Copyright 2011 IEEE. Published in the proceedings of VTC-Fall 2011, San Francisco, CA, USA, 2011. Optimal Pilot Symbol Power Allocation in LTE ˇ Michal Simko, Stefan Pendl, Stefan Schwarz, Qi Wang, Josep Colom Ikuno and Markus Rupp Institute of Telecommunications, Vienna University of Technology Gusshausstrasse 25/389, A-1040 Vienna, Austria Email: email@example.com Web: http://www.nt.tuwien.ac.at/ltesimulator Abstract—The UMTS Long Term Evolution (LTE) allows the B. Contributionpilot symbol power to be adjusted with respect to that of the In this paper, we derive analytical expressions for optimaldata symbols. Such power increase at the pilot symbols resultsin a more accurate channel estimate, but in turn reduces the power allocation for a MIMO system with Zero Forcing (ZF)amount of power available for the data transmission. In this equalizer under imperfect channel state information. We uti-paper, we derive optimal pilot symbol power allocation based lize the post-equalization SINR, as the optimization function,on maximization of the post-equalization Signal to Interference which is analogous to the throughput maximization in a realand Noise Ratio (SINR) under imperfect channel knowledge. system.Simulation validates our analytical mode for optimal pilot symbolpower allocation. The main contributions of the paper are: Index Terms—LTE, Channel Estimation, OFDM, MIMO. • By maximizing the post-equalization SINR, we deliver optimal values for the pilot symbol power adjustment in MIMO Orthogonal Frequency Division Multiplexing I. I NTRODUCTION (OFDM) systems. • The post-equalization SINR expression is derived for a Current systems for cellular wireless communication are ZF receiver under imperfect channel knowledge.designed for coherent detection. Therefore, channel estimator • We analytically derive the Mean Square Error (MSE)is a crucial part of a receiver. UMTS Long Term Evolution expression of the Least Squares (LS) channel estimator(LTE) provides a possibility to change the power radiated at utilizing linear interpolation.the pilot subcarriers relative to the that at data subcarriers. • Simulation results with an LTE compliant simulator [6, 7]Clearly, this additional degree of freedom in the system design conﬁrm optimal values for pilot symbol power.provides potential for optimization. • As with our previous work, all data, tools, as well implementations needed to reproduce the results of thisA. Related Work paper can be downloaded from our homepage . The remainder of the paper is organized as follows. In In order to optimize the pilot symbol power allocation Section II we describe the mathematical system model. Ina model that takes into account the pilot power adjusting, Section III, we derive the post-equalization SINR expressionreceiver structure and channel estimation error at the same for ZF with imperfect channel knowledge. The channel estima-time, is needed. It has been shown by simulation that pilot tors of this work are brieﬂy discussed in Section IV and theirsymbol power allocation has a strong impact on capacity . MSEs are derived. We formulate the optimization problem forAuthors of  show by simulation the impact of different optimal pilot symbol power allocation in Section V. Finally,power allocations on the system’s Bit Error Rate (BER). we present LTE simulation results in Section VI and concludeHowever, their analysis is based on Signal to Noise Ratio our paper in Section VII.(SNR) so that they only approximate the impact of imperfectchannel knowledge on BER for Binary Phase-shift Keying II. S YSTEM M ODEL(BPSK) modulation. In , optimal pilot symbol allocation In this section, we brieﬂy point out the key aspects of LTEis derived analytically for Phase-shift Keying (PSK) modula- relevant for this paper, as well as an system model.tion of order two and four, using BER as the optimization In the time domain the LTE signal consists of frames withcriterion. In  optimal pilot symbol power in Multiple Input a duration of 10 ms. Each frame is split into ten equally longMultiple Output (MIMO) system is derived based on lower subframes and each subframe into two equally long slots withbound for capacity. Authors of  investigate power allocation a duration of 0.5 ms. Depending on the cyclic preﬁx length,between pilot and data symbols for MIMO systems using post- being either extended or normal, each slot consists of Ns = 6equalization Signal to Interference and Noise Ratio (SINR) as or Ns = 7 OFDM symbols, respectively. In LTE, the subcarrierthe optimization function. However, they only approximate the spacing is ﬁxed to 15 kHz. Twelve adjacent subcarriers of oneSINR expression and their model is tightly connected with slot are grouped into a so-called resource block. The numbera Linear Minimum Mean Square Error (LMMSE) channel of resource blocks in an LTE slot ranges from 6 up to 100,estimator. corresponding to a bandwidth from 1.4 MHz up to 20 MHz.
A received OFDM symbol in frequency domain can be III. P OST- EQUALIZATION SINRwritten as In this section, we derive an analytical expression for the Nt post-equalization SINR of a MIMO system under imperfect ynr = Xnt hnt ,nr + nnr , (1) channel knowledge and ZF equalizer given by the system nt =1 model in Equation (4). In the following section, we omit thewhere the vector hnt ,nr contains the channel coefﬁcients in subcarrier index k, since the concept we present is independentthe frequency domain between the nt -th transmiter and nr -th of it.receiver antennas and nnr is additive white zero mean Gaus- If the equalizer has perfect channel knowledge available,sian noise with variance σn at the nr -th receiver antenna. The 2 the ZF estimate of the data symbol s is given asdiagonal matrix Xnt = diag (xnt ) comprises the precoded −1 ˆ = GH G s GH y. (9)data symbols xd,nt and the pilot symbols xp,nt at the nt -thtransmit antenna placed by a suitable permutation matrix P s The data estimate ˆ given by Equation (9) results in the post-on the main diagonal equalization SINR of the l-th layer given as  T xnt = P xT t xT t . (2) σs 2 p,n d,n γl = −1 , (10) σn eH (GH G) 2 l elThe length of the vector xnt is Nsub corresponding to thenumber of subcarriers. Let us denote by Np and Nd , the where the vector el is an Nl × 1 zero vector with the l-th ele-number of pilot symbols and the number of precoded data ment being 1. This vector serves to extract the correspondingsymbols, respectively. The precoded data symbols xd,nt are layer power after the equalizer.obtained from the data symbols via precoding Let us proceed to the case of imperfect channel knowledge. T T We deﬁne the perfect channel as the channel estimate plus the [xd,1,k · · · xd,Nt ,k ] = Wk [s1,k s2,k · · · sNl ,k ] , (3) error matrix due to the imperfect channel estimationwhere xd,nt ,k is a precoded data symbol at the nt -th transmit ˆ H = H + E, (11)antenna port and the k-th subcarrier, Wk is the precodingmatrix at the k-th subcarrier and snl ,k is the data symbol of where the elements of the matrix E are independent of eachthe nl -th layer at the k-th subcarrier. other with variance σe . Inserting Equation (11) in Equa- 2 Note that according to Equation (2), also the vectors ynr , tion (4), the input output relation changes tohnt ,nr and wnr can be divided into two parts corresponding ˆto the pilot symbol positions and to the data symbol positions. y = H + E Ws + n. (12) For the derivation of the post-equalization SINR, we will Since the channel estimation error matrix E is unknown atuse the input-output relation at the subcarrier level, given as: the receiver, the ZF solution is given again by Equation (9), ˆ but channel matrix H is replaced by its estimate H, which is yk = Hk Wk sk + nk . (4) known at the receiverMatrix Hk is the MIMO channel matrix of size Nr × Nt . −1 s ˆ ˆ ˆ = GH G ˆ GH y, (13)Furthermore, matrix Wk is a unitary precoding matrix of sizeNt × Nl . In LTE the precoding matrix can be chosen from a ˆ ˆ with matrix G being equal to HW. The error after theﬁnite set of precoding matrices . The vector sk comprisesthe data symbols of all layers at the k-th subcarrier. We denote equalization process using ZF is given asthe effective channel matrix that includes the effect of the −1 s ˆ ˆ ˆ − s = GH G ˆ GH (EWs + n) . (14)channel and precoding by Gk Gk = Hk Wk . (5) From Equation (14) we can compute the MSE matrix HFurthermore, let us denote the average data power transmitted MSE = E (ˆ − s) (ˆ − s) s s (15)at one layer by σs , the total data power by σx , and the average 2 2 −1pilot symbol power by σp 2 = σn + σx σe 2 2 2 ˆ ˆ GH G , 1 with σe being the MSE of the channel estimator. Equation (15) 2 σs = E 2 sl,k 2 = , (6) 2 Nl directly leads to the SINR of the individual layers Nt σx = 2 E xd,nt 2 σs 2 2 = 1, (7) γl = −1 . (16) nt =1 2 2 2 ˆ ˆ (σn + σe σx ) eH GH G el l Nt σp = 2 E xp,nt 2 2 = 1. (8) Note, that in practice variables σs , σx and σn are replaced by 2 2 2 nt =1 their estimates.
Interpolation By plugging (18) into (20), expanding the square and using the identities npi Channel gain E Δh∗i hpi = E p hp = 0, (22) xpi i c2 = 1 − c1 , (23) e p1 p2 pm i pn we obtain for the LS MSE at an interpolated position: Subcarrier index (i,i) (p ,pn ) (p ,pm )Fig. 1. Linear inter- and extrapolation of the channel on pilot subcarriers to σei = Rh,h + c2 (Rh,h 2 1 n + σn ) + c2 (Rh,h 2 2 m + σn ) 2intermediate subcarriers. (p ,pn ) (i,p ) (p ,i) +2c1 c2 Rh,h m − 2c1 Rh,hn − 2c2 Rh,h m IV. C HANNEL E STIMATION (24) In this section, we present state-of-the-art channel estimators (i,j)and derive analytical expressions for their MSE. Due to the Here, Rh,h denotes the element in the i-th row and j-thorthogonal pilot symbol pattern utilized in LTE, the MIMO column of matrix Rh,h = E hhH . For an extrapolatedchannel can be estimated as Nt Nr individual Single Input position, we utilize the two consecutive neighboring pilotSingle Output (SISO) channels. Therefore, in the following positions, p1 and p2 , to obtain the estimated channel elementsection, we omit the antenna indices. according to: ˆ ˆ e − p1 ˆ ˆA. LS Channel Estimation he = hp1 + (hp2 − hp1 ) (25) p2 − p1 The LS channel estimator [11, 12] for the pilot symbol c1positions is given as the solution to the minimization problem: 2 Then, the same Formula (24), with appropriately modiﬁed ˆp ˆ hLS = arg min yp − Xp hp = X−1 yp p (17) indices, applies for the extrapolated MSE σee as well. The 2 ˆ hp 2 MSE at an extrapolated position is generally larger than at anThe remaining channel coefﬁcients at the data subcarriers have interpolated position, because the cross correlation between theto be obtained by means of interpolation. In this work, we channels at position e and p2 is smaller than those betweenutilize linear interpolation of the pilot subcarriers on each the channels at position i and pn .antenna to obtain the channel vector h containing the channel The total MSE of LS channel estimator is given as meanelements of all subcarriers in one OFDM symbol. over all subcarriers. Due to the dense pilot symbol pattern and In order to derive the theoretical MSE, we distinguish three thus strong correlation over frequency, the total MSE can becases, as shown in Figure 1: expressed as • MSE at a pilot subcarrier pi : σep 2 σe = ce σn , 2 2 (26) • MSE at an interpolated subcarrier i: σei 2 • MSE at an extrapolated subcarrier e: σee 2 where ce is a constant. Equation (26) follows from Equa-At a pilot position pi the estimated channel element equals: tion (23) and Equation (24). ˆ yp np hpi = i = hpi + i (18) B. LMMSE Channel Estimation xpi xpi The LMMSE channel estimator requires the second order Δhpi statistics of the channel and the noise. It can be shown thatThe MSE of the channel estimator is given by the noise the LMMSE channel estimate is obtained by multiplying thevariance σep = σn . To obtain the variance at an interpolated 2 2 LS estimate with a ﬁltering matrix ALMMSE [13–15]:position consider the following equation to compute the in- ˆ ˆ ˆp hLMMSE = ALMMSE hLS (27)terpolated channel element hi from the two neighboring pilot ˆ p and hp :positions h m ˆ n In order to ﬁnd the LMMSE ﬁltering matrix, the MSE ˆ ˆ i − pm ˆ ˆ hi = hpm + ·(hpn − hpm ) (19) 2 pn − pm σe = E 2 ˆp h − ALMMSE hLS , (28) 2 c1 has to be minimized, leading toThen, the MSE can be computed as: −1 2 ALMMSE = Rh,hp Rhp ,hp + σn I 2 , (29) σi = E 2 ˆ hi − hi (20) 2 where the matrix Rhp ,hp = E hp hH is the channel autocor- p 2 =E ˆ ˆ ˆ hi − hpm + c1 · (hpn − hpm ) (21) relation matrix at the pilot symbol positions, and the matrix 2 Rh,hp = E hhH is the channel crosscorrelation matrix. p
x 106 To derive the theoretical MSE, we plug Equation (29) into 1.8Equation (28): 1.7 4x4 LS channel estimatorσe =E 2 2 ˆp h − (Rh,hp (Rhp ,hp − σn I)−1 hLS ) · (30) 1.6 1.5 H h − (Rh,hp (Rhp ,hp − 2 ˆp σn I)−1 hLS ) 1.4 f(o) 2x2 1.3After a straightforward manipulation, the average MSE at each 1.2 LMMSE channel estimatorsubcarrier is expressed as 1.1 1x1 1 −1 1 σe = 2 tr Rh,h − Rh,hp Rhp ,hp + σn I 2 Rhp ,h . -10 -5 0 5 10 Nsub poff[dB] Fig. 2. Power allocation function f (poﬀ ) for different numbers of transmit V. P OWER A LLOCATION antennas and LS (solid line) and LMMSE (dashed line) channel estimators In this section, we describe the problem of optimal pilot TABLE I VALUES OF THE PARAMETERS OF f (poﬀ ) FOR DIFFERENT NUMBER OFpower allocation in LTE based on the maximization of the TRANSMIT ANTENNAS FOR 1.4 MH Z BANDWIDTH , ITU P EDA post-equalization SINR under imperfect channel knowledge. CHANNEL MODEL , LS AND LMMSE CHANNEL ESTIMATORSAlthough the shown results are from the application to LTE, Parameter Tx = 1 Tx = 2 Tx = 4the presented concept can be applied to any MIMO OFDM Nd 960 912 864 Np 48 96 144system. If we increase the power at the pilot symbol by a factor c2 , p LSthe MSE of the channel estimator improves by the factor c2 ce 0.3704 0.3704 0.5556 p poﬀ,opt [dB] ≈5.8 ≈3.6 ≈3.5 σe 2 σe = ˜2 . (31) LMMSE c2 p ce 0.0394 0.0394 0.0544 poﬀ,opt [dB] ≈-0.7 ≈-2.8 ≈-3.2However, the power of the data symbols has to be decreased bya factor c2 in order to keep the total transmit power constant. The target is to ﬁnd an optimal value of poﬀ,opt that max- dObviously, the two factors c2 and c2 are connected. For this imizes the post-equalization SINR while keeping the overall p dpurpose, we deﬁne a variable poﬀ , expressing the power offset transmit power constantbetween the mean energy of the pilot symbols and the data maximize γl (37)symbols, and refer to it as pilot offset. The variables c2 , c2 p d poffand poﬀ are interconnected as follows: subject to Nd σx + Np σp = const 2 2 Np + N d In order to maximize the post-equalization SINR, the power cp = (32) poﬀ Nd + Np allocation function f (poﬀ ) in the denominator of Equation (35) Np + Nd has to be minimized. The minimum of the power allocation cd = Np = poﬀ cp (33) Nd + poff function f (poﬀ ) can be found by simply differentiating it and solving for 0. By these means, the optimal value of the variablePlugging in the variables c2 and c2 in Equation (16), we obtain d p poﬀ is given as solution to the following expression:the SINR expression with adjusted power of the pilot symbols −2 (poﬀ Nd + Np ) p−3 + 2 (poﬀ Nd + Np ) Nd p−2 + ce = 0. 2 oﬀ oﬀ σs c2 2 (38) γl = d . (34) σe 2 2 2 −1 σn + 2 c2 σx cd eH (GH G) l el An analytical solution for Equation (38) can be obtained by pIf we insert Equation (32) and Equation (33) into Equation (34) means of Ferrari’s solution . Figure 2 depicts f (poﬀ ) forand simplify the expression, we obtain: different numbers of transmit antennas for LTE. Typical values of parameters Nd , Np and ce are given in Table I. Note, 1 that although Nd and Np depend on the utilized bandwidth, γl = , (35) Nl σn eH (GH G)−1 el 2 the minimum of f (poﬀ ) is independent of it, since Nd and l (Nd +Np )2 f (poﬀ ) Np scale with the same constant with increasing bandwidth.for which the function f (poﬀ ) is given as The value of ce is different for four transmit antennas due to the lower number of pilot symbols at the third and fourth 2 1 f (poﬀ ) = (poﬀ Nd + Np ) + ce . (36) antenna. The last row of Table I gives the optimal values of p2 oﬀ poﬀ,opt for different numbers of transmit antennas and an ITUWe refer to it as power allocation function. Note, that it is PedA  type channel model. While utilizing the LMMSEindependent of channel realization and noise power. Therefore, channel estimator, one obtains negative values for poﬀ,opt .it is possible to ﬁnd an optimum value for pilot symbol power This means, that optimally, the pilot symbol power has toallocation independent of the SNR and channel realization. be reduced in order to maximize the post-equalization SINR.
TABLE II S IMULATOR SETTINGS FOR FAST FADING SIMULATIONS linear interpolation. Throughput simulation results validate the Parameter Value accuracy of our analytical model for the optimum pilot power Bandwidth 1.4 MHz adjustment. All data, tools and scripts are available online in Number of transmit antennas 1, 2, 4 Number of receive antennas 1, 2, 4 order to allow other researchers to reproduce the results shown Receiver type ZF in the paper . Transmission mode Open-loop spatial multiplexing Channel type ITU PedA  ACKNOWLEDGMENTS MCS 9 coding rate 616/1 024 ≈ 0.602 The authors would like to thank the LTE research group symbol alphabet 16 QAM for continuous support and lively discussions. This work has been funded by the Christian Doppler Laboratory for 3.5 Wireless Technologies for Sustainable Mobility, KATHREIN- Werke KG, and A1 Telekom Austria AG. 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Mehlf¨ hrer, J. Eilert, and D. Liu, “Implementation usymbol power adjustment for MIMO OFDM systems. As Aspects of Channel Estimation for 3GPP LTE Terminals,” in Proc.optimization criterion we utilized the post-equalization SINR European Wireless (EW 2011), Vienna, Austria, Apr. 2011.  G. A. Korn and T. M. Korn, Mathematical Handbook for Scientistsunder imperfect channel state information, that is connected and Engineers: Deﬁnitions, Theorems, and Formulas for Reference andto the system throughout. Furthermore, we derive analytical Review, Dover Publications, June 2000.expression for the post-equalization SINR under imperfect  ITU, “Recommendation ITU-R M.1225: Guidelines for evaluation of radio transmission technologies for IMT- 2000 systems,” Recommen-channel knowledge using ZF and we provide analytical ex- dation ITU-R M.1225, International Telecommunication Union, 1998.pression for the MSE of the LS channel estimator utilizing