Adaptive Windowing of Insufficient CP for Joint Minimization of ISI and ACI Beyond 5G

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1 Adaptve Wndowng of Insuffcent CP for Jont Mnmzaton of ISI and ACI Beyond 5G Berker Peköz, Student Member, IEEE, Selçuk Köse, Member, IEEE, and Hüseyn Arslan, Fellow, IEEE Department of Electrcal Engneerng, Unversty of South Florda, Tampa, FL, School of Engneerng and Natural Scences, İstanbul Medpol Unversty, İstanbul, TURKEY, E-mal: Abstract Usng mnmum, even nsuffcent guards are proposed to acheve the spectral effcency and latency requrements of cellular communcaton systems beyond 5G. Ths leads to nterference n both tme and frequency domans. In ths paper, a partal-non-orthogonal multple accessng scenaro n whch the desred user s experencng both ntersymbol nterference (ISI) due to nsuffcent cyclc prefx (CP) and adjacent channel nterference (ACI) caused by asynchronous transmtters usng non-orthogonal numerologes n adjacent bands s nvestgated. ISI and ACI depend on the power offset between desred and nterferng users, the nstantaneous channel mpulse responses of nterferng users and transmtter and recever wndow functons. Therefore, jont and adaptve utlzaton of CP requres real-tme calculaton of ISI and ACI. Analytcal expressons for expected ISI and ACI at each subcarrer of the desred user are derved to mnmze ther combnaton. Accordngly, an adaptve algorthm consstng of wndowng each subcarrer at the recever wth wndow length that mnmzes the combned nterference at that subcarrer by optmally exchangng ISI and ACI s proposed. Interference reducton performances of current, outdated and average optmal wndow length rased cosne recever wndows are assessed and compared to fxed and no recever wndowng. Wndowng reduces nterference even when CP s shorter than the channel f wndow length s determned usng the proposed desgn gudelnes. I. INTRODUCTION Conventonal orthogonal frequency dvson multplexng (OFDM) recevers are desgned assumng the cyclc prefx (CP) s longer than the maxmum excess delay (MED) of the desred users channel to not experence ntersymbol nterference (ISI). Users n adjacent bands are assumed to cause neglgble adjacent channel nterference (ACI). Ths s acheved by avodng channels wth MEDs longer than CPs by elongatng the CP duratons, such as the extended-cp opton n Long Term Evoluton (LTE). Possble ACI due to nterferers n adjacent bands are ether mtgated usng nterference cancellaton [1], avoded by ncreasng guard bandwdth untl ACI power becomes neglgble [2], or suppressed [3]. There are numerous approaches to suppress ACI, wth the most promnent one beng wndowng due to ts low computatonal complexty and effcacy. Wndowng /17/$31.00 c 2017 IEEE can be appled at the transmtter to reduce out-ofband (OOB) emsson and correspondng ACI before t eventuates [4], or at the recever [5] to reject present ACI. However, both references utlze the same wndow functon at all subcarrers, whle t s known that edge subcarrers are crtcal n OOB emssons and are more prone to present ACI. Motvated by ths property, [6] ntroduces subcarrer specfc wndow (SSW) concept at the transmtter sde whereas [7] ntroduces optmal SSW functon desgn for both transmtter and recever. Addressng conventonal systems, [6] assumes CP s longer than the MED of the channel to accommodate wndowng and lmt the wndow length to the guard nterval that s not dsturbed by multpath recepton whle [7] even allocates addtonal samples for wndowng, reducng spectral effcency. Cellular communcaton standards beyond 5G are envsoned to provde dverse servces wth varous requrements smultaneously to a myrad of devces. Increasng spectral effcency s crucal to support the projected number of devces, especally, n lower carrer frequences, favorng reduced guards [8]. Usng CP duratons shorter than users MEDs are proposed [9] to satsfy the lower latency requred by new servces n systems beyond 5G whle ncreasng spectral effcency. Thus, the augmented guards promoted by [7] asde, even the more than suffcent CP requred by [6] becomes a luxury n current trend. These conventonal approaches do not address the requrements of communcaton systems beyond 5G and therefore need to be extended. Asynchronous non-orthogonal waveforms wth dfferent parameterzatons, referred to as numerologes, are also proposed to be used n adjacent bands to provde dverse servces n future standards [8]. Although such scenaro s mentoned n [7], how the ACI caused by such non-orthogonal numerologes can be determned s not provded n any of aforementoned works. In ths paper, we utlze nsuffcent CP optmally to jontly mnmze ISI and ACI, addressng spectral effcency requrements of systems beyond 5G and the correspondng real-tme condtons adaptvely. To the best of authors knowledge, ths s the frst work propos-

2 ng wndowng n a system wth nsuffcent CP. We frst determne ncdent ISI caused by nsuffcent CP, ACI caused by dfferent numerologes n adjacent bands, and the combned nterference power for each subcarrer as t s the optmzaton metrc to be mnmzed. These analyses lay out the framework for optmal SSW functons at the recever, but we lmt the dscusson to rased cosne recever wndow lengths. We also analyze the nterference reducton performances of resultng optmal SSW and fxed length wndowng compared to no recever wndowng; wth wndow lengths determned for current and outdated channel mpulse responses (CIRs) and power delay profles (PDPs) to demonstrate the possble gans and robustness of the desgn example. II. SYSTEM MODEL A 1-ndexed algebra s used where I N s the N N dentty matrx, 0 N M s the N M zero and 1 N M s the N M ones matrx. Conjugate, transpose and Hermtan operatons are denoted by ( ), ( ) T and ( ) H, respectvely. A B s the Hadamard product of matrces A and B and A B denotes the Hadamard dvson of A to B. X 2 s the Hadamard product of matrx X wth tself.e a s the expectaton operator over varable a. dag(c 1, c 2,..., c N ) represents the N N dagonal ( matrx wth dagonal elements c 1, c 2,..., c N, toep A, B) denotes the Toepltz matrx of whch frst column s A and frst row s B,δ( ) s the Drac delta functon, N ( µ;σ 2) s the normal dstrbuton wth mean µ and varance σ 2, and flplr( ) s the functon that flps a matrx from left to rght,.e., X M,n = flplr ( ) X M,N n+1. All propertes exstng wth subscrpts u denote that the gven matrx or vector s assocated wth the uth user. Let s u C M u I u denote the modulated data symbols, where M u s number of uth user s data subcarrers and I u s the number of uth user s OFDM symbols n a frame. Q u R N u M u s uth user s subcarrer mappng matrx. A u R N u+k u N u s uth user s CP nserton matrx, consstng of [ ] 0Ku (N A u = u K u ) I Ku (1) I Nu n case of no transmtter wndowng where K u s the number of CP samples. The CP removal and wndowng matrx B L w R N u N u +K u s shown n (2), where n,,u Ln,,u w 0, 1,..., K u s the taper length of ether sde of the wndow n number of samples, used for the recepton of the nth subcarrer of th OFDM symbol of uth user and W n,,u R 1 Lw n,,u, the recever ( wndow coeffcents, are calculated usng W k; Ln,,u w = ) ( ( )) πk cos L w n,,u +1, k = 1, 2,..., Ln,,u w, whch generates rased cosne wndow coeffcents usng taper length nstead of roll-off. Note that for Ln,,u w = 0, (2) smplfes to B 0 = [ ] 0 Nu K u I Nu, whch s the CP removal matrx wthout wndowng. h,u C 1 L u denotes the CIR nvarant durng recepton of the correspondng OFDM symbol where L u s the MED uth user experences n number of samples, whch s obtaned by h 1 α,u (k) = P u u α 1 α uv(k) k where u Lu P u s the receved power of uth user s sgnal, α u s the exponental decay rate of uth user s channel and v(k) C 1 L u CN (0, 1) k 0, 1,..., L u 1 [10]. Then, h conv,u C N u+k u N u +K u s the lnear channel convoluton matrx bounded to one symbol duraton, where h ( conv [ ] T, [ toep h,u 0 1 Nu +K u L u h,u (0),u = 0 1 Nu +K u 1] ). H,u C Nu 1 s the channel frequency response (CFR) of uth user s th OFDM symbol, whch can be calculated as H,u = [ ] T N u F u h,u 0 1 Nu L u. Let us defne the ISI free condton as K u L w n,,u L u, n 1, 2,..., N u (3) Assume desred OFDM symbol s the dth OFDM symbol of 0th user. Let us frst assume the absence of the nterferng users and (3) s satsfed. In ths case the product B L w h conv :,d,0 d,0 A 0 results n the perfect crcular channel convoluton matrx h crc C N 0 N 0 d,0 shown n (4). Furthermore, F 0 B L w h conv :,d,0 d,0 A 0F H 0 results n ( dag H d,0 ), where F u C N N denotes the normalzed fast Fourer transformaton (FFT) matrx uth user uses n the generaton and recepton of OFDM symbols. Hence, gnorng the nose, the receved symbols y :,d,0 C N0 1, where y :,d,0 = F 0 B L w h conv :,d,0 d,0 A 0F H 0 Q us :,d,0 = ( ) dag H d,0 s :,d,0 = H d,0 s :,d,0. y :,d,0 s equalzed usng zero forcng (ZF) equalzaton [11] va a smlar Hadamard dvson by CFR to obtan the symbol estmates ŝ :,d,0 C N0 1 : ( ) ŝ :,d,0 = Q H u y :,d,0 Ĥ d,0 (5) where Ĥ d,0 s the desred OFDM symbol s CFR estmated at the recever. In ths work, although there would be resdual ISI as (3) s nvald, equalzaton wll stll be performed as n (5) and no nterference cancellaton technque other than recever wndowng s appled to reduce the ACI and resdual ISI. In the scenaro of nterest, the receved sgnal conssts of the dstorted desred sgnal and nterference from other sgnals, ncludng ISI from the prevous symbol, and ACI from sgnals n adjacent bands. The am of ths study s to mnmze the aggregaton of the dstorton of desred sgnal and nterference. The dstorton of the desred sgnal can be calculated by calculatng the dfference between the sgnal that

3 B L w n,,u = [ 0Nu L w n,,u K u L w n,,u 0 L w n,,u K u L w n,,u 0 Nu L w n,,u Lw n,,u dag ( flplr ( W n,,u )) I Nu L w n,,u 0 L w n,,u N u L w n,,u 0 Nu L w n,,u Lw n,,u dag ( W n,,u ) ] (2) h crc,u = toep ( [ h,u ] T, [ ([ 0 1 Nu L u h,u (0) flplr h,u (1 : L u 1) 0 1 Nu L u ])] ) (4) would have been receved f (3) was satsfed, and the actual receved sgnal. If (3) was satsfed, the channel convoluton matrx would have been perfectly crcular, and receved sgnal would be y :,d,0 = F 0 h crc d,0 FH 0 Q us :,d,0. Then, the dfference between the perfect and effectve crcular channel convoluton matrces when CP s added usng (1) and removed usng (2), forms the dstorton matrx h dst d,0 CN u N u, whch s h dst d,0 = B L w hconv A h crc. n; Hence, the dstorton n the nth subcarrer of the desred OFDM symbol s found as y dst = F 0h dst d,0 FH 0 Q us :,d,0. The ISI and ACI from all other sgnals are calculated by projectng samples of each receved OFDM symbol to the correspondng samples of the desred OFDM symbol n ths asynchronous scenaro. Each receved OFDM symbol affects a total of f 0 f u (N u + K u +(L u 1)) tme samples. The channel output, ncludng the CIR flter tal, s calculated by left multplyng the transmt samples wth h full,u C f0 fu (N u+k u +(L u 1)) N u +K u, where h full toep ( [ h,u 0 1 Nu +K u 1] T, [ h,u (0) 0 1 Nu +K u 1] ) R,,u = where R C f 0 fu (N u+k u ) N u +K u s any resamplng transform 1. Let t,u R f 0 fu (N u+k u +(L u 1)) 1 denote the tme ndces of the receved samples that contans energy from the samples of the th OFDM symbol of uth user. Then, a projecton matrx Π,u;d,0 R N 0+K 0 f 0 fu (N u+k u +(L u 1)) s formed such that the msalgned, asynchronous samples are projected onto the receved symbol: 1, t d,0 (g) = t,u (j) Π,u;d,0 (g, j) = (6) 0, o.w. Thus, the aggregate nterference on the nth subcarrer of the desred symbol s found as: y nt = y dst + u,u d,0 F 0 B L w Π,u;d,0 h full,u A uf H u Q u s :,,u (7) Usng ths formulaton, the nstantaneous nterference power s calculated easly f all parameters are known. 1 In the numercal verfcaton of ths work, samplng rates are matched usng Fourer nterpolaton, mplyng a Drchlet kernel. However, practcally, nformaton symbols of all users are unknown at the tme of recepton, and an estmate of the expected nterference power s needed. To calculate ths value, the followng statstcal conjecture s used: Conjecture 1. The symbols transmtted usng any subcarrer of any OFDM symbol of any user are ndependent from each other and the used modulaton s unt average power,.e., E s n,,u sn,,u = δ(n n )δ( )δ(u u ) n, n,,, u, u. Conjecture 1 mples that, for practcal number of subcarrers, the varance of ther sum s the sum of ther varances by the law of large numbers [12]. Each column of F H contans the phase rotaton of a normal random varable and the sum of varances of all columns yelds the total nterference power contrbuted to the symbol. Thus, the expected aggregate nterference to the nth receved subcarrer of the desred user s gven n the nth column of P nt :,d,0 + u,u d,0 = 1 1 N ( F0 h dst d,0 FH 0 2) T 1 1 N ( F0 B L w Π,u;d,0 h full,u A uq u F H u (8) 2) T where the nth column of 1 1 N X T contans the sum of all elements n the nth row of X. III. PROPOSED METHOD The recever s to solve ether of n,,u = arg mn P nt L w n,d,u n,d,u L avs n,u = arg mn L w n,,u L fx = arg mn d,u L w n,d,u L u avf = arg mn L w n,,u Es P nt n,,u E n P nt n,d,u En Es P nt n,,u (9) (10) (11) (12) subject to L w n,,u 0, 1,..., K u (13) to fnd 1) optmal SSWs lengths for known CIRs 2) average SSW lengths dependng on users PDPs

4 0-5 P nt P nt (s) P ISI P ACI P ISI+ACI Fg. 1. Post-equalzaton P nt for L w = 0 n. and P nt (s) for a realzaton, 3) optmal wndow length for conventonal fxed recever wndowng usng the same wndow lengths for all subcarrers for known CIRs 4) average fxed length dependng on users PDPs where requred computatonal complexty decreases along wth performance as we get to the bottom of the optons. The solutons to wndow length calculatons are not provded but performance gan wll be shown. Provded the solutons are known, SSW requres addtonal ( L w \ L fx 4L w + N 2 log 2 N) multplcatons and ( L w \ L fx 2L w + N log 2 N ) addtons on top of fxed wndowng, due to addtonal overlappng (frst terms) and FFT operatons (second terms). IV. NUMERICAL VERIFICATION A system wth the followng parameters was smulated to demonstrate the gans of the proposed algorthm.α u, CIRs and tme offset between users are randomzed at each run. Ĥ,u = H,u, u and h,u h =,u P u δ( ). P 1 = P 1 always, and are equal to 2P 0 n the remanng fgures except Fg. 4. There s no guard band between any user, frst subcarrer of the user wth narrower bandwdth s located at the frst null of the adjacent user s edge-most subcarrer. 2 f 1 = f 0 = f 1 /2, where user ndces dstngushes ther order n the spectrum. The rest of the varables are gven n the samplng rate of user 0. N 1,0,1 = 512, 256, 128, M 1,0,1 = 123, 127, 31, and K 1,0,1 = 36, 18, 9 whereas L 1,0,1 = 64, 32, 16. The post-equalzaton expected aggregate nterference for unknown sgnals and the actual nterference for known sgnals for a sngle realzaton of the aforementoned setup s shown n Fg. 1. The expected nterference calculatons are accurate n determnng the actual nterference, but a slght msmatch occurs due to dependance of ACI on nterferng users sgnals Fg (a) One realzaton P ISI P ACI P ACI+ISI (b) Mean of many realzatons Pre-wndow nterference power n desred user s sgnal. The ISI power (consstng of both the dstorton of the symbol n nterest and the leakage from precedng symbol of desred), ACI power and the combned nterference power at the subcarrers of the desred sgnal are shown n Fg. 2. In case of a sngle realzaton shown n Fg. 2a, the dependency to the nstantaneous channels of nterferng users can be observed by the power offset at edge subcarrers although both nterferers have the same transmt powers. As the results are averaged over many realzatons as shown n Fg. 2b, ISI becomes unform throughout the subcarrers and ACI becomes stronger at edges and weaker n nner subcarrers. The results of the grd search for optmal SSW length satsfyng (9) are shown n Fg. 3 for the same realzaton depcted n Fg. 2a, whch agrees wth channel dependency of optmal SSW lengths. As shown n Fg. 3b, longer wndow lengths are requred at edge subcarrers. The SIR gans of seven dfferent recevers over many power offsets are calculated, and the gan over no wndowng s presented n Fg. 4. SSW guarantees hgher

5 P(dB) P(dB) P w nt L :,d,0=0 P L fx nt :,d,0 P SSW nt L (a) One realzaton P nt L w =0 n,,0 P nt L fx :,,0 P nt n,,0 n; (b) Mean of many realzatons P nt for L 0, n;d,0 w = L fx d, n,d for L 0, n,,0 w = L fx, n,. n, /K and Fg. 3. (a) (b) L avs n /K, and P nt n,,0 SIR gan over no wndowng (db) Fxed (Out) Fxed (Avg) Fxed (Cur) SSW (Out) SSW (Avg) SSW (Cur) Fg. 4. SIR gan of recevers wth Ln,,u w = L fx 1,u, L u avf, L fx,u, n, 1,u, Ln,u, avs n,,u over no wndowng for dfferent nterferer power offsets. Percentage of CP used for wndowng Percentage of CP used for wndowng. gan than fxed wndowng wth current and average optmal length, and outdated lengths become robust as nterferers become more powerful. Most carrers are stll wndowed effcently albet fluctuatons around the expected nterference trend wth outdated CIRs and PDPs, but the performance recedes compared to current lengths due to the non-optmal wndowng as the CIRs of all users may have changed drastcally. V. CONCLUSIONS We have determned expected and nstantaneous nterference powers. Interference power s used to determne subcarrer specfc wndow lengths mnmzng the nterference. We lad down numerous gudelnes wth varous computatonal complextes to determne optmal wndow lengths under nsuffcent CP. The proposed subcarrer specfc wndowng scheme mproves SIR even when CP s nsuffcent. Average optmal wndow lengths depend only on PDPs, and although nstantaneous optmal wndow lengths depend on users CIRs, fluctuaton s lttle. Therefore, subcarrer specfc wndowng outperforms fxed wndowng even wth outdated wndow lengths n case of powerful nterferers. REFERENCES [1] P. Szulakewcz, R. Kotrys, M. Krasck, P. Remlen, and A. Stelter, OFDM nterferng sgnal rejecton from ac channel, n Proc IEEE 23rd Int. Symp. Personal, Indoor and Moble Rado Commun., Sydney, AU, Sep. 2012, pp [2] P. C. Chan, E. Lo, R. Wang, E. S. Au, V. N. Lau, R. Cheng, W. Mow, R. Murch, and K. Letaef, The evoluton path of 4G networks: FDD or TDD? IEEE Commun. Mag., vol. 44, no. 12, pp , Dec [3] Z. You, J. Fang, and I.-T. Lu, Out-of-band emsson suppresson technques based on a generalzed OFDM framework, EURASIP J. Adv. Sgnal Process., vol. 2014, no. 1, p. 74, Dec [4] M. Gudmundson and P. O. Anderson, Adjacent channel nterference n an OFDM system, n Proc IEEE Veh. Technol. Conf., vol. 2, Atlanta, GA, Apr. 1996, pp [5] C. Muschallk, Improvng an OFDM recepton usng an adaptve Nyqust wndowng, IEEE Trans. Consum. Electron., vol. 42, no. 3, pp , Aug [6] A. Sahn and H. Arslan, Edge Wndowng for OFDM Based Systems, IEEE Commun. Lett., vol. 15, no. 11, pp , Nov [7] E. Güvenkaya, A. Şahn, E. Bala, R. Yang, and H. Arslan, A Wndowng Technque for Optmal Tme-Frequency Concentraton and ACI Rejecton n OFDM-Based Systems, IEEE Trans. Commun., vol. 63, no. 12, pp , Dec [8] Z. Ankaralı, B. Peköz, and H. Arslan, Flexble Rado Access Beyond 5G: A Future Projecton on Waveform, Numerology Frame Desgn Prncples, IEEE Access, vol. PP, no. 99, pp. 1 1, [9] J. Lorca, Cyclc Prefx Overhead Reducton for Low-Latency Wreless Communcatons n OFDM, n Proc IEEE 81st Veh. Technol. Conf., Glasgow, SCOT, May 2015, pp [10] D. R. Morgan, Analyss and Realzaton of an Exponentally- Decayng Impulse Response Model for Frequency-Selectve Fadng Channels, IEEE Sgnal Process. Lett., vol. 15, pp , [11] R. W. Lucky, Automatc equalzaton for dgtal communcaton, The Bell System Techncal Journal, vol. 44, no. 4, pp , Apr [12] C. M. Grnstead and J. L. Snell, Introducton to Probablty: Second Revsed Edton. Amercan Mathematcal Socety, Oct