Time Delay Estimation by Bispectrum Interpolation

Transcription

1 Snsors & Transucrs, Vol. 58, Issu, Novmbr 3, pp Snsors & Transucrs 3 by IFSA Tim Dlay Estimation by ispctrum Intrpolation Xiao CHEN, Zhnlin QU Jiangsu Ky Laboratory of Mtorological obsrvation an Information Procssing, Nanjing Univrsity of Information Scinc an Tchnology, Ningliu oa 9, Nanjing, 44, China School of Elctronic an Information Enginring, Nanjing Univrsity of Information Scinc an Tchnology, Ningliu oa 9, Nanjing, 44, China Tl.: , fax: chnxiao@nuist.u.cn civ: 9 August 3 /Accpt: 5 Octobr 3 /Publish: 3 Novmbr 3 Abstract: Tim lay is an important paramtr for th charactrization of signal propagation of snsors. In orr to improv th prcision of tim lay stimation, this articl proposs an algorithm bas on bispctrum intrpolation. Th algorithm first calculats th charactristic spctrum function on th basis of th bispctrum of th two signals. Th frquncy omain intrpolation is on on th charactristic spctrum function. y conucting th invrs Fourir transform an th maxima sking mtho, th calculation formula for th tim lay of th two signals is uc. Th simulation tst rsults show that compar with th bispctrum-bas tim lay stimation mtho, th bispctrum intrpolation-bas tim lay stimation mtho has improv gratly in trms of th stimation prcision an tmporal rsolution. Th algorithm has th avantags of high prcision, small rror an strong ability to supprss th corrlativ Gaussian nois. Copyright 3 IFSA. Kywors: Tim lay, Estimation, Cross-corrlation, ispctrum, Intrpolation.. Introuction Tim lay stimation is a vry important rsarch topic in signal procssing. It is wily appli in raar, snsor, biological scincs, sonar, oil xploration, communications, gophysics, unrwatr acoustics, spch signal nhancmnt an othr scintific fils [-3]. Th cor part of tim lay stimation is th rciv targt signals. On th basis of th known rfrnc signal, th tim lay of th rciv signal in rlation to th rfrnc signal is calculat fast an accurat. Corrlation analysis is a basic tim lay stimation mtho. Tim lay stimation mthos bas on corrlation analysis inclu basic corrlation mtho an gnraliz corrlation mtho. In th basic lay stimation algorithms, th most typical algorithm is cross-corrlation algorithm, an it is also th bas of th scon-orr lay stimation algorithm. Th basic corrlation mtho is avantag by its asy ralization, but is suscptibl to nois an rquir that signal an nois shoul b inpnnt from ach othr. In orr to liminat or ruc th influnc of nois on th corrlation-bas tim lay stimation, Knappy t al propos a gnraliz corrlation mtho for tim lay stimation [4]. This mtho first prprocsss th signal contaminat with nois, to improv th SN. This mtho coul bttr yil th tim lay stimation in th cas of Articl numbr P_5 89

2 Snsors & Transucrs, Vol. 58, Issu, Novmbr 3, pp low SN. ut whn th instantanous corrlation nrgy valu of th nois xcs that of th targt signal, th algorithm woul consir th nois as vali signal, rsulting in rror. In gnral, th rlvant lay stimation mthos assum that th noiss of th signal mol ar all Gaussian whit nois an inpnnt from ach othr. Howvr, in practical applications, nois may b corrlat an non-gaussian. In this cas, th corrlation-bas mthos coul not accuratly stimat th tim lay. To solv this problm, rsarchrs hav tri to us bispctrum-bas mtho for lay stimation [5-4]. This mtho uss shiftinvarianc proprty of bispctrum. Th prformanc of convntional bispctrum-bas signal procssing mtho is also worth improving. First, if th input SN is larg nough, th bispctrum mtho will introuc spcific istortions in th rconstruct signal. What s mor, thr ar many ways to improv th prformanc of bispctrum-bas signal procssing by combining it with robust stimation an nonlinar filtring. Nikias an Pan propos th tim lay stimation mtho bas on thir-orr statistics [5]. This mtho coul stimat th tim lay of non-gaussian signals in an unknown Gaussian nois nvironmnt. Howvr, this mtho involvs th matrix invrsion. Whn th invrs matrix is absnt, th tim lay coul not b stimat. Hinich mploy th Fourir transform ovr th tim lay mtho bas on thir-orr statistics to riv th bispctrum-bas tim lay mtho [6]. obust stimation was also appli at th stag of bispctrum stimat. oth non-aaptiv an aaptiv lay stimation wr us insta of convntional avraging. Th most prominnt avantag of bispctrum-bas mtho is that it compltly liminats Gaussian nois. Howvr, th prformanc of this mtho is affct by th sampl. Whn th sampling frquncy of th signal is low, th calculat rsolution of th corrlation function is also low. In orr to improv th tim omain rsolution, this articl proposs a bispctrum intrpolation-bas lay stimation algorithm using th thir-orr cumulant. It improvs th prcision of tim lay stimation by implmnting th frquncy omain intrpolation. This algorithm combins th supprssion of th corrlat noiss an rfinmnt of frquncy omain in bispctrum-bas algorithm that improv tmporal rsolution. Th xprimntal rsults show that rlativ to bispctrum-bas tchniqu, th prcision of tim lay stimation using bispctrum intrpolation-bas tchniqu is highr, an th rror is smallr.. Mthos Suppos th transmitt signal an th rciv signal to b x ( t an x ( t rspctivly. x ( = s( + n ( x( = s( t D + n(, ( whr s( is th original signal without nois. n ( an n ( ar th noiss suprimpos on th two signals, an thy may b corrlat. D is th tim lay btwn th two signals. t is th tim. Th classical tim lay stimation is bas on analyzing th cross corrlation btwn x ( t an x ( t as th two signals of intrst an th lag that corrspons to th maximum valu of cross corrlation is th tim lay. Th cross corrlation function for x ( t an x ( t is givn as + x x ( x( x( t = t. ( In practical applications, nois may b corrlat. In this cas, th corrlation-bas mthos coul not accuratly stimat th tim lay. caus th noiss ar corrlat, th bispctrum algorithm is us for th calculation of tim lay. It is known that th bispctrum can liminat th corrlativ nois. Th thir-orr slf-corrlation function of x ( t is (, = E[ x( x( t +, (3 x ( t + ] = (, sss whr an ar tim lay variabls. Th thir-orr corrlation functions of x ( t an x ( t is (, = E[ x ( x( t +. (4 x ( t + ] = ( D, Th thir-orr slf-corrlation function an cross-corrlation functions aftr FFT transform hav th following form: + + ( w, w = (, * sss π j ( w + w + + ( w, w = (, * Thir iscrt forms ar: j π ( w + w, (5,. (6 9

3 Snsors & Transucrs, Vol. 58, Issu, Novmbr 3, pp ( k, k = ( k, k = N N m= m= jπ ( km + km N N N m= m= jπ ( km + km N ( m, m*, (7 ( m, m*, (8 As coul b sn from th abov quation, th only iffrnc is a linar phas shift btwn ( w, w an ( w, w in th frquncy omain. Charactristic function: g( + = + = 4π δ ( D ( w, w * ( w, w jπ w w, w (9 As coul b sn from th abov quation, th maximum valu point of g( is th tim lay D btwn x ( t an x ( t. Givn th iscrt systm, it can b known from th sampling thorm that th charactristic function g( with continuous tim omain an limit frquncy spctrum is sampl with th prio of Ts. Th frquncy spctrum G(k corrsponing to th squnc g(n is th prioic rptition of th frquncy spctrum G(f of th squnc g(n. Th rptition intrval is qual to th sampling frquncy f s. As long as th sampling frquncy is gratr than or twic as th highst frquncy of th signal, g(n coul b rcovr bas on G(k with no istortion. Whn th sampling frquncy is low, th calculat corrlation function rsolution is low. In orr to improv th rsolution of corrlation pak, thr ar gnrally two ways: incrasing th sampling frquncy an th corrlation pak intrpolation. Th highr th sampling frquncy, th smallr th stimation rror. ut th highr th sampling frquncy of th chip prformanc rquirmnts will b highr. For xampl, if rquirs th stimation rror of ns, clock frquncy will n to incras to GHz. Th prsnt igital signal procssing chip us is ifficult to achiv such a high procssing rat. At th sam tim, th problm of lctromagntic intrfrnc will also giv circuit boar layout, matrial slction, procssing brings highr rquirmnts. Intrpolation calculation of th corrlation pak, such as th us of th last squar mtho or thr splin intrpolation, may not only incras th amount of calculation, but also will introuc nw rrors. W prsnt an algorithm to improv th stimation mtho of bispctrum intrpolation lay accuracy by intrpolation in frquncy omain whn th sampling rat is low. If G(k is xpan in th frquncy omain, it is quivalnt to xpaning th rptition intrval of th frquncy spctrum. Th invrs transform of g(n wavform os not chang. It woul not bring nw rror, but th sampling rat of g(n coul improv at th sam tim. as on th cor ia of algorithm for spctrum intrpolation, i.. frquncy omain zro paing coul improv th rsolution of th tim-omain wavform in th charactristic function, G(k squnc with sampling lngth of N points is zro pa at th spctrum intrval of th charactristic function. N N is assum to construct G (k squnc. G ( k = G( k k =,,,... N G ( k = k = N, N +,... N N. G ( k = G(N N + k k = N N, N N +,... N ( Thn w o th invrs Fourir transform on G (k to gt g (n. It is quivalnt to improving th sampling rat of th corrlation function g (n to N fs. In orr to giv tim lay stimation with N high prcision, th sampling rat of g(k has to b maximiz. N shoul b larg nough. ut this irctly incrass th numbr of points in th invrs Fourir transform, thrby incrasing th ovrall computation work. Thrfor, th prcision an computation workloa of tim lay stimation shoul b balanc pning on th actual conitions. 3. Exprimntal sults an Discussion To vrify th ffct of th bispctrum intrpolation-bas tim lay stimation algorithm, th numrical simulation tst is prform. Th algorithm is compar with th cross-corrlation an bispctrum algorithms. Whn th signal is a banlimit signal, an SN is a constant, th tim lay is st as s. Th numbr of bispctrum intrpolation points is N = 4N. Th xprimntal rsults of th cross-corrlation-bas stimation, bispctrum intrpolation-bas mtho an bispctrum-bas stimation ar shown in Fig.. From th figur, it coul b sn that in th prsnc of nois, th tim lay positioning of bispctrum intrpolation-bas algorithm is mor accurat, with highr prcision than th bispctrum-bas stimation an crosscorrlation-bas stimation. 9

4 Snsors & Transucrs, Vol. 58, Issu, Novmbr 3, pp Margin 5 5 x 4 Cross-corrlation ispctrum ispctral stimation intrpolation oftn corrlat. This articl uss th bispctrumbas tchniqu to supprss th influnc of corrlat Gaussian nois an its autocorrlation on th ntir tim lay. Th frquncy omain intrpolation of th bispctrum is carri out to rfin th frquncy omain an incras tmporal rsolution, thrby incrasing th prcision of tim lay stimation ispctral stimation intrpolation tim lag Fig.. Comparison of thr tim lay stimation algorithms. Margin N = 4N an tim lay valu st. an.5 s. Th sampling rat is twic th original rat, an SN is st to b a constant. Th calculat tim lay is shown in Fig. an Fig. 3. As coul b sn from thos figurs, bispctrum intrpolation-bas algorithm has high stimation prcision for tim lay positioning. Givn th initial valus of svral tim lays, th tim lay stimation rsults of svral mthos ar shown in Tabl blow. It coul b sn from Tabl that th tim lay stimation obtain by IFFT opration aftr bispctrum intrpolation is mor accurat than irctly using bispctrum to calculat tim lay. Th tim lay rrors an rlativ rror calculat accoring to Tabl unr crtain SN conitions ar shown in Tabls an 3. It coul b known from Tabls that whn th tim lay valu incrass, th masurmnt prcision incrass as wll. Th prcision of th bispctrum intrpolationbas algorithm of this articl is significantly highr than that of th othr two algorithms. Th numbr of intrpolations has crtain influnc on th prcision of tim lay stimation. Whn th tim lay valu is.,.5 an.7 s, stimation prcisions of intrpolating iffrnt numbrs of valus ar shown blow in Fig. 4 to Fig. 6. As coul b sn from thos figurs, with th intrpolation points incrasing, tim lay stimation prcision incrass as wll. ut whn it rachs a crtain prcision, tim lay stimation prcision woul improv slowly. Thrfor, in th actual situation, th numbr of intrpolation points has to b rasonably chosn. 4. Conclusions In th absnc of nois or whn th noiss ar not corrlat, th classic cross corrlation-bas algorithm coul hav a goo positioning prcision for tim lay. ut in actual conitions, th noiss ar Tim-lay Fig.. Position prcision of bispctrum intrpolation-bas tim lay stimation at th st tim. s. Margin ispctral stimation intrpolation Tim-lay Fig. 3. Position prcision of bispctrum intrpolation-bas tim lay stimation at th st tim.5 s. Tabl. Tim lay comparison at iffrnt st tim. St Crosscorrlation(s intrpolation (s ispctral ispctrum (s

5 Snsors & Transucrs, Vol. 58, Issu, Novmbr 3, pp Tabl. Estimation rror comparison at iffrnt st tim. St Crosscorrlation(s ispctral intrpolation(s Tabl 3. Estimation rlativ rror at iffrnt st tim. St Tim (s Crosscorrlation (% ispctrum (% ispctral intrpolation (% Fig. 4. Tim lay stimation of bispctrum intrpolationbas algorithm at st tim. s Fig. 5. Tim lay stimation of bispctrum intrpolationbas algorithm at st tim.5 s. N N N Fig. 6. Tim lay stimation of bispctrum intrpolationbas algorithm at st tim.7 s. Acknowlgmnts This work was support by th 3 opn fun of th Jiangsu Ky Laboratory of Mtorological obsrvation an Information Procssing at Nanjing Univrsity of Information Scinc an Tchnology, th Qing Lan Projct, an th Priority Acamic Program Dvlopmnt of Jiangsu Highr Eucation Institutions. frncs []. Y. C. Elar, Tim-lay stimation from low-rat sampls: a union of subspacs approach, IEEE Transactions on Signal Procssing, Vol. 58, Issu 6,, pp []. X. Li, Y. Li, Y. Wang, Tim-lay stimation of chirp signals in th fractional Fourir omain, IEEE Transactions on Signal Procssing, Vol. 57, Issu 7, 9, pp [3]. W. Zng, H. C. So, M. Ablhak, An lp-norm minimization approach to tim lay stimation in impulsiv nois, Digital Signal Procssing, Vol. 3, Issu 4, 3, pp [4]. A. Hro, S. Schwartz, Nw gnraliz cross corrlator, IEEE Transactions on Acoustics, Spch, an Signal Procssing, Vol. 33, Issu, 985, pp [5]. C. Nikias,. Pan, Tim lay stimation in unknown Gaussian spatially corrlat nois, IEEE Transactions on Acoustics, Spch, Signal Procssing, Vol. 36, Issu, 988, pp [6]. M. Hinich, G. Wilson, Tim lay stimation using th cross bispctrum, IEEE Transactions on Signal Procssing, Vol. 4, Issu, 99, pp [7]. X. Zhnghua, Tim offst masurmnt algorithm bas on bispctrum for pulsar intgrat puls profils, Acta Physica Sinica, Vol. 57, Issu, 8, pp [8]. N. asumallick, S. Narasimhan, Improv bispctrum stimation bas on moifi group lay, Signal, Imag an Vio Procssing, Vol. 6, Issu,, pp [9]. P. Ajmra, N. Nh, D. Jahav,. Holamb, obust fatur xtraction from spctrum stimat using 93

6 Snsors & Transucrs, Vol. 58, Issu, Novmbr 3, pp bispctrum for spakr rcognition, Intrnational Journal of Spch Tchnology, Vol. 5, Issu 3,, pp []. L. Saii, F. Fnaich, H. Hnao, G-A. Capolino, G. Cirrincion, Diagnosis of brokn-bars fault in inuction machins using highr orr spctral analysis, Intrnational Socity of Automation (ISA Transactions, Vol. 5, 3, pp []. C. Png, Application of bispctrum tim-lay stimation in passiv ranging marin, Elctric an Elctronic Tchnology, Vol. 3, Issu 6,, pp []. J. Sun, W. Li, ispctrum bas mtho of timlay stimation us in ultrasonic flowmtr in th oil wll, Signal Procssing, Vol. 6, Issu,, pp [3]. X. Li, Y. Xin, Th lakag tction systm for watr piplin bas on tim-lay stimation with crossbispctrum, Journal of Shaanxi Normal Univrsity, Vol. 38, Issu 5,, pp [4]. S. Li, Th algorithms of stimating bispctrum an crossbispctrum bas on acoustic vctor signal, Tchnical Acoustics, Vol. 7, Issu 5, 8, pp Copyright, Intrnational Frquncy Snsor Association (IFSA. All rights rsrv. ( 94