A COMPARISON OF ALTERNATIVE FILTERBANK MULTICARRIER METHODS FOR COGNITIVE RADIO SYSTEMS

Transcription

1 A COMPARISON OF ALTERNATIVE FILTERBANK MULTICARRIER METHODS FOR COGNITIVE RADIO SYSTEMS Peima Amii Uiversity of Utah, Salt Lake City, UT, USA + ; pamii@ece.utah.edu), Rolad Kempter + ; kempter@ece.utah.edu), Behrouz Farhag-Boroujey + ; farhag@ece.utah.edu) ABSTRACT We itroduce ad compare three filterbak-based commuicatio methods for cogitive radio systems. I the first method, called Filtered Multitoe FMT), subcarriers are arraged such that adjacet subbads do ot overlap. While this makes FMT robust to chael imparimets, it leads to cosiderable losses i badwidth efficiecy. Next, we discuss orthogoal frequecy divisio multiplexig with offset QAM OFDM-OQAM). OFDM-OQAM operates based o quadrature amplitude modulated QAM) symbols whose i-phase ad quadrature compoets are time offset by half the symbol period. The third method, which is referred to as cosie modulated multitoe CMT), is based o cosie modulated filterbaks. I CMT, the subcarrier symbols are pulse amplitude ad vestigial side-bad VSB) modulated. We preset filter desigs for all methods ad show that OFDM-OQAM offers highest stopbad atteuatio for a fixed filter legth ad umber of subcarriers. 1. INTRODUCTION Spectrum sesig as well as data trasmissio is a challegig problem i cogitive radio CR) systems. Sice the Fast Fourier trasform FFT) may be used for spectral aalysis as well as for modulatio/demodulatio, orthogoal frequecy divisio multiplexig OFDM) has bee suggested as a cadidate for multicarrier based CR systems [3. However, OFDM has a umber of shortcomigs i the CR eviromet. For istace, Weiss et al. [3 have studied the mutual iterferece betwee primary ad secodary users i a OFDM-based CR settig, ad have show that the reductio of such iterferece is oly possible by sacrificig a sigificat portio of the trasmissio badwidth. Furthermore, for reliable detectio of spectrum holes, the chael sesig mechaism eeds to feature a high spectral dyamic rage. Ufortuately, the FFT as part of a OFDM data trasmissio system is either able to fulfill this requiremet [2, or ca it meet the FCC s evisioed out-of-bad rejectio specificatios [5. As a cosequece, it is worthwhile to ivestigate alterative multicarrier/sigal processig methods which ca overcome the limitatios of the FFT/OFDM. Recetly, filterbak multitoe has bee proposed for CR systems for data trasmissio as well as chael sesig ad it has bee show that filterbak multitoe ca serve as a ear-optimal o-parametric spectrum aalysis tool [1, [2. Over the past three decades, three classes of filterbakbased multicarrier commuicatio systems have bee itroduced. Saltzberg [7 was the first to propose a filterbak multicarrier commuicatio system usig a special quadrature amplitude modulatio QAM) techique. Prior research that iitiated this developmet was performed by Chag [6. Developmets i digital subscriber lie DSL) techologies led to two other classes of filterbak multicarrier commuicatio systems; amely, filtered multitoe FMT) [10- [12 ad discrete wavelet multitoe DWMT) modulatio [13. Out of these, DWMT has bee further developed recetly ad reamed cosie-modulated multitoe CMT), [14. I the remaider of this paper, we preset a tutorial review of FMT, CMT ad OFDM-OQAM. Our itetio is to itroduce these three methods as alterative cadidates to the widely studied covetioal OFDM that suffers from sigificat leakage problems amog differet subcarrier bads [4. A filter desig is preseted for all three methods. 2. FILTERED MULTITONE, FMT I FMT, subcarriers are arraged such that adjacet subbads do ot overlap. As such, FMT may be see as a multicarrier commuicatio techique that follows the priciple of the legacy frequecy divisio multiplexig FDM) methodology to separate a high-rate data stream ito a umber of disjoit frequecy bads. FMT s distictive feature haresses the fact that either all or at least a cluster of subcarriers are trasmitted from the same source. Hece, those carriers ca be sychroized ad budled together as oe sigal for trasmissio. Moreover, polyphase techiques ca be used for efficiet implemetatio. However, we ote that i order to keep the subcarrier bads o-overlappig, excess badwidth has to be reserved to allow for a trasitio bad for each subcarrier. I [10 ad [12, where FMT was itroduced for Digital Subscriber Lie DSL) techology, the width of the subcarrier trasitio bad was kept at a miimum at the cost of sigal distortio. To compesate, oe decisio feedback equalizer per subcarrier was required at the receiver, addig sigificat complexity. Movig from DSL to CR applicatios, we believe that the use of equalizers at the receiver may be impractical because of two techical reaso: i) the added complexity may ot be justifiable, particularly, whe oe ca use CMT ad/or OFDM-OQAM that offer higher badwidth efficiecy ad lower complexity. ii) It may be hard, if ot impossible, to implemet a adaptive equalizer of the size metioed i [12) that ca cope with the time variatio of wireless chaels. Cosequetly, we believe that i a CR settig, FMT will suffer from sigificat badwidth loss due

2 to the large roll-off factors which have to be allowed i each subcarrier bad. 3. OFDM-OQAM I OFDM-OQAM, subcarrier bads are spaced by the symbol rate, 1/T. I cotrast to FMT, this results i sigificat overlap amog adjacet bads. Successful sigal separatio is evertheless possible thaks to a specific sigalig arragemet. A itroduced orthogoality coditio betwee subcarriers guaratees that the trasmitted symbols arrive at the receiver free of itersymbol ISI) ad itercarrier iterferece ICI). Carrier orthogoality is achieved through time staggerig the i-phase ad quadrature compoets of the subcarrier symbols ad desigig proper trasmit ad receive filters. I this sectio, we elaborate o these aspects of OFDM-OQAM. I a basebad equivalet of a OFDM-OQAM system as show i Figure 1, N parallel complex data streams are passed to N subcarrier trasmissio filters. The i-phase ad quadrature compoets are the staggered i time by half a symbol period, T/2. The outputs of these filters are the modulated usig N subcarrier modulators whose carrier frequecies are 1/T -spaced apart. Suppose that we have complex iput symbols accordig to x k = a k + jb k 1) where a k ad b k are the real ad imagiary parts of the th symbol, respectively. The complex-valued basebad OFDM- OQAM modulated sigal is defied as where s m t) = l= ot) = m=0 s m t)e jm ) 2) a m l ht lt ) + jb m l ht lt T/2. 3) Aalogously, the output of the receiver, ˆx k, cosists of the real ad imagiary compoets â k ad ˆb k, ˆx k = â k + jˆb k. 4) I 4), â k is foud as the real part of the sigal at the output of the correspodig matched filter with respose ht) ad expressed as â k = R[h t) pt) t=kt [ = R pt)h kt tdt [ = R ht kt )pt)dt where deotes covolutio, ad 5) pt) = e j ) ot) 6) Fig. 1. Basebad OFDM-OQAM Trasmitter is the demodulated sigal before matched filterig. Substitutig 2) ad 6) i 5), we obtai â k = R [ ht kt ) s m t)e jm ) ) dt. m=0 Furthermore, substitutig 3) i 7), we get â k = l= m=0 ) +jb m l ht lt T/2) e This ca be further rearraged as â k = l= m=0 [ Re ht kt ) a m l ht lt ) jm ) ) [ a m l ht kt )ht lt ) 7) dt.8) cos m ) T + π 2 b m l ht kt )ht lt T/2) si m ) dt.9) Chagig the variable t kt to t ad the takig ito accout

3 that l varies from to, 9) ca be simplified to [ â k = ht lt )ht) l= m=0 a l+k cos b l+kht lt T/2)ht) si m ) m ) T + π ) 2 ) dt. 10) The block diagram of a OFDM-OQAM receiver performig these operatios is depicted i Figure 2. Fig. 2. Basebad OFDM-OQAM Receiver Similarly, the imagiary part of the sigal ˆx k, ˆb k ca be foud as ˆb k = I[h t T/2) pt) t=kt. 11) Startig with 11) ad followig the same lie of derivatios as above, we obtai [ ˆb k = a l+k + ht lt )ht + T/2) l= m=0 si m ) T + π 2 +b l+kht lt + T/2)ht + T/2) cos m ) T + π dt. 12) 2 We ow proceed to discuss the desig of the matched filter, ht). I a ideal trasmissio system, the received sigal equals the trasmitted oe ad â k = a k ˆb k = b k. 13) It follows directly from 10) ad 12), that 13) ca be met, if ht) is chose such that the equalities i 14)-17) are satisfied. For coveiece of the desig, it is commo to costrai ht) to a real ad eve i.e., symmetric aroud t = 0) fuctio of time, t. Uder these costraits, oe ca show that the itegrad i 15) is ati-symmetric aroud t = lt/2 + T/4 ad the itegrad i 16) is ati-symmetric aroud t = lt/2 T/4. These, i tur, imply that 15) ad 16) are automatically satisfied. Hece, i desigig ht), it is sufficiet to limit ourselves to the costraits imposed by 14) ad 17). Also, to further simplify the desig of ht), it is reasoable to assume that oly adjacet subcarrier bads may overlap. Whe this is the case, oly istaces of m ad where m = 0, +1, 1 i 14) ad 17) eed to be cosidered. Values of m ad where m > 1 are related to o-adjacet subcarrier bads ad thus, their multiplicatio i 14) results i values close to) zero. Also, for m = 0, ad iff ht) is a root-nyquist filter, 14) equates to ht lt )ht)dt = δ0, l) = δl). 18) Fially, for m = ±1, 14) reduces to ) ht lt )ht) si = 0 19) T for ay real ad eve choice of ht). I a similar way, ad uder the same coditios, oe ca show that 17) reduces to 18). I summary, ay realizatio of ht) which is eve ad real ad satisfies the root-nyquist coditio i 18), leads to a OFDM-OQAM trasceiver system which satisfies 13). A example of such a desig is give i Sectio 5-A. 4. COSINE MODULATED MULTITONE, CMT Figure 3 presets the structure of a CMT multicarrier system. A sythesis filterbak is used to badlimit a set of PAM symbols to vestigial sidebad sigals ad modulate them to various frequecy bads. This process is outlied graphically i Figure 4. Fudametally, vestigial sidebad filterig is performed through a frequecy shifted versio of a lowpass filter ht), cetered at f = π/2t with impulse respose ht)e j π 2T t. The trasmit sigal, correspodig to the positive part of the frequecy axis, has the form s + mt) = ) a kht T )e j π 2T t T ) e jω kt+θ k ) k 20) where ω k s represets the carrier frequecy of the kth subcarrier bad ad θ k is a phase shift whose role will be explaied below. We recall that while the trasmit sigal is s m t) = R[s + mt), for the purpose of our discussio, we limit our attetio to the use of s + mt). To extract the kth subcarrier data sequece a k, s mt) is first multiplied by e jω k+θ k ). The result is the passed

4 ht lt )ht) cos m ) ht lt T/2)ht) si m ) ht lt )ht + T/2) si m ) ht lt + T/2)ht + T/2) cos m ) dt = δm, l) 14) dt = 0 15) dt = 0 16) dt = δm, l) 17) Fig. 3. Basebad CMT Tras-multiplexer ad k 1th subcarriers. Deotig the correspodig sigal compoets by y i t), i = 0, +1, 1, we have π 2T where x 0 t) = y i t) = x i t) h t)e j π 2T t 21) a kht T )e j π 2T t T ) ), 22) Fig. 4. CMT Modulatio through a lowpass filter, whose respose is matched to the trasmit filter ht)e j π 2T t, viz. h t)e j π 2T t. Assumig that the matched filter has a high stopbad atteuatio, the matched filter output will oly have three sigificat sigal compoets. These compoets origiate from the kth, k + 1th ad x 1 t) = x 1 t) = a k+1ht T )e j π 2T t T ) ) e j[ω k+1 ω k )t+θ k+1 θ k, 23) a k 1ht T )e j π 2T t T ) ) e j[ω k 1 ω k )t+θ k 1 θ k. 24) I CMT, the phase agle θ ca be expressed as θ k = 1) k π 4 ad the subcarrier bads are spaced ω k ω k 1 = π/t apart.

5 Substitutig these i the above set of equatios, we get ) x 1 t) = ±j a k+1ht T )e j π 2T t T ) e j π T t 25) ad x 1 t) = j a k 1ht T )e j π 2T t T ) ) e j π T t. 26) Here, the + or sigs deped o k beig either eve or odd. Usig the above results, we obtai ) y 0 t) = a kht T )e j π 2T t T ) h t)e j π 2T t = a kpt T )e j π 2T t T ) 27) where pt) = ht) h t). We ote that the complex sigal at the output of the matched filter is cetered at π/2t. Hece, takig the real part of the filter output results i the origial sigal, ad samplig it at the time istats T costitutes ear-perfect recostructio without ay ISI. Also, passig x 1 t) through the matched filter h t)e j π 2T t results i y 1 t) = ±j = ±j a k+1ht T )e j π 2T t T ) ) e j π T t h t)e j π 2T t a kqt T ) 28) where qt) = ht)e j π 2T t) e j π T t h t)e j π 2T t which ca be calculated as qt) = e j π 2T t + hτ)hτ t)e j π T τ dτ. 29) It is easy to show that qt ) is real. This i tur meas that y 1 t) is strictly imagiary, ad thus has o effect o the output of the the CMT demodulator where the real part of the filter output is used oly. I much the same way, oe ca show that y 1 t), too, is a imagiary umber. As a result, with the chose θ k ad ω k, ICI ca be avoided completely. 5. A COMPARISON OF FMT, OFDM-OQAM AND CMT I OFDM-OQAM, each subcarrier bad is double sidebad modulated ad carries a sequece of QAM i.e., complex-valued) symbols. Opposed to this, i CMT, subcarrier modulatio is vestigial sidebad ad the subcarriers carry a sequece of PAM i.e., real-valued) symbols. Therefore, assumig idetical symbol duratio ad umber of sub-carriers, the CMT sigal occupies half the badwidth of OFDM- OQAM of course, oly providig half of its data rate. FMT, o the other had, itroduces guard bads betwee adjacet subcarriers which are complex modulated. The width of the guardbads depeds o the specific system implemetatio. Therefore, for a idetical umber of carriers ad idetical symbol timig, FMT requires more badwidth tha OFDM- OQAM ad CMT. This relatioship is show i Figure 5. a) FMT b) OFDM-OQAM c) CMT Fig. 5. A. Numerical Results 2π R= T 2π R= T NR π R /2= T NR1+a) NR/2 a>0 Compariso betwee CMT, FMT ad OFDM-OQAM I this sectio, we desig filters for FMT, OFDM-OQAM, ad CMT with a stopbad atteuatio of 60 db for a multicarrier system with N = 16 subcarriers. We recall that while OFDM-OQAM filters are Nyquist N filters, CMT filters are Nyquist 2N filters. We assume idetical roll-off factors of β = 1 for both, CMT ad OQAM. I the case of FMT, we use a Nyquist N filter ad assume a roll off factor of β = 0.25 to avoid loosig more tha 25% of badwidth to guardbads. As the desig method, we use a iterative least square algorithm [17. The resultig filter resposes are preseted i Figure 6. With N = 16 subcarriers ad a desired stopbad atteuatio of 60 db, the required OFDM-OQAM filter legth is 4N = 64, the CMT filter legth is 8N = 128, ad FMT requires a filter legth of 16N = 256. The reaso why the filter legth for CMT is twice that of OFDM-OQAM lies i the differet badwidth of the subcarriers, requirig half the trasitio bad ad thus twice the filter legth i the case of CMT. However, as far as FMT is cocered, a rolloff factor of β = 0.25 results i a trasitio bad which is 4 times smaller tha that of OFDM-OQAM, hece requirig 4 times its filter legth. 6. THE EFFECT OF THE CHANNEL ON FILTERBANK MULTICARRIER SYSTEMS I covetioal OFDM, a frequecy selective chael ca be coverted ito a umber of subcarrier chaels with the aid of the cyclic prefix. Each of these subcarrier chaels ca the be modeled as a flat chael, i.e., a chael that ca be f f f

6 Fig. 6. Compariso betwee CMT ad OFDM-OQAM prototype filters characterized by a costat complex) gai. Accordigly, at the receiver, distortio ca be compesated i each subcarrier bad by applyig a sigle-tap equalizer whose gai is the iverse of the chael gai. The same is true i filterbakbased multicarrier commuicatio systems. Here, by reducig the width of each subcarrier bad through icreasig the umber of subcarriers, N), subcarrier chaels ca be modeled as arrow-bad, o-frequecy selective chaels. As a cosequece, here also, equalizatio becomes trivial ad ca be performed through a sigle-tap equalizer. 7. CONCLUSION AND OUTLOOK INTO FUTURE RESEARCH We discussed ad compared OFDM-OQAM, CMT ad FMT as multicarrier commuicatio techiques for CR systems. We derived the orthogoality coditio for OFDM- OQAM ad showed that of all multicarrier methodologies, for a fixed filter legth, OFDM-OQAM achieves the highest stopbad atteuatio. However, while iferior i terms of spectral leakage, CMT offers better frequecy selectivity. For both methods, more research is ecessary to quatify the effects of imperfect timig phase ad carrier recovery o system performace. 8. REFERENCES [1 P. Amii, R. Kempter, R. R. Che, L. Li ad B. Farhag-Boroujey, Filter Bak Multitoe: A Physical Layer Cadidate for Cogitive Radios, Software Defied Radio Techical Coferece, SDR 2005, November 14-18, [2 B. Farhag-Boroujey ad R. Kempter, Multicarrier Commuicatio Techiques for Spectrum Sesig ad Commuicatio i Cogitive Radios, submitted to IEEE Commuicatios Magazie [3 T.A. Weiss ad F.K. Jodral, Spectrum Poolig: A Iovative Strategy For The Ehacemet Of Spectrum Efficiecy, IEEE Commuicatios Magazie, Vol. 42, No. 3, March 2004, pp. S8 - S14. [4 T.A. Weiss, J. Hillebrad, A. Kroh, ad F.K. Jodral, Mutual Iterferece I OFDM-Based Spectrum Poolig Systems, IEEE 59th Vehicular Techology Coferece, VTC 2004-Sprig, 2004, vol. 4, May 17-19, pp [5 Federal Commuicatios Commissio, Spectrum Policy Task Force, rep. ET Docket o , Nov [6 R.W. Chag, High-speed Multichael Data Trasmissio With Badlimited Orthogoal Sigals, Bell Sys. Tech. J., vol. 45, pp , Dec [7 B.R. Saltzberg, Performace of a Efficiet Parallel Data Trasmissio System, IEEE Trasactio o Commuicatio Techiques, vol. 15, o. 6, pp , Dec [8 B. Hirosaki, A Orthogoally Multiplexed QAM System Usig the Discrete Fourier Trasform, IEEE Trasactios o Commuicatios, Volume 29, Issue 7, Jul 1981 Pages): [9 M. Bellec, P. Pirat, OQAM Performaces Ad Complexity, cotributios, doc., IEEE /0004r0 [10 G. Cherubii, E. Eleftheriou, S. Olcer, J.M. Cioffi, Filter Bak Modulatio Techiques For Very High Speed Digital Subscriber Lies, IEEE Commuicatios Magazie., vol. 38, o. 5, pp , May [11 G. Cherubii, E. Eleftheriou, S. Olcer, Filtered Multitoe Modulatio For Very High-Speed Digital Subscriber Lies, IEEE Joural o Selected Areas i Commuicatios, vol. 20, o. 5, pp , Jue [12 G. Cherubii, E. Eleftheriou, S. Olcer, Filtered Multitoe Modulatio for VDSL, i Proc. IEEE Globecom 99, vol. 2, pp , [13 S.D. Sadberg ad M.A. Tzaes, Overlapped Discrete Multitoe Modulatio for High Speed Copper Wire Commuicatios, IEEE Joural o Selected Areas i Commuicatios, vol. 13, o. 9, pp , Dec [14 B. Farhag-Boroujey, Multicarrier modulatio with blid detectio capability usig cosie modulated filter baks, IEEE Trasactios o Commuicatios, vol. 51,o. 12, pp , Dec [15 L. Li ad B. Farhag-Boroujey, Cosie modulated multitoe for very high-speed digital subscriber lies, EURASIP, Accepted for publicatio. [16 Rossi, M., Ji-Yu Zhag, Steeaart, W., Iterative least squares desig of perfect recostructio QMF baks, IEEE Caadia Coferece o Electrical ad Computer Egieerig, 1996, [17 B. Farhag-Boroujei, A Uiversal Square-rout Nyquiest M) Filter Desig for Digital Commuicatio Systems, Software Defied Radio Techical Coferece, SDR 2006, November 13-17, 2006