Using Higher Order Cyclostationarity to Identify Space-Time Block Codes

Transcription

1 Usng Hgher Order Cyclostatonarty to Identfy Space-me Bloc Codes Marcus R. DeYoung,, Robert W. Heath, Jr.,andBranL.Evans {deyoung, rheath, Dept. of Electrcal and Computer Engneerng, he Unversty of exas at Austn, Austn, X, 787 Zeta Assocates, Inc. 3 Eaton Place, Sute 5, Farfax, VA 3 Abstract Research n cogntve rados has renewed nterest n tools, such as spectrum estmaton and modulaton dentfcaton, to characterze the rado frequency (RF) envronment. he use of multple antennas for multple-nput multple-output (MIMO) communcatons presents a new challenge n detectng and classfyng sgnals. In ths paper, we propose a cyclostatonarty-based statstcal test to detect space-tme bloc codes, focusng on the two transmtter Alamout space-tme bloc code (SBC). Our test explots a new characterzaton of the Alamout code usng fourth order cyclc frequences. he test requres only a sngle receve antenna, and does not requre any symbol synchronzaton. I. INRODUCION Cogntve rados are expected to help fll under-utlzed bandwdth by sensng for holes n the spectrum and transmttng n a manner that does not nterfere wth other sgnals []. Along wth spectrum estmaton, blnd modulaton classfcaton determnng the modulaton type of sgnal wth unnown parameters helps present a more complete pcture wth whch an ntellgent software defned rado may optmze ts transmsson scheme []. Current research ndcates that cogntve rados should be able to operate n both whte space (no sgnal energy present) and gray space (sgnal energy present) by classfyng the sgnals present and transmttng n a way that avods nterference [3]. Modulaton classfcaton algorthms have been extensvely studed for nterference characterzaton and survellance. Pror wor focuses on sngle antenna modulatons, such as M- QAM, FSK, and PSK. Emergng wreless systems are usng multple transmt and receve antennas. hese multple-nput multple-output (MIMO) communcatons systems provde large benefts n ncreased spectral effcency, thereby allowng more nformaton to be transmtted n a smaller amount of costly bandwdth [4]. here are many dfferent types of MIMO communcaton algorthms, such as space-tme bloc codng, spatal multplexng, and beamformng. Modulaton classfcaton methods are needed for each of these methods for cogntve rados to be able to operate n envronments wth MIMO communcaton. In ths paper, we propose a classfcaton method for the Alamout space-tme bloc code [5]. he Alamout code s a M. DeYoung and B. Evans were supported n part by an equpment grant from Intel Corporaton. R. W. Heath was supported n part by the Natonal Scence Foundaton under grant CCF MIMO modulaton technque for two transmt antennas, whch s used to ncrease spatal dversty n wreless lns. It s the most extensvely employed MIMO technology, beng already deployed n WCDMA and ncluded n several WMax profles. We derve the fourth order cyclc frequency of the Alamout space-tme bloc code, present a statstcal test for ts presence wth only a sngle receve antenna n a flat-fadng envronment, and provde smulaton results llustratng the effectveness of the proposed algorthm. II. CYCLOSAIONARIY he theory of cyclostatonarty has receved much attenton n the communcatons world due to ts relatvely accurate stochastc descrpton of dgtal and analog communcatons sgnals. Cyclostatonary propertes are often used when lttle can be assumed about the sgnal or nose envronment, such as wth modulaton dentfcaton [6], spectrum sensng [7], and blnd channel dentfcaton[8], [9]. A cyclostatonary process s one whose statstcs vary perodcally n tme. A process s sad to exhbt -th order cyclostatonarty f there exsts a nonlnear transform of the -th order that produces fnte strength snusod components []. Practcally speang, ths means there s a -th order perodcty nherent n the statstcs of the sgnal. he -th order tme varyng moment m x (t; τ) of a sgnal x(t) can be defned as a Fourer expanson of the -th order cyclc moment M x as [] m x (t; τ) = E{x(t)x ( ) (t + τ )...x ( ) (t + τ )} M x = lm t= m x(t; τ)e jt where ( ) denotes an optonal conjugaton and s a cycle frequency. Smlarly, the tme-varyng cumulant c x can be represented by a Fourer seres representaton of ts cyclc covarance C x (; τ) as c x (t; τ) = C x (; τ)e jt C x (; τ) = lm c x (t; τ)e jt. t= Note that the second order cumulant s just the covarance; the second and thrd order cumulants are both equvalent to ther respectve moments. Also, note that at =, the cyclc

2 moments and cumulants are equvalent to ther respectve standard (non-cyclc) forms. III. SYSEM MODEL Consder an unnown, sampled, contnuous-tme communcatons sgnal that s dgtally modulated and employs pulse shapng. Let a dgtally modulated sgnal n a flat fadng channel oversampled by a factor of P = s be represented as y[n] =h[n]e jπfepn s[]g tx [n s ]+v[n] where f e s the frequency offset, s[] s a zero-mean,..d. symbol sequence, g tx [n] s the transmtter pulse-shape, s s the symbol rate, and v[n] s addtve whte Gaussan nose (AWGN). Usng the cyclostatonary propertes of the dgtal modulaton, we can fnd the symbol rate by detectng a cyclc frequency at = s []. We wll assume for smplcty that there s no frequency offset (f e =), but the results stll hold for mperfect carrer recovery []. he receved sgnal from a MIMO transmsson wth two transmtters, one recever, and no frequency offset may be represented as y[n] =h [n] s []g tx [n s ]+ h [n] s []g tx [n s ]+v[n] where s [n] and s [n] are the source symbols from the two transmtters. Channel mparments caused by multpath may mpact the performance proposed algorthm. Frequency selectve fadng nduces addtonal repetton nto the receved sgnal, whch could ncrease the false alarm rate for sgnals other than the Alamout space-tme bloc code. Flat fadng presents problems when the scalng s very asymmetrc between channels. he man focus of ths paper s on the presence and detectablty of the unque fourth-order cyclc frequences. herefore, we wll employ the smplest channel model n the subsequent smulatons unt channel gans n addtve whte gaussan nose and leave the detaled analyss of the effects of channel mparments as an avenue for further research. IV. SAISICAL ES FOR ALAMOUI CODE Followng from the concepts of symbol rate detecton, we derve a statstcal test to dstngush the Alamout SBC from spatal multplexng or a sngle transmtter. he Alamout code s gven by [5] [ ] s s X = s s. If we smplfy and consder perfect samplng at the rate of one sample per symbol, then the receved samples of Alamout coded symbols may be wrtten as y[n] =h [n]s [n]+h [n]s [n]+v[n] y[n +]= h [n +]s [n] + h [n +]s [n]+v[n]. he cyclc autocorrelaton s the same for Alamout coded data as t s for a sngle emtter or even multple ndependent streams, as n a spatal multplexng transmsson mode. herefore, the second order cyclostatonary technques can stll be used to fnd the symbol rate, but provde no useful dstncton between Alamout coded data and a sngle transmtter or multple ndependent transmtters. However, the fourth order cyclc cumulant s dfferent wth Alamout coded sgnals, provdng the bass for the followng proposed statstcal test. A summary of the dervaton s provded n the Appendx. A. Fourth order Cyclc Frequency for Alamout Code he fourth order cyclc cumulant s defned as [3] C 4x (; τ,τ,τ 3 )=M 4x (; τ,τ,τ 3 ) M x ( β; τ )M x (β; τ )e jβτ + β A m M x ( β; τ )M x (β; τ τ 3 )e jβτ3 + M x ( β; τ 3 )M x (β; τ τ )e jβτ () where A m s the set of second order cyclc frequences. Fortunately, we can see that β A m where ( β) / A m, the second order terms go to zero and C 4x (; τ,τ,τ 3 ) = M 4x (; τ,τ,τ 3 ). he dervaton for the fourth order cyclc frequency of the Alamout code s provded n the Appendx. We fnd a cycle at = ± P = ± s. We expect there to be a relaton at half the symbol rate snce the symbols at tme t + are related to those at tme t every other set of symbols. Space-tme bloc codes wth larger dmensons also have correspondng dstngushng features wth even hgher order cyclc cumulants. For example, t can be shown that the 4 4 quas-orthogonal space-tme bloc code defned n [4] has spectral lnes at ± cumulants. B. Statstcal est 4 s when usng 8-th order cyclc Determnng the symbol rate wll be the frst step n the algorthm presented n Secton IV. Assumng that the symbol rate s nown, we now chec the presence of a cyclc frequency at = s. Dandawate and Gannas provde a method to detect the presence of a cyclc frequency n a -th order cyclc cumulant [3]. he authors defne the sample estmator for the fourth order cyclc cumulant as the true cumulant plus error as Ĉ 4x (t; τ) c 4x (t; τ)+ɛ ( ) where ɛ ( ) s the estmaton error (goes to zero as ). Wth ths formulaton, we map cycle detecton to a bnary

3 .5 x 3.5 x c 4x c 4x (a) Ĉ4x, Alamout (b) Ĉ4x, wo ndependent streams Fg.. Sample fourth order cyclc cumulants as a functon of wth τ =(,P,,P) and the delayed versons conjugated. he modulaton s QPSK, E s =5dB, =, N =, symbols. N o s 5 hypothess test as shown below H :/ A 4 {τ n } N n= c ˆ 4x (t; τ) =ɛ ( ) H : A 4 for some{τ n } N n= ĉ 4x (t; τ) =c 4x (t; τ)+ɛ ( ) where A 4 s the set of fourth-order cyclc frequences. It s ( ) shown that lm ɛ x converges n dstrbuton to a multvarate normal dstrbuton wth zero mean and asymptotc covarance Σ c. hs leads to the defnton of a test statstc for a -th order cycle c as c ĉ x (t; τ)ˆσ c ĉx(t; τ) where ĉ x (t; τ) s a vector of the real and magnary portons of the sample cyclc cumulant for a fxed set of lags {τ,...τ N } and ˆΣ c s the sample covarance of ths vector computed wth the sample spectral correlaton functons. It s shown that c asymptotcally converges to a χ random varable wthout the presence of a cycle (H ), and a Gaussan random varable wth hgher mean (H ) f the cycle exsts. herefore, a desred false alarm rate (P F ) or probablty of detecton (P D ) can be computed for a gven cyclc frequency. he reader s referred to [3] for a detaled explanaton. he followng procedure may be used detect the presence of the fourth order cycles nduced by a Alamout bloc code. ) Estmate = s usng the cyclc autocovarance test from []. ) Compute the desred probablty of detecton P D accordngtotheχ dstrbuton. 3) Compute ĉ 4x (t; τ...) for ± = ± and compare to the threshold. he complexty of the algorthm s fundamentally a functon of the sample cyclc cumulant computaton, and the computaton of the sample covarance. For the Alamout code, only a fourth order sample cyclc cumulant at a sngle frequency s requred, and may be computed as Ĉ = N N n= x[n]x [n + P ]x[n]x [n + P ]e jt he complexty of ths operaton s O(N). he reader s referred to [3] for detals on the computaton of the sample covarance matrx ˆΣ 4c. he elements are lnear combnatons of the real and magnary portons of the sample second order cyclc spectrum, whch may be estmated by usng a smoothed perodogram approach. hs operaton s O(NW n ), where W n s the length of the smoothng wndow. hus, overall the test taes O(NW n ) multplcatons. V. SIMULAION RESULS o demonstrate the effectveness of our proposed test, we run the algorthm defned n Secton IV on both the symbols generated by the Alamout code and those from a spatal multplexng scheme. In these smulatons, the spatal multplexng system s two ndependent streams of data from the two transmtters. We assume a flat-fadng envronment, and

4 ROC curve at db SNR ROC curve at 5 db SNR rue Postve Rate.6.4 rue Postve Rate Symbols 5 Symbols 5 Symbols.. Symbols 5 Symbols Symbols False Postve Rate False Postve Rate (a) db Aggregate Receved SNR (b) 5dB Aggregate Receved SNR Fg.. Recever operatng characterstc (ROC) curves for the proposed test at 5 and db (aggregate) SNR. he curves llustrate the tradeoffs between false alarm rates and attanable detecton rates n addtve whte Gaussan nose. he modulaton s QPSK wth a square root-rased cosne pulse. he data s oversampled by a rato of P =4. for smplcty we use unt channel gans. A square root-rased cosne pulse wth excess bandwdth of 5% s employed at the transmtter. he data was oversampled by a factor of P =4. A Kaser wndow wth parameter and length of taps s used to smooth the spectral correlaton functon. he receved sgnal to nose rato Es N s defned such that E s = E s + E s. In Fgure, we plot smulated recever operatng characterstc (ROC) curves for the proposed test statstc 4c for the cyclc frequency P for the Alamout code aganst two ndependent streams receved wth one antenna. From these plots, we llustrate the effects of receved SNR and number of symbols used on accurate detecton. Fgure (a) shows curves wth db of SNR receved on the sngle receve antenna usng, 5, and 5 symbols. Fgure (b) shows smlar curves when 5 db SNR s receved. In each case, we can get to very hgh probablty of detecton wth very low probablty of error. However, we note that at db aggregate SNR we can acheve ths wth 5 symbols, but at 5 db aggregate SNR we need approxmately symbols to acheve smlar results. hus, t s clear from these smulatons that the detecton ablty of the statstcal test s dependent on both the sgnal to nose rato and the number of symbols ncluded n the test. hs s one of the man problems nherent wth a hgher order statstcal test. As the order of the cyclostatonarty ncreases, the number of samples requred to obtan an accurate estmate ncreases greatly. VI. CONCLUSIONS In ths paper, we derved unque fourth order cyclc statstcs for the Alamout space-tme bloc code, and proposed a statstcal test based on the wor of Dandawate and Gannas. hrough smulatons, we llustrated the effectveness of the test wth a relatvely low number of samples at hgher SNR, as well as the drawbac of requrng many samples to acheve accurate results n lower SNR envronments. Fndng lower complexty algorthms that perform well n wth low SNR s an area for further research. It s mportant to note that ths test does not necessarly unquely dentfy the Alamout code. In the case of second order symbol rate detecton, t was seen that by weghtng the observatons we can acheve better results []. Any s sgnal that has a fourth order cyclc frequency at wll be dentfed usng ths test, ncludng possbly hgher order (e.g. 4 4) bloc codes. herefore, to accurately dstngush between many types of modulaton, we must nclude even hgher order statstcal tests. As the order ncreases, the number of samples requred rses as well, so t remans an area of further research to fnd more effcent tests. Another factor that maes realstc processng dffcult s fadng. Flat fadng provdes a challenge f the matrx s very unbalanced (e.g. h =,h =.5). Frequency selectve fadng s even more dffcult, snce multple scaled versons of the same symbols wll be receved and more cycles wll be observed. One more

5 avenue for further research s the use of multple receve antennas. It may be possble to use selecton combnng to ncrease the effectve SNR and thus ncrease the accuracy of the test or reduce the number of samples needed. APPENDIX A setch of the dervaton for the cyclc frequency at P, where P s the oversamplng factor, s now provded. From Secton IV, we see that we only need to loo at the fourth order moment m 4x (t; τ,τ,τ 3 ) for the Alamout code. Let x[] represent the receved symbol, whch s h[]s []+h[]s [] at tme and h[]s []+h[]s [] at tme +. Note that x[] and x[+] are not (necessarly) ndependent. However, n the case of a sngle transmtter or multple antenna transmsson streams sendng ndependent data, x[] s ndependent from x[+]. For notatonal smplcty, shortenng the pulse shape g tx to g, let G4 n (κ,κ,κ 3,κ 4 ) g[n κ ]g [n κ ]g[n κ 3 ]g [n κ 4 ]. () Ignorng the nose term (whch has a zero fourth order moment f Gaussan), we wrte out the fourth order moment wth τ = P as m 4y (n; τ) = E{y[n]y [n + τ]y[n]y [n + τ]} = E{( x[]g[n P ])( x [j]g [n + τ jp]) j ( x[]g[n P])( x [l]g [n + τ lp])} l = E{x[]x [j]x[]x [l]} j l G4 n (P, jp τ,p,lp τ) (3) followng from the defnton of G4 n n (). We then further expand (3) usng the property that, for Alamout coded data, x[] s ndependent from x[ + ], = + E{x[]x []x[]x []}G n 4 (P, P τ,p,p τ) E{x[]x []x[]x [ +]} G4 n (P, P τ,p,( +)P τ) E{x[ +]x [ +]x[ +]x []} G4 n (( +)P, ( +)P τ,( +)P, P τ) (4) When x[n] s a sngle transmtter wth has an ndependent source stream, only the frst summaton n (4) s nonzero snce x[n] s ndependent from x[n+] n. hs mples there s not a fourth order cyclc frequency at P for a sngle transmtter, or for a spatal multplexng stream of ndependently modulated data from multple antennas. o llustrate the P perod n the Alamout coded sgnal, we see that the expanson, startng wth the second term, s m 4y (n + P; τ) = E{x[]x []x[]x [ +]} G4 n (( )P, ( )P τ,( )P, ( +)P τ) +... (5) = E{x[j]x [j]x[j]x [j +]} j=( ) G4 n (jp,jp τ,jp,(j +)P τ)+... =m 4y (n; τ) f = ±z,z Z. (6) Due to the structure of the Alamout space-tme bloc code, x[] and x[ +] are dependent when s even and ndependent when s odd, assumng zero-ndexed data. Note that n equaton (5), f s a multple of, we are guaranteed ndependence between x[] and x[ + ]. herefore, we see that m 4x (n+p ; τ) =m 4y (n; τ), whch shows that Alamout coded sgnals have a fourth order perodcty of length P. REFERENCES [] S. Hayn, Cogntve rado: bran-empowered wreless communcatons, IEEE Journal on Selected Areas n Communcatons, vol. 3, pp., Feb. 5. [] O. Dobre, A. Abd, Y. Bar-Ness, and W. Su, Cyclostatonarty-based blnd classfcaton of analog and dgtal modulatons, n Proc IEEE. Mltary Communcatons Conf., Oct. 6, pp. 7. [3] A. Mody, S. Blatt, D. Mlls,. Mcelwan, N. hammahoune, J. Nedzwec, M. Sherman, C. Myers, and P. Fore, Recent advances n cogntve communcatons, IEEE Communcatons Magazne, vol.45, pp. 54 6, Oct. 7. [4] G. J. Foschn and M. J. Gans, On lmts of wreless communcatons n a fadng envronment when usng multple antennas, Wreless Personal Communcatons, vol. 6, no. 3, pp , March 998. [5] S. Alamout, A smple transmt dversty technque for wreless communcatons, IEEE Journal on Selected Areas n Communcatons, vol. 6, pp , Oct [6] K. Km, I. A. Abar, K. K. Bae, J.-S. Um, C. M. Spooner, and J. H. Reed, Cyclostatonary approaches to sgnal detecton and classfcaton n cogntve rado, IEEE Internatonal Symposum on New Fronters n Dynamc Spectrum Access Networs, pp. 5, Apr. 7. [7] D. Cabrc, S. Mshra, and R. Brodersen, Implementaton ssues n spectrum sensng for cogntve rados, Proc. IEEE Aslomar Conference on Sgnals, Systems and Computers, vol., pp , Nov. 4. [8] R.W. Heath, Jr. and G.B. Gannas, Explotng nput cyclostatonarty for blnd channel dentfcaton n OFDM systems, IEEE ransactons on Sgnal Processng, vol. 47, pp , Mar [9] H. Bolcse and R.W. Heath, Jr. and A.J. Paulraj, Blnd channel dentfcaton and equalzaton n OFDM-based multantenna systems, IEEE ransactons on Sgnal Processng, vol. 5, pp. 96 9, Jan.. [] W. Gardner and C. Spooner, he cumulant theory of cyclostatonary tme-seres, Part I: Foundaton, IEEE ransactons on Sgnal Processng, vol. 4, pp , Dec [] L. Mazet and P. Loubaton, Cyclc correlaton based symbol rate estmaton, Proc. IEEE Aslomar Conference on Sgnals, Systems and Computers, vol., pp. 8, Oct [] P. Cblat, P. Loubaton, E. Serpedn, and G. Gannas, Asymptotc analyss of blnd cyclc correlaton-based symbol-rate estmators, IEEE ransactons on Informaton heory, vol. 48, pp , July. [3] A. Dandawate and G. Gannas, Statstcal tests for presence of cyclostatonarty, IEEE ransactons on Sgnal Processng, vol. 4, pp , Sept [4] H. Jafarhan, A quas-orthogonal space-tme bloc code, IEEE ransactons on Communcatons, vol. 49, pp. 4, Jan..