IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY

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1 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY Low-Complexty ICI Mtgaton Methods for Hgh-Moblty SISO/MIMO-OFDM Systems Chao-Yuan Hsu, Student Member, IEEE, and Wen-Rong Wu, Member, IEEE Abstract In orthogonal frequency-dvson multplexng (OFDM) systems, t s generally assumed that the channel response s statc n an OFDM symbol perod. However, the assumpton does not hold n hgh-moblty envronments. As a result, ntercarrer nterference (ICI) s nduced, and system performance s degraded. A smple remedy for ths problem s the applcaton of the zero-forcng (ZF) equalzer. Unfortunately, the drect ZF method requres the nverson of an N N ICI matrx, where N s the number of subcarrers. When N s large, the computatonal complexty can become prohbtvely hgh. In ths paper, we frst propose a low-complexty ZF method to solve the problem n sngle-nput sngle-output (SISO) OFDM systems. The man dea s to explore the specal structure nherent n the ICI matrx and apply Newton s teraton for matrx nverson. Wth our formulaton, fast Fourer transforms (FFTs) can be used n the teratve process, reducng the complexty from O(N 3 ) to O(N log 2 N ). Another feature of the proposed algorthm s that t can converge very fast, typcally n one or two teratons. We also analyze the convergence behavor of the proposed method and derve the theoretcal output sgnal-to-nterference-plus-nose rato (SINR). For a multple-nput multple-output (MIMO) OFDM system, the complexty of the ZF method becomes more ntractable. We then extend the method proposed for SISO-OFDM systems to MIMO-OFDM systems. It can be shown that the computatonal complexty can be reduced even more sgnfcantly. Smulatons show that the proposed methods perform almost as well as the drect ZF method, whle the requred computatonal complexty s reduced dramatcally. Index Terms Fast Fourer transform (FFT), ntercarrer nterference (ICI), Newton s teraton. I. INTRODUCTION ORTHOGONAL frequency-dvson multplexng (OFDM) s known to be a successful technque for copng wth the multpath channel effect n wreless communcatons [1]. For conventonal OFDM systems, t s usually assumed that the channel s statc n an OFDM symbol perod. However, n hgh-speed moble envronments, ths assumpton no longer holds. If the channel s tme varant n an OFDM symbol perod, orthogonalty wll be destroyed. Intercarrer nterference (ICI) s nduced, and system performance s degraded. The behavor of moblty-nduced ICI has been extensvely nvestgated n the lterature [2] [7]. In [2] and [3], t s shown that the nterference on a subcarrer manly comes from neghborng Manuscrpt receved January 3, 2008; revsed October 1, Frst publshed December 22, 2008; current verson publshed May 29, Ths work was supported by the Natonal Scence Councl, Tawan, under Grant NSC E The revew of ths paper was coordnated by Dr. W. Su. The authors are wth the Department of Communcaton Engneerng, Natonal Chao Tung Unversty, Hsnchu 300, Tawan (e-mal: cyhsu.cm92g@ nctu.edu.tw; wrwu@faculty.nctu.edu.tw). Dgtal Object Identfer /TVT subcarrers. In addton, the nterference level s proportonal to the Doppler frequency. Many technques have been developed to solve the mobltynduced ICI problem. Two algorthms are well known, namely, 1) the zero-forcng (ZF) method and 2) the mnmum mean square error (MMSE) equalzaton method. Unfortunately, these methods requre the nverson of an N N ICI matrx, where N s the number of subcarrers. When N s large, the requred computatonal complexty becomes hgh. Systems wth a lot of subcarrers are not uncommon n real-world applcatons. For example, for the applcaton of dgtal vdeo broadcastng (DVB), the number of subcarrers can be as large as To solve the problem, a smpler ICI equalzer for the ZF method was developed n [8]. As mentoned, ICI on a subcarrer manly comes from a few neghborng subcarrers. Thus, ICI from other subcarrers can then be gnored. Ths method has good performance n low-moblty envronments. In hgh-moblty envronments, however, the number of nsgnfcant ICI terms wll be decreased, and the computatonal complexty wll be sgnfcantly ncreased. Successve nterference cancellaton (SIC) and parallel nterference cancellaton (PIC) are two well-known multuser nterference (MUI) cancellaton technques n code-dvson multple-access systems. Snce the characterstc of ICI s smlar to that of MUI, these methods can be drectly appled to ICI suppresson n OFDM systems. A method that combnes the MMSE method and SIC was frst proposed n [9]. Later, t was mproved wth a recursve method n [10], reducng the requred complexty further. Although good performance can be acheved wth these methods, the requred complexty s stll hgh, and the tme delay can be ntolerably large. The PIC technque was then employed to solve the problem [11] [15]. Although the processng delay s greatly reduced, the performance s dscounted as well. Other approaches use transmtter frequency-doman codng or beamformng to reduce ICI or to enhance the receved sgnal-to-nterference-plus-nose rato (SINR). Interested readers are referred to [16] [19]. Apart from the processng n the frequency doman, some researchers also explore that n the tme doman. In [20], a tme-doman flterng technque, maxmzng the sgnal-to-iciplus-nose rato, was proposed for sngle-nput sngle-output (SISO)/multple-nput multple-output (MIMO) OFDM systems. One dsadvantage of ths method s that t requres matrx operatons to solve a generalzed egenvalue problem. Another approach nvolves the use of a tme-varant tme-doman equalzer, makng the tme-varant channel less varant. Transferrng the equalzer from tme doman to frequency doman, one can obtan a frequency-doman per-tone equalzer (PTEQ) /$ IEEE

2 2756 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY 2009 The PTEQ was orgnally proposed to deal wth the nsuffcent cyclc prefx (CP) problem n OFDM systems. Lately, t has been extended to suppress ICI n SISO/MIMO-OFDM systems [21] [25]. The PTEQ s well known for ts good performance; however, ts mplementaton complexty and storage requrement can be hgh. In [26], a two-stage equalzer was proposed. In the frst stage, a tme-doman wndowng technque s used to shorten the ICI response n the frequency doman. In the second stage, an teratve MMSE method s used to suppress the resdual ICI. Although the wndowng approach s smple, the teratve MMSE processng s not trval. As mentoned, the man problem n the ZF method s the matrx nverson. Thus, conductng ths operaton effcently becomes the man concern. It s found that some teratve methods can be much more effcent than the drect nverson method. In [27], the Gauss Sedel teraton was used to conduct the matrx nverson. However, t stll needs a matrx nverse n ts teratve process. Another method, called operator perturbaton, was recently proposed [28]. Smlar to [27], ths method also requres a matrx nverse n ts teratons. Thus, the computatonal complexty for the methods n [27] and [28] s stll hgh. In [30], t was dscovered that the ICI matrx for a lnear tme-varant (LTV) channel model exhbts a specal structure, allowng the applcaton of fast Fourer transforms (FFTs) n the matrx nverson. The LTV channel model was proposed n [8], and t s orgnally desgned for tme-varant channel estmaton [29]. Explotng ths structure, Fu et al. [30] proposed a power-seres expanson (PSE) method for the ICI matrx nverson. Although the PSE method can greatly reduce the computatonal complexty, t does not perform well n hghmoblty envronments. In ths paper, we propose a low-complexty ZF method to solve the moblty-nduced ICI problem n SISO-OFDM [32] and MIMO-OFDM systems. Smlarly to [30], we explot the specal structure nherent n the LTV channel model. We frst develop a method that can mplement Newton s teraton for the ICI matrx nverson n SISO-OFDM systems. Wth our specally desgned archtecture, FFTs can be used, reducng the computatonal complexty effectvely. We also propose a method for the calculaton of ntal values. Wth those values, Newton s teraton can converge very fast, usually wthn a couple of teratons. Unlke the PSE method [30], our method works well even n hgh-moblty envronments. Smulaton results show that the performance of the proposed low-complexty ZF method can be as good as that of the drect ZF method. However, the requred computatonal complexty s reduced from O(N 3 ) to O(N log 2 N). We also analyze the convergence behavor of the proposed low-complexty ZF algorthm and derve the theoretcal output SINR. Wth a new MIMO-OFDM system formulaton, we then extend the proposed method to ICI suppresson n MIMO-OFDM systems. It s shown that n MIMO-OFDM systems, the computatonal complexty can be reduced even more sgnfcantly. For an M M system, where M s the number of transmt (receve) antennas, the proposed algorthm can reduce the computatonal complexty from O(M 3 N 3 ) to O(MN log 2 N). The rest of ths paper s organzed as follows. Secton II presents the proposed ICI mtgaton method for SISO-OFDM systems. Secton III descrbes the proposed ICI cancellaton method for MIMO-OFDM systems. Secton IV conducts the performance analyss. Smulaton results are reported n Secton V, demonstratng the effectveness of the proposed method. Fnally, conclusons are drawn n Secton VI. II. ICI MITIGATION FOR SISO-OFDM A. ZF Method Consder a moble OFDM system whose channel varaton s large such that the moblty-ntroduced ICI cannot be gnored. It was shown n [29] that the LTV channel model can be used to approxmate a tme-varant channel for normalzed Doppler frequency up to 20%, where the normalzed Doppler frequency s defned as the maxmum Doppler frequency dvded by subcarrer spacng. Usng the LTV channel model, we can approxmate the tme-varant channel n a specfc OFDM symbol perod as h j (n) =h 0,j + n h 1,j (1) where n s the tme ndex, h j (n) s the jth-tap channel response at tme nstant n, h 0,j s ts constant term, and h 1,j s ts varaton slope. We assume that n s 0 at the mdpont of an OFDM symbol. Let h 0 =[h 0,0,h 0,1,...,h 0, ] T, h 1 =[h 1,0,h 1,1,...,h 1, ] T, H 0 = cr(h 0 ), and H 1 = cr(h 1 ), where cr(c) denotes a crculant matrx wth the frst column vector beng c. In addton, we defne v 1 =[( N +1)/2, ( N +3)/2,...,(N 1)/2] T and V 1 = dag(v 1 ), where the notaton dag(d) denotes a dagonal matrx wth the dagonal vector of d. Accordng to (1), we can express the receved tme-doman sgnal n the OFDM symbol (after CP removal) as y =(H 0 + V 1 H 1 )x + z (2) where y and x are the receve and transmt tme-doman sgnals, respectvely, and z s the nose vector (addtve whte Gaussan). Let G be a untary dscrete Fourer transform (DFT) matrx wth the property that GG H = I N, where I N s an N N dentty matrx. Furthermore, let ỹ = NGy, x = NGx, z = NGz, h0 = NGh 0, h 1 = NGh 1, H 0 = dag( h 0 ), and H 1 = dag( h 1 ). Multplyng both sdes of (2) by NG, we can express the receve sgnal n the frequency doman as ỹ =[ H 0 + GV 1 G H H1 ] x + z = M x + z (3) where M = H 0 + GV 1 G H H1. Note that M can also be rewrtten as M = H 0 + Ṽ1 H 1, where Ṽ1 = GV 1 G H = [cr(ṽ 1 )] T and ṽ 1 =(1/ N)G H v 1. Denote the ZF equalzed sgnal as x ZF. Then, we can have x ZF = M 1 ỹ. From the above formulaton, t s smple to see that drect mplementaton of the ZF method wll requre hgh computatonal complexty f N s large. The PSE method was

3 HSU AND WU: LOW-COMPLEXITY ICI MITIGATION METHODS FOR HIGH-MOBILITY SISO/MIMO-OFDM SYSTEMS 2757 ntroduced n [30] to solve the problem. The dea s to express M 1 as [( ) ] 1 M 1 = I N + GV 1 G H H1 H 1 0 H0 = H 1 0 (I N P) 1 (4) where P = GV 1 G H H and H = H H Then, (I N P) 1 s expanded wth a power seres, and hgh-order terms are truncated,.e., (I N P) 1 Q =0 P, where Q s the hghest order retaned n the expanson. The convergence condton for ths expanson s that P < 1 [30], where P ndcates the p-norm of P [31]. Fnally, the equalzed x, whch s denoted 1 as x PSE, s equal to H Q 0 =0 a, where a = P ỹ.note that a +1 = Pa = GV 1 G H (Ha ). Thus, a can be recursvely calculated. Furthermore, wth the specal structure of P, FFTs/nverse FFTs (IFFTs) can be used to calculate a. Although the computatonal complexty can be reduced effectvely, the performance of the PSE method s unsatsfactory n hgh-moblty envronments. In Secton II-B, we wll present the proposed method to solve the problem. B. Proposed ICI Mtgaton Method As mentoned, the PSE method s unsatsfactory n hghmoblty envronments. To solve the problem, we seek for a more powerful teratve method for matrx nverson. Specfcally, we fnd that Newton s teraton s useful. Newton s teraton s well known for ts fast convergence [33], [34] and has been nvestgated extensvely [35] [38]. Let W k be the estmated nverse of M at the kth teraton. The (k +1)th Newton s teraton can be descrbed as follows: W k+1 =(2I N W k M)Wk, k =0, 1, 2,...,. (5) Let Rk = I N W k M represent the estmaton resdual. 2 Equaton (5) mples that I N W k M IN W 0 M k for all k. If I N W 0 M < 1, we then have a quadratc convergence [39]. From (5), we can clearly see that Newton s teraton requres matrx-to-matrx multplcatons whose computatonal complexty s hgh. As a matter of fact, ts complexty s even hgher than that of the drect ZF method when k s large. Thus, a drect applcaton of Newton s teraton for matrx nverson s not feasble. In what follows, we propose a method of solvng the problem. Iteratng (5), we obtan a sequence of matrces {W 0, W 1,...,W k }. The relatonshp between W 0 and W k can be found straghtforwardly n W k = 2 k 1 m=0 c k,m (W 0 M) m W 0 (6) where c k,m s the coeffcent of the mth summaton term n (6). Equaton (6) can be seen as an expanson form of Newton s teraton, whle (5) s an teratve form. It turns out that to obtan a low-complexty algorthm, we have to use the expanson form. Assgn c k,m s as coeffcents of a polynomal functon of z,.e., f k (z) =c k,0 z 0 + c k,1 z c k,2 k 1z 2k 1. Then, the polynomal f k+1 (z) can be derved from f k (z) as f k+1 (z) =2f k (z) z[f k (z)] 2, where f 0 (z) =1. Ths s to say that c k,m can be recursvely calculated. Note that our objectve s to obtan the equalzed result W k ỹ and not the matrx nverse W k tself. By multplyng both sdes of (6) by ỹ and lettng x k = W k ỹ and u m =(W 0 M) m W 0 ỹ,wehave the equalzed result as x k = 2 k 1 m=0 From the defnton of u m, we can then have c k,m u m. (7) u m+1 =(W 0 M)um. (8) As a result, u m can be recursvely calculated as well. Usng ths approach, we have transformed matrx-to-matrx multplcatons n (6) nto matrx-to-vector multplcatons n (7) and (8). To complete our low-complexty algorthm, we make use of the specal structure nherent n the ICI matrx. From the foregong dervaton, we know that M = H 0 + GV 1 G H H1. By usng ths structure, we can then rewrte (8) as [ u m+1 = W 0 ( H ] 0 + GV 1 G H H1 ) u m [ = W 0 H0 u m + GV 1 G H ( H ] 1 u m ). (9) Note that H0, H1, and V 1 are all dagonal matrces. If we further pose a constrant that W 0 s a dagonal matrx, we can transform matrx-to-matrx operatons nto vector-to-vector and DFT/IDFT operatons, as shown n (9). As known, DFTs/IDFTs can be effcently mplemented wth FFTs/IFFTs whose complexty s O(N log 2 N). Thus, the computatonal complexty of the proposed algorthm s O(N log 2 N). The constrant on W 0 may not always yeld satsfactory performance n all scenaros. Instead of a dagonal matrx, we may let W 0 be a low-bandwdth banded matrx. Let the (, j)th entry of a matrx B be denoted as B(, j). The banded matrx s defned as B(, j) 0,f j D, and B(, j) =0, otherwse. Here, D s the bandwdth of the banded matrx. If D =0, the banded matrx s reduced to a dagonal matrx. If D =1, the matrx wll have three nonzero dagonal vectors. Wth ths type of W 0,the computatonal complexty n (9) wll only be ncreased slghtly. For later smulatons, we wll only consder the cases of D =0 and D =1. It turns out that for D =1, the performance of the proposed algorthm s good enough. The fnal thng we have to deal wth s how to determne the ntal matrx W 0. A good ntal matrx can reduce the number of teratons sgnfcantly and provde good cancellaton performance. As known, the man functon of ZF s to nvert the ICI matrx, and n the deal case, I N W k M = 0N, where 0 N s an N N zero matrx. As a result, f I N W 0 M can be made as close to 0 N as possble, fast convergence n Newton s algorthm can be obtaned. Based on ths dea, we propose to mnmze the Frobenus norm of I N W 0 M,.e., W 0 = arg mn I N W M 2 F (10) W

4 2758 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY 2009 TABLE I COMPLEXITY COMPARISON AMONG N-ZF, PSE, AND DIRECT ZF METHODS IN A SISO-OFDM SYSTEM where R F means the Frobenus norm of R. Snce the dervaton of the optmal ntal values s tedous, we put that n Appendx A. For easy reference, we denote the proposed low-complexty ZF method as the Newton-ZF (N-ZF) method. Fnally, we summarze the requred complexty for the N-ZF method, the PSE method, and the drect ZF method for a SISO-OFDM system n Table I. III. PROPOSED ICI CANCELLATION METHOD FOR MIMO-OFDM ICI suppresson s more challengng n MIMO-OFDM systems. In the applcaton of spatal multplexng, nterantenna nterference s further ntroduced. It s possble to formulate the whole system wth a lnear model and to apply a ZF equalzer to suppress all nterference. However, the dmenson of the MIMO-OFDM ICI matrx wll become much larger than that n a SISO-OFDM system, and the requred complexty s even more ntractable. In ths secton, we extend the method proposed n Secton II to solve the problem. By consderng an M M system, where M s the number of transmt/receve antennas, and by usng the sgnal model n SISO-OFDM systems, we can express the frequency-doman sgnal for the th receve-antenna as ỹ = M M,j x j (11) j=1 where ỹ s the frequency-doman sgnal n the th receve antenna, x j s the frequency-doman sgnal n the jth transmt antenna, and M,j s the ICI matrx for the jth transmt antenna and the th receve antenna. By stackng all the receve frequency-doman sgnals from all antennas n a column vector, we have the followng sgnal model: ỹ = M x + z (12) where ỹ =[ỹ T 1, ỹ T 2,...,ỹ T M ]T s the receve frequencydoman sgnal, x =[ x T 1, x T 2,..., x T M ]T s the transmt frequency-doman sgnal, z =[ z T 1, z T 2,..., z T M ]T s the frequency-doman nose, and the frequency-doman ICI channel matrx can be expressed as follows: M 1,1 M1,2... M1,M M 2,1 M2,2... M2,M M = (13) M M,1 MM,2... MM,M For ease of descrpton, we only consder a 2 2MIMO- OFDM system n the followng dervaton. However, the proposed algorthm can be extended to a general M M MIMO-OFDM system wthout any dffculty. Smlar to the channel model n SISO-OFDM systems, we also use the LTV model for MIMO-OFDM channels. Thus, we can obtan the MIMO-OFDM ICI matrx gven by M = = = M1,1 M1,2 M 2,1 M2,2 Ã0 + GV 1 G H Ã 1 B0 + GV 1 G H B1 C 0 + GV 1 G H C1 D0 + GV 1 G H D1 Ã0 B0 GV1 G + H 0 N Ã1 B1 C 0 D0 0 N GV 1 G H C 1 D1 (14) where Ã, B, C, and D play the same role as H n a SISO-OFDM system. The derved sgnal model s obtaned by groupng together subcarrer sgnals n the same antenna. For the purpose of comparson, we consder a ZF equalzaton method, gnorng the ICI effect n (14). In ths case, the ICI channel matrx turns out to be Ã0 B0 M s =. (15) C 0 D0 1 Thus, the equalzed sgnal can be obtaned wth x s = M s ỹ. Usng the block matrx nverson formula, we know that to obtan the th subcarrer equalzed sgnal for antenna one or two, the method wll requre two weghts. For ease of reference, we denote ths equalzaton method as a two-tap frequencydoman equalzer (FEQ). Usng the derved MIMO-OFDM ICI matrx n (14), we can now apply Newton s teraton to mplement the ZF equalzer. Wth (6), we can obtan the equalzed result as x k = 2 k 1 m=0 c k,m v m (16) where x k, c k,m, and v m are defned as those n (7). Let the ntal matrx W 0 be composed of four N N matrces expressed as Wα W W 0 = β (17) W γ W ω

5 HSU AND WU: LOW-COMPLEXITY ICI MITIGATION METHODS FOR HIGH-MOBILITY SISO/MIMO-OFDM SYSTEMS 2759 TABLE II COMPLEXITY COMPARISON BETWEEN N-ZF AND DIRECT ZF METHODS IN A 2 2 MIMO-OFDM SYSTEM and v m =[v T m,1, v T m,2] T. Accordng to the defnton of v m, we can obtan v m+1 as (18), shown at the bottom of the page. Note that Ã, B, C, D, and V 1 are all dagonal matrces. It s obvous that f we let W α, W β, W γ, and W ω be low-bandwdth banded matrces, (18) can be mplemented by vector and DFT/IDFT operatons. Note that the DFT sze s N nstead of 2N. Thus, the requred computatonal complexty s O(MN log 2 N). It s straghtforward to see that the computatonal complexty of the drect ZF method s O(M 3 N 3 ). The complexty reducton acheved by the proposed method n MIMO-OFDM systems can be greater compared wth that n SISO-OFDM systems. The fnal problem s how the 2N 2N ntal matrx can be optmally determned. As we dd n SISO-OFDM systems, we use the mnmum-frobenus-norm crteron. Snce the dervaton s lengthy, we put t n Appendx B. Fnally, we summarze the computatonal complexty of the N-ZF and drect ZF methods for a 2 2 MIMO-OFDM system n Table II. Note that the proposed method can be extended to a general M T M R MIMO-OFDM system, where M T <M R. In such a system, we can use the least squares (LS) method, nstead of ZF, to cancel ICI. Usng the LS method, we can have the equalzed sgnal vector as x =( M H M) 1 MHỹ = Q 1 y (19) where Q = M H M, and y = MHỹ. The matrx Q, nhertng the propertes of M, conssts of dagonal matrces and FFT/IFFT matrces as well. As a result, the proposed method dscussed above can be appled. Snce the dervaton s smple and straghtforward, t s omtted here. Due to the applcaton of the LS method, the requred complexty n ths scenaro wll be somewhat hgher. IV. PERFORMANCE ANALYSIS For the proposed algorthm, the teraton number s usually preset. Unlke other teratve algorthms, the convergence s not a concern here. The reason we can use a preset teraton number s due to the fast convergence property of Newton s teraton, as well as our good ntal values. If the proposed algorthm converges, only a small number of teratons are necessary. On the other hand, f the proposed algorthm dverges, the preset number of teratons wll lmt the performance degradaton. As a matter of fact, even for dvergence cases, we can stll have mproved SINRs f the teraton number s set properly. We wll provde ntutve statements to explan why ths s true. It turns out that the determnaton of the teraton number s smple and straghtforward. Here, we start wth the analyss of convergence behavor. After that, we wll derve theoretcal SINRs that the proposed algorthm can provde. Note that snce the SISO/MIMO-OFDM sgnal models are smlar n form, the only dfference s the dmenson. Thus, we focus on a SISO-OFDM system, but the analyss can be extended to MIMO-OFDM systems. We frst perform the egenvalue decomposton for R 0 as follows: R 0 = UDU 1 (20) where U =[u 0, u 1,...,u ] s a matrx composed of egenvectors of R 0, and D = dag([λ 0,λ 1,...,λ ] T ) conssts of egenvalues λ. We assume that λ λ j for j. Snce R k = R 2 k 1, we can then decompose R k as R k = UD 2k U 1. (21) If λ 0 < 1, then R k 0 N as k. Thus, we can have the convergence condton for Newton s teraton as ρ( R 0 ) < 1, where ρ( R 0 ) denotes the spectral radus of R 0 ; the spectral radus ndcates the largest absolute value of all egenvalues [34]. Ths s equvalent to say that for Newton s teraton to converge, the ampltudes of all egenvalues of R 0 have to be smaller than one. For a moderate moble speed, ths condton holds for most cases. If not, the number of egenvalues wth ampltudes greater than one s small, and ther ampltudes do not devate from one too much. These results can easly be observed from smulatons, though they are dffcult to prove theoretcally. In what follows, we wll frst show that even for dvergence cases, we may stll beneft from Newton s teraton. Let U 1 =[p 0, p 1,...,p ] T, λ > 1 for =0, 1,...,P 1 and λ < 1 for = P, P +1,...,N 1. } Wα W v m+1 = β ]{[Ã0 B0 GV1 G + H 0 N ][Ã1 B1 vm,1 W γ W ω C 0 D0 0 N GV 1 G H C 1 D1 v m,2 [ Wα W = β ]{[Ã0 v m,1 + B ] [ 0 v m,2 GV1 G W γ W ω C 0 v m,1 + D + H (Ã1v m,1 + B ]} 1 v m,2 ) 0 v m,2 GV 1 G H ( C 1 v m,1 + D 1 v m,2 ) (18)

6 2760 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY 2009 By defnton, R k = I N W k M. Then, we can represent the ICI matrx as P 1 W k M = IN =0 λ 2k u p T As for W k, we can reformulate t as j=p λ 2k j u j p T j. (22) W k =(I N + R k 1 )(I N + R k 2 )...(I N + R 0 )W 0. (23) By usng (21), we can further express W k as W k = j=0 φ j,k u j p T j W 0 (24) where φ j,k = k 1 =0 (1 + λ2 j ). Wth (22) and (24), the ZF-equalzed sgnal can be expressed as P 1 x k = x =0 λ 2k u p T x j=p λ 2k j u j p T j x + j=0 φ j,k u j p T j ṽ0 (25) where ṽ 0 = W 0 ṽ. Snce the egenvectors {u 0, u 1,...,u } span the N-dmensonal space, we can decompose x and ṽ 0 usng these egenvectors. Let x = l=0 β lu l and ṽ 0 = l=0 γ lu l, respectvely. Then, we can rewrte (25) as P 1 x k = x =0 λ 2k β u j=p λ 2k j β j u j + j=0 φ j,k γ j u j. (26) Let x k =[x k,0, x k,1,...,x k, ] T and x=[ x 0, x 1,..., x ] T. Thus, the mth subcarrer sgnal after equalzaton can be expressed as x k,m = x m + f 1,k,m + f 2,k,m + f 3,k,m (27) where f1,k,m = P 1 =0 λ2k β u (m), f2,k,m = j=p λ2k j β ju j (m), and f3,k,m = j=0 φ j,kγ j u j (m). From (27), we can see that the equalzed sgnal suffers from three nterference terms. For f 1,k,m, t wll become large when k ncreases; however, for f 2,k,m, t wll become small when k ncreases. As for the nose term f 3,k,m, ts dependence on k s not strong. As mentoned, only a few egenvalues ampltudes wll be larger than one (.e., P s small), and ther ampltudes often do not devate from one too much. Then, t s easy to see that the decreasng amount of f 2,k,m wll be larger than the ncreasng amount of f 1,k,m n the early teraton. Thus, for dvergence cases, the nterference wll decrease frst and then ncrease as the teraton proceeds. If we can stop the teraton before the overall nterference ncreases, we can stll have performance gan even though the teraton dverges eventually. Addtonally, we can ncrease D to make the ntal matrx closer to the exact matrx nverse. Ths way, P may be mnmzed, and f 1,k,m wll decrease, whch makes the proposed method work better. Because of the fast convergence property of Newton s method, the number of teratons requred s small. For example, t can be as small as one or two when D =1. For dvergent cases, the overall nterference stll decreases n the frst one or two teratons. Snce the performance of an OFDM-based system depends on each subcarrer SINR, we wll analyze the subcarrer SINR of the proposed algorthm n the sequel. From (3), we can express the equalzed sgnal as W k ỹ = T k x + W k z, where T k = 2 k 1 m=0 c m+1 k,m(w 0 M) s the equalzed ICI matrx. Ideally, T k wll be an dentty matrx. Let z =[ z 0, z 1,..., z ] T, σ 2 x = E{ x 2 }, σ 2 z = E{ z 2 } (0 N 1), and ψ =(σ ). The subcarrer SINR for 2 z /σ2 x the proposed method wth k teratons n the th subcarrer, whch s denoted as SINR k,, can be shown as SINR k, = = { E j=0 j j=0 j t k,j x j t k 2, t k,j 2 + ψ j=0 { t } k E, x 2 2} { + E j=0 wk,j z j 2} w k 2 (28),j where t k,j = T k(, j), and w k,j = W k(, j). For comparson, we also calculate the SINR n the th subcarrer before equalzaton, whch s denoted as SINR,as E { m, x 2} SINR = { } E 2 j=0 m,j x j + E { z 2 } j m, 2 = j=0 m,j 2 + ψ. (29) j As a result, we must calculate each element n T k and W k for SINR k,. To make the followng dervaton more compact, we rewrte the ICI matrx as M = R r=0 Ṽr H r, where Ṽ0 = I N, and R =1. Snce M = R r=0 Ṽr H r, the equalzed ICI matrx T k can be expanded as (30), shown at the bottom of the next page, where Ām = m f=0 W 0Ṽr Hrm,f m,f, and Ā 1 = I N. Usng the same way, we also expand W k as W k = 2 k 1 m=0 c k,m R R r m,0 =0 r m,1 =0... R r m,m 1 =0 B m (31) where B m = Ām 1W 0. We then calculate each element Ā m ( m,j m ). For brevty of presentaton, we consder the case of D =0. To compact and smplfy the notaton, we redefne h r =[ h r 0, h r 1,..., h r ]T and ṽ r =[ṽ0, r ṽ1,...,ṽ r r ]T, and we let w = w,. Snce Ṽr m,f s a crculant matrx, t can be expressed as Ṽr m,f =[cr(ṽ rm,f )] T. Furthermore, from the defnton, we have H rm,f = dag( h rm,f ).Form =0,wehave Ā 0 ( 0,j 0 ) as Ā 0 ( 0,j 0 )=w 0 ṽ r 0,0 j 0 0,N h r 0,0 j 0. (32)

7 HSU AND WU: LOW-COMPLEXITY ICI MITIGATION METHODS FOR HIGH-MOBILITY SISO/MIMO-OFDM SYSTEMS 2761 Then, we can obtan Ā 1 =(W 0 Ṽ r1,1 Hr1,1 )Ā0 and Ā 2 =(W 0 Ṽ r2,2 Hr2,2 )Ā1 accordngly. By repeatng ths process, we can obtan a formula for m 1. Form 1, we have Ām( m,j m ), as n (33), shown at the bottom of the page. Wth M = R r=0 Ṽr H r, we can expand m,j as m,j = R ṽ j,n h r r j. (34) r=0 From (57) and (34), we can express w as w ( R r=0 ṽr 0 h r ) j Ω. (35) R r=0 ṽr j,n h r j 2 Snce we have derved ṽ j and h j, we can then calculate w. Thus, we can have Ām( m,j m ). By usng (30) and Ā m ( m,j m ), we can obtan T k. By usng the relaton of B m = Ā m 1 W 0, (33), and (35), we can compute B m ( m,j m ) as ( R Ā m 1 ( m,j m ) r=0 B ṽr 0 h r j m m ( m,j m )= R j Ω r=0 ṽr j j m,n h r j 2 (36) where Ω = j m S : j m + S, N. From (31) and (36), we can then calculate W k.havngt k and W k ready, we can fnally evaluate SINR k, n (28). Wth (34), we can further express SINR n (29) as SINR = j=0 j R R r=0 ṽr 0 h r 2 r=0 ṽr j,n h r j ) 2. (37) + ψ As for the case of D 0, t can be derved the same way. V. S IMULATION RESULTS In ths secton, we report smulatons to demonstrate the effectveness of the proposed method. We consder OFDM systems wth N = 128 and the CP sze of 16. The modulaton scheme s 16-quadrature ampltude modulaton (16-QAM). The channel length s set to L =15, and the power delay profle s characterzed by an exponental functon,.e., σl 2 = e l/l / L 1 =0 e /L, where l s the tap ndex. Each channel tap s generated by Jakes model [41]. Here, we assume that the channel response s exactly known for the drect ZF method. For the proposed method, the parameters of the LTV channel model are obtaned by LS fttngs. Defne the normalzed Doppler frequency as f d = f d NT s, where f d s the maxmum Doppler frequency, and T s s the samplng perod. For a MIMO-OFDM system, ts settngs are the same as those n a SISO-OFDM system. Specfcally, we consder a 2 2 MIMO-OFDM system. Snce the N-ZF method wth S =2 have the smlar performance to that wth S = N/2 1,weset S =2for all smulatons. Frst, we evaluate the valdty of the analytc output SINRs for the proposed method. Two cases are consdered: case 1 meets the convergence condton derved n Secton IV, whereas case 2 does not. We consder a SISO-OFDM system, and let f d =0.1, sgnal-to-nose rato (SNR) =35 db, S =2, and D =0. Fg. 1 shows the analytc subcarrer SINRs for case 1. Snce the smulated SINRs are almost dentcal to the analytc SINRs, they are not shown n the fgure. In ths fgure, we fnd that each subcarrer exhbts a dfferent SINR due to the characterstc of the frequency-selectve channel. In addton, subcarrer SINRs are all ncreased as the number of teratons s ncreased. The performance of the proposed method wth two teratons s smlar to that wth three teratons. We can also fnd that the output SINRs of the proposed method are very close to those of the drect ZF method. Fg. 2 shows analytc subcarrer SINRs for case 2. We see that even n ths dvergence case, T k = = = 2 k 1 m=0 2 k 1 m=0 2 k 1 m=0 c k,m c k,m c k,m R r m,0 =1 R r m,0 =0 r m,1 =0 R W 0 Ṽ rm,0 Hrm,0 R R r m,0 =0 r m,1 = R R r m,1 =1 r m,m =0 f=0 R r m,m =0 W 0 Ṽ rm,1 Hrm,1... m W 0 Ṽ rm,f Hrm,f Ā m R r m,m =1 W 0 Ṽ rm,m Hrm,m (30) Ā m ( m,j m )= k m 1 =0 k m 2 =0,..., k 0 =0 w k0 w k1,...,w km 1 w m ṽ r m,m j m k m 1,N ṽ r m,m 1 k m 1 k m 2,N,...,ṽr m,1 k 1 k 0,N ṽr m,0 k 0 m,n h r m,m j m hr m,m 1 k m 1,..., h r m,0 k 0 (33)

8 2762 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY 2009 Fg. 1. SINR analyss of the N-ZF method (D =0and S =2) for case 1, where f d =0.1, and SNR =35dB. Fg. 3. BER comparson among one-tap FEQ, PSE, N-ZF (D =0 and S =2), drect ZF, and drect MMSE methods n a SISO-OFDM system; f d =0.1, and 16-QAM. Fg. 2. SINR analyss of the N-ZF method (D =0and S =2) for case 2, where f d =0.1, and SNR =35dB. SINRs are stll ncreased for the frst two or three teratons. For the fourth teraton, subcarrer SINRs start to fall. The result for D =1s smlar to that for D =0, except that the requred number of teratons s reduced to one or two. Next, we consder the performance comparson for the proposed and conventonal methods n a SISO-OFDM system. Here, f d s set to 0.1. Specfcally, the bt error rate (BER) s used as the performance measure. For the benchmarkng purpose, we also show the result of the drect MMSE method and that of the drect ZF method wth f d =0(ICI-free). From extensve smulatons, we also fnd that the performance of the PSE method cannot be further enhanced when Q>2. For ths reason, we only show the result for Q =2.Fg.3showsthe BER performance comparson for D =0. As we can see, the performance of the PSE method s lmted and has an error floor phenomenon. The N-ZF method outperforms the PSE method, even wth one teraton only. As mentoned, there s a convergence condton for the PSE method. Ths condton s Fg. 4. BER comparson among one-tap FEQ, N-ZF (D =1and S =2), drect ZF, and drect MMSE methods n an SISO-OFDM system; f d =0.1, and 16-QAM. totally dependent on the channel. The N-ZF method also has ts convergence condton. However, t depends on the ntal matrx as well as the channel. It s then possble to reduce the probablty of dvergence through the determnaton of W 0. Ths s the man reason the N-ZF method can outperform the PSE method. The requred complexty of the N-ZF method s lower than that of the PSE method (ths can be seen later). Moreover, the performance of the N-ZF method wth three teratons can approach that of the drect ZF method. Here, the performance of the drect ZF s only slghtly worse than that of the drect MMSE method. Fg. 4 shows the BER performance comparson for D =1. It s obvous that the N-ZF method can quckly approach the drect ZF method wth one or two teratons. Note that the N-ZF method wth two teratons can even perform slghtly better than the drect ZF method. Ths behavor s nterestng and can be explaned as follows.

9 HSU AND WU: LOW-COMPLEXITY ICI MITIGATION METHODS FOR HIGH-MOBILITY SISO/MIMO-OFDM SYSTEMS 2763 Fg. 5. BER comparson among one-tap FEQ, N-ZF (D =1and S =2), drect ZF, and drect MMSE methods n an SISO-OFDM system; f d =0.2, and 16-QAM. The N-ZF method only terates Newton s method two or three tmes, and t may not converge completely n all cases. As known, the drect ZF method has a nose enhancement problem. It s then possble that the nose enhancement caused by the N-ZF method s smaller. As a result, the performance of the N-ZF method can be better than that of the drect ZF method. If the convergence condton s met and the number of teratons s hgh enough, the performance of the N-ZF and drect ZF methods wll be the same. Ths phenomenon has been verfed by smulatons, but the result s not shown here. Compared wth Fg. 3, we see that the N-ZF method wth D =1has better performance than that wth D =0, and t can approach the drect ZF method more quckly. To test the lmtaton of all algorthms, we consder a more severe case n whch f d =0.2. Fg. 5 shows the smulaton result. In ths fgure, we see that the N-ZF method can stll work, but ts performance s degraded snce ICI s much larger than that n the prevous cases. In addton, we can see that the degradaton of the MMSE method s smaller and that the performance gap between the ZF and MMSE methods become larger. We also conduct smulatons to evaluate the robustness of all algorthms to the varaton of the normalzed Doppler frequency. Fg. 6 shows the results for f d, varyng from 0 to 0.2. Here, the SNR s set to 30 db. In ths fgure, we can see that the performance of all methods s degraded as the normalzed Doppler frequency s ncreased. Furthermore, the MMSE method s the most robust method, followed by the N- ZF method. Table III summarzes the requred computatonal complexty of the drect ZF method, the PSE method, and the N-ZF method for the smulaton settng shown above. In ths table, the ratos nsde parentheses ndcate the number of operatons requred for the N-ZF method dvded by those for the drect ZF and PSE methods. The frst rato s for the drect ZF method, and the second rato s for the PSE method. From the above smulatons, we can say that for D =0, the requred teraton number for the N-ZF method s two or three, whereas for D =1, t s one or two. In Table III, we can see that the N-ZF method wth D =0 Fg. 6. BER comparson among one-tap FEQ, N-ZF (D =1and S =2), drect ZF, and drect MMSE methods n an SISO-OFDM system; f d =0 0.2, 16-QAM, and SNR =30dB. can save tremendous computatons compared to the drect ZF method. Wth two (three) teratons, ts multplcaton complexty s only (0.015) of that of the drect ZF method. As for the case of D =1, t also saves a lot of computatons compared to the drect ZF method. We can fnd that the multplcaton complexty of the N-ZF method wth one teraton (two teratons) s only (0.013) of that of the drect ZF method. As to addtons, the complexty ratos are smlar to those of multplcatons. As to dvsons, the ratos are and for D =0and D =1, respectvely. Compared to the PSE method (Q =2), the N-ZF method (k =1and D =0) also has lower complexty and better performance. Its requred multplcatons (addtons/dvsons) s 0.85 (0.745/0.5) of those of the PSE method. Fnally, we report the smulaton results for a 2 2 MIMO-OFDM system. From prevous smulatons, we can see that the computatonal complexty of the N-ZF method wth D =0 and D =1 s smlar when the requred number of teratons s taken nto account. However, the N-ZF method wth D =1 tends to have better results. Thus, we wll only consder the N-ZF method wth D =1. Here, we consder two envronments,.e., f d =0.05 and 0.1. Fg. 7 depcts the BER performance for the proposed method, the drect ZF method, and the drect MMSE method for f d =0.05. In ths fgure, we see that the two-tap FEQ method has an rreducble error floor due to ICI. It s obvous that the N-ZF method can effectvely suppress the ICI and that ts performance can quckly approach that of the drect ZF method. As we can see, the teraton number can be as small as one. In addton, note that the drect MMSE method does not gve too much performance enhancement compared wth the drect ZF method. Fg. 8 llustrates the BER performance for the case of f d =0.1. The N-ZF method can stll effectvely suppress the nterference, and ts performance can approach that of the drect ZF method quckly. In ths case, one or two teratons are suffcent for the N-ZF method. Note further that wth two teratons, the N-ZF method can outperform the drect ZF method. Table IV summarzes the requred computatonal complexty of the drect

10 2764 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY 2009 TABLE III COMPLEXITY COMPARISON AMONG N-ZF, PSE, AND DIRECT ZF METHODS IN AN SISO-OFDM SYSTEM (N = 128 AND S = 2) TABLE IV COMPLEXITY COMPARISON BETWEEN N-ZF AND DIRECT ZF METHODS IN A 2 2 MIMO-OFDM SYSTEM (N = 128 AND S =2) Fg. 7. BER comparson among two-tap FEQ, N-ZF (D =1and S =2), drect ZF, and drect MMSE methods n a 2 2 MIMO-OFDM system; f d = 0.05, and 16-QAM. Comparng SISO-OFDM and MIMO-OFDM systems, we fnd that the reducton n computatonal complexty s greater n a MIMO-OFDM system than that n a SISO-OFDM system. In a SISO-OFDM system, the rato of multplcatons s 0.013, whle that n a MIMO-OFDM system s (D =1and k =2). Ths s because the complexty of the drect ZF method s proportonal to O(M 3 N 3 ), whereas that of the N-ZF method s proportonal to O(MN log 2 N). As a result, we can save more computatons as M ncreases. In addton, when N gets larger, the reducton n computatons becomes more apparent as well. Another mportant property of the proposed N-ZF method s that t can trade the desred performance for the requred complexty. However, the drect ZF method does not have such a choce. Ths property wll make the N-ZF method a more effcent method snce t can adapt tself to varous SNR envronments. If the operated SNR s not hgh, the teraton number can be reduced. For example, n Fg. 3, t only requres one teraton to approach the drect ZF method when SNR s less than 25 db. Fg. 8. BER comparson among two-tap FEQ, N-ZF (D =1and S =2), drect ZF, and drect MMSE methods n a 2 2 MIMO-OFDM system; f d = 0.1, and 16-QAM. ZF and N-ZF methods. Wth one teraton, the multplcaton (addton) complexty of the N-ZF method (k =1) s only (0.005) of that of the drect ZF method. Wth two teratons, the complexty ratos of multplcatons and addtons s and VI. CONCLUSION In hgh-moblty envronments, the quas-statc channel assumpton for SISO/MIMO-OFDM systems no longer holds, and ICI tends to become the performance bottleneck. The drect ZF equalzaton method, relyng on the nverson of the ICI matrx, may requre very hgh computatonal complexty. In ths paper, we have proposed a low-complexty ZF algorthm to solve the problem. By explorng the structure nherent n the ICI matrx, we develop a method that employs Newton s

11 HSU AND WU: LOW-COMPLEXITY ICI MITIGATION METHODS FOR HIGH-MOBILITY SISO/MIMO-OFDM SYSTEMS 2765 Fg. 10. Example of A for N =8and D =1. Note that A overlaps wth A 1 and A +1. Fg. 9. Example of the structure of a banded ntal matrx for N =8and D =1. The elements n the shaded area are nonzeros, whle the others are zeros. teraton for the matrx nverson. By usng our formulaton, we effectvely use FFTs/IFFTs n the teratve process. Smulatons show that whle the proposed algorthm can have comparable performance wth the drect ZF method, ts computatonal complexty s much lower. We also analyze the convergence behavor of the proposed algorthm and derve the theoretcal output SINRs. Note that FFT/IFFT operatons are avalable n an OFDM-based transcever. Thus, the proposed algorthm only requres lmted extra crcuts, and the mplementaton complexty of the proposed method can be lower, facltatng ts real-world applcatons. Wth the smlar dea, t s possble to derve a low-complexty MMSE equalzaton method that employs Newton s teraton. In some scenaros, the MMSE method can sgnfcantly outperform the ZF method. Investgaton n ths topc s now underway. APPENDIX A DERIVATION OF THE INITIAL VALUE FOR SISO-OFDM SYSTEMS Let W be a banded matrx wth bandwdth D. Then, the mnmum-frobenus-norm crteron s gven by W 0 = arg mn w I N W M 2 F. (38) Before the dervaton of the optmal soluton n (38), we frst observe a property n a banded matrx. Fg. 9 shows an example of a banded ntal matrx for N =8and D =1. In the fgure, only the data n the shaded area are nonzeros. Note that the number of the nonzero elements n each row may not be the same. For the zeroth and seventh rows, the number of the nonzero elements s two. For the rest of rows, the number of the nonzero elements s three. For a general case, the number of the nonzero elements n the th row frst ncreases, remans the same, and fnally decreases (as ncreases). Due to ths property, we need to consder the three cases when solvng (38). Defne m,j = M(, j), w,j = W 0 (, j), and a,j = n=0 m,n m j,n. By dfferentatng (38) wth respect to w,j and settng the result to zero, we can obtan the followng equaton: A w = m, =0, 1,...,N 1 (39) where w s the th row of the optmum W 0. A, w, and m for the aforementoned three cases are defned as follows. 1) For = D, D +1,...,N 1 D, wehave a D, D a D,+D A =..... (40) a +D, D a +D,+D w =[w, D,w, D+1,...,w,+D ] T (41) m =[ m D,, m D+1,,..., m +D,] T. (42) 2) For =0, 1,...,D 1, wehave A = A D (0 : D +, 0:D + ) (43) w =[w,0,w,1,...,w,+d ] T (44) m = [ m 0,, m 1,,..., m T +D,] (45) where C( 1 : 2,j 1 : j 2 ) ndcates a submatrx of C, obtaned from the 1 th row to the 2 th row and from the j 1 th column to the j 2 th column of C. 3) For = N D, N D +1,...,N 1, wehave A = A D ( N +1+D :2D, N +1+D :2D) (46) w =[w, D,w, D+1,...,w, ] T (47) m = [ m D,, m D+1,,..., m T,]. (48) Note that A n the second case s an upper left submatrx of A D n (40), whle that n the thrd case s a lower rght submatrx of A D n (40). Now, we can obtan the optmum m. To clearly understand the structure of A, we show an example n Fg. 10 (N =8and D =1). In ths fgure, we can see that A 0 s the upper left 2 2 submatrx of A 1.For =1, 2,...,5, the lower rght 2 2 submatrx of A s exactly the same as the upper left 2 2 submatrx of A +1. The lower rght 2 2 submatrx of A 6 s A 7. By usng ths property, we can obtan a recursve algorthm for fast computaton of A 1. soluton for (38) by w = A 1

12 2766 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY 2009 Snce a j, = a,j, A s a Hermtan matrx. For = D, D + 1,...,N 2 D, we can further partton A nto the followng form: s s A = H (49) s and A +1 nto the followng form: U ũ A +1 = +1 ũ H (50) +1 ũ +1 where s and ũ +1 are scalars, s and ũ +1 are column vectors, and U s a square matrx whose dmenson s smaller than that of A by one. Snce A s a Hermtan matrx, we can wrte ts nverse as A 1 v v = H (51) v V where v s a scalar, v s a column vector, and V s a square matrx wth dmenson smaller than that of A 1 by one. From the block matrx nverson formula [42], we can obtan A 1 +1 from A 1 wth the followng formula: [ U A = + b ] +1 bh +1 β+1 b+1 β+1 b H β (52) β+1 where β+1 =(ũ +1 ũ H +1 U 1 ũ +1 ) 1, b+1 = ũ +1, and U 1 U 1 U = V v v H v. (53) For =0, 1,...,D 1, A +1 ncludes A as ts submatrx. We then have A ũ A +1 = +1 ũ H. (54) +1 ũ +1 Consequently, A 1 +1 can be obtaned by (52), where U 1 = A 1.For = N D,...,N 1, A becomes a submatrx of A 1, whch s gven by s s A 1 = H. (55) Thus, A 1 s A can be obtaned wth (53) as follows: A 1 = V 1 v 1v H 1 v 1. (56) Thus, we only have to conduct one matrx nverson explctly,.e., A 1 0, and ts dmenson s (D +1) (D +1). To further reduce the complexty, we can make an approxmaton when calculatng a,j. From the defnton, we have a,j = n=0 m,n m j,n. We can reduce the number of terms ncluded n the summaton. We let a,j n Ω m,n m j,n, where Ω= S : + S, N j S : j + S, N, and S s the number of one-sded ICI terms taken nto consderaton (0 S N/2 1). The notaton : j, N denotes a sequence of { N /N,+1 N ( +1)/N,...,j N j/n } (here, and j are ntegers and j). Wth ths approach, a,j s approxmately evaluated, and so s A n (39). The value of S then determnes the accuracy of the soluton n (39). A small S can greatly reduce the complexty but results n low accuracy of the soluton. Recall that ICI on a subcarrer manly comes from a few neghborng subcarrers. As a result, we can always fnd a small S affectng the fnal result only slghtly. For the determnaton of S, t depends on the value of ICI; the larger the ICI, the larger S we should use. In our smulatons, the largest S that we use s two. As mentoned, f D =0, W 0 wll become a dagonal matrx. In ths case, the ntal values can be approxmated as w, m, n Ω m,n 2 (57) where Ω = S : + S, N. There s an nterestng property n (57). If we take only the dagonal terms of the ICI matrx nto account (.e., S =0), the ntal values wll degenerate nto the coeffcents of the conventonal one-tap FEQ. If there s no ICI, Newton s teraton wth (57) wll stop after ntalzaton (k =0). APPENDIX B DERIVATION OF THE INITIAL VALUE FOR MIMO-OFDM SYSTEMS As we dd n the SISO-OFDM systems, we use the mnmum-frobenus-norm crteron to obtan the ntal value. Note that the ntal matrx for the MIMO-OFDM scenaro s no longer a banded matrx. Instead, t s a matrx composed of four banded submatrces. Let α,j = W α (, j), β,j = W β (, j), γ,j = W γ (, j), and ω,j = W ω (, j). Furthermore, defne q,j = n=0 (ã,nãj,n + b,n b j,n ), r,j = n=0 ( c,n c j,n + d d,n j,n ), and s,j = n=0 (ã,n c j,n + b d,n j,n ), where ã,j = M 1,1 (, j), b,j = M 1,2 (, j), c,j = M 2,1 (, j), and d,j = M 2,2 (, j). It turns out that we can obtan the optmum ntal values by solvng the followng equaton: T b = c, =0, 1,...,2N 1 (58) where b s the th row of the optmum W 0. Defntons of T, b, and c may be dfferent for dfferent s. Note that we have two sets of banded matrces to deal wth: One s for = 0, 1,...,N 1, and the other s for = N,N +1,...,2N 1. We then need to consder three cases for each set. Fortunately, they are smlar. For the set of =0, 1,...,N 1,wehavethe followng. 1) For = D, D +1,...,N 1 D, wehave Q S T = S H (59) R q D, D q D,+D Q = (60) q +D, D q +D,+D b =[α, D,...,α,+D,β, D,...,β,+D ] T (61) c = [ ã D,,...,ã +D,, c D,,..., c T +D,]. (62)

13 HSU AND WU: LOW-COMPLEXITY ICI MITIGATION METHODS FOR HIGH-MOBILITY SISO/MIMO-OFDM SYSTEMS 2767 Matrces R and S can be obtaned by replacng q m,n n Q wth r m,n and s m,n, respectvely. 2) For =0, 1,...,D 1, wehave Q S T = (63) S H R Q = Q D (0 : D +, 0:D + ) (64) R = R D (0 : D +, 0:D + ) (65) S = S D (0 : D +, 0:D + ) (66) b =[α,0,...,α,+d,β,0,...,β,+d ] T (67) c = [ ã 0,,...,ã +D,, c 0,,..., c +D,] T. (68) 3) For = N D, N D +1,...,N 1, wehave Q S T = S H R Q = Q D ( N +1+D :2D, R = R D ( N +1+D :2D, S = S D ( N +1+D :2D, (69) N +1+D :2D) (70) N +1+D :2D) (71) N +1+D :2D) (72) b =[α, D,...,α,,β, D,...,β, ] T (73) c = [ ã D,,...,ã,, c D,,..., c,] T. (74) Note that Q, R, and S n the second case correspond to the upper left submatrces of Q D, R D, and S D, respectvely. In addton, Q, R, and S n the thrd case correspond to the lower rght submatrces of Q D, R D, and S D, respectvely. For = N,N +1,...,2N 1, we can use the same equatons shown n (59) (74). However, we have to let T = T N and replace α,j and β,j n b wth γ,j and ω,j, respectvely, and ã,j and c,j n c wth b,j and d,j, respectvely. Snce the ntal matrx s no longer a banded matrx, we are not able to obtan a recursve relatonshp n solvng (58). Gaussan elmnaton may be a good choce for ths problem. As mentoned, T = T N ; we need only to construct T and perform Gaussan elmnaton of T for =0, 1,...,N 1 once. Furthermore, T s a Hermtan matrx, makng Gaussan elmnaton even less complex. To further reduce the computatonal complexty, we can make some approxmatons n the calculaton of q,j, r,j, and s,j. We can let q,j n Ω (ã,nãj,n + b,n b j,n ), r,j n Ω ( c,n c j,n + d,n d j,n ), and s,j n Ω (ã,n c j,n + b,n dj,n ), where Ω= S : + S, N j S : j + S, N. Agan, ths approxmaton makes use of the property that elements n M,j close to the man dagonal have larger values than the others. REFERENCES [1] J. A. C. Bngham, Multcarrer modulaton for data transmsson: An dea whose tme has come, IEEE Commun. Mag., vol. 28, no. 5, pp. 5 14, May [2] M. Russell and G. L. Stüber, Interchannel nterference analyss of OFDM n a moble envronment, n Proc. IEEE VTC, Jul , 1995, vol. 2, pp [3] P. Robertson and S. Kaser, The effects of Doppler spreads n OFDM(A) moble rado systems, n Proc. IEEE VTC Fall, Sep , 1999, vol. 1, pp [4] Y. L and L. J. Cmn, Jr., Bounds on the nterchannel nterference of OFDM n tme-varyng mparments, IEEE Trans. Commun., vol. 49, no. 3, pp , Mar [5] J. L and M. Kavehrad, Effects of tme selectve multpath fadng on OFDM systems for broadband moble applcatons, IEEE Commun. Lett., vol. 3, no. 12, pp , Dec [6] 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. 12, pp , Dec [7] Y. Zhang and H. Lu, MIMO-OFDM systems n the presence of phase nose and doubly selectve fadng, IEEE Trans. Veh. Technol., vol. 56, no. 4, pp , Jul [8] W. G. Jeon, K. H. Chang, and Y. S. Cho, An equalzaton technque for orthogonal frequency-dvson multplexng systems n tme-varant multpath channels, IEEE Trans. Commun., vol. 47, no. 1, pp , Jan [9] Y.-S. Cho, P. J. Voltz, and F. A. Cassara, On channel estmaton and detecton for multcarrer sgnals n fast and selectve Raylegh fadng channels, IEEE Trans. Commun., vol. 49, no. 8, pp , Aug [10] X. Ca and G. B. Gannaks, Boundng performance and suppressng ntercarrer nterference n wreless moble OFDM, IEEE Trans. Commun., vol. 51, no. 12, pp , Dec [11] W.-S. Hou and B.-S. Chen, ICI cancellaton for OFDM communcaton systems n tme-varyng multpath fadng channels, IEEE Trans. Wreless Commun., vol. 4, no. 5, pp , Sep [12] A. Gorokhov and J.-P. Lnnartz, Robust OFDM recevers for dspersve tme-varyng channels: Equalzaton and channel acquston, IEEE Trans. Commun., vol. 52, no. 4, pp , Apr [13] S. Tomasn, A. Gorokhov, H. Yang, and J.-P. Lnnartz, Iteratve nterference cancellaton and channel estmaton for moble OFDM, IEEE Trans. Wreless Commun., vol. 4, no. 1, pp , Jan [14] X. Huang and H.-C. Wu, Robust and effcent ntercarrer nterference mtgaton for OFDM systems n tme-varyng fadng channels, IEEE Trans. Veh. Technol., vol. 56, no. 5, pp , Sep [15] W.-G. Song and J.-T. Lm, Channel estmaton and sgnal detecton for MIMO-OFDM wth tme varyng channels, IEEE Commun. Lett., vol. 10, no. 7, pp , Jul [16] Y. Zhao and S.-G. Haggman, Intercarrer nterference self-cancellaton scheme for OFDM moble communcaton systems, IEEE Trans. Commun., vol. 49, no. 7, pp , Jul [17] Y. Zhang and H. Lu, Frequency-doman correlatve codng for MIMO-OFDM systems over fast fadng channels, IEEE Commun. Lett., vol. 10, no. 5, pp , May [18] K. W. Park and Y. S. Cho, An MIMO-OFDM technque for hgh-speed moble channels, IEEE Commun. Lett., vol. 9, no. 7, pp , Jul [19] S. Das and P. Schnter, Beamformng and combnng strateges for MIMO-OFDM over doubly selectve channels, n Proc. IEEE ACSSC, Oct./Nov. 2006, pp [20] A. Stamouls, S. N. Dggav, and N. Al-Dhahr, Intercarrer nterference n MIMO OFDM, IEEE Trans. Sgnal Process., vol. 50, no. 10, pp , Oct [21] I. Barhum, G. Leus, and M. Moonen, Equalzaton for OFDM over doubly selectve channels, IEEE Trans. Sgnal Process., vol. 54, no. 4, pp , Apr [22] I. Barhum, G. Leus, and M. Moonen, Tme-varyng FIR equalzaton for doubly selectve channels, IEEE Trans. Wreless Commun., vol. 4, no. 1, pp , Jan [23] I. Barhum, G. Leus, and M. Moonen, Tme-doman and frequencydoman per-tone equalzaton for OFDM over doubly selectve channels, Sgnal Process., vol. 84, no. 11, pp , Nov [24] K. Van Acker, G. Leus, M. Moonen, O. van de Wel, and T. Pollet, Per tone equalzaton for DMT-based systems, IEEE Trans. Commun., vol. 49, no. 1, pp , Jan

14 2768 IEEE TRANSACTIONS ON VEHICULAR TECHNOLOGY, VOL. 58, NO. 6, JULY 2009 [25] G. Leus and M. Moonen, Per-tone equalzaton for MIMO OFDM systems, IEEE Trans. Sgnal Process., vol. 51, no. 11, pp , Nov [26] P. Schnter, Low-complexty equalzaton of OFDM n doubly selectve channels, IEEE Trans. Sgnal Process., vol. 52, no. 4, pp , Apr [27] X. Qan and L. Zhang, Interchannel nterference cancellaton n wreless OFDM systems va Gauss Sedel method, n Proc. IEEE ICCT, Apr. 9 11, 2003, pp [28] A. F. Molsch, M. Toeltsch, and S. Verman, Iteratve methods for cancellaton of ntercarrer nterference n OFDM systems, IEEE Trans. Veh. Technol., vol. 56, no. 4, pp , Jul [29] Y. Mostof and D. C. Cox, ICI mtgaton for plot-aded OFDM moble systems, IEEE Trans. Wreless Commun., vol. 4, no. 2, pp , Mar [30] J. Fu, C.-Y. Pan, Z.-X. Yang, and L. Yang, Low-complexty equalzaton for TDS-OFDM systems over doubly selectve channels, IEEE Trans. Broadcast., vol. 51, no. 3, pp , Sep [31] G. H. Golub and C. F. Van Loan, Matrx Computatons. London, U.K.: Johns Hopkns Unv. Press, [32] C.-Y. Hsu and W.-R. Wu, A low-complexty ICI mtgaton method for hgh-speed moble OFDM systems, n Proc. IEEE ISCAS, May 21 24, 2006, pp [33] G. Schulz, Iteratve berechnung der rezproken matrx, Z. Angew. Math. Mech., vol. 13, pp , [34] A. S. Householder, The Theory of Matrx n Numercal Analyss. New York: Dover, [35] A. Ben-Israel, A note on teratve method for generalzed nverson of matrces, Math. Comput., vol. 20, no. 95, pp , Jul [36] A. Ben-Israel and D. Cohen, On teratve computaton of generalzed nverses and assocated projectons, SIAM J. Numer. Anal., vol. 3, no. 3, pp , Sep [37] V. Y. Pan and R. Schreber, An mproved Newton teraton for the generalzed nverse of a matrx, wth applcatons, SIAM J. Sc. Stat. Comput., vol. 12, no. 5, pp , Sep [38] V. Y. Pan and J. Ref, Fast and effcent parallel soluton of dense lnear systems, Comput. Math. Appl., vol. 17, no. 11, pp , [39] V. Y. Pan, Y. Ram, and X. Wang, Structured matrces and Newton s teraton: Unfed approach, Lnear Algebra Appl.,vol.343/344,pp , Mar [40] S. J. Leon, Lnear Algebra Wth Applcatons. Englewood Clffs, NJ: Prentce Hall, [41] M. F. Pop and N. C. Beauleu, Lmtatons of sum-of-snusods fadng channel smulators, IEEE Trans. Commun., vol. 49, no. 4, pp , Apr [42] D. A. Harvlle, Matrx Algebra From a Statstcan s Perspectve. New York: Sprnger-Verlag, communcaton. Chao-Yuan Hsu (S 08) receved the B.S. degree n electrcal engneerng n 1999 from Natonal Sun Yat-Sen Unversty, Kaohsung, Tawan, and the M.S. degree n communcaton engneerng n 2003 from Natonal Chao Tung Unversty, Hsnchu, Tawan, where he s currently workng toward the Ph.D. degree n communcaton engneerng wthn the Department of Communcaton Engneerng. Hs research nterests nclude ntercarrer nterference cancellaton n orthogonal frequency dvson multplexng-based systems and dgtal Wen-Rong Wu (M 89) receved the B.S. degree n mechancal engneerng from Tatung Insttute of Technology, Tape, Tawan, n 1980 and the M.S. degree n mechancal engneerng n 1985, the M.S. degree n electrcal engneerng n 1986, and the Ph.D. degree n electrcal engneerng n 1989 from the State Unversty of New York, Buffalo. Snce August 1989, he has been a faculty member wth the Department of Communcaton Engneerng, Natonal Chao Tung Unversty, Hsnchu, Tawan. Hs research nterests nclude statstcal sgnal processng and dgtal communcaton.