )|, but its analytic expression is difficult to be obtained. The value |χ(0, 0, 0, f3 )| is shown in Fig. 3 with the integration timeT = 1.28s. It is shown that δf3 can be determined by the numerical experiments from the fitted results versus T ∈ [0.64s, 10.24s], and the experimental resolution can be expressed as δf3 =

5.7 T3

(64)

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C. Parameters Bound Settings and Local Maximum Location Method

For the practical implementation of third-order PFT, the search bounds of different order phase coefficients, i.e., fi min and fi max , i = 1, 2, 3, should be determined beforehand. From the derived (40)–(42), the search bounds of f1 , f2 , f3 can be directly obtained according to the known system parameters and possible velocities and accelerations of real aircrafts. Therefore, the residual problem of the proposed flowchart in Section III-A is how to automatically locate the local maximum in the 3D parameter space of third-order PFT or third-order polynomial GRT, i.e., G(f1 , f2 , f3 ) and H (f1 , f2 , f3 ), respectively. In this paper, a simple automatic method called watershed algorithm [46, 47] is used. It is noted that the original watershed algorithm treats the 2D magnitude image as a three-dimensional (3D) landscape under the water. In this paper, this watershed method is generalized into a case of 3D images, i.e.,G(f1 , f2 , f3 ) and H (f1 , f2 , f3 ), and they are regarded as the four-dimensional (4D) landscape as well for the proposed method. It is known that the local maximals of landscape surfaces will appear when the water is drained and they are assigned as new segments. The segments will “grow” when the water is drained until they meet other segments. Therefore, each segment corresponds to a certain local maximal of the region. As a consequence, the revised 3D watershed algorithm is proposed to accomplish step 5 of the third-order PFT as well as the final step of the third-order polynomial GRT. D. Computational Complexity Analysis

In order to compare the computational loads of third-order PFT and third-order polynomial GRT, it is assumed that the LCIT is composed of M pulses. For the third-order PFT method, the computational load is mainly decided by two parts. The first is a loop operation including steps 2, 3, and 4. The second is step 5 to automatically locate the local maximums. Step 2 needs M times of complex multiplications (Mc). Step 3 needs M times of complex addition (Ac) and one absolute operation. As we know, one absolute operation needs 12 times of Mc. Thus, the first part needs L1 L2 L3 (M + 12) times of Mc and L1 L2 L3 M times of Ac, where L1 , L2 and L3 are the searching grid number of f1 , f2 , and f3 , respectively. The second part is to search the local maximum and the watershed method needs L1 L2 L3 times of comparison as well as 0.5L1 L2 L3 times of Ac for searching. For the third-order polynomial GRT method, the computation is mainly composed of three parts, the first is to obtain the STFT of W (t, f ), the second part is to obtain the matrix G(f1 , f2 , f3 ), and the third part is to automatically locate the local maximums. The first part is composed of M times of N-point FFT where N = Tw fs is the sampling point in a window Tw . It is known that one time weighted N-point fast Fourier transform (FFT) needs

TABLE II Computation Loads of Third-Order PFT/Third-Order Polynomial GRT Third-Order PFT Mc Ac Total loads in Ac

Third-Order Polynomial GRT

L1 L2 L3 (M + 12) M(0.5N log2 N + N) + 12MN 0.5L1 L2 L3 (M + 1) L1 L2 L3 (M + 0.5) L1 L2 L3 (4M + 36.5) 3M(0.5N log2 N + N) + 36MN +0.5L1 L2 L3 (M + 1)

Fig. 4. Computation ratio between third-order PFT and third-order polynomial GRT. TABLE III System Parameters used in the Simulation Experiment Parameters Value

Fc

B

Tp

PRF

Fs

700 MHz

2 MHz

200 us

400 Hz

4 MHz

N times complex multiplications (Mc) for windowing and N-point FFT. The N-point FFT needs 0.5Nlog2 N times of Mc and Nlog2 N times of Ac. Then, the image of time-frequency distribution can be obtained by an absolute operation, which needs MN times of absolute operation. It is known that one time complex absolute needs nearly 12 times of Mc operations. Thus, the first part needs M(0.5Nlog2 N + N) + 12MN times of Mc and MNlog2 N times of Ac. The second part is a loop operation to detect a curve in a 3D plane. For the L1 × L2 × L3 3D search grids, M times of real additions (Ar) are needed for each grid. Thus, the second part needs 0.5L1 L2 L3 M of Ac. The third part is to search the local maximum of G(f1 , f2 , f3 ) by the watershed method, which needs 0.5L1 L2 L3 times of Ac. For comparison, the Mc operation is transformed into the Ac operation. Normally, one Mc operation is equal to four real multiplications (Mr) and two Ar. In the digital signal processing (DSP)-based real-time system, the Mr and Ar operation are all accomplished in a single machine cycle. Therefore, the one Mc operation is equal to six Ar operations or three Ac operations. The total loads in Ac of third-order PFT and third-order polynomial GRT are shown in Table II, and the computation load ratio between third-order PFT and third-order polynomial GRT can be obtained, accordingly. For comparison, some typical parameters are given as N = M/8, L1 = 1000,L2 = 500 and L3 = 500 and M changes from 512 to 8192. The

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TABLE IV MAF Parameters used in the Simulation Experiment Formation Target R0 (km) X0 (m) Y0 (m) vx,0 (m/s) vy,0 (m/s) ax (m/s2 ) ay (m/s2 ) Tp,min (s) Tc,min (s) T4,mim (s)

1 i

2 ii

i

300 10 408 1 0 0 7.8782

0 0 408 1 68.6 1 7.8782



3.0578

60 0 0 408 1 0 0 7.8782

ii

i

300 10 408 1 68.6 1 7.8782

0 0 200 10 0 0 16.0714

3.0578



60

0.1050 ∞

3

4 ii

i

100 10 340 4 0 0 9.4538

0 0 408 8 58.8 0.5 7.8782



3.3028

60

0.1050

ii 60

0.3782

200 10 510 2 68.6 0 6.3025 0.1261 3.0578

computation load is shown in Fig. 4. It is found that the computational loads ratio between third-order PFT and third-order polynomial GRT is nearly eight times. Therefore, in the real application, some computational burden should be added for the third-order PFT. Fortunately, the complexity increase compared with GRT is mainly caused by the complex operations of PFT different from the real operations of GRT. Therefore, third-order PFT has the same complexity order as third-order polynomial GRT and it is easy for implementation via the modern processors like DSP, field-programmable gate array (FPGA), and so on due to the natural parallelism of 3D search. Furthermore, for the real applications of third-order PFT and third-order polynomial GRT, the heuristic search methods like the particle swarm optimizer (PSO) method can be used to effectively reduce the computation complexity [5, 48]. IV. NUMERIC EXPERIMENTS OF MAF PROPERTIES AND THE THIRD-ORDER PFT METHOD A. Experiments of MAF Signal Model Properties

1) Numerical Experiments Based on Simulation Data: It assumed that a narrowband coherent radar transmits an LFM signal as shown in Fig. 1, the system parameters are shown in Table III, where Fc , B, Tp , PRF, and Fs are the carrier frequency, bandwidth, duration, pulse repetition frequency, and sampling frequency, respectively. The integration time is T = 2.56s. Each MAF contains two aircraft labeled as i and ii, and the target geometry center location and motion parameters are shown in Table IV, where the definitions of parameters, such as R0 , X0 , Y0 , vx,0 , vy,0 , ax , and ay can be found in Fig. 1.The aircraft is composed of three scatterers with 2 m intervals along x axis and y axis, respectively. Thus, the two edge point interval along the x axis is xp = 4m, the Tp,min and Tc,min are calculated in Table IV according to (30) and (32), respectively. It is obvious that the integration time satisfies T < Tp,min . Thus, each target can be regarded as a point target during the integration time T = 2.56s. The SNR of the range profile is about 20dB after pulse compression. From Table IV, it is shown that MAF 1, 2130

Fig. 5. Simulated MAF echoes and their STFT results. (a) Type I range profiles. (b) STFT of Type I. (c) Type II range profiles. (d) STFT of Type II. (e) Type III range profiles. (f) STFT of Type III. (g) Type IV range profiles. (h) STFT of Type IV.

MAF 2 belong to Type I and Type II while MAF 3 and MAF 4 belong to Type III and Type IV, respectively. The four figures in the left column of Fig. 5 are the MAF range profiles after range compression and the right column are

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their STFT results, respectively. The window of STFT is Tw = 0.32s which is much smaller than the value of Tp min in Table V. From the STFT results, it is obvious that the STFT result of Type I has parallel lines with the same chirp rate. Type II has parabolas with the same chirp rate and cubic rate. Type III has lines with different chirp rates and Type IV has parabolas with different chirp rates and cubic rates. Besides, due to the phase interferometry among multiple components, all types of MAFs display the “interference fringes”, i.e., fringes with periodically dark and light distributions, as discussed in Section II. It is shown that the proposed signal model in Section II is an effective model to describe different MAFs. 2) Numerical Experiments Based on Real Data: The real data were obtained by an experimental prototype radar system with similar system parameters as Table IV. The MAF aircrafts flew with different structures and different aircraft numbers. The background of MAF echoes are noise and clutter free. A large amount of data was obtained and the results proved that the MAF can be modeled as the four types as shown in Table I. Furthermore, four typical formations are selected to demonstrate the effectiveness of the MAF classification given in Section III. The results are shown in Fig. 6. The figures in the left column are range profiles after range compression while those in the right column are their STFT results. The first two rows of figures are the results of the MAF with rigid structure as Type I, Type II, respectively. The third and fourth rows are the results of MAF with nonrigid structure as Type III and Type IV, respectively. Notably, there are some differences on the rates of Doppler change for Type III and Type IV, i.e., Fig. 5(g)–5(h) and Fig. 6(g)–6(h), for the simulated and measured data. The first reason is that we do not know the exact information of the MAF in the real data and cannot guarantee the completely identical JTFA distributions of the simulated data and real data. The second reason is that there are some real modulations caused by radar systems, such as beam patterns, that have been omitted for simulation. Therefore, the appearances of the simulated data and real data have some differences on the JTFA plane.

Fig. 6. Real MAF echoes and their STFT results. (a) Type I range profiles. (b) RS of Type I. (c) Type II range profiles. (d) STFT of Type II. (e) Type III range profiles. (f) STFT of Type III. (g) Type IV range profiles. (h) STFT of Type IV.

B. MAF Identification Experiments Based on Real Data

The real MAF data with two aircraft and their time-frequency distributions are shown in Fig. 7 to verify the third-order PFT. The peaks, i.e., local maximums, can be located by the proposed watershed method in G(f1 , f2 , f3 ) and H (f1 , f2 , f3 ) for third-order polynomial GRT and third-order PFT, respectively. Furthermore, the 2D slices of the peaks are used to show the identification results. For example, when there exists one peak located at (10, 20, 30) in G(f1 , f2 , f3 ), the first image gives the f2 − f3 slice as G(10, f2 , f3 ), the second image gives the f1 − f3 slice as G(f1 , 20, f3 ), and the third image gives the f1 − f2 slice as G(f1 , f2 , 30). The three slices of the third-order polynomial GRT are given in Fig. 8(a), Fig. 8(c), and Fig. 8(e), respectively. Also, the three slices

Fig. 7. STFT result of two-aircraft MAF echoes.

of the third-order PFT are provided in Fig. 8(b), Fig. 8(d), and Fig. 8(f), respectively. The peak located by the watershed method is marked by black ‘ + ’. There is only one local maximum located by the third-order polynomial GRT method at (119.12 m/s, 0.13 m/s2 , −1.46 m/s3 ). Thus, the aircraft number identified by third-order polynomial GRT is K = 1, which is a mistake due to the low resolution of third-order polynomial GRT. However, there are two local maximums for the third-order PFT method located at (118.34 m/s, 0.13 m/s2 , −1.46 m/s3 ) and (122.24 m/s, −5.97 m/s2 , 1.26 m/s3 ) from

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Fig. 8. Target discriminated results by third-order polynomial GRT and third-order PFT. (a) Third-order polynomial GRT in f2 − f3 plane. (b) Third-order PFT in f2 − f3 plane. (c) Third-order polynomial GRT in f1 − f3 plane. (d) Third-order PFT in f1 − f3 plane. (e) Third-order polynomial GRT in f1 − f2 plane. (f) Third-order PFT in f1 − f2 plane.

Fig. 9(a)–Fig. 9(c), respectively. The target locations are marked by black symbol ‘ + ’ as shown in Fig. 8. Therefore, the aircraft number K = 2. is estimated. In Fig. 9, the peaks’ normalized profiles of third-order polynomial GRT and third-order PFT are also shown along different 2D planes. Because two peaks of the third-order PFT versus certain parameters are obtained when the other parameters vary, the two peaks of the third-order PFT are represented with two curves in Fig. 9(a)–Fig. 9(c), respectively. Accordingly, the MAF number is estimated as K = 1 for the third-order polynomial GRT method and K = 2 can be determined for the third-order PFT method. Third-order PFT can identify two aircraft in each 2D slices while third-order polynomial GRT cannot. It is clear that the third-order PFT method has much higher parameter resolution than the third-order polynomial GRT method, which is essential for MAF identification. Besides, the accuracies of GFT have been well discussed in [29] for different order motion parameter estimation, which can approach the Cramer-Rao bounds (CRBs) in AWGN environment. That is, as a special case of GFT, third-order PFT is also a statistically optimal estimator with high accuracies than other existing methods. 2132

Fig. 9. Peaks’ normalized profiles of third-order polynomial GRT and third-order PFT. (a) Normalized profiles along f1 . (b) Normalized profiles alongf2 . (c) Normalized profiles alongf3 .

V. CONCLUSIONS

In this paper, MAF identification is investigated for narrowband coherent radar in LCIT. At first, the signal model is established by dividing MAFs into four types under the limitations of integration time and formation structures. Interestingly, it is shown that each type can be modeled as mcPPS with order lower than fourth order. Furthermore, an effective third-order PFT method is applied to obtain the aircraft number as well as motion parameters for each target. Also, an existing third-order polynomial GRT method is used for comparison. The performance analysis is given on parameter estimation resolution and computational complexity to show that third-order PFT has a higher parameter estimation

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We have

# $   | | > 0.25πT 2 sin c 0.25πT 2 f2

(70)

In the same way, the integration along the circumscribed circle can be expressed as  π  T /√2   r exp −j 2πf2 r 2 drdθ < −π 0  

sin π2 T 2 f2 j πT 2 f (71) = exp − f2 2 We have Fig. 10. Integration area of the ambiguity function in terms of f2 .

resolution than the existing methods with some additional computational complexity. Finally, some numerical experiments are also provided to demonstrate the effectiveness of the third-order PFT method by both simulated and real-measured data.

# $   | | < 0.5πT 2 sin c 0.5πT 2 f2

(72)

Therefore, the ambiguity function varied long f2 can be expressed as #   0.25πT 2 sin c 0.25πT 2 f < |χ (0, 0, f2 , 0)| #   (73) < 0.5πT 2 sin c 0.5πT 2 f

APPENDIX

The ambiguity function in terms of f2 can be expressed as  T /2   χ (0, 0, f2 , 0) = exp −j 2πf2 t 2 dt

REFERENCES [1]

(65)

−T /2

Let’s define

2  T /2   2 exp −j 2πf2 t dt =  =

−T /2 T /2

−T /2





exp −j 2πf2 x dx 2



[2]

[3]

T /2

−T /2

  exp −j 2πf2 y 2 dy

[4]

(66) [5]

We have χ (0, 0, f2 , 0) =

$

| |

(67)

The integration area in (66) is a square shown in Fig. 10. We define y  $ (68) r = x 2 + y 2 , θ = arctan x The integration in (66) can be approximated by using circle integration. We know that the maximum integration area is a circumscribed circle of the square and the circle can be expressed as r ∈ [0, T2 ], θ ∈ [−π, π]. The minimum integration area is an inscribed circle of the square and the circle can be expressed as r ∈ [0, √T2 ], θ ∈ [−π, π]. For the integration with the inscribed circle, (66) can be expressed as  π  T /2   > r exp −j 2πf2 r 2 drdθ −π 0  

sin π4 T 2 f2 j πT 2 f (69) exp − = f2 4

[6]

[7]

[8]

[9]

[10]

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[11]

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Shi-bao Peng was born in Jiangxi, China, in 1982. He received the B.S. and Ph.D. degrees from Radar Academy of Air Force, Wuhan, China, in 2004 and 2011, respectively. Currently, he is a postdoctor in the Electronic Engineering Department of Tsinghua University, Beijing, China. His current research interests are in the areas of SAR/ISAR, ground moving target indication, and target recognition. Jia Xu (M’05—SM’15) was born in Anhui Province, P. R. China, in 1974. He received the B.S. and M.S. degrees from Radar Academy of Air Force, Wuhan, China in 1995 and 1998, and the Ph.D. degree from Navy Engineering University, Wuhan, China, in 2001. He was an Associate Professor in the Radar Academy of Air Force during 2006–2009 and was an Associate Professor in Tsinghua University during 2009–2012. Currently, he is a full professor in the School of Information and Electronic, Beijing Institute of Technology (BIT), China. His current research interests include detection and estimation theory, SAR/ISAR imaging, target recognition, array signal processing, and adaptive signal processing. Dr. Xu has authored or coauthored more than 150 papers. He received the Outstanding Post-Doctor Honor of Tsinghua University in 2004 and the second-order National Invention Award in 2007. He was supported by the Outstanding Youth Teacher Training Plan of BIT in 2012. He was supported by the New Century Excellent Talents Supporting Plan of Ministry of Education of China in 2013. He is a Fellow of IET. He is a senior member of the Chinese Institute of Electronics (CIE) and member of the Academic Committee of CIE. He is Associate Editor of four Chinese journals all about radar and its applications. PENG ET AL.: MAF IDENTIFICATION FOR NARROWBAND COHERENT RADAR IN LCIT

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Xiang-Gen Xia (M’97—S’00—F’09) received his B.S. degree in mathematics from Nanjing Normal University, Nanjing, China, and his M.S. degree in mathematics from Nankai University, Tianjin, China, and his Ph.D. degree in electrical engineering from the University of Southern California, Los Angeles, in 1983, 1986, and 1992, respectively. He was a Senior/Research Staff Member at Hughes Research Laboratories, Malibu, CA, during 1995–1996. In Sept. 1996, he joined the Department of Electrical and Computer Engineering, University of Delaware, Newark, where he is the Charles Black Evans Professor. His current research interests include space-time coding, MIMO and OFDM systems, digital signal processing, and SAR and ISAR imaging. Dr. Xia is the author of the book, Modulated Coding for Intersymbol Interference Channels (Marcel Dekker, 2000). He received the National Science Foundation (NSF) Faculty Early Career Development (CAREER) Program Award in 1997, the Office of Naval Research (ONR) Young Investigator Award in 1998, and the Outstanding Overseas Young Investigator Award from the National Nature Science Foundation of China in 2001. He also received the Outstanding Junior Faculty Award of the Engineering School of the University of Delaware in 2001. He is currently serving and has served as an Associate Editor for numerous international journals including IEEE Transactions on Signal Processing, IEEE Transactions on Wireless Communications, IEEE Transactions on Mobile Computing, and IEEE Transactions on Vehicular Technology. He is Technical Program Chair of the Signal Processing Symposium, Globecom 2007 in Washington, D.C., and the General Co-Chair of ICASSP 2005 in Philadelphia.

Feng Liu was born in Beijing, P. R. China. in 1978. He received the B.S. and Ph.D. degrees from Beijing Institute of Technology (BIT) in 1999 and 2004, respectively. He became an Associate Professor of the School of Information and Electronics of BIT in 2011. He was the General Manager of Beijing Racobit Electronic Information Technology Co., Ltd. from 2009 to 2011. His research is mainly about real time digital signal processing with applications to radar and navigation systems.

Teng Long (M’10—SM’13) was born in Fujian Province, P. R. China in 1968. He received the B.S. degree from University of Science and Technology of China (USTC) in 1989, and M.S. and Ph.D. degrees from Beijing Institute of Technology (BIT) in 1991 and 1995. He became a full professor in the Electrical Engineering Department of BIT in 2000. Since 2008 he has been the Dean of the School of Information and Electronics. He has been a visiting scholar at Stanford University and University College of London in 1999 and 2002, respectively. His research is mainly about SAR systems and real-time digital signal processing, with applications to radar and communication systems. Dr. Long has authored or coauthored more than 200 papers. He is a fellow of the Institute of Electronic and Technology (IET) and Chinese Institute of Electronics (CIE). He has received many awards for his contributions to research and invention in China. 2136

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Jian Yang (M’98–SM’02) received the B.S. and M.S. degrees from Northwestern Polytechnical University, Xi‘an, China, in 1985 and 1990, respectively, and the Ph.D. degree from Niigata University, Niigata, Japan, in 1999. In 1985, he was with the Department of Applied Mathematics, Northwestern Polytechnical University. From 1999 to 2000, he was an Assistant Professor with Niigata University. Since April 2000, he has been with the Department of Electronic Engineering, Tsinghua University, Beijing, China, where he is currently a professor. His research interests include radar polarimetry, remote sensing, mathematical modeling, optimization in engineering, and fuzzy theory. Dr. Yang is the Chairman of the Institute of Electronics, Information, and Communication Engineers in the Beijing area and the Vice-Chairman of the IEEE Aerospace and Electronic Systems Society Beijing Chapter.

Ying-Ning Peng (M’93—SM’97) was born in Sichuan Province, P.R. China, in 1939. He received the B.S. and M.S. degrees from Tsinghua University, Beijing, China, in 1962 and 1965, respectively. Since 1993, he has been with the Department of Electronic Engineering, Tsinghua University, where he is now a Professor and Director of the Institute of Signal Detection and Processing. He has worked with real-time signal processing for many years. His recent research interests include processing, parallel signal processing and radar polarimetry. Prof. Peng has published more than 200 papers in the areas of his research interests. He is a fellow of Chinese Institute of Electronics. He has received many awards for his contributions to research and education in China. PENG ET AL.: MAF IDENTIFICATION FOR NARROWBAND COHERENT RADAR IN LCIT

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