Performance Analysis of LTE Downlink under Symbol Timing Offset

Performance Analysis of LTE Downlink under Symbol Timing Offset Qi Wang, Michal Šimko and Markus Ru Institute of Telecommunications, Vienna University of Technology Gusshausstrasse 25/389, A-1040 Vienna, Austria Email: {qwang, msimko, mru}@nt.tuwien.ac.at Web: htt://www.nt.tuwien.ac.at/research/mobile-communications Abstract In this aer, we evaluate the erformance of a standardized OFDM system, namely Long Term Evolution (LTE) downlink, with imerfect symbol timing. A closed form exression of the ost-equalization Signal to Interference and Noise Ratio (SINR) is derived and comared with results obtained from a standard comliant simulator. Also, we analyzed the channel estimation erformance when Symbol Timing Offset (STO) occurs. This work reveals the imact of imerfect synchronization on the link erformance. It also allows an accurate and realistic modeling of the hysical layer behavior, which can be alied to reroduce results from time-consuming link level simulations. I. INTRODUCTION Orthogonal Frequency Division Multilexing (OFDM) has become a dominant hysical layer technique of modern wireless communication systems thanks to its high sectrum efficiency and robustness to frequency selective fading channels [1]. In order to eliminate Inter-Symbol Interference (), a guard interval Cyclic Prefix (CP) is aended at the beginning of each OFDM symbol. This, to some extent, rovides the system a certain tolerance to symbol timing errors. However, the valid symbol timing region shrinks as the maximum channel delay increases. Plenty of techniques can be found in literature on symbol timing error estimation [2 5]. Their estimation erformance are usually evaluated in terms of lock-in robability, namely the robability that the estimated symbol timing falls within the valid symbol timing region. In [6], a mathematical analysis of the imact of timing synchronization error was resented. However, the evaluation is in terms of the SINR after the Fast Fourier Transform (FFT) at the receiver. Although this metric reflects the erformance loss, in order to evaluate the link erformance of a realistic communications system, other asects in the receiver need to be considered, e.g., channel estimation and equalization. In this work, we consider the downlink of LTE with a STO at samling eriod level and analytically derive a closed form exression of the ost-equalization SINR as a function of the symbol timing error. Additionally, the imact of the STO on the channel estimation erformance is investigated. Afterwards, these results are validated using standard comliant simulations [7]. The aer is organized as follows. In Section II, we describe a mathematical model as a basis of the analysis. A ostequalization SINR model is resented in Section III. In Section IV, we discuss the estimation erformance of a Least Squares (LS) channel estimator under STO. Numerical results are rovided in Section V. Section VI concludes the work. II. SYSTEM MODEL In this section, a system model is described. We consider an STO of θ samling eriods, i.e., θ Z. The samling time index is referred to as n, the OFDM symbol index as l and the subcarrier indices as k and. Among the N subcarriers, N tot are occuied by data symbols. Given a multi-ath channel h n L c τ δ K (n τ), (1) τ0 the data transmission in the OFDM symbol l in the frequency domain for LTE Single-Inut Single-Outut (SISO) downlink can be described by with R l P H F W H (t) MF H P X l + (2) + P H F W H (t) MF H P X }{{ int +V } l, AX l + BX int + V l, (3) A P H F W H (t) MF H P, C Ntot Ntot, (4) B P H F W H (t) MF H P, C Ntot Ntot. (5) where X l denotes the desired data symbol vector, X int the interfering signal vector, V l the Additive White Gaussian Noise (AWGN) vector and R l the received data symbol vector. Matrices A and B can be interreted as the transfer functions for the desired signal X l and the interfering signal X int. Secifically, zero subcarriers are added by the matrix P of size N N tot to eliminate inter-band interference. Matrices F (F H ) is the corresonding N N FFT (IFFT) matrix. The insertion of the CP is fulfilled by the matrix M of size (N + N g ) N where N g denotes the CP length.

The convolution in the time domain is described as a Toelitz matrix comosed by the channel coefficients in Equation (1) c 0 0 0 c 1 c 0 0 0 H (t)........... (6). 0.. 0 cl c L 1 0 0 c L which has a dimension of (N + N g + L) (N + N g ). On the receiver side, the signal vector that is fed into the IFFT unit is extracted by a windowing matrix W and W of size N (N + N g + L). In the ideal timing case θ 0, W [ 0 N Ng I N N ] 0 N L, W 0 N (N+Ng+L). (7) We denote the entries in matrix A as a ij and in B as b ij. Given Equation (3), the received signal on subcarrier k in OFDM symbol l can be written as R l,k a X l,k + a X l, + b X int, +V l,k. (8) }{{}}{{} In case of ideal symbol timing, transfer matrix A becomes diagonal and matrix B all-zero. Therefore, neither Inter- Carrier Interference () nor occurs. In the following, we extend Equation (8) to a N R N T Multile-Inut Multile-Outut (MIMO) case. The transfer functions A (m,q) and B (m,q) from Transmitter (TX) q to Receiver (RX) m are constructed using the corresonding channel coefficients h (m,q) n. Thus, we obtain a vector-matrix form r l,k a x l,k + a x l, + b x int, +v l,k, (9) }{{}}{{} where r l,k (v l,k ) denotes the N R 1 received signal (noise) vector, x l,k the N T 1 transmitted signal vector. The transfer functions for subcarrier k are of dimension N R N T, given as a (1,1) a (1,NT) a..... a (NR,1) a (NR,NT) b (1,1) b (1,NT) b..... b (NR,1) b (NR,NT), (10). (11) We distinguish two cases, namely STO to the right (late timing) and STO to the left (early timing). Corresondingly, X int is either X l+1 or X l 1. A. STO to the right (θ < 0) In the late timing case, the FFT window shifts to the right and takes in the interference from the successive OFDM symbol. Such a rocess can be modeled by alying W [ ] 0 N (Ng θ) I N N 0 N (L+θ) (12) [ W 0 0 I ( θ) ( θ) 0 to Equations (2), (4), and (5). B. STO to the left (θ > N g L) ] (13) When STO occurs to the left, no interference is induced as long as the FFT window does not embrace the tail of the revious OFDM symbol. Otherwise, there are d θ + L N g samles in the desired OFDM symbol corruted by the revious one. Similarly, this can be modeled by alying W in Equation (12) and [ ] W 0 Id d. (14) 0 0 III. POST-EQUALIZATION SINR The ost-equalization SINR is a metric of imortance for a transmission system which emloys linear satial equalizers at the receiver, because it directly determines the theoretically ossible throughut I via Shannon s formula: I log 2 det(1 + SINR). (15) In this section, a closed form exression of the ostequalization SINR of a LTE downlink transmission under symbol timing offset is derived. In an LTE system with coherent detection, channel estimation is erformed with the hel of the standardized reference signal defined in [1]. Nevertheless, given the received signal in Equations (8) and (9), a simle linear channel estimator is merely able to estimate the diagonal elements of the effective channel matrix A, namely a. Starting with a simle assumtion of erfect channel knowledge, the Zero Forcing (ZF) equalizer can be exressed as g l,k a (ah a ) 1 a H. (16) The outut signal of such an equalizer can be written as ˆx l,k g l,k r l,k (17) x l,k + g l,k a x l, + g l,k b x int, + g l,k v l,k. }{{} noise Define σ 2 E a x l, 2 F σs a 2 2 F, (18) { } σ 2 E b x int, 2 F σs b 2 2 F, (19)

we obtain a closed form exression of the SINR of the outut signal from the equalizer, exressed as E { } x l,k 2 2 SINR l,k E{ ˆx l,k x l,k 2 2 } σs 2 N T { } (σ 2 + σ2 + σ2 v) tr g l,k gl,k H σs 2 N T (σ 2 + σ2 + σ2 v) tr { (a H a ) 1}, (20) where σ 2 s and σ 2 v denote the signal and the noise ower. Although the STO is not a time-variant distortion, it also introduces and which degrades the link erformance significantly. IV. IMPACT ON CHANNEL ESTIMATOR In the revious analysis, a erfect channel knowledge was assumed for simlicity. Only and that is imosed to the received signal itself was taken into account. In fact, another significant effect of these interference is that they also degrade the erformance of the channel estimator. Afterwards, in the overall SINR exression, the imact of this estimation error is further enhanced. In this section, the estimation erformance of a tyical LS channel estimator is investigate in the context of LTE downlink. In Rel-8 [9], cell-secific Reference Signals (RSs) are utilized in channel estimation for both demodulation and feedback calculation. Let K denote the set of RS ositions, a simle LS channel estimator on these osition (l, k ) K can be exressed as ĥ LS l,k arg min ĥ l,k r l,k ĥl,k x l,k 2 2 r l,k x l,k. (21) Given the received signal shown in Equation (9), on the RS ositions, the Mean Squared Error (MSE) of the LS estimation is given as σe,rs 2 E { h l,k ĥl,k 2} a 2 + }{{} b 2 + σ2 v σ 2 s N T. (22) On the data symbol ositions, channel estimates are obtained by linear interolation. Their theoretical MSE erformance has been rovided in [8] as σ 2 e,data c e σ 2 e,rs + d, (23) where c e is a scalar determined by the RS structure. The factor d deends on the channel autocorrelation matrix as well as the RS structure. Knowing that the and in Equation (22) are deendent on the channel Power Delay Profile (PDP) and the STO, a saturation in the MSE erformance at the higher Signal-to-Noise Ratio (SNR) regime is exected. TABLE I SIMULATION PARAMETERS Parameter Value Bandwidth 1.4 MHz FFT size (N) 128 Number of data subcarriers (N tot) 72 CP length (N g) normal [9] Samling frequency 1.92 MHz Subcarrier sacing 15 khz Transmission setting 1 1, 2 2 OLSM Modulation & coding adative Channel model ITU PedB [10] Channel state information erfect/ls Equalizer ZF Channel Quality Indicator (CQI) feedback otimal TABLE II PDP OF ITU PEDESTRIAN B CHANNEL MODEL [10] Excess ta delay (ns) 0 200 800 1200 2300 3700 Relative ower (db) 0-0.9-4.9-8.0-7.8-23.9 In addition, when there is an STO, the Linear Minimum Mean Squared Error (LMMSE) estimator which normally outerforms the LS estimator needs to be treated with care. Given the effective channel transfer function in Equation (4), the second order statistics of the effective channel becomes STO-deendent. Certain adjustments in the LMMSE estimator are necessary. V. NUMERICAL RESULTS In this section, we validate our analytical solution by comaring results with those from the standard comliant simulations using the Vienna LTE Link Level simulator [7]. Simulation arameters are listed in Table I and Table II. A. Post-equalization SINR In order to validate Equation (20), a Monte Carlo simulation was carried out using 500 LTE subframes. With these 500 channel realizations, the ost-equalization SINRs were calculated using the closed form exression. Since the system erformance is more sensitive to the interference at high SNR region, SNR was chosen to be fixed at 30 db. A series of STO θ [ 5, 15] was introduced. Corresonding results are shown in Figure 1 in terms of the so-called wide-band SINR which is averaged over 72 data subcarriers. In Figure 1, the curve obtained from the closed form exression agrees well with the one from the Monte Carlo simulation. In the regard of θ < 0, namely a late timing occurs, a shar dro can be observed in ost-equalization SINR; while on the other hand, where an early timing occurs, the CP alleviate the situation. Although the CP has a length of 4.7 µs, corresonding to 9 samling eriods in this case, given the channel PDP in Table II, arises at the head of the CPs. Therefore, an SINR degradation aears after merely five samles. The 95 % confidence intervals are relatively large which can be exlained by the strong frequency selectivity induced by the multi-ath channel.

SNR 30 db SNR 30 db 24 ITU Pedestrian B channel 6 ITU Pedestrian B channel, SISO ost-equalization SINR [db] 20 16 12 8 4 0 2x2 calculated simulated 1x1-5 0 5 10 15 symbol timing offset θ (samling eriod) coded throughut [Mbits/s] 5 4 3 2 1 erfect channel knowledge LS channel estimator -5 0 5 10 15 symbol timing offset θ (samling eriod) Fig. 1. Comarison of the calculated and simulated ost-equalization SINR under STOs. Fig. 3. Coded throughut of a SISO transmission under STOs. Mean Squared Error 10 0 10-1 10-2 10-3 calculated simulated 10-4 0 5 10 15 20 SNR [db] ITU Pedestrian B channel, SISO 25 STO 8 STO -5 STO -2 STO 4 STO 0 Fig. 2. MSE erformance of an LS channel estimator under various STOs: θ { 5, 2, 0, 4.8}. B. Channel Estimation Performance Figure 2 resents the MSE erformance of an LS channel estimator with linear interolation under different levels of STOs. Both late and early timing were considered. As indicated in Equation (22), a saturation can be observed due to the and. Interestingly, it is noticed that for θ 4, namely a timing offset of four samling eriods within the -free region, a saturation also occurs due to the. When ZF equalization is erformed using imerfect channel estimates, the energy from the estimation error could be further enhanced. This effect was investigated by a throughut simulation at 30 db SNR. Results are shown Figure 3. Comared to the erfect channel knowledge case, a further loss can be found. In order to model this effect analytically, an aroriate mathematical model for the channel estimation error must be chosen. VI. CONCLUSION In this work, we analytically derived an exression of the ost-equalization SINR for the LTE downlink with imer- 30 fect symbol timing synchronization. A comarison is made between statistical simulation results and calculation results using a closed form exression in which the data and noise realizations are averaged. This work allows an accurate and realistic modeling of the hysical layer behavior, which can be alied to system level simulations [11]. Additionally, the investigation on the channel estimation erformance shows that the STO is a significant source of as well, although it is not a time-variant distortion. In order to model the SINR loss due to the channel estimation error, a valid characterization of the imerfect channel knowledge needs to be chosen carefully, which is considered to be a future work. ACKNOWLEDGMENTS Funding for this research was rovided by the fforte WIT - Women in Technology Program of the Vienna University of Technology. This rogram is co-financed by the Vienna University of Technology, the Ministry for Science and Research and the fforte Initiative of the Austrian Government. Also, this work has been co-funded by the Christian Doler Laboratory for Wireless Technologies for Sustainable Mobility. REFERENCES [1] 3GPP, Technical secification grou radio access network; (E-UTRA) and (E-UTRAN); overall descrition; stage 2, Tech. Re., Se. 2008. [Online]. Available: htt://www.3g.org/ft/secs/html-info/36300.htm [2] T. Schmidl and D. Cox, Robust frequency and timing synchronization for OFDM, IEEE Transactions on Communications, vol. 45, no. 12,. 1613 1621, Dec. 1997. [3] H. Minn, V. K. Bhargava, and K. B. Letaief, A robust timing and frequency synchronization for OFDM systems, IEEE Trans. on Wireless Com., vol. 2, no. 4,. 822 839, Jul. 2003. [4] H. Bölcskei, Blind estimation of symbol timing and carrier frequency offset in wireless OFDM systems, IEEE Trans. on Communications, vol. 49, no. 6,. 988 999, Jun. 2001. [Online]. Available: htt://www.nari.ee.ethz.ch/commth/ubs//synch [5] Q. Wang, M. Šimko, and M. Ru, Modified symbol timing offset estimation for OFDM over frequency selective channels, in Proceeding of IEEE 74th Vehicular Technology Conference (VTC2011-Fall), San Francisco, USA, Se. 2011. [6] Y. Mostofi and D. C. Cox, Mathematical analysis of the imact of timing synchronization errors on the erformance of an OFDM system,

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