IDENTIFICATION OF DIGITAL MODULATION TYPES USING THE WAVELET TRANSFORM

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1 IDENTIFICATION OF DIGITAL MODULATION TYPES USING THE WAVELET TRANSFORM Lig Hog K.C. Ho Deprtmet of Electricl Egieerig, Uiversity of Missouri-Columbi, Columbi MO 65, USA ABSTRACT Automtic idetifictio of the digitl modultio type of sigl hs foud pplictios i my res, icludig electroic wrfre, surveillce d thret lysis. This pper studies the use of wvelet trsform to distiguish QAM sigl, PSK sigl d FSK sigl. The pproch is to use wvelet trsform to extrct the trsiet chrcteristics i digitl modultio sigl, d pply the distict ptter i wvelet trsform domi for simple idetifictio. The relevt sttistics for optimum threshold selectio re derived uder the coditio tht the iput oise is dditive white Gussi. The performce of the idetifictio scheme is ivestigted through simultios. Whe CNR is greter th 5 db, the percetge of correct idetifictio is bout 97% with 5 observtio symbols.. INTRODUCTION Automtic idetifictio of the digitl modultio type of sigl is rpidly evolvig re []. It hs foud wide pplictios i my res, icludig electroic wrfre, surveillce d thret lysis. A vriety of techiques hve bee proposed to distiguish qudrture mplitude modultio(qam) sigl, phse shift keyig(psk) sigl d frequecy shift keyig(fsk) sigl. Hsue d Solim [] proposed to use the vrice of the zero-crossig itervl sequece s well s the frequecy d phse differece histogrm for idetifictio. Beids d Weber [3] proposed clssifier bsed o the time domi higher-order correltios. Hero et l. [4] proposed method usig eigedecompositio of momet mtrices. Huo d Dooho [5] used the couts of sigls fllig ito differet prts of the sigl ple to idetify the digitl modultio types. These methods re either computtiolly itesive or require high crrier-to-oise rtio (CNR). Uder the ssumptio tht some commuictio prmeter such s crrier frequecy d symbol rte re vilble, Polydoros d Kim [6] derived pproximte likelihood rtio modultio clssifier to seprte BPSK d QPSK. The clssifier works well eve t low CNR. Aother pproch for digitl modultio types idetifictio is to use wvelet trsform (WT) [7]. Differet digitl modultio sigls coti differet trsiets i mplitude, frequecy or phse. The WT hs cpbility to extrct trsiet iformtio d thereby llowig simple methods to perform modultio idetifictio. Li d Kuo [8] pplied Morlet wvelet to detect the phse chges, d used the likelihood fuctio bsed o the totl umber of detected phse chges s feture to clssify M-ry PSK sigl. Ho et l. [9], o the other hd, proposed method to idetify PSK d FSK sigls usig the Hr WT without the eed of y commuictio prmeter of modulted sigl. I idel cse, the Hr WT mgitude(jhwtj) of PSK sigl is costt d tht of FSK sigl is multistep fuctio. Hece the vrice of jhwtj of iput sigl is used s feture to clssify the two sigls. This pper exteds the pproch of [9] to iclude QAM sigl i the sigl set for idetifictio. Compred to PSK d FSK sigls, oe distictio i QAM sigl is tht it does ot hve costt mplitude. The jhwtj of QAM sigl is multi-step fuctio similr to tht of FSK sigl becuse of the chge i mplitude s symbol chges. O the other hd, if we ormlize QAM sigl to uity mplitude, it will behve like PSK sigl d its jhwtj will be costt. Amplitude ormliztio hs little effect o the jhwtj of PSK d FSK sigls sice both of them hve costt mplitude. Hece we c distiguish QAM, PSK d FSK sigls by computig the vrices of jhwtj with d without pplyig mplitude ormliztio. We shll elborte i detil the methodology for digitl modultio idetifictio usig WT i the ext sectio. Sectio 3 gives the simultio results. Filly, coclusio is mde i Sectio 4.. MODULATION IDENTIFICATION BY WT Let the received wveform r(t), t T be described s r(t) =s(t) (t), where(t) is complex white Gussi oise. The sigl s(t) c be represeted i complex form s s(t) =~s(t)e j(!ctc) () where! c is the crrier frequecy d c is the crrier phse /99/$. (c) 999 IEEE

2 For QAM sigl, ~s QAM (t) = i= (A i jb i )u T (t, it ); A i ;B i fm,, M;m =; ; :::; M g: () For PSK sigl, p X N ~s PSK (t) = S e j'i u T (t, it ); i= ' i f (m, );m=; ; :::; M g: (3) M For FSK sigl, p X N ~s FSK (t) = S e j(!iti) u T (t, it ); i=! i f! ;! ; :::;! M g; i (; ): (4) I (), (3) d (4), S is the sigl power, N is the umber of observed symbols, T is the symbol durtio d u T (t) is the stdrd uit pulse of durtio T. I this study, we ssume tht the three sigls hve the sme symbol durtio. It c be see from ()-(4) tht symbol chges will give rise to trsiets i the modulted sigls. The trsiets re creted idepedetly i the chges of mplitude, phse d frequecy respectively. WT c chrcterize these trsiets effectively d llowig simple method for idetifictio of the three sigls. The cotiuouswt of sigl s(t) is defied s [7] CWT(; ) = = Z s(t) Z p s(t) (t)dt t, ( )dt: (5) where is the scle, is the trsltio d the superscript * deotes complex cojugte. The fuctio (t) is the mother wvelet d the bby wvelet (t) comes from time-sclig d trsltio of the mother wvelet. The choice of mother wvelet depeds o its pplictio. Some widely used wvelets re Morlet, Hr d Sho [7]. Due to its simple form d ese of computtio, we shll cosider the Hr wvelet oly. The followig lysis derives the Hr WT of QAM, PSK d FSK sigls. I idel cse, whe the Hr wvelet is withi symbol time, the Hr WT of QAM sigl is CWT(; )= p (, Z, Z (A i jb i )e j(!c(t)c) dt (A i jb i )e j(!c(t)c ) dt) = 4p S i j p! c si (! c 4 )ej(!cc i) : (6) p where S i = A i Bi is the mplitude of the ith symbol d i = t, (B i =A i ) is the phse of the ith symbol. Tkig the mgitude of (6) to elimite the ukow crrier phse gives jcwt QAM (; )j = 4p S i p!c si (! c 4 ) (7) Whe the Hr wvelet covers symbol chge, the Hr WT of QAM sigl is CWT(; )= p ( =, Z d Z Z d, (A i jb i )e j(!c(t)c ) dt (A i jb i )e j(!c(t)c ) dt (A i jb i )e j(!c(t)c ) dt) j p! c e j(!cci) [S i (e j!cd, e,j!c ) S i e j (, e j!cd, e j!c )]: (8) where S i d S i re the mplitudes of the ith d (i )th symbols, i d i re the phses of ith d (i )th symbols, = i, i is the phse chge d d is the time istce whe the symbol chges reltive to the ceter of the wvelet. It is ssumed to be egtive i (8). For positive d, we c derive similr formul s (8). It is observed from (6)-(8) tht whe the wvelet is withi symbol period, the jhwtjof QAM sigl without oise is costt idepedet of the trsltio. Sice the mplitude of the QAM sigl S i is vrible, the jhwtj ptter of QAM sigl is multi-step fuctio. There will be distict peks i the jhwtj resulted from phse chges t the times where the wvelet covers symbol chge. The jhwtj of PSK sigl d FSK sigl hve bee derived from [9] whe the Hr wvelet is withi symbol time: jcwt PSK (; )j = 4p S p si (! c!c 4 ) (9) 4 p S jcwt FSK (; )j = p (!c! i ) si [ (! c! i ) ] 4 () Note tht! i is fixed withi symbol. It is cler from (9) d () tht i idel cse, the jhwtj of PSK sigl is costt, while the jhwtj of FSK sigl is multi-step fuctio sice the frequecy is vrible. Similr to QAM sigl, there will be distict peks i the jhwtj of the two sigls whe the Hr wvelet covers symbol chge. Figure shows the jhwtj of QAM, PSK d FSK sigls /99/$. (c) 999 IEEE

3 of QAM sigl without mplitude ormliztio of QAM sigl with mplitude ormliztio of PSK sigl without mplitude ormliztio of PSK sigl with mplitude ormliztio of FSK sigl without mplitude ormliztio of FSK sigl with mplitude ormliztio Figure : jhwtj of QAM, PSK d FSK sigls. We defie mplitude ormliztio of sigl ~s(t) s follows: s(t) = ~s(t) () j~s(t)j From ()-(4), we hve the mplitude ormlized modultio sigls: s QAM (t) = i= s PSK (t) = s FSK (t) = i= e j rct B i A i u T (t, it ) () i= e j'i u T (t, it ) (3) e j(!iti) u T (t, it ) (4) Amplitude ormliztio does ot ffect the modultio type of the PSK sigl d FSK sigl sice they re costt mplitude modultio. O the other hd, the mplitude vritios i QAM sigl dispper fter mplitude ormliztio. s QAM (t) hs the sme form s s PSK (t). As result, QAM sigl with mplitude ormliztio will hve costt jhwtj with peks due to phse chge. However, the chrcteristics of the jhwtj of mplitude ormlized PSK sigl d FSK sigl will remi the sme. Figure shows the jhwtj of the three sigls fter mplitude ormliztio. To differetite QAM, PSK d FSK sigls, we must fid commo feture d select criterio bsed o differeces. If we igore the peks, comprig Figure d Figure : jhwtj of QAM, PSK d FSK sigls with mplitude ormliztio. Figure revels tht the jhwtj of QAM sigl is multistep fuctio. It becomes dc whe mplitude ormliztio is pplied to the sigl. Agi, igorig the peks the jhwtj of PSK sigl is costt d tht of FSK sigl is multi-step fuctio, o mtter their mplitudes re ormlized to uity or ot. The vrice of costt is zero while tht of multi-step fuctio is lrger th zero. Hece we c distiguish QAM sigl, PSK sigl d FSK sigl by computig the vrices of the jhwtj with d without mplitude ormliztio, fter removig the peks by medi filterig. Figure 3 is the WT modultio idetifier for QAM, PSK d FSK sigls. The idetifier cosists of two brches d decisio block. Oe brch is without mplitude ormliztio d the other is with mplitude ormliztio. The idetifier first fids the jhwtj of iput sigl. After removig the peks of jhwtj by medi filter, the idetifier computes the vrice of the medi filter outputs. The decisio block compres the two vrices with two thresholds to decide the modultio type of iput sigl. If the vrice produced by the brch without mplitude ormliztio is lrger th the threshold while the other vrice is smller th the threshold, the iput is clssified s QAM sigl. If both vrices re lower th the thresholds, the iput is clssified s PSK sigl. If both vrices re higher th the thresholds, the iput is clssified s FSK sigl. Uder the coditio tht the iput oise is white Gussi, the relevt sttistics for optimum threshold selectio c be evluted. First, we lyze the theoreticl vrice /99/$. (c) 999 IEEE

4 Iput Normlize mplitude to uity Medi filter Medi filter Compute vrice Compute vrice - - TH TH Decisio Block QAM > < PSK < < FSK > > i (5) should be replced by = E[jr(t)j ],wherejr(t)j is the mplitude of iput sigl. If the probbility of misclssifictio of PSK sigl is give, we c determie the optimum thresholds for the two brches of the idetifier. 3. SIMULATION Figure 3: Block digrm of the digitl modultio idetifier. of jhwtj of the brch without mplitude ormliztio. Deote rdom vrible r s the jhwtj d ssume the oise power is. It c be show tht the Hr WT of white Gussi oise is orml distributed with the sme me d vrice s the white Gussi oise sice WT is lier trsform. Hece the probbility desity fuctio (pdf) of r is []: f R (r) = r expf, r A gi ( Ar ) (5) where A is the true jhwtj without oise. I () is the modified Bessel fuctio of the first kid d zero order. The pdf reduces to Gussi t high CNR. If the iput is PSK sigl, the theoreticl vrice of the medi filter output c be clculted becuse the output is costt with rdom oise dded to it. O the other hd, if the iput is QAM sigl or FSK sigl, the theoreticl vrice of the medi filter output is ukow becuse it depeds o the ukow modultio prmeters such s symbol rte, mplitude d crrier frequecy. Whe the medi filter iput is the jhwtj of PSK sigl d whe the CNR is high, the pdf of the medi filter output c be pproximted by Gussi [], with the me v equls to the theoreticl medi d vrice equls to v = 4fL v, [4f R (v) ], gf R (v) (6) where f R (v) is the pdf vlue i (5) evluted t r =v. L v is the legth of the medi filter. It c be rbitrry s log s it is greter th 3. v c be pproximted by vergig the medi filter output. Hece uder the hypothesis tht the iput is PSK sigl, the smple vrice of the medi filter output c be evluted d it hs Chi-squre distributio. At high CNR, the theoreticl vrice of jhwtj i the brch with mplitude ormliztio c be evluted usig the sme method s before. It c be show tht the smple vrice of the medi filter output, uder the hypothesis tht the iput is PSK sigl, is lso Chi-squre distributio. Becuse of mplitude ormliztio, the oise power I this sectio, we preset the simultio results of the proposed digitl modultio idetifier show i Figure 3. We used the QAM6 sigl, QPSK sigl d QFSK sigl s exmple for QAM, PSK d FSK sigls. The crrier frequecy ws khz d the smplig frequecy ws khz. The symbol rte ws 6 symbols per secod. Except for the CNR d oise power, o other prior kowledge ws ssumed for the idetifier. These two qutities c be estimted from received sigl. The oise ws white Gussi. The biry dt strem for modultio ws obtied from rdom umber geertor. The umber of symbols i ech sigl sequece ws 5. Tble gives the percetge of correct idetifictio for QAM, PSK d FSK sigls. The results were the verge of idepedet esemble rus. The legth of the medi filter ws 5. The scle of the WT ws 4 for the brch without mplitude ormliztio d 6 for the other oe. Choosig lrger scle i the brch with mplitude ormliztio c reduce oise level of QAM sigl icresed by the mplitude ormliztio. We pplied dow-smplig i WT to reduce the computtio lod. The two thresholds were set by the Chi-squre distributiowith. probbility of PSK misclssifictio i the upper brch d QAM misclssifictio i the lower brch of the idetifier. The computtio of v from (6) requires f R (v). f R (v) is depedet o A s show i (5). The vlue A ws geerted by vergig the jhwtj of the iput sigl. Tble : Percetge of Correct Idetifictio for QAM, PSK d FSK sigls. (%) i/out CNR=dB CNR= QAM PSK FSK QAM PSK FSK QAM QPSK QFSK i/out CNR=dB CNR= QAM PSK FSK QAM PSK FSK QAM QPSK QFSK Tble clerly shows tht the proposed idetifier hs high ccurcy. Whe CNR is greter th 5 db, the per /99/$. (c) 999 IEEE

5 cetge of correct idetifictio for FSK sigl is %. The percetge of correct idetifictio for QAM sigl is higher th 98.7%. The icrese of the percetge of correct idetifictio for QAM sigl t CNR is probbly becuse of the limited umber of esemble rus. Whe CNR is lower th db, the misclssifictio percetge for PSK sigl is lrger th the expected vlue of.. It is suspected tht the error is due to fiite dt legth d the iccurcy i estimtig the true jhwtj vlue A i (5) sice peks were lso verged. Aother criterio to evlute the performce of the proposed modultioidetifierisby the receiver opertig chrcteristics (ROC) curves. It is plot of probbility of detectio (PD) s fuctio of the probbility of flse lrm (PFA). We clculted the probbilities usig esemble rus t differet thresholds d geerted curves ccordig to the results. The fster the curve rises, the better is the performce of the idetifier. Figure 4 shows the ROC curves for the idetifier whe CNR is equl to, d db. probbility of detectio.5 ROC for PSK d QAM sigls db ROC for PSK d FSK sigls db ROC for QAM d FSK sigls db probbility of flse lrm Figure 4: ROC of the proposed idetifier. Whe CNR is, PD is % idepedet of PFA. Whe CNR is d the PFA is., the PD betwee PSK d QAM is 97% d tht betwee QAM d FSK is 95%. Whe CNR is db d the PFA is smller th.4, the PD drops rpidly. This is becuse the hypothesisof moderte CNR used to obti the optimum threshold will o loger be vlid. 4. CONCLUSIONS We hve proposed d studied i this pper digitl modultio idetifier usig wvelet trsform for QAM, PSK d FSK sigls. The idetifier cosists of two brches d decisio block. It computes the jhwtj of iput sigl with d without mplitude ormliztio, uses medi filters to remove the peks i the jhwtj s, clcultes the vrices of the medi filter outputs, d mkes the decisio of the iput modultio type by comprig the vrices from the two brches with thresholds. The relevt sttistics for optimum threshold selectio were derived. Simultios showed tht the percetge of correct idetifictio is higher th 97% with 5 observtio symbols, whe CNR is ot lower th 5 db. 5. REFERENCES [] E.E.AzzouzdA.K.Ndi,Automtic Modultio Recogitio of Commuictio Sigls, Kluwer Acdemic Publishers, 996. [] S.-Z. Hsue d S.S. Solim, Automtic modultio clssifictio usig zero crossig, i IEE Proceedigs F (Rdr d Sigl Processig), vol. 37, o. 6, pp , December 99. [3] B.F. Beids d C.L. Weber, Higher-order correltiobsed pproch to modultio clssifictio of digitlly modulted sigls, i IEEE Jourl o Selected Are i Commuictios, vol. 3, o., pp. 89-, Jury 995. [4] Alfred O. Hero III d Hfez Hdiejd-Mhrm, Digitl modultio clssifictio usig power momet mtrices, i Proceedigs IEEE ICASSP-98, My 998. [5] Xiomig Huo d Dvid Dooho, A simple d robust modultio clssifictio method vi coutig, i Proceedigs IEEE ICASSP-98, My 998. [6] A. Polydoros d K. Kim, O the detectio d clssifictio of qudrture digitl modultios i brod-bd oise, i IEEE Trsctios o Commuictios,vol. 38, o. 8, pp. 99-, August 99. [7] Y. T. Ch, Wvelet Bsics, Kluwer Acdemic Publishers, 995. [8] Y.C. Li d C.-C. Jy Kuo, Modultio clssifictio usig wvelet trsform, i Proceedigs SPIE, vol. 33, pp [9] K.C. Ho, W. Prokopiw d Y.T. Ch, Idetifictio of M-ry PSK d FSK sigls by the wvelet trsform, i Proceedigs IEEE Militry Commuictios Cof., pp , Clifori, November 995. [] M. Schwrtz, Iformtio Trsmissio, Modultio, d Noise, McGrw-Hill, New York, 97. [] B.I. Justusso, Medi filterig: sttisticl properties, i Two-Dimesiol Digitl Sigl Processig, Topics i Applied Physics, Spriger-Verlg, New York, /99/$. (c) 999 IEEE