Optimal Pilot Symbol Power Allocation in LTE

Transcription

1 Otimal Pilot Symbol Power Allocation in LTE Michal Šimko, Stefan Pendl, Stefan Schwarz, Qi Wang, Jose Colom Ikuno and Markus Ru Institute of Telecommunications, Vienna University of Technology Gusshausstrasse 5/389, A-040 Vienna, Austria Web: htt:// Abstract The UMTS Long Term Evolution LTE allows the ilot symbol ower to be adjusted with resect to that of the data symbols. Such ower increase at the ilot symbols results in a more accurate channel estimate, but in turn reduces the amount of ower available for the data transmission. In this aer, we derive otimal ilot symbol ower allocation based on maximization of the ost-equalization Signal to Interference and Noise Ratio SINR under imerfect channel knowledge. Simulation validates our analytical mode for otimal ilot symbol ower allocation. Index Terms LTE, Channel Estimation, OFDM, MIMO. I. INTRODUCTION Current systems for cellular wireless communication are designed for coherent detection. Therefore, channel estimator is a crucial art of a receiver. UMTS Long Term Evolution LTE rovides a ossibility to change the ower radiated at the ilot subcarriers relative to the that at data subcarriers. Clearly, this additional degree of freedom in the system design rovides otential for otimization. A. Related Work In order to otimize the ilot symbol ower allocation a model that takes into account the ilot ower adjusting, receiver structure and channel estimation error at the same time, is needed. It has been shown by simulation that ilot symbol ower allocation has a strong imact on caacity []. Authors of [] show by simulation the imact of different ower allocations on the system s Bit Error Rate BER. However, their analysis is based on Signal to Noise Ratio SNR so that they only aroximate the imact of imerfect channel knowledge on BER for Binary Phase-shift Keying BPSK modulation. In [3], otimal ilot symbol allocation is derived analytically for Phase-shift Keying PSK modulation of order two and four, using BER as the otimization criterion. In [4] otimal ilot symbol ower in Multile Inut Multile Outut MIMO system is derived based on lower bound for caacity. Authors of [5] investigate ower allocation between ilot and data symbols for MIMO systems using ostequalization Signal to Interference and Noise Ratio SINR as the otimization function. However, they only aroximate the SINR exression and their model is tightly connected with a Linear Minimum Mean Square Error LMMSE channel estimator. B. Contribution In this aer, we derive analytical exressions for otimal ower allocation for a MIMO system with Zero Forcing ZF equalizer under imerfect channel state information. We utilize the ost-equalization SINR, as the otimization function, which is analogous to the throughut maximization in a real system. The main contributions of the aer are: By maximizing the ost-equalization SINR, we deliver otimal values for the ilot symbol ower adjustment in MIMO Orthogonal Frequency Division Multilexing OFDM systems. The ost-equalization SINR exression is derived for a ZF receiver under imerfect channel knowledge. We analytically derive the Mean Square Error MSE exression of the Least Squares LS channel estimator utilizing linear interolation. Simulation results with an LTE comliant simulator [6, 7] confirm otimal values for ilot symbol ower. As with our revious work, all data, tools, as well imlementations needed to reroduce the results of this aer can be downloaded from our homeage [8]. The remainder of the aer is organized as follows. In Section II we describe the mathematical system model. In Section III, we derive the ost-equalization SINR exression for ZF with imerfect channel knowledge. The channel estimators of this work are briefly discussed in Section IV and their MSEs are derived. We formulate the otimization roblem for otimal ilot symbol ower allocation in Section V. Finally, we resent LTE simulation results in Section VI and conclude our aer in Section VII. II. SYSTEM MODEL In this section, we briefly oint out the key asects of LTE relevant for this aer, as well as an system model. In the time domain the LTE signal consists of frames with a duration of 0 ms. Each frame is slit into ten equally long subframes and each subframe into two equally long slots with a duration of 0.5ms. Deending on the cyclic refix length, being either extended or normal, each slot consists of N s =6 or N s =7OFDM symbols, resectively. In LTE, the subcarrier sacing is fixed to 5 khz. Twelve adjacent subcarriers of one slot are groued into a so-called resource block. The number of resource blocks in an LTE slot ranges from 6 u to 00, corresonding to a bandwidth from.4 MHz u to 0 MHz.

2 A received OFDM symbol in frequency domain can be written as y nr = X nt h nt,n r + n nr, where the vector h nt,n r contains the channel coefficients in the frequency domain between the n t -th transmiter and n r -th receiver antennas and n nr is additive white zero mean Gaussian noise with variance σn at the n r -th receiver antenna. The diagonal matrix X nt = diag x nt comrises the recoded data symbols x d,nt and the ilot symbols x,nt at the n t -th transmit antenna laced by a suitable ermutation matrix P on the main diagonal x nt = P [ x T,n t x T d,n t ] T. The length of the vector x nt is N sub corresonding to the number of subcarriers. Let us denote by N and N d, the number of ilot symbols and the number of recoded data symbols, resectively. The recoded data symbols x d,nt are obtained from the data symbols via recoding [x d,,k x d,nt,k] T = W k [s,k s,k s Nl,k] T, 3 where x d,nt,k is a recoded data symbol at the n t -th transmit antenna ort and the k-th subcarrier, W k is the recoding matrix at the k-th subcarrier and s nl,k is the data symbol of the n l -th layer at the k-th subcarrier. Note that according to Equation, also the vectors y nr, h nt,n r and w nr can be divided into two arts corresonding to the ilot symbol ositions and to the data symbol ositions. For the derivation of the ost-equalization SINR, we will use the inut-outut relation at the subcarrier level, given as: y k = H k W k s k + n k. 4 Matrix H k is the MIMO channel matrix of size N r N t. Furthermore, matrix W k is a unitary recoding matrix of size N t N l. In LTE the recoding matrix can be chosen from a finite set of recoding matrices [9]. The vector s k comrises the data symbols of all layers at the k-th subcarrier. We denote the effective channel matrix that includes the effect of the channel and recoding by G k G k = H k W k. 5 Furthermore, let us denote the average data ower transmitted at one layer by σ s, the total data ower by σ x, and the average ilot symbol ower by σ σs = E { s l,k =, 6 N l σ x = σ = E { x d,nt =, 7 E { x,nt =. 8 III. POST-EQUALIZATION SINR In this section, we derive an analytical exression for the ost-equalization SINR of a MIMO system under imerfect channel knowledge and ZF equalizer given by the system model in Equation 4. In the following section, we omit the subcarrier index k, since the concet we resent is indeendent of it. If the equalizer has erfect channel knowledge available, the ZF estimate of the data symbol s is given as ŝ = G H G G H y. 9 The data estimate ŝ given by Equation 9 results in the ostequalization SINR of the l-th layer given as [0] σs γ l = σne H l G H G, 0 e l where the vector e l is an N l zero vector with the l-th element being. This vector serves to extract the corresonding layer ower after the equalizer. Let us roceed to the case of imerfect channel knowledge. We define the erfect channel as the channel estimate lus the error matrix due to the imerfect channel estimation H = Ĥ + E, where the elements of the matrix E are indeendent of each other with variance σe. Inserting Equation in Equation 4, the inut outut relation changes to y = Ĥ + E Ws + n. Since the channel estimation error matrix E is unknown at the receiver, the ZF solution is given again by Equation 9, but channel matrix H is relaced by its estimate Ĥ, which is known at the receiver ŝ = ĜH Ĝ ĜH y, 3 with matrix Ĝ being equal to ĤW. The error after the equalization rocess using ZF is given as ŝ s = ĜH Ĝ ĜH EWs + n. 4 From Equation 4 we can comute the MSE matrix { MSE = E ŝ sŝ s H 5 = σ n + σ xσ e Ĝ H Ĝ, with σ e being the MSE of the channel estimator. Equation 5 directly leads to the SINR of the individual layers σ s γ l =. 6 σn + σeσ x e H l ĜH Ĝ el Note, that in ractice variables σ s,σ x and σ n are relaced by their estimates.

3 Channel gain Interolation By lugging 8 into 0, exanding the square and using the identities E { { Δh ni i h i = E h i =0, x i c = c, 3 e m i n Subcarrier index Fig.. Linear inter- and extraolation of the channel on ilot subcarriers to intermediate subcarriers. IV. CHANNEL ESTIMATION In this section, we resent state-of-the-art channel estimators and derive analytical exressions for their MSE. Due to the orthogonal ilot symbol attern utilized in LTE, the MIMO channel can be estimated as N t N r individual Single Inut Single Outut SISO channels. Therefore, in the following section, we omit the antenna indices. A. LS Channel Estimation The LS channel estimator [, ] for the ilot symbol ositions is given as the solution to the minimization roblem: ĥ LS y = arg min X ĥ = X y 7 ĥ The remaining channel coefficients at the data subcarriers have to be obtained by means of interolation. In this work, we utilize linear interolation of the ilot subcarriers on each antenna to obtain the channel vector h containing the channel elements of all subcarriers in one OFDM symbol. In order to derive the theoretical MSE, we distinguish three cases, as shown in Figure : MSE at a ilot subcarrier i : σ e MSE at an interolated subcarrier i: σ e i MSE at an extraolated subcarrier e: σ e e At a ilot osition i the estimated channel element equals: ĥ i = y i x i = h i + n i x i {{ Δh i 8 The MSE of the channel estimator is given by the noise variance σe = σn. To obtain the variance at an interolated osition consider the following equation to comute the interolated channel element ĥi from the two neighboring ilot ositions ĥ m and ĥ n : ĥ i = ĥ m + i m ĥ n n ĥ m 9 {{ m c Then, the MSE can be comuted as: { hi σi = E ĥi 0 { hi = E ĥm + c ĥ n ĥ m we obtain for the LS MSE at an interolated osition: σe i = R i,i + c R n,n + σn+c R m,m + σn +c c R R m,n c R R i,n c R R m,i 4 Here, R i,j denotes the element in the i-th row and j-th column of matrix R = E { hh H. For an extraolated osition, we utilize the two consecutive neighboring ilot ositions, and, to obtain the estimated channel element according to: ĥ e = ĥ + e ĥ ĥ 5 {{ c Then, the same Formula 4, with aroriately modified indices, alies for the extraolated MSE σe e as well. The MSE at an extraolated osition is generally larger than at an interolated osition, because the cross correlation between the channels at osition e and is smaller than those between the channels at osition i and n. The total MSE of LS channel estimator is given as mean over all subcarriers. Due to the dense ilot symbol attern and thus strong correlation over frequency, the total MSE can be exressed as σ e = c e σ n, 6 where c e is a constant. Equation 6 follows from Equation 3 and Equation 4. B. LMMSE Channel Estimation The LMMSE channel estimator requires the second order statistics of the channel and the noise. It can be shown that the LMMSE channel estimate is obtained by multilying the LS estimate with a filtering matrix A LMMSE [3 5]: ĥ LMMSE = A LMMSE ĥ LS 7 In order to find the LMMSE filtering matrix, the MSE { h σe = E ALMMSEĥLS, 8 has to be minimized, leading to A LMMSE = R Rh,h + σ ni, 9 where the matrix R h,h = E { h h H is the channel autocorrelation matrix at the ilot symbol ositions, and the matrix R = E { hh H is the channel crosscorrelation matrix.

4 To derive the theoretical MSE, we lug Equation 9 into Equation 8: { σe =E h R R h,h σni ĥ LS 30 H h R R h,h σni ĥ LS After a straightforward maniulation, the average MSE at each subcarrier is exressed as σe = { tr R R Rh,h N + σni Rh,h. sub V. POWER ALLOCATION In this section, we describe the roblem of otimal ilot ower allocation in LTE based on the maximization of the ost-equalization SINR under imerfect channel knowledge. Although the shown results are from the alication to LTE, the resented concet can be alied to any MIMO OFDM system. If we increase the ower at the ilot symbol by a factor c, the MSE of the channel estimator imroves by the factor c σ e = σ e c. 3 However, the ower of the data symbols has to be decreased by a factor c d in order to kee the total transmit ower constant. Obviously, the two factors c and c d are connected. For this urose, we define a variable off, exressing the ower offset between the mean energy of the ilot symbols and the data symbols, and refer to it as ilot offset. The variables c, c d and off are interconnected as follows: c = N + N d 3 off N d + N c d = N + N d = off c 33 N d + N off Plugging in the variables c d and c in Equation 6, we obtain the SINR exression with adjusted ower of the ilot symbols σ γ l = sc d. 34 σn + σ e c σxc d e Hl G H G e l If we insert Equation 3 and Equation 33 into Equation 34 and simlify the exression, we obtain: γ l =, 35 N l σn eh l GH G e l f N d +N off for which the function f off is given as f off = off N d + N off + c e. 36 We refer to it as ower allocation function. Note, that it is indeendent of channel realization and noise ower. Therefore, it is ossible to find an otimum value for ilot symbol ower allocation indeendent of the SNR and channel realization. fo.8 x LMMSE channel estimator LS channel estimator 4x4 x x off [db] Fig.. Power allocation function f off for different numbers of transmit antennas and LS solid line and LMMSE dashed line channel estimators TABLE I VALUES OF THE PARAMETERS OF f off FOR DIFFERENT NUMBER OF TRANSMIT ANTENNAS FOR.4 MHZ BANDWIDTH, ITUPEDA [7] CHANNEL MODEL, LSAND LMMSE CHANNEL ESTIMATORS Parameter Tx= Tx= Tx=4 N d N LS c e off,ot [db] LMMSE c e off,ot [db] The target is to find an otimal value of off,ot that maximizes the ost-equalization SINR while keeing the overall transmit ower constant maximize off γ l 37 subject to N d σx + N σ = const In order to maximize the ost-equalization SINR, the ower allocation function f off in the denominator of Equation 35 has to be minimized. The minimum of the ower allocation function f off can be found by simly differentiating it and solving for 0. By these means, the otimal value of the variable off is given as solution to the following exression: off N d + N 3 off + offn d + N N d off + c e =0. 38 An analytical solution for Equation 38 can be obtained by means of Ferrari s solution [6]. Figure deicts f off for different numbers of transmit antennas for LTE. Tyical values of arameters N d, N and c e are given in Table I. Note, that although N d and N deend on the utilized bandwidth, the minimum of f off is indeendent of it, since N d and N scale with the same constant with increasing bandwidth. The value of c e is different for four transmit antennas due to the lower number of ilot symbols at the third and fourth antenna. The last row of Table I gives the otimal values of off,ot for different numbers of transmit antennas and an ITU PedA [7] tye channel model. While utilizing the LMMSE channel estimator, one obtains negative values for off,ot. This means, that otimally, the ilot symbol ower has to be reduced in order to maximize the ost-equalization SINR.

5 TABLE II SIMULATOR SETTINGS FOR FAST FADING SIMULATIONS Parameter Value Bandwidth.4 MHz Number of transmit antennas,, 4 Number of receive antennas,, 4 Receiver tye ZF throughut [Mbit/s] Transmission mode Oen-loo satial multilexing Channel tye ITU PedA [7] MCS 9 coding rate 66/ symbol alhabet 6 QAM x 4x4 x LS channel estimator LMMSE channel estimator off,ot off [db] Fig. 3. Throughut of LTE downlink over ilot symbol ower Such result is intuitively clear, with imroving quality of the channel estimator, it is sufficient to use less ower on the ilot symbols to obtain the same quality of the channel estimate. VI. SIMULATION RESULTS In this section, we resent simulation results and discuss the erformance of LTE system using different ilot symbol owers. All results are obtained with the LTE Link Level Simulator version r8 [6], which can be downloaded from All data, tools and scrits are available online in order to allow other researchers to reroduce the results shown in the aer [8]. Table II resents the most imortant simulator settings. Since what is to be observed is the otimal value of ilot symbol ower adjustment, an SNR of 4 db has been chosen, at which the system is not saturated for this secific Modulation and Coding Scheme MCS. Simulation results showing throughut erformance for,, and 4 4 antenna configurations are shown in Figure 3 for LS and LMMSE channel estimators. The maximum throughut values corresond excellently with off,ot shown in Table I, thus the simulation results match with the analytical results. Note, that although we show throughut results for secific SNR value, the otimal oint is indeendent of it. VII. CONCLUSION In this aer, we have analytically derived the otimal ilot symbol ower adjustment for MIMO OFDM systems. As otimization criterion we utilized the ost-equalization SINR under imerfect channel state information, that is connected to the system throughout. Furthermore, we derive analytical exression for the ost-equalization SINR under imerfect channel knowledge using ZF and we rovide analytical exression for the MSE of the LS channel estimator utilizing linear interolation. Throughut simulation results validate the accuracy of our analytical model for the otimum ilot ower adjustment. All data, tools and scrits are available online in order to allow other researchers to reroduce the results shown in the aer [8]. ACKNOWLEDGMENTS The authors would like to thank the LTE research grou for continuous suort and lively discussions. This work has been funded by the Christian Doler Laboratory for Wireless Technologies for Sustainable Mobility, KATHREIN- Werke KG, and A Telekom Austria AG. The financial suort by the Federal Ministry of Economy, Family and Youth and the National Foundation for Research, Technology and Develoment is gratefully acknowledged. REFERENCES [] C. Novak and G. Matz, Low-comlexity MIMO-BICM receivers with imerfect channel state information: Caacity-based erformance comarison, in Proc. of SPAWC 00, Morocco, June 00. [] E. Alsusa, M. W. Baidas, and Yeonwoo Lee, On the Imact of Efficient Power Allocation in Pilot Based Channel Estimation Techniques for Multicarrier Systems, in Proc. of IEEE PIMRC 005, Set. 005, vol., [3] J. Chen, Y. Tang, and S. Li, Pilot ower allocation for OFDM systems, in Proc. of IEEE VTC Sring, Ar. 003, vol., vol.. [4] B. Hassibi and B.M. Hochwald, How much training is needed in multile-antenna wireless links?, IEEE Transactions on Information Theory, vol. 49, no. 4, , Ar [5] Jun Wang, Oliver Yu Wen, Hongyang Chen, and Shaoqian Li, Power Allocation between Pilot and Data Symbols for MIMO Systems with MMSE Detection under MMSE Channel Estimation, EURASIP Journal on Wireless Communications and Networking, Jan. 0. [6] C. Mehlführer, M. Wrulich, J. Colom Ikuno, D. Bosanska, and M. Ru, Simulating the Long Term Evolution Physical Layer, in Proc. of EUSIPCO 009, Glasgow, Scotland, Aug [7] C. Mehlführer, J. Colom Ikuno, M. Šimko, S. Schwarz, M. Wrulich, and M. Ru, The Vienna LTE Simulators - Enabling Reroducibility in Wireless Communications Research, EURASIP Journal on Advances in Signal Processing, 0, acceted. [8] LTE simulator homeage, [online] htt:// ltesimulator/. [9] 3GPP, Evolved Universal Terrestrial Radio Access E-UTRA; Physical channels and modulation, TS 36., 3rd Generation Partnershi Project 3GPP, Set [0] Hedayat A., Nosratinia A., and Al-Dhahir N., Linear Equalizers for Flat Rayleigh MIMO Channels, in Proc. of IEEE ICASSP 005, Mar. 005, vol. 3,. iii/445 iii/448 Vol. 3. [] J. J. van de Beek, O. Edfors, M. Sandell, S. K. Wilson, and P. O. Borjesson, On Channel Estimation in OFDM Systems, in Proc. of IEEE VTC 995, 995, vol., [] M. Šimko, C. Mehlführer, T. Zemen, and M. Ru, Inter carrier interference estimation in MIMO OFDM systems with arbitrary ilot structure, in Proc. 73rd IEEE Vehicular Technology Conference VTC0-Sring, Budaest, Hungary, May 0. [3] S. Omar, A. Ancora, and D.T.M. Slock, Performance Analysis of General Pilot-Aided Linear Channel Estimation in LTE OFDMA Systems with Alication to Simlified MMSE Schemes, in Proc. of IEEE PIMRC 008, Set. 008,. 6. [4] M. Šimko, C. Mehlführer, M. Wrulich, and M. Ru, Doubly Disersive Channel Estimation with Scalable Comlexity, in Proc. of WSA 00, Bremen, Germany, Feb. 00. [5] M. Šimko, D. Wu, C. Mehlführer, J. Eilert, and D. Liu, Imlementation Asects of Channel Estimation for 3GPP LTE Terminals, in Proc. Euroean Wireless EW 0, Vienna, Austria, Ar. 0. [6] G. A. Korn and T. M. Korn, Mathematical Handbook for Scientists and Engineers: Definitions, Theorems, and Formulas for Reference and Review, Dover Publications, June 000. [7] ITU, Recommendation ITU-R M.5: Guidelines for evaluation of radio transmission technologies for IMT- 000 systems, Recommendation ITU-R M.5, International Telecommunication Union, 998.