Analysis of an FBMC/OQAM scheme for asynchronous access in wireless communications

Transcription

1 Mattera et al. EURASIP Joural o Advaces i Sigal Processig (2015) 2015:23 DOI /s RESEARCH Ope Access Aalysis of a FBMC/OQAM scheme for asychroous access i wireless commuicatios Davide Mattera 1, Mario Tada 1* ad Maurice Bellager 2 Abstract The OFDM/OQAM trasceiver belogs to the filter-bak-based multicarrier (FBMC) family ad, ulike OFDM schemes, it is particularly able to meet the requiremets of the physical layer of cogitive radio etworks such as high level of adjacet chael leakage ratio ad asychroous commuicatios. The paper proposes ad aalyzes a ew implemetatio structure, amed frequecy spreadig, for the OFDM/OQAM trasceiver. O flat chaels, it is equivalet to the stadard oe i terms of iput-output relatios, though more complex. O multipath chaels, it offers a crucial advatage i terms of equalizatio, which is performed i the frequecy domai, leadig to high performace ad o additioal delay. With its flexibility ad level of performace, the aalyzed scheme has the potetial to outperform OFDM i the asychroous access cotext ad i cogitive radio etworks. Keywords: OFDM/OQAM; Multicarrier systems; Prototype filter; FBMC; Asychroous access; Cogitive radio 1 Itroductio The cogitive radio trasmissio cotext exhibits a umber of specific features which make it sigificatly differet from the covetioal trasmissio eviromet 1,2. First, the available badwidth is likely to be fragmeted, i.e., it is made of o-adjacet spectrum chuks that have to be exploited joitly for high speed data commuicatios. The, the sectios of the spectrum that are ot available might be occupied by a primary user ad a high level of protectio must be provided. Specifically, the trasmissio system must guaratee a high level of adjacet chael leakage ratio (ACLR) a. Next, the trasmissio bad is likely to be chagig o short otice or eve without otice. O the exploitatio side, the total badwidth available might be dedicated to a sigle user requirig high bit-rates or it ca be dyamically shared by several users i proportio to their istataeous capacity eeds. If opportuistic operatio is cotemplated, these users have the freedom to show up ad disappear as they wish. I such coditios, a rigid commuicatio procedure, where each user must be aliged before the trasmissio ca start, is iadequate. I fact, asychroous operatio is ecessary to reach a acceptable *Correspodece: tada@uia.it 1 Dipartimeto di Igegeria Elettrica e delle Tecologie dell Iformazioe, Uiversità degli Studi di Napoli Federico II, via Claudio 21, Napoli, Italy Full list of author iformatio is available at the ed of the article level of spectral efficiecy. Clearly, to cope with such a cotext, a appropriate physical layer is required. The spectrum graularity offered by multicarrier trasmissio techiques has prove its efficiecy for spectrum exploitatio, ad the most popular techique, orthogoal frequecy divisio multiplexig (OFDM), has bee widely used i commuicatios for more tha a decade ow. However, for the cogitive radio cotext as described above, it lacks flexibility ad it is likely to lead to poor spectral efficiecy, eve with the itroductio of additioal processig 3. Therefore, a ehaced multicarrier techique is eeded, as poited out i 4, where it is show that a filter-bak-based multicarrier (FBMC) physical layer ca meet the ACLR requiremets 2,4. I particular, FBMC/OQAM ca overcome the limits of OFDM provided that we impose a costrait o the cut-off frequecy of the prototype filter, which caot exceed the sub-chael spacig as poited out i 5. High stop-bad atteuatio filters have bee proposed that do ot satisfy this costrait, such as isotropic orthogoal trasform algorithm (IOTA) 6 ad Hermite filters. These filters are associated with sigle-tap equalizers as metioed i 7: such a sceario does ot allow the exploitatio of the potetialadvatagesoffbmc/oqamsystemsforcogitive ad, therefore, they caot compete with CP-OFDM. O the cotrary, the high performace equalizatio objectivecabemetiftheprototypefilter(employedithe 2015 Mattera et al.; licesee Spriger. This is a Ope Access article distributed uder the terms of the Creative Commos Attributio Licese ( which permits urestricted use, distributio, ad reproductio i ay medium, provided the origial work is properly credited.

2 Mattera et al. EURASIP Joural o Advaces i Sigal Processig (2015) 2015:23 Page 2 of 22 FBMC/OQAM scheme) is desiged usig the frequecy samplig techique itroduced i 8 ad developed i 5. With this simple approach, the coefficiets are derived from a few samples of the filter frequecy respose, which makes the implemetatio of the filter bak i the frequecy domai practical. Whe compariso betwee such FBMC/OQAM system ad OFDM system is cosidered, it appears that a key advatage of OFDM with cyclic prefix (CP) is the capability to achieve perfect chael equalizatio, as log as the chael impulse respose remais shorter tha the guard time provided by the CP. Thus, i order to be accepted, the FBMC/OQAM approach must have a high performace equalizatio capability, particularly i the asychroous cotext, characterized by the fact that the system must compesate simultaeously the timig offset, the frequecy offset, ad the chael distortio. I the absece of CP, the equalizatio capability of the FBMC system rests o the sub-chael equalizers, which caot be sigle-tap as i OFDM, but must be multitap to reach similar performace. However, the use of multitap equalizers implies a icrease of the receiver latecy; this motivates the search for a equalizer structure that does ot itroduce such a disadvatage. The mai cotributio of the paper lies i the proposal of a ew trasceiver structure for FBMC/OQAM systems that is able to provide satisfactory performace without icreasig the trasceiver delay ad acceptig the presece of sigificat timig ad frequecy offsets amog the users that are performig the multiple access, as it is commo i a cogitive radio sceario. The cocept has bee preseted i 9,10, alog with prelimiary performace results, uder the ame frequecy spreadig (FS)-FBMC, but a rigorous aalysis of the correspodig scheme is still missig. A objective of the preset paper is to provide such a aalysis ad prove the equivalece of FS-FBMC with the covetioal polyphase etwork (PPN)-FFT scheme i both trasmitter ad receiver. This equivalece is importat because it opes the way to mixed implemetatios. For example, i uplik trasmissio, the distat user ca be equipped with the covetioal IFFT-PPN trasmitter, while the high performace but more complex FS-FBMC receiver is implemeted at the base statio. May advaces i the applicatios of FBMC to various scearios will be able to take advatages from the proposed trasceiver structure. I particular, the capability to use multiple ateas at the trasmitter ad/or at the receiver, which sigificatly icreases badwidth efficiecy, ca be easily carried out alog the lies itroduced i 11-13, which however do ot take ito accout the frequecy-despreadig structure at each receiver; further works is eeded to defie the details of the MIMO extesio of the proposed structure. Alterative structures are also uder cosideratio for achievig the same goal of operatig o multipath chael with a miimum implemetatio complexity. For example, the fastcovolutio structure 14, which is curretly uder study for its extesio o the multipath chael, is superior to the proposed structure i terms of computatioal complexity. It is equivalet i terms of flexibility (e.g., it shows a similar capability to easily compesate a time-offset i the frequecy-domai as suggested i 15), while the fast-covolutio trasceiver latecy is larger tha that achieved by the frequecy-despreadig system 14. The orgaizatio of the paper is as follows. I Sectio 2, the FS-FBMC scheme for the trasmitter is described ad the proof of the equivalece with the stadard FBMC, amely the IFFT-PPN cascade, is provided. Sectio 3 is dedicated to the receiver structures ad, agai, the proof of the equivalece betwee FS-FBMC ad stadard FBMC, amely the cascade of PPN ad FFT, is provided; moreover, i Sectio 3, it is show that the FS- FBMC structure is computatioally more complex while i Sectio 4, it is show that o a multipath chael, it offers a crucial advatage i terms of equalizatio, which is performed i the frequecy domai, just like OFDM, leadig to high performace ad o additioal delay. I Sectio 5, the performace of the proposed scheme is illustrated ad cotrasted with the results obtaied for OFDM ad stadard FBMC whe the sub-chael equalizer has a sigle-tap. I Sectio 6, the mai aspects of FS-FBMC are summarized ad the potetial impact is discussed. Notatio: j = 1, superscripts (), () T,ad() H deote the complex cojugatio, the traspose, ad the cojugate traspose, respectively, R istherealpart,log is the base-2 logarithm, is the liear covolutio, δkis thekroeckerdelta,ceilx is the smallest iteger larger tha or equal to x, ad deotes the time average, i.e., x = 1 lim N= N N + 2N+1 x ad mod M (l) = l qm with q such that mod M (l) {0, 1,..., M 1}. Moreover, we deote with DFTxthevector x whose kth compoet ca be writte as x k = N 1 N 1 x ie j 2π N ki ad with IDFT x thevector x whose kth compoet ca be writte as ˆx k = N 1 x ie j 2π N ki where x i is the ith compoet of the N 1iputvectorx. Fially, lowercase boldface letters deote colum vectors, the compoetwise product betwee two vectors ad, fially, 0 deotes the ull vector. 2 The trasmitter with stadard ad frequecy-spreadig structures Let us cosider a FBMC system usig offset QAM modulatio, ofte desigated by OFDM-OQAM 16. We

3 Mattera et al. EURASIP Joural o Advaces i Sigal Processig (2015) 2015:23 Page 3 of 22 assume that the umber M of subcarriers be eve; the low-pass trasmitted sigal s(t) ca be writte as with s(t) = s R (t) + js I (t T/2) (1) s R (t) = s I (t) = N b +N s 1 =0 N b +N s 1 =0 k A k A a R ejk( 2π T t+ π ) 2 g(t T) (2) a I ejk( 2π T t+ π ) 2 g (t T) (3) where T is the multicarrier symbol iterval, A {0, 1,..., M 1} is the set of active subcarriers whose size is M u, the sequeces a R ad ai idicate the real ad imagiary parts of the complex data symbols trasmitted o the kth subcarrier durig the th QAM symbol, N b is the umber of traiig symbols, N s is the umber of payload symbols, while g(t) is the prototype filter. It is assumed that the data symbols a R ad ai are statistically idepedet with zero-mea ad variace σa 2. The discrete-time low-pass versio si = s(t) t=its of the trasmitted sigal (T s = T/M is the samplig iterval) ca be writte as si = s R (it s ) + js I ((i M/2)T s ). (4) I the ext subsectios, we report the derivatio of a efficiet geeratio procedure for the sigal s R (t). A aalogous derivatio ca be straightforwardly obtaied for the sigal s I (t). Sice the cotiuous time sigal is geerated by D/A coversio, we cosider the geeratio of its discrete-time samples s R i = s R (it s ) = N b +N s 1 =0 k A a R ejk( 2π M i+ π ) 2 g i M where we have used Equatio 2 ad the followig defiitio (5) gi = g(it s ). (6) The geeratio of the sequece s R i is equivalet to the geeratio of the sequece of M 1vectorsd (R) whose kth compoet d (R) is equal to sr M + k fork {0, 1,..., M 1}. I the followig, we cosider two implemetatio structures ad their implemetatio complexities: though the stadard implemetatio structure based o a IFFT over M poits exhibits a reduced computatioal complexity, the frequecy-spreadig structure based o a IFFT over a larger umber of poits provides useful isights ito the structure of the trasmitted sigal. 2.1 Stadard trasmitter structure The stadard structure, ofte amed the polyphase etwork, for the implemetatio of the OFDM/OQAM trasceiver has bee first proposed i 17,18. To make clear its compariso with the proposed alterative, we briefly recall its derivatio here. The kth compoet d (R) s R M + kofd (R) ca be writte as Nb+Ns 1 d (R) = where b (R) =0 = k A = b (R),k g k + ( )M k {0, 1,..., M 1} (j k a R ) e jk 2π M k k {0, 1,..., M 1} (8) which is the IDFT of the sequece j k a R with respect to the idex k. If we defie the vector b (R) as the M 1vector whose kth compoet (for k {0, 1,..., M 1})isb (R) i Equatio 8, we ca compactly write b (R) = IDFT w a (R) where IDFT deotes the IDFT operator o the iput vector ad, for k A, thekth compoet w k of the M 1-vector w is w k = j k (10) ad the kth compoet of the vector a (R) is the symbol a R i Equatio 2 while, for k / A, w k = 0adthecompoets of a (R) are irrelevat. Note that Equatio 9 is oly defied for {0, 1,..., N b +N s 1}, but we ca straightforwardly exted it to ay provided that we assume that a (R) (7) (9) b (R) 0 / {0, 1,..., N b + N s 1}. (11) We ca compactly write Equatio 7 as N b +N s 1 d (R) = b (R) g (12) =0 where the vector g is defied so that its kth compoet g is g = gk + M k {0, 1,..., M 1}. (13) Therefore, Equatio 13 defies the polyphase compoets of g. The prototype filter g(t) satisfies the followig property g(t) 0 t / 0, KT) (14)

4 Mattera et al. EURASIP Joural o Advaces i Sigal Processig (2015) 2015:23 Page 4 of 22 where K is the overlap parameter. The property i Equatio 14 implies that the vector g is oull oly for {0, 1,..., K 1}. Cosequetly, Equatio 12 ca be rewritte as d (R) = g 0 b (R) + g 1 b (R) g K 1 b (R) (K 1). (15) Aalogously, the geeratio of the sequece s (I) i i Equatio 4 is equivalet to the geeratio of the sequece of vectors d (I) defied as the output of the PPN: d (I) with b (I) = g 0 b (I) +g 1 b (I) g K 1 b (I) (K 1) (16) = IDFT w a (I) where the kth compoet (k A) ofthevectora (I) symbol a I i Equatio 3. Agai, we have assumed a (I) (17) is the b(i) 0 / {0, 1,..., N b + N s 1}. (18) 2.2 Frequecy-spreadig structure I the preset subsectio, we derive a alterative structure for implemetig the OFDM/OQAM trasmitter, amed the frequecy-spreadig structure. By usig Equatios 7 ad 8, i Appedix A it is show that the kth compoet d (R) of d(r) ca be writte as: d (R) = h + h 1,k+M + h 2,k+2M + h 3,k+3M h (K 1),k+(K 1)M (19) wherewehavedefied h m,p = g p m,k e j 2π M pk ad,k = { j k k A a R,k p {0, 1,..., K}. (20) {0, 1,..., N b + N s 1} ad k A 0 / {0, 1,..., N b + N s 1} ad k (21) Equatio 19, which has to be applied for k {0, 1,..., }, is similar to that preset i the classic fast-covolutio procedure, usually amed overlap-ad-add structure; here, it operates o the sequece of KM 1vectorsh, defied as follows: h m = hm,0 h m,1...h m,k T (22) More specifically, let us first defie the vectors h (i) m such that h (i) m = h m,im h m,im+1 h m,im+2... h m,im+ T (23) i {0, 1,..., K 1}. From Equatios 22 ad 23, it follows that T h m = h (0)T m h(1)t m... h (K 1)T m. (24) Usig Equatios 23 ad 24, Equatio 19 ca be re-writte as d (R) = h (0) + h(1) 1 + h(2) 2 ++h(k 1) (K 1). (25) We have ow to study the structure of the vector h i Equatio 24 for a geeral time-step i order to simplify its geeratio. From Equatio 20, it follows that its kth compoet h ca be writte as h = g k = = k A K 1 k = (K 1) K 1 k = (K 1) e j 2π M kk k {0, 1,..., KM 1} { } G k e j KM 2π kk e j KM 2π Kkk k A { K G k e j KM 2π kk m=0 } c (KR) 2π,m ej KM km (26) where we have assumed that the prototype filter has bee desiged accordig to a frequecy-samplig approach so that its Fourier trasform G(F) = K satisfies the property G(F) F= k KM gi e j2πfi (27) = 0 k {K, K + 1,..., KM K}. (28) The Fourier coefficiets i Equatio 26 are give by 1 G k = KM G(F) k. (29) F= k KM Note that the 2K 1 oull values {G k } K 1 k= (K 1) are the free parameters of the prototype filter whe it is desiged accordig to the frequecy-samplig procedure used i 5. Furthermore, i Equatio 26, we have also itroduced the followig defiitio of the K-times upsampled versio of i Equatio 21: { c (KR),m = m = kk 0 otherwise. (30) From Equatio 26, it follows that h = IDFTG IDFTc = IDFTz (31)

5 Mattera et al. EURASIP Joural o Advaces i Sigal Processig (2015) 2015:23 Page 5 of 22 where the vectors G ad c i Equatio 31 are defied as follows: G =G 0 G 1 G 2... G K T (32) = G 0 G 1...G K KM 2K+1 G KM (K 1)...G K T c =,0 0 1 (K 1),1 0 1 (K 1),2 0 1 (K 1)..., 0 1 (K 1) (33) T (34) ad the vector z i Equatio 31 for the well-kow property of the DFT ad IDFT operators is the circular covolutio of the two KM 1vectorsGad c.wehave used the property i Equatio 28 to write Equatio 33. I Appedix B, it is show that the compoets of the vector z i Equatio 31 ca be writte as z,pk+k = j p a R,p G k + ja R,mod M (p+1) G k K (1 δk ) (35) where p {0, 1,..., M} ad k {0, 1,..., K 1}. Therefore, the frequecy-spreadig structure requires: 1. to use the iput symbols a R to calculate (for l {0, 1,..., KM 1}) the compoets z,l of z accordig to Equatio 35. Note that the symbol a R,p is spread over 2K 1 compoets of the vector z ad for this reaso, the structure is amed FS-FBMC; i fact, each compoet of z is depedet o two adjacet symbols ad each symbol a,p, accordig to Equatio 35, ot oly determies the compoet pk of the frequecy-domai vector z but also spreads its effect, weighted by the frequecy respose of the prototype filter, o the differet compoets of the same vector ragig from pk (K 1) up to pk + K 1; 2. to determie h startig from z by performig the IDFT over KM poits i the right-had side of Equatio 31; 3. to evaluate d (R) by the overlap-ad-add processig defied i Equatios 24 ad Complexity compariso of the two structures The stadard trasmitter structure requires to calculate (a) the IFFT over M samples accordig to the defiitio accordig to the PPN (Equatio 15). The frequecy-spreadig structure requiremets have bee just summarized. Iastructurewithasigleprocessor,thecomplexity compariso is equivalet to the cout of the umber of flops required by the two structures. The umber of complex flops for calculatig the IFFT over M samples ca be i Equatio 9 the to calculate the vector d (R) writte as 1.5M log(m) while the umber of real flops ca be writte b as 4M log(m) 6M + 8 as i the split-radix. The umber of complex multiplicatios for calculatig IFFT over M samples ca be writte as 0.5M (log ) while the umber of real multiplicatios ca be writte as M log(m) 3M by removig most of the trivial operatios ad usig three real multiplicatios per complex multiplicatio. accordig to the PPN (Equatio 15), the followig umber of real-valued flops are ecessary: 2M real-valued multiplicatios for each of the K terms ad M complex-valued additios for each of the K 1 couples of vectors to be summed. With the frequecy-spreadig structure, we eed 2KM M real-valued multiplicatios for calculatig z accordig to Equatio 35 while the IFFT for calculatig h by usig Equatio 31 requires 4KM log(km) 6KM + 8 real-valued flops. Fially, (K 1)M complex-valued additios are eeded by the overlap-ad-sum structure i Equatio 25. To calculate the vector d (R) Therefore, for geeratig each vector d (R) stadard structure, we eed C (ST) f real-valued multiplicatios where C (ST) m C (ST) f = 4M log(m) 6M MK + 2M(K 1) }{{}}{{} (9) (15) = 8 + M 4log(M) + 4K 8 C (ST) m, with the real-valued flops or = M log(m) 3M MK }{{}}{{} = M log(m) + 2K (9) (15) (36) while with the frequecy-spreadig structure, we eed C (FT) f real-valued flops or C m (FT) real-valued multiplicatios where C (FT) f C (FT) m = 2KM M + 4KM log(km) 6KM + 8 }{{}}{{} (35) = 8 + M = 2KM M }{{} (35) (31) { 4K log(km) 0.5 } 3 + KM log(km) 3KM + 4 }{{} (31) { = M K log(k) + log(m) 1 } Note that C (FT) f MK 4log(M) + 4log(K) 2 MK 4log(M) + 4K 8 = KC (ST) f. + 2(K 1)M }{{} (25) (37) (38) The first approximatio is obvious while the secod approximatio holds provided that M is sufficietly large ad K sufficietly small (e.g., for K = 4adM = 1024, we

6 Mattera et al. EURASIP Joural o Advaces i Sigal Processig (2015) 2015:23 Page 6 of 22 have a ormalized approximatio error of 6.1%). Moreover, C m (ST) /C m (FT) is aroud 58% for K = 2, 35% for K = 4, ad 24% for K = 8, maily idepedetly of M {512,1024,2048,4096}. Therefore, the frequecyspreadig structure is about K times more complex whe implemeted usig a structure with a sigle processor. 3 The receiver for stadard ad frequecy-spreadig structures Accordig to Equatios 4 ad 5, assumig perfect sychroizatio o the flat chael (i.e., the chael with uit respose i the frequecy domai) ad eglectig the presece of the oise, the received sigal ri cabe writte as follows ri = α N b +N s 1 =0 k A { a R ejk( 2π M i+ π ) 2 g i M + ja I ( 1)k e jk( 2π M i+ π ) } 2 g i M M/2 = αa R ejk( 2π M i+ π ) 2 gi M +o (R) i (39) = jαa I ( 1)k e jk( 2π M i+ π ) 2 g i M M/2 + o (I) i (40) where 0 < α < 1 deotes the chael gai ad o (R) i deotes terms preset i the trasmitted sigal that do ot deped o a R while o(i) i deotes terms preset i the trasmitted sigal that do ot deped o a I,with {0, 1,..., N b + N s } ad k A. We cosider at the receiver the decisio variables: â R â I { } =R D (R) { } =R D (I) D (R) = 1 A D (I) = j A + i= + i= ri e jk( 2π M i+ π ) 2 gi M ri ( 1) k e jk( 2π M i+ π ) 2 gi M M/2 (41) (42) where A is a proper costat amplitude, defied so that â (R) = ar whe o(r) i = 0adâ (I) = ai whe o(i) i = 0. Cosequetly, A = αe g (43) with E g = K g 2 i. Equatio 41 is motivated by a proper desig of the prototype filter g thatguaratees a egligible projectio of the iterferece terms o (R) i o the matched filter, as usual i FBMC receiver 16,20,21: { + R i= o (R) i e jk( 2π M i+ π ) 2 gi M } 0. (44) I fact, accordig to Equatio 39, o (R) i deotesthe additive sigals preset i the received sigal ri thatdo ot deped o the useful symbol a R ; the coditio i Equatio 44 therefore implies that such additive sigals do ot iterfere with the useful sigal whe the matchedfilter projectio (desiged accordig to the useful term) is performed (i.e., the result of the matched-filter projectio is idepedet of the iterferig sigals); o the other had, the matched-filter projectio is optimum (i the maximum-likelihood sese) whe the iterferece sigals are ot preset ad oly the oisy versio of the useful term is take ito accout. Cosequetly, the decisio variable i Equatio 41 operatig o the flat chael without sychroizatio error is optimum (i the maximumlikelihood sese) for estimatig statistically idepedet iformatio symbols; aalogously, the same optimality holds for the decisio variable i Equatio 42. Obviously, the receiver implemeted accordig to Equatio 41 has to be modified i order to operate o a multipath chael. However, before discussig such modificatios, we first eed to describe the two structures implemetig Equatio 41. Cosequetly, we recall the stadard structure for implemetig Equatio 41 i Subsectio 3.1 ad we itroduce a alterative structure i Subsectio 3.2; moreover, we compare their complexities i Subsectio The stadard receiver structure I the preset subsectio, we briefly recall the derivatio of the stadard receiver structure. From Equatios 41 ad 14, it follows that D (R) = 1 A j k rm + i e jk 2π M i gi + r ( + 1)M + i e jk 2π M i gm + i + r ( + 2)M + i e jk 2π M i g 2M + i r ( + K 1)M + i e jk 2π M i g (K 1)M + i. (45) Let us itroduce the vector r whose ith compoet r,i (i {0, 1,..., M 1}) is defied as follows r,i = rm + i. (46) By usig also Equatio 13, let us cosider the vector r (g,) +l = r +l g l l {0, 1,..., K 1} (47)

7 Mattera et al. EURASIP Joural o Advaces i Sigal Processig (2015) 2015:23 Page 7 of 22 ad let us deote with r (g,) +l,i its ith compoet for i {0, 1,..., }. Usig such defiitios, from Equatio 45, it follows that Aj k D (R) = q=0 + r,q (g,) e jk 2π M q + q=0 Defiig the vector D (R) q=0 r (g,) +1,q e jk 2π M q r (g,) +2,q e jk 2π M q q=0 r (g,) +(K 1),q e jk 2π M q. (48) = D (R) T,0 D(R),1... D(R), cotaiig the decisio variables o the vector a (R), it follows from Equatio 48 that A D (R) w = M DFT or, equivaletly, D (R) + M DFT r (g,) + M DFT r (g,) +(K 1) K 1 = M DFT r +l g l l=0 = M w A DFT K 1 r (g,) (49) r +l g l. (50) l=0 Aalogously, i Appedix C, it is show that K 1 D (I) = M w ja DFT r (I) +l g l where the vector D (I) the vector a (I) D (I) = : l=0 (51) cotais the decisio variables o D (I) T,0 D(I),1... D(I), (52) while the ith compoet r (I),i (i {0, 1,..., M 1}) of the vector r (I) is defied as follows r (I),i = r M + M/2 + i. (53) 3.2 The frequecy-despreadig receiver structure I the preset subsectio, we derive the receiver couterpart of the frequecy-spreadig trasmitter described i Subsectio 2.2. From Equatio 41, it follows that for k {0, 1,..., M 1} D (R) = 1 K A j k rm + i e jk 2π M i = 1 K 1 A j k k = (K 1) K Gk K 1 k = (K 1) G k e j 2π KM ik 2π j rm + i e KM i(kk+k ) (54) where we have take ito accout the properties of the prototype filter (see Equatios 14, 28, ad 29) ad the fact that it is real ad, cosequetly, G k = Gk. Let us itroduce the KM 1vectorr (K) whose kth compoet r (K) is defied as r (K) = rm + k k {0, 1,..., KM 1} (55) ad the vector R,whosekth compoet is deoted with R, defied as the DFT over KM samples of the vector r (K) : R = DFT r (K). (56) Usig such defiitios, Equatio 54 ca be re-writte as D (R) = 1 A j k KM K 1 k = (K 1) G k R,mod KM (kk+k )k {0, 1,..., M 1}. (57) Let us defie the (2K 1) 1vectorR (FS) (k {0, 1,..., M 1}) R (FS) ad the vector T R,modKM (kk+(k 1)) = R,modKM (kk (K 1))... R,modKM (kk)... (58) G (K) = G (K 1)...G 1 G 0 G 1...G K 1 T. (59) Takig ito accout Equatios 58 ad 59, Equatio 57 becomes D (R) = KM A j k G (K)H R (FS). (60) Aalogously, i Appedix D, it is show that the decisio variable o a (I) ca be writte as D (I) = KM A j k 1 G (K)H R (I,FS) (61) where we defie the (2K 1) 1vectorR (I,FS) (k {0, 1,..., M 1}) as follows R (I,FS) The vector R (I) R (I) = R (I),mod KM (kk (K 1))... R(I),mod KM (kk)... = DFT R (I),mod KM (kk+(k 1)) T. r (I,K) (62) (63)

8 Mattera et al. EURASIP Joural o Advaces i Sigal Processig (2015) 2015:23 Page 8 of 22 whose kth compoet is deoted with R (I) adusedi Equatio 62 is defied as the DFT over KM samples of the vector r (I,K) r (I,K) whose kth compoet r (I,K) is defied as = r M + M/2 + k k {0, 1,..., KM 1}. (64) Therefore, Equatios 60 ad 61 defie the followig receiver structure, illustrated i Figure 1: 1. Collect the samples of the received sigal r toform the KM 1 vector r (K) i Equatio 55 (ad half period later the vector r (I,K) i Equatio 64); 2. Calculate the vector R i Equatio 56 by FFT over KM poits (ad half period later the vector R (I) i Equatio 63); 3. Perform M differet projectios of the vector R accordig to Equatios 60 ad 58 i order to obtai each decisio variable D (R) for k {0, 1,..., M 1} (ad half period later accordig to Equatios 61 ad 62 to obtai D (I) ). Thus, to obtai the datum a(r) that i the trasmitter has bee spread over 2K 1 compoets of the vector z (see the first poit of the setece after Equatio 35), the same compoets of the vector R are exploited by usig as weights the (cojugate) Fourier coefficiets Gk with k { (K 1),..., K 1}. For this reaso, such a structure is called frequecy-despreadig receiver: i fact, it collects all the compoets of the frequecy-domai vector R depedet o the useful symbol a R, due to the spreadig performed at the trasmitter, ad weights them accordig to the frequecy respose of the prototype filter, achievig the despreadig of the useful symbol. The importace of the proposed structure is ot limited by the assumptio i Equatio 28 because, whe it is ecessary to itroduce a possible mismatch (i.e., to use at the trasmitter a prototype filter that does ot satisfy Equatio 28), it ca be maaged with very margial performace loss (i.e., the frequecy despreadig receiver ca be still employed at the receiver, with its advatages cosidered i the paper ad without appreciable disadvatages due to the presece of a mismatch). 3.3 Complexity compariso of the two structures The stadard receiver structure (see Equatio 50) requires to calculate (a) MK multiplicatios, (b) K 1 additios of M 1 vectors, ad (c) a FFT over M samples, which is the same requiremet of the stadard trasmitter structure. Therefore, the overall umber of real-valued flops C SR f C SR m C ST m ad the overall umber of real-valued multiplicatios are equal to their trasmitter couterpart CST f ad i Equatio 36. The frequecy-despreadig structure Figure 1 Frequecy despreadig. The frequecy despreadig structure at the receiver side. O a flat chael H m = 1 for ay m while o a multipath chael, the values of H m depeds o the chael frequecy respose ad are used to equalize the chael distortios.

9 Mattera et al. EURASIP Joural o Advaces i Sigal Processig (2015) 2015:23 Page 9 of 22 is described i Equatios 60, 58, ad 56. With referece to such a structure, the overall umber of real-valued flops ad the overall umber of real-valued multiplicatios C FR f C FR m ca be writte as C FR f = 4KM log(km) 6KM + 8 }{{} C FR m FFT i Equatio (2K 1) }{{} + 2(2K 2) multiplicatios i Equatio 60 M }{{} k {0,1...,} i Equatio 60 }{{} additios i Equatio 60 = 8 + M 4K log(km) + 2K 6 (65) = KM log(km) 3KM + 4 }{{} FFT i Equatio 56 M }{{} k {0,1...,} i Equatio 60 = 4 + M K log(km) + K 2. Note that C FR f + 2(2K 1) }{{} multiplicatios i Equatio 60 = MK 4log(M) + 4log(K) M MK 4log(M) + 4K 8 KCf SR (66) where the first ad the secod approximatios are obvious while the third oe holds provided that M is sufficietly large ad K is sufficietly small (e.g, for M = 1024 ad K = 4, 4 log(m) + 4log(K) + 2 = 50 while 4 log(m) + 4K 8 = 48). Therefore, uder the same assumptios used at the trasmitter side, we ca obtai the followig approximatio: Cf FR = KCf SR ;moreover,cm SR/CFR m is about 50% for K = 2, 30% for K = 4, ad 21% for K = 8, idepedetly of M {512, 1024, 2048, 4096}. Thus,alsofor the receiver case, the frequecy-spreadig structure has a computatioal complexity about K times larger. The complexities of the differet cosidered structures are summarized i Table 1. Note that the aalysis shows that the complexity icrease is due to the fact that the frequecy-spreadig structure requires to evaluate the DFT over KM poits istead of the DFT over M poits required by the stadard structure. Sice may solutios exist for implemetig FFT processig, the result of the compariso depeds o the particular solutio for its implemetatio; however, a K-fold icrease of the implemetatiocostscabecosideredaupper-boud;ot always, however, the istalled processig power ca be optimized to the effective eeds ad, cosequetly, the icrease of the actual costs (eeded to perform the FFT over a larger umber of poits) ca be much smaller depedig o the effective implemetatio details. 4 Adaptig the frequecy-despreadig structure to the multipath chael I the preset sectio, we first defie the adaptatio of the frequecy-despreadig structure to the multipath chael, the we recall a stadard approach to adapt the stadard structure to the multipath chael ad we fially compare their performace. 4.1 Frequecy-despreadig structure operatig o multipath chael Whe the multipath ature of the chael is take ito accout, the received sigal ricabewritteas L h ri = hl si l +ηi (67) l=0 where the complex-valued sequece hi oflegthl h + 1 models the multipath chael. I Equatio 67, the oise term ηi is assumed to be the ith sample of the low-pass equivalet of a white Gaussia radom process with ull average ad power spectral desity equal to N 0 /2 i the sigal badwidth; the aalog low-pass equivalet has ull average ad power spectral desity equal to 2N 0. The low-pass equivalet is sampled with samplig period T s ; assumig a ideal atialiasig filter with badwidth equal to 1/(2T s ),theradomvariable ηi ideally sampled at the filter output has ull average ad variace VARηi= 2N 0 /T s.moreover,the sequece η i is zero-mea, white, Gaussia adcircularly symmetric; thus, E η η m = 2N 0 T s δm E η η m 0. (68) Table 1 Structure complexity Structure Number of flops Number of multiplicatios Stadard trasmitter 8 + M 4log(M) + 4K 8 M log(m) + 2K Stadard receiver 8 + M 4log(M) + 4K 8 M log(m) + 2K { FS trasmitter 8 + M 4K log(km) 0.5 } { 3 M K log(k) + log(m) 1 } FS receiver 8 + M 4K log(km) + 2K M K log(km) + K 2 The umber of flops ad the umber of multiplicatios of the two trasceiver structures for each half multicarrier symbol period.

10 Mattera et al. EURASIP Joural o Advaces i Sigal Processig (2015) 2015:23 Page 10 of 22 O a multipath chael, a equalizatio stage has to be icluded i the structures already described i Sectio 3. I particular, whe the frequecy-despreadig receiver (see Equatio 57) is cosidered, the followig modified structure is cosidered D (R) = 1 K 1 A j k KM k = (K 1) F (k) k R,modKM (kk+k ) k {0, 1,..., } (69) where the complex-valued coefficiets F (k) k i Equatio 69 replace the coefficiets G k i Equatio 54; therefore, the coefficiets F (k) k ca be set to G k obtaiig the structure for a flat chael; o a multipath chael, we ca set them i order to equalize the chael improvig the receiver performace. The value of α to be used i Equatio 43 is give by α = P r /P s where P r = E hi si 2 represets the average power of the useful compoet of the received sigal ri adp s = E s i 2 represets the average power of the trasmitted sigal si. The decisio variable D (R) i Equatio 69 ca be writte as follows: D (R) = I(R,k) 0,0 a R + I + η (70) where I (R,k) 0,0 represets the coefficiet of the useful term a R, η describes the effect of the backgroud oise, ad I describes the itersymbol ad itercarrier iterfereces of the symbols a R m,k q ((m, q) = (0, 0)) ad a I m,k q o the useful symbol ar. Such iterfereces would be egligible o a flat chael but they become sigificat o the multipath chael (see Equatio 67). It is easy to prove that I cabewritteasaliear combiatio of the iterferig symbols; the coefficiets of such a combiatio ca be writte as I (R,k) m,q =R {( 1) m ( v (k)h 2m,q f k ) } I (I,k) m,q =R { ( 1) m ( v (k)h 2m 1,q f k ) } (71) where I m,q (R,k) ad I m,q (I,k) are the coefficiets of the symbols of a R m,k q ad ai m,k q, respectively. This shows that the coefficiets F (k) k of the receiver structure, collected i the (2K 1) 1vector f k = F (k) (K 1)... F(k) 1 F(k) 0 F (k) 1... F K 1 (k) T (72) ifluece the useful coefficiet ad the iterferece power; we ca, therefore, set the vector f k i order to equalize the effects of the multipath chael i Equatio 67. The oise term η i Equatio 70 is a zero-mea complex-valued Gaussia radom variable with variace 2 f k 2 σa 2 where γ is defied as the sigal-to-oise ratio per γ G (K) 2 subcarrier, i.e., γ = E s N 0 where E s is defied as the eergy of the useful term of the received sigal i a multicarrier symbol period that is dedicated to each active subcarrier. O a flat chael (h = δ), we deote with v (FLAT,k),q the vector v (k),q i Equatio 71; its k th compoet v (FLAT,k),q,k (k {( K 1),..., 1, 0, 1,..., K 1})cabewritteas v (FLAT,k) 1,q,k = G (K) 2 j q ( 1) k k jπ e KM+M/2 1 K i=m/2 gi e qk+k j2π KM i (73) The, the properties of the prototype filter ad the choice f k = G (K) for settig the receiver coefficiets guaratee that the useful coefficiet I (R,k) 0,0 1 ad the iterferece coefficiets are practically ull; i practice, the achieved sigal-to-iterferece-ratio is aroud 65 db, i.e., v (FLAT,k)H 0,0 G (K) 1 { R v (FLAT,k)H m,q G (K)} 1 (m, q) = (0, 0) (74) The satisfactio of the coditio i Equatio 74 is equivalet to the coditio i Equatio 44 ad it cocers the desig of the prototype filter. Moreover, it guaratees the optimality (i the maximum-likelihood sese) of the receiver structure o the flat chael. Whe all the compoets of the vector v (FLAT,k) m,q oull, the vector v (k) m,q ca be writte as are v (k) m,q = v(flat,k) m,q w (k) m,q (75) where the k th compoet w (k) defied as w (k) L h m,q,k = l=0 h l KM+mM/2 l 1 i=mm/2 l KM+mM/2 1 i=mm/2 m,q,k gi e of the vector w (k) m,q is qk+k j2π KM i qk+k j2π gi e KM i e j2π kk+k KM l (76) Whe v (FLAT,k),q,k = 0, the the deomiator of Equatio 76 is ull ad Equatio 75 is still valid provided that we replace with uit both the deomiator of Equatio 76 ad the same quatity i Equatio 73. From Equatios 71, 74, ad 75, it follows that the choice f k = G (K) /w (k) m,q (77) where we deote with / the compoet-wise divisio of the two vectors ad (m, q) = (0, 0),implies c that R { v (k)h m,q f k} 1 (78) However, a sigle vector f k has to be chose ad, cosequetly, Equatio 78 ca be satisfied for a sigle value of (m, q). A robust choice usually doe for settig f k lies i

11 Mattera et al. EURASIP Joural o Advaces i Sigal Processig (2015) 2015:23 Page 11 of 22 usig Equatio 77 with (m, q) = (0, 0), which implies that the useful coefficiet is practically set to uit: } R {v (k)h 0,0 f k 1 (79) Such a derivatio shows that also the optimum choice of f k may ot able to achieve the optimum performace achieved o a flat chael, uless the vector w (k) m,q exhibits aweakdepedeceo(m, q) so that a sigle vector f k ca approximately satisfy the coditio i Equatio 77 for ay value of (m, q) or at least for the most importat values of (m, q). Such a weak depedece is preset whe L h M; i such a case, the quatity i Equatio 76 ca be approximated as follows: KM+mM/2 l 1 i=mm/2 l KM+mM/2 1 i=mm/2 gi e qk+k j2π KM i 1 qk+k j2π gi e KM i l L h (80) ad, cosequetly, the vector w (k) m,q ca be approximated by the followig vector h, which is idepedet of (m, q): T h k = HkK (K 1)... H kk 1 H kk H kk+1... H kk+k 1. (81) where H m deotes the multipath chael frequecy respose at the frequecy F = m/(km). Cosequetly, the choice i Equatio 77 with the use of the expressio h k for w (k) m,q, i.e., f k = G (K) /h k F(k) k = G k HkK+k. (82) guaratees to the structure the same performace achieved o flat chael provided that L h M (see Figure1foraschemeoftheFSstructureiEquatio69 whe Equatio 82 is chose). 4.2 Recallig the stadard approach The stadard approach cosists i icludig a equalizer stage i cascade with the structure described i Subsectio 3.1. The effects of the multipath chael ca be equalized by usig a sigle-tap structure 7. I this case, the kth etry of the DFT output (see Equatios 50 or 51) is multiplied by 1/H(F k ) with F k = M k i the stadard structure for k {0, 1,..., M 1}. More sophisticated multitap structures could be used ad have also bee proposed with referece to the stadard structure 22. Sice they operate i the frequecy domai ad at twice the multicarrier symbol rate, they itroduce a additioal delay proportioal to the umberoftaps.wecosiderthesigle-tapequalizeriboth structures sice it maitais limited the overall latecy of the trasceiver. It may appear that the FS equalizer be equivalet to a multitap sub-chael equalizer followig the stadard structure ad therefore that the cosidered compariso be ufair. However, they are ot equivalet for two reasos: (a) because the delay itroduced by the two structures is differet ad obtaiig the miimum delay is importat i a trasceiver, like the OFDM/OQAM oe, with a already larger delay i compariso with the OFDM system; (b) the PPN-FFT scheme performs equalizatio after samplig rate reductio which itroduces a iterpolatio operatio. The distictio of the two structures i terms of samplig rate reductio lies i the fact that the FS structure performs equalizatio i its iteral behavior ad therefore before samplig rate reductio while the PPN structure performs equalizatio after samplig rate reductio ad cosequetly eeds to use the sigle-tap equalizer to ot icrease the trasceiver delay. 4.3 Comparig the sigal-to-iterferece-ad-oise ratios of the two structures The equivalece of the two structures described i Sectio 3 o a flat chael implies that the behavior of the sigle-tap equalizer ca be described with the same relatios itroduced i Subsectio 4.1 provided that the coefficiets expressio i Equatio 82 is replaced by the followig oe F (k) k = G k HkK. (83) I other terms, the choice i Equatio 83 makes the FS receiver equivalet to the PPN structure equipped with the sigle-tap equalizer of coefficiet 1/H kk. Therefore, by comparig Equatios 82 ad 83, we ca obviously ote that the advatage of the frequecy-despreadig equalizer lies i its capability of usig the coefficiet H kk+k istead of the costat term H kk.sicethefsstructure i Equatio 69 first extracts the DFT of the iput vector accordig to Equatio 56 ad subsequetly uses the coefficiet F (k) k to equalize the chael effect at frequecy (kk + k )/(KM) = k/m + k /(KM), the FS structure with the choice i Equatio 82 uses the right coefficiet (i.e., H kk+k ) to equalize the chael respose at frequecy (kk + k )/(KM) while the FS structure with the choice i Equatio 83, which is equivalet to the PPN structure equipped with the sigle-tap equalizer, always uses the same coefficiet H kk to equalize the chael resposes r (K) at the differet frequecies (kk + k )/(KM) for k { (K 1),..., 1, 0, 1,..., K 1}. I other terms, differetly from the PPN structure equipped with the sigle-tap equalizer, the FS structure is able to equalize with differet coefficiets the differet parts of the subcarrier bad. Sice the effect of a offset τ i timig sychroizatio, perfectly compesated however i the frequecy domai,

12 Mattera et al. EURASIP Joural o Advaces i Sigal Processig (2015) 2015:23 Page 12 of 22 ca be obtaied by settig hl = δl τ, our aalysis shows that a performace improvemet of the FS equalizer, which for τ = 0 (flat chael) is equivalet to the sigle-tap equalizer, appears whe larger values of τ determie faster variatios of the chael frequecy respose so that o-egligible variatios appear withi the subcarrier bad; i such a case, the FS structure is able to use differet coefficiets to equalize each part of the subcarrier bad ad ca therefore achieve improved performace i compariso with the PPN structure equipped with sigle-tap equalizer. Such a superior capability of the FS structure is irrelevat i the presece of a flat chael; therefore, the two structures show the same performace o the flat chael or o chaels where the variatios o the subcarrier bad (of legth 1/M)aremargial. Furthermore, a uified expressio of the sigal-toiterferece-ad-oise ratio (SINR) o the kth subcarrier, deoted as SINR k, of the two structures o the kth subcarrier ca be writte as follows: SINR k = f k 2 γ G (K) 2 + max = mi, =0 { } R 2 v (k)h 0,0 f k { R 2 v (k)h,0 f k }+ max = mi q=1 { } R 2 v (k)h,q f k (84) where the SINR of the frequecy-despreadig receiver is determied by the use of the vector f k described i Equatio 82 whereas the sigle-tap equalizer is determied by the use of the vector f k i Equatio 83. Note that the sigal-to-iterferece ratio SIR k ca be obtaied by employig Equatio 84 without the first term at the deomiator. I order to plot the resultig SINRs for the two structures, we have to set the prototype filter g : we set it to the filter desiged i 5. For the two structures, we report the SIR ad the SINR for a choice of the subcarrier idex k = 56, M = 1024, ad by usig 21 subcarriers ad 21 time-iterval values to approximate the sum i Equatio 84. The chaels employed to determie the receiver SINR are 1,000 radom realizatios of the ITU- R Vehicular B 23 ad the SINR values, evaluated from Equatio 84 ad reported i Figures 2, 3, ad 4, have bee ormalized to H kk 2 i order to separate the effects due to the equalizer capability from that due to the fadig of the kth subcarrier. Figure 5 shows that the frequecy-despreadig receiver is able to get the same SIR performace that it achieves o the flat chael o a large fractio of the cosidered differet chaels, radomly selected ad ordered for decreasig values of output SIR achieved by the frequecy-despreadig receiver. The multipath chaels where it caot achieve the SIR achieved o the flat chael are those with very fast variatios of the chael frequecy respose o the cosidered subcarrier. The sigle-tap shows poorer performace maily because, o may chaels realizatios, the chael frequecy respose is ot flat i the subcarrier bad of size 1/M. From the SIR evaluatio, it follows that, whe we set the sigal-to-oise ratio γ betwee the SIR achieved by the Normalized Output SINR Frequecy spreadig equalizer Sigle tap equalizer Chael Number Figure 2 Samples of output SINR whe γ is 30 db. The ormalized output SINRs for the two structures o 1,000 differet chael realizatios accordig to the ITU-R Vehicular B model whe γ is 30 db.

13 Mattera et al. EURASIP Joural o Advaces i Sigal Processig (2015) 2015:23 Page 13 of 22 Normalized Output SINR Frequecy spreadig equalizer Sigle tap equalizer Chael umber Figure 3 Samples of output SINR whe γ is 20 db. The ormalized output SINRs for the two structures o 1,000 differet chael realizatios accordig to the ITU-R Vehicular B model whe γ is 20 db. sigle-tap equalizer ad that achieved by the frequecyspreadig oe, the SIR limitatio implies also a SINR limitatio ad, therefore, a performace advatage of the frequecy-spreadig structure is preset. I fact, from Figures 2, 3, ad 4, we ca otice a sigificat advatage of the frequecy-despreadig equalizer whe we set the value of γ at 30 db; such advatage is reduced but it is still sigificat at γ equal to 20 db while the oise limitatio becomes domiat for γ equal to 10 db ad the frequecy-despreadig structure caot take advatage from its superior iterferece-rejectio capability. 5 Performace compariso of the two structures I the preset sectio, we assess via computer simulatios the equalizatio performace achieved by usig the frequecy-spreadig structure ad compare it with Normalized Output SINR Frequecy spreadig equalizer Sigle tap equalizer Chael Number Figure 4 Samples of output SINR whe γ is 10 db. The ormalized output SINRs for the two structures o 1,000 differet chael realizatios accordig to the ITU-R Vehicular B model whe γ is 10 db.

14 Mattera et al. EURASIP Joural o Advaces i Sigal Processig (2015) 2015:23 Page 14 of Normalized Output SIR Frequecy spreadig equalizer Sigle tap equalizer Chael Number Figure 5 Samples of output SIR. Samples of the output SIRs for the two structures o 1,000 differet chael realizatios accordig to the ITU-R Vehicular B model. that achieved by the stadard structure. We have also icluded i the performace compariso the classical OFDM system that is ofte cosidered for opportuistic trasmissios because it is a classical scheme employig the multicarrier approach where may practical difficulties have already bee resolved; this has a strog impact o the overall cost. However, for the cogitive radio cotext, it lacks flexibility ad it is likely to lead to poor spectral efficiecy. Sice the latecy d of the FBMC receiver is K times larger tha that of the OFDM receiver with the same umber of subcarriers, we have set the umber of subcarriers i the OFDM trasceiver K times larger i order to compare two structures with the same latecy. Moreover, with such a choice, the OFDM receiver ad the FS-FBMC receiver perform the FFT procedure o the same size, though FS-FBMC has still to perform it to a rate 2K times larger. I particular, we have used 2,048 subcarriers for OFDM while we have used oly 512 subcarriers for FBMC trasceiver ad we have used K {2, 3, 4} i order to verify the effect of the overlap factor. Aumberof10 4 Mote Carlo trials has bee performed uder the followig coditios: 1. The cosidered FBMC ad OFDM systems have a badwidth T 1 s = 11.2 MHz; 2. The trasmitted symbols are the real ad imagiary parts of 64-QAM symbols; 3. The cosidered multipath fadig chael model is the ITU-R Vehicular B 23; 4. The used prototype filter is that proposed i 5. Actually, ay type of prototype filter ca be implemeted with a exteded FFT, due to the equivalece betwee time ad frequecy domais. However, i order to be practical, the umber of frequecy domai filter coefficiets must be the smallest possible, which is the case of the used filter; 5. The chael is fixed i each ru but it is idepedet from oe ru to aother; 6. The residual timig offset (RTO) ad the ormalized residual carrier frequecy offset (RCFO) are cotrolled as simulatio parameters; 7. Both systems exploit a oe-tap subcarrier equalizer with perfect kowledge of the chael ad of the residual timig error, i.e., whe simulatig the presece of the timig offset τ,wehaveused exp( j2π τ k/m)/h(k/m) istead of 1/H(k/M) as coefficiet of the sigle-tap equalizer i the stadard structure ad exp( j2π τ k/km)/h(k/km) as coefficiet of the sigle-tap equalizer for the frequecy-despreadig structure; 8. The effect of the RCFO o the phase of each decisio variable i the frequecy domai, which icreases 20 liearly with time, is ot compesated; therefore, the BER is depedet o the specific multicarrier symbol iterval cosidered for equalizatio. I order to maitai sufficietly limited the effects of such oideal receiver behavior, we evaluate the BER o the data trasmitted i oe of the first multicarrier symbol itervals, the eighth oe;

15 Mattera et al. EURASIP Joural o Advaces i Sigal Processig (2015) 2015:23 Page 15 of I order to use the same badwidth i both FBMC ad OFDM, which exhibits a larger spectral leakage, we have set the percetage of active subcarriers i OFDM trasceiver as 82% of the overall umber of subcarriers while we have set to 89% this percetage i OFDM/OQAM trasceiver; 10. The legth of the cyclic prefix is 1/8 of the OFDM multicarrier symbol period (ote that sice i the FBMC system the cyclic prefix is ot used, i the cosidered case a icrease of the bit-rate early equal to 11.1% with respect to the OFDM system is obtaied). Note that, i cosequece of the choices reported at the poits 9 ad 10, the data rate of the OFDM system is about 82% of the data rate of the FBMC system. We first cosider a simulatio sceario where o timig or frequecy offset is itroduced; Figure 7 reports the results of the experimet. We ca verify that the two FBMC structures, which are equivalet i the flat chael, perform differetly o multipath chael because of the differece betwee the equalizatio mechaisms; more specifically, the frequecy-spreadig structure provides improved performace with respect to the stadard structure ad this improvemet icreases as the overlappig factor icreases. Moreover, both structures exhibit a performace floor at large values of E b /N 0. However, while the performace achieved by the stadard FBMC receiver is very poor, i the rage 0 db E b /N 0 20 db, the performace of the FS-FBMC system is practically equivalet to that of the OFDM trasceiver employig 2,048 subcarriers for a sufficietly large value of the overlap parameter K. Sice the chael coherece time may impose a smaller umber of subcarriers also to the OFDM trasceiver, it is iterestig to test the OFDM system with oly 512 subcarriers; the results ot reported for clarity i Figure 7 show a poor performace, practically equivalet to the stadard structure, sice it is uable to equalize the cosidered chael (the legth of the cyclic prefix (512/8)T s is smaller tha the ITU-R Vehicular B chael legth). We have also performed other simulatio experimets to verify the performace o the less hostile ITU-R Vehicular A chael: here, the coditio L h M is better satisfied. I fact, the correspodig results, show i Figure 6, report that the two structures ad the OFDM system are practically equivalet o chael A. Oly for larger values of E b /N 0 we ca observe some differece; i particular, we ote that the sigle-tap equalizer for K = 2 provides the worst performace; moreover, the three dashed-lie curves, correspodig to the sigle-tap equalizer for K = 3adK = 4aswellasthefrequecy spreadig structure for K = 2. Oly for K = 3ad K = 4 the frequecy despreadig structure behaves practically equivalet to the OFDM system. Therefore, we ca coclude that similar performace is achieved by OFDM ad FS-FBMC trasceivers also if the latter uses oly 512 subcarriers while the former uses 2,048 subcarriers BER tap K=4 1 tap K=3 1 tap K=2 FS K=4 FS K=3 FS K=2 OFDM E b /N 0 Figure 6 Compariso wrt E b /N 0 (ITU-R Vehicular B chael). The effect of the oise o the performace of the OFDM trasceiver (with 2,048 subcarriers) ad of the two structures for FBMC trasceiver (with 512 subcarriers) o ITU-R Vehicular B chael. Three possible values of the overlap parameters are cosidered.