Matchd filtr basd algorithm for blind rcognition of OFDM systms Abdlaziz Bouzgzi, Philipp Ciblat, and Pirr Jallon CEA-LETI, MIATEC - Grnobl, Franc (Email: abdlaziz.bouzgzi@ca.fr, pirr.jallon@ca.fr) EST, Paris, Franc (Email: philipp.ciblat@nst.fr) Abstract In th Cognitiv Radio contxt, snsing procss is a crucial task. Th cognitiv dvis has to b abl to dtct and to idntify svral radio systms. As mor standards ar now basd on OFDM modulation th papr aims with th paramtrs stimation of such a modulation. All actual OFDMbasd standards diffr from thir subcarrir spacing thrfor it will b ingnious to focus on this paramtr to idntify ths systms in a non data aidd contxt. W propos a nw fficint algorithm basd on th matchd filtr principl. Th bhavior of th proposd approach will b studid in th contxt of practical impairmnts lik frquncy and/or tim offsts and a multipaths fading channl. Finally, th prformanc of th proposd algorithm will b valuatd in contrast with th stat of art mthods by mans of computr simulations. I. ITRODUCTIO Th cognitiv radio concpt has bn first introducd by [] and consists for a radio in carry out additional functionalitis to adapt its transmission paramtrs in rgard of its spctral nvironnmnt. Consquntly, a cognitiv trminal nds to b abl to dtct th accss points of diffrnt systms in it nighbourhood and to rcogniz thir usd standards. Th Opportunistic Radio is on of th multipl Cognitiv Radio applications. It aims that th opportunistic dvic has to b abl to dtct th unusd spctrum bands and to adapt its transmission paramtrs in ordr to transmit within ths bands. Indd, dtrmining th frquncy band usd by ach systm can not b considrd as an fficint idntification tool and thus nw blind tchniqus sm to b ndful. Most of popular standards us OFDM modulations (.g. WiFi, WiMax, DVB, 3GPP/LTE). vrthlss, thir intrcarrir spacing paramtrs nabl to distinguish thm from ach othrs. In fact, th intrcarrir spacing is qual to 5.65kHz, 4kHz, 3.5kHz,.6kHz, 5kHz for Fixd WiMAX [], Mobil WiMAX [], WiFi [3], DVB-T [4], 3GPP/LTE [5] rspctivly. Th problm of th systm rcognition issu hnc boils down to th stimation of th intrcarrir spacing of th rciv signal. W focus on this problm in this papr. Obviously, th proposd stimation algorithm also applis to military contxts. Only fw rsults about this problm can b found in th litratur, mainly, in [6], [7], [8] which ar all basd on th corrlation proprty of th cyclic-prfixd OFDM signals. In fact, th cyclic prfix is addd on ach OFDM symbol by copying a part of th usful part at th bginning of th symbol. OFDM signals hnc xhibit a corrlation pak at a tim lag qual to thir usful tim. As th usful tim is also qual to th invrs of th intrcarrir spacing, th systm rcognition can b don accordingly. vrthlss, it is straightforward to undrstand that if th cyclic prfix is short (in rgard to th usful tim of th OFDM symbol), th corrlation pak is low and th corrlation basd mthods fail. In practic, ths contxts occur at last for DVB-T signals and WiMax signals for which th cyclic prfix may b vry short. Morovr, this approach is snsibl to multipaths channls and falls down if anothr schm of OFDM signals is usd (x: ZP-OFDM). In this papr w propos an altrnativ mthod basd on th matchd filtr principl. W prov th ffctivnss of th proposd cost function and w show its robustnss to any schm of OFDM signals (short CP, ZP-OFDM). Furthrmor, w show that this approach rsists in a multipaths channl contxt. Th papr is organizd as follows: In Sction II, w driv th signal modl. In Sction III w giv th guidlins of th proposd mthod. In Sction V w show how th proposd algorithm can b modifid dal with a carrir offst. Finally, Sction V is dvotd to numrical illustrations whr a comparison with th autocorrlation basd mthod is also drawn. II. SIGAL MODEL In Additiv Whit Gaussian ois channl, th OFDM rciv signal sampld at a rat /T writs: y(m) = K k= n= a k,n n(mt DTc kts) iπ g a (mt kt s ) + b(m) () whr K is th numbr of transmit symbols, is th numbr of subcarrirs and whr /T c is th information symbol rat in absnc of guard intrval. To kp all th information of th rciv signal, w assum that T satisfis th Shannon condition, i.. T < T c. Th intrcarrir spacing is qual to /T c. Th lngth of th cyclic prfix is st to DT c. Th duration of a whol OFDM symbol is T s = ( + D)T c. Th squnc {a k,n } rprsnts th transmit unknown data symbols at subcarrir n and OFDM block k and assumd to b indpndnt and idntically distributd. Th shaping filtr g a (t) is assumd to b qual to if t < T s and othrwis. Th complx-valud nois b(m) is assumd to b indpndnt and idntically distributd, circularly-symmtric
zro-man whit Gaussian nois. Its varianc is qual to pr ral dimnsion. Th rciv sampls in Eq. () can also b writtn as th rsult of a linar transformation. If th vctor y stands for [y(),, y(m )] T whr (.) T is th transposition oprator and M is th numbr of th rcivd sampls, Eq. () indd rwrits: y = F θ a + b () whr a k = [a k,,, a k, ] T of siz a = [a T,,aT K ]T of siz K b = [b(),, b(m )] T of siz M In Eq. (), th matrix F θ dpnds on θ = [, DT c, T c ]. Thanks to th trm g a (mt kt s ) in Eq. (), it is straightforward to chck that F θ is dfind by block sinc: g a (mt kt s ) mt kt s < T s which implis that m T T s < k m T T s. Consquntly, for a givn m, it xists only an uniqu valu of k, dnotd by k m, such as g a (mt kt s ). F θ is thn composd by null componnts xcpt th nxt ons F θ (m, k m + n) = T DTc iπnm iπn(km+) for m =,...,M and n =,,. In th considrd contxt, th trminal just has th knowldg of {y(m)} M m=, M and T and wishs to stimat th valu of T c, th invrs of th intrcarrir spacing. In this papr, w propos to do that by stimating F θ which also dpnds on and DT c. III. MATCH FILTER BASED ALGORITHM In this sction w propos to build an stimator of th unknown paramtrs basd on th match filtr approach. Th main ida consists of sarching th st of paramtrs θ = [Ñ, DT c, ÑT c] which maximizs th following cost function J(θ) = E{ FH θ y } F θ F H θ F whr suprscript (.) H stands for th hrmitian oprator, x is th Euclidian norm of th vctor x and A F is th Frobnius norm of th matrix A quals to Tr(A H A). Th ffctivnss of th proposd cost function is basd on th following thorm: Thorm : In noislss contxt, th following quality holds: J(θ) J(θ ) = F θ F H θ F and th quality is rachd if and only if θ = θ. Th proof of Thorm is givn using th two following propositions. Lt us considr a flat fading channl and a noislss contxt. Using th fact that th data symbols ar assumd to b i.i.d w not that E{aa H } = I whr I is th idntity matrix. Thrfor, w gt E{ F H θ y } = Tr(F H θ F θ F H θ F θ ). Th first proposition is statd as follows: Proposition : By dnoting A = F θ F H θ and B = F θ F H θ w gt Tr(A H B) Tr(AA H )Tr(BB H ) and th quality holds if and only if F θ F H θ = F θ F H θ Proof: Th inquality of Proposition can b rwrittn as A H B F AH A F B H B F (3) To prov this inquality, an xplicit form of A H B F is givn. Using th dfinition of th Frobnius norm, this trm can b xprssd as following KÑ A H B F = K M (A H ) il B lj (4) i= j= M M KÑ K = B lj B l j (5) l = i= A lia l i j= whr (.) stands for th complx conjugat and K is th stimatd numbr of OFDM symbols within th rciv signal duration, it is givn by MT /(ÑT c + DT c ). For a sak of simplicity lt us introduc th matrix V = AA H and W = BB H of which th componnts ar xprssd as follows K K V l l = A lia l i W ll = B lj B l j i= j= for all l =,..., M and l =,..., M. By rplacing ths xprssions in Eq. (5) w gt A H B F = M l = M V l lw ll (6) A first application of th Cauchy-Schwartz inquality to th sum of indx l in Eq. (6) writs ( M M M M ) ( ) M V l lw ll V l l W ll (7) l = l = W considr th following notations ( M l = v l = l = V l l ) ( M w l = l = W ll ) W rplac ths trms in th RHS trm of Eq. (7) and w apply th Cauchy-Schwartz inquality to th sum of indx l ( M M ) ( ) M v l w l v l w l (8) By rplacing v l and w l by thir rspctiv xprssions w dduc that L L v l = A H A F and w l = B H B F
Finally, using Eq. (7) and Eq. (8), it is straightforward to dduc th rsult of Eq. (3). Proposition : Th quality F θ F H θ = F θ F H θ holds if and only if θ = θ Bfor giving th proof of Proposition, it will b worthwhil to study th bhavior of th proposd algorithm in a tim missynchronization contxt which occurs if th bginning of th rciv signal dos not match with th bginning of an OFDM symbol. Indd, this problm can distort th ffctivnss of th proposd cost function. In this contxt, th rciv signal can b xprssd as follows ỹ(m) = y(m τ) whr y is th prfctly synchronizd signal of Eq. (). To dal with this impairmnt, th proposd algorithm has to b xpandd by introducing th nw matrix modl F θ whr θ = [T c, DT c,, τ]. This matrix is obtaind by th truncation of th first τ/t rows of F θ. Using this nw modl matrix, w propos to maximiz th following cost function according to th nw vctor of th unknown paramtrs θ = [ÑT c, DT c, Ñ, τ]: J( θ) = E{ FH θy } F θf H θ F ot that Proposition is still vrifid in this contxt, morovr, Proposition can b asily xpandd as follows: Proposition 3: Th quality F θf H θ = F θ F H θ is rachd if and only if th signal is corrctly synchronizd and th OFDM paramtrs ar wll stimatd, i.., θ = θ Th proof of Proposition can b dducd from th proof of Proposition 3 which can b sktchd as following. Proof: W rcall that is a block matrix of siz F θ (M τ/t ) ˆK ˆ. Th matrix F F θ H θ is thn composd by squar blocks matrics as follows: th first block is of siz (T c + DT c )/T τ/t. th siz of th othr blocks (xcpt th last on) is (T c + DT c )/T Consquntly, if th quality F θf H θ = F θ F H θ holds th two sids matrics will hav th sam block structur. Assuming that T ÑT c + DT c, it is straightforward to dduc that ÑT c + DT c = T c + DT c and τ/t = τ/t ot that th block structur of F θ F H θ is sufficint to show th tim synchronization condition. vrthlss, th condition about T c and DT c rquir th xprssion of th quality of ach componnt of F θ F H θ. To xprss th componnts of F θ F H θ lt us assum a prfct tim sychronization (τ = ) and an intgr numbr of symbols within th rciv signal. Undr ths assumptions and aftr som mathmatical calculations th componnts F F θ H θ can b xprssd as follows: sin π T ˆ (l l) ˆ iπ(l l )T sin π ( F θf H θ ) T (l l) if l l T ÑT c + DT c and l l l,l = if l = l if l l T > ÑT c + DT c for l =,..., M and l =,..., M. By assuming that F θf H θ = F F θ H θ, lt w considr a nonnull, non-diagonal componnt with indxs l and l such as l l T ÑT c + DT c and l l. By prforming th phas and modulus qualitis w can writ: (l l )T = (l l )T T c T c + k sin π T (l l) T (l l)) sin π T (l l) = sin(π sin(π T (l l)) whr k Z. It is straightforward to show that if ths two qualitis hold th rsult can b asily dducd i., ÑT c = T c and DT c = DT c. IV. IMPACT OF CARRIER FREQUECY OFFSET In a practical contxt, a frquncy offst can altr th rcivd signal. Consquntly, th study of this paramtr sms to b crucial in ordr to valuat its impact on th proposd approach. W considr a non-null componnt of F θ in th modl of Eq. () with a normalizd frquncy offst dnotd δf : F θ (m, k m + n) = n+δf DTc iπmt iπn(km+) whr m =,...,M, k m =,...,K and n =,,. W propos to xtnd th proposd algorithm to ovrcom th ffct of th frquncy offst. Indd, w introduc an additional loop to sarch jointly th unknown OFDM paramtrs and th frquncy offst stimat. Thorm can b modifid in ordr to tak into account th unknown carrir offst. Accordingly, w xprss th following thorm Thorm : By dnoting θ = [, DT c, T c, δf] th following inquality still vrifid: J(θ) J(θ ) and th quality holds if and if θ = θ. Th proof of Thorm can b built using Proposition which rmains valid and using Proposition which can b xtndd asily to rsolv th problm of th stimation of th carrir offst. According to this xtnsion, th ovrall algorithm can b xprssd as follows: fix th tstd valu of th intrcarrir spacing (/ÑT c) fix th tstd valu of th guard tim duration DT c whr DT c ÑT c {/, /4, /8, /6, /3, }
.. Matchd Filtr basd stimation.3.6.93.5.56.87.9 Corrct Dtction Rat.. Matchd Filtr basd stimation 8 6 4 4 6 SR (db) Fig.. Corrct dtction rat vs. D/ (SR=dB) Fig.. Corrct dtction rat vs. SR (D/ = /8) fix th valu of th numbr of carrirs Ñ compnsat th carrir offst with a tstd valu of th normalizd frquncy offst δf: z(m) = y(m) iπmt δf drop th first τ T sampls of z, such as τ < ÑT c+ DT c valuat th cost function dfind by: J(ÑT c, DT c, Ñ, δf, τ) = E{ FH θ z } F θ F H θ F updat all paramtrs and rstart th solution is th vctor θ = [ÑT c, DT c, Ñ, δf, τ] which maximizs th function J. V. SIMULATIOS W propos to valuat th prformanc of th proposd algorithm by mans of som computr simulations. W rcall that th tratd problm is th rcognition of OFDM basd systms in th Cognitiv Radio contxt. As sn in th introduction, an OFDM basd systm can b idntifid using th intrcarrir spacing. Consquntly, w will focus on th stimation of this paramtr. Furthrmor, as w trat a dtction problm w do not nd a tight stimation of this paramtr but w only nd an stimation up to % (cf. Sction I) of th right valu. Th prformanc of our algorithm will b compard to th stat of art tchniqu basd on th corrlation inducd by th cyclic prfix whr th following cost function is usd to xtract th paramtr of intrst: { T c = argmax E T c { y(t + ÑT c)y (t) }} To valuat ths two algorithms, OFDM signals hav bn gnratd with = 64, T c = µs and th sampling tim is chosn to b qual to th half of th chip tim T = T c /. Th numbr of availabl OFDM symbols is. Th rciv discrt-tim signal is writtn as y(m) = L h l x(m l) + b(m) l= whr L is th lngth of th channl impuls rspons and ach componnt of this rspons is assumd to b Gaussian distributd with zro man and sam varianc. Unlss othrwis statd, th discrt-tim channl disprsion tim is fixd to b a quartr of th cyclic prfix duration. b(m) is th nois sampls. Unlss othrwis statd, Signal-to-ois Ratio has bn fixd to b qual to db. As prformanc masur, w considr th prcntag whr th stimatd intrcarrir spacing matchs with th corrct on up to %. Consquntly, w hav considrd a grid of valus of ÑT c of stp of µs. For ach considrd ÑT c, th cyclic prfix valu DT c taks valus in th st ÑT c {/, /4, /8, /6, /3,}, and Ñ taks valus in th st {56, 8, 64, 3}. Th prformanc curvs hav bn drawn by avraging th good dtction rat ovr runs. First, w propos to valuat th impact of th ratio btwn th usful and th cyclic prfix durations on th prformanc of both algorithms. Th masurd corrct dtction rat is showd in Figur. As xpctd, th corrlation-basd dtction fails for small cyclic prfixs bcaus th corrlation is strongly dcrasd. In contrast, th proposd algorithm maintains th sam prformanc whatvr th valu of th usd cyclic prfix. Consquntly, th proposd algorithm is mor appropriat for th Cognitiv Radio which on of svral problms is how to trat systms using diffrnt CP lngths. Morovr, notic that within som systms thr xist diffrnt mods with diffrnt valus of CP duration (.g., WiMax []). In th following, w fix th CP lngth as D/ = /8. W inspct th impact of th Signal to ois ratio on th prformanc of ach algorithm. Figur shows that th matchd
Corrct Dtction Rat. MF with a frquncy offst stimation. MF with a tim offst stimation 8 6 4 4 6 SR (db) Corrct Dtction Rat.. Matchd Filtr basd stimation. Channl lngth / CP duration Fig. 3. Corrct dtction rat vs. SR including ithr tim offst or frquncy offst (D/ = /8) Fig. 4. Corrct dtction rat vs. th channl rspons lngth (D/ = /8) filtr basd tchniqu outprforms strongly th corrlationbasd stimation. Indd, th proposd approachs nsur % of good dtctions until an SR of 4dB. Obviously, th prfct frquncy and tim synchronization ar a non ralistic hypothsis and thus w nd to tst th robustnss of th proposd approach in a practical situation. W start by valuating th ffct of th carrir offst. To do that, an OFDM signal has bn gnratd by introducing a random frquncy offst. As mntiond in Sction IV, an additional loop must b addd to prform th joint stimation of both th frquncy offst and th OFDM paramtrs. Figur 3 shows that th proposd approach still work wll and rsist to th frquncy offst impairmnt. Morovr, it can nsur a good stimation of th carrir offst which allows th rcivr to prform data dcoding and thn xtract mor information about th rcivd systm. ow w propos to study th ffct of th tim offst which occurs if th bginning of th rcivd signal dos not match with th bginning of an OFDM symbol. Using similar simulations, w show how w can ovrcom th missynchronization drawbacks. An OFDM signal has bn gnratd and a random numbr of sampls is droppd. Th prformanc of this approach is shown in Figur 3 in contrast with th stat of art mthod. As xpctd th corrlation-basd tchniqu is insnsitiv to a tim and a frquncy missynchronizations. vrthlss, it is still lss fficint than th modifid matchd filtr-basd approach which th prformanc is as good as th prfct contxt (cf. Figur ). Finally, Figur 4 displays th prformanc of both mthods vrsus th lngth of th channl rspons. Th match filtr basd approach shows a good rsistanc to multi-paths channl in contrast with th stat of art mthod. of OFDM modulations. W hav provd th ffctivnss of th cost function in th contxt of frquncy and tim offsts. Obviously, th ndd multidimnsional optimization inducs an additional cost on complxity. vrthlss, this drawback can b justifid by th robustnss and th adaptability of th proposd approach. Indd, w hav shown by mans of computr simulations that th nw mthod outprforms th stat of art approach in trms of signal to nois ratio and th rsistanc to multi-paths fading channls. Morovr, th proposd algorithm nsurs th sam prformanc whatvr th usd cyclic prfix. Consquntly, it is mor appropriat to th Cognitiv Radio applications and to othr totally blind contxts lik military applications. REFERECES [] J. Mitola, Cognitiv Radio : an Intgratd Agnt architctur for Softwar Dfind Radio, Phd thsis, Royal Institut of Tchnology (Stockholm, Swdn),. [] L. uaymi, Wimax: tchnology for broadband wirlss accss, John Wily, 7. [3] C. Smith and J. Myr, 3G wirlss with Wimax and Wi-Fi : 8.6 and 8., McGraw-Hill, 5. [4] ETSI, Digital vidl broadcasting (DVB) framing structur, channl coding and modulation for digital trrstrial tlvision, in ETSI Rport, ovmbr 4. [5] H. Holma and A. Toskala, WCDMA for UMTS: HSDPA volution and LTE, John Wily, 7. [6] P. Liu, B.-B. Li, Z.-Y. Lu, and F.-K. Gong, A blind tim-paramtrs stimation schm for OFDM in multi-path channl, in Intrnational Confrnc on Wirlss Communications, tworking and Mobil Computing, Spt. 5, pp. 4 47. [7] B. Wang and L. G, Blind idntification of OFDM signal in rayligh channls, in Intrnational Confrnc on Wirlss Communications, tworking and Mobil Computing, Dc. 5, pp. 95 954. [8] H. Ishii and G.W. Wornll, OFDM blind paramtr idntification in cognitiv radios, in IEEE Intrnational Confrnc on Prsonal, Indoor and Mobil Radio Communications, Spt. 5, pp. 7 75. VI. COCLUSIO In this papr, w hav introducd a nw matchd filtr basd algorithm to prform th stimation of th intrcarrir spacing