Coherent Integration for Targets with Constant Cartesian Velocities Based on Accurate Range Model
CCoherent Integration for Targets with Constant CartesianVelocities Based on Accurate Range Model
Gongjian Zhou, Zeyu Xu, and Yuchao Yang
School of Electronics and Information Engineering, Harbin Institute of Technology, Harbin, ChinaKey Laboratory of Marine Environmental Monitoring and Information Processing, Ministry of Industry and InformationTechnology, ChinaCorresponding author: Gongjian ZhouEmail: [email protected]
Abstract
Long-time coherent integration (LTCI) is one of the most important techniques to improve radardetection performance of weak targets. However, for the targets moving with constant Cartesianvelocities (CCV), the existing LTCI methods based on polynomial motion models su ff er fromlimited integration time and coverage of target speed due to model mismatch. Here, a novelgeneralized Radon Fourier transform method for CCV targets is presented, based on the accuraterange evolving model, which is a square root of a polynomial with terms up to the second orderwith target speed as the factor. The accurate model instead of approximate polynomial modelsused in the proposed method enables e ff ective energy integration on characteristic invariant withfeasible computational complexity. The target samplings are collected and the phase fluctuationamong pulses is compensated according to the accurate range model. The high order rangemigration and complex Doppler frequency migration caused by the highly nonlinear signal areeliminated simultaneously. Integration results demonstrate that the proposed method can notonly achieve e ff ective coherent integration of CCV targets regardless of target speed and coherentprocessing interval, but also provide additional observation and resolution in speed domain. Keywords:
Long-time coherent integration, high-speed target detection, constant Cartesianvelocities, accurate range model, speed estimation.1 a r X i v : . [ ee ss . SP ] F e b . Introduction To improve the detection performance of targets in a large coverage of position and tar-get characteristic via pulse integration is a hot topic in the area of radar technology [1–10]. It isknown that coherent integration can produce better performance than the noncoherent integrationby compensating phase fluctuation among sampling pulses [11–13]. The moving target detec-tion (MTD) method [14, 15] has been widely used in modern radar systems to suppress strongbackground clutter and integrate energy of target echo signal with unknown Doppler frequency.However, the performance gain of the MTD is limited by the target’s resident in a range cell. Therange migration (RM) and Doppler frequency migration (DFM) e ff ects may occur when targetperforms maneuvers, e.g., high velocity and acceleration, during the integration interval. Thisprevents the MTD method from improving detection performance through long-time coherentintegration (LTCI).In order to realize e ff ective detection of low signal-to-noise ratio (SNR) targets (e.g., far-range or stealth targets) and high maneuvering targets (e.g., ballistic missiles, jet fighters), LTCImethods with ability of RM correction and / or DFM compensation have been under intensiveinvestigation for the past decade. The related works can be classified into three categories basedon the target motion model considered in the methods.In the first category, the target motion model is formulated by a primary polynomial withrespect to slow time. In this case, the e ff ect of RM should be eliminated to obtain e ff ectiveintegration for a quite long time. The most popular methods may be keystone transform (KT)[16–18] and Radon Fourier transform (RFT) [19–21]. The KT method corrects the RM e ff ect byrescaling the time axis for each frequency, while the RFT method realizes coherent integration for Preprint submitted to Elsevier February 23, 2021 oving targets with RM e ff ect by a two-dimensional searching scheme in range-Doppler domain.Some other typical methods include sequence reversing transform (SRT) [22], axis rotation MTD(AR-MTD) [23], scaled inverse Fourier transform (SCIFT) [24], frequency-domain deramp-KT(FDDKT) [25], modified location rotation transform (MLRT) [26], etc. In these methods [16–26], the target is assumed to move with a constant radial velocity during the integration interval,which is actually a very strict assumption and will be violated easily in practical applications.In the second category, the target motion model is described by a quadratic polynomial,where the target is assumed to move with a constant radial acceleration. In this case, not only theRM caused by target’s radial velocity, but also the DFM and range curvature (RC) induced bytarget’s radial acceleration would occur. The improved axis rotation-fractional Fourier transform(IAR-FRFT) [27] and KT with Lv’s distribution (KT-LVD) [28] were proposed to eliminate thelinear RM and DFM e ff ects. To further improve the detection performance of target with constantradial acceleration, some more e ff ective methods have been proposed, including radon-fractionalFourier transform (RFRFT) [29], KT and matched filtering process (KT-MFP) [30], Radon-Lv’sdistribution (RLVD) [31], two steps scaling and fractional Fourier transform (TSS-FRFT) [32],etc. In these methods [29–32], the RM induced by both the target’s radial velocity and radialacceleration are corrected.In the third category, more complex motion is considered by formulating the target motionwith constant radial jerk, where the target motion model is expressed by a cubic polynomial.RM e ff ects up to third order and DFM e ff ects up to second order can be induced by this motion.In dealing with LTCI problem in this case, the generalized RFT (GRFT) [33] and polynomialRadon-polynomial Fourier transform (PRPFT) [34] could obtain optimal integration gain, butmay be computationally prohibitive due to the high dimensional searching strategy. The adja-cent cross correlation function (ACCF) [35–37] is more computationally e ff ective at a cost of3erformance degradation. Some other methods, such as generalized KT and generalized dechirpprocess (GKTGDP) [38], KT and generalized dechirp process (KTGDP) [39], KT and matchedfiltering processing [40], Radon modified Lv’s distribution (RMLVD) [41], Radon-high-ordertime-chirp rate transform (RHTRT) [42], discrete polynomial-phase transform and Lv’s distri-bution [43], the method based on series reversion [44] were also proposed, where the tradeo ff between integration performance and computational complexity is considered.In the aforementioned methods [16–44], the target motion model is formulated straightly inrange coordinate, based on finite order polynomial with respect to slow time. However, the timeevolving function of a real target moving in the Cartesian coordinates is highly nonlinear, forexample the motion with constant Cartesian velocities (CCV), due to the nonlinearity betweenCartesian and polar coordinates. The polynomial motion models mentioned above may not beable to provide accurate description of a target with the common CCV motion. Actually, in someliteratures [34, 41–43], approximations based on Taylor series expansion are explicitly employed,resulting in the polynomial models. To achieve accurate formulation, the polynomial model withinfinite orders is desired, but this is impractical. The existing LTCI methods based on finite orderpolynomial models cannot guarantee optimal integration performance due to model mismatchand inaccurate energy accumulation.In this paper, a novel generalized Radon Fourier transform method is presented by formulat-ing the signal model directly by the accurate range evolving equation corresponding to the CCVmotion, without approximations based on Taylor series expansion. The target samplings are ex-tracted according to the accurate range evolving model and the phase fluctuation among di ff erentpulses is compensated, which eliminates the high-order RM and complex DFM simultaneously.Since there is no error in the range model and the factor of highest order in the model is relatedto the invariant speed of the CCV target, the proposed GRFT based on accurate range evolving4odel (AREM-GRFT) can achieve e ff ective coherent integration regardless of target speed andcoherent processing interval (CPI). Moreover, the AREM-GRFT method can provide additionalobservation and resolution of the target in speed domain by focusing on the motion characteris-tic invariant, which is of significance for both target detection and tracking. Simulation resultsdemonstrate the e ff ectiveness and superiority of the proposed AREM-GRFT method in terms ofintegration performance and detection ability.The remainder of this paper is organized as follows. In Section 2, the signal model is es-tablished. The AREM-GRFT method is presented in Section 3. In Section 4, some numericalsimulation experiments are provided to evaluate the performance of the proposed method, fol-lowed by conclusions in Section 5.
2. Problem formulation
Suppose that the radar transmits the linear frequency modulated (LFM) signal, i.e., s T (cid:0) ˜ t (cid:1) = rect (cid:32) ˜ tT p (cid:33) exp (cid:16) j πµ ˜ t (cid:17) exp (cid:0) j π f c ˜ t (cid:1) (1)where rect( u ) = (cid:40) , | u | ≤ / , | u | > / t denotes the fast time, f c denotes the carrier frequency, T p denotes the pulse duration, µ = B / T p represents the frequency modulation rate, and B denotesthe signal bandwidth.The received baseband echo signal of a moving target can be expressed as [19, 33] s r (cid:0) t m , ˜ t (cid:1) = A rect (cid:32) ˜ t − τ T p (cid:33) exp (cid:104) j πµ (cid:0) ˜ t − τ (cid:1) (cid:105) exp ( − j π f c τ ) (2)where A is target reflectivity, τ = r ( t m ) / c is the time delay, c is the speed of light, r ( t m ) is the5nstantaneous slant range between the moving target and the radar, t m = mT r ( m = , , . . . , M − M and T r represent the number of integration pulses and pulse repetitiontime, respectively.After pulse compression (PC), the compressed signal of the moving target can be written as[19, 37] s (cid:0) t m , ˜ t (cid:1) = A sinc (cid:34) B (cid:32) ˜ t − r ( t m ) c (cid:33)(cid:35) exp (cid:32) − j πλ r ( t m ) (cid:33) (3)where sinc( x ) = sin( π x ) / ( π x ) denotes the sinc function, A denotes the signal amplitude afterPC, and λ = c / f c denotes the wavelength.Let ˜ t = r / c , (3) can be further written as [19] s ( t m , ˜ r ) = A sinc (cid:34) B (cid:32) r − r ( t m )) c (cid:33)(cid:35) exp (cid:32) − j πλ r ( t m ) (cid:33) (4)where ˜ r is the range corresponding to the fast time ˜ t .It is seen from (4) that the envelope position and phase of the target compressed signal aretotally determined by the target range. Due to the target movement, the signal envelope wouldbe shifted away from its original position with the increase of the integration time. When theo ff set of the signal envelope exceeds the range resolution ρ r = c / (2 B ), the RM e ff ect wouldoccur [45]. In addition, the change of range rate will also cause the o ff set of Doppler spectrum(the instantaneous Doppler frequency can be expressed as f d ( t m ) = − λ d r ( t m )d t m ). When the o ff setof the Doppler spectrum exceeds the Doppler resolution ρ d = / ( MT r ), the DFM e ff ect wouldhappen [29]. The migration e ff ects would make it di ffi cult to coherently integrate target’s signalenergy, which may deteriorate target detection performance. In order to eliminate the RM andDFM e ff ects due to target movement, it is crucial to establish an accurate range evolution model,6 Y O x r y q v ( , ) x y v v v ( , ) x y v v Fig. 1.
Target motion with constant Cartesian velocities. which needs to match the actual movement of the target. As mentioned in the previous section,the existing LTCI methods di ff er in the order of the polynomial model of r ( t m ). Nevertheless,the first-order, the second-order polynomial models and even the higher order polynomial mod-els may not match the CCV motion well, which makes it di ffi cult for LTCI methods based onthese polynomial models to achieve e ff ective coherent integration of CCV targets. Therefore, theaccurate range evolving model instead of approximate polynomial model is used in this paper toformulate the echo signal, corresponding to the target moving with constant Cartesian velocities.
3. Long time coherent integration via AREM-GRFT
As shown in Fig. 1, suppose that there is a target moving with constant Cartesian velocities (cid:16) v x , v y (cid:17) , starting from position ( x , y ) in Cartesian coordinates. A radar is observing the target atthe origin of the coordinates. 7ccording to the assumption of constant velocity motion in Cartesian coordinates, the instan-taneous Cartesian position of the target in x and y directions can be expressed as x ( t m ) = x + v x t m y ( t m ) = y + v y t m (5)Then, the instantaneous slant range r ( t m ) between the target and radar during the observationtime (i.e., 0 ≤ t m ≤ CPI) can be given by r ( t m ) = (cid:113) [ x ( t m )] + (cid:2) y ( t m ) (cid:3) = (cid:113) x + y + (cid:16) x v x + y v y (cid:17) t m + (cid:16) v x + v y (cid:17) t m (6)The initial slant range between the target and radar is r = (cid:113) x + y (7)The instantaneous radial velocity (also called range rate or Doppler velocity) of the targetrelative to the radar can be expressed as˙ r ( t m ) = d r ( t m )d t m = x ( t m ) ˙ x ( t m ) + y ( t m ) ˙ y ( t m ) r ( t m ) (8)where ˙ x ( t m ) and ˙ y ( t m ) denote the instantaneous Cartesian velocities of the target in x and ydirections, respectively. For CCV motion, the Cartesian velocities are time-invariant. Therefore,we denote ˙ x ( t m ) and ˙ y ( t m ) by v x and v y , respectively.8imilarly, the initial radial velocity of the target relative to the radar is˙ r = x v x + y v y (cid:113) x + y (9)Substituting (7) and (9) into (6), the time evolution of target range can be expressed as r ( t m ) = (cid:113) r + r ˙ r t m + v t m (10)where v = (cid:113) v x + v y denotes the invariant speed of the CCV target.It can be seen from (10) that the time evolution of range for a CCV target is a highly nonlinearfunction with the square root over the second order polynomial. Note that although the accuraterange evolving equation in (10) is derived in a two-dimensional Cartesian coordinates, the sameequation can be obtained in the three-dimensional Cartesian coordinates. The polynomial mo-tion models, such as the first-order polynomial model, the second-order polynomial model, failto accurately describe the model in (10). Actually, in order to represent the model in (10) bypolynomial, Taylor series expansion with infinite terms is required as r ( t m ) = ∞ (cid:88) l = r ( l ) (0) l ! t lm = r + ˙ r t m + v − ˙ r r t m + ˙ r − ˙ r v r t m + r v − v − r r t m + · · · (11)where r ( l ) (0), l =
0, 1, 2, . . . , denotes the l th-order derivative of r ( t m ) at t m = r ( t m ), as illustrated in (10) and (11), will cause high-order RM andcomplex DFM e ff ects, which may not be handled well by the LTCI methods based on polynomialmodel with finite terms. The discarding of any high order in (11) may lead to model mismatch9nd performance degradation, especially in the case of high speed and extremely long integrationtime. In addition, since the target range and radial velocity are time variant, the factor of eachterm in (11) di ff ers in di ff erent integration intervals, which is not good for data association andtarget tracking. The factor of the second-order term in (11) is not a real range acceleration. Inother words, for a CCV target, the second-order polynomial based LTCI methods [27, 29, 30]may not provide a parameter estimation with a physical meaning of range acceleration. On thecontrary, the factor of the second order of the polynomial under the square root in (10) has aphysical meaning of speed square and is time invariant for CCV targets.In this paper, the accurate time evolution of range in (10) is directly used to formulate thesignal model without approximations to eliminate the problem of model mismatch. And a GRFTbased method is presented to realize e ff ective coherent integration for CCV targets and providerange-Doppler measurement as well as speed measurement, which comes from focusing on thetime invariant speed. Substituting (10) into (4), the compressed signal of a target with CCV motion can be repre-sented as s ( t m , ˜ r ) = A sinc B (cid:18) ˜ r − (cid:113) r + r ˙ r t m + v t m (cid:19) c exp (cid:32) − j πλ (cid:113) r + r ˙ r t m + v t m (cid:33) (12)As shown in (12), the envelope position and target signal phase change complicatedly withtime due to the comprehensive e ff ect of target’s initial range, initial radial velocity and speed,which will cause high-order RM and complex DFM within the integration time. To achievee ff ective energy accumulation under this complex conditions, the AREM-GRFT method is pre-10ented, which is defined as G ( r s , ˙ r s , v s ) = (cid:90) T s (cid:18) t m , (cid:113) r s + r s ˙ r s t m + v s t m (cid:19) exp (cid:32) j πλ (cid:113) r s + r s ˙ r s t m + v s t m (cid:33) dt m (13)where r s , ˙ r s and v s denote respectively the searching slant range, searching radial velocity andsearching speed, T denotes the coherent integration time. The AREM-GRFT can be interpretedas employing the accurate range evolving model to extract the target signal and compensate forthe phase fluctuation. Since the accurate range evolution model is used, the complex RM andDFM e ff ects can be eliminated and e ff ective energy accumulation can be achieved.As is known, the definition of RFT is [19] RFT ( r s , ˙ r s ) = (cid:90) T s ( t m , r s + ˙ r s t m ) exp (cid:32) j πλ ˙ r s t m (cid:33) dt m (14)In the above, the assumption that the target moves with a constant radial velocity in range coor-dinate is used. However, this cannot be satisfied for most CCV targets, since the target’s radialvelocity usually varies and is di ff erent from target’s speed. Actually, the RFT method is a spe-cial case of the AREM-GRFT method. When the target’s speed equals its radial velocity, that is˙ r = v , the AREM-GRFT is reduced to RFT. The proposed AREM-GRFT method is valid forall CCV targets, while the assumption of the RFT is very restrict. In addition, when encounter-ing multiple CCV targets with the same initial slant range and radial velocity, the RFT methodcannot distinguish among these targets due to insu ffi cient reflection of the motion characteristicfor CCV target. Because of the introduction of the extended parameter (i.e., v s ) in AREM-GRFTmethod, these moving targets can be distinguished in the speed domain, as will be illustrated inthe numerical simulation experiments. 11or convenience, the searching trajectory based on the accurate range model is denoted as r s ( t m ), i.e., r s ( t m ) = (cid:113) r s + r s ˙ r s t m + v s t m (15)Substituting (4) and (15) into (13), the coherent integration output via AREM-GRFT can beexpressed as G ( r s , ˙ r s , v s ) = (cid:90) T A sinc (cid:34) B (cid:32) r s ( t m ) − r ( t m )) c (cid:33)(cid:35) exp (cid:32) j πλ ( r s ( t m ) − r ( t m )) (cid:33) dt m (16)As illustrated in (16), the coherent integration output via AREM-GRFT with respect to di ff er-ent searching motion parameter pairs ( r s , ˙ r s , v s ) can be obtained. The e ff ective energy integrationis based on the fact that the RM elimination and DFM compensation are accomplished only whenthe searching parameters are consistent with the true motion parameters of the target.When r s = r , ˙ r s = ˙ r and v s = v , we have r s ( t m ) − r ( t m ) = (cid:113) r s + r s ˙ r s t m + v s t m − (cid:113) r + r ˙ r t m + v t m ≡ r s ( t m ) − r ( t m ) does not change with slow time and is constantlyequal to zero. Substituting (17) into (16), the corresponding coherent integration output viaAREM-GRFT can be rewritten as G ( r s , ˙ r s , v s ) = (cid:90) T A sinc (cid:34) Bc (cid:35) exp (cid:32) j πλ (cid:33) dt m = A T (18)As shown in (18), the high-order RM and complex DFM are eliminated simultaneously when12he searching motion parameters match the target’s motion parameters (i.e., r s = r , ˙ r s = ˙ r and v s = v ). As a result, the signal energy distributed in multiple range and Doppler cells is totallyextracted and coherently integrated. The target samplings in the range and slow time plane are extracted according to the accu-rate range evolving model with preset searching parameters. Then the phase fluctuation amongdi ff erent pulses is simultaneously compensated by the determined searching trajectory. Whenthe searching trajectory determined by target’s initial range, initial radial velocity and speed isthe same as the true trajectory of the target, the signal energy of target would be focused in therange-Doppler-speed domain, which can eliminate the complex RM and DFM e ff ectively. Theflowchart of long-time coherent integration via AREM-GRFT is given in Fig. 2, and the mainsteps are illustrated as follows. Step 1: Perform demodulation and pulse compression on the raw echo data.
During the observation time, the raw echo data are sampled and stored as a two-dimensionalmatrix in the range and slow time domain. After demodulation and pulse compression, thecompressed echo data s ( m , n ) for coherent integration are then obtained, where m and n denotethe pulse index and discrete sampling range cell index, respectively. Step 2: Set coherent integration parameters for AREM-GRFT.
Based on the prior information and the motion characteristics of the target, the searchingscopes of range, radial velocity and speed can be determined as [ r min , r max ], [˙ r min , ˙ r max ] and[ v min , v max ], respectively. According to the determined radar system parameters, the searching13ntervals of the range, radial velocity and speed can be respectively set as [29] ∆ r = c / (2 B ) (19) ∆ ˙ r = λ/ (2 T ) (20) ∆ v = λ/ (2 T ) (21)Accordingly, the discrete values of the searching range, searching radial velocity and search-ing speed can be respectively represented as r s , i = r min : ∆ r : r max , i = , , . . . , N r (22)˙ r s , j = ˙ r min : ∆ ˙ r : ˙ r max , j = , , . . . , N ˙ r (23) v s , q = v min : ∆ v : v max , q = , , . . . , N v (24)where N r , N ˙ r and N v denote the searching number of the range, radial velocity and speed, re-spectively. Step 3: Apply the AREM-GRFT operation on the compressed data.
Determine the searching trajectory according to the preset searching parameters. Each timea set of searching parameter pair (cid:16) r s , i , ˙ r s , j , v s , q (cid:17) is determined, a searching trajectory based on the14ccurate range evolving model is determined, which can be expressed as r s ( mT r ) = (cid:113) r s , i + r s , i ˙ r s , j mT r + v s , q m T r (25)Extract the target samplings in the range and slow time plane according to the determinedsearching trajectory. Only when the searching motion parameters are identical to the true motionparameters of the target, the target signal along the actual moving trajectory can be extracted andthus the highly nonlinear RM is eliminated. The extracted samplings X M ( mT r ) can be expressedas X M ( mT r ) = s (cid:32) m , round (cid:32) r s ( mT r ) ∆ r (cid:33)(cid:33) = s m , round (cid:113) r s , i + r s , i ˙ r s , j mT r + v s , q m T r ∆ r (26)where round ( · ) represents the integer operator, ∆ r = c / (2 f s ) is the range sampling cell, f s is thesampling frequency.The extracted samplings X M ( mT r ) are compensated and summed to eliminate the complexDFM and integrate target energy. Each set of searching parameter pair (cid:16) r s , i , ˙ r s , j , v s , q (cid:17) has a corre-sponding integration output G (cid:16) r s , i , ˙ r s , j , v s , q (cid:17) , which can be expressed as G (cid:16) r s , i , ˙ r s , j , v s , q (cid:17) = M − (cid:88) m = X M ( mT r ) exp (cid:32) j πλ r s ( mT r ) (cid:33) = M − (cid:88) m = s m , round (cid:113) r s , i + r s , i ˙ r s , j mT r + v s , q m T r ∆ r exp (cid:32) j πλ (cid:113) r s , i + r s , i ˙ r s , j mT r + v s , q m T r (cid:33) (27)15t can be seen from (27) that the ideal integration gain of the discrete AREM-GRFT methodis the number of integration pulses. Step 4: Go through all searching parameters to obtain the integration output in range-Doppler-speed domain.
Repeat step 3 for all the searching range, radial velocity and speed to obtain the output matrixin the three-dimensional domain. When a target exists, a peak will appear at the position withthree parameters close to the target’s initial range, initial radial velocity and speed, respectively.
Remark 1:
Since the absolute value of the target radial velocity is less than or equal to theabsolute value of the velocity, the case that the searching speed is less than the absolute value ofsearching radial velocity can be skipped in the AREM-GRFT processing.
Step 5: Perform constant false alarm rate (CFAR) detection under a given false alarm prob-ability. (cid:12)(cid:12)(cid:12)(cid:12) G (cid:16) r s , i , ˙ r s , j , v s , q (cid:17)(cid:12)(cid:12)(cid:12)(cid:12) H ≷ H η (28)where η is the CFAR detection threshold [29]. Use the amplitude of AREM-GRFT output as teststatistic and perform CFAR detection to confirm targets. The adaptive detection threshold canbe obtained by the reference cells in the output matrix under the given false alarm probability. Ifthe test statistic is less than the detection threshold, there is no target to be declared. If the teststatistic is larger than the detection threshold, a target is detected. Step 6: Estimate target motion parameters.
After the target is confirmed, the corresponding peak located in range-Doppler-speed domaincan achieve the estimation of target motion parameters. Based on the initial range, initial radialvelocity and speed corresponding to the coordinate of peak in range-Doppler-speed domain, theestimated parameters (cid:16) ˆ r , ˆ˙ r , ˆ v (cid:17) can be obtained. In addition, the trajectory of the moving target16 able 1 Computational complexity.
Methods Computational complexity Search dimensionRFT O ( N r N ˙ r M ) 2-D searchKT-MFP O (cid:0) N k N a MN r log M (cid:1) O ( N r N ˙ r N a N ˙ a M ) 4-D searchProposed O ( N r N ˙ r N v M ) 3-D searchcan be achieved by ˆ r ( t m ) = (cid:113) ˆ r + r ˆ˙ r t m + ˆ v t m (29) In this subsection, the computational complexity of the proposed method is discussed andcompared against those of RFT [19], KT-MFP [30], and GRFT [33]. The number of integratedpulses, searching range, searching radial velocity, searching speed, searching radial acceleration,searching radial jerk and searching fold factor are denoted by M , N r , N ˙ r , N v , N a , N ˙ a and N k ,respectively.Table 1 gives the computational complexity of RFT [19], KT-MFP [30], GRFT [33] and theproposed method. RFT requires two-dimensional (2-D) parameter search to correct the first-order RM caused by radial velocity, whose computational cost is O ( N r N ˙ r M ) [19]. KT-MFP fur-ther eliminates the RM and DFM caused by radial acceleration with KT operation and matchedfiltering process. The computational complexity of KT operation is in the order of M N r andthe computational complexity of matched filtering process is in the order of N k N a MN r log M .Hence, the computational burden of KT-MFP is O (cid:0) N k N a MN r log M (cid:1) [30]. As to the proposedmethod, its computational complexity is O ( N r N ˙ r N v M ) because of three-dimensional (3-D) pa-rameter search in range-Doppler-speed domain. GRFT involves four-dimensional (4-D) param-eter search (i.e., search of range, radial velocity, radial acceleration and radial jerk), whose com-17 aw echo data Demodulation and pulse compressionDetermine the searching trajectory based on the accurate range evolving modelExtract the target samplingsGo through all the searching parameters
Integration output Y NCompensate for the phase fluctuation and integrate target energy Y N CFAR detection s j s q r v £ s j s q r v £ s j s qs q £ Searching parameters settings
Range
Radial velocity
Speed [ ] , min max : , , s i r r r r D [ ] , min max : , , s q v v v v D [ ] , min max : , , s j r r r r D [ ] , min max : , , s i r r r r D [ ] , min max : , , s q v v v v D [ ] , min max : , , s j r r r r D Searching parameters settings
Range
Radial velocity
Speed [ ] , min max : , , s i r r r r D [ ] , min max : , , s q v v v v D [ ] , min max : , , s j r r r r D ] , min max [ s i , m, m [ ] [ ] , min max [ : , , [ ] , min max, min [ s j , m, m r r r r , m : , ,: , [ ] , min max, min, min ma [ s j , m, m D Parameter estimation target exists h‡ (cid:222) no target h< (cid:222)
Fig. 2.
Flowchart of long-time coherent integration via AREM-GRFT method. Y O Target IITarget I (cid:3)
Target III (cid:3) ( , ) x y ( , ) x y ( , ) x y r r / v r q XY O Target IITarget I (cid:3)
Target III (cid:3) ( , ) x y ( , ) x y ( , ) x y r r rr / v r // rr // rr q r r q r r v v Fig. 3.
Relationship of target motion parameters in Cartesian and range coordinates. putational complexity is O ( N r N ˙ r N a N ˙ a M ) [33].In summary, the computational complexity of the proposed method is higher than that of RFTand KT-MFP, but lower than GRFT’s. Normally, for the polynomial model based LTCI methods,extending the order of Taylor polynomial model could approximate the optimal integration gain,but it also means a sharp increase in computational complexity, which makes the polynomialmodel based LTCI methods face to the contradiction between integration gain and computationalcomplexity. On the contrary, the proposed method can achieve better integration performancethan RFT, KT-MFP and GRFT for LTCI problem of CCV target with feasible computationalcomplexity, as will be illustrated in the numerical simulation experiments.
4. Numerical simulation experiments
In order to evaluate the performance of the proposed method, comprehensive experiments arepresented in this section. Before the specification of the scenario parameters, the relationship oftarget motion parameters in Cartesian and range coordinates is discussed, as illustrated in Fig. 3.19or a CCV target (e.g., Target I), its motion can be summarized by the initial Cartesian position( x , y ) and the Cartesian velocity vector (cid:126) v . Given these parameters in Cartesian coordinates, thecharacteristic motion parameters in range coordinate of the target, initial range r , initial radialvelocity ˙ r , and speed | (cid:126) v | , can be uniquely determined. On the other hand, given the set of initialtarget range r , initial radial velocity ˙ r , and speed | (cid:126) v | , there will be innumerable possible CCVtargets, such as Target I and Target II in Fig. 3. Since Target I and Target II have the same initialrange, initial radial velocity and speed, they cannot be distinguished by the proposed methodand there will be only one peak in the range-Doppler-speed domain. When the absolute valueof the target’s radial velocity is equal to target’s speed, the CCV motion is reduced to a constantradial velocity motion (see Target III in Fig. 3). In this case, both the existing polynomial modelbased LTCI method and the proposed method can achieve e ff ective energy integration. However,the target moves with constant radial velocity is just a special case of CCV motion. Most CCVtargets violate the constant radial velocity assumption and any polynomial models with finiteorders. This problem becomes serious in the case of long integration time and target with highspeed. In these cases, the transverse component of target velocity, i.e., the di ff erence betweenradial velocity and velocity, cannot be ignored.To verify the e ff ectiveness and superiority of the proposed method, simulation experimentsare performed for both constant radial velocity motion and common CCV motion, where thecases with long integration time and high speed are explored. Meanwhile, the multi-target in-tegration performance and detection performance of the proposed method are also evaluated.Several typical LTCI methods are used for comparison, including first-order polynomial modelbased method (i.e., RFT) [19], second-order polynomial model based method (i.e., KT-MFP)[30], and high-order polynomial model based method (i.e., GRFT) [33].Note that the PC gain of the pulse-compressed signal is normalized in the simulation exper-20 able 2 Radar parameters.
Parameters Value (Unit)Carrier frequency 1.5 (GHz)Pulse duration time 10 (us)Signal bandwidth 20 (MHz)Range sampling frequency 50 (MHz)Pulse repetition frequency 200 (Hz)Pulse number 500iments, so that the value of the output amplitude of the target reflects the e ff ective integratedpulse number. If a method can provide an integration result whose amplitude is consistent withthe pulse number, it means that the method achieves the ideal integration gain.The radar parameters are listed in Table 2. In this subsection, three scenarios are considered to evaluate the performance of the pro-posed AREM-GRFT method: 1) constant radial velocity motion, which is a special case of CCVmotion; 2) CCV motion for long integration time; 3) CCV motion for target with high speed.
When the absolute value of the target’s radial velocity is equal to target’s speed, the CCVmotion is reduced to a constant radial velocity motion. In this scenario, the target parameters areset as: initial range r =
25 km, initial radial velocity ˙ r =
800 m / s, speed v =
800 m / s, and SNRafter PC is 6 dB. The radar parameters are the same as those in Table 2.The corresponding simulation results are shown in Fig. 4. Fig. 4(a) illustrates the pulsecompression result. The integration results of RFT, KT-MFP, and GRFT are shown in Fig. 4(b)–Fig. 4(d), respectively. The integration results of the proposed method in the slice of range-21
500 8000 8500 9000 9500Range cell100200300400500 P u l s e nu m b e r (a) (b)(c) (d)(e) (f)(g) Fig. 4.
Simulation results in case of constant radial velocity motion. (a) Result after pulse compression.(b) RFT. (c) KT-MFP. (d) GRFT. (e) Range-Doppler slice of the AREM-GRFT at speed of 800 m / s. (f)Doppler-speed slice of the AREM-GRFT at range of 25 km. (g) Speed-range slice of the AREM-GRFT atradial velocity of 800 m / s. ff erent direction than the radar line of sight.When the transverse component cannot be ignored (that is, the di ff erence between the absolutevalue of radial velocity and speed is large) [19], the approximate polynomial motion modelscannot match the CCV motion well. Moreover, the model mismatch will be enlarged in thecase of long integration time and high speed, further resulting in performance degradation. Incontrast, the proposed method is able to achieve e ff ective LTCI without limitation of integrationtime and target speed, since signal extraction and compensation are performed according to anaccurate range model. In the following, another two scenarios are considered to further verifythe e ff ectiveness of the proposed method. To evaluate the performance of the proposed method for long integration time, the pulsenumber in the radar parameters changes to 800, and other radar parameters are the same as thosein Table 2. The parameters of the target with CCV motion are set as: initial range r =
25 km,initial radial velocity ˙ r =
60 m / s, speed v =
800 m / s, and SNR after PC is 6 dB.The simulation results in case of long integration time are illustrated in Fig. 5. It can be seenfrom the result of signal after pulse compression in Fig. 5(a) that the trajectory of the target is23ighly nonlinear, which implies that the RM caused by the comprehensive e ff ect of range, radialvelocity, and speed is complex. The integration results of RFT, KT-MFP, and GRFT are shown inFig. 5(b)–Fig. 5(d), respectively. The integration results of the proposed AREM-GRFT methodin the slice of range-Doppler, Doppler-speed, and speed-range are given in Fig. 5(e)–Fig. 5(g),respectively.It can be seen from Fig. 5(b) that the target energy resulted by the RFT is unfocused. This isbecause the RFT method could only correct the linear RM and has no capability of dealing withnonlinear RM and complex DFM. The result of KT-MFP is slightly better than that of RFT, andthe GRFT is superior to the KT-MFP, as illustrated in Fig. 5(c) and Fig. 5(d). It is because thatKT-MFP method considers second-order RM and first-order DFM, and GRFT method furtherconsiders third-order RM and second-order DFM during the integration period. Normally, thehigher the order of the polynomial model considered, the better the integration performance.However, the target energy resulted by the KT-MFP method still spreads over di ff erent range andDoppler cells, and the amplitude (i.e., e ff ective integrated pulse number) of the GRFT method inFig. 5(d) is 257.1, which indicates that the coherent integration gain of GRFT is about 5 dB lessthan the ideal integration gain (i.e., 800). The integration results of KT-MFP method and GRFTmethod are not well-focused, since the CCV motion is highly nonlinear in long integration time.In summary, for the long integration time, these LTCI methods (i.e., RFT, KT-MFP and GRFT)cannot e ff ectively eliminate the highly nonlinear RM and complex DFM caused by CCV motion.On the contrary, it can be seen from Fig. 5(e)–Fig. 5(g) that the proposed AREM-GRFTmethod can provide an e ff ective integrated amplitude of 800.3 and achieve the ideal integrationgain, thanks to the range evolving model accurately matching the CCV motion. In addition,the estimated parameters (i.e., corresponding to the position of the peak) are consistent withthe actual motion parameters. Note that the speed parameter can also be observed from the24
250 8350 8450 8550Range cell100200300400500600700800 P u l s e nu m b e r (a) (b)(c) (d)(e) (f)(g) Fig. 5.
Simulation results in case of long integration time. (a) Result after pulse compression. (b) RFT. (c)KT-MFP. (d) GRFT. (e) Range-Doppler slice of the AREM-GRFT at speed of 800 m / s. (f) Doppler-speedslice of the AREM-GRFT at range of 25 km. (g) Speed-range slice of the AREM-GRFT at radial velocityof 60 m / s. able 3 Parameters of the multiple targets.
Parameters (Unit) Target 1 Target 2 Target 3 Target 4Initial slant range (km) 21 21 21.15 21.21Initial radial velocity (m / s) -10 -10 17 -20Speed (m / s) 1150 1300 1300 1200SNR after PC (dB) 0 0 0 0integration result of the AREM-GRFT method in addition to the range and Doppler parameters.This extra measurement is beneficial to tracking filter, data association and further applications. To evaluate the coherent integration performance of the proposed method for target with highspeed, the target parameters are set as: initial range r =
25 km, initial radial velocity ˙ r = / s, speed v = / s, and SNR after PC is 6 dB. The radar parameters are the same as thosein Table 2.The corresponding simulation results in case of high speed are shown in Fig. 6. In the casewith the high speed, the compressed target signal also appears highly nonlinear with respect totime, as shown in Fig. 6(a). The RFT, KT-MFP, and GRFT su ff er from serious performance lossand fail to produce e ff ective integration of target energy, as illustrated in Fig. 6(b)–Fig. 6(d).On the contrary, the proposed AREM-GRFT method provides e ff ective integration output. Apeak with amplitude of 499.5, close to the pulse number, is formed at the location correspondingto the true target parameters, as shown in Fig. 6(e)–Fig. 6(g). The proposed method obtains analmost ideal integration gain in this scenario. 26
200 8300 8400 8500 8600Range cell100200300400500 P u l s e nu m b e r (a) (b)(c) (d)(e) (f)(g) Fig. 6.
Simulation results for high speed. (a) Result after pulse compression. (b) RFT. (c) KT-MFP. (d)GRFT. (e) Range-Doppler slice of the AREM-GRFT at speed of 1500 m / s. (f) Doppler-speed slice of theAREM-GRFT at range of 25 km. (g) Speed-range slice of the AREM-GRFT at radial velocity of 60 m / s.
950 7000 7050 7100 7150 7200Range cell100200300400500 P u l s e nu m b e r (a) P u l s e nu m b e r Target 4Target 3Target 2Target 1 (b)(c) (d)(e)
Fig. 7.
Simulation result of the proposed method for multiple targets. (a) Result after pulse compression.(b) Result after pulse compression without noise. (c) Doppler-speed slice of the AREM-GRFT at range of21 km. (d) Speed-range slice of the AREM-GRFT at radial velocity of 17 m / s. (e) Range-Doppler slice ofthe AREM-GRFT at speed of 1200 m / s. To evaluate the LTCI performance of the proposed method for multiple targets, four targetswith CCV motion are considered in this scenario, whose parameters are given in Table 3. Target 1and target 2 have the same initial range and radial velocity, but di ff erent speeds. The initial rangeand radial velocity of target 3 are di ff erent from those of target 2, and target 4 has parameters28i ff erent from the parameters of all the other targets. The radar parameters are the same as thosein Table 2.Fig. 7 shows the simulation results of the AREM-GRFT method for multiple targets. Thesignal after pulse compression is shown in Fig. 7(a). It can be seen that the signal energy isalmost submerged in noise. In order to clearly show the trajectories of the targets, the pulsecompression result without noise is shown in Fig. 7(b). The slice of Doppler-speed at range of21 km, identical to the initial range of target 1 and target 2, is illustrated in Fig. 7(c). Two peakscan be observed, and the peak positions are consistent with target 1 and target 2, respectively.Note that these two peaks have the same radial velocity. They can only be distinguished in thedirection of speed. In Fig. 7(d), the slice of speed-range with radial velocity 17 m / s, identical toTarget 3 is provided. In Fig. 7(e), the slice of range-Doppler at speed of 1200 m / s is illustrated.In each of these two figures, only one peak is observed. This is because only one target has thesame parameter corresponding to the slice. The peak in Fig. 7(d) is target 3 and that in Fig.7(e) is target 4. Based on the integration results in Fig. 7, it can be concluded that the proposedAREM-GRFT method could not only achieve e ff ective coherent integration for multiple targets,but also provide additional observation and resolution in speed domain.It should be noted that like some parameter-search algorithms such as RFT [19] and GRFT[33], the blind speed sidelobe (BSSL) also exists in the proposed method (see Fig. 4(e), Fig. 5(e),Fig. 6(e) and Fig. 7(e)). When the searching radial velocity di ff ers from the radial velocity ofthe target by an integer times of blind speed, the target energy would also be partially integrated.Some e ff ective methods [20, 46] have been proposed for BSSL suppression of RFT. To suppressthe BSSL in the proposed method is an important topic needs to be further studied.29
40 -30 -20 -10 0 10 20 30SNR (dB)00.20.40.60.81 D e t ec ti on p r ob a b ilit y ProposedGRFTKT-MFPRFTMTD
Fig. 8.
Detection probability of MTD, RFT, KT-MFP, GRFT, and the proposed method.
In this subsection, the detection performance of the proposed AREM-GRFT method andseveral typical coherent integration methods (i.e., MTD, RFT, KT-MFP, GRFT) are investigatedby Monte Carlo experiments. The radar parameters and motion parameters of the target are thesame as those in Section 4.1.2. The CFAR detector is used for the five coherent integrationmethods, i.e., MTD, RFT, KT-MFP, GRFT, and the proposed AREM-GRFT method. The falsealarm probability is set as P f a = − and the Gaussian noises are added to the target echoes.The SNR after pulse compression varies from −
40 dB to 30 dB and the step size is 1 dB. In eachcase, 1000 times Monte Carlo experiments are carried out.The detection probability under di ff erent SNR levels is shown in Fig. 8. It can be seenthat the detection performance of the proposed method is superior to MTD, RFT, KT-MFP, andGRFT. Particularly, for the detection probability P d = / /
20 dB lower than that of GRFT / KT-MFP / RFT, respectively. It is because that theproposed method can achieve e ff ective energy integration thanks to the ability to match CCVmotion accurately, while MTD, RFT, KT-MFP, and GRFT fail to deal with the highly nonlinear30M and complex DFM due to model mismatch and inaccurate energy accumulation.
5. Conclusion
In this paper, a novel generalized Radon Fourier transform method for target moving withconstant Cartesian velocities (CCV) was presented, based on the accurate range evolving model,which is a square root of a polynomial with terms up to second order with target speed as the fac-tor. The accurate model instead of approximate polynomial models used in the proposed methodenables e ff ective energy integration on characteristic invariant with feasible computational com-plexity. The target samplings are collected and the phase fluctuation among di ff erent pulsesis compensated according to the accurate range evolving model. The highly nonlinear rangemigration and complex Doppler frequency migration caused by CCV motion are eliminated si-multaneously. Numerical simulation experiments demonstrated that the proposed method canachieve better integration gain and detection performance than several typical polynomial modelbased coherent integration methods (i.e., RFT, KT-MFP and GRFT). Additionally, the proposedmethod can not only achieve e ff ective coherent integration for CCV targets regardless of the cov-erage of target speed and coherent processing interval, but also provide additional observationand resolution in speed domain. References [1] J. Su, M. Xing, G. Wang, and Z. Bao, “High-speed multi-target detection with narrowband radar,”
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