TENSOR-BASED FRAMEWORK FOR THE PREDICTION OF FREQUENCY-SELECTIVE TIME-VARIANT MIMO CHANNELS

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1 TENSOR-BASED FRAMEWORK FOR THE PREDICTION OF FREQUENCY-SELECTIVE TIME-VARIANT MIMO CHANNELS Marko Miojević, Giovanni De Gado, and Martin Haardt Imenau University of Technoogy, Communications Research Laboratory, P.O. Box 1565, Imenau, Germany {marko.miojevic, ABSTRACT In this contribution we propose a tensor-based framework for the prediction of time-variant frequency-seective Mutipe- Input Mutipe-Output (MIMO) channes from noisy channe estimates. This method performs the prediction in a transformed domain obtained via the higher order singuar vaue decomposition (HOSVD), namey on the transformed tensor eements. This is foowed by the inverse transformation of the predicted transformed tensor eements onto a basis corresponding to the signa subspace. To verify our strategy, we compare the resuts in terms of the normaized mean square error using a known prediction method, e.g., a Wiener fiter, appied to the transformed tensor eements with the identica method appied directy to the channe coefficients. The resuts of our investigation show that the tensor-based prediction method outperforms the direct prediction method. Athough we concentrate in this contribution on the prediction in the time domain, this framework can aso be used for the estimation in other domains. 1. INTRODUCTION Channe prediction has been studied in numerous contributions. It is an important too to enhance the performance of wireess communication systems beyond 3G [1], where time variant frequency seective MIMO channes are expoited with OFDM techniques. Channe prediction combats efficienty feedback deay and is beneficia for adaptive moduation, power contro, MIMO precoding, and muti-user scheduing. It is shown in [2, 3] that imperfect channe state information degrades the OFDM system performance, and that channe prediction improves the system performance. Performance of channe prediction with inear fiters and quadratic Voterra fiters is discussed in [4]. It is found that inear fiters have a imited performance for ong range prediction. On the other hand, the east squares soution for quadratic fiters reies on accurate estimates of fourth order moments, which imits its use. A noninear channe predictor using the Mutivariate Adaptive Regression Spines (MARS) is proposed in [5]. It is abe to detect the statistica dependence of time-varying channes better than inear fiters, yieding a onger prediction interva. The modified Root-MUSIC and ESPRIT agorithms are used to mode and predict the time-varying channes in [6] and [7], respectivey. In [7] it is shown that channe prediction at different frequencies jointy performs better than channe prediction over every singe frequency separatey. In [8] a ow compexity short range channe prediction using poynomia approximation is proposed. The MMSE channe predictor is introduced and improved in [9] and [1], respectivey. The adaptive normaized east-mean square (NLMS) and recursive east squares (RLS) channe predictors for OFDM systems are introduced in [1]. The NLMS agorithm showed ower computationa compexity but a worse performance than the RLS agorithm. A ower bound on the channe prediction error for MIMO channes is derived using the Cramer-Rao ower bound in [11]. Moreover, it is concuded that MIMO channes can be predicted onger in the future than Singe-Input Singe- Output (SISO) channes. Channe prediction schemes based on Kaman fitering are proposed in [12, 13, 14]. In [15] the channe prediction based on poarimetric modeing is studied. The time-variant fat-fading channe prediction based on discrete proate spheroida sequences is studied in [16]. Reference [17] introduces subspace-based channe prediction. In contrast to the methods reviewed above, in this contribution we present a time variant frequency seective MIMO tensorbased channe prediction framework: the prediction is performed in a transformed domain, namey on the transformed tensor eements, instead of the channe prediction performed directy on the channe coefficients. Different fiter prediction structures can be used, e.g., inear fiters, non-inear fiters, adaptive prediction fiters, etc. In this paper we combine a tensor-based prediction framework with a simpe inear fiter, in order to compare it with the direct channe prediction. The paper is organized as foows: the tensor-based channe prediction framework is described in Section 2, simuation resuts for specific yet reaistic scenarios are presented in Section 3, and Section 4 draws the concusions /8/$25. c 8 IEEE 147

2 2. TENSOR BASED CHANNEL PREDICTION FRAMEWORK The higher order singuar vaue decomposition (HOSVD) of an N th-order tensor is defined for every compex tensor H of size (I 1 I 2... I N ). According to [18], we write the tensor H as H = S 1 U 1 2 U 2... N U N, (1) where n denotes the n-mode product of a tensor by a matrix, each U n is a unitary (I n I n ) matrix consisting of n-mode singuar vectors, and S C I1 I2... IN is the core tensor. The tensor S has two significant properties: i) Orthogonaity: if a b two sub-tensors S in=a (the n- th dimension is set to the fixed vaue a) ands in=b are orthogona for a possibe vaues of n, a, andb. ii) Ordering: the higher-order norms of a tensor S in= 1 H, =1, 2,...I n, denoted aso as σ (n),aren-mode singuar vaues of H with the foowing property: S in=1 H S in=2 H... S in=i n H for a vaues of n. The n-mode product H n U of a tensor H C I1 I2... IN by a matrix U C Jn In is an (I 1 I 2... I n 1 J n I n+1... I N ) tensor whose eements are cacuated as (H n U) i1...i n 1j ni n+1...i N = i n H i1i 2...i N u jni n, (2) where the subscripts i 1...i N denotes the indices of the corresponding tensor dimensions (e.g., U i1i 2 is an eement of the matrix U with row index i 1 and coumn index i 2 ). The HOSVD decomposition in case of a two dimensiona tensor (matrix) reduces to the SVD decomposition. For more detais on the HOSVD see [18, 19, ]. The samped channe impuse response (CIR) of a timevariant frequency-seective MIMO channe can be represented by a four-dimensiona tensor H (exact) C MR MT Nf NT, where M R and M T are the number of antenna eements at the receiver and at the transmitter, whereas N f and N T are the number of snapshots samped in the frequency and the time domain, respectivey. We assume that the past noisy channe estimates are avaiabe. If we group the ast N T noisy time snapshots for a antennas and frequencies, they form the estimated channe tensor H C MR MT Nf NT.Inthis paper we focus on the short term prediction in the time domain for a antenna eements and frequencies. Let h r,q,i,j 1 The scaar product of two tensors S, B C I 1 I 2... I N is symboized by S, B and computed by summing the eement-wise product of S and B over a indices, where denotes compex conjugation. The scaar product aows us to define the higher-order norm of a tensor S as. p S H = S, S. denote the tensor eement corresponding to the channe between the r-th receive and q-th transmit antenna at baseband frequency f i = iδf and time t j = jδt, whereδt and Δf are the time samping interva and the subcarrier spacing of the system under consideration, respectivey. Let H(t n ) (t n denotes the time index of the ast known sampe) be the sub-tensor of the tensor H, containing the eements h r,q,i,j, where r {1, 2,..., M R }, q {1, 2,..., M T }, i F := { Nf Nf 2 +1, 2 +2,..., N f 2 },andj T := {n NT s +1,n N T s +2,..., n}. Here the current time and the number of sub-tensor time snapshots are denoted by t n and NT s, respectivey. The vaues N T s and N f shoud be chosen such that NT s Δt is cose to the coherence time, whie N fδf is cose to the coherence bandwidth of the channe. In Fig. 1 the 4-D tensor channe representation is depicted. MT MR f f f f 2 tn t MT MR tn Past (Known) Fig D tensor channe representation. t Future (Predicted) Knowing N T N f noisy MIMO channe estimates from the past, we predict the exact channe coefficients h (exact) r,q,i,p, where p = n + P is the index of the future time snapshot at the prediction horizon. Prediction at time instances that are not mutipes of a time samping interva Δt can be performed via an interpoation. The vaue of the prediction horizon P depends on system specific parameters. A reaistic exampe is discussed in Section 3. By using the HOSVD we carry out the prediction on the transformed tensor coefficients instead of the channe coefficients. To cacuate the transformed tensor, first the matrices U n =[u (n) 1 u (n) 2... u (n) I n ], n =1, 2,..., N, consisting of n-mode singuar vectors u (n), =1...I n,which efficienty describe the tensor signa subspace, must be determined. They can be cacuated as the matrix of eft singuar vectors of the n-th unfoding, n =1, 2,..., N [18]. For the tensor-based prediction we operate within the coherence time and coherence bandwidth of the channe, whie the antenna array eements are fixed. Therefore, we assume that the bases U m,m k, wherek is the prediction or estimation dimension (here time) are constant in the vicinity of the observed present channe coefficients h r,q,i,n. After obtaining the matrices U m, the transformed tensor A can be computed as A = H 1 U1 H... k 1Uk 1 H k+1uk+1 H... N UN H = H 1 U1 H 2U2 H... N UN H ku k = S k U k, (3) Internationa ITG Workshop on Smart Antennas (WSA 8)

3 where ( ) H denotes the Hermitian transpose and k is the prediction dimension. Note, that the equation (3) is generaized to an N dimensiona tensor, athough we dea here with the 4-D tensor. Let A(t j ) denote the transformed tensor from (3) cacuated from the known estimated channe subtensor H(t j ) with the use of matrices U n, n k obtained from the whoe tensor H, andeta r,q,i,j (t j ) be the corresponding transformed tensor eements. By doing this, the channe changes in the time domain are represented in tensors A(t j ). As an exampe, we use a inear prediction fiter. To perform prediction at the time p = n + P with the inear fiter, we cacuate the transformed tensors A(t n N+1), A(t n N+2),..., A(t n ) to track its change in time, where N is the number of sub-tensors. Let us denote the prediction fiter function as F ( ). Now we can predict the transformed tensor eements at the future time p = n + P based on the knowedge of the transformed tensor eements from the past: â r,q,i,j (t p )=F(a r,q,i,j (t j )) (4) for every vaue of r, q, i and j, wherej {n N +1,n N +2,..., n}, and( ) ˆ denotes the prediction in the future. For comparison, the same prediction function is appied directy on the channe coefficients: ĥ r,q,i,p = F (h r,q,i,j ) (5) for every vaue of r, q and i, wherej {n N +1,n N +2,..., n}. If the time window T used for the tensor cacuation is comparabe to the coherence time of the channe and if the channe is composed of muti-path components whose directions of arrivas and directions of departures change sowy in time, the channe wi span a common signa subspace so that a reduced number r n, n =1, 2,...N, of basis vectors per tensor dimension n is sufficient to represent it. Here <r n I n. Therefore, it is possibe to predict H (exact) (t p ) using a ow rank approximation of Â: Ĥ (exact) (t p )=Â[s] (t p ) 1 U [r1] 1... k 1 U [rk 1] k 1 k+1 U [rk+1] k+1... N U [rn ] N. (6) Here U n [rn] C In rn, n =1, 2,...N, is the matrix consisting of r n n-mode singuar vectors (defining the signa subspace) corresponding to the r n strongest n-mode singuar vaues, and Â[s] r1 r2... rn (t p ) C is the predicted truncated transformed tensor consisting of eements corresponding to the signa subspace singuar vectors. We denote this method as tensor based channe prediction. The prediction performed directy on the estimated channe coefficients is denoted as direct channe prediction. The prediction of the transformed tensor eements rather than the channe eements themseves has the advantage of a reduced time variance and an inherent spatia noise fitering. That has been shown in [21] for its matrix vaued counterpart. For the tensor antenna dimensions (the first and the second dimension) ony the dominant basis vectors map the strongest muti-path components, whie the remaining basis vectors map either the diffuse mutipath component or the estimation noise. Severa fiter structures F ( ) can be used for the prediction on the future transformed tensors coefficients or for the direct prediction of the channe coefficients ĥr,q,i,p: inear fiters [4], non-inear fiters [5], adaptive prediction fiters [1, 12, 13] etc. To make a fair comparison between tensor-based and direct channe prediction, we perform tensor-based prediction and direct channe prediction, as formuated in (4) and (5), using identica fiter structure F ( ). Thus, the prediction of a time variant frequency seective MIMO channe is partitioned into M R M T N f SISO channe predictions. In this contribution we use a Wiener fiter, described in the foowing. Let y(j) be the vaue of the compex variabe y at time instant jδt. Wepredicty at time (n + P )Δt, nameyŷ(n + P ), in the future based on the known sampes of y up to the present time nδt. Let the vector φ(n) contain a set of known N P sampes: φ(n) =[y(n P ) y(n P 1)... y(n N +1)] T, (7) where ( ) T denotes the transpose operator. vector w(n) is then cacuated as: The weighting w(n) =(φ(n) H φ(n)) 1 φ(n) y(n) C N P 1, (8) where ( ) denotes conjugation. Let the vector θ(n) contain the ast N P known sampes of y θ(n) =[y(n) y(n 1)... y(n N + P +1)] T. (9) Finay, the predicted vaue of y at time (n+p )Δt is obtained as: ŷ(n + P )=θ(n) T w(n). (1) 3. SIMULATION RESULTS In this section we test the vaidity of our tensor-based prediction framework. We compare a prediction method defined in equations (7)-(1) carried out on each of M R M T N f SISO channe inks in the time dimension, and the same prediction method carried out on the transformed tensor eements a r,q,i,j (t j ) (M R M T N f NT s predictions) as defined in (4) foowed by the step in equation (6) from predicted tensor, by means of the Normaized Mean Square Error (NMSE). The NMSE of the tensor Â, which is the estimate of A,isdefined as ɛ = Â A 2 H A 2. (11) H The same OFDM simuation parameters as defined in [1] are used. The basic OFDM time-frequency resource unit, named chunk, asts for.3372 ms and has a bandwidth of khz. In our anaysis we use synthetic CIRs generated with the ImProp, a geometry based channe modeing too for wireess communications [22]. Fig. 2 depicts a bridge-to-car 8 Internationa ITG Workshop on Smart Antennas (WSA 8) 149

4 scenario, the propagation environment which we use here for the discussion of the simuation resuts. In this scenario the 4-eement Uniform Linear Array (4-ULA) mobie termina is moving with a speed of 5 km/honthehighway,whieasecond 4-ULA, acting as base station (BS), is mounted on the bridge crossing over the highway. The channe parameters (deay spread, Dopper spread, etc.) of the channes created with the ImProp match very cosey measured channe parameters in the corresponding scenario [23]. In [24, 25], it BS Fig. 2. A bridge-to-car propagation scenario generated with the ImProp. has been shown that a channe prediction with an NMSE of.1 eads to a minor degradation in the attained spectra efficiency. An upper imit of.15 for the aowed NMSE has been chosen in [1]. In investigations presented in [26], the channe estimation error is approximated by white Gaussian noise. The power σe 2 of the normaized mean square channe estimation error is modeed as: { ρ 13 db, ρ < 17 db σe 2 = (12) 3 db, ρ 17 db, where ρ is the Signa to Noise Ratio (SNR) at the input of the channe estimator in db. The noisy channe estimates H can be modeed as H = H (exact) + E, (13) where E is the noise term introduced by the channe estimator containing independent and identicay distributed compex Gaussian random numbers with average power σe 2. Let ɛ ten (t p ) be the predicted channe s normaized mean square error using the tensor-based method and ɛ dir (t p ) the predicted channe s NMSE empoying the prediction directy on the channe coefficients, both averaged over frequency and antenna dimensions, at future time snapshot t p. The term ɛ ten (t p ) depends on the size of T and F, and on the vaues r n, n =1...N, from (6). In our simuations we used vaues N T Δt=5.4 ms and N f Δf=1.25 MHz, which are cose to the coherence time and coherence bandwidth of the channe, respectivey. Fig. 3 shows the absoute vaues of the tensor n- mode singuar vaues in a dimensions for the bridge-to-car M1 scenario, averaged over a other dimensions but the observed dimension n. In a 4 dimensions the first n-mode singuar vaue, corresponding to the strongest singuar vector in that dimension, is significanty higher than the other singuar vaues in that mode for this specific scenario. Singuar vectors corresponding to very weak singuar vaues, can be negected (e.g. in the 3 rd dimension (frequency) the singuar vectors starting from the fourth) in the equation (6). Channes with r n significanty ower than I n are quite common in reaity. An approach to determine the number of singuar vectors r n that shoud be taken into account in (6) to predict the channe coefficients is to use the first r n singuar vectors corresponding to the r n argest singuar vaues in the signa subspace. Since the noise is white Gaussian (equation (12)) it is expected that the noise contributes the same vaue to a eigenvaues defining the eigenvaue threshod: ony r n eigenvectors corresponding to the eigenvaues higher than this threshod are used in (6). Since the n-th tensor dimension has I n n-mode singuar vaues, the eigenvaue threshod σ (n) th for the n-th tensor dimension can be estimated as: σ (n) th = σ 2 e I n. (14) Note that singuar vaues σ (n) have to be scaed for every dimension so that the tota channe power is equa to 1. That step is not done for the vaues in Fig. 3. (1) E[σ ] (3) E[σ ] (2) E[σ ] (4) E[σ ] Fig. 3. Singuar vaues σ (n) of the tensor corresponding to the n-th dimension of the channe. The index n corresponds to the foowing channe dimensions: 1 - Receive antenna domain, 2 - Transmit antenna domain, 3 - Frequency domain, 4 -Timedomain In Fig. 4, we compare ɛ ten (t p ) and ɛ dir (t p ) for different SNR vaues and different prediction horizons (t p ). Additionay, we show the performance of the matrix-based subspace 15 8 Internationa ITG Workshop on Smart Antennas (WSA 8)

5 prediction presented in [17]. Its NMSE averaged over frequency and antenna dimensions at the future snapshot t p is denoted as ɛ mat (t p ). For the observed scenario ɛ ten (t p ) is ower since the tensor based prediction incudes the singuar vectors corresponding to the strongest singuar vaues (signa subspace) whie negecting most of the noise subspace. Moreover, the tensor-based subspace channe prediction performs better than the matrix-based subspace channe prediction since it is abe to assess better the properties of the timevariant frequency-seective MIMO channes [27] especiay for arge prediction horizons and high SNR vaues. The tensor based prediction method outperforms the direct prediction method when the noise eve is we determined. The performance difference is bigger for ow SNR vaues, where the proposed method reduces the NMSE significanty. If there were no channe estimation errors, the best resut woud be obtained by the use of a singuar vectors. If the noise eve vaue is not determined appropriatey, we might negect a part of the signa subspace or incude the noise and therefore increase prediction errors. In Fig. 4 the asymptotic behavior of two prediction methods can be noticed: at high SNRs and ow prediction horizons the prediction error ɛ dir (t p ) approaches ɛ ten (t p ). The tensor-based prediction compexity is significanty higher than the direct prediction compexity, mosty due to the HOSVDs and a mutipe n-mode products of a tensor with a matrices that have to be performed. The ow rank approximation (6) reduces the computationa compexity twofod: 1) the n-mode products have ess mutipications and additions, and 2) not a but r 1 r 2... r k 1 r k+1... r N transformed tensor coefficients in (4) have to be predicted. Additionay, since the core tensor is approximatey constant for N + P sub-tensors it does not have to be cacuated at each time step. Simiar investigations have been performed for Non Line-Of-Sight (NLOS) mobie scenarios where we have richer scattering than in the LOS environments. There the tensor-based prediction again outperforms the direct prediction, with the performance gap being ower as in the LOS case. 4. CONCLUSIONS In this contribution we propose a tensor-based prediction framework for time-variant frequency-seective MIMO channes from noisy channe estimates. This method performs the prediction on the transformed tensor eements whie predicting ony the signa subspace of the channe. The proposed framework is compared with a direct channe prediction technique with respect to the normaized mean square errors of the predicted channes. The tensor-based prediction framework outperforms the direct prediction method. In the anayzed scenarios, the NMSE improvement is higher for ower SNR regimes of the estimated channe. Moreover, the tensor-based subspace prediction framework for time-variant frequency-seective MIMO channes outperforms its matrix vaued counterpart Fig. 4. Resuting NMSE as a comparison between the tensorbased and matrix-based subspace channe predictions and the direct channe prediction because it captures better the mutidimensiona nature of the channe, especiay for arge prediction horizons and high SNR vaues. 5. REFERENCES [1] D2.4: Assessment of adaptive transmission technoogies, Tech. Rep. IST (avaiabe at WINNER, February 5. [2] S. Ye, R. Bum, and L. Cimini, Adaptive moduation for variabe rate OFDM systems with imperfect channe information, in Proc. Vehicuar Techno. Conference, vo. 2, Birmingham, AL, 2. [3] M. R. Sourya and R. L. Pickhotz, Adaptive moduation with imperfect channe information in OFDM, in Proc. IEEE ICC, 1. [4] T. Ekman, G. Kubin, M. Sternad, and A. Ahén, Quadratic and inear fiters for mobie radio channe prediction, in Proc. IEEE Vehicuar Technoogy Conference, VTCFa1999, Amsterdam, The Netherands, [5] T. Ekman and G. Kubin, Noninear prediction of mobie radio channes: measurements and MARS mode designs, in Proc. IEEE Internationa Conference on Acoustics, Speech and Signa Processing, Phoenix, AR, [6] J. Hwang and J. Winters, Sinusoida modeing and prediction of fast fading processes, in Proc. IEEE GLOBECOM, [7] L. Dong, G. Xu, and H. Ling, Prediction of fast fading mobie radio channes in wideband communication systems, in Proc. IEEE GLOBECOM 1, San Antonio, TX, 1. [8] Z. Shen, J. G. Andrews, and B. L. Evans, Short range wireess channe prediction using oca information, in Proc. IEEE Asiomar Conf. on Signas, Systems, and Computers, Pacific Grove, CA, USA, 3. [9] D. Schafhuber, G. Matz, and F. Hawatsch, Predictive equaization of time-varying channes for coded OFDM/BFDM systems, in Proc. IEEE GLOBECOM, San Francisco, November, pp [1] D. Schafhuber, G. Matz, and F. Hawatsch, Adaptive prediction of time-varying channes for coded OFDM systems, in Proc. ICASSP 2, Orando, FL, USA, May 2. 8 Internationa ITG Workshop on Smart Antennas (WSA 8) 151

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