Mathmatical Mthods and chniqus in Enginring and Environmntal Scinc A simpl automatic classifir of PSK and FSK signals using charactristic cyclic spctrum ANONIN MAZALEK, ZUZANA VANOVA, VOJECH ONDYHAL, VACLAV PLAENKA Communication and Information systms Dpartmnt Univrsity of Dfnc Brno, Kounicova 65, 66 10 CZECH EPUBLIC {antonin.mazalk zuzana.vranova vojtch.ondryhal vaclav.platnka}@unob.cz http://www.vojnskaskola.cz/school/ud/fmt/structur/k09/pags/popl.aspx Abstract: - his articl dals with currnt issus of automatic modulation rcognition. As a suitabl mthod, classification invstigating charactristic shap of th signal cyclic spctrum was chosn. First, a thortical analysis of th issu of calculating of cyclic spctrum stimation is mad using Strip Spctral Corrlation Algorithm (SSCA). In anothr part, important findings larnd during xprimnts ar prsntd and conditions for obtaining th charactristic cyclic spctrum ar dtrmind. Aftr that, a simpl automatic classifir of modulations is dsignd and optimizd by using a sris of xprimnts. h classifir dsignd was tstd by using a sris of random modulatd signal ralizations with AWGN nois. Ky-Words: - automatic modulation rcognition, modulation classifir, PSK, FSK, cyclic spctrum, cyclostationarity, Matlab 1 Introduction Partial, but vry complx problm of radio signal analysis is rcognition of th modulation typ usd. Difficulty of a task is basd on th varity of possibl modulation typs usd and thir paramtrs, and also on th spcifics of radio channls, such as fading or multipath propagation. Contmporary trnd is automatization of modulation rcognition. h issu of automatic rcognition of modulatd signals rcordd considrabl progrss in th last two dcads. his is du to both tchnological dvlopmnt and digitization of rang of branchs and particularly dynamic growth in th fild of pattrn rcognition. Originally, th ara of modulation rcognition blongd primarily to th domain of spcializd civilian institutions daling with managmnt and us of frquncy spctrum and slctd scurity forcs (polic, army, intllignc srvics). wadays, it is also usd in commrcial communications systms, such as cognitiv radio [1]. In this articl, w dscrib a simpl classifir of digitally modulatd PSK and FSK signals basd on xamining th cyclic spctrum shap. h scond chaptr of th articl dals with thortical analysis of cyclostationary signals. In th third chaptr, a simpl classifir will b dsignd and a st of xprimntal rsults will b proposd. Chaptr 4 contains a summary of th rsults achivd. Estimation of signal cyclic spctrum A random procss t) is said to b Nth ordr cyclostationary in th strict sns if its Nth ordr distribution function xhibits priodicity in tim with priod [] F ( x1, x,..., xn; t1, t,..., tn) = (1) F( x1, x,..., xn ; t1 + m, t + m,..., tn + m ). In practic it is oftn sufficint to us only scond ordr statistics, which lad to th dfinition of scond-ordr cyclostationarity in wid sns. h ky scond ordr statistical charactristic is instantanous autocorrlation function (t,τ). So, th procss t) is said to b wid sns cyclostationary if its autocorrlation function is priodic in tim with priod. h instantanous autocorrlation function is dfind as ( t, τ ) = t + τ / ) x ( t τ / ), () whr τ is th tim lag and rprsnts th complx conjugat of th signal t). Bcaus th instantanous autocorrlation function is priodical ISBN: 978-1-61804-046-6 435
Mathmatical Mthods and chniqus in Enginring and Environmntal Scinc in tim (for all τ) it can b xpandd as Fourir sris (th convrgnc of sris is assumption) + n= i t t = π (, τ ) ( τ ), (3) whr = n / ar calld cyclic frquncis, and (τ ) ar th Fourir cofficints of th instantanous autocorrlation function that ar also rfrrd to as cyclic autocorrlation function 1 i πt ( τ ) = ( t, τ ) dt. (4) h cyclic autocorrlation function prsnts th statistical dscriptor of cyclostationary signals in two dimnsion tim-frquncy domain. Whn w rxprss th quation (4) by substitution (), w obtain [3] 1 1 ( τ ) = t + τ / ) x ( t τ / ) iπt iπ( t+ τ / ) + iπ( tτ / ) [ t + τ / ) ][ t τ / ) ] { u( t) v( t) } dt = dt = (5) E w it is possibl to bring out two othr, but quivalnt, dfinitions of cyclostationary signals [3]. Firstly, th signal t) contains scond ordr priodicity (is cyclostationary signal) if and only if th powr of spctral dnsity of th dlay-product signal () for som dlays τ contains spctral lins at som nonzro frquncis 0, that is if and only if ( τ ) 0 is satisfid. π i t Scondly, th multiplication with ± shifts th signal t) in th frquncy domain of ± /. So th cyclic autocorrlation function of a signal t) is cross-corrlation function of th frquncy shiftd vrsions u(t) and v(t) of th sam signal t) which lads to third dfinition. h signal t) xhibits scond ordr cyclostationary if and only if frquncy shiftd vrsions of t) ar corrlatd with ach othr, that is if E { u( t) v( t) } = ( τ ) is not idntically zro as a function of τ for som 0. Cyclic autocorrlation function is a tim domain dscriptor. For many applications it is a mor usful and convnint frquncy domain dscriptor. It is wll known that th Winr-Khinchin thorm that dtrmins th powr spctral dnsity by applying Fourir transforms on autocorrlation function. So whn w apply th Fourir transform on th cyclic autocorrlation function, w obtain so calld spctral corrlation dnsity function S ( f ) (SCD), which rprsnts th ky dscriptor of cyclostationary signals in frquncy domain S { ( τ )} = ( f ). (6) i τ = τ π f F ( ) dτ h form (7) is convntionally usd du to rlation to practic calculation of SCD t / 1 1 S( f ) = limlim X t, f + X t, f dt, (7) t t / t whr X ( t, f ) = t + / t / u) iπfu du, (8) is a spctral componnt of t) at frquncy f with bandwidth 1/..1 SCD Estimation using SSCA Algorithm A numbr of algorithms which ar abl to calculat th signal cyclic spctrum stimation wr publishd [4]. With rgard to th computational fficincy th Strip Spctral Corrlation Algorithm (SSCA) was chosn. A block diagram of th SSCA calculation is shown in Figur 1. Fig. 1.: A block diagram of calculation of th SCD stimation using th SSCA ISBN: 978-1-61804-046-6 436
Mathmatical Mthods and chniqus in Enginring and Environmntal Scinc h SSCA algorithm can b mathmatically xprssd as formula f + q f k q S k m, = N 1 t iπ qn N X ( m + n, fk ) x ( m + n) g( n) N n=, (9) whr k is a multipl of frquncy rsolution f, q is a multipl of cyclic rsolution, g(n) is a rctangular window of lngth t, x ( m ) is a complx conjugat unfiltrd input signal and X ( m, f ) is a complx dmodulation of th discrt signal x[m] dscribd by th rlationship X ( m, f ) = a( k) m k) 1 k = k = 1 a( k) m + k) s iπ f ( mk ) vz iπfk iπfm s = (10) whr a(k) is th Hamming window of lngth N o and s is a sampling priod. Cyclic rsolution dpnds on th numbr of sampls of signal = 1 / t = 1/ N and frquncy rsolution is dtrmind by th lngth of xamind sctions of th signal f = 1/ N [4]. o Fig..: SCD PSK4 with paramtrs N=104, N o =16, fc = 11 khz, vm = 500Bd, fs = 44 khz 3 Problm solving In th litrary sourcs, it is possibl to find how idal SCD of various signals looks lik (including th drivd analytical rlationships) [], [3], [5]. Although, it is oftn statd that SCD is suitabl for automatic signal classification, publications focusing on this topic wr issud just a fw. As an xampl w mntion [6]. hrfor, w dcidd to tst th simpl automatic classifir. h digital modulation typs that can b classifid ar: PSK, PSK4, and FSK FSK4. As programming nvironmnt, th tool Matlab was usd. In xprimnts with PSK and FSK modulations, it was found that th shap of th obtaind cyclic spctrum strongly dpnds on th modulation paramtrs, namly on th ratio of modulation spd v m and frquncy of carrir wav f c. Optimal spctrum - asy to us for classification - can b obtaind only whn th frquncy of th carrir wav is much highr than th modulation spd of transfr data. his fact is illustratd in th Figurs and 3. Fig. 3.: SCD PSK4 with paramtrs N=104, N o =16, fc = 11 khz, vm = 5500Bd, fs = 44 khz On this basis and with rgard to th frqunt subsqunt procssing on standard PC w hav slctd paramtrs listd in abl 1 to simulat th chosn modulation. abl 1: Paramtrs of simulatd modulation Modulation f s [khz] f c [khz] v m f [Bd/s] [Hz] PSK 44 15 500 - PSK4 44 15 500 - FSK 44 15 500 500 FSK4 44 15 500 500 ISBN: 978-1-61804-046-6 437
Mathmatical Mthods and chniqus in Enginring and Environmntal Scinc On th basis of xamination of th shap of SCD rprsntativ ralization of PSK and FSK modulation th simpl classifir, th block diagram of which is shown in Figur 4, has bn dsignd. Fig. 6.: h curvs bounding th optimal dcision lvls about th numbr of carrir wavs h digital modulation classifir dsignd was tstd on simulatd signals. For ach typ of modulation th lvl of nois was st in th rang of th SN = 30 to -5 db in stps of 1 db. For ach valu of nois 00 random signal ralizations with AWGN nois with lngth of 4416 sampls wr gnratd. h succss rat of th classifir masurd is plottd in th graph in Figur 7. Fig. 4.: A block diagram of th automatic classifir of PSK and FSK modulations For optimal sttings of dcision lvls of classifir, a sris of hundrds xprimnts was carrid out. h valus wr dtrmind xprimntally from th masurd curvs rfrrd to th graphs in Figurs 5 and 6. Fig. 7.: Succss rat of th automatic classifir Fig. 5.: Graph for dtrmining th dcision lvl of amplitud of a PSK signal carrir wav 4 Conclusion In this articl, th simpl automatic classifir of PSK and FSK digital modulation was dsignd. Basd on a sris of xprimnts its optimization was conductd. It is known that th mthod of classification basd on cyclic spctrum is vry rsistant against AWGN nois. h classifir dsignd has confirmd this hypothsis, as it has ISBN: 978-1-61804-046-6 438
Mathmatical Mthods and chniqus in Enginring and Environmntal Scinc provd corrct rcognition of th modulation up to lvl of nois SN = db. h xprimnts dmonstratd dpndnc of th spctrum shap on signal modulation paramtrs. Of grat importanc is th siz of th carrir frquncy and modulation rat of data. h rsults show, that during th classification of signals that ar clos to th paramtrs of th signals on which th classifir has bn dbuggd, th succss rat of th classification rachs 100 %. Modulation classification mthod basd on cyclostationarity of signals is vry promising. Upon furthr rsarch w will focus on tsting of th classifir on ral signal sampls and on xtnsion of th st of classifid signals. frncs: [1] Sbsta, V., Estimating a Spctral Corrlation Function undr th Conditions of Imprfct lation btwn Signal Frquncy and a Sampling Frquncy, adionginring, 010, vol. 19, no. 1. ISSN 110-51. [] Gardnr, W. A., Statistical Spctral Analysis. A nprobabilistic hory, Nw Jrsy: PENICE HALL, 1988. ISBN 0-13-84457-9. [3] Gardnr, W. A., Exploitation of Spctral dundancy in Cyclostationary Signals, IEEE Signal Procssing Magazin. April 1991, vol. 8, no., p. 14-36. ISSN 1053-5888. [4] obrts,., Brown, W. A., Loomish, H., Computationally Efficint Algorithms for Cyclic Spctral Analysis, IEEE Signal Procssing Magazin. April 1991, vol. 8, no., p. 38-49. ISSN 1053-5888. [5] Antoni, J., Cyclic Spctral Analysis in Practic, Mchanical Systms and Signal Procssing. Fbruary 007, vol. 1, no., p. 597-630. ISSN 0888-370. [6] Lik, E., Chakravarthy, V. D., atazzi, P., Wu, Z., Signal Classification in Fading Channls Using cyclic Spctral Analysis. EUASIP Journal on Wirlss Communications and Ntworking. Vol. 009, Articl ID 87981, 14 pags, 009. doi:10.1155/009/87981. ISBN: 978-1-61804-046-6 439