A Survey of Automatic Modulation Classification Techniques: Classical Approaches and New Trends


 Moris Wade
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1 Survey of utomatc Modulaton Clafcaton Technque: Clacal pproache and New Trend Octava. Dobre, l bd, Yeheel BarNe and We Su 3 Faculty of Engneerng and ppled Scence, Memoral Unverty of Newfoundland St. John, NL B 3X5, Canada Dept. of Electrcal and Computer Engneerng, New Jerey Inttute of Technology, Newar, NJ 7, US 3 US rmy RDECOM Fort Monmouth, NJ, 773, US btract The automatc recognton of the modulaton format of a detected gnal, the ntermedate tep between gnal detecton and demodulaton, a major ta of an ntellgent recever, wth varou cvlan and mltary applcaton. Obvouly, wth no nowledge of the tranmtted data and many unnown parameter at the recever, uch a the gnal power, carrer frequency and phae offet, tmng nformaton, etc., blnd dentfcaton of the modulaton a dffcult ta. Th become even more challengng n realworld cenaro wth multpath fadng, frequencyelectve and tmevaryng channel. In th paper we provde a comprehenve urvey of dfferent modulaton recognton technque, n a ytematc way. unfed notaton ued to brng n together, under the ame umbrella, the vat amount of reult and clafer, developed for dfferent modulaton. The two general clae of automatc modulaton dentfcaton algorthm are dcued n detal, whch rely on the lelhood functon and feature of the receved gnal, repectvely. The contrbuton of numerou artcle are ummarzed n compact form. Th help the reader to ee the man charactertc of each technque. However, n many cae, the reported reult n the lterature have been obtaned under dfferent condton. So, we have alo mulated ome major technque under the ame condton, whch allow a far comparon among dfferent methodologe. Furthermore, new problem that have appeared a a reult of emergng wrele technologe are outlned. t the end, open problem and poble drecton for future reearch are brefly dcued. Keyword: utomatc modulaton clafcaton, utomatc modulaton recognton, Clafer performance, Lelhood functon, Maxmum lelhood, Feature extracton, Preproceng ta, Model mmatche, Probablty of correct clafcaton. Part of th wor wa publhed n a prelmnary form and preented at the Sarnoff Sympoum, Prnceton, NJ, US, 89 prl 5, under the ttle Blnd modulaton clafcaton: a concept whoe tme ha come.
2 . INTRODUCTION utomatc modulaton clafcaton (MC) an ntermedate tep between gnal detecton and demodulaton, and play a ey role n varou cvlan and mltary applcaton. Implementaton of advanced nformaton ervce and ytem for mltary applcaton, n a crowded electromagnetc pectrum, a challengng ta for communcaton engneer. Frendly gnal hould be ecurely tranmtted and receved, wherea hotle gnal mut be located, dentfed and jammed. The pectrum of thee gnal may range from hgh frequency (HF) to mllmeter frequency band, and ther format can vary from mple narrowband modulaton to wdeband cheme. Under uch condton, advanced technque are requred for realtme gnal ntercepton and proceng, whch are vtal for decon nvolvng electronc warfare operaton and other tactcal acton. Furthermore, blnd recognton of the modulaton format of the receved gnal an mportant problem n commercal ytem, epecally n oftware defned rado (SDR), whch cope wth the varety of communcaton ytem. Uually, upplementary nformaton tranmtted to reconfgure the SDR ytem. Blnd technque can be ued wth an ntellgent recever, yeldng an ncreae n the tranmon effcency by reducng the overhead. Such applcaton have emerged the need for flexble ntellgent communcaton ytem, where the automatc recognton of the modulaton of a detected gnal a major ta. mplfed bloc dagram of the ytem model hown n Fg.. The degn of a modulaton clafer eentally nvolve two tep: gnal preproceng and proper electon of the clafcaton algorthm. Preproceng ta may nclude, but not lmted to perform ome or all of, noe reducton, etmaton of carrer frequency, ymbol perod, and gnal power, equalzaton, etc. Dependng on the clafcaton algorthm choen n the econd tep, preproceng ta wth dfferent level of accuracy are requred; ome clafcaton method requre prece etmate, wherea other are le entve to the unnown parameter. Regardng the econd tep, two general clae of MC algorthm can be crytallzed, lelhoodbaed (LB) [][6] and featurebaed (FB) [7][88] method, repectvely. The former baed on the lelhood functon of the receved gnal and the decon made comparng the lelhood rato agant a threhold. oluton offered by the LB algorthm optmal n the Bayean ene, vz., t mnmze the probablty of fale clafcaton. The optmal oluton uffer from computatonal complexty, whch n many cae of nteret naturally gve re to uboptmal clafer. In the FB approach, on the other hand, everal feature are uually employed and a decon made baed on ther oberved value. Thee feature are normally choen n an adhoc way. lthough a FBbaed method may not be optmal, t uually mple to mplement, wth nearoptmal performance, when degned properly. Once the modulaton format correctly dentfed, other operaton, uch a gnal demodulaton and nformaton extracton, can be ubequently performed. In general, MC a challengng ta, epecally n a noncooperatve envronment, where n addton to multpath propagaton, frequencyelectvty and tmevaryng nature of the channel, no pror nowledge of the ncomng gnal avalable. In recent year, new technologe for wrele communcaton have emerged. The wrele ndutry ha hown great nteret n orthogonal frequency dvon multplexng (OFDM) ytem, due to the effcency of OFDM cheme to tranmt nformaton n frequency electve fadng channel, wthout complex equalzer
3 [89][9]. Multplenput multpleoutput (MIMO) ytem have alo receved conderable attenton, due to the gnfcant capacty ncreae they offer. Such emergng technologe n wrele communcaton have raed new challenge for the degner of gnal ntellgence and SDR ytem, uch a, dcrmnatng between OFDM and ngle carrer modulaton [9], dentfcaton of gnal tranmtted from multple antenna ytem, and o on. Reearch on automatc clafcaton of both dgtal and analog modulaton ha been carred out for at leat two decade [][88]. Partal urvey of algorthm for dentfyng dgtally modulated gnal are gven n [9] and [93]. Of coure, many technque have been developed, whch are dfferent from each other when t come to detal. However, general tructure that connect a varety of apparently dfferent technque can be dentfed. In th paper, we provde a unfed comprehenve overvew of what ha been accomplhed o far n th area, hghlghtng the bottlenec and challengng ue whch need to be addreed by further reearch. comparon among the performance of dfferent LB and FB algorthm alo carred out, emphazng the advantage and dadvantage of dvere technque. The ret of the paper organzed a follow. In Secton II the gnal model and clafer performance meaure are dcued. Secton III and IV are devoted to LB and FB method, repectvely. Numercal performance aement and comparon are provded n Secton V, and ome concludng remar are gven n Secton VI.. SIGNL MODEL ND PERFORMNCE MESURES MC algorthm propoed n the lterature employ nformaton extracted from ether the receved baeband waveform [][9], [4][8], [4][54], [58][65] or ntermedate frequency [3][38], [55], [88]. general expreon for the baeband receved complex envelope gven by rt () = t (; u ) + nt (), () where ( ) (; ) j π ft j θ K jφ = ( ( ) ) = ε t u ae e e g t T T, t KT () the noefree baeband complex envelope of the receved gnal. In () we have a = E σ E, j =, ( ) p wth M ( ) E a the baeband gnal energy, σ ( ) = M m a the varance of the th zeromean gnal m= contellaton, M a the number of equprobable pont n the th gnal contellaton, Ep ptx( t) dt = a the pule energy, wth ptx () t a the tranmtter pule hape, f the carrer frequency offet, θ the tmenvarant carrer phae, { } K () φ = repreent the phae jtter, { } K = are K complex tranmtted data ymbol taen from the th fntealphabet modulaton format, T the ymbol perod, ε denote the tmng offet wth repect to (w.r.t.) the recever reference cloc, uch that ε<, and gt () = p () t ht (), wth ht ( ) a the channel mpule repone and a the convoluton. For example, for a lowlyvaryng fadng channel wth P TX ndependent path, ht P p () e jϕ = αp δ( t τp), wth δ (.) a the Drac delta functon, uch that { α } P p p=, { ϕ } P p p= p= 3
4 and { τ } P are the ampltude, phae and delay of the path, repectvely. We adopt the notaton tu (; ) to p p= tre the gnal dependence on the unnown quantte,.e., u = [ a f θ T ε g( t) { φ } { } ], wth a K () K = = the tranpoe. The term unnown quantte refer to unnown parameter, uch a the carrer frequency offet, a well a unnown data ymbol. Wthout lo of generalty, unt varance contellaton wll be condered n the equel, obtaned by normalzng the gnal contellaton. For example, for M ary ampltudehft eyng (SK) the ymbol are gven by, {(m M) / σ, m=,..., M}, =,..., K, ( MSK) ( MSK) ( M SK) =, I, I (MSK) for rectangular M ary quadrature ampltude modulaton (QM) = + j, ( MQM) ( MQM) ( MQM), I, Q, {(m M )/ σ, m=,..., M }, =,..., K, for M ary phaehfteyng (PSK) ( MQM) ( MQM) / / I, Q, ( M QM ) j m = e θ, θm { π mm, m=,..., M }, =,..., K and for M ary frequencyhfteyng (FSK) ( M PSK) =, f {(m M) f, m=,..., M}, where the ubcrpt I and Q repreent the nphae (real) ( M FSK) j fmt e π m d and quadrature (magnary) part, repectvely, σ ( ) the varance of the contellaton before normalzaton, M a power of, and f d the frequency devaton or pacng between any two adjacent FSK contellaton pont, whch for orthogonal gnalng a multple of /T (ee, for example, [94] Ch. 4). Note that the data ymbol depend on t for FSK. For other, uch a SK, PSK, QM, contant for each perod ( ) T to T. To mplfy the () () notaton, ubequently we ue wthout t alo for FSK gnal, unle otherwe mentoned. Note that nt ( ) n () the aggregate baeband complex noe,.e., recever noe, a well a cochannel nterference and jammer. clafer uppoed to correctly chooe the modulaton format of the ncomng gnal from a pool of canddate modulaton, denoted by the nteger mod N mod =,..., N, or to decde that the modulaton format cannot be recognzed. The latter cae not dcued here, a not addreed n the lterature. Due to the lac of pace, we focu on algorthm for SK, PSK, QM, and FSK clafcaton. a bac performance meaure, let P denote the (clafcaton) probablty to declare that the ' th gnal (' ) c format ha been ent, when the modulaton format of the ncomng gnal. For, ' =,..., N, thee mod probablte can be arranged a a N mod N mod confuon matrx, where the dagonal element P ( ) c the probablty of correct clafcaton for the th modulaton. In clafyng average probablty of correct clafcaton defned by N mod equprobable modulaton, the Nmod ( ) mod = c P = N P. (3) cc Obvouly, one can ue the complementary probablte a a performance meaure,.e., the probablty of error for the th modulaton, defned a P = P, and the average probablty of error, defned a P = P. Mot () ( ) e c e cc of the MC wor ued P cc, or equvalently, P e, a a performance meaure. However, by ung the confuon matrx, one gan more nght nto the clafer behavor. 4
5 Clearly, a derable clafer hould provde a hgh probablty of correct clafcaton n a hort obervaton nterval, partcularly for a large range of gnaltonoe rato (SNR). In addton, t hould atfy thee requrement: capablty to recognze many dfferent modulaton n envronment wth dvere propagaton charactertc, robutne to model mmatche, realtme functonalty, and low computatonal complexty. 3. LIKELIHOODBSED PPROCH TO MC Wthn the LB framewor, MC a multple compote hypothetetng problem. The dea behnd the LBMC that the probablty denty functon (PDF) of the oberved waveform, condtoned on the embedded modulated gnal, contan all nformaton for clafcaton. Dependng on the model choen for the unnown quantte, three LBMC technque are propoed n the lterature: average lelhood rato tet (LRT) [][3], [], [3], generalzed lelhood rato tet (GLRT) [4], [7], [8] and hybrd lelhood rato tet (HLRT) [4] [6], [9][]. Qua LRT [3][5], [7], [9][3] and qua HLRT [][] are alo propoed n the lterature. LRT Th approach treat the unnown quantte a random varable (r.v. ) wth certan PDF. So, the lelhood functon (LF) under the hypothe H, repreentatve of the th modulaton, =,..., N, gven by () Λ [()] rt = Λ[() rt v, H] p( v H) dv, (4) where Λ[ rt ( ) v, H] the condtonal LF of the noy receved gnal rt ( ) under unnown vector v, and p( v H) the a pror PDF of v under mod H, condtoned on the H. The nown PDF of v enabled u to reduce the problem to a mple hypothetetng problem by ntegratng over v. For a baeband complex addtve whte Gauan noe (WGN) n (), the condtonal LF gven by (ee, for example, [94] Ch. 6) KT KT * Λ [() rt v, H] = exp N Re rt () (; t ) dt N t (; ) dt, u u (5) where N the twoded power pectral denty (PSD) of WGN n W/Hz, wth the autocorrelaton * E{ ntn ( ) ( t )} N ( ) = [ N ] +τ = δ τ uch that E{.} expectaton and * denote the complex conjugate. Furthermore, here v u, and Re{.} tand for the real part. If the choen p( v H ) the ame a the true PDF, LRT reult n an optmal clafer n the Bayean ene. GLRT In th approach the unnown parameter are treated a unnown determntc. The bet performance acheved by the ocalled unformly mot powerful (UMP) tet [95]. For a neceary and uffcent condton for the extence of an UMP tet ee, for example, [95] Ch.. When an UMP tet doe not ext or hard to derve, a logcal procedure to etmate the unnown quantte, aumng H true, and then ue thee etmate n a lelhood rato tet, a f they were correct. If maxmum lelhood (ML) ued for etmate, the tet called 5
6 GLRT. Obvouly, GLRT treat the unnown quantte (ncludng both the parameter and data ymbol) a determntc unnown, and the LF under H gven by () Λ [ rt ( )] = max Λ[ rt ( ) v, H]. G v (6) HLRT Th a combnaton of the aforementoned approache, for whch the LF under H v H gven by Λ () [()] rt = max Λ[() rt v, v, H] p( v H) dv, (7) where v = [ v v ] and, v and v are vector of unnown quantte modeled a unnown determntc and r.v., repectvely. Uually, v and v cont of parameter and data ymbol, repectvely. Note that LRT requre a multdmenonal ntegraton, wherea GLRT requre a multdmenonal maxmzaton. The dffculty of performng a multdmenonal ntegraton for a large number of unnown quantte and the need for nowng the pror PDF may render the LRT mpractcal. On the other hand, maxmzaton over the unnown data ymbol n GLRT can lead to the ame value of the LF for neted gnal contellaton, e.g., BPSK and QPSK, 6QM and 64QM, [4], [96] Ch. 6, whch n turn yeld ncorrect clafcaton. veragng over the unnown data ymbol n HLRT, however, remove the neted contellaton problem of GLRT. Fnally we emphaze that the etmate of the unnown quantte, a byproduct of GLRT and HLRT, are of nteret for data demodulaton. In a twohypothe clafcaton problem, the decon made accordng to H () () > l [()] rt l [()] rt < l H Λ Λ η, l = (LRT), G (GLRT), H (HLRT), (8) where η l a threhold. The lefthand de referred to a the lelhood rato and the tet called average lelhood rato tet (LRT), generalzed lelhood rato tet (GLRT) and hybrd lelhood rato tet (HLRT), repectvely, dependng on the method employed to compute the LF. Extenon of (8) to multple clae traghtforward (ee, for example, [95] Ch. and [96] Ch. 3 and 6). Equvalently, the log functon can be appled to both member of the nequalty (8). ccordngly, the term loglelhood rato and loglelhood rato tet are ued. Table I lt everal LBMC algorthm propoed n the lterature, emphazng the type of modulaton, unnown parameter, and channel ued. mplfed gnal model of that gven n () wa condered n the lterature, a follow. The tranmt pule hape wa aumed rectangular,.e., p () t = u () t, where u () t = for t < T and zero otherwe. So, Ep = T and wth σ ( ) =, one obtan a = S, n whch S = E / T the gnal jϕ power. Wth all th and ht () =αe δ () t, wth α and ϕ contant over the K ymbol nterval, () can be wrtten a j ( θ+ϕ) j π ft K jφ ( ) = T (; t u ) =α Se e e u ( t ( ) T εt), (9) TX T T 6
7 where α= and ϕ= when there no fadng. We defne the oberved perymbol SNR γ =Ω ST N, n whch Ω= when there no fadng. Otherwe, ampltude. Under fadng, S et equal to one. 3.. LRTbaed lgorthm Ω = E{ α } the average fadng power, wth α a the channel In th ecton, optmal and uboptmal LRTbaed algorthm applcable to dentfy both lnearly modulated and FSK gnal under varou condton wll be preented, a well a LRTbaed clafer pecfc to lnear modulaton and FSK, repectvely. Suboptmal clafer are obtaned baed on the approxmaton of the LF at low SNR. Interetngly, everal FB clafer are hown to be mplfed veron of uch uboptmal tructure [5]. Hence, decon theory can be perceved a a rgorou framewor that jutfe the electon of feature n ome FB method LRTMC for lnearly modulated and FSK gnal LRTbaed clafer Wth all parameter perfectly nown,.e., v () K = [{ } = ], LRT lead to a tructure whoe performance better than all the other, whch have to deal wth ome unnown parameter. Therefore, the performance of th clafer can be condered a a benchmar. The data ymbol () { } K = are treated a ndependent and dentcally dtrbuted (..d.) r.v.. The LF under hypothe H computed by averagng over the contellaton pont correpondng to the th modulaton format. Th done by ubttutng (5) nto (4) (ee (37), ppendx ) { { } } () K () () Λ [()] rt = E ( ) expα SN Re R STN, = α () where E ( ){.} nothng but a fnte ummaton over all the M poble contellaton pont of the th modulaton, dvded by M, for the th nterval. Furthermore, T T () ()* ()* = ( ) T = T ( ) T,,..., R rt () () tu( t ( ) Tdt ) rt () () tdt = K. () Note that for lnear modulaton () () t contant over the perod ( ) T to T and thu, R = r, wth () ()* r T = r() t dt the output of the receve matched flter at t = T. ( ) T Wth multple antenna at the recever, WGN and bloc fadng, and all parameter perfectly nown,.e., () K [{ } = ] v =, the LF gven by (ee, e.g., [95] Ch.3) Due to the lac of pace, n the equel we gve detal of ome of the algorthm, epecally thoe ued n the comparatve tudy of the MC algorthm n Secton V, wherea we only menton other. From now on, we et the nown parameter to ome fxed numercal value. In WGN channel, wth all parameter K K perfectly nown, θ= f =ε=ϕ= { φ } = = and α =, wherea n a bloc fadng channel f =ε= { φ } = =, S =, and θ ncluded nto ϕ. Of coure, the unnown wll be put nto the vector v. 7
8 jϕ {, } L K () () () r t E ( ) SN e R STN = = Λ [()] = expα Re{ } α, () where L the number of antenna at the recever de, j e ϕ α the fadng proce on each branch, =,..., L, and () T ()* = T R r, () t () t dt ( ), =,..., K, =,..., L, wth r () t = (; t u ) + n () t a the receved gnal on the th branch, (; t u ) a the noe free envelope, and n () t a the zeromean WGN, wth the PSD N. The expreon for the noe free envelope on the th branch, (; t u ), can be ealy wrtten mlar to (9) 3. Note that both the noe { ( )} L j L n t = and fadng procee { e ϕ α } = among the L dverty branche are aumed to be ndependent. ctually, a maxmal rato combnng (MRC) wa ued here to combne the receved gnal. In fadng channel uch a tructure tae advantage of the array gan, a well a dverty gan (ee, for example, [97] Ch. 5), and thu, performance mprovement expected when compared wth a ngle antenna clafer. However, a one can ealy notce, when n addton to the unnown data ymbol, there are other unnown parameter, e.g., L L { α } = and { ϕ } =, ntegraton over thee parameter become more dffcult and, the mplementaton of a multantenna LRTbaed clafer turn out to be even more complex. In WGN, wth v and unform dtrbuton for θ over [ π, π ), repreentng no pror () K =θ [ { } = ] nowledge of the tmenvarant phae, the LF can be hown to be [] (ee (38), ppendx ) ( ) STN ηk { K } Λ [()] rt = E e I( SN ξ ), (3) () () ( ) K { } = where the notaton E () K {.} { } = emphaze that the averagng performed over K data ymbol, η =, () K () K = I (.) the zeroorder modfed Beel functon of the frt nd, and () K () ξ K = R. Obvouly, uch a clafer K dffcult to mplement, a requre M data equence to compute the LF under the hypothe H. In WGN, under the aumpton of perymbol phaencoherence due to phae jtter,.e., θ = and v = [{ φ } { } ], wth { φ } K a..d. unform r.v., t can be ealy hown that the LF gven by K () K = = (ee (39), ppendx ) = ( ) { } () rt K e STN I SN R () ( ) = Λ [()] = E ( ). (4) In a lowlyvaryng flat Raylegh fadng channel, characterzed va a Rayleghdtrbuted α and unform = ϕ, uch that v, the LF gven by [] (ee (4), ppendx ) () K = [ α ϕ { } = ] 3 Wth a multantenna clafer, n WGN channel and all parameter perfectly nown, we et L L K θ= f = { ε } = = { ϕ } = = { φ } = = and { } L L K α = =, wherea wth a flat blocfadng channel f = { ε } = = { φ } = =, S = and θ ncluded nto ϕ, =,...,L. 8
9 ( ) () ΩN ξ K Λ [()] rt = E ( ) K exp. { } ( ) ( ) = +ΩTN η K +ΩTN ηk (5) One can notce that the LF depend on the average fadng power Ω, aumed perfectly nown. The LRTbaed clafer are mplemented by replacng the expreon of the LF gven n (), (), (3), (4) and (5), repectvely, n (8), wth η = ; uch a clafer called the ML clafer. Performance analy of the ML clafer Theoretcal performance analy of the ML clafer wa performed n [] for no unnown parameter, when dentfyng lnearly modulated gnal n WGN, wth the LF gven n (). The probablty of error under hown to be [] H () e = ( ) Km, Km, Km b b,,..., N, + Km mod, Nmod P p H d Nmod =, (6) where b = ( a Km )/ K, a = [ a,... a, a, +... a, N ], mod a r r, j, K () ( ), [ln( ( )) ln( j j= Λ ( ))] = Λ Λ ( r ) = E {exp[ SN Re{ r } STN ]}, m = [ m,... m, m, +... m, N ], and mod () ()* () ( ) m = Λ r Λ r H, j. The () ( j), j E{ln( ( )) ln( ( )) j} ( Nmod ) vector b hown to be the um of K..d. random vector whch atfe the multvarate central lmt theorem f K large. Therefore, p( b H ) wa condered a a multvarate Gauan denty, wth zero mean and covarance matrx, bb E{ H}, dependng on () ( j) the frt and econd order tattc of ln( Λ ( r )) ln( Λ ( r )) []. The ntegral n (6) wa numercally calculated for V. 9, 6QM, 3QM and 64QM, for K =, and []. Note that V.9 a pecal QM modulaton, wth 6 pont n the gnal contellaton []. Qua LRTbaed clafer ynchronou clafer ( ε= ) can be mply tranformed nto an aynchronou one, wth the tmng offet ε a a unformly dtrbuted r.v. over [,), ung the followng approxmaton of the LF [9], [], [3] () [()] D rt D [() rt, ], d d H = (7) Λ Λ ε where D the number of level to whch the tmng offet quantzed and ε = d/ D, d =,..., D. For uch a d cenaro, () R, a defned n (), need to be replaced by T dt ()* ε d = +ε ( ) T+ε T T εd R ( ) rt () () tu ( t ( ) T Tdt ). Th d approxmaton mprove a D, nce the ummaton converge to an ntegral. The value of D drectly determne the clafer complexty, a ntroduce more term n (7). Note that a mlar approxmaton can be alo ued when the carrer phae θ unnown, by dcretzng the range of t value. 9
10 3... LRTMC for lnearly modulated gnal In the equel we preent varou clafer for lnear modulaton clafcaton under dfferent condton, uch a a dfferental LRT algorthm degned for unnown carrer phae, qua LRT clafer degned alo for unnown carrer phae, a well a unnown carrer phae/ tmng offet, etc. Dfferental LRT wth unnown carrer phae dfferental data oluton wa propoed n [] to clafy lnearly modulated gnal n WGN channel, wth the unnown carrer phae unformly dtrbuted over [ π, π ),.e., v () K = [ θ { } = ]. The jont PDF of the magntude r and phae dfference ψ = ψ ψ, uch that tan ψ mod = r, Q / r, I and r = r, I + jr, Q, wa + π ued to derve the LF. Under the aumpton that the gnal ampltude and phae dfference are approxmately ndependent r.v. at large SNR, the LF gven by [] { + } Λ [ ( )] = E (, ) ( ψ,, ), (8) () rt K p r () H p () () H ( ) ( ) = {, } + n whch r wa taen a Rceandtrbuted and, for large SNR, ψ wa approxmated by a Gauan PDF. Due to the lac of pace we omt here the expreon for the PDF of r and ψ. For detal ee [], p. 9. The advantage of ung the phae dfference ntead of phae telf that the effect of a tmenvarant phae offet wll be mtgated. However, the clafer performance can tll be degraded due to the phae jtter. The dfferental LRTbaed clafer wa mplemented wth the decon rule gven n (8) ( η = ) and the expreon of the LF gven n (8). Qua LRT wth unnown carrer phae 4 Wth v, where θ unformly dtrbuted over [ π, π ), approxmaton of the LF were () K =θ [ { } = ] developed n [3][5], [7] for WGN channel, leadng to uboptmal clafcaton tructure. low SNR approxmaton of the LF for PSK and QM gnal gven by 5 n n / { ( ) ( ) ˆ n= q=, n, q,, } (9) () Λ [()] rt exp ( SN ) K υ q n q!( n q )! m m r n q( n ), () n q ()* q where m ( ) = E ( ) {( ) ( ) } the n thorder/ q conjugate moment of the th contellaton,, n, q mˆ ( ) = K r ( r ) the ample etmate of the n thorder/ q conjugate moment at zerodelay vector K n q * q rnq,, n = n 6,. denote roundng up to the nearet nteger, and fnally υ n q f q= n/ and f q< n/. Eq. (9) 4 The qua LRT clafer wa orgnally derved a an LB method. However, t can be condered a an FB technque a well. 5 Eq. (9) can be ealy obtaned from eq. (5) gven n [7], by ung the gnal moment. 6 For the defnton of the n thorder moment/ q conjugate, mrnq,,( τ n ) and the cumulant crnq,,( τ n ) of a tatonary random proce, a well the relaton between the moment and cumulant, ee, e.g., [98] and [99] Ch.. n delay vector an ( n ) vector, wth all the element equal to zero.
11 can be further mplfed, a for ymmetrc contellaton the odd order moment are equal to zero [63]. The rghthand de of (9) actually a meaure of the correlaton between the theoretcal moment of the th contellaton and ample etmate. Theoretcal value of the n th order/ q conjugate moment m (, wth n =,4,6,8 and ), nq, q =,..., n/, are gven n Table II for dfferent gnal contellaton. Thee value were computed a enemble average over the dealnoe free contellaton under the contrant of unty varance and the aumpton of equprobable ymbol. one can notce from Table II, the lowet order tattc to dtnguh between M PSK and M ' PSK ( M ' > M ) the M thorder / zeroconjugate moment, m (. On the other hand, alo from ),, Table II, th property doe not hold for QM gnal. general rule for QM gnal that m = ( when n ),, a multple of four ( n = 4 and 8 n Table II) and q odd or n not a multple of four ( n = 6 n Table II) and q even. By reortng to only the lowet order tattc, mall n, uboptmal but mplementatonally manageable clafer were propoed to dcrmnate PSK and QM n [3][5] and [7], repectvely. Baed on the aforementoned property of moment for PSK gnal, Polydoro et al. propoed a bnary decon tree clafer for PSK gnal [3][5], where the decon at each node wa made by comparng the followng metrc v K M M, PSK r = M n q = () agant a threhold, denoted here by η ( M PSK, M ' PSK) vm. an example, ee ppendx B for the dervaton of () for BPSK/QPSK clafcaton. The decon rule nvolvng the approxmaton of the LF requre approprate threhold. In order to maxmze the probablty of correct clafcaton when dcrmnatng between the M PSK and M ' PSK modulaton, the threhold η wa choen to atfy ( M PSK, M ' PSK) vm p ( η ) = p ( η ), () ( M PSK) ( M PSK, M ' PSK) ( M ' PSK) ( M PSK, M ' PSK) vm vm where ( M p PSK) ( v M ) and ( M ' p PSK) ( v M ) are the PDF of the metrc v M under the hypothe that M PSK and M ' PSK are the modulaton format of the ncomng gnal, repectvely. cloedform oluton wa derved n [5] for clafyng PSK gnal wth unnown carrer phae. Under the aumpton of a large number of avalable ymbol K, th threhold wa approxmated by 7 η v = KS T /. () ( M PSK, M ' PSK) M / M M one can notce, the threhold depend on the gnal power, S. In the equel we denote t by η, Q TE, where Q and TE tand for qualrt and theoretcal, repectvely. n example of a bnary decon tree clafer ued for PSK gnal hown n Fg.. 7 The orgnal preentaton n [4] ued r / NT ntead of r n the metrc n (). To account for th dfference, the threhold we gve n () ndeed ( NT /) M tme the threhold of [4].
12 prevouly explaned, the n th order/ q conjugate moment for QM gnal do not have the attractve property ued to deve a bnary decon tree for PSK gnal dentfcaton. Ung (9) and the reult gven n Table II, t can be ealy notced that the lowet order tattc whch can be ued for QM gnal clafcaton of order n = 4 ( q =,). Followng the ame procedure a n the example gven n ppendx B for BPSK and QPSK gnal, t can be ealy hown that the lowet order metrc whch can be ued to dtnguh between any two QM gnal gven by, (3) K r 4 B K r 4 QM = = ν 4, = + where the coeffcent and B depend on the theoretcal value of the fourthorder/ zero and twoconjugate moment of the QM gnal, repectvely. Such a metrc wa ued n [7] to dcrmnate between 6QM and V.9, wth =.35 and B =.46. The decon wa made by comparng the metrc agant a threhold, whch wa emprcally et 8. We denote th threhold by η, Q E, where Q and E tand for qualrt and emprcal, repectvely. By comparng () and (3) wth (3), one can ay that the complexty of a qualrt clafer much le than that of the LRT clafer, a t need nether an averagng operaton nor the computaton of the Beel functon. Qua LRT wth unnown carrer phae and tmng offet In WGN, wth v () K = [ θ ε { } = ], where the carrer phae θ and tmng offet ε are unformly dtrbuted over [ π, π ) and [,), repectvely, the followng tattc wa ued to dtnguh between M PSK and M ' PSK [5] D K M M, PSK r d= = ( d/ D) ν =, (4) wth T dt ε d = +ε ( ) T+ε T T εd r ( ) r ( t ) u ( t ( ) T T ) dt. Smlar to Fg., a bnary decon tree clafer wa employed d for PSK gnal clafcaton, wth (4) compared agant a threhold. The threhold wa emprcally choen, followng the htogram method 8. LRT wth unnown gnal level n LRT algorthm wa developed n [8] to dentfy PSK gnal n WGN, wth v () K = [ α { } = ], where the gnal level α a Rayleghdtrbuted r.v. The decon rule gven n (8) wa employed, wth the threhold η et to one. Mcellaneou clafer Numercal calculaton of the ntegral n LRT ung a Marov chan Monte Carlo method wa performed n [4]. In the algorthm prevouly decrbed, MC wa treated a a hypothe tetng problem wth a fxed 8 Th threhold et to maxmze the average probablty of correct clafcaton over a large number of data and noe realzaton. It aumed that uch mulaton can be run offlne and the threhold can be tored a a functon of the noe and gnal parameter. In a practcal mplementaton, the threhold therefore obtaned from a looup table.
13 number of receved ymbol. By formulatng the MC problem a a varable ample ze hypothe tetng problem, an algorthm baed on the equental probablty rato tet wa propoed n [5], [6] LRTMC for FSK gnal LRTbaed clafer under the aumpton of perymbol phaencoherence In WGN, under the aumpton of perymbol phaencoherence,.e., a..d. unform r.v., the LF of (4), wth () () v [{ } K { } K = φ = = ], wth { φ } K = =, wa derved n [9]. one can ealy notce from () and (4), the mplementaton of uch clafer requre the explct calculaton of the Fourer pectrum of the receved waveform at a et of BW M canddate frequence [9]. The gnal bandwdth of an = M T, wth T a the ymbol perod under the hypothe M FSK gnal defned a H. Optmal tructure were developed for the three poble cae: the ame gnal bandwdth and dtnct ymbol perod (equal M T, wth dfferent T ), the ame ymbol perod and dtnct bandwdth (dfferent M T, wth equal T ), and both the ymbol perod and gnal bandwdth dfferent (dfferent M T, wth dfferent T ) [9]. The decon wa made baed on the rule gven n (8), wth η =. Qua LRTbaed clafer under the aumpton of perymbol phaencoherence 4 On the other hand, wth v = [{ φ } { } ], by ung a power ere expanon of the modfed Beel K () K = = functon n (4), an approxmaton of the LF baed on hgherorder correlaton (HOC) wa gven n [9][]. For a ngle ymbol nterval th gven by T T () *, r T, r 3 T, r, r Λ [()] rt + Cc () + C c () τ dτ+ C c () τ c () τ dτ, (5) + C c ( τ) dτ+ C c ( τ) c ( τ) dτ+... T T * 4 T, r 5 T, r 3, r where c (), τ the gnal autocorrelaton, defned a r c ( ) ( +τ), τ<,, ( ) ( ), . T τ * r t r t dt T r, () τ = T τ * r t + τ r t dt T τ< and crn, () τ the n thorder ( n ) correlaton, defned a the autocorrelaton of c () rn, τ, c rn, n T τ * n c n rn, t crn, t dt T T +τ τ< τ = n T τ * n c n T rn, t crn, t dt T + τ τ< () ( ) ( ),,. ( ) ( ),. The coeffcent C, C, C3, C 4, and C 5 are gven by γ ( NT BW), γ (6 NT BW), γ (88 NT BW), γ (96 NT BW) and γ (468 NT BW), repectvely. When the receved gnal nclude K..d. ymbol, the LF mply the product of K LF, each one gven by (5). To ncreae the accuracy of the LF approxmaton n (5), one need to ue hgherorder correlaton. Th reult n a better but 3
14 more complex clafer. The decon wa made by comparng the metrc agant an emprcal threhold, η HOC, E, where HOC and E denote hgherorder correlaton and emprcal, repectvely. n FSK gnal clafer baed alo on an approxmaton of the LF wa propoed n [3] for flat Raylegh fadng channel. Uually, n the lterature only one ncomng gnal aumed at the recever. The cae of multple FSK gnal at the recever wa dcued n [3]. Ung (7), the prevouly dcued LRT baed and qualrt ynchronou algorthm were tranformed nto aynchronou clafer, where the tmng offet ε wa aumed unformly dtrbuted over [,) [], [], [3]. 3.. GLRT and HLRTbaed lgorthm the LRT algorthm uffer from hgh computatonal complexty n mot practcal cae, GLRT and HLRT algorthm have been nvetgated a poble oluton to dentfy lnear modulaton [4][]. In WGN and wth Λ v, the LF for GLRT and HLRT are repectvely gven by [4] () K = [ θ { } = ] K θ { ( = ) ( ) } [ rt ( )] = max max Re[ re ] ST, (6) () ()* j () G θ K θ { ( ) { = }} Λ [ rt ( )] = max E exp SN Re[ re ] STN. (7) () ()* j () H θ Thee relaton can be ealy derved accordng to (6) and (7), repectvely, after ubttutng (9) nto (5). The emprcal htogram method 8 wa ued to et the threhold wth the GLRT and HLRT tet [4]. Here we denote thee threhold a η GE, and η H, E, repectvely. GLRT dplay ome mplementaton advantage over LRT and HLRT, a t avod the calculaton of exponental functon and doe not requre the nowledge of noe power to compute the LF. However, t uffer from the neted contellaton problem dcued earler. Note that HLRT doe not have th problem. Other GLRT and HLRTbaed clafer nvetgated n the lterature are a follow. The MC problem wa examned n an nterymbol nterference (ISI) envronment, where the gnal wa condered to be degraded by WGN and ISI [7], [8]. The LF wa computed ung the ML etmate of the data equence and channel coeffcent { g } P =, wth the perurvvor proceng technque employed for etmaton [8]. Obvouly, th p p GLRT wth v = [ { g } { } ]. The threhold ued for decon wa emprcally et 8. Wth P () K p p= = u = [ θ S { } = ] () K and unnown PSD N,.e., v = [ θ S { } N ], an HLRT clafer wa explored n [5], where ML etmate () K = of S and N were ued, together wth the approxmate LF n (9), obtaned by averagng over θ and () { } K =. In () K other word, the LF computed baed on (7), wth v =θ [ { } ] and v = [ SN]. Both emprcal and = theoretcal threhold were ued. n HLRTbaed multantenna clafer wa developed for BPSK/QPSK n WGN [6], wth v () K = [ ϑ { } = ], where ϑ an unnown phae hft between two adjacent antenna element, whch appear due to ther patal eparaton. The decon rule gven n (8) wa employed, wth the threhold et to one. HLRTbaed clafer were developed for lnear modulaton dentfcaton n flat bloc fadng channel n 4
15 [9][], wth v and () K = [ α ϕ { } = ] v = [ α ϕ { } N ], repectvely. The LF wa computed by averagng () K = over the data ymbol and ung the ML etmate of the unnown parameter. The threhold η H ued for decon wa et to one. HLRT ha the advantage over LRT that no pror PDF of the channel parameter are needed, and therefore, t applcable to dfferent envronment, e.g., Rcan and Raylegh fadng [9][]. Qua HLRT clafer HLRT doe not eem to be a good oluton wth an ncreaed number of unnown parameter, a fndng ther ML etmate can be very tme conumng. QuaHLRT clafer, whch ue lowcomplexty yet accurate etmate, can be ued ntead. Such clafer were propoed n [][] to dentfy lnear dgtal modulaton n bloc fadng, wth v and () K = [ α ϕ { } = ] v = [ α ϕ { } N ], repectvely. quahlrt clafer wa alo () K = propoed n [] to dcrmnate QM gnal n WGN channel, wth v = [ f { } = ]. () K an example, to dcrmnate QM gnal n bloc fadng channel, the etmator ued n [] for the channel ampltude and phae are gven, repectvely, by ˆ α T = [ mˆ ()  mˆ ( )]( m m ), (8) 4 4 r,, r,4, 3 ( ),, ( ),4, and ˆ 4 angle( K 4 ϕ = ) = r. (9) Then, ung thee etmator, the LF wa calculated accordng to ϕˆ { ˆ ˆ } Λ = α α. (3) () ()* () [ rt K ( )] E ( ) exp N Re[ re j qua H ] TN = The decon rule gven n (8), wth the threhold et to one. multantenna quahlrt clafer wa alo propoed n [], wth (8) and (9) ued to etmate the channel phae and ampltude on each branch, repectvely. The threhold ued for decon wa et to one. 4. FETUREBSED PPROCH TO MC The degn of a FB algorthm frt need ome feature for data repreentaton and then decon mang []. Example of feature are the correlaton between the nphae and quadrature gnal component [7], the varance of the centered normalzed gnal ampltude, phae and frequency [8], the varance of the zerocrong nterval [3], [33], the varance of the magntude of the gnal wavelet tranform (WT) after pea removal [36][38], the phae PDF [44][46] and t tattcal moment [47][49], moment, cumulant, and cyclc cumulant of the gnal telf [4][43], [53], [54], [58][66], etc. The entropy [67], [68], fuzzy logc [69], [7], a moment matrx technque [7], [7] and a contellaton hape recovery method [73] were alo ued for MC. Dfferent method were employed for decon mang, uch a PDFbaed [4][53], the Hellnger dtance [74], [75], the Eucldan dtance [6][65] and unuperved cluterng technque [76], [77]. Table III ummarze mot of the FBMC wor, emphazng the elected feature, type of modulaton, channel and the unnown parameter. In t frt part, algorthm whch employ nformaton extracted from the ntantaneou ampltude, 5
16 phae and frequency of the receved gnal are preented. The econd and thrd part nclude clafer baed on the wavelet tranform and gnal tattc, repectvely. Fnally, a clafer baed on pectral properte of FSK gnal mentoned. In what follow, thee FBMC algorthm are preented wth a herarchcal approach n mnd,.e., the modulaton cla of the ncomng gnal frt dentfed (e.g., SK, PSK, QM, FSK), and then the modulaton order M wthn the recognzed cla. ngle gnal n WGN, wth the parameter perfectly nown, and a rectangular pule hape u () t were aumed, except otherwe mentoned. In addton to SK, T PSK, QM and FSK, the dentfcaton of other modulaton wa examned n the lterature, e.g., MSK [66], [78][8], OQPSK [79], contnuouphae FSK (CPFSK) [8]. 4.. FB lgorthm to Dtnguh between Dfferent Clae Intantaneou ampltude, phae and frequencybaed algorthm The mot ntutve way to dentfy the modulaton cla of the ncomng gnal to ue the nformaton contaned n t ntantaneou ampltude, phae and frequency. To extract uch nformaton, dfferent method were appled n the lterature [8][4]. The followng dfference between gnal clae were employed for clafcaton n [8][3]: FSK gnal are characterzed by contant ntantaneou ampltude, wherea SK gnal have ampltude fluctuaton, and PSK gnal have nformaton n the phae. The maxmum of the dcrete Fourer tranform (DFT) of centered 9 normalzed ntantaneou ampltude wa ued a a feature to dtnguh between FSK and SK/ PSK clae, SK and BPSK gnal have no nformaton n the abolute phae, wherea M PSK ( M > ) ha. The varance of abolute centered 9 normalzed phae wa ued to dtnguh between M PSK ( M > ) and realvalued contellaton, BPSK and SK, SK gnal have no phae nformaton by ther nature, wherea BPSK ha. Varance of drect (not abolute) centered 9 normalzed phae wa ued to dtnguh between BPSK and SK clae. bnary decon tree tructure wa employed to dcrmnate between clae, and furthermore, wthn each cla, a we wll brefly menton n Secton 4. and 4.3. t each node of the tree, the decon wa made by comparng a tattc agant a threhold 8. In [3] and [33], the varance of the zerocrong nterval wa ued a a feature to dtnguh FSK from PSK and the unmodulated waveform (UW). The zerocrong nterval a meaure of the ntantaneou frequency, and t a tarcae functon for FSK gnal, wherea a contant for UW and PSK gnal. The MC treated a a two hypothe tetng problem: H for FSK and H for UW and PSK. The hypothee are formulated baed on the Gauan aumpton for the etmated feature,.e, N, =,, wth the hypothedependent mean ( µ H, σ ) H µ H and varance σ H. n LRT ued for decon, whch due to the Gauan aumpton mplfed to the comparon of the feature agant a threhold η, derved from the LRT. For any two cla problem, aumng equal pror, the average probablty of error then gven by 9 The term centered pecfe that the average removed from the data et. The mean actually the theoretcal value of the feature under H, wherea the varance etmated under each hypothe. 6
17 P = [erfc(( η µ ) / σ ) + erfc(( µ η) / σ )]/, (3) e H H H H / where erfc(.) the complementary error functon, defned a erfc( x) = ( π) exp( u / ) du. x The varance of the ntantaneou frequency wa alo employed n [34], [35] to dcrmnate FSK from UW and PSK. In fact, the autoregreve pectrum modelng wa ued to extract the ntantaneou frequency. The decon wa made by comparng the feature agant a threhold 8. Wavelet tranformbaed algorthm The utlty of the wavelet tranform to localze the change n the ntantaneou frequency, ampltude and phae of the receved gnal wa alo tuded for MC. The dtnct behavor of the Haar WT (HWT) magntude for PSK, QM and FSK gnal wa employed for cla dentfcaton n [36][38]. For a PSK gnal th a contant, wth pea occurrng at phae change. On the other hand, becaue of the frequency and ampltude varaton n FSK and QM, repectvely, the HWT magntude a tarcae functon wth pea at phae change. Thee pea do not provde ueful nformaton for noncontnuou phae FSK gnal. If only the phae retaned for a QM gnal, t behave le a PSK gnal and thu, the HWT magntude contant. On the other hand, a PSK and FSK gnal are of contant ampltude, ampltude normalzaton ha no effect on ther HWT magntude. fter pea removal, the varance of the HWT magntude wth ampltude normalzaton wa ued to dcrmnate FSK from PSK and QM. Furthermore, the varance of the HWT magntude wthout ampltude normalzaton wa employed to dtnguh between QM and PSK. The decon were made by comparng the feature agant ome threhold, choen baed on the tattcal analy of the feature, to mnmze the probablty of error for PSK gnal [36][38]. Neural networ (NN) were alo ued for clafcaton n [8][3], [35]. The WgnerVlle dtrbuton wa ued n [8] to dtnguh between PSK and FSK gnal. Sgnal tattcbaed algorthm To dcrmnate among BPSK, SK, M PSK ( M > ) and QM, the cumulantbaed feature c ( )/ c () wa propoed n [4], where crnq,,( n ) 6 the n thorder/ q conjugate cumulant of the output r,4, 3 r,, of the matched flter { r } K =, at the zero delay vector. For decon, an LRT baed on the PDF of the ample etmate of the feature wa formulated to acheve mnmum probablty of error. The momentbaed feature m ( )/ m () wa ued n [4], where mrnq,,( n ) the n thorder/ q conjugate moment of the output of the 3 r,6,3 5 r,, matched flter { r } K =, at the zero delay vector. The goal wa to dtnguh between PSK and QM. jont power etmaton and clafcaton wa performed n [4]. The decon wa made baed on the mnmum abolute value of the dfference between the ample etmate and precrbed value of the feature. Reference [43] combned everal normalzed moment and cumulant for tranng a NN, to dentfy FSK, PSK and QM n multpath envronment. 4.. FB lgorthm for Lnearly Modulated Sgnal Clafer ummarzed n Table III, whch can be appled to dentfy the modulaton order M of lnear modulaton, are dcued n the equel. 7
18 Intantaneou ampltude and phaebaed algorthm Informaton extracted from the ntantaneou ampltude and phae of the receved gnal wa exploted for lnear modulaton recognton, a follow. The varance of the abolute value of the normalzed centered 9 ntantaneou ampltude wa ued to dtnguh between SK and 4SK, a for the former the ampltude change between two level, equal n magntude and oppote n gn, o, t ha no nformaton n the abolute ampltude, wherea t ha for the latter [8][3]. The tattc wa compared agant a threhold for decon mang 8 at a tree node, a part of the bnary decon tree clafer mentoned n Secton 4.. The phae PDF and t tattcal moment were nvetgated for PSK gnal recognton n [44][5]. The phae PDF multmodal, and the number of mode provde nformaton for the PSK order dentfcaton. In the hghsnr regon, M PSK exbt M dtnct mode, whle when the SNR decreae or M ncreae, the pea mear off and fnally the PDF converge to a unform PDF [48]. Specfcally for PSK gnal clafcaton, an approxmaton ung the Tchonov PDF and a Fourer ere expanon of the phae PDF were employed n [44][46], wth a loglelhood rato tet for decon. By ung thee method to compute the phae PDF, cloedform expreon for the phae tattcal moment were derved, and the PDF of the ample etmate of the moment were ued for decon mang [47][5]. The dtrbuton of the ample etmate of the n thorder moment wa aumed to be Gauan, ( µ, σ ) N nh, nh,, where the mean nh, µ and varance σ nh, depend on the hypothe H and n. The decon crteron wa further reduced to comparng the ample etmate of the phae moment wth a threhold. The htogram of the phae dfference between two adjacent ymbol wa ued n [3], [33], [39] for PSK order dentfcaton, wth the decon made baed on the comparon of the htogram agant partcular pattern. The perodc component of the phae PDF were analyzed for PSK order dentfcaton n [5], ung the DFT of the phae htogram. In other word, the emprcal charactertc functon of the phae wa exploted for clafcaton n th wor. Furthermore, n [5] the algorthm wa extended to QM gnal clafcaton, by explotng the addtonal nformaton provded by the magntude of the receved gnal. Other feature extracted from the ntantaneou ampltude and phae were nvetgated for PSK and QM dentfcaton n [4], [78], [83], [84], uch a the urto of the ampltude. Wavelet tranformbaed algorthm Dfferent PSK gnal gve re to dfferent et of pea value n the magntude of the Haar wavelet tranform. The htogram of the pea magntude wa employed to dentfy the order of a PSK gnal n [37], wth the decon made by comparng the htogram wth the theoretcal PDF correpondng to dfferent order. Sgnal tattcbaed algorthm Cumulantbaed feature were propoed n [4] to dentfy the order of SK, PSK, and QM modulaton, a follow: the normalzed cumulant of fourthorder/ twoconjugate, c ( )/ c (), for SK, the magntude of r,4, 3 r,, the normalzed cumulant of fourthorder/zeroconjugate, c ( )/ c (), for PSK ( M > ), and the normalzed r,4, 3 r,, cumulant of fourthorder/zeroconjugate, c ( )/ c (), for QM. The theoretcal value of the n thorder/ r,4, 3 r,, q conjugate cumulant c, ( q =,..., n /, n even, for everal lnear modulaton are gven n Table IV. Thee ),, nq 8
19 value were computed ung the moment to cumulant formula 6, n whch the nthorder moment were calculated a enemble average over the noefree untvarance contellaton wth equprobable ymbol. Note that due to the ymmetry of the gnal contellaton condered, the nthorder moment for n odd are zero and hence, ung the moment to cumulant formula, t eay to how that the nthorder cumulant for n odd are alo zero. On the other hand, for n even we have c () = c (). n LRT wa formulated baed on the PDF of the ample etmate of, nq,, nn, q feature, whch are Gauan,.e., N ( µ H, σ ) H. Wth a mplfyng approxmaton,.e., equal varance under all the hypothee, the decon wa further reduced to comparng the ample etmate of the choen feature ˆω agant a threhold, wth ω a any of the cumulantbaed feature prevouly mentoned. For an N mod hypothe tetng problem, wth the hypothee ordered uch that µ H <µ... H < <µ H N, the decon rule to chooe H f mod ( µ +µ )/ <ω< ˆ ( µ +µ )/, (3) H H H H+ where µ H = and µ =. H N mod + Note that the cumulantbaed feature c ( )/ c () and r,4, 3 r,, c ( )/ c () do not depend on a fxed carrer r,4, 3 r,, phae θ, a for q = n/ the exponental factor whch depend on θ cancel each other, wherea for q n/ the phae dependency dropped by tang the magntude. Th wor wa extended n [53] to clafy lnear modulaton n frequencyelectve channel. The blnd alphabetmatched equalzaton algorthm (M) [], whch wa ued for equalzaton, wa alo employed for clafcaton. Some other cumulantbaed feature were added [3] to the et of feature extracted from the ntantaneou ampltude, phae and frequency [8][9], to nclude QM gnal n the et of canddate modulaton to be recognzed. Sgnal moment were appled to dtnguh between QPSK and 6QM n [54]. Specfcally, a lnear combnaton of the fourthorder/twoconjugate moment and the quared econdorder/oneconjugate moment were employed, wth the coeffcent and the delay vector optmzed to maxmze the probablty of correct clafcaton. et of feature wa choen for certan value of the delay vector, and clafcaton wa made baed on the correlaton between the ample etmate and theoretcal feature vector. The gnalmoment feature m ( )/ m () wa employed to dentfy the order of QM gnal n [4], wth the decon made baed on 3 r,6,3 5 r,, the mnmum abolute value of the dfference between the ample etmate and precrbed value of the feature. Sgnal cyclotatonarty wa alo exploted for lnear modulaton dentfcaton [55][65], va two approache: pectral lne generaton when pang the gnal through dfferent nonlnearte [55][57], and perodc fluctuaton wth tme of cumulant up to the n thorder [58][65]. We note that the n thorder cycle frequence (CF) are gven by ( n q) f + m/ T, wth m an nteger [6], [63]. The n thorder CF formula alo hold for an IF gnal, where f replaced by the IF frequency, f IF. Wth th property, the cyclotatonarty of the receved gnal wa exploted for MC through a pattern of newave frequence n gnal polynomal tranformaton. For example, the f IF and 4 f IF nuod that appear n the econd and fourth power of the receved gnal, repectvely, were ued n [55] to dtnguh between BPSK and QPSK. In [56], [57] the ame property wa 9
20 explored for a baeband gnal. By ncreang the order of the nonlnear gnal tranformaton beyond fourth power, th argument can be extended to dentfy modulaton of order hgher than QPSK. Note that the quaoptmal algorthm derved wthn the LB framewor for PSK gnal clafcaton alo explot uch a property, by ung the nformaton extracted n tme doman [3][5]. However, the gnal cyclotatonarty not exploted n th wor, a the amplng performed at the ymbol rate T. Cyclccumulant (CC) baed feature of dfferent order were nvetgated for modulaton clafcaton n [58][65]. feature baed on fourthorder/twoconjugate and econdorder/one conjugate CC at the CF equal to the ymbol rate, mlar to the one that ued moment [54], wa propoed n [58] and [59], to dentfy the order of QM modulaton. The ame decon crteron a n [54] wa employed. In [6] a generc algorthm wa propoed to explot gnal cyclotatonarty for clafcaton. feature vector wa propoed, whoe component were the magntude of the CC up to the n thorder, raed to the power of /n, when n goe to nfnty, and computed at all poble CF and delay vector. pparently, uch a clafer hard to mplement. Note that rang the n thorder CC magntude to the power of /n force the feature to tae value wthn the ame order of magntude. Therefore, the clacal Eucldan dtance can be ued for decon. For lnear modulaton, the n th order/ q conjugate CC of rt (), where rt () gven n (), wth φ =, =,..., K, g() t a a raed cone pule hape, and nt () a the WGN, and the et of CF are gven by [6], [63] c ( ; ) a c T e e e g ( t) g ( t ) e dt, q=,..., n, (33) () ( ) n j f ( ) n j πβεt j n q θ π u uτu (*) n = n (*) u jπtβ rnq,, γ τn = ( ), n, q +τ u= u () { : ( n q) f, T, nteger, c ( τ ) } κ = γ γ=β+ β= γ;, (34) nq, rnq,, n where γ a CF, (*) u repreent a poble conjugaton of the u th term, u =,..., n, uch that the total number of conjugaton q, and ( ) u the mnu gn aocated wth the poble conjugaton (*) u, u =,..., n. Snce nt () a tatonary, zeromean Gauan proce, t cumulant are tme ndependent and nonzero only for the econd order. Therefore, WGN doe not have any contrbuton to the hgherorder ( n 3) CC of rt ( ). One can ealy notce that by tang the magntude of the n th order CC, a feature robut to the carrer phae and tmng offet obtaned. In [6][6] and [6], the magntude of the CC up to the fourth and xthorder, at a CF equal to γ= ( n q) f + / T and a delay vector for whch a maxmum reached (.e., τ  =  [63]), were nvetgated a feature, repectvely. Baed on thee feature, a feature vector wa propoed n [6], a () () () () () () () F = [ c,,( )...,,( ),4,( 3 )...,4,4( 3 ),6,( 5 )... r γ; cr γ; cr γ; cr γ; cr γ; cr,6,6( γ; 5) ], =,..., Nmod. (35) The CCbaed feature are etmated from Kρ ample, taen over the oberved K ymbol nterval []. Note that the receved gnal overampled n order to explot gnal cyclotatonarty. The amplng frequency equal to ρ /T, wth ρ a potve nteger, called the overamplng factor. The decon made by comparng the ample etmate wth precrbed feature vector from a looup table, = F F ˆ, (36) ˆ arg mn d( (), ) Nmod n n
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