A Micro-Doppler Modulation of Spin Projectile on CW Radar

Transcription

1 ITM Web of Confeences 11, (2017) DOI: / tmconf/ A Mco-Dopple Modulaton of Spn Pojectle on CW Rada Zh-Xue LIU a Bacheng Odnance Test Cente of Chna, Bacheng , P. R. Chna Abstact. To obtan the spn speed of pojectle effectvely, a mco-dopple modulaton model of otatng pojectle measued by contnuous-wave ada (CW ada) s ntoduced. Hgh spn speed of pojectle bngs mco-dopple modulaton on echoes of CW ada, and thee ae many mco-dopple modulaton hamonc waves n the zeo ntemedate fequency (ZIF) echoes. The fequency nteval of the adjacent hamonc waves s seveal tmes of otatonal fequency, but the ntegal multple s unknown. The smulaton esults pove coectness of the poposed mathematc model. 1 Intoducton The spn speed s a key test paamete of motve pojectle and dectly elated to the desgn of vaous pojectles, gudance devces and fuses etc. The man methods to measue pojectle spn speed n ealy stage, such as telemety method and optcal method, have some defects 1. To the telemety method, some sensos must be added n the pojectle, so t wll change the stuctue of pojectle. The optcal method could only measue a lttle pat of the ballstc tajectoy and t s affected geatly by the envonment. In fact, hgh speed spn of pojectle bng modulaton on echoes of CW ada, and thee ae many hamonc waves aound Dopple sgnal, whch called mco-dopple effect [2]. In ecent yeas, spn speed s measued by contnuous wave Dopple ada, though goovng the bottom of the pojectle, whch can stengthen the mco-dopple sgnal [3]. Compaed wth optcal method, ths method can measue the spn speed of pojectle along wth tme, but t also changes the stuctue of pojectle as the telemety method. In ode to measue spn speed wth unboken pojectle, a method based on dgtal demodulaton s poposed by extactng the mco-dopple sgnal n the echo of CW ada, whch s the fst applcaton of mco-dopple technology on unboken pojectle [4]. But the mathematcal model poposed by Refeences 4 s ncomplete. The method poposed n Refeences 4 s ncomplete, and the extacted spn speed s ambguous. So, an emendatoy mco-dopple modulaton of otatng pojectle on CW ada echoes s poposed, whch explans the eason of spn speed ambguous. 2 Modulaton of otatng pojectle The stuctue of uncontolled pojectle, such as cannonballs, ockets and etc, s symmetcal. Wthout loss of genealty, t s supposed that thee ae N scatteng pontes p 1,, p,, p N, evenly a Coespondng autho: luzx09@qq.com The Authos, publshed by EDP Scences. Ths s an open access atcle dstbuted unde the tems of the Ceatve Commons Attbuton Lcense 4.0 (

2 ITM Web of Confeences 11, (2017) DOI: / tmconf/ dstbuted on the bottom edge, whch ae l away fom the cente of the bottom shown as Fgue 1. In the coodnate system of pojectle axs, the coodnates of p s X =(0, l snθ, l cosθ ) T, whee θ =2πf t+2π/n, f s the spn fequency of pojectle. y p 1 p 2 p 3 z x Fgue 1. Rotaton dagam of pojectle Rada measuement coodnate system O XYZ and efeence coodnate system O d X N Y N Z N s establshed shown as Fgue 2. The ogn of ada measuement coodnate system s the ada, and the O X axs paallels the fng azmuth of the gun. The ogn of efeence coodnate s the cente of the pojectle bottom, and theo d X N axs paallels the decton of velocty. Refeence coodnate system moves wth the pojectle. In the pocess of launchng, the O X axs of ada measuement coodnate system almost paallels the decton of velocty. So the angula alttude β of velocty n Fgue 2 s equal to the elevaton E l of ada n measuement coodnate system, and the azmuth ψ of velocty s equal to azmuth A z of ada. Usually, ada s placed close to the fng azmuth of the gun, so the measung azmuth A z of ada s vey small, and so t also can be consdeed A z =0 hee. Afte twce otatons, the coodnates of p n efeence coodnate system can be tansfomed nto the coodnates of p n the coodnate system of pojectle axs. Fst, efeence coodnate system otates aound the new Y axs wth ψ, and then otates aound the Z axs wth β. The β s elevaton of the pojectle axs and ψ s azmuth of the pojectle axs. Theefoe the coodnates of scatteng ponte p n efeence coodnates s shown as Eq. 1. x cos El sn El 0 y sn E cos E 0 = X z l l (1) Y Z O El p Az RO y Y N P x z Od X Z N Q X S N Fgue 2. The elatonshp of coodnate systems The slant ange between p and ada s shown as Eq. 2. 2

3 ITM Web of Confeences 11, (2017) DOI: / tmconf/ = [( Rcos E cos A l sn E sn θ ) 2 l z l + ( Rcos E sn A + l cos E sn θ ) l z l + ( RsnE + l cos θ ) ] l 2 1/2 = [ R + 2 R( a cosθ + b sn θ ) + l ] (2) Whee, a= lsn E l ; b= lcos Elsn( Az + El). Wthn a cetan tme, azmuth A z, elevaton E l and β vay slowly compaed wth mco-moton, so they can be consdeed as constant. Because R>>l, can be appoxmated as Eq. 3 based on Taylo expanson. R+ ( a cosθ + b sn θ )/2 = R+ dcos( θ + ξ) (3) 2 2 1/2 Whee, d = ( a + b ) /2; ξ = ctge sn( A + E ). l z l The contnuous wave ada tansmts a sngle tone shown as Eq. 4. s () t = cos(2 π ft) (4) T In Eq. 4, f s the fequency of sngle tone. Whle the sngle tone llumnates the otatng pojectle, the echo fom the scatteng ponte p s shown as Eq. 5. s () t = m cos[2 π f( t 2 c)] (5) 0 In Eq. 5, m 0 s the ampltude of the echo fom the scatteng ponte p. Put Eq.3 nto Eq.5, then s () t = m cos{2π ft 4 π fr/ c+δ ϕ} (6) 0 Whee 2πft s the phase vayng wth tme of sngle tone, 4πftR/c s the phase etadaton caused by the ange between the ada and the pojectle, Δφ s the phase shft caused by otaton of pojectle and shown as Eq.7. Δ ϕ = 4π fd cos( θ + ξ)/ c = 4π fd cos(2π f t + 2 π / N + ξ)/ c (7) In Eq. 7, the fom of Δφ s same wth the phase of fequency modulaton, whch means that the phase shft s the fequency modulaton [5]. Accodng to the fequency modulaton theoy, whle Δφ >π/6, the echo fom the scatteng ponte p can be expanded to a tgonometc sees shown as Eq. 8, whch coeffcents ae Bessel functon. + s () t = m J ( β )cos(2π ft 0 k FM 4 π fr / c + 2πkf t + 2 πk / N + kξ) (8) In Eq. 8, J k (β FM ) s the kth Bessel functon, β FM =4πfd/cf s the fequency modulaton coeffcent. The ZIF echoes ae shown as Eq. 9. 3

4 ITM Web of Confeences 11, (2017) DOI: / tmconf/ x () t = m J ( β ) k FM exp[ j2 π( f + kf ) t+ 2 πk/ N] d (9) In Eq. 9, m s the ampltude of the ZIF echoes fom p, f d the s the Dopple fequency. The ZIF echoes fom all scatteng pontes ae shown as Eq. 10. xt () = Mk exp[ j2 π ( fd + kf)] t (10) In Eq. 10, M k s the ampltude of mco-dopple modulaton hamonc waves and shown n Eq. 10. ( β ) N M = J m exp( j2 πk/ N) (11) k k FM = 1 Whle the scatteng ae equal, the ampltude of echoes fom dffeent scatteng ponts ae equal, then m 1 = m 2 = = m N. The ampltude of hamonc wave, whch fequency s kf, s shown n Eq.11. M k NJ = k ( FM ) m1 β k = NK 0 k NK (12) In Eq.11, K s an ntege, and the hamonc wave whch fequency s not Nf dsappeaed. The mathematc model n Eq. 10 shows that the ZIF echoes fom the otatng pojectle contan the mco-dopple modulaton hamonc wave. The fequency nteval of the adjacent hamonc waves s the otatonal fequency, and the fequency nteval between the adjacent hamonc waves to the Dopple sgnal s also seveal tmes of the otatonal fequency. In geneal, all hamoncs should exst as the Eq. 11 shows, but n some case some hamonc ae dsappeaed as the Eq. 12 shows. Compaed Eq. 11 and Eq. 12, the fequency nteval of the adjacent hamonc wave s ndefnte n dffeent condtons, and the spn fequency cannot be extacted dectly fom the fequency nteval between the adjacent hamonc waves o the hamonc fequency. 3 Pocessng of mco-dopple sgnal In the ZIF echoes fom the otatng pojectle, the mco-dopple modulaton hamonc wave s weake than the Dopple sgnal, and the fequency nteval between the hamonc waves and the Dopple sgnal s also vey small, so the mco-dopple modulaton hamonc wave ae submeged by nose n the powe spectum calculated by the Foue tansfom. In the pactcal engneeng, zoom- FFT s used fo analyzng the spectum of mco-dopple modulaton hamonc wave at hgh esoluton. Afte the Dopple fequency f d s obtaned, a local oscllato can be stuctued as Eq. 13. () d s1 t j2π f t = e (13) Usng s 1 (t), then the ZIF echoes x(t) can be demodulated to a new sgnal shown n Eq. 14. x() t = xts () () t = xte () 1 j2π fd t = Mk exp[ j2 π ( kf) t] (14) 4

5 ITM Web of Confeences 11, (2017) DOI: / tmconf/ In the new sgnal x (t), the Dopple sgnal s shfted to zeo fequency, and the mco-dopple modulaton hamonc wave, whch fequency s f d+ kf, ae shfted to kf.. Then the new sgnal x (t), can be named as the mco-dopple sgnal. To mpove the spectum esoluton and estan the fequency alasng, zoom-fft adopts a low ate e-samplng and a low pass flte. Anothe Foue tansfom can calculate the spectum of x (t), and the fequency peak n spectum s the fequency of mco-dopple modulaton hamonc wave. 4 Expements In ths secton, the coectness of the poposed mathematc model s poved though two smulaton expement. The contol paametes used n ou smulaton ae summazed n Table 1. Table 1. Paametes used n the smulaton example. Cente fequency Hz Sample nteval 20μs Radal velocty t m/s Spn speed t+0.05*t 2 Hz Radal ange Eq. 2 Scatteng pont 2 o 4 In the fst expement, the ampltude of the echo fom scatteng ponte s a andom numbe that obeys the (1, 0.5) nomal dstbuton. The spectogam and powe spectal densty ae shown n Fgue 3. In Fgue 3 (a), the spectogam of the mco-dopple sgnal fom the otatng pojectle wth 2 scatte s shown n Fg. 3 (a). Fg. 3 (b) shows the powe spectum cuve at the fst tme pont n Fgue 3 (a). In Fgue 3 (c), the spectogam of the mco-dopple sgnal fom the otatng pojectle wth 4 scatte s shown. Fgue 3 (d) shows the powe spectum cuve at the fst tme pont. It s shown fom Fgue 3 that, the fequency nteval of the adjacent hamonc waves n the mco-dopple sgnal fom scattes wth andom scatteng coeffcent s the otatonal fequency. The fequency nteval of the adjacent hamonc waves s seveal tmes of the otatonal fequency, but the ntegal multple s unknown. (a) spectogam of 2 scatte (b) spectum on 1 st pont of 2 scatte (c) spectogam of 4 scatte (d) spectum on 1 st pont of 4 scatte Fgue 3. The Spectogam and spectum of andom scatteng 5

6 ITM Web of Confeences 11, (2017) DOI: / tmconf/ In the second expement, the ampltude of the echo fom scatteng ponte s equal nstead of the andom dstbuton. The esults ae shown n Fgue 4. (a) spectogam of 2 scatte (b) spectum on 1 st pont of 2 scatte (c) spectogam of 4 scatte (d) spectum on 1 st pont of 4 scatte Fgue 4. The spectogam and spectum of equal scatteng It s shown fom Fgue 4 that, the fequency nteval of the adjacent hamonc waves n the mco- Dopple sgnal fom scattes wth dentcal scatteng coeffcent s seveal otatonal fequences. The fequency nteval of the adjacent hamonc waves s the ntege tmes of otatonal fequency, and the ntegal multple s the numbe of scattes. Compae the esult of Fg. 3 and Fg. 4, the fequency nteval of the adjacent hamonc waves s ndefnte n dffeent condtons, t cannot be extacted dectly fom the ada echoes. Conclusons The mco-dopple modulaton model of the pojectle otaton wth mult-scattes s establshed based on the elatonshp between the ZIF echoes of pojectle measued by CW ada and pojectle otaton. It s deduced that the fequency nteval of the adjacent hamonc waves n the mco-dopple sgnal s seveal tmes of the spn speed of pojectle, and the tmes of the fequency nteval to spn speed cannot be pocessed dectly fom the mco-dopple sgnal. In the pactcal, the multple between the fequency nteval and the spn speed can be calculated wth the help of othe equpment. Fom the esults of smulaton, t s concluded that the mco-dopple modulaton model poposed n the pape s coect. Refeences 1. Han Zpeng. Pojectle and Rocket Exteo Ballstcs [M]. Bejng: Bejng Unvesty of Aeonautcs and Astonauts pess, 2008, V C Chen, F L,S Ho,et al. Mco-Dopple effect n ada-phenomenon, model and smulaton study [J]. IEEE Tansactons on Aeospace and Electonc Systems, 2006, 42(1): Wang Yuan-qn et al. A obust RCS peodcty estmaton algothm fo ballstc taget[j]. Aeospace Electonc Wafae, 2008, 24(2): Zhang Wan-jun Wu Xao-yng Leng Xue-bng, et al. Testng Method of Pojectle Rotatng Speed Based on Mco-Dopple Effect of CW Radas [J]. Jounal of Academy of Amoed Foce Engneeng. 2012, 26(5): Poaks, J. G., Dgtal Communcatons, 3d ed., New Yok, McGaw-Hll,