A Pre-FFT OFDM Adaptive Antenna Array with Eigenvector Combining

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1 A Pre- OFDM Adaptive Antenna Array ith Eigenvector Cobining Shinsuke Hara y, Montree Budsabathon y, and Yoshitaka Hara y Departent of Electronic, Inforation and Energy Engineering, Graduate School of Engineering, Osaka University, Osaka, Japan, e-ail: hara@eie.eng.osaka-u.ac.jp Mitsubishi Electric Inforation Technology Centre Europe B.V. (ITE), Rennes, France Abstract This paper proposes a novel pre- type OFDM adaptive array antenna called Eigenvector Cobining. The eigenvector cobining is a realiation of a post- type OFDM adaptive array antenna through a pre- signal processing, so it can achieve excellent perforance ith less coputational coplexity and shorter training sybols. uerical results deonstrate that the proposed eigenvector cobining shos excellent bit error rate perforance close to the loer bound just ith to training sybols. I. ITRODUCTIO Orthogonal frequency division ultiplexing (OFDM), hich is a pre-distortion or an equaliation technique at transit side in a sense, is an efficient technique for high-speed digital transission over severe ultipath fading channels[], and recently has been considered to be a proising technique for next generation obile counications systes[2]. OFDM schee inserts a guard interval in every OFDM sybol so as to be robust to delayed signals ithin the guard interval, hoever, once delayed signals beyond the guard interval (naely, co-channel interfering signals) are introduced in a channel ith larger delay spread, inter-sybol (co-channel) interference causes a severe degradation in the transission perforance. To aintain high-speed reliable ireless counications systes, the use of ultiple antennas at receive side has been considered as an effective tool not only for gain enhanceent, increased spectral efficiency[3] but also for interference suppression[4]. Although a post-fast Fourier transfor () subcarrier-by-subcarrier cobining OFDM adaptive array antenna is optiu in ters of axiiing signal-tointerference-and-noise poer ratio (SIR), it requires the increased nuber of processors and heavy coputations, hich increase ith the nuber of antennas and subcarriers, and a quite long training signal[5]. On the other hand, a pre- type OFDM adaptive array antenna, hich requires only one processor, can drastically reduce the coputational coplexity by tolerating soe perforance degradation[6], [7]. This paper proposes a novel pre- type OFDM adaptive array antenna called Eigenvector Cobining[8]. The eigenvector cobining is a realiation of a post- type OFDM adaptive array antenna through a pre- signal processing[9], so it has the characteristics of the to types of the array antennas, naely, excellent perforance inherent in the post- type and less coputational coplexity and shorter training sybols proper to the pre- type. # r * * (a) Post- Type Fig.. Fig. 2. OFDM adaptive array antennas. Signal burst forat. p [sybols] Preable Cyclic Prefix L g [saples] # r * * (b) Pre- Type D [sybols] Payload OFDM Sybol Useful Sybol L u [saples] y II. SYSTEM MODEL Figures (a) and (b) shos the block diagras of a post- type OFDM adaptive array antenna and a pre- type OFDM adaptive array antenna, respectively. Each antenna is equipped ith antenna eleents. Figure 2 shos a signal burst forat. The signal burst is coposed of preable and payload. The preable is coposed of p training OFDM sybols and D data OFDM sybols, and one OFDM sybol is coposed of L g saplelong cyclic prefix and L u saple-long useful sybol. III. PROS AD COS OF PRE- AD POST- TYPE ARRAY ATEAS First, let us pay attention to the post- array antenna as shon in Fig. (a). The received signal vector r ( ) is coposed of the desired signal vector x ( ), the interfering signal vector i ( ) and the noise vector n ( ): x, i and n are utually uncorrelated: r = x + i + n: () r = [r ] T (2) x = [x x ] T (3) i = [i i ] T (4) n = [n n ] T (5)

2 BER Fig Angle Spread =20 deg. Interfering Signal Desired Signal Post -Type Post -Type 20 Training Sybols Pre -Type 2 Training Sybols Average E b / 0 [db] BER of post- and pre- type array antennas. here T and l denote transpose and sapling index, respectively. Define as the output vector ( ) for the -th subcarrier: =[ ]T : (6) The eight control criterion for the post- array antenna is given by iniie E jp ; H j 2 ( = M) (7) here E[] denotes expectation, H denotes Heritian transpose, M is the nuber of subcarriers, p is the training sybol at the -th subcarrier and is the array eight vector ( ):? =[ ]T : (8) Equation (7) can be solved by ell knon adaptive algoriths. Hoever, the conventional algoriths require at least several tens of sybols for good convergence, in other ords, they require several tens of training OFDM sybols in the preable (note that one training OFDM sybol gives only one reference point at each subcarrier). Figure 3 shos the BER (Bit Error Rate) of a post- type array antenna (ith M = 52, D = 0, L g = 6 and L u =64) for the case of three desired signals and one interfering signal (see the spatio-teporal odel in the sae figure, or see Section VII). Here, the Saple Matrix Inversion (SMI) algorith is used for array eight calculation[4], so the coputational coplexity is O( 3 p (L g + L u ) 3 M ). When setting p =2, the BER is poor, on the other hand, hen setting p =20, the BER is excellent, but the burst efficiency is 0.33 (=0/(20+0)). ext, let us pay attention to the pre- array antenna as shon in Fig. (b). The pre- type eight control is based on not OFDM sybol but saple-driven algorith, so to OFDM sybol-long preable ( p =2, p (L p + L u )= 60 saples) is long enough to get good convergence. Figure 3 also shos the BER of a pre- array antenna (see Section IV for the detail algorith). Here, the SMI algorith is used for array eight calculation, so the coputational coplexity is O( 3 p (L g +L u ) 3 ).Ascopared ith the post- array antenna ith the sae preable length, the pre- type can achieve a uch better BER perforance, hereas its perforance is still inferior to that of the post- type ith enough preable length. Therefore, a key issue in the paper is ho to ake the perforance of a pre- array antenna be close to that of the post- array antenna, keeping the shorter preable length and the less coputational coplexity. IV. PRICIPLE OF PRE- TYPE ARRAY ATEA Define the OFDM signal vector (L g ) and the channel ipulse response vector (L g ) atthe n-th antenna eleent as s = [s() s(;l g )] T (9) h n = [h n h nl g ]T (0) here s is the transitted OFDM signal sapled at the l- th sapling instant. In addition, define the channel ipulse response atrix ( L g )as With Eqs.(9) and (), x is ritten as The input is ritten as H =[h h ] T : () r = Hs: (2) y = H r (3) here is the array eight vector ( ): =[ ] T : (4) The SIR of the input can be calculated as[7] SIR = H C x H C rec : (5) C x and C rec are the correlation atrices ( ) ofthe desired signal and the received signal: C x = E[xx H ]=E[Hss H H H ]= 2 s HHH (6) C rec = E[rr H ] (7) here 2 s is the average poer of the OFDM signal. For derivation of Eq.(6), the folloing iportant property of OFDM signal is used: E[ss H ]= 2 s I Lg (8) here I L g is the identity atrix (L g L g ). The eight control criterion for the pre- array antenna: axiie SIR = H C x H C rec (9)

3 r r # # *, *, *, K y yk α * K α K * *, K (a) Proposed Eigenvector Cobining Fig. 4. Eigenvector cobining. r # k= *, k α * k (b) Equivalent Post- Type Representation K Σ K *, k α * Σ k k= leads to the solution of the folloing generalied eigenvalue proble: find hich satisfies C x = ax C rec (20) here ax is the first largest generalied eigenvalue. When setting the array eights so as to satisfy Eq.(20), the largest SIR given by ax is realiable at the input. after being cobined ith a set of k (k = K) are coherently cobined on subcarrier-by-subcarrier basis (also after operation) in a axiu ratio cobining (MRC) anner. ote that to perfor the MRC properly, a linear constraint needs to be iposed to Eq.(2): subject to j k j =: (22) The signal cobined by k at the k-th virtual array branch is ritten as y k = H k r (l = L g L g + L u ) (23) and the input vector (L u )atthe k-th virtual branch is ritten as y k = [y k (L g ) y k (L g + L u )] T = ; H k [r(l g ) r(l g + L u )] T = ; H k R T (24) here R is the received signal atrix ( L u ). Defining d =[ e ;j2=lu e ;j2(lu;)=lu ] T as the L u -point vector (L u ) for the -th frequency (subcarrier) coponent, the -th subcarrier output at the k-th virtual array branch can be ritten as k = yt k d : (25) On the other hand, the coplex envelope for the -th subcarrier coponent at the k-th virtual array branch is given by V. EIGEVECTOR COMBIIG When the angle spread of received signals is sall enough, the first largest generalied eigenvalue is doinant, so the pre- array antenna can utilie alost all part of the received signal poer. In this case, there is no proble ith the pre- array antenna. Hoever, hen the angle spread becoes large, the other eigenvalues such as the second and third largest ones becoe larger (naely, their values becoe coparable ith the value of the first largest one), so the pre- array antenna shos a poer utiliation inefficiency in the transission perforance. Utiliation of the array eights associated ith the other larger eigenvalues can iprove the transission perforance. This is the principle of the eigenvector cobining. Equation (20) can be reritten as C x k = k C rec k (k = K) (2) here there are assued to be K non-ero generalied eigenvalues in the decreasing order, naely, = ax 2 K. Figure 4 (a) shos the structure of the proposed pre- array antenna ith eigenvector cobining, hich virtually has K sets of pre- array antennas. The SIR of the k-th pre- array antenna is given by k, here the poer of interfering signals can be ell suppressed. Therefore, the K signals k = H k H0 d (26) here H 0 is the ipulse response atrix ( L u ) obtained by (L u ; L g )-ero padding to the ro vector in H. Therefore, the MRC-cobined -th final subcarrier output is ritten as v = k k k= = d H H 0H k= ( k H k )Rd : (27) The advantages of the proposed eigenvector cobining is clear, that is, it requires only fe training OFDM sybols for array eight calculation, and can utilie alost all part of received signal poer. When the SMI algorith is used for array eight calculation, the coputational coplexity is still O( 3 p (L g + L u ) 3 ). VI. RELATIOSHIP BETWEE EIGEVECTOR COMBIIG AD POST- TYPE The operation is linear, so ultiplication of a eight at the input is equal to that of the sae eight at all the outputs. Taking into consideration this input/output relationship, the structure of the proposed eigenvector cobining

4 pdf shon in Fig.4 (a) can be changed into that of the very post- type shon in Fig.4 (b). Therefore, the eigenvector cobining is a realiation of a post- array antenna through a pre- signal processing. The equivalence beteen the for the case of no interference can be proved as follos. When there is no interference, the generalied eigenvalue proble given by Eq.(20) becoes just an eigenvalue proble, here the eigenvectors are utually orthogonal. Therefore, taking the noise subspace into consideration, the suation ter up to in Eq.(27) is siplified into so Eq.(27) becoes k= ( k H k )=I (28) v =(H 0 d ) H (Rd ) : (29) Equation (29) shos the equivalence beteen the eigenvector cobining and the post- type, here the -th final subcarrier output is the su of coplex envelope-eighted -th outputs. Fig. 5. Envelope: Rayleigh distributed Arrival Tie: uniforly distributed ithin guard interval (Desired Signal) uniforly distributed ithin useful sybol (Interfering Signal) Interfering Signal { A spatio-teporal channel odel Desired Signal DoA: uniforly distributed in [0 deg., 360 deg.) 8 Waves Angle Spread: 0, 20 or 30 deg. Average E b / 0 =6 db Angle Spread=0 deg. Angle Spread=30 deg. λ 3 λ 2 VII. UMERICAL RESULTS AD DISCUSSIOS For coputer siulation, an eight-eleent circular array ith half-avelength adjacent spacing is eployed. A coherent quadrature phase shift keying (QPSK) schee is assued ith a half-rate convolutional encoding/viterbi decoding ith a constraint length (K) of7and a (2 8) block interleaver. One OFDM sybol is coposed of 80 saples, here the guard interval length is 6 saples and the useful sybol length is 64 saples. Here, the OFDM sybol is generated ith the 64-point inverse (I), here only 48 subcarriers convey inforation, 4 subcarriers are knon pilot signals and the other 2 subcarriers are virtual subcarriers. The one signal burst is coposed of 2 OFDM sybol-long preable and 0 OFDM sybol-long payload. In the preable, one training OFDM sybol is used for estiation of channel ipulse response and the other training OFDM sybol estiates the correlation atrix of the received signal. We restrict our attention to an uplink transission, here the position of a base station (BS) is sufficiently high so that fe local scatterings occur. Therefore, spatial fading at the BS ill be correlated ith an exact correlation depending on the BS antenna eleent spacing, the carrier frequency used, and the angle spread observed at the BS array[0]. Figure 5 shos a spatio-tepral channel odel, here there are three desired signals arriving ithin the guard interval and as a cochannel interfering signal, there is one delayed signal beyond the guard interval. Each signal arrives foring a cluster ith an angle spread, hich is coposed of eight aves, the envelope is Rayleigh-distributed ith the sae poer and the DoA is uniforly distributed in [0 deg., 360 deg.). The arrival tie of the desired signal is uniforly distributed ithin the guard interval, hereas that of the interfering signal is uniforly distributed ithin the useful sybol interval beyond Fig Eigenvalue oralied by λ Eigenvalue distribution. the guard interval. Furtherore, the channel fading is slo enough so that the channel ipulse response does not significantly change over one signal burst. Figure 6 shos the siulated probability density function (pdf) of eigenvalues 2 and 3 noralied by the first largest eigenvalue, here the ratio of the average received energy per bit to the hite noise poer spectral density per antenna (E b = 0 )is6db and outcoes are eployed. The occurrence of large k substantially increases hen the angle spread increases (here, fro 0 deg. to 30 deg.). This suggests us that the BER perforance of the eigenvector cobining can becoe better as the angle spread increases. Figure 7 shos the BER versus the average E b = 0 of the proposed eigenvector cobining. Here, a threshold paraeter th is nely defined, here the eigenvectors associated ith k th are eployed for cobining. This eans that the eigenvector cobining ith th = is equivalent to the conventional pre- array antenna. The figure clearly shos that the BER becoes better hen th decreases and the angle spread increases (in the other ords, the fading correlation aong antenna eleents decreases), as expected intuitively. Figure 8 shos the BER versus th of the proposed schee. The BER perforance is iproved as the th decreases, i.e., ore eigenvectors are eployed for cobining. o threshold effect is observed hich is typical in subspace approaches such as MUSIC.

5 BER BER BER Angle Spread=0 deg. λ th =.0 (Pre- Type) Average E b / 0 =2 db Angle Spread=0 deg. Average E b / 0 =6 db Fig λ th =0. (Proposed Eigenvector Cobining) Average E b / 0 [db] BER perforance. Finally, Figure 9 shos the BER coparison beteen the eigenvector cobining and the post- array antenna. When there is interference, it has not been copleted to theoretically prove the equivalence beteen the to schees, but the figure clearly shos that the perforance of the eigenvector cobining ith just to training sybols is alost the sae as that of the post- array antenna ith tenty training sybols. According to our calculation, the achievable SIR of the eigenvector cobining is saller than that of the post- array antenna by less than %. VIII. COCLUSIOS This paper proposed a novel pre- type OFDM adaptive array antenna called Eigenvector Cobining. The eigenvector cobining is a realiation of a post- type OFDM adaptive array antenna through a pre- signal processing, and has the advantageous characteristics of the to types of array antennas: excellent perforance due to poer efficiency inherent in post- type and less coputational coplexity and shorter training sybols due to pre- signal processing proper to pre- type. uerical results shoed that, for the case of three desired signals and one co-channel interfering signal, ith to training sybols, the post- array antenna cannot ork ell at all, on the other hand, the proposed eigenvector cobining shos excellent BER perforance close to the loer bound. REFERECES [] J.A.C. Bingha, Multicarrier odulation for data transission: An idea hose tie has coe, IEEE Coun. Mag., vol.28, pp.5 4, May 990. [2] J. Chuang and. Sollenberger, Beyond 3G: ideband ireless data access based on OFDM and dynaic packet assignent, IEEE Coun. Mag., vol.38, no.7, pp.78 87, July [3] J.H. Winters, et al., The ipact of antenna diversity on the capacity of ireless counication systes, IEEE Trans. Coun., vol.42, no.4, pp , Apr [4] R.T. Copton, Adaptive Antennas, Prentice-Hall, Engleood Cliffs, e Jersey, Average E b / 0 =0 db Eigenvector Cobining Threshold (λ th ) Fig. 8. BER perforance versus th. Fig Post -Type λ th =0. (Proposed Eigenvector Cobining) 0-5 Post -Type Training Sybols Average E b / 0 [db] BER coparison beteen eigenvector cobining and post- type. [5] Y.(G.) Li and.r. Sollenberger, Adaptive antenna arrays for OFDM syste ith cochannel interference, IEEE Trans. Coun., vol.47, no.2, pp , Feb [6] M. Okada and S. Koaki, Pre-DFT cobining space diversity assisted COFDM, IEEE Trans. Veh. Technol., vol.50, no.2, pp , Mar 200. [7] M. Budsabathon, Y. Hara, and S. Hara, Optiu beaforing for pre- OFDM adaptive antenna array, Proc. IEEE WCC 03, in CD- ROM, Mar [8] M. Budsabathon, Y. Hara, and S. Hara, An OFDM adaptive array antenna ith eigen-vector cobining, Technical Report of the IEICE, AP , pp.45-48, Feb [9] D. Huang and K. B. Letaief, Pre- processing for coded OFDM systes ith receive space diversity, Proc. IEEE WCC 03, in CD-ROM, Mar [0] R.B. Ertel, et al., Overvie of spatial channel odels for antenna array counication systes, IEEE Personal Coun., no.2, pp.2 22, Feb. 998.