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1、2852IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.57,NO.7,JULY 200913 M.Kawamoto,K.Kohno,and Y.Inouye,“Eigenvector algorithms in-corporated with reference systems for solving blind deconvolution ofMIMO-IIR linear systems,”IEEE Signal Process.Lett.,vol.14,pp.996999,Dec.2007.14 E.Moreau and J.-C.Pesquet,
2、“Generalized contrasts for multichanelblinddeconvolutionoflinearsystems,”IEEESignalProcess.Lett.,vol.4,pp.182183,Jun.1997.15 E.Moreau,J.-C.Pesquet,and N.Thirion-Moreau,“Convolutive blindsignal separation based on asymmetrical contrast functions,”IEEETrans.Signal Process.,vol.55,no.1,pp.356371,Jan.20
3、07.16 C.Simon,P.Loubaton,and C.Jutten,“Separation of a class of con-volutive mixtures:A contrast function approach,”Signal Process.,no.81,pp.883887,2001.17 L.Tong,Y.R.Liu,V.Soon,and Y.Huang,“Indeterminacy and iden-tifiability of blind identification,”IEEE Trans.Circuits Syst.,vol.38,pp.499509,May 19
4、91.18 J.K.Tugnait,“Identification and deconvolution of multichannel linearnon-Gaussian processes using higher order statistics and inverse filtercriteria,”IEEETrans.SignalProcess.,vol.45,no.3,pp.658672,Mar.1997.Time-Delay Estimation of Chirp Signals in theFractional Fourier DomainRan Tao,Xue-Mei Li,
5、Yan-Lei Li,and Yue WangAbstractA new time-delay estimator is presented in this paper.It isevaluated based on the delay property of the fractional Fourier transformwith less computation and is suitable for chirp signals.The statistical anal-ysis in terms of signal-to-noise ratio(SNR)and estimation ac
6、curacy for thisestimation is also studied.The proposed method for the time delay yields avariance which is theoretically equal to the CramrRao lower bound.Thevalidity of this estimation method is demonstrated via simulations.Index TermsCramrRao lower bound,fractional Fourier transform,time-delay est
7、imation.I.INTRODUCTIONThe time-delay estimation(TDE)is a fundamental issue in manysignal processing areas including radar,sonar,communications and ul-trasonics 1,2,10,11,1416,21.The objective of the TDEistodeterminethedelayofthecorruptedreceivedsignalsfromatarget.Conventional TDE techniques are base
8、d on matched filtering or crosscorrelation which can be seen as the optimal and useful algorithm 2,Manuscript received October 08,2008;accepted February 02,2009.Firstpublished March 31,2009;current version published June 17,2009.The as-sociate editor coordinating the review of this manuscript and ap
9、proving it forpublication was Dr.Peter J.Schreier.This work was supported in part by the NSFC under Grant 60890072;theMOST under Grant 2009CB724003;and the NSFC for Distinguished YoungScholars under Grant 60625104.R.Tao,Y.-L.Li and Y.Wang are with the Department of ElectronicEngineering,Beijing Inst
10、itute of Technology,Beijing 100081,China(e-mail:).X.-M.Li with the Department of Electronic Engineering,Beijing Instituteof Technology,Beijing 100081,China.He is also with the Department of Elec-tronicandInformationEngineering,BeijingElectronicScienceandTechnologyInstitute,Beijing 100070,China(e-mai
11、l:li-).Digital Object Identifier 10.1109/TSP.2009.202002810,21.The accuracy of the TDE has been presented in differentways1,15,17.Several othertime-delayestimation methodsasso-ciated with the cross correlation for the subsample resolution have alsobeen studied 10,11,16.Also reported in the conventio
12、nal Fourierdomain is an estimation method for separating closely spaced narrow-bandsignals15.WiththedevelopmentofthefractionalFouriertrans-form(FRFT),the delay estimation based on the FRFT has receivedmuch attention in the signal processing community.Pei and Ding ap-plied discrete fractional correla
13、tion to determine the same object lo-cated in a different place for the pattern recognition 9.Sharma andJoshi proposed several estimators of the TDE using the FRFT to en-hance the performance of the time delay 14.However,their analysesare presented without considering the mathematical expressions of
14、 theoutput signal-to-noise ratio(SNR)and the accuracy for TDE associ-ated with the FRFT.In recent time,the FRFT has received much attention in many areassuchasquantumphysics,opticalsystemsandsignalprocessing39,1214,19,20.Its operation indicates a rotation of a signal?withanangle?inthetime-frequencyp
15、lane.TheFRFTofasignal?with an angle?is defined as 3?(1)where?isthetransformationkernel,?,and?an integer.The FRFT reduces tothe conventional Fourier transform(FT)when?,i.e.,?,where?denotes the FT.Also,the FRFTdevelops from its original function to the FT in the time-frequencyplane when?varies from 0
16、to?.In this paper,?is not a multipleof?and?.The FRFT has a number of useful properties whichcan be seen as the generalization of those in the conventional Fouriersense 3.One important property is the delay property.If a signal?is delayed by?,the FRFT with angle?of the delayed signal?can be written a
17、s?(2)In contrast to the case of the conventional FT,the delay property of theFRFT can behave with respect to the location of the extremum of thefractional Fourier spectrum.Thus,the TDE is converted into a param-eter estimation problem in the fractional Fourier domain(FRFD).In many applications such
18、as target location and SAR imaging,etc.,the signals are usually the chirp forms.Since the FRFT has a fine con-centration for chirp signals,the time-delay estimation for chirp signalsbecomes feasible by applying the delay property of the FRFT.In thispaper,we present a FRFT based TDE method which requ
19、ires less cal-culation and also different from the classical correlation based methodand thosereported by Pei 9and Sharma14.Wealso present astatis-tical analysis in terms of SNR and the estimation accuracy which canyield the same form as that of the CramrRao lower bound(CRLB)for TDE.Simulations are
20、given to verify the validity of our proposedestimation method.1053-587X/$25.00 2009 IEEEIEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.57,NO.7,JULY 20092853II.TIME-DELAYESTIMATIONWITH THEFRFTConsider a referenced signal?with the duration?in an active system,where?otherwise(3)?is the real magnitude,?the
21、modulated rate and?the center fre-quency.Without loss of generality,we assume?in this paper.The received signal delayed by?in the presence of noise?can bewritten as?(4)It is known that a finite duration chirp signal can be concentrated max-imally in the FRFD with an angle?which is determined by the
22、mod-ulated rate?of the signal satisfying the relation 5?(5)According to(2),when the delayed signal?is concentrated inthe FRFD,the maximum magnitude of the fractional Fourier spectrumappears at?,which produces a measure of the delayed time?scaled by?,where?is the angle of the FRFT.When?,thepeak value
23、 of?appears at the zero point of the coordinate axisin the FRFD,and shifts to another location when?.Thus,we canestimate the delayed time?via the maximum value of the fractionalFourier spectrum.The estimation of the delayed time for the receivedsignal?which is a composite of the delayed signal?and t
24、henoise?can therefore be given as?(6)where?is the FRFT of the received signal?.It can be foundfrom(6)that the proposed TDE employs the FRFT to the signal?only once before searching for the maximum value of?whereas the classical correlation method employs the FFT twice andthe IFFT once followed by th
25、e search for the maximum magnitude.Since the computation of the FRFT does not take much longer com-pared with that of the conventional FT 5,the proposed method canbe implemented at less computational cost.If the received signal isclean without noise,the estimator?in(6)gives the real delayed time?.Bu
26、t in the presence of noise,the value of the estimator?will moveto the point?,where?is the time-delay estimation error.Let us consider a study case where the noise is assumed to be a zeromean,complex Gaussian noise with variance?,uncorrelated withthe useful signal.We evaluate the performance of our m
27、ethod via theoutput SNR and the accuracy of the TDE in the FRFD as follows.A.The Output SNRThe output SNR in the FRFD can be given asSNR?(7)where?denotes?and?is the FRFT of the signal?.The extremum of?in the FRFD with angle?can bepresented as?.The expected value and the secondorder moment of?can be
28、written as?(8)and?(9)where?denotes the conjugation operator.Combining expressions(8)and(9),we can obtain the variance of?as?(10)Combining(7),(10)and the extremum of?,we can present themaximal output SNR asSNR?SNR?(11)where SNR?is defined as?.Expression(11)reveals that theoutput SNR in the FRFD is de
29、pendent on?and SNR?.Thus,theoutput SNR can be improved as?increases.B.The AccuracyWe evaluate the accuracy of our method by a perturbation anal-ysis using the fractional energy spectrum 3,13.It can be seenthat the zero value of?is at the same point as that of?in the FRFD.In the absence of noise asso
30、ciated withthe received signal?,the first derivative of the fractional energyspectrum?can be written as?(12)The location of the zero value in(12)gives the true time-delay esti-mation,i.e.,?whereas in the presence of noise,a deviation ofa random perturbation?,which makes the estimated delay?excursefr
31、omthetruedelay?willbeproduced.Consideringthesituationwherethe noise is small or insignificant,we can write the fractional energyspectrum?as?(13)2854IEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.57,NO.7,JULY 2009where?is a random distur-bance depending on the noise and its interaction with the useful si
32、gnal.Substituting(13)into(12),we obtain?(14)By representing the first derivative in the previous equation around thevalue?,we obtain?(15)Since the first item in the above expression is always equal to zero,theestimation error?can be obtained as?(16)where?.Next,we evaluate the expectation and the var
33、i-ance of the time-delay error?.The expectation of?can be writtenas?(17)which implies that?.The variance of?can be given as?(18)where the constant?in the denominator of(18)canbe given as?(19)Combining the relation?21,the random item inthe numerator of(18)can be obtained as?(20)Substituting(19)and(20
34、)into(18),we obtain?SNR?(21)Noting that the angle?of the FRFT is associated with the modulatedrate?of the chirp signal in(5),the variance of the time-delay errorcan be written as?(22)Itis known thatthe bandwidth ofa chirpsignal with the modulated rate?and the observation time?can be approximated to?
35、when?21.Thus,(22)can be expressed as?(23)where?is the energy of the signal?.It can be seen that the ex-pression(23)has the same form as that of the CRLB for TDE 1,17which shows that the variance of the TDE is related with the energyand bandwidth of the chirp signal,likewise the variance of the noise
36、.That is to say the variance of the proposed TDE is theoretically equalto the CRLB validating the efficiency of the proposed TDE method.III.SIMULATIONRESULTSIn this section,we evaluate the performance of the proposed frac-tional time-delay estimator(FRTDE).We select a chirp signal whichhas uniform a
37、mplitude with a modulated rate of?and a band-width of?;according to(5),the angle?of the FRFT is?.Fig.1 shows the FRFT of the delayed signal under dif-ferent noises for SNR?10 dB.Simulation results are got from 10independent Monte Carlo runs.The zero-mean Gaussian noise is con-sidered in Fig.1(a)whic
38、h reveals that the FRFT of the white noisecan not be grasped in the FRFD.In Fig.1(b),we apply a nonstationarynoisewhichisthe resultfromthe productofthe whitenoiseand achirpIEEE TRANSACTIONS ON SIGNAL PROCESSING,VOL.57,NO.7,JULY 20092855Fig.1.FRFT of the signal with different noise.(a)With zero mean
39、Gaussiannoise.(b)With a chirp noise.Fig.2.Variance of time-delay estimation as a function of SNR.signal.Although the effect of the nonstationary noise is severe com-pared to that of the white noise,we can still obtain the delayed time asshown in Fig.1(b).The next simulation is to evaluate the varian
40、ce of?compared withtheDCestimator2andtheCRLB1.Thenoiseisassumedtobezeromean,complex Gaussian noise.Simulation results are obtained from1000 independent Monte Carlo runs.Fig.2 shows the performance ofthe variance of?.It can be seen from Fig.2 that the variance hassevere performance degradations at lo
41、wer SNR because of the verynoisy condition.As the SNR is increasing,the variance approaches theCRLB.Thus,our method is applicable for estimating the time delay.It should be noted that the proposed TDE implemented in the FRFDproduces different quantization errors compared to those from the DCbased es
42、timation and leads to the deviation of the variance from theCRLB.IV.CONCLUSIONInthis paper,atime-delayestimatorbased onthe FRFTis presented.This estimation is carried out by the search for the peak value of theFRFT of the observed signal,which needs less computation,especiallyin the active system.We
43、 evaluate the performance of this method intermsoftheoutputSNRandtheaccuracy.TheoutputSNRintheFRFDcan be improved as?increases.The estimation accuracy is analyzedthrough a perturbation analysis.The variance of the time-delay errorin expression(23)is proportional to the energy and bandwidth of thesig
44、nal and has the same form as the lower bound given in 2.Thus,the proposed TDE technique can give reliable time-delay estimationresults for chirp signals,which makes it an efficient method for targetlocation or SAR imaging etc.The validity of this estimation method isdemonstrated via simulations.REFE
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