Narrow-Scope Searching FrFT Method Applied in the Elimination of Sonar Ranging Error

Transcription

1 International Journal of Infoation an Electronic Engineering, Vol. 4, No. 5, Setember 14 Narrow-Scoe Searching FrF Metho Alie in the Elimination of Sonar Ranging Error Yonghou Yang, Shiliang Fang, an Shuai Yao Abtract For the quetion that the comutation of earching eak i large in ignal arameter etimation uing FrF metho, a new metho of narrow-coe earching FrF i rooe in thi aer. Accoring to the oible ee range of the target motion, the change range of receive LFM ignal arameter caue by Doler Effect can be reicte. By etimate the FrF otimal orer in a very mall range accurately, the earch coe an comutation amount will be greatly reuce. An thi metho wa uccefully alie to eliminate the active onar ranging error. heoretical analyi an imulation have confie mall oeration amount, high accuracy an goo real-time erfoance for thi reente metho. Inex e Active onar, error elimination, fractional fourier tranfo (FrF), linear frequency moulation (LFM) an arameter etimation. I. INRODUCION he time length an broaban of Linear Frequency Moulation (LFM) ignal can ajut iniviually. It i a comlex an comreible ignal, an ha wiely ue in raar, onar, Acoutic Doler Current Profiler (ADCP) [1]-[3] an other ytem. he tatitical roertie of LFM ignal change with time, then it i a tyical of non-tationary ignal comare to the traitional Fourier tranfo. With the eveloment of non-tationary ignal roceing theory, a variety of time-frequency analyi metho are emerging, uch a hort time Fourier tranfo (SF), wavelet tranfo (W) [4], Wigner-Ville Ditribution(WVD) [5], WVD-Hough ranfo(wh) [6], [7], Fractional Fourier ranfo (FrF) [8] an o on. he obervation winow of SF i very narrow, an the with of the W time winow i varie. hee characteritic all affect the ignal reolution in the time-frequency omain. he WVD ue on the LFM ignal ha a goo time-frequency gathere erfoance, but becaue it i non-linear tranfoation, when ealing with multi-comonent LFM ignal it will be affecte by cro-te then the erfoance lowing [9]. WH can effectively ure cro-te interference, but the calculation conume too much time an the initial hae cannot be etimate. FrF i a linear tranfoation, 1 without cro-te interference, an i very uitable for roceing the LFM ignal. he traitional Fourier Manucrit receive March 6, 14; revie May 9, 14. hi work wa uorte in art by the Funamental Reearch Fun for the Central Univeritie (114 7, ). Yonghou Yang, Shiliang Fang, an Shuai Yao are with Southeat Univerity, Nanjing, 196 China ( yyang@eu.eu.cn). tranfo can be een a ignal ecomoition on haonic wavelet, when the ignal i a haonic comonent, the energy of the Fourier tranfo will gather on the correoning haonic frequency. Similar to the Fourier tranfo, FRF alo can be een a ignal ecomoition bae on the Chir, when the LFM of the orer with the FrF into a certain relationhi, when the LFM moulation frequency an FrF orer have a certain relationhi, the energy of converion reult will focu on a articular oint. In the roagation of the acoutic wave, the henomenon that the ignal frequency hifte ue to the Doler Effect i known a the Doler frequency hift. here are two main reaon for Doler hift i generate on unerwater acoutic channel, one i the relative movement of the ignal receive an tranmitte an the other i the movement of the meia. Becaue of the Doler hift, the receive ignal of LFM ignal ha change in the time cale tranfoation. When the center frequency, wavelength an hae of LFM ignal change, the DOA etimation accuracy will be inevitably affecte. Bae on the above, a comenation metho of the Doler hift mimatch in the active onar i rooe. he arameter of the echo ignal i etimate by FrF in orer to ajut the arameter of the ule comreion ignal an reuce the avere effect of the Doler frequency hift. In orer to reuce the comutation an imrove the real-time, a metho that reuce FrF eak earch range accoring to the oible velocity range of the target boy i rooe. he imulation reult how that the mimatch comenation effect i obviou, an the comutation i reuce a lot than common FrF, an the racticality of the metho i obviou. II. ANALYSIS OF HE IMPAC OF DOPPLER SHIF ON LFM PULSE COMPRESSION LFM ule ignal with noie can be exree a: t x( t) a rect( )ex( j j ft jt ) w( t) (1) Wherein, reectively, a f an B/ i the amlitue, the initial hae, the center frequency an the moulation frequency of the LFM ignal. rect(t/) i a -with rectangle function an wt () i an aitive white Gauian noie. When there i relative movement between the meauring robe an the target evice, the frequency of the receive ignal will change an the center frequency alo change. For a ingle-frequency ignal f t, the moving ee of tranmitting ie i aume a v, the moving ee of the DOI: /IJIEE.14.V

2 International Journal of Infoation an Electronic Engineering, Vol. 4, No. 5, Setember 14 receiving en i aume a v R. he frequency f r of the receive ignal can calculate by (). c- v fr kf, k, f ( k -1) f () c- v Wherein, v, vr are vector, an they take tranmit to the receiver ie a the oitive irection, their unit are m/. he unit of f an f R are Hz, c tan for the oun ee in eawater, an generally 15 m/ [1], the coefficient k i efine a ratio between the receive ignal frequency an the frequency of the tranmitte ignal, calle the Doler factor. f i the Doler hift frequency. Equation () how that cale tranfoation on time omain occurre on the LFM ule ignal affecte by Doler Effect. Aume that the tranmiion ignal i xt () hown in (1), the Doler Effect make the ignal become x( kt - ) at the receiving en. Wherein, k i the Doler factor, i the receiving elay. he center frequency of receive ignal become kf from f, frequency moulation rate become k from. In active onar ytem, array element acing i aume fixe, the DOA of an acquiition target i θ, an the elay of element i remain at τ i. he hae ifference φ i between two ajacent array element a (3) how. Doler Effect make the hae ifference of receive ignal between two ajacent array element from φ i to kφ i. R ( i-1) i - in f (3) c he ignal after matche filter of a LFM ule ignal without Doler Effect how in (4) [1]. On the contrary, a LFM ule ignal with Doler Effect become y( t - f / ) after the matche filtering. Wherein, f tan for the Doler hift frequency an i the frequency moulation rate of LFM ignal. In other wor, Doler Effect make the ignal enveloe after ule comreion hift t f / in the timeline. Ranging error caue by thi i S ct / cf / ( ). in[ B( t - t )] y() t a B (4) B( t - t ) In (4), a i ule amlitue, tan for ule with, B i FM with an t i the aitional elay coure by match filter. III. COMPENSAE RANGING ERROR USING FR A. FrF Energy Concentration Characteritic on LFM Signal FrF i a generalize fo of Fourier tranfo. If Fourier tranfo i a ignal rotate counterclockwie π/ from the time axi to the frequency axi in the time-frequency lane, FrF can be regare a a ignal rotate counterclockwie angle α from the time axi to the u axi. P-orer FrF of ignal xt () i efine a (5). F X ( u) F [ x( t)] K ( u, t) x( t) t Wherein, R i the orer of FrF, F [] i FrF - oerator an K ( u, t) i FrF kernel function. A In (6), ex[ ( - )] A j u ut t n K ( u, t) ( u - t) n ( u t) (n 1) (5) (6) cot, cc, / ( R) an 1 jcot. FRF can be een a the ecomoition of a ignal bae on Chir bae. When the frequency moulation rate of a LFM ignal an FrF orer have a certain relationhi, the energy of converion reult will focu on a articular oint [11]. For a noie-free LFM ignal x( t) a ex( j f t jt ), when it center frequency meet f an the FrF orer meet arc cot( ) /, it FrF i efine by (7). F [ x( t)] X ( u) A ( u) (7) In (7), A i a -relate factor, δ(u) i imule function. Foula (7) how that the converion energy aroximately focue on a oint. hi i calle the otimal orer. B. Comenation Metho of Ranging Error A hown in Section II, the Doler Effect will generate a target ranging error S cf /. o thi en, an error comenation metho i rooe: to etimate the center frequency an FM rate of receive LFM ule ignal uing FrF, then ue them to correct the matche filtering ignal arameter in real time, in orer to reuce the ranging error. C. Narrow-Scoe Searching FrF Metho A LFM ignal with noie can be efine by the (8). ( ) ex( ) ( ) x t a j j f t jt w t (8) Wherein, reectively, a f an B/ i the amlitue, the initial hae, the center frequency an the moulation frequency of the LFM ignal. he roce of etecting an etimating above LFM ignal uing FrF metho can be ecribe a (9) an (1)., u arg max X ( u) (9), u cot f u cc X ( u ) arg A ex(j u cot ) X ( u ) a ta (1) In orer to reuce comutation an imrove thi metho uability, we narrow FrF eak earching range in a mall coe [, ] which can be etimate by the own u 385

3 International Journal of Infoation an Electronic Engineering, Vol. 4, No. 5, Setember 14 oible region of[ v, v ] relative velocity between the etection equiment an tet objective. Comaring to the full-coe earching FrF metho, the narrow-coe earching metho can exonentially imrove the etimation accuracy in the cae of enuring the ame amount of comutation, alo can alo exonentially reuce the amount of comutation in the cae to enure the ame etimation accuracy. Firtly, get the range of Doler factor k by k ( c - v ) / ( c - v ) an v v [ v, v ] hown in (). R R c v c v k c v c v (11) hen, to euce the region of receive ignal FM rate accoring to the concluion hown in the Section II: FM rate of receive ignal i k time tranmitte ignal FM rate. c v c v c v c v (1) Before oing FrF of a LFM ignal, thi LFM ignal houl be imenionle noalize. here i a relationhi between ignal frequencie an FM rate before an after noalization [1]. In (13), ' f ' f / f, ' / f (13) f i noalize frequency an ' noalize FM rate. herefore, the forecating range of noalize FM rate how a follow. c v c v / f ' / f c v c v i (14) hen, accoring to the (1) an the firt line of (1), the earch range of FrF rotation angle i erive a follow. c v c v arc cot[ / f] arc cot[ / f] c v c v (15) he calculation comlexity of (9) i greatly reuce ue to c>>v m an the earch range the rotation angle in (15) i limite to a mall range. Here i an examle, auming c 15 m /, v 3 m / an 1 Hz /, the range of rotation angle can be calculate a or he reence of two interval i ue to the erioic of the anti-cotangent function. In cae of enuring ame earching accuracy, the calculation comlexity of thi metho i only about 1/6 full-coe earching metho ( ). Dicrete FrF can be calculate by uing Ozakta algorithm [13] in ractical engineering alication. hi algorithm can be imlemente uing FF, an it ha the ame comutational comlexity ( lg ) O N N with FF algorithm. Before calculating, the ignal houl necearily be imenionle noalize. In orer to make the range of ignal time omain rereentation i efine on [- /, / ] an the range of it frequency omain rereentation i efine on [- F /, F / ]. he fo of icrete FrF calculation can be efine a (16). n X ( m) F [x( )] F N A j ( m mn n )/(F) n e x( ) F F nn (16) Wherein, X ( m) i the DFrF of ignal xt (), cot, cc, /. D. Correction of Matche Filtering Parameter Baing the (9) an (1) in Section A, we can etimate the frequency an FM rate of receive ignal, an the etimate value reectively are f an. he relationhi between them an the tranmitte ignal one are hown a follow. f kf k, (17) In the cae of none comenation for the ranging error, the frequency an FM rate of the coy ignal uring the matche filtering of receive ignal are reectively f an. Due to the Doler Effect, an error S cf / ( ) i generate uring the active onar ranging. o eliminate thi error, frequency an FM rate of a coy ignal uing in matche filtering are ajute to f,. he theoretical analyi in the Section II of thi aer how that thi metho can effectively eliminate ranging error. Meanwhile, becaue of the mall amount of calculation, arameter ajutment can be one in real time. IV. SIMULAION ANALYSIS In orer to verify the valiity an efficiency of etimating a LFM ignal center frequency an FM rate uing narrow-coe earching FrF metho, a well a the valiity of correcting the ranging error caue by the Doler effect uing matching arameter correction metho, the following imulation analyi i given. In an active onar ytem, the tranmitte ignal i a LFM ule ignal, ule with i.5, amling frequency f = Hz, the number of amling oint i 11, center frequency i f 5Hz, FM rate i 1 Hz /. Interference i Gauian white noie, SNR i B. Auming meaurement equiment i fixe an meauring target frontally aroache at the ee of 6 knot (3m / ). Doler factor k 1.8 can be calculate by uing the equation (). heory value of receive ignal frequency an FM rate reectively are fr kf 51Hz an R 14 Hz /. Before FrF calculation for the receive ignal, it nee to have the receive ignal be imenionle noalize. Proce i a follow: Firtly, et the time origin at the mioint of ignal obervation time to make the ignal time-omain locate on the interval [-.5,.5], frequency-omain locate on the interval [-1Hz, 1Hz], o the cale factor S / f =.5 an the noalize with [11] x f = 1. Seconly, change the ignal time-omain to the interval [-.5/ S,.5/ S], 386

4 International Journal of Infoation an Electronic Engineering, Vol. 4, No. 5, Setember 14 change the amling rate to x, change the center frequency to f S an change the FM rate to S. Both time-omain an frequency-omain of the noalize ignal locate on the interval [-x/, x/] i.e., [-5, 5]. Accoring to the analyi in Section III-B Part A, the forecating range of FrF otimum rotation angle for the receive ignal i or (ouble interval i ue to erioic of anti-cotangent function), correoning to the forecating range of the orer i or he comarion of two FrF metho imulation reult are hown a follow. () Sie view of Fig. (c). Fig. 1. he contrat of two FrF metho etimation reult on ame ignal. Firtly, o full-coe earching FrF whoe arameter are 4 an teer =.1 for the receive LFM ignal, the imulation reult i hown in Fig. 1. he evaluation reult i = 3.17 an u.54. he time thi comutation conume i t F he etimation value of the analyze ignal center frequency an FM rate before noalization reectively are f 58 Hz, 168 Hz/ accoring to (1) an (13). 1. (a) Full-coe ( 4)FrF earching reult. 1 theoretical oition meaure otion X: 356 Y: 1.14 X: 361 Y: Amlitue Ditance(m) (a) Delay between filtering reult an theoretical wavefo when filter arameter in t uate. 1. (b) Sie view of Fig. (a). 1 X: 356 Y:.9967 X: 356 Y: Amlitue Ditance(m) (b) filtering reult matche the theoretical wavefo when filter arameter i uate. Fig.. Ranging error comenation by uating matche filter arameter. (c) Narrow-coe ( )FrF earching reult. Seconly, o narrow-coe earching FrF whoe arameter are an teer =.1 for the 387

5 International Journal of Infoation an Electronic Engineering, Vol. 4, No. 5, Setember 14 receive LFM ignal, the imulation reult i hown in Fig.. he evaluation reult alo i = 3.17 an u.54. he time thi comutation conume i t F.757. he etimation value of the analyze ignal center frequency an FM rate before noalization reectively are f 58 Hz, 168 Hz/ accoring to (1) an (13). he evaluation reult of narrow-coe earching FrF metho an full-coe earching FrF are ame, But the foer time ent i only one-thouanth the latter. hirly, uate the coy ignal center frequency an FM rate uring matche filtering uing the etimation value of the receive ignal arameter. Fig. how the contrat of matche filtering reult between the no-uate an uate ituation. It eay to ee, when the arameter are NO uate, there i an obviou time elay between the matche filtering reult an theoretical wavefo on the timeline. he elay i the reaon for ranging error. Differently, when the arameter are uate, there i no time elay baically between the matche filtering reult an theoretical wavefo on the timeline. he ranging error i comenate uccefully. V. CONCLUSION hi aer focue on uing the narrow-coe earching FrF metho to etimate the arameter of LFM ignal an comenating the active onar ranging error caue by the Doler Effect by uating matche filter arameter with etimation value in real time. heoretical analyi an imulation have both confie that the narrow-coe earching FrF metho can etimate LFM ignal arameter accurately, an it comutation amount reuce three orer of magnitue at leat than the full-coe earching FrF metho. Uate matche filter arameter metho can comenate meter-level ranging error of active onar ytem baing on ignal arameter accurate etimation on FrF metho. Goo characteritic of having mall comutation amount an goo real-time erfoance make narrow-coe earching FrF metho ha high value in engineering alication. ACKNOWLEDGMEN he author wih to acknowlege the helful comment an uggetion from our team member in Southeat Univerity, Nanjing, China. An thi aer i financial uorte by the Funamental Reearch Fun for the Central Univeritie (114 7, ). REFERENCES [1] P. M. Djuric an S. M. Kay. "Parameter etimation of chir ignal," IEEE ranaction on Acoutic, Seech an Signal Proceing, vol. 38, no. 1, , 199. [] B. Sergio an V. Petrone, "Analyi of olynomial hae ignal by the integrate generalize ambiguity function," IEEE ranaction on Signal Proceing, vol. 45, no., , [3] Y. C. Mao an Z. Bao, "Parameter etimation of chir ignal with time-varying amlitue uing Cyclotationary aroach," Act a Electronica Sinica, vol. 7, no. 4, , [4] A. M. Haimovich, C. D. Peckham, an J. G. eti Jr, "SAR imagery of moving target: Alication of time-frequency itribution for etimating motion arameter," in Proc. SPIE' International Symoium on Otical Engineering an Photonic in Aeroace Sening, International Society for Otic an Photonic, [5] P. Rao an F. J. aylor, "Etimation of intantaneou frequency uing the icrete Wigner itribution," Electronic Letter, vol. 6, no. 4, , 199. [6] B. Sergio, "Analyi of multicomonent LFM ignal by a combine Wigner-Hough tranfo," Signal Proceing, IEEE ranaction on, vol.43, no. 6, , [7] X. H. ian, G. S. Liao, an Y.. Wu. "Joint etimation of Doler an multiath time elay of overlaing echoe for LFM ule raar," Act a Electronica Sinica, vol. 3, no. 6, ,. [8] X. G. Xia, "Dicrete chir-fourier tranfo an it alication to chir rate etimation," Signal Proceing, IEEE ranaction on, vol. 48, no. 11, ,. [9] H. I. Choi an W. J. William. "Imrove time-frequency rereentation of multicomonent ignal uing exonential kernel," IEEE ranaction on Acoutic, Seech an Signal Proceing, vol. 37, no. 6, , [1] H. J. Hu, "Range error analyi of range-oler couling effect to LFM raar," Moem Raar, vol. 33, no. 1,. 47-5, 11. [11] R. ao, B. Deng, an Y. Wang, Fractional Fourier ranfo an it Alication, inghua Univerity Pre, 9. [1] X. H. Zhao, B. Deng, an R. ao, "Dimenional noalization in the igital comutation of the fractional fourier tranfo," ranaction of Beijing Intitute of echnology, vol. 5, no. 4, , 5. [13] H. M. Ozakta et al., "Digital comutation of the fractional Fourier tranfo," IEEE ranaction on Signal Proceing, vol. 44, no. 9, , Yonghou Yang wa born in 1985 in Hefei City, P.R. China. He receive the B.S. egree in electronic engineering from Anhui Univerity, Anhui Province, China in 6 an the M.S. egree in ignal an infoation roceing from Guilin Univerity of Electronic an echnology, Guangxi Province, China in 13. During hi M.S. tuy, he wa a member of water-etecting raar reearching team an tuie the roceing of raar ignal. He i currently a PhD tuent in the School of Infoation Science an Engineering at Southeat Univerity, Nanjing, China. Hi current reearch interet are in time-frequency analyi, onar ignal roceing an tatitical ignal roceing. Shiliang Fang wa born in 1959 in Changzhou, P. R. China. He receive the B.S egree an M.S egree from Southeat Univerity, Nanjing, China, in 1983 an 1986, all in electronic engineering. After hi grauate tuie, he joine Southeat Univerity, Nanjing, China, where he i reently a rofeor of electronic engineering. He i a enior member of Acoutical Society of China an the author or coauthor of over 16 referee journal article an numerou conference reentation an aer. Hi acaemic interet inclue tatitical ignal roceing, array ignal roceing an UWA (Unerwater Acoutic) electronic engineering. Shuai Yao wa born in Zao-zhuang City, China, on March 13, he eucational backgroun i a follow. Firt, PhD tuent of engineering, Southeat Univerity, Nanjing, China, 11-reent. he major i infoation an communication engineering; Secon, Mater of Engineering, Southeat Univerity, Nanjing, China, he Major i infoation an communication engineering; hir, bachelor of engineering, Ocean Univerity of China, Qingao, China, 5-9, the major i electronic infoation engineering. He ha been working in Key Laboratory of Unerwater Acoutic Signal Proceing of Minitry of Eucation a PhD tuent from 11 till now. Current an reviou reearch interet i onar ignal roceing an tatitical ignal roceing. Mr. Yao ha ublihe five aer on onar ignal roceing an tatitical ignal roceing, wrote four invention atent fo 11 till now. 388