Pilot Designs with Guard Bands for Channel Estimation of MIMO OFDM Systems with Frequency-Dependent I/Q Imbalances

Transcription

1 Plot Desgns wth Guard Bands for Channel Estmaton of MMO OFDM Systems wth Frequency-Dependent / mbalances Danel Munoz and Hlang Mnn Department of Electrcal Engneerng, Unversty of Texas at Dallas E-mal: {djm072000, hlang.mnn}@utdallas.edu Abstract All practcal OFDM systems nsert null guard bands to facltate easy flter mplementaton and emsson spectrum control. The null guard bands ntroduce a constrant for optmal plot desgns n MMO OFDM systems. Semconductor downscalng and the use of hgh modulaton order have sgnfed the mportance of plot desgns for OFDM systems n the presence of / mbalances, but exstng desgns have not addressed the use of guard bands, the overhead effcency, the estmaton optmalty, and general applcablty n plot data multplexed format and/or MMO setup. Ths paper proposes a plot desgn for channel estmaton of MMO OFDM systems wth guard bands and both transmt and receve frequency-dependent / mbalances, whch overcomes the lmtatons of the exstng desgns. Smulaton and numercal results corroborate better performance, overhead advantage and applcablty n plot-data-multplexed MMO setup of the proposed desgn over the exstng MMO OFDM plot desgn wth / mbalances. ndex Terms Channel estmaton, Guard band, / mbalance, MMO, OFDM, Plot desgn. NTRODUCTON Channel estmaton, crucal for coherent detecton, s typcally accomplshed by the ad of plot tones n practcal OFDM systems. Plot desgns wth performance optmalty and mnmal overhead are much desrable especally for multnput mult-output MMO) or multple antennas systems snce MMO systems are more senstve to estmaton errors and they have a multtude of channel parameters. There exst several plot desgns for MMO OFDM systems n the lterature e.g., [1], [2] and references theren). However, these desgns do not consder / mbalances msmatches between the nphase and the quadrature branches of the transcever). The detrmental effects of / mbalances become more sgnfcant due to the semconductor downscalng and the use of hgh order modulaton [3]. Ths mandates development of effcent plot desgns for channel estmaton of MMO OFDM systems wth / mbalances. All practcal OFDM systems exclude some subcarrers at the band edges from use, as seen n [4] where for a FFT sze of 128, only 73 subcarrers are allocated for data and plot use. Ths ntroduces an addtonal constrant n the plot desgn. Many papers have addressed the / mbalance ssue and proposed plot desgns whch ether estmate the / mbalance parameters and try to remove ts effects from the channel see [5] [9], or estmate the combned response of the channel and / mbalance e.g. [10] [17]). n ths paper, we consder the latter approach. Among the exstng desgns, only [17] addresses MMO systems. However, t does not consder guard bands n the desgn. Although [16] apples a guard band n ts smulaton, ts plot desgn does not consder guard bands, and t s only for SSO systems. n ths paper we develop a plot desgn for MMO OFDM systems wth guard bands n the presence of transmt and receve frequency-dependent / mbalances. The proposed desgn s applcable to MMO OFDM systems wth guard bands and both transmt and receve frequency-dependent / mbalances whle mantanng estmaton performance optmalty, mnmal overhead, and generalty n transmsson format e.g., plot-data multplexed setup), hence overcomng the lmtatons and dsadvantages of the exstng methods. The rest of the paper s organzed as follows. Secton descrbes sgnal model n the presence of frequency-ndependent and -dependent transmt and receve / mbalances n OFDM systems. Secton presents plot desgn crtera. n Secton V, we present our proposed plot desgn. Performance evaluaton and smulaton results are dscussed n Secton V, and conclusons are provded n Secton V.. SGNAL MODEL Frst, consder a sngle antenna system wth both transmt and receve / mbalances where {a t,a t } and {θt,θ t } represent frequency-ndependent gans and phase offsets of the and branches at the transmtter. The equvalent pulse shapng flters the overall mpulse responses ncludng DAC, amplfer, pulse shapng, and frequency-dependent / mbalances) for the and branches of the transmtter are denoted by gt t) and g t t). The recever parameters are defned n the same manner wth the subscrpt t replaced by r. Then the transmt system can be vewed as the summaton of two systems namely drect system whose nput s the same as the transmtter nput sgnal st) = s t) +js t) and mrror system whose nput s s t). The mpulse responses of the drect and mrror systems at the transmtter are denoted by gt Dt) and gm T t), and those at the recever are gd R t) and gr M t), respectvely. They are related to the / mbalance parameters and the equvalent pulse shapng flters as gt D t) =[a t ejθ t g t t)+a t ejθ t g t t)]/2 1) gt M t) =[a t e jθ t g t t) a t e jθ t g t t)]/2 2) gr D t) =[a r e jθ r g r t)+a r e jθ r g r t)]/2 3) g M R t) =[a re jθ r g r t) a r e jθ r g r t)]/2. 4) The channel mpulse response ht) s the low-pass-equvalent of the bandpass channel h bp t) /09/$ EEE

2 n MMO systems, dfferent RF chans correspondng to dfferent antennas may gve rse to dfferent / mbalances. The above mpulse responses correspondng to the th transmt antenna and the jth receve antenna are denoted by gt, D t), gt, M t), gd R,j t), andgm R,j t), respectvely. A smplfed lowpass-equvalent sgnal model for MMO s presented n Fg. 1 for the jth receve antenna where the mpulse responses of the overall drect channel p,j t) and the overall mrror channel q j t) correspondng to the th transmt antenna and the jth receve antenna read as p j t) =gt, D t) h jt) gr,j D t) 5) +gt,t)) M h jt) gr,jt) M q j t) =gt,t) M h j t) gr,jt) D 6) +gt, D t)) h j t) gm R,j t). The receve flter output sgnal correspondng to the receve antenna j and transmt sgnals {s t)} s r j t) = N Tx 1 =0 s t) p j t)+s t) q j t)) + n j t) 7) where the complex Gaussan nose n j t) s gven by n j t) =wt) gr,j D t)+w t) gr,j M t). 8) Fg. 1. A smplfed equvalent lowpass MMO system model wth frequencyndependent and frequency-dependent / mbalances When MMO channels are ndependent or ther jont statstcs are unknown, the logcal approach to the estmaton of the equvalent drect and mrror channels s to estmate them at each receve antenna ndependently. Hence, we just need to descrbe them for one receve antenna. n the followng we drop the receve antenna ndex j. We consder an OFDM system wth a cyclc prefx CP) nterval N CP samples) longer than the maxmum span L samples) of {p t)} and {q t)}. Now, let us consder a low-pass-equvalent dscrete-tme OFDM system wth N subcarrers. The channels are assumed to be constant over K symbol ntervals. The dscrete-tme transmt tranng sgnal from the th transmt antenna durng the lth symbol duraton s denoted by s l) [k] wth the nteger ndex k [ N CP,N 1], and the CP samples s l) [m] = [N m] for m [ N CP, 1]. Smlar notatons apply s l) to data sgnal x l) [k]. The dscrete-tme versons of p t) and q t) are denoted by L 1 vectors p and q, respectvely. The tme-doman N 1 receved sgnal vector after CP removal at each receve antenna for the lth OFDM symbol s denoted by r l whch can be expressed n a general plot-data multplexed setup whch ncludes plot only or data only symbols as specal cases) as r l = N Tx 1 =0 {S [l]+x [l])p +S [l]+x [l])q } + n l where m, k)th element of the N L sgnal matrx S [l] or X [l]) ss l) [m k] or x l) [m k]) wth m [0,N 1] and k [0,L 1]. The receved sgnal vector for K OFDM symbols s gven by 9) r = Sp + S q + Xp + X q + n 10) where r =[r T 0 rt 1...rT K 1 ]T, p =[p T 0 pt 1...pT N Tx 1 ] T, q = [q T 0 qt 1...qT N Tx 1 ] T, n = [n T 0 nt 1...nT K 1 ]T, the k, )th submatrce of S and X are respectvely gven by S [k] and X [k], wth k [0,K 1] and [0,N Tx 1]. Thecomplex Gaussan nose vectors {n l } are gven by n l = G R,D w l + G R, w l 11) where {w l } are ndependent and dentcally dstrbuted d) random vectors, each consstng of d crcularly-symmetrc complex Gaussan random varables wth the varance σw 2.Let λ denote the maxmum of the number of taps of gr D [k] and g R [k]. Then, G R,D and G R, are N N + λ) matrces wth ther frst rows gven by [gr D[0], gd R [1],..., gd R [λ], 0 N 1] and [g R [0], g R [1],..., g R [λ], 0 N 1], respectvely, and ther next kth rows are cyclcally k-rght-shft versons of ther correspondng frst rows.. MMO OFDM PLOT DESGN CRTERA For coherent detecton, the drect channel p and the mrror channel q need to be estmated at the recever. n practcal systems, the statstcs of the channel and the transcever mperfectons are unknown and they can be non-statonary as well. Ths leads to the practcal choce of least-squares type channel estmators as consdered n ths paper. The estmates of the drect and mrror CR vectors are gven by 1 ˆp = S S) H S H r 12) ˆq = S T S ) 1 S T r. 13) Our plot desgns wll be based on mnmzng the mean-square error MSE) of the channel estmaton. We observe the followng condtons for the plot desgns: 1) Estmaton dentfablty Condton: The dentfablty of p and q estmaton requres that S H S s of full-rank. 2) Zero Cross Channel nterference Condton: The nterference between the drect chanel and the mrror channel s completely removed when S H S = 0 LNTx.

3 3) Zero Data nterference Condton: The random data nterference s completely suppressed when S H X = 0 LNTx and S H X = 0 LNTx. 4) Whte Nose Optmalty Condton: For the scenaro where the equvalent receve-flter s a square-root Nyqust flter, the MSE due to the nose s mnmzed when S H S = E K LNTx where E K s the total energy of the transmt sgnal vector from a transmt antenna over K symbols excludng CPs). The frequencyndependent recever / mbalance wth a square-root rased cosne receve flter represents ths scenaro. 5) For the scenaro wth frequency-dependent recever / mbalance, the nose covarance matrx s unknown a pror, and hence t s nfeasble to develop optmal plot desgns. However, frequency selectvty of the recever / mbalance s typcally very small. A practcal approach n ths case s to assume frequency-flat recever / mbalance n the plot desgns whch leads to the requrement of S H S = E K LNTx. When the dentfablty and zero data nterference condton are satsfed, the MSEs of ˆp and ˆq becomes 1 ) 1 MSE p =Tr[ S S) H S H S E[qq H ]S T S S H S 1 ) ] 1 + S S) H S H C n S S H S 14) [ MSE q =Tr S T S ) 1 S T SE[pp H ]S H S S T S ) 1 + S T S ) 1 S T C n S S T S ) 1 ] 15) where the nose covarance matrx C n s gven by C n = σ 2 w G R,DG H R,D + G R,G H R, ) K. 16) f the zero cross channel nterference condton s also met, the above MSE expressons become 1 ) ] 1 MSE p =Tr[ S S) H S H C n S S H S 17) MSE q =Tr[ S T S ) 1 S T C n S S T S ) 1 ]. 18) Addtonally, f the demodulator output nose samples are whte.e., C n = σn), 2 [ the MSE expressons smplfy to ) ] 1 MSE p =MSE q = σntr 2 S H S. The MMO OFDM plot desgn crteron satsfyng estmaton dentfablty, zero data nterference condton, zero cross channel nterference, and whte nose optmalty reads as ) S H S = E K NTxL & S H X = 0 NTxL S H S ) = 0 NTxL & S H X = 0 NTxL } 19) wth the constrant that the plot tones do not occupy the guard bands. The correspondng MSE becomes MSE p =MSE q = σ2 w EK 2 Tr [ S H G R,D G H R,D +G R, G H R, ) K ) ) ] 1 S. 20) V. PLOT DESGNS For a typcal FFT sze N a power of 2), defne L 0 = 2 log2l), L = 2 L 0, and M = N/L. The plot tone ndex set for antenna ndex and OFDM symbol ndex l s represented by J whch conssts of non-zero plot tone ndex set J plot and the null plot tone ndex set J null. The unon of J across all transmt antennas s denoted by J l. The tone ndex of the null guard band s defned by J guard f the DC subcarrer s not used, we nclude t n ths set), and the unon of data tone ndex sets from all transmt antennas for OFDM symbol ndex l s denoted by l.letc [k] denote the plot symbol on the kth subcarrer of the lth OFDM symbol from the th transmt antenna. A. Reference Desgn For a comparson, the desgn n [17] wll be used as the reference. Ths plot desgn uses a Hadamard sequence structure for channel estmaton n the presence of / mbalance to avod both nter-carrer and nter-antenna nterferences. n our notaton, the ndex sets are gven by J plot = {1, 2,...N 1}\J guard for =0, 1,...N Tx and l =0, 1...2N Tx 1 21) = for =0, 1,...N Tx 1 and l =0, 1...2N Tx 1 22) J plot = for =0, 1,...N Tx 1andl 2N Tx 23) = {1, 2,...N 1}\J guard for =0, 1,...N Tx 1andl 2N Tx. 24) [ ] 1 1 Defne H to be the Hadamard matrx.defnec 1 1 to be the set of plots requred for +1 antennas. The plots are gven by c 0,0 [k] { 1, 1} for k J plot 0,0 25) c 0,1 [k] =c 0,0 [k] for k J plot 0,1 and k<n/2 26) c 0,1 [k] = c 0,0 [k] fork J plot 0,1 and k>n/2 27) C = H C 1. 28) For example, C 0 = [c 0,0 c 0,1 ] where c s a row vector comprsng [ of {c [k], k}. ] WhenN Tx =2,thenC 1 = c0,0 c 0,1 c 0,0 c 0,1 where the columns represent c 0,0 c 0,1 c 0,0 c 0,1 plots for seperate symbols and the rows represent the plots for seperate antennas. n ths reference desgn, excludng subcarrers wth ndex 0 and N/2 volates the plot desgn condtons from Secton. n addton, the use of null guard bands causes more devaton from our plot desgn condtons, yeldng further performance degradaton for the reference desgn.

4 B. Proposed Desgn n the proposed desgn, plots of dfferent antennas are decoupled by means of tme dvson multplexng TDM) whle mrror tone nterferences are suppressed by means of null tones. All antennas transmt on the same set of subcarrers, but each antenna transmts durng a dfferent OFDM symbol. Thus, K = N Tx OFDM symbols are requred to meet the desgn requrements, whle data can be transmtted on other subcarrers n a plot-data-multplexed format. Defne T n,k [k, k + M n,k +2M n,...,k + N M n ] whch conssts of cyclcally equ-spaced L n ndexes from [0,N 1]. Frst, obtan T n,τ such that {T n,τ J guard } = 29) τ {{1, 2,...,M n 1}\{M n /2}}, 30) τ M n τ. 31) Defne Tn,τ {N T n,τ } modulo N. Then the ndex sets are gven by { J plot Tn,τ, f = l = 32), else { Tn,τ J null, f = l = T n,τ T 33) n,τ, else l = {0, 1, 2,...,N 1}\{T n,τ T n,τ J guard }. 34) The non-zero plot tones can take on arbtrary phases as c, [k] =e jφ,k, arbtrary φ,k, k J plot,, {0, 1,...,N Tx 1}. 35) The assgnment of the plot, null and data ndces n 32), 33), and 34) guarentee that the zero data nterference condton s met. Due to TDM, there s no mrror tone nterference across dfferent antennas.e., mrror tone nterference suppresson through TD) so the estmaton dentfablty and the whte nose optmalty condtons are met. The null mrror plot tones ensure the zero cross channel nterference condton s met. To mnmze overhead, M n should be set to M 0. Wth the DC subcarrer typcally set to null, the number of band-edge null guard tones that can be supported by the proposed desgn wthout any performance loss s M 0 3. V. SMULATON RESULTS AND DSCUSSONS System parameters n the smulaton are N = 128, 13 null guard tones at the band edges, 16-AM, and a Raylegh fadng channel wth 4 uncorrelated channel taps havng an exponental power delay profle 3 db per tap decay factor). The DC subcarrer s not used as typcal n practcal OFDM systems. The frequency-ndependent / mbalances are set to α = a t = a r = db), and Δθ = θ a t a t θ t r = θr θr =3. The frequency-dependent / mbalances are modelled by 3-tap flters wth dscrete-tme mpulse responses of [0.01, , 0.01] and [0.015, , 0.01] for the transmt and branches, and [0.012, , 0.018],and [0.01, , 0.02] for the receve and branches. As a TABLE PLOT OVERHEAD COMPARSON FOR A MMO SYSTEM WTH N Tx TRANSMT ANTENNAS Desgn Overhead # of Tones) Proposed 2N TxL 0 Reference [17] 2N TxN used reference, we evaluate the desgn proposed n [17] wth null guard tones accordng to the above settng. n all methods, the maxmum lkelhood detecton s appled. n the reference method, the energy of each non-zero plot tone s kept the same as that of data tone. For a far comparson, total plot energes of the two methods are set the same. The MMO system conssts of two transmt antennas and two receve antannas. Fg. 2 shows the total channel estmaton MSE MSE p + MSE q ) for the proposed desgn and the reference as well as the theoretcal MSEs from Eqns. 14), 15), and 20). The theoretcal and smulaton MSEs are essentally the same. Due to the null guard tones, the reference desgn loses estmaton optmalty slghtly. n Fg. 3, the theoretcal channel MSE s plotted aganst the number of guard tones. Snce the reference desgn depends on all subcarrers beng utlzed, the performance degrades from the optmum MSE performance acheved by the propsed desgn as the number of guard tones ncreases. Fnally, n Fg. 4, the uncoded) BER performances of the two plot desgns are compared. The proposed desgn slghtly outperforms the reference desgn due to the better estmaton performance. Table compares plot overhead of the proposed desgns and the reference desgn. The proposed desgns requre only 2N Tx L 0 tones whle the reference desgn uses 2N Tx N used tones where N used s the total number of subcarrers excludng null guard tones and the DC tone. Snce L 0 s typcally sgnfcantly less than N used wth our smulaton parameters, L 0 =8 and N used = 114), ths demonstrates a substantal reducton n overhead for the proposed desgn, whle provdng better BER performance and applcablty n plot-data-multplexed format. V. CONCLUSONS The presence of more guard tones causes larger performance degradaton to the exstng reference plot desgn for MMO OFDM systems wth / mbalance, but for typcal system settngs, ts performance loss s small. Hence, ts man lmtatons le wth large plot overhead and napplcablty to plot-datamultplexed format. Formulatng n a plot-data-multplexed sgnal model, we have presented a new plot desgn for estmatng the channel n MMO OFDM systems affected by transmt and receve frequency-dependent / mbalances n a multpath fadng channel. Ths plot desgn s applcable to plot-data multplexed systems and provdes mnmal overhead wthout loss of optmal channel estmaton for systems wth up to M 0 3) band-edge null guard tones. Other plot desgns e.g. codng across tme) are possble to acheve our desgn crtera wth null guard tones and more detals are under development.

5 REFERENCES [1] H. Mnn and N. Al-Dhahr, Optmal tranng sgnals for MMO OFDM channel estmaton, EEE Trans. Wreless Commun., vol. 5, no. 5, pp , May [2]. Barhum, G. Leus, and M. Moonen, Optmal tranng desgn for MMO OFDM systems n moble wreless channels, EEE Trans. Sgnal Process., vol. 51, no. 6, pp , Jun [3] B. Razav, RF Mcroelectroncs, Prentce-Hall, NJ, [4] The Moble Broadband Evoluton: 3GPP Release 8 and Beyond: HSPA+, SAE/LTE and LTE-Advanced, 3G Amercas, 2009 [5] C. L. Lu, mpacts of / mbalance on PSK-OFDM-AM detecton, EEE Trans. Consum. Electron., vol. 44, no. 3, pp , Aug [6] M. Valkama, M. Renfors, and V. Kovunen, Compensaton of frequency-selectve / mbalances n wdeband recevers: models and algorthms, EEE SPAWC, Mar. 2001, pp [7] J. Tubbax, B. Come, L. Van der Perre, S. Donnay, M. Engels, M. Moonen, and H. De Man, Jont compensaton of mbalance and frequency offset n OFDM systems, RAWCON, 2003, pp [8] A. Targhat and A. H. Sayed, Jont compensaton of transmtter and recever mparments n OFDM systems, EEE Trans. Wreless Commun., vol. 6, no. 1, pp , Jan [9] D. Tandur and M. Moonen, Jont adaptve compensaton of transmtter and recever mbalance under carrer frequency offset n OFDMbased systems, EEE Trans. Sgnal Process., 2007, [10] W. Krkland and K. Teo, / dstorton correcton for OFDM drect converson recever, Electroncs Letters, 2003, vol. 39, pp [11] L. Brotje, S. Vogeler, K-D. Kammeyer, Estmaton and correcton of transmtter-caused / mbalance n OFDM systems, Proc. 7th ntl. OFDM Workshop, Sept. 2002, pp [12] Y. Egashra, Y. Tanabe, and K. Sato, A novel mbalance compensaton method wth plot-sgnals for OFDM system, EEE VTC Fall, 2006, pp [13] A. Targhat, R. Bagher, and A. Sayed, Compensaton schemes and performance analyss of mbalances n OFDM recevers, EEE Trans. Sgnal Process., 2005, vol. 53, pp [14] R. Chrabeh and S. Solman, mbalance mtgaton va unbased tranng sequences, Proc. EEE GLOBECOM, 2007, pp [15] E. Lopez-Estravz, S. De Rore, F. Horln, and L. Van der Perre, Optmal tranng sequences for jont channel and frequency-dependent mbalance estmaton n OFDM-based recevers, EEE CC 2006, vol. 10, pp [16] E. Lopez-Estravz, S. De Rore, F. Horln, and A. Bourdoux, Plot desgn for jont channel and frequency-dependent transmt/receve mbalance estmaton and compensaton n OFDM-based transcevers, EEE CC 2007, pp [17] T. Schenk, P. Smulders, and E. Fledderus, Estmaton and compensaton of frequency selectve TX/RX mbalance n MMO OFDM systems, EEE CC 2006, vol. 1, pp Channel Estmaton MSE SNR =15 db Proposed Ref # of guard tones Fg. 3. Estmaton MSE comparson varyng number of guard tones n a 2 2 MMO OFDM system BER AM Proposed Ref E b /N 0 db) Channel Estmaton MSE AM Proposed Theo) Ref. Theo) Proposed Smu) Ref. Smu) Fg. 4. BER comparson of dfferent plot desgns for a 2 2 MMO OFDM system wth 16-AM E K /N)/N 0 db) Fg. 2. Estmaton MSE comparson for dfferent plot desgns n a 2 2 MMO OFDM system