Progress In Electromagnetics Research M, Vol. 20, , 2011

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1 Progress In Electromagnetcs Research M, Vol. 20, , 2011 DETECTION AND ESTIMATION OF MULTI- COMPONENT POLYNOMIAL PHASE SIGNALS BY CONSTRUCTING REGULAR CROSS TERMS X. Zhang *, J. Ca, L. Lu, and Y. Yang School of Communcaton and Informaton Engneerng, Unversty of Electronc Scence and Technology of Chna, Chengdu, Schuan , Chna Abstract A regular cross terms algorthm s derved for the parameter estmaton of the mult-component polynomal phase sgnals n addtve whte Gaussan nose. The basc dea s frst to separate ts phase parameters nto two sets by nonlnear proceduresand then each set has half of the parameters n ts auto-terms. Furthermore, usng two lnear transforms to deal wth the two sgnals respectvely, the phase coeffcents of cross terms can be regulated for the dentfcaton and elmnaton of false peaks caused by the cross terms. Smulatons are presented to llustrate the performance of the proposed algorthm. 1. INTRODUCTION Polynomal phase sgnal (PPS) s a very mportant model to deal wth nonstatonary sgnals, whch have consderable technologcal applcatons n radar, wreless communcatons, sesmologyneuroethologyetc. For example, n synthetc aperture radar (SAR) [1 3], the relatve radar-target moton can cause a tme-varyng phase n the transmtted sgnal. The contnuous varaton of the dstance between radar and target leads the nstantaneous phase shft to a contnuous functon of tme. Its phase estmaton has receved consderable attenton n the feld of sgnal processng, as the Weerstrass theorem mples that the contnuous nstantaneous phase can be well approxmated by a fnte-order polynomal wthn a fnte nterval. There are many methods to estmate the parameters for a PPS [4 15]. The maxmum-lkelhood (ML) [4], and nonlnear nstantaneous Receved 11 July 2011, Accepted 19 August 2011, Scheduled 29 August 2011 * Correspondng author: Xhu Zhang (seaharm yeah@163.com).

2 144 Zhang et al. least squares (NILS) [7] estmators can provde very hgh estmaton accuracy, whose estmator varances acheve Cramér-Rao lower bound (CRLB) asymptotcally under addtve whte Gaussan nose [8]. But ML and NILS methods result n a P-dmensonal search wth O(N P log 2 N) operatons when the order of phase s P. In [9], the observed PPS wth the order more than three s frst converted to another sequence by a sgnal transformaton procedure, and then the parameter search dmenson s reduced by about half. The polynomalphase transform or the hgh-order ambguty functon (HAF) [9, 10] or Polynomal Wgner-Vlle dstrbuton (PWVD) [14] can estmate the PPS parameters by one-dmensonal search va multple nonlnear operatons on the receved sgnal. For mult-component polynomal phase sgnals (mc-ppss), the cross terms between the components gve rse to undesred snusods n the hgh-order nstantaneous moment (HIM), whch s the man problem strongly to affect algorthms based on frequency estmaton. The prncple of demodulaton of monocomponent PPS no longer works wth mc-ppss [16]. Many methods for mc-ppss analyss are derved from the prevous methods for mono-component PPS or generalze the ones. The product hgh-order ambguty functon (PHAF) [13], whch s the extenson of HAF, mproves the dentfcaton of the hghest order polynomal phase coeffcents by usng hgh-order multple transform technque and proper scalng. PWVD s also usng hgh-order multple transform smlarly. There have been a number of methods to generalze exstng classes of multlnear functons wth a vew to mprovng the power and flexblty of analyss for PPS [16], such as the generalzed representaton of phase dervatves (GRPD) [17], the generalzed hghorder phase functons (GHOPF) [18]. For the cross term nterference problem of mult-component, the man methods used n the exstng lteratures are compressng the cross term to reduce the nterference; however, t can t remove the nterference completely. Partcularly, when two components share hgh-order coeffcents and other coeffcents of the same order are dfferent from each other, some of the cross terms can also be converted to snusods n HIM and they are ndstngushable from the auto-terms. Then, the dentfablty problem occurs serously. In ths paper, cross terms are constructed regularly n order to acheve completely the dentfcaton of auto-terms from cross terms. The phase parameters of mc-ppss are separated nto two sets by nonlnear proceduresand then each set has half of the parameters n ts auto-terms. And we ntroduce two lnear transforms smlar to chrpz transform to estmate the parameters of the two sets respectvely by searchng the spectrum peak. If arbtrary two peaks satsfy some

3 Progress In Electromagnetcs Research M, Vol. 20, regular characterstc, for example, the symmetry around arbtrary axs, they can be dentfed as cross terms. The proposed algorthm can well detect mc-ppss wthout the cross term nterference completely, especally for the case of 3rd or 4th order. 2. REGULAR CROSS TERMS It wll lead to serous cross term nterference sometmes that hgh-order mc-ppss have been deal wth by nonlnear algorthm. For example, consder the estmaton of mult-component cubc phase sgnals by HAF: x(t) = N A e j(a b tc t 2 d t 3) n(t), t/2 t t/2 (1) where A, a, b, c, d, = 1, 2,... M denote the ampltude and phase parameters correspondng the th-component and n(t) s the addtve complex whte Gaussan nose wth zero-mean. The frst order of HIM can be expressed as P 1 (t, τ) = x(t τ)x (t τ) = N 1,k N, k A2 e j[(2b τ2d τ 3 )4c τt6d τt 2 ] A A k e j[φ 0φ 1 tφ 2 t 2 (d d k )t 3 ] n (t) (2) where the frst summaton part denotes the auto-terms and the second summaton part denotes the cross terms. The frst three orders of phase coeffcents of cross terms can be expressed as φ 0 = (a a k ) (b b k )τ (c c k )τ 2 (d d k )τ 3 φ 1 = b b k 2c τ 2c k τ 3d τ 2 3d k τ 2 φ 2 = c c k 3d τ 3d k τ. From (2), t can appear that the number of the hghest-order phase n cross term s same wth the auto-terms when d d k = 0. As a result, ths cross term wll be treated as an auto-term n the next order of HAF and can not be dentfed. We ntroduce a sgnal transformaton procedure to convert mc- PPSs to new functons whch have symmetrc cross terms. It s worth mentonng that the smlar sgnal transformaton procedure s used to smplfy the phase of mono-pps n [7]. Consder two new functons x 1 (t) and x 2 (t): x 1 (t) = x(t)x( t) (3) x 2 (t) = x(t)x ( t) (4)

4 146 Zhang et al. where * denotes the conjugate operator. Substtute (1) nto (3) and (4), then x 1 (t) = N x 2 (t) = N where A2 e 2j(a c t 2 ) 1,k N, k A2 e 2j(b td t 3 ) 1,k N, k A A k e j[a a k (b b k )t(c c k )t 2 (d d k )t 3] n 1 (t)(5) A A k e j[a a k (b b k )t(c c k )t 2 (d d k )t 3] n 2 (t)(6) n 1 (t) = n( t) N A e j(a b tc t 2 d t 3 ) n(t) N A e j(a b tc t 2 d t 3) n(t)n( t) n 2 (t) = n ( t) N A e j(a b tc t 2 d t 3 ) n(t) N A e j(a b tc t 2 d t 3) n(t)n ( t). In Equaton (5), the frst summaton ncludes N number of auto-terms and the second summaton ncludes N(N 1) number of cross terms. In order to estmate the 0th order and second order phase coeffcents, we dvde the dstrbuton of the cross terms nto three cases accordng to the relatonshp of the frst order and thrd order coeffcents of dfferent components. (c1) All the thrd order coeffcents are dfferent from each other (d k d, k, = 1,..., N, k ). All cross terms n ths case have a cubc phase that s dfferent from the second order phase n the autoterms. As a result, the cross terms would not affect the detecton of auto-terms. (c2) Some of the thrd order coeffcents concde (d k = d, b k b, k ) and the correspondng second order coeffcents are dfferent (b k b, k ). In ths case, the k cross terms have the same order phase wth the auto-terms, whch would affect the detecton of auto-terms. Snce the frst order phase coeffcents of the two the cross terms,.e., (b k b ) and (b b k ), are opposte numbers of each other, the cross terms appears n pars wth respect to the frst order phase, whch can be removed by the symmetry. (c3) Some of the hghest order and the frst order coeffcents also concde (d k = d, b k = b, k ). In ths case, the two cross terms of k combne nto one term and do not appear n the there s no

5 Progress In Electromagnetcs Research M, Vol. 20, symmetrcal property to use. However, the hghest order coeffcent of the k cross term s (c c k ) that s the average of the hghest order coeffcents of the two auto-terms, so the dentfcaton of these cross terms need refer to the phase relatonshp of the k cross term and the correspondng auto-terms. Smlarly, three cases are consdered to estmate the 1 order and 3rd order phase coeffcents accordng to the 0th order and second order coeffcents of dfferent components. (I) All the second order coeffcents are dfferent from each other (c k c m, k, m = 1,..., N, k m). (II) Some of the second order coeffcents concde (c k = c m, k m) and the correspondng 0th order coeffcents are dfferent (a k a m, k m). (III) Some of the second order and the 0th order coeffcents also concde (c k = c r, a k = a r, k r). For the case of (c2), (c3), (II) and (III), by usng approprate transforms, the relatonshps of the phase coeffcents between autoterms and cross terms can be converted nto the poston relatonshps of the correspondng peaks n the new transform doman. The nterferences of cross terms can be completely elmnated by the symmetry and the estmaton algorthm s dscussed n the next secton. 3. ESTIMATION ALGORITHMS We ntroduce dfferent transforms X 1 (u, α) and X 2 (u, β) to deal wth x 1 (t) and x 2 (t) respectvely, whch can be defned as X 1 (u, α) = X 2 (u, β) = x 1 (t)e j(utt2 cot α) dt (7) x 2 (t)e j(utt3 cot β) dt (8) The kernel of X 1 (u, α) contans a quadratc phase that s sutable for estmate the parameters of auto-terms n x 1 (t). Substtute (5) nto (7), and then X 1 (u, α) = N A2 e 2ja 1,k N, k e j[ut(cot α2c )t 2] dt A A k e jϕ(t,u,α) dt n 1 (t)e j(utt2 cot α) dt (9)

6 148 Zhang et al. where ϕ(t, u, α) = a a k (b b k u)t(c c k cot α)t 2 (d d k )t 3. In (9), we can see that X 1 (u, α) would appear peaks n the place of cot α = 2c, u = 0, whch are correspondng the N number of auto-terms. The cross terms can not appear peaks on condton of (c1). There can appear peaks n pars wth respect to the u axs on condton of (c2). However, the par of peaks combnes nto one peak on condton of (c3). If there exst two peaks whose parameters satsfy d k = d r and b k = b r, there must exst a cross term peak whose parameters satsfy c = (c k c r )/2, a = (a k a r )/2 and A = A k A r. The smlar analyss for X 2 (u, β) on condton of (I), (II) and (III) can be obtaned by the followng equaton: X 2 (u, β) = N A2 1,k N, k e j[(u2b )t(cot β2d )t 3] dt A A k e jϕ(t,u,β) dt n 2 (t)e j(utt3 cot β) dt (10) where ϕ(t, u, β) = a a k (b b k u)t(c c k )t 2 (d d k cot β)t 3. The flowchart of parameter estmaton s shown n Fg. 1 as follows. If a = a a k r and c = c c, k r abandon a and c x1( t) X ( u, α ) 1 Elmnate symmetrcal peaks Estmate a and c f ( t) f ˆ( t) x ( t) 2 X ( u, β ) 2 Elmnate symmetrcal peaks Estmate b and d If b = b b k r and d = d d, k r abandon b and d Fgure 1. Parameter estmaton flowchart. For the 3rd or 4th order mc-pps, the proposed algorthm needs a

7 Progress In Electromagnetcs Research M, Vol. 20, dmensonal maxmzaton. When the order of mc-ppss s N(N 1), the correspondng parameter search dmenson s P /2 1, where denotes the celng operator. 4. RESULT AND DISCUSSION In ths secton, some smulatons have been carred out to evaluate the mult-ppss parameter estmaton performance of the proposed algorthm n the presence of whte Gaussan nose. In the frst experment, we consder the mult-ppss as sum of two cubc sgnals wth dfferent coeffcents correspondng (c1) and (I) f 1 (t) = 2 A exp [ j ( a b t c t 2 d t 3)] n(t) (11) where n(t) denotes complex whte Gaussan nose wth SNR = 0 db. The parameters are A 1 = 1, a 1 = 0, b 1 = 1.5, c 1 = 1.98, d 1 = 2.8; A 2 = 1, a 2 = 0, b 2 = 0, c 2 = 1.58, d 2 = 1.8. The tme range s ( 1.25π, 1.25π) wth 401 samples. Fg. 2 and Fg. 3 show the 3-D graphc of X 1 and X 2 respectvely. As the hghest order coeffcents of cross terms n X 1 or the 2nd order coeffcents of cross terms n X 2 are not zero, we can see that all peaks n Fg. 2 and Fg. 3 result from auto-terms. However, other methods lke HAF, PHAF or PWVD, can not appear false peaks from cross terms effect. Fg. 4 shows the spectrum of the 2nd order HAF of f 1 (t), where τ = 2. From Fg. 4, we can see that there are two peaks correspondng two auto-terms and no false peaks from cross terms. In Fg. 5, we compare the root mean square error (RMSE) performance of HAF and the proposed method Fgure 2. X 1. The 3-D graphcs of Fgure 3. X 2. The 3-D graphcs of

8 150 Zhang et al. HAF f (Hz) Fgure 4. HAF of the sum of two cubc PPSs whose phase parameters have dfferent thrd order phase. The RMSE for d1 (db) CRLB HAF proposed method SNR (db) Fgure 5. RMSE versus SNR for the parameters of d 1. Fgure 6. The 3-D graphcs of X 1 of f 2 (t). Fgure 7. The 3-D graphcs of X 2 of f 2 (t). n parameter estmaton of d 1. As the proposed method nvolves only second-order nonlneartes that s less than the fourth- or hgher order nonlneartes n the HAF algorthm, we can see that the performance of proposed method s better than HAF n low SNR. In the second experment, we consder the mult-ppss as sum of two cubc sgnals whch ther phase coeffcents satsfy d 1 = d 2 and b 1 b 2 correspondng (c2) f 2 (t) = 2 A exp [ j ( a b t c t 2 d t 3)] n(t). (12) The parameters are A 1 = 1, a 1 = 0, b 1 = 50, c 1 = 1.98, d 1 = 3.9; A 2 = 1, a 2 = 0, b 2 = 0, c 2 = 1.58, d 2 = 3.9. The 3-D graphc of X 1 and X 2 are shown n Fg. 6 and Fg. 7 respectvely. In Fg. 6, we can see that there appear four peaks, two of

9 Progress In Electromagnetcs Research M, Vol. 20, HAF f (Hz) Fgure 8. HAF of the sum of two cubc PPSs whose phase parameters have the same thrd order phase. Fgure 9. The 3-D graphcs of X 1 of f 3 (t). Fgure 10. The 3-D graphcs of X 2 of f 3 (t). whch are symmetrcal about the axs. Accordng to the symmetry of false peaks, the cross terms can be easly dentfed and the remanng two peaks are correspondng to the auto-terms. Generally, HAF, PWVD, PHAF or other nonlnear algorthms have not obvous regular characterstc to dentfy the cross terms. Fg. 8 shows the spectrum of the 2nd order HAF of f 2 (t) wth τ = 3.4, where we can see three peaks nstead of one peak correspondng to the two overlappng auto-terms. In the thrd experment, we consder the mult-ppss as sum of two cubc sgnals f 3 (t) = 2 A exp [ j ( a b t c t 2 d t 3)] n(t) (13) where the phase coeffcents satsfy A 1 = 1, a 1 = 0, b 1 = 20, c 1 = 1.38, d 1 = 3.9;

10 152 Zhang et al. A 2 = 1, a 2 = 0, b 2 = 20, c 2 = 2.2, d 2 = 3.9. The 3-D graphc of X 1 and X 2 are shown n Fg. 9 and Fg. 10 respectvely. As the frst order phase coeffcents concde, we can see that the two symmetrc false peaks combne nto a peak at the u-axs n Fg. 9. There s one peak n Fg. 10, for the 1st and 3rd order coeffcents concde. 5. CONCLUSION A regular cross terms based method for the parameter estmaton of mult-ppss n whte Gaussan nose has been proposed. The phase parameters are separated nto two new sgnals by smple nonlnear procedures. Both the two new sgnals have regular cross terms that can be dentfed by two lnear transform. In partcular, the parameter estmaton of mult-component cubc sgnals s analyzed n detal. Smulatons llustrate that the proposed algorthm can well dentfy and elmnate the cross term nterference n the parameter estmaton. REFERENCES 1. Lm, K.-S. and V. C. Koo, Desgn and constructon of wdeband Vna ground-based radar system wth real and synthetc aperture measurement capabltes, Progress In Electromagnetcs Research, Vol. 86, , Sabry, R. and P. W. Vachon, Advanced polarmetrc synthetc aperture radar (SAR) and electro-optcal (Eo) data fuson through unfed coherent formulaton of the scattered EM feld, Progress In Electromagnetcs Research, Vol. 84, , Zhao, Y. W., M. Zhang, and H. Chen, An effcent ocean SAR raw sgnal smulaton by employng fast Fourer transform, Journal of Electromagnetc Waves and Applcatons, Vol. 24, No. 16, , Grouffaud, J., P. Larzabal, A. Ferréol, and H. Clergeot, Adaptve maxmum lkelhood algorthms for the blnd trackng of tmevaryng multpath channels, Internatonal Journal of Adaptve Control and Sgnal Processng, Vol. 12, No. 2, , Ikram, M. Z. and G. Tong Zhou, Estmaton of multcomponent polynomal phase sgnals of mxed orders, Sgnal Processng, Vol. 81, No. 11, , Nov Ferrar, A., C. Theys, and G. Alengrn, Polynomal-phase sgnal analyss usng statonary moments, Sgnal Processng, Vol. 54, No. 3, , Nov

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