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A Low complexity LMMSE channel estimation method in

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A Low complexity LMMSE channel estimation method in

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    A Low complexity LMMSE channel estimation method in

    LTE downlink

    Wang Subin, Peng Tao

    5 (School of Information & Communication, Beijing University of Posts & Telecommunications,

    Beijing 100876)

    Abstract: Channel estimation is critical to receiver performance in the long term evolution (LTE)

    system. Reference signals are interspersed with data signals among the subcarriers to aid channel

    estimation. Linear minimum mean square error (LMMSE) channel estimation is an efficient approach 10 to estimate the knowledge of channel with high complexity. This paper proposed a channel estimation

    approach based on singular value decomposition (SVD) for downlink channel in LTE system.

    Compared to the conventional algorithm, the proposed approach investigates the channel

    autocorrelation matrix respectively and substitutes it by small submatrices to reduce the complexity.

    The simulation result demonstrates that the complexity is significantly reduced with negligible 15 performance degradation, so it could be a candidate of the channel estimation algorithm for downlink

    channel estimation in LTE system.

    Keywords: Communication; LTE; Channel Estimation; LS; LMMSE; SVD

    0 Introduction

    20 Channel Estimation is a critical part of modern wireless communication of receiver

    performance whose quality has a direct impact on the whole system performance, especially in

    Long Term Evolution (LTE) system adopts Orthogonal Frequency Division Multiplexing (OFDM)

    and Multiple Input Multiple Output (MIMO). For certain pilot symbol structure, how to keep the

    optimum balance between the complexity and performance of channel estimation is still well 25 studied. Traditional channel estimation techniques for OFDM systems in terms of the criteria on

    [1]which they are based can be categorized as follows: (1) Least Squares (LS). (2) Linear

    [2]Minimum Mean Square Error (LMMSE). LS estimator has low complexity, but suffers from a

    high mean square error (MSE). On the other hand, LMMSE estimator has high complexity, but

     LMMSE estimator for pilot-symbol-aided system requires channel gives good performance.

    30 statistics information such as channel frequency correlation and the matrix inversion, which are

    usually unknown or difficult to get in practical scenarios. In order to reduce the complexity of this

    method, a new approach computes an approximated LMMSE estimation using optimal rank

    [3] reduction by singular value decomposition (SVD)has been proposed, it partitions system into

    subsystems to reduce high computational complexity of the LMMSE estimation. But there is no 35 thorough investigation on the performance of partitioning in LTE system.

    This paper discusses method to reduce the complexity of LMMSE algorithm of channel

    estimation by improving channel autocorrelation matrix calculation and SVD method. Its

    performance is similar with the traditional LMMSE method. However, the complexity of channel

    estimation will greatly be reduced. The method mainly includes two steps to be implemented. 40 Firstly, it gains the channel autocorrelation matrix through channel model instead of pilot

    autocorrelation matrix. Secondly the improved SVD method will be used to implement channel

    estimation in sub-carriers with pilot. At last, every sub-carrier of channel estimation will be gained

     channel estimation value after frequency interpolation.

    Brief author introduction:Wang Subin, (1988- ),Female,Postgraduate,Wireless communication.

    Correspondance author: Peng Tao, (1977-), Male, Vise professor, Wireless communication. E-mail:

    pengtao@bupt.edu.cn

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     1 LTE downlink description

45 1.1 LTE downlink frame structure

    A general description of physical layer of the LTE system is provided in [4]. The

    time-frequency resources are subdivided as illustrated in figure 1. The largest unit of time is the

    10ms radio frame, which is further subdivided into ten 1 ms subframes, each of which is split into

    two 0.5 ms slots. For normal cyclic prefix (CP), each slot contains seven OFDM symbols and in

     50 case of extended CP, six OFDM symbols are slotted in each slot.

    One radio frame, T= 307200T= 10 ms f s

    One slot, T= 15360T= 0.5 ms slot s

    #0 #1 #2 #3 #18#19

     One subframe

     Fig. 1 Frame Structure of LTE

    55 In this paper, our research is based on Physical Downlink Shared Channel (PDSCH), which

    is the main data-bearing downlink channel in LTE system. It is used for all user data, as well as for

    broadcast system information which is not carried on the Physical Broadcast Channel (PBCH),

    and for paging messages.

    1.2 LTE downlink pilot structure

    60 The cell-specific reference signal (RS) used in channel estimation are QPSK modulated (a

    constant modules modulation). This property ensures that the Peak-to-Average Power Ratio of the

     transmit waveform is kept low. These signals are defined by 1 1 1 -j 1 -r (m ) =(2 ?c(2m ) (2 ?c(2m1) (1)))l,ns 2 2

    m l Where is the index of the cell-specific RS, is the OFDM symbol number within the nC (×) s is the pseudo-random sequence with is the slot number within a radio frame,65 slot, initialization defined in (2). 10 cell cell (2) c= 2? ( 7 ? ( n+ 1) + l + 1) ? 2 ? N + 1+ 2 ? N + N ( ) init s CPID ID max,DL N RB is the largest downlink bandwidth configuration. For time direction, cell-specific RS

    are mapped in the first and the third last elements of resource block (RB), whereas cell-specific RS

     are inserted over every six subcarriers in frequency domain. The calculations of the next sections 70

    are based on each OFDM symbol containing reference signal.:

    2 Channel estimation approach design

    In this section, we present the LMMSE estimate of the channel frequency response H at the

    RS position from the received RS Y and the transmitted RS X . The complexity reduction of the

    75 LMMSE estimation consists of three separate steps. In the first step, we modify the LMMSE by

    using the characteristic of RS, obtaining a simplified estimation; in the second step, we improve

    [5] the channel autocorrelation matrixcalculation with given channel model, which can be

    implemented in advance since it is independent of pilot position channel estimation matrix; in the

    [6] third step, we partition the channel frequency autocorrelation matrix into several submatricesin

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     80 order to efficiently realize in practice; in the fourth step. 2.1 LS method

    H LS method is the simplest way in channel estimation, its channel transfer function can be k

    Y X gained by the receiver RS devided the transmitted RS symbol . The relationship between kk

    X Y known transmitted RS symbol and receivedis k k

    H X + Z (3) Y =k k k k 85

    H Z Where is the channel transfer function at the k-th subcarrier, andis the noise. For k k

    Q known pilot symbol of OFDM with known carriers, the observations can be expressed in as

    follows:

     0 0 0 XH"Z? ?? ? ?? 11 1 Y ? ? 1 ? ? ? ? ? ?0 H 0 X Z" 222 ? ? ? ? ? ? ? ?Y 2 ? ? ? 0 0 % 0 ? ? #? ? # ?"= + (4) ? # ? ?? ? ? ? ?# # % # X Z ? Q ?1 ?1 ?Q ? ? ? ? ? ?Y Q ? ? " 0 0 H X Z0Q Q Q

    = HX + Z Y 90

    Therefore for an OFDM symbol of LTE with known pilot symbols, the LS method of channel

     estimation can be estimated for a simple division as shown equation (5). Y YY YQ - 1Q 1 2 H = [" ] (5)LSX X X X 1 2Q - 1 Q

    2.2 LMMSE method

    95 The LMMSE estimation tries to minimize the mean square error between the actual and

    estimated value. The estimated channel response at the RS position can be written as 2 H - 1 + s (X X ) )HH = R (R(6)LMMS E HH HH LS

    H 2LS s Where is the channel response at the RS position gained by LS estimation method, RHH is the variance of the additive channel noise.is the channel autocorrelation matrix of RS. H 100 R= E[ HH ] (7) HH IH Q XX due to theis constant and equal to the identity matrix In LTE downlink, the term

     generation of cell-specific RS sequence according to (1). We have 2 ?1 (8) H = R(R+ σ I)H LMMSE HH HH Q LS

     We can see that LMMSE channel estimation requires knowledge of the channel 105 2Rwhich can be expressed by Signal-to-Noice Ratio (SNR). In correlation and the operating HH s practice, the size the channel correlation is too large to further compute the result according to (8). Thus, to overcome high complexity of LMMSE, new autocorrelation matrix calculation method

     and low rank approximation was proposed as follows. 2.3 Improved channel autocorrelation matrix calculation method [7] Regarding multipath fading channel, its channel impulse response (CIR)can express as 110

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     M ?1 g (τ ) = αδ (τ ? τ T) (9)? iis i =0 Where M is the CIR length,ais the Gaussian random variable and mutual independent, i ?(τ /τ ) k rms with q(t ) θ (τ ) = Cet as its power delay spectrum, and.subjects to uniform i i k

     distribution as equation (10) shows

     1/ Lτ ? [0, L]k 115f (τ ) = { (10)k τ k else0

    Where L is the largest length of channel impulse response, t is the average delay of each rm s

     π k 2M ?1 ? j τi N H= αechannel. The k-th subcarrierof CIR can express as , so the channel k ? i =0

    i

     autocorrelation matrix is H R= E{HH } = [r] (11) HHm,n

     π mM ?12π n2M ?1 ? j τ τ ik* 120NN r= E{H H } = E{ αe α e } m,n m n ? i ? k ? j

    ? L((1/τ )+ 2π j (m ? n) / N )rms 1 ? ei =0 k = 0 = ? L /τ rms τ (1 ?e )(1 / τ + 2π j(m ? n) / N )rms rms t rm s L N In practice, always equals CP length,equals a quarter of the CP length and can

    change its value according to the channel condition.

    2.4 Improved SVD approach

     Adopting SVD, the channel autocorrelation matrix in (8) can be expressed as H R= U L U (12) HH 125

    U L N Whereis a unitary matrix, and is a diagonal matrix containing singular values

    l (0) l (1) l (M "1) 0 . By substituting (12) into (8) and using unitary matrix property,

     the SVD LMMSE estimation can be written as H H 2H ?1 (13) H = U ΛU (U ΛU + σ UU )H LMMSE LS H = Udiag (?,", 0)U H LS Where130 l (1) l (0) l (M - 1) D = diag(,, ,)"222 l (0) + s l (1) + s l (M -1) + s To further lower the complexity of this estimation due to the large bandwidth available in

     LTE downlink, we proceed with reduced complexity estimation based on above-mentioned working. We subdivide the channel vector into subvectors depending on the actual bandwidth T T T T 135 H = [H , H ,", H ](14) 1 2 G G = Q / D . In Suppose we denote the size of each subvector by D, thus in (14), we have g = 1, ,G T H addition, . H = [H ,", H ]is the gth subvector of , forg D( g ?1) +1D( g ?1)+ D

    From (8), we have

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    RR " R ?? H, H H,HH,H1 21 11 G ? ?R RR " H ,HH , H H ,H 2 12 2 G ??R = (15) 2HH ?# # % # ? ? ?RRR" H ,HH ,H H ,H ? ?G 1G 2G G

    140 For the channel autocorrelation property, highly correlated elements have a great impact on

    [8]the performance of channel estimation, while the effect by lowly correlated elements is slight.

    Since the relatively low correlated channel correlation components outside the coherence

    bandwidth have little effect on the performance of channel estimation, we can set these channel RHH as a banded matrix coefficients of correlation matrix to zero. Hence, we can approximate

    which can be expressed in terms of 145

    R? diag(R, R, , R) (16) "HH H ,H H ,H H ,H 1 1 2 2 G G

    From (8), (13) and (16), the approximate LMMSE estimation becomes H H ? ?Udiag (?,, 0)UH"? ?1, LMMSE 11 11, LS ? ?? ?H ^ HU diag ? U H "LMMSE 2, 2 2, LS 2? ( , , 0) ? ? ? 2 = = (17)

    H H LMMSE H diag (?,, 0)U H U "? ?G, LMMSE ? ? G G G G, LS ???? ? # ? # ? ?? ? Where H 150 (18) H = U diag (? ,, 0)U H "g ,LMMSE g g g

     g ,LS

     We have 1) / D ] + 1, 1 q Q g = floor[(q - And T YY? ?D( g ?1)+ L D( g ?1) +1 H = ,",g ,LS ??

    X X D( g ?1)+1D( g ?1)+ L ? ?

    155 Is the gth LS estimation.

     Thus, the LMMSE channel estimation vector using the approximate banded approach is T T T H = [H , H , , H ] " 1,MMS E 2,MMSE G ,MMS E LMMSE ,ban ded 3 Simulation results To demonstrate the performance of the proposed channel estimator based on improved

    channel autocorrelation matrix calculation method and banded SVD approach, computer 160

    simulations are evaluated for the different channel estimation algorithm. The simulation

    parameters of LTE system are listed in table I.

    Tab. 1 System Parameters

    Bandwidth20MHz

    Frame structureTDLTE

    Subcarrier spacing15KHz

    TX antennas 2

    RX antennas2

    PRB number100

    CP typeNormal CP

    FFT Size2048

    Channel modeEVA 70

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Simulation iterations10000

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     165 Fig. 2 MSE Performance comparison among channel estimation methods

     Figure 2 shows the comparison of the MSE versus the SNR for three channel estimation methods. From the figure, the improved channel autocorrelation matrix calculation LMMSE

    algorithm has a remarkable improvement in compare with the other two methods. 170

     175 Fig. 4 MSE Performance among all kinds of channel Fig. 3 MSE Performance among different values of estimation approach mentioned in this paper D for banded SVD LMMSE algorithm Figure 3 depicts the MSE performance for banded SVD LMMSE estimation with different

    value of D, i.e. D takes 10,20,25 and 40 to determinate the fine vale. This figure demonstrates that 180

     the size of channel autocorrelation matrix should be determined by bandwidth available to ensure that the number of submatrices is almost 10.

     Figure 4shows the comparison of MSE versus SNR for different channel estimation method

     mentioned in this paper. It can be seen that improved LMMSE and SVD LMMSE have the best 185 performance, banded LMMSE goes between improved LMMSE and LMMSE, LS estimation has

    the worst performance among all the channel estimation methods. We compare the complexity of

     the five algorithms in Figure 4 in terms of multiplication operation since its difficulty in 3 2 Q realization. Thus, the complexity of LS estimation is , traditional LMMSE , Q + 2Q 2 3 2 improved LMMSE and banded SVD LMMSE (Q + 2)M + QM , SVD Q + Q

    LMMSE 190 2 respectively. It is clear that traditional LMMSE is the most G [(D + 2)(M / G )+ DM / G ] complex method. Hence, take our simulation as example, the complexity number for banded SVD

     approach is approximately 1/150 of multiplications for SVD LMMSE. After comparison of both complexity and performance of the proposed channel estimation techniques with those of the full

     SVD LMMSE channel estimations and its other optimal versions, the proposed estimation has 195 performance degradation as compared with full LMMSE estimation, its complexity is significantly

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     reduced.

     4 Conclusion This paper aims at proposing a LTE downlink channel estimation approach which is suitable for realization on hardware. The key idea in developing this approach is making use of channel

    model to get channel autocorrelation matrix and partitioning it in full LMMSE with SVD into 200

     small submatrices by banded techniques. The proposed estimations used small submatrices with the suitable size determined by actual bandwidth to reduce the complexity of full LMMSE estimation with SVD. We compared both complexity and performance of the proposed channel estimation techniques with those of the full SVD LMMSE channel estimation and its other

    optimal versions. Though the proposed estimation has performance degradation as compared with 205

     full LMMSE estimation, its complexity is significantly reduced.

     References [1] E.Dahlman, H.Ekstron, A.Furuskar, Y.Jading, J.Karlsson, M.Lundevall, and S.Parkvall, The 3G Long-Term 210 l.1, pp. 137-141, Evolution-Radio Interface Concepts and Performance Evaluation. IEEE VTC 2006 Spring, vo

     2006. [2] L Bolliger, HA Loeliger, C Vogel, "Simulation, MMSE Estimation, and Interpolation of sampled Continuous-time signals using factor graphs" Information theory & Application workshop(ITA), UCSD, La Talla, Feb.2010.

    215 [3] Y. Li, Simplified channel estimation for OFDM systems with multiple transmit antennas. Wireless

     Communications, IEEE Transactions on, vol.1, no. 1, pp. 67-75, 2002. [4] 3GPP TS 36.213: Evolved Universal Terrestrial Radio Access(E-UTRA), Physical channels and modulation. Release 8, (2009-09). [5] Over Edfor et al., Analysis of DFT-based Channel Estimators for OFDM. Wireless Personal Communication,

    Page(s):55-70, Kluwer Academic Pub220 lishers, 2000.

     [6] Y. Li, L. J. Cimini, and N.R.Sollenberger, "Robust channel estimation for OFDM systems with rapid dispersive fading channels", IEEE Transactions on Commun, vol. 46, no. 7, pp. 902-915, July 1998. [7] O. Edfors, M. Sandell, J.van de Beek, S. Kate Wilson, and P. OlaBorjesson, "OFDM channel estimation by singular value decomposition". IEEE Transactions on Commun, vol.46, no. 7, pp. 931-939, July 1998. 225[8] M. Noh, Y. Lee, and H. park, Low complexity LMMSE channel estimation for OFDM, IEEE Process.

    Commun, vol. 153, no. 5, October 2006.

     一种基于 SVD 分解的 LMMSE 信道估计算法 230

     王素斌,彭涛 ?北京邮电大学信息与通信工程学院,北京 100876 摘要!信道估计是 LTE 系统接收端至关重要的模块。本文针对采用梳状导频的下行物理信

     道,提出了一种基于奇异值分解的 LMMSE 算法,与传统的 LMMSE 算法相比,该算法有 两个方面的改进!一是改进了信道自相关矩阵的计算方法,直接根据信道模型得到信道自相 235 关矩阵并预先存储,在硬件实现起来更方便,二是改进了 SVD 分解的算法,将信道自相关 矩阵分解为若干子矩阵,分别进行计算,降低了算法的复杂度。仿真结果表明,该算法在极 大减少算法复杂度的情况下,可以保证优良的信道估计性能,因此可以作为 LTE 下行信道

     估计算法的较佳备选方案。 关键词!通信技术,LTE,信道估计,LSLMMSESVD240 中图分类号!TN929.53

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