Analytical Link Performance Evaluation of LTE Downlink with Carrier Frequency Offset

Analytical Link Performance Evaluation of LTE Downlink with Carrier Frequency Offset Qi Wang and Markus Rupp Institute of Telecommunications, Vienna University of Technology Gusshausstrasse 5/389, A-4 Vienna, Austria Email: {qwang, mrupp}@nt.tuwien.ac.at Web: http://www.nt.tuwien.ac.at/research/mobile-communications Abstract In this paper, we evaluate the link performance of a practical OFDM system, namely Long Term Evolution (LTE) downlink, with imperfect frequency synchronization. The analytical expression of the post-equalization Signal to Interference and Noise Ratio (SINR) is derived and compared with results obtained from a standard compliant simulator. An excellent agreement is obtained. Due to the fact that equalization is carried out based on the constant channel knowledge within one subframe, other than the inter-carrier interference, the common phase error introduced by the residual carrier frequency offset degrades the SINR severely. This implies that in order to eliminate the link performance loss, frequency synchronization of rather high accuracy is required. I. INTRODUCTION Orthogonal Frequency Division Multiplexing (OFDM) has become a dominant physical layer technique of modern wireless communication systems thanks to its high spectrum efficiency and robustness to frequency selective fading channels [1]. However, these advantages are guaranteed only when the orthogonality between subcarriers is ideally preserved. In practical systems, due to the cost limitation, local oscillators on the user equipment side have a typical frequency stability tolerance of ± ppm. This introduces a Carrier Frequency Offset (CFO) that destroys the orthogonality between subcarriers and degrades the system performance considerably [, 3]. Plenty of literatures on CFO estimation can be found [, 4 9]. There, the performance is evaluated in terms of the estimation error, in other words, the Mean Squared Error (MSE). On the other hand, the performance of the transmission system is evaluated in terms of SINR, Bit Error Ratio (BER) and most importantly the coded throughput. Therefore, not only the estimation performance becomes of interest, but also the effect of the estimation error on the overall link performance. In the past decade, efforts have been made in evaluating the performance degradation caused by the CFO in terms of BER or Symbol Error Ratio (SER) analytically [ 14]. Some consider the effect of CFO only and give analytical expressions in terms of uncoded BER for the Additive White Gaussian Noise (AWGN) channel [, 11]. This work is extended to frequency selective fading channels in [1]. The authors of [13, 14] further develop the calculation for uncoded BER and link capacity under the aggregate effect of time-variant impairments, namely CFO, imperfect channel knowledge and I/Q imbalance. In this work, we consider the downlink of LTE with a residual CFO after the frequency synchronization and analytically derive the post-equalization SINR with the CFO impairment. The resulting expression is compared with standard compliant simulations. This evaluation in terms of the post-equalization SINR, on one hand, indicates the impact of the frequency synchronization error on the link performance; on the other hand, can be further applied to derive corresponding system performance measures, e.g., Block Error Ratio (BLER) and coded throughput. The paper is organized as follows. In Section II, we describe the evaluation scenario and address the assumptions. In Section III, the mathematical model of the analysis is defined. The analytical derivation of the link performance in terms of the post-equalization SINR is found in Section IV. The result is validated by a standard compliant simulation in Section V. Section VI concludes the work. II. EVALUATION SCENARIO In practice, most OFDM systems utilize a frame based transmission. One frame is defined as a certain number of OFDM symbols with a fixed length in time. Block fading is assumed within the time duration of one frame so that signal processing at the receiver is performed on a frame basis. Specifically in LTE, as shown in Figure 1, the data transmission is based on 1 ms subframes. Each subframe contains 14 OFDM symbols. The reference signals used for channel estimation and feedback calculation are defined on a frequency time grid basis with 1 subcarriers over one subframe. In the following analysis, we assume: the channel coherence time greater than one subframe (1 ms) frequency selective fading channel multiple antennas served by one common local oscillator at TX(RX) residual CFO after the imperfect frequency synchronization

1 frame ms 1 subframe 1ms 1 3 4 5 6 7 8 reference signal 1 3 4 5 6 7 1 3 4 5 6 7 1 subcarriers The transmit signals in x l,k are mapped onto N L layers and precoded by the matrix W k of size N T N L. The channel H k is described by an N R N T matrix. For simplicity, we denote the channel matrix incorporated with the precoding matrix by H k. Thus, when the system is impaired by the CFO, there is r l,k = I(, ε, l) H k x l,k + I(p k, ε, l) H p x l,p + v l,k. Given the distortion I(, ε, l), the effective channel I(, ε, l) H k is dependent on the OFDM symbol index l, in other words, becomes time-variant. (6) 1 slot 14 OFDM symbols Fig. 1. LTE frame structure. 14 OFDM symbols III. MATHEMATICAL MODEL In our system model, a CFO is normalized to the subcarrier spacing, denoted by ε [.5,.5]. We denote the OFDM symbol index within one subframe by l and the subcarrier index by k and p. Correspondingly, the transmitted signal is referred to as X l,k, the received signal as R l,k and the Gaussian noise as V l,k. Given the block fading assumption, the channel frequency response is represented by H k. Thus, a Single-Input Single-Output (SISO) connection with a residual CFO ε can be expressed as R l,k = I(, ε, l) H k X l,k + H p X l,p I(p k, ε, l) +V l,k, } {{ } I l,k with sin(πε) I(, ε, l) = N sin(πε/n) πε(n 1) ej N e jφ(ε,l), () sin[π(p k + ε)] I(p k, ε, l) = N sin[π(p k + ε)/n] (1) e j π(p k+ε)(n 1) N e jφ(ε,l), (3) e jφ(ε,l) = e j πεl(n+ng) N, (4) where N is the Fast Fourier Transform (FFT) size and N g the Cyclic Prefix (CP) length. The effect introduced by the imperfect frequency synchronization is represented by I(p k, ε, l). The phase term Φ(ε, l) is referred to as the common phase error which is identical for all subcarriers in one OFDM symbol. In summary, the CFO has a twofold effect, namely a distortion I(, ε, l) to the desired signal, Inter-Carrier Interference (ICI): I l,k. In the following analysis, we consider a Multiple-Input Multiple-Output (MIMO) signal model with precoding at the subcarrier level, expressed as r l,k = H k W k x l,k + v l,k = H k x l,k + v l,k. (5) IV. POST-EQUALIZATION SINR In this section, we derive an analytical expression of the post-equalization SINR using the mathematical model defined in Section III. If the perfect channel state information is available at each subframe, a Zero Forcing (ZF) equalizer at the subcarrier k is given as G k = ( H H k H k ) 1 HH k. (7) Given the equalization performed on a subframe basis, the estimated data symbol after equalization becomes ˆx l,k = G k r l,k = I(, ε, l) x l,k + + G k I(p k, ε, l) H p x l,p } {{ } i l,k + G k v l,k ṽ l,k = I(, ε, l) x l,k + i l,k + ṽ l,k. (8) Therefore, the SINR of the subcarrier k in the l-th OFDM symbol can be found by E { x l,k } SINR l,k = E{ ˆx l,k x l,k }. (9) We denote the average signal power on each subcarrier and each layer by σ s and the corresponding noise power by σ n. Plugging Equation (8) into Equation (9), we obtain a closed form expression of the post-equalization SINR, expressed in Equation (). It is noted that the interference and noise terms in the denominator consist of not only the ICI and enhanced noise but also the distorted signal term σs N L I(, ε, l) 1 (11) =σs N L sin(πε) N sin(πε/n) πε(n 1) ej N e jφ(ε,l) 1. Given the block fading assumption, the ZF equalizer in Equation (7) is constructed using the constant channel state information H k over 14 OFDM symbols within one subframe. However, due to the residual CFO, the common phase error Φ(ε, l) increases with the OFDM symbol index l within one subframe, leaving the effective channel I(, ε, l) H k varying over time. Without taking this into account, the received signal

E { x l,k } SINR l,k = E{ i l,k } + E{ ṽ l,k } + I(, ε, l) 1 E{ x l,k } σs N L = σs } I(p k, ε, l) tr {G H k G k Hp HH p + σn tr { G H } () k G k +σs N L I(, ε, l) 1 noise ICI TABLE I SIMULATION PARAMETERS post-equalization SINR [db] - 1 3 ε=.1 SISO AWGN channel, SNR=dB ε=.1 4 5 6 7 8 9 11 1 13 14 OFDM symbol index Parameter Value Bandwidth 1.4 MHz FFT size (N) 18 Number of data subcarriers 7 CP length (N g) normal [18] Subcarrier spacing 15 khz Transmission setting 1 1, Modulation & coding adaptive Channel model AWGN / ITU PedB [19] Channel state information perfect Equalizer ZF Fig.. variation of the post-equalization SINR over OFDM symbols within one subframe. frequency sync. Fig. 3. CP removal FFT post-fft SINR channel estimation equalization demapper decoder post-equalization throughput SINR an evaluation chain of the frequency synchronization. in fact is equalized using outdated channel knowledge. Hence, the constellation of the equalized signal is rotated. In Figure, we plot the post-equalization SINR over the OFDM symbol index within one subframe. With a smaller CFO (ε =.1), the post-equalization decreases slowly from one OFDM symbol to the next. With a larger CFO (ε =.1), it drops rapidly then increases because of the periodic characteristics of the common phase error. V. NUMERICAL RESULT In this section, we validate the mathematical analysis by a cross comparison with the results obtained from a standard compliant LTE link level simulator [15, 16]. Corresponding parameters are listed in Table I. These intervals are indicated by the vertical bars in the simulated curves. Performance measures are observed at three positions in the receiver chain as shown in Figure 3. Using the bootstrap algorithm [17], we calculated the 95% confidence intervals for all simulated curves, expressed by vertical lines. In the first experiment, a simulation is carried out in a SISO AWGN channel. Twenty logarithmically spaced CFOs between 4 and 1 subcarrier spacings are introduced. In SINR [db] 5 15 5-5 Fig. 4. -4 calculated simulated -3 SISO AWGN channel, SNR=dB loss due to the residual common phase error post-equalization SINR - ICI loss post-fft SINR CFO normalized to the subcarrier spacing ( ) -1 ε SINR at two evaluation positions in a SISO AWGN channel. order to visualize the ICI, the Signal-to-Noise Ratio (SNR) is fixed at db. We measure the SINR after the FFT demodulation and the ZF equalizer. The post-fft SINR is calculated according to Equation (1), which indicates the impact of the ICI itself on the received signal. As shown in Figure 4, the calculated curves agree well with those acquired from simulations. Compared to the ICI, the residual common phase error has much greater impact on the link performance in terms of the post-equalization SINR. At lower CFO levels where the loss in post-fft SINR is invisible, an apparent drop can be observed. In the second experiment, we repeat the simulation in frequency selective fading channels, namely ITU Pedestrian B [19]. Besides the SISO case, a MIMO channel is considered. As illustrated in Figure 3, we measure the postequalization SINR and the coded throughput. Resulting curves can be found in Figure 5.

post-equalization SINR [db] coded throughput [Mbit/s] 5 15 5-5 8 6 4-4 x MIMO -3 1x1 SISO x MIMO 1x1 SISO ITU Pedestrian B channel, SNR=dB - calculated simulated CFO normalized to the subcarrier spacing ( ) -1 ε Fig. 5. post-equalization SINR and the corresponding coded throughput in the ITU Pedestrian B channel. In the upper figure, the calculated and simulated SINR obtain excellent agreement. The SINR for the SISO and MIMO case exhibit a similar behavior. Due to the interference between two layers, the SINR of the MIMO transmission is slightly lower than that of the SISO case. From the coded throughput curves shown in the lower figure, we arrive at a conclusion that a lossless transmission at db SNR requires a CFO estimation error lower than ε < 3. Specifically in LTE, this corresponds to approximately a precision of 15 Hz. For the low SNR cases, since the major degradation comes from the distorted signal term shown in Equation (11) which is independent from the noise power, SINR curves with a similar trend can be expected. VI. CONCLUSION In this work, we analytically derived the expression of the post-equalization SINR for the LTE downlink with imperfect frequency synchronization. In such systems where instant channel knowledge is not available at each OFDM symbol, the common phase error introduced by the residual CFO has a severe impact besides the ICI. Given a time invariant equalizer constructed using outdated channel state information, the link performance decreases dramatically. This result implies that in order to eliminate the link performance loss, frequency synchronization of rather high accuracy is required in reality. ACKNOWLEDGMENTS Funding for this research was provided by the fforte WIT - Women in Technology Program of the Vienna University of Technology. 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@InProceedings{Asilomar11_Qi, author = {Qi Wang and Markus Rupp}, title = {Analytical Link Performance Evaluation of {LTE} Downlink with Carrier Frequency Offset}, booktitle = {Conference Record of the Fourtyfifth Asilomar Conference on Signals, Systems and Computers, 11 ({Asilomar-11})}, month = nov, year = 11, address = {Pacific Grove, USA}, }