Research on Frequency Estimation Based on LS-SVC in Unknown Noise

Transcription

1 Joural of Commucatos Vol. 8, o. 10, October 013 Research o Frequecy Estmato Based o LS-SVC Ukow ose Xueqa Lu Zhegzhou Iformato Scece ad echology Isttute, Zhegzhou 45000, Cha Emal: jxdlxq@16.com Abstract Whe ose model s already kow, maxmum lkelhood estmator (MLE) s asymptotcally the most optmum oe. However, the truth s just the opposte that, ose s ukow burst commucato systems. Amg that, ths paper utlzes the trasmtted symbols of burst commucatos whch are elmated frstly commo methods ad proposes a data-aded (DA) frequecy estmato algorthm based o least squares support vector classfcato (LS-SVC). By researchg o statstcal learg theory (SL), we costruct a structural rsk mmzato (SRM) fucto wth respect to frequecy, ad covert the estmato problem to dervg the extremum value of a classfcato fucto. Cosequetly, support vector classfcato (SVC) s good learg ad geeralzato capabltes are completely explored ad employed. Expermetal results show that the proposed algorthm s close to MLE the case of Gaussa ose, ad also exhts good performace o-gaussa codto. Idex erms frequecy estmato, least squares support vector classfcato (LS-SVC), ukow ose, gaussa dstrbuto, -stable dstrbuto, support vector classfcato (SVC), structural rsk mmzato (SRM) I. IRODUCIO Frequecy estmato of susodal sgal wth ose s a classcal problem sgal processg, whch has started from 1970s [1]-[8]. Ad we usually suppose that the ose models are Gaussa dstrbuto oes. However, as a matter of fact, the assumpto of that s mpractcal may scees. Resultly, frequecy estmato algorthms the codto of o-gaussa ose are addressed [9]- [13]. Eve though wth that beg the case, ther dstrbutos are always kow, such as -stable ad pulse oes. Dfferetly, frequecy estmato uder the assumpto of ukow ose dstrbuto s dscussed ths paper. At ths momet, the tradtoal methods based o a pror kowledge are vald. Ad we utlze the mache learg method to obta model formato as much as possble. he relatoshps betwee puts ad outputs are assumed already kow artfcal eural etwork (A), oly wth ukow parameters. Hece, over- Mauscrpt receved July 17, 013, revsed October 8, 013. hs work was supported by the atoal ature Scece Foudato of Cha uder Grat o Correspodg author emal: jxdlxq@16.com do: /jcm matchg ad local mmum problems always exst. Dstctly, statstcal learg theory (SL) whch s specalzed the research of small sample codto, tres to get er coectos oly by depedet ad detcally dstrbuted (..d.) data samples [14]. As ts cocrete mplemet, support vector classfcato (SVC) exhbts good performaces geeralzg, hghdmesoal processg ad olear processg. Where least squares support vector classfcato (LS-SVC) has the mprovemets: equalty costrats are substtuted by equalty oes; a squared loss fucto s take for the error varable. he rest of ths paper s orgazed as follows. Secto II brefly troduces the basc theory of SL ad LS-SVC. Frequecy estmato algorthm based o LS-SVC s proposed Secto III, ad the proper choces of LS- SVC s parameters are also dscussed, takg bary phase shft keyg (BPSK) sgal for example. he Secto IV, smulatos ad expermets verfes the feasblty ad valdty of the proposed algorthm. At last, coclusos of the paper are gve Secto V. A. SL II. SL AD LS-SVC Amg at bary classfcato, the jot probablty desty fucto (PDF) of set S x, y 1,, s ukow, whch s deoted as Px, y, where x s dmeso characterstc vector, 1,1 y s category label. ow, we try to select a optmzg oe from the respose set of learg mache f xw, to mmze the expected rsk,.e.:,,, R w L y f x w dp x y (1) where,, L y f xw s the loss fucto used to show the dfferece betwee respose of trag mache f xw,. Accordg to bary y ad learg mache classfcato, t s deoted as L y, f xw, 0f y f xw, 1 f y f xw, 013 Egeerg ad echology Publshg 643

2 Joural of Commucatos Vol. 8, o. 10, October 013 where w s the argumet of f xw,. Ad, the dfferet type of learg problem has the dfferet L y, f xw, whch ca udoubtly mpact the ts learg performace. It s proved that f s a radom umber durg the rage 0,1, the equato s satsfed as follows, wth probablty 1 [14]: emp emp h R( w) R ( w) h R ( w) l h 1 l( 4) where Remp ( ) L y, f x, 1 () 1 w w s emprcal rsk fucto, h s cofdece lmt ad determes ts geeralzato capablty, h s Vapk-Cheroeks (VC) dmeso ad deotes ts complexty. As R ( ) emp w s drectly proportoal to h, h s versely proportoal to h, structural rsk mmz- R w. It eto (SRM) rule s preseted to mmzg has dvded S f x, w to a seres of subset: S1 S... Sk... S; S S (3) where ther VC dmeso s arraged by: h1 h... h k... (4) I each subset, the rule tres to fd the fucto havg the smallest emprcal rsk. At the same tme, t cosders the compromse betwee emprcal rsk ad cofdece lmt, ad ultmately derve the smallest expected rsk, whch s depcted Fg. 1. expected rsk w x b 1 f y 1 w x b 1 f y 1 Maxmzg the marg betwee two sorts whch equals w ad gettg the optmal hyperplae: 1 m J( w, b) w s.t. y b w x 1 0, 1,, By trouducg error varables e ad LS method, (6) s coverted to: 1 C m J w, b w e 1 s.t. y b w x 1 e, 1,, (5) (6) (7) Mmzg the frst tem of J( w, b) meas that the marg of (5) ow s the largest, ad mmzg the secod oe meas that both of the boudares ow are the ceter of ther sort as much as possble. Pealty factor C s a postve costat ad ca cotrol the pealty degree of fttg errors. It meas that, SRM s troduced to LS-SVC to make compromse cofdece lmt ad emprcal rsk. Equato (7) s a strct covex quadratc programmg (QP) problem optmzato theory. Usg Lagrage multpler method, where s Lagrage multpler: 1 w y ( x ) (8) 1 y 0 (9) Ce (10) Combg (7), (8), (9) ad (10), ad replacg ( x ) ( x ) wth kerel fucto K( x, x ) : j j h1 S 1 hsrm S SRM S cofdece lmt emprcal rsk h 0 Y b 0 1 = Y PQP I α E C where,,, 1,,1, y,, y (11) α 1 E Y 1, P s a dagoal matrx whose ma dagoal elemets are y,, 1 y, Q s amed as kerel fucto matrx. Beacause we select radus bass fucto (RBF) ths study, so the, j th elemet of Q s: Fgure 1. Structure rsk mmzato B. LS-SVC Supposed that, S ca be dvded by a hyperplae ( w( x )) b 0 wthout errors, where () s a er product operato ad () s a olear mappg from low to hgh dmeso characterstc space. he hyperplae ca be ormalzed, ad the boudary samples are satsfed by: x x j Qj K x, x j exp (1) h Ultmately, the dscrmat fucto s obtaed as f( x) sg w x b sg y K x, x+ b 1 (13) 013 Egeerg ad echology Publshg 644

3 Joural of Commucatos Vol. 8, o. 10, October 013 III. FREQUECY ESIMAIO ALGORIHM BASED O LS-SVC Accordgly, ts varace ad power are both meagless, ad the ga of sgal-to-ose rato (GSR) s defed as A. Sgal Model he susodal sgal polluted by ose s modeled as j f0 s 0 r a Ae w, 0,, 1 (14) where a s a depedet symbol, M s the modulato order; A 0, f0[ 0.5,0.5), 0[, ) are ampltude, determstc but ukow frequecy ad tal phase, respectvely; S s the sample perod, s the sample sze; w s a depedet complex ose wth zero-mea ad ukow PDF. For the sake of smplcty, we frstly set s 1 ad take bary phase shft keyg (BPSK) sgal for example. As a commo o-gaussa dstrbuto, -stable oe oly has ufed characterstc fucto [15]: t exp j t t 1 j sg twt, (15) where ta / f 1 wt,, / l t f 1 where 0, s characterstc expoet ad descrbes the thckess of tals, whe, t s Gaussa dstrbuto; whe 1, 0, t s Cauchy dstrbuto. 1,1 s skewess parameter, whe 0, t s symmetrcal about ad called S S for short. 0, s scale parameter ad smlar wth the varace of Gaussa dstrbuto., are shft parameter, whe 1, s the mea value; whe 0 1, s the termedate value. We set 1.5, 0, 1, 0 ths study, ad plot the dstrbuto of 1000 real samples Fg.. where S 1 A 10lg C g S exp l w, 0 ad Cg 1.78 s a postve costat. (a) f 0.09 (b) f 0.1 Fgure 3. Costellatos of ' r wthout ose Fgure. Real -stable dstrbuto wth 1.5, 0, 1 Cotemporarly, o lmted secod-order momet s exstg fractoal low-order -stable dstrbuto. (a) f Egeerg ad echology Publshg 645

4 Joural of Commucatos Vol. 8, o. 10, October 013 (b) f 0.1 (c) f 0.13 Fgure 5. Costellatos of r' whe ose s -stable dstrbuto (GSR s 6dB) B. Frequecy Estmato Algorthm Based o LS-SVC We costruct b e j f after settg a freque- cy f 0.5,0.5 ad phase,. Lettg r' r b r e j f (16) j f f 0 j f Aa e 0 w e (16) meas, the fluece of determstc frequecy f ad tal phase are removed from receved (c) f 0.13 sgal. We dvde r' to two sorts by a 1 or a ' Fgure 4. Costellatos of r whe ose s Gaussa dstrbuto,.e., we costruct the trag set S (SR s 6dB) x f,,, x f or Re r' f, or Im ', Re, Im are ormr f, y a, where or y 1, alzed, takg real ad mage part operators, respectvely. We make use of LS-SVC ad derve the SRM fucto wth respect to f : f m J f, w, b w,b (17) Itegratg (7) ad (8), (17) s coverted to: 1 f α f PQ f P α f 1 α f α f C (a) f 0.09 (18) where α f s the soluto of (11). Searchg the hole terval of f, ad gettg the estmato value of frequecy: fˆ arg m ( f ) f 0.5,0.5 (19) At frst, we cosder the uosy codto, ow r Aa e ' j f0 f 0. Settg f0 0.1, 0 0, A 1, 3, 0.5, the costellatos of r' wth dfferet f are llustrated Fg. 3. We ca see that oly f (b) f Egeerg ad echology Publshg 646

5 Joural of Commucatos Vol. 8, o. 10, October 013 f f0, r' are two dscrete pots; otherelse, r' s dfferet wth. As a result, eve f both two sorts are completely separable ad the secod tem of (18) s equal to 0 wth the dfferet f, 0.1 s stll the 1 (5) fˆ arg m α f PQ f P α f [ 0.5,0.5] As (4) ad (5) show, ow C barely flueces α( f ) ad fˆ whch s oly determed by Q f. Acco- mmzed value because of the frst oe. At the same tme, e j meas the clockwse rotato of a fxed agle costellato. herefore, the value of wll ot have a mpact o the proposed algorthm. he expermetal smulatos Part IV are cosstet wth ths cocluso, so 0 s set ths study. he, we take the osy codto to accout. Settg 0, SR or GSR 6dB, others are as Fg. 3, Fg. 4 ad Fg. 5 llustrate the costellatos of r' wth dfferet f whle oses are depedet rdgly, the estmato accuracy of the proposed algorththm decreases. Everythg s as Fg. 5 other tha settg the wdth of RBF h 1, ad the umber of Mote Carlo expermets s 1000, Fg. 6 llustrates the mea square error (MSE) curves of the proposed algorthm wth that C are 0.1, 1, 1000 ad 10000, resectvely. Fg. 6 s cosstet wth all above aalyses, ad we select C 1000 ths study. complex Gaussa dstrbuto ad -stable dstrbuto, respectvely, where Gaussa dstrbuto has zero-mea ad varace, the parameter settg of -stable dstrbuto s the same as Fg.. From them, we kow that whether the ose dstrbuto s, 0.1 keeps the mmzed oe. C. Parameter Settgs of LS-SVC From (11): 1 Y PQP I E C b 1 Y PQP I Y C (0) Fgure 6. Impact of C o MSE 1 YY PQP I 1 C α PQP I I E C 1 Y PQP I Y C (1) Whe C decreases rapdly, PQP 1 I 1 I, the (1) C C ad (19) are smplfed as YY YY α f C I E C I Y Y E 1 fˆ arg m α f α f f 0.5,0.5 C () (3) From () ad (3), we ca kow that fˆ wth Fgure 7. Impact of h o MSE dfferet f0 f are same ad very small, for the reaso of the equal α( f ) wth dfferet f0 f. As a result, the proposed algorthm fals. Summarzgly, we must select C as large as possble. However, too large C wll lead Accordg to (1), the ma dagoal elemets of Q keep 1 all the tme. Whe h decreases, other elemets gradually approach to 0, thus Q I. O the cotrary, whe h creases, these elemets are gradually close to 1, thus Q s approxmately a all-oe matrx. Durg the both codtos, α( f ) ad fˆ keep the same wth that PQP 1 I PQP, the (1) ad (19) are smplfc ed as α f PQ f P YY PQ f P I E Y PQ f P Y 013 Egeerg ad echology Publshg dfferet f0 f all the tme. I the sequel, the estmato accuracy of the proposed algorthm decreases. Everythg (4) 647

6 Joural of Commucatos Vol. 8, o. 10, October 013 s as Fg. 6 other tha C 1000, Fg. 7 llustrates the MSE curves of the proposed algorthm wth that h are 0.1, 1, 10 ad 100, resectvely. Fg. 7 s cosstet wth all above aalyses, ad we select h 1 ths study. IV. SIMULAIOS AD EXPERIMES Frstly, the mpact of o the estmato performace s cosdered. Everythg s as Fg. 6 except that C 1000, Fg. 8 llustrates the MSE curves of the proposed algorthm wth that s 8, 16, 3 ad 64, resectvely. It s show, MSE performace s mproved as creases. ext, the mpact of o the estmato performace s cosdered. Everythg s as Fg. 6 except that C 1000, Fg. 9 llustrates the MSE curves of the proposed algorthm wth that are , respectvely. It s show, MSE performaces of dfferet are almost the same. Rao lower boud (CRLB). he proposed algorthm also searches extremum durg frequecy rage, so t s compared wth FDP, the umber of FF pots s We tegrate coarse ad fe search ths study, whose steps are ad 1e 5, respectvely. he cocrete searchg method s Gauss-ewto. Everythg s as Fg. 4 ad Fg. 5 except that C 1000, h 1, ad the umber of Mote Carlo expermets s 1000, the mea of both two algorthms wth dfferet f 0 are plotted Fg. 10 ad Fg. 11. Dstctly, whe ose s -stable dstrbuto, the ubased performace of the proposed algorthm s better tha MLE oe s the codto of low SR, ad they are almost the same uder other codtos. (a) SR s -4dB Fgure 8. Impact of o MSE (b) SR s 0dB Fgure 10. Mea whe ose s Gaussa dstrbuto Fgure 9. Impact of o MSE he, the mea performace s worked out. Uder the assumpto of Gaussa ose, maxmum lkelhood estmator (MLE) realzed by fast Fourer trasform (FF) [1] s the best oe because of ts MSE ca reach Cramer 013 Egeerg ad echology Publshg (a) GSR s -4dB 648

7 Joural of Commucatos Vol. 8, o. 10, October 013 V. COCLUSIOS (b) GSR s 0dB Fgure 11. Mea whe ose s -stable dstrbuto Last, the estmato performace s take to accout. Everythg s as Fg. 10 ad Fg. 11 except that f 0.1, the MSE curves of both two algorthms are plotted Fg. 1 ad Fg. 13, respectvely. Fgure 1. MSE whe ose s Gaussa dstrbuto o some extet, SVC whch s based o SL ca solve the problems about curse of dmesoalty ad overlearg. hs paper coverts estmato problem to classfcato oe durg the searchg part, ad takes use of LS-SVC to estmate frequecy wth ukow os-e dstrbuto. Also, from vews of qualtatve aalyses ad expermet results, we dscuss the choce of LS-SVC s parameters. At last, we verfy the feasblty ad valdty through smulatos. I ths paper, BPSK s take for example. If sgal type s QPSK, 8PSK or other modulato mode, bary classfcato s exteded to multclass problem LS-SVC by the same way. At the same tme, emphaszed that, ths paper takes Gaussa dstrbuto ad -stable dstrbuto as two cases of ukow ose model, but the proposed algorthm s ot oly oret to these two cases. Classcal algorthms of frequecy estmato prmarly try to chage receved sgals to sgle-toe oes. O the cotrary, the proposed algorthm completely cosders effect of symbol formato ad drectly estmates the frequecy. Hece, amg at sgle-toe sgals, we ca estmate frequecy by meas of addg kow symbol sequece to receved sgals. It s a ovel ad coverse thought, ad wll be valuable. Resolvg covex QP problem havg equalty costrats wll brg large computatoal load ad tme cosumg to SVC. Although LS-SVC s preseted, the proposed algorthm wll search mmum value durg whole rage, uavodably. As a result, there wll be a tradeoff betwee estmato accuracy ad computatoal complexty. At the same tme, we oly obta the proper rage of LS-SVC s parameters. Ad as a ext step, how to select them exactly s a mportat research pot. REFERECES Fgure 13. MSE whe ose s -stable dstrbuto As show Fg. 1 ad Fg. 13, whe ose s Gaussa dstrbuto, the MSE curves of the proposed algorthm s observed to le very close to that of MLE the codto of kow dstrbuto model. Whe ose s -stable dstrbuto, MLE s effectve as ose model s ukow. evertheless, the proposed algorthm stll keep ts estmato accuracy, eve though that threshold effects are exstg t. [1] H. Fu ad P. Y. Kam, Kalma estmato of sgle-toe parameters ad performace comparso wth map estmator, IEEE rasactos o Sgal Processg, vol. 56, o. 9, pp , 008. [] R. G. Mckllam, B. G. Qu, I. V. L. Clarkso, ad B. Mora, Frequecy estmato by phase uwrappg, IEEE rasactos o Sgal Processg, vol. 58, o. 6, pp , 010. [3] D. Kudu, et al., Super effcet frequecy estmato, Joural of Statstcal Plag ad Iferece, vol. 141, o. 8, pp , 011. [4]. Raïss, Relable ampltude ad frequecy estmato for based ad osy sgals, Commucatos olear Scece ad umercal Smulato, vol. 16, o. 11, pp , 011. [5] Y. Cao ad G. We, A oteratve frequecy estmator wth ratoal combato of three spectrum les, IEEE rasactos o Sgal Processg, vol. 59, o. 10, pp , 011. [6] G. We, Y. Cao, ad F. J. Che, Closed-Form frequecy estmator based o arrow-bad approxmato uder 013 Egeerg ad echology Publshg 649

8 Joural of Commucatos Vol. 8, o. 10, October 013 osy evromet, Sgal Processg, vol. 91, pp , 011. [7] Y. Cao, G. We, ad F. J. Che, A Closed-Form expaded autocorrelato method for frequecy estmato of a susod, Sgal Processg, vol. 9, pp , 01. [8] H. Fu ad P. Y. Kam, Phase-Based, mg-doma estmato of the frequecy ad phase of a sgle susod AWG the role ad applcatos of the addto observato phase ose model, IEEE rasactos o Iformato heory, vol. 59, o. 5, pp , 013. [9] S. A. Kassam, Sgal Detecto o-gaussa ose, ew York: Sprger-Verlag, [10] X. Y. Ma ad C. L. kas, Jot estmato of tme delay ad frequecy delay mpulsve ose usg fractoal lower order statstcs, IEEE ras o Sgal Processg, vol. 44, o. 11, pp , ov [11] K. Barma ad M.. Arvd, A mmum varace sgle frequecy estmator usg recursve least squares estmate of ose statstcs, Proc Mdwest Symposum o Crcuts ad Systems, otre Dame, Frace, [1] S.. L, H. Wa, Y. Q. Huag, ad W. J. Huo, Precse measuremet of RCF colored gaussa ose, Proc. 007 Iteratoal Coferece o Wreless Commucatos, etworkg ad Moble Computg, Shagha, Cha, 007. [13] S. P. Lu, X. We, ad Z. G. Wag, Least square sgletoe frequecy estmator o-gaussa colored ose, Joural of Commucatos, vol. 9, o. 7, pp , July 008. [14] V.. Vapk, he ature of Statstcal Learg heory, ew York: Sprger-Verlag, [15] G. Samorodtsky ad M. S. aqqu, Stable o-gaussa Radom Processes: Stochastc Models wth Ifte Varace, ew York: Chapma & Hall, Xueqa Lu receved hs B.S. ad M.S. degree both from Zhegzhou Iformato Scece ad echology Isttute sgal processg 006 ad formato hdg 009, respectvely. He s curretly pursug the Ph.D. degree commucato egeerg at Zhegzhou Iformato Scece ad echology Isttute. Hs research terests clude dgtal sgal processg, mage processg ad formato hdg. 013 Egeerg ad echology Publshg 650