RANGE ALIGNMENT AND MOTION COMPENSATION FOR MISSILE-BORNE FREQUENCY STEPPED CHIRP RADAR

Transcription

1 Progress In Eletromagnetis Researh, Vol. 136, , 213 RANGE ALIGNMENT AND MOTION COMPENSATION FOR MISSILE-BORNE FREQUENCY STEPPED CHIRP RADAR Bo Liu * and Wenge Chang Shool of Eletroni Siene and Engineering, National University of Defense Tehnology, Changsha, Hunan 4173, China Abstrat One of the diffiulties for frequeny stepped hirp radar (FSCR) is to resolve the range-doppler oupling due to relative motion between the radar and the target. Motion ompensation is usually adopted to solve the problem in realizing syntheti high range resolution profile (HRRP) for a moving target. For missileborne FSCR, the range migration of target eho during a oherent proessing interval, whih is resulted from the high speed motion of missile, is serious and will affet target detetion and syntheti high range resolution profile. Therefore, range migration orretion and motion ompensation are very important for missile-borne FSCR signal proessing. In the paper, with the bakground of terminal guidane anti-ship FSCR seeker, the range alignment is aomplished in frequeny domain during the proess of real-time digital pulse ompression. Then an effetive veloity estimation algorithm based on the waveform entropy of the Doppler amplitude spetrum of target ehoes is addressed and the veloity estimation auray is derived. Finally, the simulation indiates that the new method an estimate the radial veloity aurately and reonstrut the distorted HRRP suessfully. In addition, the method has good anti-noise performane and works in the senario of multi-target with different veloities as well. 1. INTRODUCTION High range resolution (HRR) radars use wide-band waveforms to resolve individual satterers within the target [1, 2]. Frequeny stepped hirp radar (FSCR) is a kind of HRR radar and is widely used in reent Reeived 8 November 212, Aepted 17 January 213, Sheduled 23 January 213 * Corresponding author: Bo Liu (liubo198312@163.om).

2 524 Liu and Chang years [3 6], for it an ahieve high range resolution while still retaining the advantages of narrow instantaneous reeiver bandwidth and low analog-to-digital (AD) sampling rate. FSCR transmits a hirp train with frequeny stepped arriers, and ahieves high range resolution by synthesis wide-band tehnique. For HRR radar, the multiple sattering entres of a target may appear in a number of isolated range ells, so the target is alled a range-spread target [7, 8]. The high range resolution profile (HRRP) of the target is used for target reognition and high auray traking [9 11], whih is very important for missile to improve the traking ability and attaking auray. However, the disadvantage of FSCR is the ompliation aused by range-doppler oupling, due to relative motion between the radar and the target, whih results in irular shift and the spreading of HRRP [12]. In reent years, some investigations have been proposed to mitigate the distortion of HRRP aused by motion [13 15]. In [13], two suessive stepped-frequeny pulse trains were transmitted to eliminate the phase errors for the moving target. In [14], a method using multiple stepped-frequeny pulse trains and the robust phase unwrapping theorem to estimate the range and the veloity of a target was introdued. However, the methods in [13, 14] need to transmit additional assistant signal, whih leads to the amount of available information redued, and lowers the data rate of the radar. In [15], an algorithm based on the maximum likelihood (ML) estimation is proposed, without altering the onventional stepped-frequeny waveform. The algorithm estimates the target veloity aurately; however, the omputation onsumption is large, so it is diffiult to implement in real time. Furthermore, all these methods mentioned above do not onsider the range migration of target eho during a oherent proessing interval (CPI) [16]. The terminal veloity of missile is so high that the range migration of target eho during a CPI is serious and will affet target detetion and syntheti HRRP [17]. Hene, range alignment is neessary for missile-borne FSCR. In this paper, an effetive veloity estimation algorithm based on the waveform entropy (WE) of the Doppler amplitude spetrum of the target ehoes is developed for FSCR whih is assumed to work in anti-ship seeker. Firstly, the target eho model of missile-borne FSCR is established, and the problem of motion ompensation is analyzed. Range alignment is aomplished in frequeny domain during the proess of digital pulse ompression. Seondly, the veloity estimation algorithm based on the WE of the Doppler amplitude spetrum is addressed, and the veloity estimation auray is derived. Finally, the simulation indiates that the algorithm an aurately estimate the radial veloity between the radar and the target and reonstrut

3 Progress In Eletromagnetis Researh, Vol. 136, the distorted HRRP suessfully. The paper is organized as follows. In Setion 2, the signal model of moving targets for FSCR is formulated. In Setion 3, we address the veloity estimation method and derive the veloity estimation auray. The performane of the proposed method is assessed in Setion 4. At last, in Setion 5, some onlusions are given. 2. PROBLEM STATEMENT 2.1. Radar Eho Model The transmitted signal of FSCR is expressed as: s(t) = M 1 i= A i u i (t)e j2π(f +i f)t where t is time variable, u i (t) = ret[(t it r )/T p ] exp[jπµ(t it r ) 2 ] the omplex envelop of the i-th hirp sub-pulse, and µ = B /T p the frequeny slope of eah hirp sub-pulse, A i : amplitude of i-th hirp sub-pulse, T r : pulse repetition interval (PRI), T p : pulse width of hirp sub-pulse, and T p < T r, B : band width of eah hirp sub-pulse, f : nominal arrier frequeny, f: frequeny step, f < B, M: number of hirp sub-pulse. Assuming the radial veloity between the radar and the target remains onstant during a CPI, the relative radial veloity is v, and the initial range between the radar and the target is R. Demodulated with its orresponding arrier frequeny, the target eho is expressed as: M 1 r(t) = A i u i [t τ i ] e j2π(f +i f+f di )τ i i= [ ] t itr τ i A i ret exp jπµ(t it r τ i f di /µ M 1 = i= T p }{{} range Doppler oupling e j2π(f +i f+f di )τ i (2) where τ i = 2(R ivt r )/, is the speed of light, τ i the delay of the i-th hirp sub-pulse, and f di = 2v(f + i f)/ the Doppler frequeny (1) ) 2

4 526 Liu and Chang of the i-th hirp sub-pulse eho. From Equation (2), it an be seen that the relative radial veloity between the radar and the target produes range-doppler oupling to hirp signal [18], whih results in mismathing between the target eho and the mathed filter. After mathed filtering, a group of ompressed pulses an be obtained: y(i, t) = [ t itr A i µt p 2 τ i ret M 1 i= ( 1 t it r τ i T p T p ] sin [πb (t it r τ i f di /µ)] πb (t it r τ i f di /µ) ) e jπ/4 e j2π[(f +i f+f di )τ i µ(t itr τ i) 2 ], i =, 1, 2,..., M 1.(3) Due to mismathing, envelops of the pulse ompression results in Equation (3) are not Sin funtions any more. Aording to [18], the amplitude loss in Equation (3) is negligible, but the time shift resulting from the time-frequeny oupling of hirp signal is signifiant and will affet the range measurement auray. Additionally, for missileborne radar, the radial veloity v between the radar and the target produes serious range migration during a CPI. From Equation (3), the migration time between the adjaent pulses an be written as: Range migration fator P is defined as: t = 2v(T r f/u)/ (4) P = B (M 1) t = B 2v(M 1)(T r f/µ)/ (5) The range migration fator P represents the number of the range resolution ell that the target eho spreads in the oarse resolution domain during a CPI. Sampling at t = it r τ i, the phases of the M samples from the ompressed pulses in Equation (3) are expressed as: φ i = π 4 4πf R 8πf R v ( 2 π + 4πf vt r 8πR v f 2 i 4π fr i + 4π fvt r i 2 i + 8πv2 T r f i + 8πv2 f T r 2 2 i 2 4 4πf R 4π fr ) ( 8πf R v i }{{} 2 4πf vt r i 4π fvt ) r i 2, }{{} inherent phase terms additional phase terms i =, 1, 2,..., M 1. (6) The expansion of the phase omponent φ i an be seen as two parts: the inherent phase terms and the additional phase terms resulting from radial motion.

5 Progress In Eletromagnetis Researh, Vol. 136, If v =, the syntheti HRRP of the target an be obtained by taking the M-point inverse disrete Fourier transform (IDFT) of the olleted M samples [19], whih is given by: [ ( )] sin πm f l M f z(l) = 2R [ ( )] M sin π f l M f 2R, l =, 1,..., M 1. (7) However, if v, the additional motion phase terms in Equation (6) will result in target distortion in the syntheti HRRP [12]. Preisely, the linear omponent of the additional phase term 4πf vt r i/ auses the irular range shifting, while the quadrati omponent 4π fvt r i 2 / produes spreading for HRRP [2]. Both of them are signifiant distortions for HRRP. Partiularly, for missileborne FSCR, the target spreading is onsiderably more signifiant than the target shifting beause target spreading degrades the desired range resolution and dereases the signal-to-noise ratio (SNR), whih will influene target reognition and high auray traking. Nevertheless, one the radial veloity is able to be estimated, the additional motion phase term an be ompensated and the HRRP an be reonstruted [2]. The veloity ompensation auray for the quadrati phase is given by [2]: v < / [ 8(M 1) 2 ] ft r (8) And the veloity ompensation auray for the linear phase is given by: v < /(4Mf T r ) (9) Additionally, if v, the range migration of eho data during a CPI in Equation (4) will affet the following signal proessing of FSCR. Firstly, the range migration degrades the range measurement auray. Seondly, the range migration during a CPI disperses the energy of the target eho, whih makes it more diffiult to detet target. And finally, it affets the syntheti HRRP of the target [2]. The parameters of a FSCR are listed in Table 1, whih are assumed aording to the priniple in [1]. Heneforth, without loss of generality, the simulation and illustration both adopt the parameters in Table 1. The range migration fators resulting from radial veloity between the radar and the target are listed in Table 2. Figure 1 shows the Table 1. Parameters of a FSCR. f B f M T p T r 35 GHz 1 MHz 2 MHz µs 2 µs

6 528 Liu and Chang Table 2. Range migration fators resulting from radial veloity between the radar and the target. missile veloity (m/s) P (oarse range resolution ell) Amplitude (linear) P= P=1 P=2 P=3 P= Range (high range resolution ell) Figure 1. HRRPs at different range migration fator P. Normalized energy (db) P= P=1 P=2 P=3 P= Range (m) Figure 2. Inoherent integration results at different range migration fator P. HRRPs of a point-like target with different range migration fators, amusing that the additional motion phase terms in Equation (6) has been ompensated thoroughly. As an be seen from Figure 1, the range migration dereases the amplitude of the HRRP, the more the range migration, the more the amplitude loss will be. While the range migration fator P = 2, the amplitude of HRRP is about a half of that of the ideal HRRP. In other words, the amplitude loss is about 3 db when the range migration fator P = 2. The inoherent integration results of radar eho with different range migration are shown in Figure 2, from whih we an see that the range migration spreads the energy of target eho, whih is harmful to target detetion. In addition, the position of the target moves, whih degrades the range measurement auray of FSCR. Assume that it is required that the range migration during a CPI is no more 1/4 oarse range resolution ell [2]. Aording to Equation (5), the radial veloity between the radar and the target should satisfy the following equation: v < (1) 8B (T r f/µ)(m 1)

7 Progress In Eletromagnetis Researh, Vol. 136, Range Alignment For missile-borne radar, Equation (1) usually annot be satisfied. Therefore, it is neessary to orret the range migration between the HRRPs before target detetion and synthesizing HRRP. Assume that the veloity of missile v m an be obtained by the missile-borne inertial navigation system (INS) [21]. Setting the veloity measurement error by the INS is v m. If we adopt v m as a oarse estimation of the radial veloity between the radar and the target, the estimation error an be written as: ε = v m + v t (11) where v t is the radial veloity of the target ship. For missile-borne FSCR, the oarse estimation of the radial veloity above is not aurate enough to be up to the auray of motion phase ompensation [22]. However, for range migration orretion, the oarse estimation of the radial veloity is adequate. Therefore, we adopt the oarse estimation of the radial veloity v m to align the range migration during a CPI. Utilizing the time-frequeny symmetry properties of Fourier transform, range alignment an be aomplished in frequeny domain. Firstly, transform the pulse ompression results in Equation (3) to frequeny domain by the fast Fourier transform (FFT). Then multiply them by the orresponding frequeny domain phase terms with respet to the radial veloity. Finally, transform the produts to time domain again by the inverse fast Fourier transform (IFFT) and the range alignment result an be obtained: {y(i, } f) e j4πv m[(f +i f)/µ it r ]f/ y (i, t)=ifft [ t itr A i µt p 2 2R / ret T G ] sin [πb ( t it r 2R )] e jπ/4 e j2π[(f +i f+f di )τ i µ(t it r 2R /) 2 ] + C (t), i =, 1, 2,..., M 1. (12) where y (i, t) is the ompressed pulses after range alignment, y(i, f) = FFT[y(i, t)]. Although the proess of the range alignment above needs abundant omputation, it does not take muh additional time in the signal proessing, beause it ould be aomplished onurrently with digital pulse ompression. Figure 3 illustrates the approah of range alignment during the pulse ompression with digital mathed filter. Supposing that the veloity of missile v m provided by the INS is 2 m/s, the real radial veloity between the missile and the target v = 225 m/s, and the error of the oarse estimation of the radial veloity ε = 25 m/s. The initial range of a point-like target is 4 m.

8 53 Liu and Chang Input FFT signal IFFT Pulse ompression result Mathed filtering fator (a) Mathed filtering fator Input signal FFT IFFT Result of pulse ompression after range alignment Range alignment fator (b) Figure 3. Range alignment during the digital pulse ompression. (a) Conventional digital pulse ompression with mathed filter. (b) Range alignment during the digital pulse ompression. (a) (b) Figure 4. Top view of target ehoes after pulse ompression. (a) Without range alignment. (b) After range alignment. Aording to Equation (5) and the radar parameters in Table 1, the range migration fator P is 3.36 before range alignment and.4 after range alignment with the oarse estimation of the radial veloity v m. The top view of target ehoes are shown in Figure 4, where Figure 4(a) is the target ehoes without range alignment and Figure 4(b) the target ehoes after range alignment. Figure 5 shows the benefits of range

9 Progress In Eletromagnetis Researh, Vol. 136, Normalized amplitude (db) After range alignment Before range alignment Range (high range resolution) (a) Normalized energy (db) After range alignment Before range alignment Range (m) Figure 5. The benefits of range alignment to HRRP and inoherent integration. (a) HRRPs before and after range alignment. (b) Inoherent integration results before and after range alignment, initial range of the target R = 4 m. (b) alignment to HRRP and inoherent integration, where Figure 5(a) ompares the HRRP before and after range alignment, as an be seen that the amplitude of HRRP inreases about 7 db after range alignment. Figure 5(b) illustrates the inoherent integration results before and after range alignment. We an see that the signal-tolutter ratio (SCR) has about 5 db improvement after range alignment. Moreover, the range error aused by the range migration and timefrequeny oupling of hirp signal has been orreted effetively. 3. VELOCITY ESTIMATION AND MOTION COMPENSATION 3.1. Radial Veloity Estimation The veloity estimation method based on the Doppler shift has high auray [23]. However, missile-borne FSCR usually works with a low pulse repetition frequeny (LPRF) waveform whih is unambiguous in range but mostly ambiguous in Doppler. Aording to the radar theory [24], the most unambiguous Doppler veloity of FSCR is: v max = f prf / (2f ) (13) where f prf is the pulse repetition frequeny (PRF), f prf = 1/T r. The veloity resolution ell is: v ell = v max /M = f prf / (2Mf ) (14) Let s have a loser look at the phase omponent φ i in Equation (6).

10 532 Liu and Chang Setting f d = 2vf /, Equation (6) an be rewritten as: ( π φ i = 4 4πf R 8πf ) ( R v 4π fr 2 +2πf d T r i i 4π fvt ) r i 2, }{{}}{{} phase terms of pulsed Doppler radar additional phase terms by f i =, 1, 2,..., M 1. (15) The phase omponent φ i an be seen as another two parts: the phase terms of a pulsed Doppler radar with arrier-frequeny f and the additional phase terms. For pulsed Doppler radars, oherent proessing is implemented by the disrete Fourier transform (DFT) as a mathed filter [24]. Proessing the sampled eho data of the same arrier-frequeny via DFT, the Doppler amplitude spetrum will yields at the Doppler frequeny point. From Equation (11), we an see that the frequeny step f and the initial range R introdue an additional phase (4π fr /)i, whih will result in a irular shift of the Doppler frequeny within the unambiguous DFT window after oherent proessing, whih is alled the range-doppler oupling effet [25]. The final term is a quadrati phase 4π fvt r i 2 /, whose effets are shown in widening and distortion of the main lobe of the Doppler amplitude spetrum. In other words, this term results in energy dispersion of the Doppler amplitude spetrum along the Doppler axis. Consequently, if the initial range R is known, the range-doppler oupling term (4π fr /)i an be removed by phase ompensation. We all it resolving the range-doppler oupling. In addition, the Doppler ambiguity an be resolved by using the effets of the quadrati phase term (4π fvt r /)i 2 on the Doppler amplitude spetrum. The initial range of the target R has been obtained by target detetion in the oarse resolution domain. After resolving the range-doppler oupling and motion pre-ompensation with the oarse veloity estimation v m to Equation (6), the phase omponent of target eho an be rewritten as: φ i = π 4 4πf R 8πf R v 2 +2πf d T r i 4π f R i+ 4π f vt r i 2, i =, 1, 2,..., M 1. (16) where R is the range error and v the veloity error. Aording to the radar parameters in Table 1, the most unambiguous veloity v max = m/s. Setting the radial veloity between the radar and the target is 2 m/s, the Doppler frequeny f d = Hz. Figure 6 shows the effets of the residual quadrati phase term (4π f vt r /)i 2 on the Doppler amplitude spetrum.

11 Progress In Eletromagnetis Researh, Vol. 136, Amplitude ( linear) x v = m/s v = m/s v = m/s v = m/s v = m/s v = m/s v = m/s Waveform entropy Doppler frequeny (Hz) x 1 5 Figure 6. Quadrati phase effets on Doppler amplitude spetrum v (m/s) Figure 7. The variation urve of the Doppler amplitude spetrum s WE with the pre-ompensation veloity error. Entropy is a measure of the unertainty of random variables [26]. In [27 3], the waveform entropy (WE) is introdued and adopted to survey the dispersion degree of a waveform s energy with respet to its parametri axis. Supposing the sampled signal waveform is denoted as A(n), n = 1, 2,..., N, setting: p(n) = A(n) / A N, n = 1, 2,..., N. (17) A = A(n) n=1 Then the WE of A(n) is defined by: WE [A(n)] = N p(n) log 2 p(n) (18) n=1 Aording to the definition above, the waveform entropy has the following properties: (1) E[A(n)], when p(n), the sparser A(n) is, the smaller WE[A(n)] will be. (2) E[A(n)] log 2 N, the equation omes into existene when p(n) = 1/N, n =, 1,..., N 1, the more homogeneous the distribution of A(n) s energy is, the larger WE[A(n)] will be. If the energy of A(n) distributes uniformly along its parametri axis, the WE reahes the maximum. On the ontrary, if the energy onentrates only on one sampling point, the WE is the minimum. Aordingly, the WE, served as a measure funtion, an be employed to

12 534 Liu and Chang weigh the effets of the quadrati phase term on the Doppler amplitude spetrum. The WE of the Doppler amplitude spetrum with different pre-ompensation veloity error v are shown in Figure 7. As we an see, while the veloity error v =, the WE of the Doppler amplitude spetrum is the minimum. As a result, a veloity estimation based on the WE of the Doppler amplitude spetrum an be realized in the following three steps: Firstly, resolve the range-doppler oupling of FSCR by the initial range of the target R estimated by target detetion. Seondly, make a preliminary estimate of the radial veloity. The maximum of the oarse veloity measurement error ε is usually known. In the veloity interval [v m ε max, v m + ε max ], where ε max is the maximum of ε, with a step of v max, the preliminary veloity estimate ˆv 1 an be ahieved by searhing for the position on the v axis, where the minimum value of the WE is loated. And finally, make an aurate estimation of radial veloity. The radial veloity between the radar and the target an be expressed as: v = k v max + n v ell (19) where k is the Doppler ambiguity number and n the index of the Doppler axis, whih an be obtained by oherent proessing [24]. It is easy to find k whih satisfies the following equation: (k v max + n v ell ) ˆv 1 < [(k + 1) v max + n v ell ] (2) Using k v max +n v ell and (k +1) v max +n v ell to ompensate the motion quadrati phase, respetively, whihever makes the EW of the Doppler amplitude spetrum smaller is hosen as the final estimation of the radial veloity. In other words, the aurate estimation of the radial veloity ˆv an be given as: ˆv = arg min {WE [(k v max + n v ell ) v], k = k, k + 1} (21) where WE[ ] is the WE of the Doppler amplitude spetrum, whih is a funtion of the veloity estimation error The Veloity Estimation Auray and Targets Imaging After ompensation to the quadrati phase by the preliminary veloity estimate ˆv 1, the effet of the motion quadrati phase on the Doppler amplitude spetrum an be negligible. As a result, the veloity estimation error mainly depends upon the range-doppler oupling shift resulting from the range measure error R. Equation (16) an be rewritten as: ϕ i= π 4 4πf R 8πf ( R v 2 +2π f d 2 f R ) f prf T r i+ 4π f vt r i 2, i =, 1, 2,..., M 1. (22)

13 Progress In Eletromagnetis Researh, Vol. 136, After range migration orretion by v m, the oarse veloity error ε an also produe slight range migration between adjaent pulses and time-frequeny oupling to hirp pulses. Aording to Equation (5), the range migration indued by the oarse veloity measurement error ε is written as: P = 2B ε(m 1)(T r f/µ)/ (23) The range measurement error aused by P an be written as: r 1 = P /(2B ) = ε(m 1)(T r f/µ) (24) The range measurement error aused by time-frequeny oupling of the hirp sub-pulse is: r 2 = ε f /µ (25) Therefore, the total range measurement error is: R = r 1 + r 2 (26) By substituting R into Equation (22), the veloity estimation error of the proposed method an be obtained: e = f /µ (M 1)(T r f/µ) fε f T r < f B Tp T r ε (27) Substituting the radar parameters of Table 1 into the above equation, the radial veloity estimation error an be obtained: e =.129ε (28) After ompensation to the additional phase in Equation (6) by the veloity estimation ˆv, the syntheti HRRP of the target an be obtained by the IDFT [22]. It should be pointed out that the initial range information of the targets has been obtained by target detetion before veloity estimation, therefore, the veloity estimation and motion ompensation an be arried out just to the target eho data instead of the data of whole frame [14]. Consequently, the proposed method an work in the senario of multiple-target with different veloities. Figure 8 is the flow hart of the proposed motion ompensation and target imaging method. 4. PERFORMANCE ASSESSMENT In order to investigate the effetiveness of the proposed method, a simulation is arried out based on the radar parameters in Table 1. The sea lutter is assumed to be muh greater than the reeiver thermal noise, thus thermal noise an be ignored [31, 32]. Meanwhile, the lutter is assumed to be Gaussian distributed with zero-mean, and

14 536 Liu and Chang Target eho Digital mathed filtering and range alignment target1 Veloity estimation Motion ompensation Synthesis HRRP HRRP1 Inoherent integration target2 Veloity estimation Motion ompensation Synthesis HRRP HRRP2 Target detetion... Target parameters extration target n Veloity estimation Motion ompensation Synthesis HRRP HRRP n Figure 8. Flow hart of the proposed algorithm. Normalization energy (db) range (m) Figure 9. Pulse ompression result of the target eho (SCR = 3 db). The estimation of veloity (m/s) The veloity estimation of the proposed method 15 The veloity provided 14 by INS The real radial veloity (m/s) Figure 1. Radial veloity estimation results (SCR = 3 db, ε = 3 m/s). independent from range ell to range ell. Setting the SCR of the target eho (after pulse ompression) is 3 db, as shown in Figure 9. Figure 1 ompares the veloity estimation result of the proposed method and the oarse veloity provided by missile-borne INS, where the oarse veloity error ε = 3 m/s. The estimation error is shown in Figure 11, it an be seen that the estimation error of the proposed method is muh smaller than the oarse estimation error ε. Equation (27) indiates that the estimation error of the proposed method e is proportional to the oarse veloity error ε, Figure 12 shows the veloity estimation errors with different oarse veloity errors. From Figures 1 12, we an see that the proposed algorithm an estimate the radial veloity between the radar and the target effetively, whih is aurate enough to be up to the quadrati motion phase ompensation for FSCR [22].

15 Progress In Eletromagnetis Researh, Vol. 136, Estimation error (m/s) The proposed method The oarse veloity error The real radial veloity (m/s) Figure 11. Veloity estimation error (SCR = 3 db, ε = 3 m/s). Wave form ent ropy SCR=3 db SCR=1 db SCR=5 db Ambiguity times Figure 13. Resolving Doppler ambiguity under different SCRs. Estimation error (m/s) ε=2 m/s ε=4 m/s ε=8 m/s The real radial veloity (m/s) Figure 12. Veloity estimation errors with different oarse veloity errors (SCR = 3 db). Probability of orretly resolve ambiguity SCR (db) Figure 14. Curve of the antinoise performane. Resolving Doppler ambiguity is the key of the proposed algorithm. Assuming the radial veloity v = 55 m/s, aording to the radar parameters in Table 1, the most unambiguous veloity v max = m/s. Therefore, the radial veloity an be written as: v = 25 v max m/s (29) Therefore, the ambiguity number k = 25. The result of resolving Doppler ambiguity results under different SCR are shown in Figure 13. It an be seen that the waveform entropy of the Doppler amplitude spetrum runs to the minimum while the ambiguity number is 25. Figure 14 gives the anti-noise performane of the proposed method. The horizontal is the SCR of the target eho, and the ordinate is the probability of orretly resolving ambiguity. From Figure 14, the proposed method will ahieve a reliable ambiguity resolving as long as the SCR is more than 1 db. Hene, the anti-noise performane of the proposed algorithm is fine. Finally, the veloity ompensation result of a moving target is simulated by omputer. Suppose that the range migration of target

16 538 Liu and Chang Normalized intensity (db) Relative range (high range resolution ell) (a) Normalized intensity (db) HRRP 1 HRRP 2 HRRP Relative range (high range resolution ell) (b) Figure 15. Syntheti HRRP. (a) HRRP without motion ompensation. (b) HRRPs after motion ompensation. (HRRP1: the ideal HRRP; HRRP2: ompensated by the veloity estimation with the proposed method; HRRP3: ompensated by the oarse veloity provided by the INS.). eho has been orreted with the oarse veloity estimation v m by the range alignment method proposed in Setion 2. The target has four sattering enters with the same radial veloity, and the real radial veloity between the radar and the target is 55 m/s. The HRRP of the target, as shown in Figure 15(a), dispersed seriously without veloity ompensation. The oarse veloity v m provided by the missileborne INS is 52 m/s, the aurate radial veloity estimation ˆv obtained by the proposed algorithm is m/s. Figure 15(b) shows the ideal HRRP of the target and the HRRPs ompensated by ˆv and v m, respetively. Where HRRP1 is ideal, HRRP2 is the result ompensated by ˆv, and HRRP3 is the result ompensated by v m. It an be seen that ompared to HRRP1, HRRP3 has resolution and amplitude loss, and HRRP2 has higher amplitude and range resolution than HRRP3 and almost no distortion. The differene between HRRP2 and HRRP1 is that HRRP2 has a shift of about two high resolution ells. 5. CONCLUSION The disadvantage of FSCR is the ompliation aused by range- Doppler oupling, due to relative motion between the radar and the target, whih results in irular shift and the spreading of HRRP. Motion ompensation is generally adopted to solve the problem in realizing syntheti HRRP for a moving target. An effetive veloity estimation algorithm based on the waveform entropy (WE) of the Doppler amplitude spetrum of the target ehoes is developed for

17 Progress In Eletromagnetis Researh, Vol. 136, FSCR whih is assumed to work in anti-ship radar seeker. The target eho model of missile-borne FSCR is established, and the range alignment is aomplished in frequeny domain firstly. Then, the veloity estimation algorithm based on the WE of the Doppler amplitude spetrum is addressed, and the veloity estimation auray is derived. The simulation indiates that the algorithm an aurately estimate the radial veloity between the radar and the target and reonstrut the distorted HRRP suessfully. In fat, Doppler frequeny estimation for FSCR is the main idea behind the proposed algorithm, whih is aomplished by resolving the range-doppler oupling and the Doppler ambiguity. It should be pointed out that the proposed method also works in the senario of multi-target with different veloities and has good anti-noise performane. Moreover, the method needs not to transmit and reeive assistant waveform, and the omputational onsumption is aeptable for missile-borne radar signal proessing. However, the oarse veloity estimation (provided by the missile-borne INS) is needed in the proposed method, and the sea lutter is assumed to be Gaussian distributed, whih is an insuffiieny for the paper. Further study on more adaptive veloity estimation algorithm in non-gaussian sea lutter [33, 34] for FSCR will be disussed in our next study. ACKNOWLEDGMENT The authors wish to thank the editors, reviewer 1, reviewer 2, reviewer 3, and reviewer 4, whose omments and suggestions helped to improve the ontents of the paper. REFERENCES 1. Wehner, D. R., High-resolution Radar, Chapter 4 Chapter 5, Arteh House, Boston, Xu, H.-Y., H. Zhang, K. Lu, and X.-F. Zeng, A holly-leaf-shaped monopole antenna with low RCS for UWB appliation, Progress In Eletromagnetis Researh, Vol. 117, 35 5, Park, S.-H., H.-T. Kim, and K.-T. Kim, Stepped-frequeny ISAR motion ompensation using partile swarm optimization with an island model, Progress In Eletromagnetis Researh, Vol. 85, 25 37, Chua, M. Y. and V. C. Koo, FPGA-based hirp generator for high resolution UAV SAR, Progress In Eletromagnetis Researh, Vol. 99, 71 88, 29.

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