PRECODER AD EQUALIZER DESIG FOR MULTI-USER MIMO FBMC/OQAM WITH HIGHLY FREQUECY SELECTIVE CHAELS Yao Cheng, Leonardo G. Batar, Martin Haardt, and Josef A. osse Communications Research Laboratory Imenau University of Technoogy P.O. Box 100565, D-98684 Imenau, Germany {y.cheng, martin.haardt@tu-imenau.de Institute for Circuit Theory and Signa Processing Technische Universität München 8090 Munich, Germany {eo.batar, osef.a.nosse@tum.de ABSTRACT In this contribution we propose two new designs of transmit and receive processing for muti-user mutipe-input-mutipe-output (MIMO) downin systems that empoy fiter ban based muticarrier with offset quadrature ampitude moduation (FBMC/OQAM). Our goa is to overcome the imits on the channe frequency seectivity and/or the aowed number of receive antennas per user termina that are imposed on the state-of-the-art soutions. In the first method the design of precoders and equaizers is iterative and minimum mean square error (MMSE) based. The second is a cosed-form design based on the signa-to-eaage ratio (SLR). Via numerica simuations we evauate the performance of both methods and demonstrate their superiority over two other approaches in the iterature. Index Terms FBMC/OQAM, muti-user MIMO downin, precoding 1. ITRODUCTIO Fiter ban based muti-carrier moduation (FBMC) is widey nown as a promising aternative to orthogona frequency division mutipexing with the cycic prefix insertion (CP-OFDM). Thans to the use of spectray we-contained synthesis and anaysis fiter bans at the transmitter and at the receiver [1, ], FBMC features a concentrated spectrum and a much ower out-of-band radiation compared to CP-OFDM. Consequenty, it is beneficia to choose FBMC over CP-OFDM for asynchronous scenarios [3, 4], or to achieve an effective utiization of spectrum hoes [5, 6]. Moreover, in systems where fiter ban based muti-carrier with offset quadrature ampitude moduation (FBMC/OQAM) is empoyed, the insertion of the CP is not needed as in CP-OFDM based systems, eading to a higher spectra efficiency. To expoit the benefits of both this advanced muti-carrier scheme and the muti-user downin with space division mutipe access (SDMA), there have been severa proposas in the iterature that shed ight on appropriate designs of transmission strategies for FBMC/OQAM based muti-user downin settings. Most of these state-of-the-art soutions rey on the assumption that the channe on each subcarrier can be treated as fat fading. Among them, the spatia Tominson Harashima precoder in [7] aows ony one receive antenna at each user termina and is more prone to a high computationa compexity compared to inear precoders. On the other hand, a boc diagonaization (BD) based inear precoder has been deveoped in [8] for the FBMC/OQAM based muti-user The authors gratefuy acnowedge the financia support by the European Union FP7-ICT proect EMPhAtiC (http://www.ict-emphatic.eu) under grant agreement no. 31836. MIMO downin. It adopts the centra idea of BD [9] to mitigate the muti-user interference (MUI) and then uses the zero forcing based approach [10] to dea with the intrinsic interference canceation for the resuting equivaent singe-user transmissions. Furthermore, the coordinated beamforming based transmission strategies devised in [11, 1] have the advantage of aeviating the dimensionaity constraint such that the number of receive antennas is not restricted. evertheess, the vioation of the assumption that the channe on each subcarrier is fat fading resuts in performance degradation of the aforementioned techniques. Focusing on the case of highy frequency seective channes, the inear precoder in [13] has a structure of a fiter appied on each subcarrier and its two adacent subcarriers at twice the symbo rate. In [14], two different minimum mean square error (MMSE) based approaches have been devised for FBMC/OQAM based muti-user mutipe-input-singe-output (MISO) downin systems aso considering highy frequency seective channes. A cosed-form soution is provided in the first scheme, where one compex-vaued fractionay spaced muti-tap precoder for each user on each subcarrier is appied at the transmitter and a singe-tap rea-vaued weight at the receiver. The second scheme invoves a oint transmitter and receiver design via an iterative procedure, where now the equaizer at the receiver side is aso compex-vaued fractionay spaced muti-tap. Simiar to [13], the two methods in [14] are restricted to the case where each user is equipped with a singe receive antenna. Stemming from the probem formuation in [14], we first deveop an iterative approach for FBMC/OQAM based muti-user MIMO downin systems under highy frequency seective propagation conditions. MMSE based muti-tap precoders are designed to effectivey mitigate the MUI, inter-symbo interference (ISI), and inter-carrier interference (ICI). At each user termina equipped with mutipe antennas, ony a receive spatia fiter is appied. Via an iterative design, the precoders and the receive spatia fiters are ointy optimized. Then, we further propose a nove cosed-form signa-to-eaage (SLR) based design of the precoders that can be convenienty extended to the case of mutipe spatia streams per user. Via numerica resuts, it is observed that the proposed schemes achieve a very promising performance in case of highy frequency seective channes. They significanty outperform the state-of-the-art approaches that suffer from impractica restrictions on the channe frequency seectivity. By aowing mutipe antennas at each user termina, the benefits of the MIMO technoogy are expoited.. SYSTEM MODEL In the downin setting of an FBMC/OQAM based muti-user MIMO system, one base station equipped with T transmit an- 978-1-4673-6997-8/15/$31.00 015 IEEE 49 ICASSP 015
tennas serves U users simutaneousy. The th user has R receive antennas, whereas the tota number of receive antennas of a users is denoted by R = U =1 R. One data stream is transmitted to each user. Muti-tap precoding fiters at the base station and equaizers at the user terminas are designed to mitigate the MUI, ISI, and ICI as we as to recover the desired signas. After the OQAM staggering [1], the signa transmitted to the sth user on the th subcarrier is denoted byx s [n],s = 1,,...,U and has the foowing structure [ T, α s [m] β[m] s α s [m 1] ] is odd, x s [n] = [ T, β[m] s α s [m] β[m 1] ] s is even, whereα s [m] andβ[m], s with variance as one, represent the rea part and the imaginary part of the QAM moduated data symbo. Moreover, the impuse response of the precoding fiter for x s [n] is represented byb i,s [n],i = 1,,...,T, whieb denotes its ength. The resuting signas are mutipexed by a synthesis fiter ban (SFB). Throughout this wor, highy frequency seective channes are considered. At the receiver, signas from the subchannes are separated by an anaysis fiter ban (AFB). For the rth receive antenna of the th user on the th subcarrier, an equaizer is appied, and its impuse response is represented by g,r [n], where = 1,,...,U, and r = 1,,..., R. Assuming that the signa on the th subcarrier is contaminated by interference from the ( 1)th and the (+1)th subcarrier [14, 15], the recovered signa of the th user on the th subcarrier can be written as R ˆx [n] = ( T ( +1 i=1 = 1 g,r [n] h i,,r [n] U s=1 ( ) ) ) b i,s [n] xs [n] +ˆη,r [n], (1) where ˆη,r [n] denotes the fitered additive white Gaussian noise, and h i,,r [n] represents the equivaent channe impuse response for the signa that is transmitted on the th subcarrier from the ith transmit antenna to the rth receive antenna of the th user and passed through the anaysis fiter for the th subcarrier. It incudes the effects of the transmit fiter, the propagation channe, the receive fiter as we as the upsamping and downsamping operations. The resuting number of taps of this equivaent channe is (LP 1)+L ch Q =, () M/ wherel ch andl P denote the ength of the channe impuse response and the ength of the prototype fiter, respectivey. Here for the prototype fiter we choose a root raised cosine (RRC) with ro-off one and ength L P = KM + 1 [14, 15], where K represents the time overapping factor, andm denotes the number of subcarriers. 3. ITERATIVE DESIG OF MMSE BASED PRECODER AD REAL-VALUED RECEIVE SPATIAL FILTER The precoder is designed to mitigate the MUI, ISI, and ICI. At each user termina, ony a rea-vaued singe-tap spatia fiter is appied. The recovered desired signa ˆα [m] is expressed in a matrix-vector formuation as foows R {( ˆα [m] = +1 ) g,r Re U x st [n] H,r bs + ˆη,r [n], = 1 s=1 (3) where [ b s = b 1,sT b,st ] T b T,s T C T B contains the T coefficients of the precoding fiter for the sth user on the th subcarrier, and [ ] H,r = H 1,,r H,,r H T,,r C (B+Q 1) T B with H i,,r C (B+Q 1) B, i = 1,,..., T, representing a Toepitz matrix of the equivaent channe coefficients h i,,r [n]. The data vector x s [n] C B+Q 1 contains consecutive data symbos, whereas g,r denotes the rea-vaued coefficient of the spatia fiter for the signa on the th subcarrier received by the rth receive antenna of the th user. In case of ˆβ [m], it is obtained by taing the imaginary part of (1). Since both cases are equivaent to each other, we focus on the case of ˆα [m] in the seque. In this iterative design, the precoders and the spatia fiters are updated aternatey. First given the spatia fiter, the expression of the estimated desired signa of the th user on the th subcarrier can be written as ˆα [m] =Re +1 U = 1 s=1 x st [n] H bs + R g,r ˆη,r [n], where H = R g,r H,r is a short-hand notation. The precoder for the th user on the th subcarrier that minimizes the mean square error and the eaage can be obtained via the foowing optimization probem [14] b = argmin E b { ˆα [m] α [m ν] +c +u (4), (5) whereν = B+Q 1 is the (integer) deay of the system. The terms c and u measure the interference caused by the signa of the th user from the th subcarrier on the adacent subcarriers and on the other users. They tae the foowing forms R c = +1 = 1, u = U s=1,s + Re Im {x T [n] H,r b ( +1 = 1, {x T [n] Hs,r, b Let us define x RB+Q 1 such that Im {x T [n] Hs,r b (6) ). (7) x [n] = J x [n], (8) where the diagona matrix J C (B+Q 1) (B+Q 1) has 1 and aternatey appearing on its diagona. Simiary as in [14], the foowing inear expressions for the rea part and the imaginary part of the interference terms in (6) and (7) can be obtained = x T Re Im {x T [n] Hs,r, b {x T [n] Hs,r b [n] Ψs,r, ξ (9) = x T [n] Φs,r ξ (10) 430
where { Ψ s,r, [Re = J H s,r, { Φ s,r [Im = J H s,r [ { ] ξ Re b = Im { b { ] Im J H s,r, { ] Re J H s,r (11) (1). (13) The inear formuation for the operation of taing the rea part in (4) can be simiary obtained via defining { { ] Ψ [Re = J H Im J H. (14) Assume that data symbos are uncorreated, and the desired signa and the noise are uncorreated. After inserting (4), (6), and (7) into the optimization probem defined via (5), it can be further reformuated using (9) (14). Then, taing the derivative with respect to ξ and setting it to zero yied ( U ξ = Ψ T, Ψ, + + U s=1,s Rs +1 Φ s,rt Φ s,r s=1 = 1, Ψ s,rt, Ψs,r, ) 1 T Ψ, eν, (15) wheree ν C B+Q 1 is a unit vector with itsνth eement as one. To sove the probem that the matrix might be i-conditioned, we add α I T B in the matrix inversion invoved in (15) withα > 0 as a reguarization factor. Then, we normaize the precoders as in [14] to imit the transmit power with the reation ˆξ = ξ / ξ,t ξ. After computing the precoders, we turn to update the spatia fiters assuming that the MUI, ISI, and ICI have been canceed competey, i.e., ˆα [m] taes the foowing form in the noiseess case ˆα T [n] R [m] = Re g,r H,r, b. (16) x Define a vectorg R R that contains the R coefficients of the spatia fiter of the th user on the th subcarrier, i.e., ] g [g =,1 g, g, R T. (17) The maxima-ratio combining (MRC) is used as the criterion for the spatia fiter. Therefore, g can be obtained as g = ȟ,, (18) where the rth eement of ȟ,,r = 1,,...,R, is given by { ȟ,r, = Re e T ν H,r, b. (19) In this proposed iterative scheme,g s (s = 1,,...,U) that contains the rea-vaued coefficients of the receive spatia fiter of the sth user on the th subcarrier is initiaized randomy. To determine the termination of the iterative procedure, we propose to use the foowing stopping criterion. The term (g) that tracs the change of the receive spatia fiters is defined as (g) = U g s(p) s=1 g s(p 1), (0) where g s(p) and g s(p 1) R contain the coefficients of the receive spatia fiter of the sth user computed in the pth iteration and the (p 1)th iteration, respectivey. A threshod denoted by ǫ is set to 10 5 in the simuations. At the end of each iteration, (g) is cacuated and compared to ǫ. If (g) < ǫ, the agorithm terminates, and the precoders are obtained. Otherwise, the iterative procedure continues, and the precoders as we as the receive fiters are further updated. 4. SIGAL-TO-LEAKAGE (SLR) BASED PRECODER AD REAL-VALUED RECEIVE SPATIAL FILTER Simiar to the iterative design, we again consider a rea-vaued spatia fiter at each user node. Instead of ointy and iterativey updating the precoder and the spatia fiter, we propose a cosed-form SLR based inear precoder. A MRC based spatia fiter is empoyed at each user node. The effective channe with respect to therth receive antenna of the th user for the th subcarrier is given by h (eff),,r, = Ψ,rT, eν R T B. (1) The ISI for the signa of the th user on the th subcarrier is measured via R ISI = E x T [n] Ψ(int),,r, ξ where Ψ (int),,r, = ξ T R Ψ (int),,rt, Ψ (int),,r, ξ, () = J (ν) int Ψ,r, R(B+Q 1) T B. (3) Here J (ν) int R (B+Q 1) (B+Q 1) is constructed by repacing the νth row of a (B + Q 1)-by-(B + Q 1) identity matrix by an a-zeros vector. The interference that is eaed to the adacent subcarriers and other users by the signa for the th user on the th subcarrier can be represented as E { c +u, where c and u are given by (6) and (7), respectivey. Consequenty, the SLR on the th subcarrier for the th user denoted by SLR has the form given in (4) on the next page. The precoder for the th user on the th subcarrier can be obtained via The soution is given by ξ = argmax SLR ξ. (5) ξ = P{ C 1 A, (6) where P { represents the operator of computing the principa eigenvector of a matrix which corresponds to the maxima eigenvaue. Simiary as for the iterative design, we introduce a reguarization factor α in the matrix inversion invoved in (6). ote that the extension of this SLR based scheme to the case of mutipe spatia streams per user can be convenienty conducted based on a simiar phiosophy introduced in [16], which is not eaborated in this wor due to imited space. As the MUI, ISI and ICI are mitigated by the SLR based precoders at the base station, each user termina then appies the MRC based spatia fiter given by (18) and (19). 431
SLR = ( U ξ T Rs +1 s=1 = 1, ξ T { A { R Φ s,rt Φ s,r + U h (eff),,r, s=1,s h (eff),,rt, ξ R Ψ s,rt, Ψs,r, + Ψ (int),,rt, Ψ (int),,r, {{ C ) ξ (4) 5. SIMULATIO RESULTS In this section, we evauate the performance of the proposed transmitter and receiver designs in various muti-user MIMO downin settings. The tota number of subcarriers is 18, whereas the number of subcarriers with data symbos is 7. The subcarrier spacing is set to 15 Hz, and the bandwidth is 1.4 MHz. The data symbos are drawn from the 16 QAM consteation. We empoy the WI- ER Phase II spatia channe mode based on the 3GPP as in [14]. The maximum channe impuse response ength is approximatey L ch = sampes. For a exampes, 1000 channe reaizations have been performed with 1000 symbos per subcarrier, and a singe spatia stream is sent to each user. Fig. 1 depicts the BER performances of the MMSE based iterative design and the cosed-form SLR based design. The former sighty outperforms the atter with the price of a higher computationa compexity due to the iterative procedure. Sti, the average number of iterations required for convergence is ony around three. It is aso worth noting that via numerica simuations, we have observed that this number is barey affected by the ength of the precoders or the MIMO settings. On the other hand, even with ony a singe tap, i.e., B = 1, the performance superiority of the both proposed schemes over the BD based technique [8] is significant. uncoded BER 10 0 10 1 10 10 3 10 4 8 4, B = 1, iterative 8 4, B = 3, iterative 8 4, B = 1, SLR based 8 4, B = 3, SLR based 8 4, BD based [8] 4 4, B = 1, iterative 4 4, B = 1, SLR based 4 4, BD based [8] 15 10 5 0 5 10 15 E b / 0 Fig. 1. BER performance in different muti-user MIMO downin settings where U =, R1 = R =, T = 8, or 4, and α = 0.05 Via Fig., the impact of the number of taps and the choice of the reguarization factor is investigated. Longer precoders contribute to a superior capabiity in mitigating the MUI, ICI, and ISI. Hence, the BER performance of the proposed iterative approach becomes better, as the ength of the precoding fiters increases. evertheess, by increasing the ength from five to nine, the performance improvement is aready very sma. In addition, it is observed that a arger vaue for the matrix inversion reguarization factor eads to a better performance in the ow signa-to-noise-ratio (SR) regime, whie there is an error foor in the high SR regime. The reason is that a arger vaue for the reguarization factor resuts in arger residua ISI and residua eaage due to approximation errors induced by the reguarization procedure. Unie the ow SR regime, in the high SR regime the residua interference (residua ISI and eaage) dominates instead of the noise. Hence, this observation inspires the design of an SR-reated reguarization factor as future wor. For a the vaues of the engths of the precoders and the choices of the reguarization factor considered in this exampe, the iterative scheme achieves a much better performance compared to the IIM-CBF 1 scheme in [1] and the BD based technique [8]. uncoded BER 10 0 10 1 10 10 3 10 4 10 5 iterative design, B = 9, α = 0.01 iterative design, B = 5, α = 0.01 iterative design, B = 1, α = 0.01 iterative design, B = 5, α = 0.1 IIM CBF 1 [1] BD based [8] 15 10 5 0 5 10 15 E / [db] b 0 Fig.. BER performance in a muti-user MIMO downin setting where U =, R1 = R = 3, T = 6 6. COCLUSIOS In this wor we have proposed two designs of precoders and equaizers for FBMC/OQAM based muti-user MIMO downin systems. The first one is an iterative approach where the MMSE based precoders and the MRC based receive spatia fiters are ointy computed. In the second scheme, we have devised cosed-form SLR based precoders for the base station and a simiar MRC based receive processing for each user termina. Both schemes are deveoped to effectivey mitigate the MUI, ISI, and ICI in critica highy frequency seective propagation conditions. From the numerica resuts it can be concuded that the first approach provides a sighty better performance compared to the second method, at the cost of a higher compexity due to the oint transceiver design. Moreover, both approaches show a considerabe improvement over two state-of-the-art schemes that do not consider the channe frequency seectivity inside each subcarrier. 43
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