Performance Analysis and Optimization of DCT-Based Multicarrier System on Frequency-Selective Fading Channels

Receved January 3, 018, accepted February 1, 018, date of publcaton February 14, 018, date of current verson March 19, 018. Dgtal Object Identfer 10.1109/ACCESS.018.806318 Performance Analyss and Optmzaton of DCT-Based Multcarrer System on Frequency-Selectve Fadng Channels CHAG HE 1, LEI ZHAG, JUQUA MAO 1, AIJU CAO 3, PEI XIAO 1, (Senor Member, IEEE), AD MUHAMMAD ALI IMRA, (Senor Member, IEEE) 1 Insttute for Communcaton Systems, Unversty of Surrey, Guldford GU 7XH, U.K. School of Engneerng, Unversty of Glasgow, Glasgow G1 8QQ, U.K. 3 ZTE R&D Center, 164 51 Stockholm, Sweden Correspondng author: Le Zhang (le.zhang@glasgow.ac.uk) ABSTRACT Regarded as one of the most promsng transmsson technques for future wreless communcatons, the dscrete cosne transform (DCT)-based multcarrer modulaton (MCM) system employs cosne bass as orthogonal functons for real-modulated symbols multplexng, by whch the mnmum orthogonal frequency spacng can be reduced by half compared wth dscrete Fourer transform-based one. Wth a tme-reversed pre-flter employed at the front of the recever, nterference-free one-tap equalzaton s achevable for the DCT-based systems. However, due to the correlated pre-flterng operaton n tme doman, the sgnal-to-nose rato s enhanced as a result at the output. Ths leads to reformulated detecton crteron to compensate for such a flterng effect, renderng mnmum-mean-square-error, and maxmum lkelhood detectons applcable to the DCT-based multcarrer system. In ths paper, followng on the pre-flterngbased DCT-MCM model that bulds n the lterature work, we extend the overall system by consderng both transcever perfectons and mperfectons, where frequency offset, tme offset, and nsuffcent guard sequence are ncluded. In the presence of those mperfecton errors, the DCT-MCM systems are analyzed n terms of desred sgnal power, nter-carrer nterference, and nter-symbol nterference. Thereafter, new detecton algorthms based on zero forcng teratve results are proposed to mtgate the mperfecton effect. umercal results show that the theoretcal analyss matches the smulaton results, and the proposed teratve detecton algorthms are able to mprove the overall system performance sgnfcantly. IDEX TERMS Dscrete cosne transform (DCT), fadng channels, ntercarrer nterference (ICI), performance optmzaton. I. ITRODUCTIO In addton to some famous waveforms such as orthogonal frequency dvson multplexng (OFDM) [1], unversal fltered mult-carrer (UFMC) [] [4], generalzed frequency dvson multplexng (GFDM) [5], flter bank mult-carrer (FBMC) [6], [7], fltered OFDM [4], [8], the dscrete cosne transform (DCT) based multcarrer modulaton (MCM) s one of the spectrally-effcent transmsson canddates for the next generaton wreless communcatons. Under the restrcton that only real-valued sgnals are transmtted, t supports the mnmum subcarrer spacng at 1/(T ) Hz nstead of 1/T n OFDM by adoptng cosnusodal orthogonal functons cos(π kt/(t )) [9], where T s the symbol perod and k s the subcarrer ndex respectvely. Correspondngly, the multplexng and de-multplexng operatons for all nformaton carred subcarrers can be smply performed by dscrete cosne transform (DCT) par, whch s smlar to what dscrete Fourer transform (DFT) par dd n OFDM systems. However, the fast DCT algorthms proposed n [10] and [11] provde some mprovements n computaton complexty when compared wth fast Fourer transform (FFT) algorthms. But the complexty reducton s lmted, compared wth the order of n general for a length- sequence [1]. Addtonally, smlar to other real-transform technques for multcarrer systems (e.g. DHT-based MCM [13] [15]), the DCT uses only real arthmetc along wth real-valued sgnals. As a result, the employed system does not need a quadrature modulator VOLUME 6, 018 169-3536 018 IEEE. Translatons and content mnng are permtted for academc research only. Personal use s also permtted, but republcaton/redstrbuton requres IEEE permsson. See http://www.eee.org/publcatons_standards/publcatons/rghts/ndex.html for more nformaton. 13075

at the transmtter, thus avodng the n-phase/quadraturephase (I/Q) mbalance problem. In addton to avodng IQ mbalance, t also reduces phase nose senstvty. Ths can further reduce both the sgnal-processng complexty and power consumpton at the transmtter mplementaton [10], whch s very attractve for massve machne-type communcatons (mmtc) where transmtter devces for power effcency and low complexty are always desred to support more user equpments [16]. The study n [1] shows analytcally that n the presence of frequency offset, nter-carrer nterference (ICI) coeffcents are more concentrated around the central coeffcents n DCT-MCM than n DFT-MCM, brngng enhanced robustness aganst frequency offsets. Ths renders DCT-MCM very promsng for some hgh-speed moblty communcaton scenaros such as ralway dedcated moble communcaton systems where hgh Doppler shfts have a major mpact on the overall system performance [17]. Despte of those merts, one major desgnng dffculty that DCT-MCM schemes encounter n practcal desgn s that when the system s employed under multpath fadng channels, the crcular convoluton property by DCT does not hold and, therefore the cyclc-prefx (CP) based approaches used n conventonal OFDM systems do not apply any more [18], [19]. The convoluton property by DCT has been demonstrated n semnal work n [0], whch mples the fact that the equvalent channel matrx can only be dagonalzed by DCT when the channel mpulse response (CIR) s symmetrc. The chromatc dsperson channel n sngle-mode fbres (SMFs) [1] s one of the examples whch meets the specfed symmetry condton, but t only works n optcal communcatons. In the case of generc crcumstances, such as wreless multpath fadng channels, the transmttng sgnals and CIR n tme doman s not symmetrcally convoluted, and the channel cannot be compensated by smple one-tap equalzers explotng CP accordngly. In the lterature, varous attempts exst tryng to sdestep ths problem. The earler proposed method n [] extends the post-dct sgnal symmetrcally n double length to address the symmetry ssue, however, losng the data rate by half as an expense. A more effectve method wthout data rate sacrfce s based on zero-paddng algorthm [1], by whch guard sequence s zero nserted to elmnate ISI caused by multpath. But t destroys the orthogonalty and the resdual ICI lmts overall system performance. Another alternatve soluton s called cosne modulated multtone (CMT) based FBMC, n whch ICI ssue s mtgated and made less crucal through a bank of well-desgned flters than wth other schemes [6]. However, the sgnfcantly ncreased complexty and some other practcal system confguraton problems hnders ts practcal deployment. Besdes the above desgnng approaches, an optmzed soluton s nspred by Al-Dhahr et al. [18], where one-tap equalzaton s enabled n the cosne doman by usng a tme-reversed flter at the front of recever so that CIR symmetry s acqured after flterng. Unlke conventonal DFT-MCM system where CP s normally added at each transmsson block, ths optmsed method has to ntroduce both prefx and suffx as symmetrcally extended sequence from the nformaton block and the nformaton symbol should be one-dmensonal formatted. Wth these two constrant condtons satsfed, a one-tap equalzer s applcable to acheve the optmum performance for DCT-MCM under wreless frequency-selectve channels. To the best of our knowledge, prevous work descrbed n [18] and [3] have bult fundamental framework for the preflterng based DCT-MCM systems n the form of matrx expressons. System optmzatons for one-tap equalzaton procedure has also been nvestgated. In order to provde n-depth nsghts nto the preflterng mechansm for DCT-MCM, especally ts effects on the output SR performance, an alternatve approach to system mplementaton s developed to process the sgnals n tme doman. In ths paper, followng the fundamentals that lad out n the prevous work n [4], we complement the theoretcal bass for the employment of DCT-MCM system over frequency-selectve channels and systematcally extend our study by consderng both transcever perfectons and mperfectons. Correspondng theoretcal results are derved as well n formula. The noveltes and contrbutons of ths paper are summarzed as follows: The output SR expresson per subcarrer s derved by takng the pre-flterng effect nto account. Wth the provson of addtonal analyss on the lower boundary for the mnmum receved sgnal power, the output SR gan s verfed between among all subcarrers, compared to DFT-MCM system. In addton to conventonal zero forcng (ZF) detecton technque that has been employed n most of the DCT-MCM systems, we reformulate two mproved detecton methods. Both of them could effectvely compensate the coloured nose effect after preflterng at the recever. Comparsons are conducted and ther BER performance are presented as well. In the presence of transcever mperfectons for DCT-MCM system, we mplement an analytcal expresson n terms of desred sgnal, ICI and nter-symbol nterference (ISI) by consderng carrer frequency offset (CFO), tmng offset (TO) and nsuffcent guard sequence between symbols. Based on the framework, zero forcng based (ZF-based) teratve channel detecton algorthms are proposed and hence provde sgnfcant gan compared wth conventonal detecton methods n terms of BER performance. The rest of the paper s organzed as followng sectons. We begn wth Secton II by demonstratng the constrant condtons and the approprate transmsson formulatons for proposed optmum DCT-MCM systems. Secton III mproves the system model to matrx formulatons and gves analyss on the pre-flterng effect on both transmtted sgnal power and coloured nose varance. In Secton IV, by takng the pre-flterng effect nto account, we present reformulated crteron for conventonal detecton methods. In the presence of consdered mperfectons, we complete the receved 13076 VOLUME 6, 018

sgnal expresson n terms of CFO, TO and nsuffcent guard sequence and propose correspondng effectve detecton algorthm to combat nterference n Secton V. Lastly, we draw our conclusons n Secton VI. otatons: operator s denoted as a lnear convoluton of two vectors. I and J stand for dentty matrx and reversal matrx n dmenson, respectvely. And 0 M s the zero matrx of sze M. E[ ] means takng the expectaton of random varable. [ ] H and [ ] T refer to hermtan conjugate and transpose operaton, respectvely. dag( ) s defned to return the vector of the man dagonal elements of the operated matrx. II. COSTRAIT CODITIOS O DCT-MCM SYSTEM In ths secton, we start by ntroducng the basc constrant condtons for DCT-MCM based systems and ts approprate formaton to support data transmsson. In conventonal DFT-MCM systems lke OFDM, the absolute orthogonalty that mantaned among subcarrers s acheved by the complex exponental functon sets and the mnmum subcarrer frequency spacng F s 1/T accordngly. However, by transmttng only one-dmensonal symbols and employng a sngle set of cosnusodal functons as orthogonal sets, ths mnmum subcarrer frequency spacng can be reduced to F = 1/T n the sense that: T 0 cos (πk Ft) T cos (πm Ft)dt T = { 1, k = m 0, k = m Partcularly, n DCT-MCM systems, modulator and demodulator can be easly realzed wth the IDCT and DCT operaton, respectvely. In order to mplement a smple equalzer structure by means of a bank of scalars at the recever, the symmetrc convoluton-multplcaton property of the DCT s exploted to perform an element-by-element multplcaton n the correspondng dscrete trgonometrc doman. Therefore, the guard sequence s to be duplcated as prefx and suffx. An addtonal tme-reversed preflter s ntroduced to enforce the equvalent channel mpulse response to be symmetrc as requred. For nstance, assumng a complex equvalent channel mpulse after preflterng denoted by h(t) = h R (t) + jh I (t), then the real and magnary components of t both satsfy the symmetrc CIR constrant: h R (t) = h R ( t) and h I (t) = h I ( t). In general, there exst eght types of DCT [5]. By explotng ther correspondng type of symmetry n the symbol sequence, the constrant on channel condtons and sutable symbol structure formatons to use any type of DCT are presented for data transmsson n [6]. Ther correspondng channel estmaton technques, based on the use of tranng symbols, also takng nto account ther specal symmetrc condtons n the tme doman for any knd of DCT, are presented n [3]. Snce the results ndcate that every knd of DCT acheves dentcal performance [3], [6] wthout (1) loss of generalty, here we only consder the type-ii DCT par for multplexng and demultplexng for smplcty as t earns great popularty n most of the practcal DCT-based transcever mplementatons [5]. The baseband modulated dscrete-tme sgnal x m n a subcarrer DCT-MCM system s then represented by πk(m + 1) x m = a k β k cos [ ], m = 0, 1,, 1. k=0 where a k s ampltude-shft keyng (ASK) modulated nformaton symbol that transmtted on the kth subcarrer and the parameter coeffcent β k s defned as 1, k = 0 β k = (3), k = 1,,, 1 To completely avod the ICI and ISI problems, the length of the prefx and suffx v should be at least equal to the channel delay spread. [18] has gven the symmetrc extenson for the prefx and suffx as follows: x n = x n 1, 1 n v () x +n = x n, 1 n v (4) Assumng the real component vector of CIR wth L taps s f r = [f 0, f 1,..., f ]. Its correspondng tme-reversed flter vector s thus n the reverse form g r = [f, f L,..., f 0 ], resultng n the overall effectve symmetrc channel vector: h r = f r g r = [f 0, f 1,..., f ] [f,..., f 1, f 0 ] = [h L+1, h L+,..., h 0,..., h L, h ] (5) where h r = [h L+1, h L+,..., h 0,..., h L, h ] s denoted as the effectve CIR vector for real branch wth delay spread at length L 1. By whch the element h n n the vector s represented as a convoluton result as L n f k f k+ n f L + 1 n L 1; h n = (6) k=0 0 others. Apparently, the symmetry property arses wth the condton h n = h n mpled n the aforementoned. The effectve channel response coeffcents correspondng to those one-tap per subcarrer equalzers s obtaned n [6]: H k = h n cos [ πkn ] + h 0, k = 0, 1,, 1. n=1 Wth the above condton constrants provded, we are now able to formulate the pre-flterng method model and present further system analyss for the DCT-MCM system n multpath fadng envronments. (7) VOLUME 6, 018 13077

FIGURE 1. Baseband system model for pre-flterng DCT-MCM systems. III. SYSTEM MODEL AD OUTPUT SR AALYSIS Due to the correlated pre-flterng n tme doman, the sgnal power s no longer equally spread among subcarrers. For the purpose of analysng the pre-flterng effect on DCT-MCM systems over general multpath fadng channels, the baseband equvalent system model s proposed n the form of matrx representaton. Mathematcal representatons for ergodc output sgnal power and coloured nose varance are then provded, respectvely. ote that n DCT-MCM systems, the symbols that are ready for transmsson are all n real format after multplexng by IDCT. Consequently, the convoluton wth a complexvalued CIR can be regarded as two ndependent processes that the sgnal s convolved wth the real component and magnary component of the CIR at the same tme, wth each allocatng one-half of the total sgnal power. As to the pre-flterng, the receved sgnals after channel are frstly separated by quadrature. Then both the real and magnary components of receved sgnals are fed to ther correspondng preflter vectors respectvely as ndcated n Fg. 1 [4]. Due to the orthogonalty mantaned between n-phase and quadrature branches, the detector s able to recover the nformaton bts from ether real part or magnary part of receved sgnals. To make full use of the nformaton, the two ndependently fltered data streams are combned n complex form for detecton. Snce the two streams experence dentcal sgnal processng steps n quadrature, wthout loss of generalty, the followng dervatons are for n-phase branch and we focus on real sgnals processng mechansm. The total output sgnal power thus can be obtaned by doublng. A. SYSTEM MODEL Assumng ASK modulated symbols are allocated to subcarrers at the transmtter. The general transmsson system model on n-phase branch n the dagram s expressed as [3], [4] y = DRP R H R CD H a + DRP R n R (8) where y s the demodulated sgnal vector to be equalzed, a R 1 s nformaton symbol vector wth normalzed sgnal power (σs = 1). D R s type-ii DCT matrx that has been power normalzed and ts (l, m) entry s gven by 1)(m 1)π cos((l ), l > 1 d l,m = (9) 1, l = 1. C R (+v) s a prefx and suffx nsertng matrx at length v whch s organzed as C = [J v, 0 v ( v) ; I ; 0 v ( v), J v ]. Generally, we assume the prefx and suffx have the same guard sequence length and the whole block length s L 1 = + v. The multpath channel convoluton matrx on n-phase branch H R R L 1 L 1 s mplemented as a Toepltz matrx wth the frst row and frst column beng [f, f L,..., f 0, 0 1 (L1 L)] and [f, 0 1 (L1 1)] T respectvely. In order to acheve free ISI transmsson, suffcent guard sequence length should be met by L v [18]. The pre-flter matrx P R R L 1 L 1, whch stands for tme-reverse flterng on n-phase branch, s also mplemented by a Toepltz matrx wth ts frst row and column beng [f, 0 1 (L1 1)] and [f, f L,..., f 0, 0 1 (L1 L)] T, respectvely. The guard sequence dscardng operaton s defned by R R L 1 and we buld t n the form as R = [0 v, I, 0 v ]. (10) Accordng to the aforementoned constrant condtons, wth the employment of a pre-flter and nserton of symmetrc guard sequence, the DCT-MCM s optmsed for nformaton carryng wthout ICI and ISI problems. Ths renders the output of DRP R H R CD H ntegraton equvalent to a dagonal matrx by whch the relaton s gven by H eff,r = DRP R H R CD H (11) where H eff,r R L 1 L 1 s the effectve channel matrx for n-phase branch and the elements on the dagonal vector s 13078 VOLUME 6, 018

defned as dag(h eff,r ) = [H 0, H 1,..., H ] (1) where H k s the effectve frequency response coeffcent that has been provded n Eq. (7). In conventonal DFT-MCM systems, t s assumed that the channel frequency responses subject to Gaussan dstrbuton wth equal power for all subcarrers f a typcal Raylegh fadng channel s consdered. However, due to the correlated preflterng, the output sgnal power s not dentcally dstrbuted among subcarrers n DCT-MCM systems. In order to get an n-depth look at the output SR, we present our analyss on output sgnal power and nose varance as below, respectvely. B. ERGODIC OUTPUT SIGAL POWER In practce, many wreless multpath fadng channels are modelled as Raylegh fadng channels havng an mpulse response represented by a tapped delay lne where coeffcents are Gaussan random process [7]. As a consequence, here we denote the n-phase component of channel coeffcents as f k (0, 0.5σk ). Consderng the channel s already normalsed wth power, we get σk = 1 (13) k=0 Upon the assumpton that channel taps are ndependent to each other, we have E[f f j ] = 0 f = j (14) obtaned by the ntal subcarrer at ndex k = 0,.e., E[H0 ] = E[h 0 ] + 4 E[h n ] n=1 = 3 σp 4 4 + 3 p t=0 σp σ p+t = 0.75 (17) Ths ndcates that, for a power normalsed channel case, E[H0 ] s rrelevant to channel characterstcs and equals to a constant value at 0.75. Accountng for the other half of dstrbuted sgnal power on quadrature branch whch experence smlar sgnal processng, the total output sgnal power on the ntal subcarrer s reached to the upper bound at gan of 1.5. The power gan for other subcarrers are below the upper boundary,.e., E[Hk ] E[H 0 ]. The mnmum gan, on the other hand, s varable and subject to the channel type and ts channel parameters. However, In Appendx B, we have proved the lower bound for the mnmum gan s 1 + L 1 when the channel has L taps. Our smulaton results are n agreement wth ths analyss as ndcated n Fg. where we gve an example of typcal IEEE 80.11 channels [8] wth dfferent samplng frequency and root mean square (RMS) delay spread. When the RMS s 10ns and samplng frequency s 10MHz, whch s a sngle-tap case, the mnmum power gan s thus 1.5. As the channel tap ncreases (ether by ncreasng the samplng frequency or the RMS delay spread), ths mnmum gan wll gradually reduce and fnally concde wth the lower bound at a certan pont to 1, whch s dentcal wth analyss shown above. By combnng Eq. (6) and Eq. (14), the expected correlaton results for effectve channel coeffcents s obtaned as (see Appendx A) E[h h j ] 1 σp 4 σ L +p = ±j = 0; = 3 σp 4 4 + 1 p σp σ p+t = j = 0; t=0 0 others. (15) By substtutng Eq. (15) to Eq. (7), the ergodc output sgnal power on n-phase branch at kth subcarrer s E[Hk ] = E[h 0 ] + 4 E[h n ] cos [ π nk] (16) n=1 From Eq. (16), t s obvous to see that the subcarrers are subject to dverse gan and we are able to defne the boundares for the gan varaton. As depcted apparently n the sgnal gan equaton, the sgnal power wll be maxmzed f the multplyng term cos [ π nk] acheves the maxmum value at 1, whch s always FIGURE. The mnmum sgnal power gan for dfferent samplng frequency and RMS delay n a typcal IEEE 80.11 channel. C. COLOURED OISE VARIACE The pre-flterng operaton also affects the coloured nose varance as well. As the pre-flterng s operated ndependently on n-phase and quadrature branches, after flterng, the coloured nose v s combned by real and magnary components whch s expressed as v = DR(P R n R + jp I n I ) (18) VOLUME 6, 018 13079

where P R and P I are for real and magnary parts pre-flterng; whereas n R and n I are the components of the AWG channel nose vector n wth varance σn n quadrature. The expectaton for the coloured nose covarance matrx E[vv H ], proven n [4], s n the form of an dentcal matrx multplyng a constant factor, whch s represented as E[vv H ] = 0.5σ n I f L v (19) Ths reveals that by ndependently flterng the n-phase and quadrature branches, the resulted coloured nose varance after combner s reduced by half. The key to ths varance reducton les to the fact that modulated symbols after IDCT are n one-dmensonal format for transmsson, hence become robust aganst the complexed nose. D. OUTPUT SR GAI The ergodc output SR can be represented n terms of nput SR, output sgnal power and coloured nose varance. The nput SR s defned as SR n = σ s σ n for all subcarrers (0) Accountng for the pre-flterng effect and power loss form guard sequence nserton, the output SR at kth subcarrer s n the form as SR out (k) = + v E[H k ] σ s 0.5σn = 4 E[H k ] SR n (k) (1) + v Snce 1 E[Hk ] 1.5, we get a proposton for the acheved output SR gan. Proposton: Consder a DCT-MCM system, the preflterng process wll acheve a output SR gan between +v and 3 +v for all subcarrers. Consequently, the exact gan η of output SR obtaned for arbtrary subcarrer ndex s η(k) = 4 + v (E[h 0 ] + 4 E[h n ] cos [ π nk]) () n=1 Eq. () mples the output SR gan s subject to employed subcarrer number and channel characterstcs. We then can take a look at the exact SR gan at the output of the recever. Fg. 3 demonstrates the smulaton results for output SR gan wth respect to three wdely employed channels, whch are named by, Extended Pedestran A model (EPA), Extended Typcal Urban model (ETU) and Extended Vehcular A model (EVA), respectvely. In practce, the channel knowledge s obtaned through the ntroducton of plot symbols and the channel state wll not change wthn one transmtted symbol n slowly fadng envronments [3], [9]. Here we gnore the channel mperfecton ssues and assume nformaton s perfectly known at the recever. The samplng rate for smulaton s assumed at f s = 0MHz and we assgn FIGURE 3. The output SR gan acheved n three typcal channels. (a) EPA =64. (b) EVA =64. (c) ETU =64. (d) ETU =56 [4]. 64 and 56 subcarrers separately to the system for comparson. Consderng the SR loss from prefx/suffx nserton s fxed and not relevant to the pre-flterng process, wthout +v loss of generalty, we neglect the effect form the term n the smulaton. As can be seen n Fg. 3, for all subcarrers, our smulaton results on the lnear gan η for all knds of channels concur wth the prevous analyss n Eq. (). The subfgures (a),(b) and (c) gve very dstnct varaton trends under dfferent channel characterstcs. Generally, the output SR gan vares n the range between to 3 folds among all subcarrers, whch s verfed n our proposton. In partcular, as depcted n the fgures, the gan varaton curves fluctuate sharply along wth ncreased allocated subcarrer number and channel taps. Fg. 3(a), due to the flat fadng case n the sense of a EPA channel at f s = 0MHz, n partcular, acheves a constant gan at η = 3 for all subcarrers. The cosne term, as ndcated on the rght n Eq. (), brngs the symmetry property to the gan varaton curves, whch s clearly observed at the mddle subcarrer ndex on all accounts. IV. REFORMULATED CRITERIO FOR DCT-MCM SYSTEM DETECTIOS In ths secton, three commonly used detecton technques are dscussed to be ntegrated wth DCT-MCM systems. Among of the three, the zero forcng (ZF) one-tap equalzer s easy and effcent to buld at the recever [30] wthout the need for nose nformaton. However, at frequences where channel frequency response (CFR) s severely attenuated, the nose power s proportonally ncreased when the equalzer attempts to boost the sgnal power. As another alternatves, mnmum-mean-square-error (MMSE) and maxmum lkelhood (ML) detectons could sdestep the SR 13080 VOLUME 6, 018

degradaton problem but requre the nformaton of CIR and the nose statstc property [31]. However, as ndcated n the aforementoned secton, the pre-flterng process renders the coloured nose correlated among subcarrers, therefore devates from the Gaussan dstrbuton. Ths motvates us to compute the coloured nose power at each nstant and reformulate the crteron for MMSE and ML methods to make them adaptable for DCT-MCM so as to brng performance enhancement. A. REFORMULATED MMSE DETECTIO As an alternatve detecton method, the MMSE equalzer s consdered to compress the mean square error to the mnmum [30]. To facltate dervaton, the post-dct symbol vector s represented n terms of effectve channel matrx for real part H eff,r and magnary part H eff,i. Combnng wth effectve pre-flterng matrx G R and G I mposed on the nose, the receved sgnal vector n Eq. (14) s smplfed to y = (H eff,r + jh eff,i )a + G R n R + jg I n I = H eff a + G R n R + jg I n I (3) Where G R = DRP R and G I = DRP I. And H eff s the complex effectve channel matrx. In ZF detectons, as the equalzer represented n matrx form by W ZF = Heff H / H eff only requres estmated CIR, conventonal detecton crteron can be drectly used for DCT-MCM systems. However, n the case of MMSE detecton method, accountng for the pre-flterng effect, the fltered nose vector G R n R + jg I n I n Eq. (3) becomes coloured by multplyng a correlaton matrx on both the real and magnary components. In what follows, we provde the optmum and suboptmum MMSE detectons for DCT-MCM, respectvely. Snce the nose correlaton matrces G R and G I are non-dagonal nherently, the optmum MMSE detecton therefore s non-lnear and the overall complexty grows exponentally wth the matrx sze. On the other hand, the suboptmum subcarrer-wse MMSE case s a lnear detcton and attans the same complexty order as the ZF case. Frstly, we assume the MMSE equalzer matrx whch can effectvely compensate the correlaton effect s formed as W op, by takng tre W op y a W op = 0, we have W op = σ s H H eff σ s H eff H H eff + 1 σ n (G RG H R + G I G H I ) (4) In order to reduce the general detecton complexty from subcarrer-wse perspectve, we take the overall correlaton effect from the neghbourng subcarrers as a whole factor and calculate the nose power at each nstant for the desre subcarrer. Then, the receved symbol on kth subcarrer from Eq. (3) can be extracted smply to: y k = H k a k + g k,t n r,t + j q k,t n,t (5) t=1 t=1 where n r,t and n,t are the n-phase and quadrature components of nose vector n on subcarrer ndex t; whereas g k,t and q k,t are the entres from the kth row and tth column of G R and G I, respectvely. It s noted that, the coeffcents g k,t and q k,t n the pre-flterng matrx are tme-reversed transforms from CIR. In practce, the nose changes much faster than the channel and we can assume the channel coherent tme s long over symbols. Upon that assumpton, g k,t and q k,t are regarded as nstant values and the coloured nose on arbtrary subcarrer ndex s n the addtons of Gaussan nose varables wth power spectrum o / ( 0 = σn ) from all subcarrers. The coloured nose varance V k at subcarrer ndex k s then calculated as V k = 0 ( g k,t + q k,t ) (6) t=1 t=1 Wth the provson of the nstantaneous coloured nose varance V k, we yeld the reformulated expresson of the suboptmum MMSE equalzer for kth subcarrer Wk sub σs = H k H σs H k (7) + V k As evdenced by the above equaton, the pre-flterng effect s compensated by the MMSE equalzer for DCT-MCM systems, as the coloured nose power vares at dfferent nstants. B. MAXIMUM LIKELIHOOD DETECTIO The ML detecton s an outstandng soluton that obtans optmal performance among the three. The detector consders all potental realzatons by searchng for all possble sgnal constellaton. By assumng the raw bt value can be ether zero or non-zero, a log-lkelhood rato (LLR) based detector s employed to gve the logarthm of the rato of a posteror probabltes of the modulated symbols. In our case, the condtonal probablty densty functon of y k gven a k s n the form as p(y k a k ) = 1 πvk exp( y k H k a k V k ). Ths holds for a specfc nformaton data frame for whch the V k stands for nstantaneous coloured nose varance. In general, soft nformaton n the form of the LLR ndcates the confdence of demappng decson. In ths regard, accordng to [3], the soft nformaton contaned at the th bt s represented as a k A (1) p(a k y k ) L = ln a k A (0) p(a k y k ) = ln a k A (1) p(y k a k ) a k A (0) p(y k a k ) a k A (1) = ln exp( y k H k a k V k ) a k A (0) exp( y k H k a k V k ) (8) where A (λ) s the set of all possble canddate ASK symbols correspondng to b = λ, λ {0, 1}. Wth the provson of log-sum-exponental approxmaton property: log exp(φ ) = max (φ ), after takng out the CFR effect and assumng the ZF equalzed symbol on the kth subcarrer VOLUME 6, 018 13081

s z k = y k /H k, and the LLR equaton s smplfed to L ln max ak A (1) max ak A (0) exp[ H k (z k a k ) /V k ] exp[ H k (z k a k ) /V k ] = V k H k { max [ (z k a k ) ] max [ (z k a k ) ]} a k A (1) a k A (0) = V k { mn (z H k k a k ) + mn (z k a k ) } (9) a k A (1) a k A (0) As ndcated, the pre-flterng effect on both sgnal and nose V components are compensated on the part k n Eq. (34), H k leadng to optmum system performance for DCT-MCM systems. the performance of the optmum MMSE whle reducng the detecton complexty sgnfcantly. On the other hand, snce typcally the detecton of less dense symbols s more robust to nose, DCT-MCM systems can result n mproved errorrate performance when operatng far from the capacty lmt. Ths s verfed by the superorty gap of ZF at 4dB at a BER of 10 and MMSE at db at a BER of 10 3 between these two systems. However, t s reduced to around 1dB at a BER of 10 4 f the optmum ML approach s used for the two systems, whch s seen n the fgure. V. IMPERFECTIO AALYSIS AD EQUALIZATIO ALGORITHM In the prevous secton, we focus on the performance analyss on the DCT-MCM system wth perfect transcever and suffcent guard sequence. However, n practce, due to the hardware mparments and naccurate synchronzaton errors, a certan level of CFO and TO wll always be ntroduced at the transcevers. Moreover, suffcent prefx and suffx length s not always guaranteed n order to mprove the bandwdth effcency of the system. In ths secton, we wll frst derve the system model by takng all aforementoned mperfectons nto consderaton. The system performance s analysed n terms of power of desred sgnal, ICI, ISI and nose. Fnally, new teratve equalzaton algorthms are proposed to mtgate the effect of nterference and nose. FIGURE 4. BER performance for DCT-MCM and DFT-MCM systems n three detecton methods. C. PERFORMACE COMPARISO WITH DFT-MCM SYSTEMS The BER performance of DCT-MCM systems s plotted n Fg. 4 wth respect to the above three reformulated detecton methods under a typcal IEEE 80.11 channel. As ndcated n [1] that the bandwdth of a DCT-MCM system by transmttng one-dmensonal formatted sgnals can be only half of that requred by a classc DFT-MCM system wth the same number of subcarrers, n order to make a far comparson wth DFT-MCM systems n the sense of same bandwdth effcency, we assgn 18 number of 8ASK modulated subcarrers to DCT-MCM and 64 number of 64 quadrature ampltude modulaton (64QAM) subcarrers to DFT-MCM systems respectvely. A CP length of 1 s guaranteed for DFT-MCM whereas t s doubled n DCT-MCM. Addtonally. the channel s coded wth the polynomal (133,171) code at constrant length of 7 and rate of 1/ for both systems. As shown n Fg. 4, due to the output SR gan that explaned n proposton 4, DCT-MCM systems outperform DFT-MCM based ones on all stuatons. It can also be observed that our proposed suboptmum MMSE equalzer can approach A. SYSTEM MODEL I THE PRESECE OF IMPERFECTIO In the presence of TO effect and nsuffcent guard tme sequence, the dervaton for DCT-MCM system n Secton III usng matrx operatons are no longer sutable for mperfecton analyss. Therefore, we wll use dscrete seres n tme doman to express the generalzed model n ths secton. The dscrete tme sequence of a transmtted block s clearly shown n Eq. (). Consderng the ntroduced nter-block nterference due to system mperfecton, the correspondng dscrete tme seres s extended from sngle block case to consecutve ones, by whch the mth sample n the th transmtted block s gven by πn(m + 1) x (m) = a,n β n cos [ ], n=0 m = v,,, + v 1. (30) For each transmtted block of + v tme-doman samples passng through the channel, the recever chooses consecutve useful data samples n the mddle to be processed further and dscards the other v guard nterval samples from the front-end sde equally. Assumng the frame synchronzaton error s presented as samplng tme offset k, the ndces of the remanng samples correspondng to the jth receved block are {k k + j( + v) k = 0,, 1}. In that case, the remanng samples r(k) of the jth receved block after channel convoluton and guard symbols removng s 1308 VOLUME 6, 018

wrtten as r(k) = + +v 1 = m= v x (m)h(k m ( + v)) + n(k), k = k + j( + v),, k + j( + v) + 1. (31) It s noted that, wthout loss of generalty, we ntegrate channel convoluton process wth the pre-flterng operatons n Eq. (31) and take h(k) as a general effectve symmetrc CIR after pre-flterng. As the correlaton property for h(k) has already been proved n Eq. (15), for smplcty, we denote the effectve channel autocorrelaton functon as R(k) = E[h k h k ] and yeld the general correlaton functon: E[h k1 h k ] = [δ(k 1 k ) + δ(k 1 + k ) δ(k 1 )δ(k )]R(k 1 ) (3) Here, the unt sample sequence δ(k) acheves the unt value of 1 f k = 0 but remans 0 for k = 0. In the practcal desgn of a typcal OFDM based system, a tranng sequence s usually employed n front of each symbol for the sake of estmatng the parameters of mperfectons. Ths knd of mperfecton detecton mechansm could be appled to DCT-MCM as well. In the followng, the knowledge for CFO and TO s assumed to have been acqured from the tranng symbols. B. POWER OF DESIRED SIGAL, ICI, ISI AD OISE In the followng process, we focus on the data symbols detecton wthn the ntended receved block (j = 0). Due to fadng effect caused by multpath, the output of DCT performed as demultplexng s corrupted by nterference plus the channel nose. Assumng the data symbols are statstcally ndependent wth unt average energy and the transmtted average energy per symbol equals to E s. The power of desred sgnal receved at the nth output of the DCT can be expressed n terms of desred sgnal, ISI, ICI, and coloured nose as follows: P x (n) = E s + v (P U (n) + P ICI (n) + P ISI (n)) + V n (33) The useful power P U denotes the contrbuton from the desred symbol a 0,n. The ICI power P ICI contans the contrbuton from the other symbols transmtted n the consdered transmsson block ( = 0), whereas the ISI power P ISI contans the contrbuton power from all subcarrers wthn other transmsson blocks ( = 0). Fnally, V n denotes the contrbuton from the coloured nose defned n Eq. (6). Assumng the neghbourng subcarrers are modulated, we obtan P U (n) = E[ γ (n, 0, n) ] P ICI (n) = P ISI (n) = l=0,l =n + =, =0 l=0 E[ γ (n, 0, l) ] E[ γ (n,, l) ] (34) where γ (n,, l), gven by γ (n,, l) = β l β n k=0 cos +v 1 m= v cos π(m + 1)(l + f ) π(k + 1)n h(k k m ( + v) (35) denotes the sgnal component for the demultplexed output at nth subcarrer durng the ntended receved block j = 0, caused by the nformaton symbol a,l multplexed on the lth subcarrer wth carrer offset f consdered durng the th transmsson block. In order to dstngush the sgnal component from desred subcarrer and nterference subcarrers separately, let us denote γ (n, 0, n) ( = 0, l = n) as the desred sgnal power coeffcent and the rest of γ (n,, l) are the nterference power coeffcents. For the calculaton of the component E[ γ (n,, l) ], after substtutng Eq. (3) nto (35), we provde a computatonal effcent expresson for E[ γ (n,, l) ] by dvdng t nto sx sub-components (see Appendx B): E γ (n,, l) = E γ 1 (n,, l) + E γ (n,, l) + E γ 3 (n,, l) + E γ 4 (n,, l) + E γ 5 (n,, l) E γ 6 (n,, l) (36) In the presence of nterference for the non-synchronzed DCT-MCM system, the sgnal-to-nterference rato (SIR) of the nth subcarrer can be wrtten as: P u (n) SIR(n) = (37) P ICI (n) + P ISI (n) It s well known that DCT owns the advantage of energy compacton capabltes [1]. That s, the sgnal energy s manly dstrbuted n a few DCT coeffcents, whle the remanng coeffcents are neglgbly small. In ths regard, [1] has shown that the DCT operaton dstrbutes more energy to the desred subcarrer and less energy to the ICI than the DFT operaton under AWG channel. Ths superorty can be also verfed for multpath envronments n Fg. 5 where the smulaton channel comples wth former IEEE 80.11 one. To hghlght the the power spectrum dstncton between DCT-MCM and DFT-MCM, we choose the 10th subcarrer as an example for the desred subcarrer n a 64-subcarrer system and show ts power spectrum coeffcents E[ γ (n, 0, 10) ] from subcarrer ndex n = 4 to 16. The remanng coeffcents are suffcently small to be gnored, and are not plotted n the fgure. In the presence of normalsed frequency offset, ft = 0.05, 0.1 and 0. are presented for power spectrum comparson, respectvely. As can be seen n Fg. 5, for smaller absolute frequency offset, DCT-MCM shows better energy compacton property than DFT-MCM. Due to correlated flterng, the central desred subcarrer power can even be amplfed to 1.4 tmes whereas DFT-MCM can only close to unt. However, by ncreasng the normalsed frequency offset, the SIR for DCT-MCM wll dramatcally decrease and fnally suffers VOLUME 6, 018 13083

FIGURE 6. Power of P U and P ICI n dfferent subcarrers for DCT-MCM and DFT-MCM n the presence of transcever mperfecton. (a) CFO = 0., TO = 0. (b) CFO = 0, TO =. FIGURE 5. Power spectrum (n lnear) on the 10th subcarrer for DCT-MCM and DFT-MCM systems. (a) ft=0.05. (b) ft=0.1. (c) ft=0.. hgher ICI than that of DFT-MCM, whch concurs wth the results gven n [1] under AWG case. In terms of the desre sgnal power P U and ICI power P ICI, Fg. 6 demonstrates the nterference effect ntroduced by CFO and TO ndvdually. In the case of CFO = 0.05, P U n DCT-MCM stll see non-lnear power dstrbuton among subcarrer ndex due to flterng effect. Ths dstrbuted desred sgnal power s hgher n DCT-MCM whle the nterference nose P ICI are almost under the same level for both two systems. On the other hand, Fg. 6(b) depcts the power varaton property from TO perspectve. As shown n the fgure, the P U curve has very smlar dstrbuton trend wth cosne waves and the P ICI s the other way around. Some senstve subcarrer ndexes (e.g n = 18, 49) experence deep fadng wth maxmum ICI nose. evertheless, n practcal system desgn, snce channel codng and nterleavng technques are nvolved, ths knd of mparment can be remtted wthout much BER performance degradaton. C. EFFICIET EQUALIZATIO ALGORITHM Wth the provson of a complete sgnal model takng all the CFO, TO and nsuffcent guard sequence nto consderaton for DCT-MCM system under multpath envronments, we can get the expresson for the desred symbol on the nth subcarrer of the ntended receved block (j = 0) y 0,n = γ (n, 0, n)a 0,n + + l=0,l =n + =, =0 l=0 γ (n, 0, l)a 0,l γ (n,, l)a,l + V n (38) As can be seen n Eq. (38), wth certan level of mperfectons ntroduced the receved symbol y 0,n wll suffer from nner-block and nter-block nterference plus coloured nose. The coloured nose has been analysed on the effect of 13084 VOLUME 6, 018

pre-flterng and correspondng compensaton methods have been proposed for optmum performance. But the nterference wll render the channel equalzaton matrx nondagonal by whch huge computatonal complexty has to be nvolved due to the nverson of channel equalzaton matrx especally when s farly large. On the other hand, the overall effectve nose s no longer channel nose domnated, resultng ML method out of use n ths case. In the sequel, we frst modfy our channel equalzaton functon for the ZF method. The other two performance enhanced methods MMSE and ML are then updated by the teraton results from a ZF equalzer. By regardng the correlated nterference term as an addtonal part of the overall effectve nose, we can avod very complex jont-detecton process. Consequently, the desred symbol a 0,n that transmtted on the nth subcarrer n the ntended receved block j = 0 can be recovered through the modfed ZF equalzer by the expresson n terms of ts desred sgnal component coeffcent as W ZF 0,n = γ (n, 0, n)h / γ (n, 0, n) (39) However, for MMSE and ML equalzers, snce the nterference s correlated wth desred subcarrer, drect equalzng operaton ncurs huge computatonal burden due to the destroyed orthogonal condton. And the channel equalzer matrx s no longer strctly dagonally domnant. To acheve optmum error rate performance as well as reduced equalzaton complexty, a ZF-based teratve technque s employed to address ths problem. Wth the provson of a typcal ZF equalzer functon n Eq. (39), we yeld the equalzed symbol x 0,n = y 0,n W0,n ZF. Supposng the recovered symbol from x 0,n after a de-mapper s ã 0,n, we can estmate the nterference term by feedng back the recovered symbols ã 0,n wth ther correspondng nterference power coeffcents n IFC 0,n = l=0,l =n γ (n, 0, l)ã 0,l + + =, =0 l=0 γ (n,, l)ã,l (40) As aforementoned energy-compacton property of DCT, the desred sgnal power s hghly concentrated on the modulated subcarrer and ts few neghbourng subcarrers. Ths means most of the nterference power coeffcents are not sgnfcant and there s no need to calculate every nterference term. Hence, we may relax the constrant to approxmate the overall nterference nose by makng the followng defnton: { = γ (n,, l) f γ (n,, l) > ξγ (n, 0, n); γ (n,, l) = (41) 0 f γ (n,, l) ξγ (n, 0, n). where ξ s the nterference power mpact factor whch allows flexblty for the balance between complexty and accuracy. Extensve tested results ndcate a value of 3% for ξ s applcable for most cases where good approxmaton of nterference FIGURE 7. BER performance for 8ASK DCT-MCM n normal and teratve detecton methods n the presence of mperfectons. nose can be obtaned wth very few terms. The approxmate nterference nose s thus formulated as ñ IFC 0,n = l=0,l =n γ (n, 0, l)ã 0,l + + =, =0 l=0 γ (n,, l)ã,l (4) By elmnatng the nterference nose before equalzer, the sparse effectve channel equalzaton matrx now becomes dagonally domnant. Consequently, upon nvokng the MMSE crteron n Eq. (7), the equalzed symbol can be yeld by x 0,n MMSE = (y 0,n ñ IFC 0,n ) σs γ (n, 0, n)h σs γ (n, 0, (43) n) + V n For the ML method, equalzaton functon n Eq. (9) s updated to: a 0,n A (1) exp( ỹ 0,n γ (n,0,n)a 0,n V n ) L = ln a 0,n A (0) exp( ỹ 0,n γ (n,0,n)a 0,n V n ) V n = { mn ( z γ (n, 0, n) 0,n a 0,n ) a 0,n A (1) + mn a 0,n A (0) ( z 0,n a 0,n ) } (44) where ỹ 0,n = y 0,n ñ IFC 0,n and z 0,n = ỹ 0,n /γ (n, 0, n). However, ths equalzaton algorthm only works well when the nterference s moderate. In the case of hardware mparment where ICI and ISI account for sgnfcant power loss, n order to estmate the nterference accurately, t s beneft to update our new demodulated results and terate them over Eq. (43) and (44) for several tmes. The ZF-based teratve nterference cancellaton detectons are verfed by BER performance under comparson wth normal detecton methods. The smulaton envronment comples wth what we assumed n Secton III. In the presence VOLUME 6, 018 13085

FIGURE 8. BER performance at E b / o = 16dB for MMSE and ML detectons for 8ASK DCT-MCM n the presence of mperfectons (a) CFO = 0.05, TO = 4, v = 7. (b) CFO = 0., TO = 4, v = 7. of mperfectons, we smulate the DCT-MCM system wth CFO = 0., TO = and reduce the prefx and suffx length from 1 to 7 (channel length s 11). As can be seen n Fg. 7, the new proposed teratve detecton methods can successfully cancellate the ICI and ISI nterference and therefore sgnfcantly mprove the system performance by only one teraton. A db gan s acheved by the teratve MMSE equalzer whle the teratve ML method shows even better attractve mprovements, whch demonstrates around 3 db gan superorty than the teratve MMSE equalzer. Fg. 8, on the other hand, shows the convergence property at E b / o = 16dB. oted that the number of teratons at zero means the results are ZF based and has not yet terated to MMSE and ML cases. As can be seen from the fgure, f the nterference level s moderate (e.g. CFO = 0.05, TO = 4, v = 7) to the transmtted sgnal power, sngle teraton s suffcent to acheve the optmum performance, whch s clearly dentfed n Fg. 8(a). However, when mperfecton errors play domnant role (e.g. CFO = 0., TO = 4, v = 7), ncreasng the number of teratons may be approprately adjusted for strkng a flexble trade off between the performance gan obtaned and the complexty burden mposed, especally for ML method as verfed n Fg. 8(b). VI. COCLUSIO In ths paper, we frst modelled the DCT-MCM systems wth several constrant condtons satsfed to acheve nterference free transmsson. The boundares for output SR gan between two to three folds s verfed for all subcarrers. As to the detectors, crteron for MMSE and ML methods are reformulated for DCT-MCM systems respectvely. Smulaton results llustrate the performance superorty of DCT-MCM than DFT-MCM systems when operatng far from the capacty lmt. In the presence of transcever mperfectons, we show that the advantage of energy compacton property by DCT stll holds over multpath channels. In order to combat the nterference arsng from mperfecton errors, new teratve equalzaton algorthms are proposed to effectvely mprove the performance of one-tap channel equalzaton algorthms. In the case of severe condtons where ICI and ISI are domnated, the proposed algorthms are able to mtgate the nterference effect by adjustng the number of teraton tmes. An nterestng topc for future research s to derve the exact theoretcal BER performance for the preflterng DCT-MCM systems. APPEDIX A Consderng the effectve channel coeffcents are n symmetrc after pre-flterng: h k = h k, we frstly consder E[h h j ] for the coeffcents on the rght sde where 0, j 0. The results then can be easly extended to the whole case. 1) when = j, t s easy to verfy E[h ] and E[h j ] are uncorrelated to each other. Consequently, we yeld: E[h h j ] = 0. ) when = j = 0, we have: E[f fj ] = E[f ] E[fj ] = 0.5σ σj. In that case, E[h h j ] = E[fp f L +p ] = E[fp ] E[f L +p ] = 1 σp 4 σ L +p. 3) when = j = 0, the expected value on fourth order for Gaussan varable s calculated by: E[f 4 ] = 3 ( 0.5σ ) 4 = 3 4 σ 4. Wthout loss of generalty, E[h h j ] 13086 VOLUME 6, 018

then can be gven by: E[h h j ] = E[f 4 p p ] + t=0 = 3 σp 4 4 + 1 E[f p f p+t ] p t=0 σ p σ p+t. (45) By combnng the above three condtons and extendng them to the whole effectve channel coeffcents case, we obtan the followng: E[h h j ] 1 σp 4 σ L +p = ±j = 0; = 3 σp 4 4 + 1 p σp σ p+t = j = 0; t=0 0 others. (46) APPEDIX B The output sgnal power on the n-phase branch at kth subcarrer s represented as: E[Hk ] = E[h 0 ] + 4 E[h n ] cos [ π nk] (47) whch equals to n=1 E[Hk ] = E[h 0 ] + E[h n ] (cos[π nk] + 1) (48) n=1 The functon above now can be consdered as a convex functon optmzaton problem and t s easy to fnd E[Hk ] wll acheve the lower peak value at the ponts k/ = 1/4 by takng ts dfferentaton functon E[H k ] k = 0. As a consequence, the mnmal value of E[Hk ] s mn{e[hk ]} = 3 σp 4 4 + 1 p t=0 σp σ p+t ( + ( )p 1 ( 1)p ) (49) Consderng σp subject to the exponental power delay dstrbuton n the Raylegh fadng channel, we have mn{e[hk ]} 3 σp 4 4 + = 1 4 σp 4 p t=0 σ p σ p+t + 0.5 (50) Recallng the Cauchy-Schwarz nequalty theorem: n=1 a 1 n ( n =1 a ), we obtan the followng 1 σp 4 4 1 4 1 L ( σ p ) = 1 4L (51) Combnng the above results, we have mn{e[hk ]} 0.5 + 1 4L. Accountng the other half of dstrbuted sgnal power on quadrature branch, the lower bound of the mnmal sgnal power gan s 1 + L 1. APPEDIX C Let us consder the calculaton of the component E γ (n,,l). Wthout loss of generalty, the general form for E γ (n,, l) s expressed as: E γ (n,, l) = β l β n +v 1 k,k =0 m,m = v 1 4 [cos π(l + f )(m + m + 1) + cos π(l + f )(m m ) ] [cos πn(k + k + 1) + cos πn(k k ) ] [δ(k k m + m ) + δ(k + k m m k) δ(k m k) δ(k m k)]r(k m k) (5) For smplcty, we make new summaton varables: m + m = u 1, m m = u, k + k = u 3 and k k = u 4. By defnng the nner weght functon: A l,n (u 1, u, u 3, u 4 ) = 1 4 [cos π(l+ f )(u 1+1) + cos π(l+ f )u ] [cos πn(u 3+1) + cos πnu 4 ], we obtan: E γ (n,, l) = βl β n A l,n (u 1, u, u 3, u 4 ) u 1 u u 3 u 4 [δ(u 4 u ) + δ(u 3 u 1 k) δ(k m k) δ(k m k)]r(k m k) = βl β n [ A l,n (u 1, u, u 3, u ) u 1 u u 3 + A l,n (u 3 k, u, u 3, u 4 ) u u 3 u 4 A l,n (u 3 k, u, u 3, u )] R(k m k) u u 3 = βl β n [ A l,n (u 1, u, u 3, u ) u 3 u =0 u 1 1 + A l,n (u 1, u, u 3, u ) u = () u 1 u 3 + A l,n (u 3 k, u, u 3, u 4 ) u 3 =0 u u 4 + k 1 + A l,n (u 3 k, u, u 3, u 4 ) u 3 = u u 4 + A l,n (u 3 k, u, u 3, u 4 ) u 3 =+ k u u 4 A l,n (u 3 k, u, u 3, u )] R(k m k) u u 3 = E γ 1 (n,, l) + E γ (n,, l) + E γ 3 (n,, l) + E γ 4 (n,, l) + E γ 5 (n,, l) E γ 6 (n,, l) (53) VOLUME 6, 018 13087

whereas the sx components of E γ (n,, l) are gven by: E γ 1 (n,, l) = β l β n A l,n (u 1, u, u 3, u ) R(k m k) u =0 u 1 u 3 +v u 1 A l,n (u 1, u, u 3, u ) A l,n (u 3 k, u, u 3, u 4 ) = βl β n u =0 u 1 = v A l,n (u 1 + u, u, u, u ) u 3 =u R(u 3 u u 1 k) (54) E γ (n,, l) = βl β n 1 u = () u 1 u 3 R(k m k) 1 +v+u 1 = βl β n A l,n (u 1 u, u, u 3 u = +1 u 1 = v u 3 = u + u, u ) R(u 3 + u u 1 k) (55) E γ 3 (n,, l) = βl β n u 3 =0 u u 4 R(k m k) u 3 v+u 3 k = βl β n A l,n (u 3 k, u u 3 v u 3 =0 u 4 =0 u =0 + k, u 3, u 4 u 3 ) R(u 4 u + v k) E γ 4 (n,, l) + k 1 = βl β n A l,n (u 3 k, u, u 3, u 4 ) u 3 = u u 4 R(k m k) + k 1 u 3 v+u 3 k = βl β n A l,n (u 3 k, u u 3 = u 4 =0 u =0 u 3 v + k, u 3, u 4 + + u 3 ) R(u 4 + u 3 u + v + 1 k) (56) E γ 5 (n,, l) = β l β n u 3 =+ k A l,n (u 3 k, u u 4 u, u 3, u 4 ) R(k m k) u 3 = βl β n u 3 =+ k u 4 =0 +v u 3 + k u =0 A l,n (u 3 k, u v + + u 3 k, u 3, u 4 + + u 3 ) R(u 4 u + v + k) (57) E γ 6 (n,, l) = βl β n A l,n (u 3 k, u, u 3, u ) R(k m k) = β l β n u u 3 u 1 =0 u =0 + cos π(l + f )(u 1 u ) ] [cos πn(u 1 + u + 1) 1 4 [cos π(l + f )(u 1 + u k + 1) + cos πn(u 1 u ) ] R(0) (58) ACKOWLEDGMET Ths paper was presented n part at the 016 IEEE Internatonal Symposum on Wreless Communcaton Systems. 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[4] C. He, L. Zhang, J. Mao, A. Cao, P. Xao, and M. A. Imran, Output SR analyss and detecton crtera for optmum DCT-based multcarrer system, n Proc. Int. Symp. Wreless Commun. Syst. (ISWCS), Sep. 016, pp. 59 64. [5] V. Sanchez, P. Garca, A. M. Penado, J. C. Segura, and A. J. Rubo, Dagonalzng propertes of the dscrete cosne transforms, IEEE Trans. Sgnal Process., vol. 43, no. 11, pp. 631 641, ov. 1995. [6] F. Cruz-Roldan, M. E. Domnguez-Jmenez, G. S. Vdal, P. Amo-Lopez, M. Blanco-Velasco, and A. Bravo-Santos, On the use of dscrete cosne transforms for multcarrer communcatons, IEEE Trans. Sgnal Process., vol. 60, no. 11, pp. 6085 6090, ov. 01. [7] H. Steendam and M. Moeneclaey, Analyss and optmzaton of the performance of OFDM on frequency-selectve tme-selectve fadng channels, IEEE Trans. Commun., vol. 47, no. 1, pp. 1811 1819, Dec. 1999. [8] IEEE Standard for Informaton Technology Telecommuncatons and Informaton Exchange Between Systems Local and Metropoltan Area etworks-specfc Requrements Part 11: Wreless LA MAC and PHY Specfcatons, IEEE Standard 80.11-1997, 1997, p. I-445. [9] M. K. Ozdemr and H. Arslan, Channel estmaton for wreless OFDM systems, IEEE Commun. Surveys Tuts., vol. 9, no., pp. 18 48, nd Quart., 007. [30] A. Trmeche,. Boukd, A. Sakly, and A. Mtbaa, Performance analyss of ZF and MMSE equalzers for MIMO systems, n Proc. 7th Int. Conf. Desgn Technol. Integr. Syst., May 01, pp. 1 6. [31] C.-Y. Hung and W.-H. Chung, An mproved MMSE-based MIMO detecton usng low-complexty constellaton search, n Proc. IEEE Globecom Workshops, Dec. 010, pp. 746 750. [3] J. Mao, M. A. Abdullah, P. Xao, and A. Cao, A low complexty 56QAM soft demapper for 5G moble system, n Proc. EuCC, Jun. 016, pp. 16 1. CHAG HE receved the master s degree n wreless communcatons from the Unversty of Southampton. He s currently pursung the Ph.D. degree n electronc engneerng wth the Insttute for Communcaton Systems, Unversty of Surrey, U.K. Hs current research nterests nclude dscrete cosne transform-based mult-carrer transcever desgn and ts applcatons on ndex modulaton schemes. LEI ZHAG receved the B.Eng. degree n communcaton engneerng and the M.Sc. degree n electromagnetc felds and mcrowave technology from orthwestern Polytechnc Unversty, Chna, and the Ph.D. degree from the Unversty of Sheffeld, U.K. He was a Research Engneer wth the Huawe Communcaton Technology Laboratory and a Research Fellow at the 5G Innovaton Centre, Insttute of Communcatons, Unversty of Surrey, U.K. He s currently a Lecturer wth the Unversty of Glasgow. He holds over ten nternatonal patents on wreless communcatons. Hs research nterests broadly le n the communcatons and array sgnal processng, ncludng rado access network slcng, new ar nterface desgn (waveform, frame structure, and so on), Internet of Thngs, mult-antenna sgnal processng, cloud rado access networks, massve MIMO systems, full-duplex, and so on. He s an Assocate Edtor of the IEEE ACCESS. VOLUME 6, 018 JUQUA MAO receved the B.Eng. degree n computer scence and technology from Qngdao Unversty, Chna, n 003, and the M.Sc. degree n computer scence and technology from the Bejng Unversty of Posts and Telecommuncatons, Chna. He s currently pursung the Ph.D. degree wth the Insttute for Communcaton Systems, Unversty of Surrey, U.K. He was a Techncal Engneer wth Huawe Technologes Company Ltd., from 006 to 01. In 01, he joned the Department of Electrcal Engneerng, London Southbank Unversty, as a Lecturer. Hs research nterests nclude 5G new waveforms, physcal-layer network slcng, and non-orthogonal multple access. AIJU CAO has over 18 years of experence n wreless communcatons research and development from baseband processng to network archtecture, ncludng the desgn and optmzaton of commercal UMTS/LTE base-staton and handset products, Hetet and small cell enhancement, and so on. He has also been nvolved n standardzaton works and contrbuted to several 3GPP techncal reports. He s currently a Prncpal Archtect wth the ZTE R&D Center, Sweden (ZTE Wstron Telecom AB). He s also actve n academc and ndustral workshops and conferences related to the future wreless networks as a panelst or a (co-) author of publshed papers n refereed journals and nternatonal conferences. In addton, he holds over 50 granted or pendng patents. Hs current research nterests nclude 5G technologes related to the new energy-effcent unfed ar-nterface and network archtecture, ncludng new waveform desgn, nonorthogonal multple access schemes, random access challenges, and nnovatve sgnalng archtecture for 5G networks. PEI XIAO (SM 11) was wth ewcastle Unversty and Queen s Unversty Belfast. He also held postons at oka etworks n Fnland. He s currently a Professor wth the Insttute for Communcaton Systems, 5G Innovaton Centre (5GIC), Unversty of Surrey. He s also the Techncal Manager of 5GIC, leadng the Research Team at on the new physcal-layer work area, and coordnatng/supervsng research actvtes across all the work areas wthn 5GIC. He has publshed extensvely n the felds of communcaton theory and sgnal processng for wreless communcatons. MUHAMMAD ALI IMRA (SM 1) receved the M.Sc. (Hons.) and Ph.D. degrees from Imperal College London, U.K., n 00 and 007, respectvely. He has over 18 years of combned academc and ndustry experence, where he was prmarly nvolved n the research areas of cellular communcaton systems. He s currently the Vce Dean wth the Glasgow College, UESTC, and a Professor of communcaton systems wth the School of Engneerng, Unversty of Glasgow. He s also an Afflate Professor wth the Unversty of Oklahoma, USA, and a Vstng Professor wth the 5G Innovaton Centre, Unversty of Surrey, U.K. He holds 15 patents, has authored/co-authored over 300 journal and conference publcatons. He has supervsed over 30 successful Ph.D. graduates. He s a Senor Fellow of the Hgher Educaton Academy, U.K. He receved an Award of Excellence n recognton of hs academc achevements, conferred by the Presdent of Pakstan, the IEEE Comsoc s Fred Ellersck award 014, the FEPS Learnng and Teachng Award 014, and the Sentnel of Scence Award 016. He was twce nomnated for the Tony Jean s Inspratonal Teachng Award. He was a shortlsted Fnalst for the Wharton-QS Stars Awards 014, the QS Stars Remagne Educaton Award 016 for nnovatve teachng, and the VC s Learnng and Teachng Award from the Unversty of Surrey. 13089