4th Iteratioal Coferece o Advaced Materials ad Iformatio Techology Processig (AMITP 06) Directio of Arrival Estimatio Method i Uderdetermied Coditio Zhag Youzhi a, Li eibo b, ag Hali c Naval Aeroautical Egieerig Istitute, Yatai, Chia, 64000 a webblee009@sia.com, b yrerzyz@63.com, c gyoei@qq.com Keywords: -disoit orthogoality; Uderdetermied; Directio of arrival estimatio Abstract. Because some sigals have -disoit orthogoality, this paper proposes a sigal directio of arrival estimatio method i uderdetermied coditios. This method is maily based o the priciple that sigals of differet directios lead to differet time delay betwee the sesors, costruct a potetial fuctio by receivig sigal phase differece, ad local maximum value locatio of the potetial fuctio correspods to the sigal directio of arrival. The simulatio results show that the method ca still obtai better estimatio results i the case of oly usig two sesors. Itroductio Directio of arrival (DOA) estimatio is oe of the importat cotets of array sigal processig ad it has bee widely used i radar, soar, commuicatio, seismic exploratio, radio astroomy, ad other fields. Mai approach to estimate the directio of arrival is beam-formig techology. But for beam-formig, because of the limitatio of Rayleigh limit, after array legth is determied, its resolutio is sure []. Traditioal beam-formig ofte requires more sesors to discer the two close goals i the directio. Although some high resolutio algorithms such as MUSIC ad ESPRIT ca break through the limitatio of Rayleigh limit, trasmissio chael i array sigal processig is completely ukow ad time-varyig, which limits the use of high-resolutio directio estimatio i actual eviromet [] I recet years, the directio of arrival blid estimatio method developmet eables directio of arrival estimatio uder the coditio of ukow chael characteristics, which has good applicatio prospect. But uder uderdetermied coditios, the method does ot work. The so-called "uderdetermied" amely is that the umber of sesors is less tha source sigals. Uder uderdetermied coditio, certai features of the source sigals must be used to realize mixig matrix estimatio ad source sigal separatio, such as sparsity or -disoit orthogoality, etc. Scott Richard ad others put forward DUET (Degee Umixig ad Estimatio Techique) method [3] for part sigal s -disoit orthogoality, which ca use two sesors to achieve ay umber of mixed sigals separatio. O the basis of the method, this paper proposes a ew sigal directio of arrival estimatio method, which ca make use of two sesors to achieve multiple sources sigal directio of arrival estimatio. Sigal mixed models ad source hypothesis Sigal mixed model. If ot cosiderig the ifluece of sigal s amplitude atteuatio, multiplicity ad oise betwee sesors, sesor sigals will be the sum of the source sigals with differet time delay. For two adacet sesors, the mixed model is: x( t) s( t) s ( t) () x ( t) s ( t ) s ( t ) s I it, is the time delay of source sigal s () t betwee the two sesors. Set source sigal () t s directio of arrival, the sigal delay shall satisfy equatio (). 06. The authors - Published by Atlatis Press 540
si / c () I it, is the spacig betwee two sesors ad c is sigal trasmissio speed. After get the source sigal s time delay estimatio, we ca get the sigal s directio of arrival estimatio. Source hypothesis. Assume that the source sigal satisfies the followig coditios: ) Source sigal belogs to arrowbad sigal. ) Source sigal s short-time Fourier trasformatio pair wise orthogoal, amely there is - disoit orthogoality betwee source sigals. Give a widow fuctio (t), sigal s () t s short-time Fourier trasform (STFT) [4] is: t F ( s())(, ) ( t) s( t) e dt (3) I Literature [4] the short time Fourier trasforms is called decompositio. It is simply writte as S (, ) i this paper. Narrow-bad sigal s time delay is much smaller tha the badwidth reciprocal. Its effect is equivalet to that basebad sigal produces a phase shift, ad its short-time Fourier trasform should satisfy the followig equatio [5]: F ( s ( ))(, ) e F ( s ( ))(, ) (4) If sigal s (), t s () t s short-time Fourier trasform is orthogoal, we call s(), t s() t -disoit orthogoal. The ituitive sese of so-called "-disoit orthogoality" is that ay poit i timefrequecy plae oly has up to oe sigal which is effective, ad other sigals ifluece ca be eglected. Its mathematical expressio is: S, ) S (, ) 0,, (5) Directio of arrival estimatio ( If source sigal strictly meets -disoit orthogoal, sigals received at ay oe poit i time ad frequecy domai at the maximum is a mixture of source sigals. That is, for ay (, ), if (, ) 0, there is a source sigal s which meets: S (, ) 0, ( k,, ; k ) Sk (, ) 0, At this poit, (, ) (, ) e e e S (, ) S (, ) S (, ) (6) As a result, (, ) e (, ) (7) 54
(, ) This makes the plural phase (, ) s agle ˆ(, ). If the distace betwee two sesors is less tha half wavelegth of the sigal, its delay should satisfy equatio (8). (, ) (8) ˆ(, ) (, ) So, (, ). If (, ) 0, the ˆ(, ),, (9) So if source sigals strictly meet -disoit orthogoality, classify/ ˆ (, ) ad get ay time delay parameter estimatio. Ad this also ca complete source sigal separatio. But i practice, source sigals ca oly be approximate to meet -disoit orthogoal hypothesis, addig to the oise ifluece, / ˆ (, ) ca oly be ear time delay parameters accordig to probability distributio. I order to complete source sigals directio of arrival estimatio, this paper adopts Literature [6] s potetial fuctio method. ( ) l(, ) ( ˆ (, )) (0), ˆ I it, there is ˆ si ( (, )) ( (, )) exp( ) adl(, ) (, ) (, ). The locatio s local maximum value of potetial fuctio is correspodig to each source sigal s directio of arrival. Simulatio experimet ad result aalysis Experimet I order to test the algorithm effectiveess, aalyze algorithm performace i simulatio uderwater acoustic eviromet. Choose three sigle frequecy sigals as source sigals whose frequecies are 3 k, 3.5 k ad 4 k. Sigal sources agle differece betwee two is idetical, i which the middle sigal source is located right i frot of the sesor array. Set array elemets spacig to 5 cm, soud velocity i water is set to 430 m/sec, samplig frequecy 6 KHZ ad data legth 60000. idow fuctio uses Hammig widow with 04 poits, there are 5 poits overlap each other betwee adacet two widow fuctios. Noises are respectively -5 db, 0 db, 5 db, 0 db, 5 db, ad 0 db additive white Gaussia oise. Do 50 times simulatio test for each case, ad the algorithm s recogitio i each case ad the average are show i Table. From the experimet results, we ca see that whe the agle differece of source sigal is greater tha 3 degrees, the average of this method s directio of arrival estimatio uder the coditio that sigal-to-oise ratio is higher tha 5 db is 0.09 degrees (The is maily due to potetial fuctio discretizatio), ad the recogitio is 00%. Ad the Agle differece betwee source sigal certai circumstaces, with the loss of the sigal to oise ratio, the recogitio of the proposed algorithm is reduced, is icreased. he SNR is reduced to a certai degree, the algorithm ca't distiguish the directio of the three sources, ad oly produce a crest ear 0 degree. It ca be see from Table, oise is the mai factor affectig the algorithm, ad the algorithm ca obtai higher precisio estimatio uder certai sigal to oise ratio. 54
Table Agle estimatio s average ad recogitio uder various circumstaces Agle differece Noise itesity -5db 0db 5db 0db 5db 0db 3 5 7 9 Recogitio Recogitio Recogitio Recogitio Recogitio.3 0.47 0.09 0.09 40% 80% 00% 00%.3 0.89 0.4 0.09 0.09 % 56% 9% 00% 00%. 0.78 0. 0.09 0.09 4% 64% 00% 00% 00%. 0.3 0.09 0.09 0.09 5% 94% 00% 00% 00% 3..5 0.8 0.09 0.09 0.09 % 56% 00% 00% 00% 00% Table Array elemet umber required to distiguish three sigal source directios uder the coditio whe oise is 0 db Agle differece Covetioal method Self-adaptive method Array space high resolutio method Capo Batlette LMS LCMV MUSIC ESPRIT 3 35 35 3 8 8 5 9 8 5 7 5 5 4 0 5 4 9 9 5 4 Experimet Calculate array elemet umber required to estimate three source sigal directio of arrival uder the coditio of differet agle differece by all kids of directio of arrival estimatio 543
methods. Set sigal-to-oise ratio as 0 db, other parameters settigs are the same of experimetal oe. Uder the coditio of differet agle differece, distiguish three source sigal directio of arrival s required miimum array elemet umber, which is show i Table. It ca be see from Table that at preset the most commoly used method of directio of arrival estimatio eed more array elemet umber to distiguish the directio of the three goals. Ad i the same coditio, the text method oly eeds two array elemets. Figure Directio estimate figure of all directio of arrival estimatio method Figure shows directio estimate figure of all directio of arrival estimatio whe sigal source agle differece is 7 ad array elemet umber is 6. The method oly uses two sesors array elemets. It ca be see from the diagram that although this method oly uses two arrays elemets data, it ca still get a good estimatio result, especially relative to covetioal beam formig. Coclusio Some sigals have -disoit orthogoality. Based o DUET uderdetermied blid source separatio method, a ew method of sigal directio of arrival estimatio is proposed. The method estimates directio of arrival of sigal source by potetial fuctio costructed by phase differece receivig sigals, which ca break through the limitatio of covetioal beam formig s Rayleigh limit, ad oly two sesors array elemets are eeded to complete multiple source sigal directio of arrival estimatio. Simulatio results show that uder certai coditio, the sigal-to-oise ratio of this method ca get better precisio estimatio. Refereces [] Zhag iada, Bao Zheg. Commuicatio sigal processig [M]. Beiig: Natioal Defece Idustry Press, 000,, 36. [] Jia Yogkag. Array sigal processig robustess research. Ph.D. Dissertatio, i 'a Uiversity of Electroic Sciece ad Techology, 996, 5. [3] A. Jourie, S. Rickard, ad Ö. Yılmaz, Blid separatio of -disoit orthogoal sigals: Demixig N sources from mixtures, i Proc. IEEE It. Cof. Acoust., Speech, Sigal Process., vol. 5, Istabul, Turkey, Jue 5 9, 000, pp. 985 988. [4] I. Daubechies, te Lectures o avelets Philadelphia, PA: SIAM, 99. [5]R. Bala, J. Rosca, S. Rickard, ad J. O Ruaaidh, The ifluece of widowig o time delay estimates, i Proceedigs of the 35th Aual Coferece o Iformatio Scieces ad Systems (CISS000), Priceto, NJ, March 5-7 000, vol.,pp. P (5 7). [6] P. Bofill, M. Zibulevsky, Uderdetermied blid source separatio usig sparse represetatios, i sigal processig 8(00), 353-36. 544