Iterative Phase Noise Mitigation in MIMO-OFDM Systems with Pilot Aided Channel Estimation

Iterative Phase Noise Mitigation in MIMO-OFDM Systems with Pilot Aided Channel Estimation Steffen Bittner, Ernesto Zimmermann and Gerhard Fettweis Vodafone Chair Mobile Communications Systems Technische Universität Dresden, D-01062 Dresden, Germany Email: {bittner, zimmere, fettweis}@ifn.et.tu-dresden.de Abstract The use of multiple transmit and receive antennas in combination with multicarrier modulation, e.g. MIMO-OFDM, is a very promising technique for future wireless communication systems. In this work we investigate preamble based channel estimation under the presence of phase noise. Neglecting the influence of phase noise in the design of the preamble will lead to a significant loss in accuracy of the channel estimation. The solution involves an analysis of the mean square error approximation of the channel estimation and incorporating the result in the phase noise mitigation and data detection step. I. INTRODUCTION Using multiple antennas at transmitter and receiver (MIMO) allows to achieve high spectral efficiency by spatially multiplex several data streams into the same time frequency bin. For broadband communication systems a very promising technique is to combine the MIMO concept with multicarrier techniques such as orthogonal frequency division multiplexing (OFDM), leading to the MIMO-OFDM idea. The advantages of such MIMO-OFDM systems combination is that well studied narrowband based MIMO receivers techniques can be applied to every subcarrier individually. Phase noise (PN) causes a multiplicative phase distortion during upconversion at the transmitter and downconversion at the receiver. This impairment is mainly caused by RF imperfections e.g. imperfect oscillator. One of the main problem in MIMO-OFDM receiver design is the channel estimation (CE) part if PN is present. This renders CE a challenging problem, because PN leads to a coupling of the unknown channel and transmitted symbols. While the individual effects of PN and CE have been well studied, the joint problem has still potential for further research. Phase noise correction algorithms based on perfect channel state information at the receiver have been well investigated for the SISO and MIMO case [1], [2]. However, together with CE the problem especially for multiple antennas is only rarely discussed [3]. In this contribution we analyze the influence of PN on different orthogonal preamble design methods and its impact on PN correction algorithms. The PN correction algorithm itself is based on an improved iterative linear minimum mean square error (LMMSE) estimation of the Fourier coefficients of the PN realization [4], [5]. Here we Part of this work has been performed in the framework of the IST project IST-4-027756 WINNER II, which is partly funded by the European Union. The authors would like to acknowledge the contributions of their colleagues in WINNER II, although the views expressed are those of the authors and do not necessarily represent the project. include the PN statistics and the MSE approximation of the CE error into the correction algorithm leading to performance improvement. The remainder of this paper is organized as follows. In Sec. II the MIMO-OFDM model extended by Wiener PN is introduced, as well as the foundations of preamble based CE methods. In Sec. III we give a detailed description of the iterative receiver processing including LMMSE estimation and MIMO detection. Performance for a HiperLAN A channel is presented in Sec. IV. Conclusions are drawn in Sec. V. II. FOUNDATIONS A. MIMO-OFDM and Phase Noise Model We consider MIMO (multiple-input, multiple-output) OFDM transmission with N T x transmit (Tx), N Rx receive (Rx) antennas and N C subcarriers. Furthermore we use uppercase letters to describe frequency domain and lowercase letters for time domain signals. Let V be a vector of information bits which are encoded by an outer code and interleaved. The resulting code bit stream is partitioned into blocks X containing N T x N C M independent binary digits. Here M represents the number of bits per symbol and hence allows to distinguish between 2 M different constellation points. As part of the transmission process, every single block is mapped onto a N C 1 signal vector S = [S 1,, S l,, S NC ] T, whose components S l = [S l1,, S ltx,, S lnt x ] T denote the N T x 1 frequency domain MIMO transmit vectors for each subcarrier. Per antenna, the vector is transformed into time domain by performing an inverse fast Fourier transform (IFFT). Before transmission a cyclic prefix (CP) of length N G is added. This is assumed to be longer than the channel impulse response, to avoid possible intersymbol interference caused by frequency selective channels. The phase noise is modeled by a Brownian motion or Wiener Process resulting in a Lorentzian power density spectrum. However, the PN correction algorithm is also valid for other PN models (e.g. PLL). Furthermore we restrict ourselves to the case of phase noise at the receiver only since transmitter phase noise can be approximated by an effective receiver phase noise as presented in [6]. The phase noise variable at the (n) th sample is related to the previous one as φ(n) = φ(n 1)+w, where w is a Gaussian distributed random variable, with zero mean and variance σw 2 = 4π 2 fc 2 ct s. In this notation, T s describes the sample interval, c determines the oscillator quality and f c is the carrier frequency. Related to the 3dB single side bandwidth f 3dB of

the Lorentzian spectrum, c is given by c = f 3dB /(πf 2 c ) [7]. With f sub as the subcarrier spacing of an OFDM system, it is common to use the single relevant performance parameter δ 3dB as the relative oscillator linewidth with respect to the subcarrier spacing given by δ 3dB = f 3dB /f sub. The time domain signal at receive antenna rx in the presence of phase noise can be expressed as: y rx (n) = N T x tx=1 (s tx (n) h rx,tx (n)) e jφ(n) + ξ rx (n), (1) where stands for convolution and ξ rx (n) represents additive white Gaussian noise with variance σn. 2 In this notation, h rx,tx (n) describes the time domain channel impulse response between transmit antenna tx and receive antenna rx. After removing the CP, a discrete Fourier transform is performed per antenna, transforming the received signal back into frequency domain. The overall transmission chain including Fourier transforms is given by the following vector matrix notation: ( ) Y = (F I Rx )Ψυ hθ(f 1 I T x )S + ξ. (2) The symbol denotes the Kronecker Product, I is the identity matrix and F represents the N C N C Fourier matrix. The cyclic prefix is added by multiplication with the matrix Θ and removed by the multiplication with the matrix Ψ. The phase noise process is represented by υ = diag(e 0,, e Ntot 1) I NRx with e n = e jφ(n). Furthermore the channel is given by the matrix h of channel impulse responses. Performing standard matrix manipulation the received signal in the frequency domain can be written as: Y = ΥHS + η (3) where we applied the fact that the circular block matrix ΨhΘ can be diagonalized by the IFFT and FFT operation resulting in the N C N Rx N C N T x block diagonal matrix H which is the frequency domain representation of the channel matrix h. Phase noise at the receiver results in intercarrier interference between received symbols during downconversion, which is modeled in base band by the circular matrix Υ: E 0 E 1 E (Nc 1) Υ =......... I Rx (4) E Nc 1 E Nc 2 E 0 with E k = 1 e j2πkn/n C e jφ(ng+n) as the frequency N C N C 1 n=0 domain representation of the phase noise process. The statistical properties of the transformed (i.i.d) AWGN noise stay the same. The main tasks are: firstly to compensate the influence of the phase noise under the assumption of non perfect channel state information at the receiver and secondly to solve the spatial multiplexing detection problem. B. Preamble based Channel Estimation In order to enable coherent demodulation of the transmitted signals, the receiver has to acquire adequate channel state information. In this work we consider a preamble based channel estimation where multiple OFDM symbols are used for training. At the receiver side a least squares channel estimator is used. For each transmit antenna a total number of N P re (tx) subcarriers are used as pilots in the preamble. For the case of MIMO-OFDM systems, the channel impulse responses for all N T x N Rx antenna links have to be estimated. Let N CIR denote the length of the channel impulse response (CIR), in samples. In the following we are assuming the same number of samples for all transmit receive antenna links. Evidently, N P re (tx) N CIR must hold for each link in order for the estimation problem to be (over-)determined. Furthermore, the pilot design must take into account the fact that the signals emitted from different transmit antennas superpose at the receive antennas. This leads to the necessity to use an orthogonal design at the transmitter. This can be achieved either in time domain, frequency domain or code domain by choosing appropriate pilot patterns (e.g. Hadamard sequences) as given in Fig. 1. The performance of all approaches in subcarrier Time Code Frequency Fig. 1. Orthogonal preamble design Antenna Preamble terms of mean square error (MSE) is the same as long as the orthogonality is guaranteed [8]. We assume that each subcarrier N P re (tx) at transmit antenna tx is transmitted with power E s (tx). In the sequel we will include phase noise in the channel estimation. In the presence of phase noise a common rotation to all subcarriers will always be considered as a part of the channel. Although the phase noise realizations are based on a sampled Wiener process with mean zero, the average phase rotation per OFDM symbol, common to all subcarriers, is not zero and is called common phase error (CPE). The CPE changes from symbol to symbol. In the following we will call the CPE occurring in the preamble symbols as and E 0Data for the CPE occurring in the data symbols respectively. An extended approximation of the MSE on the estimated channel coefficients including PN is now given by [8]: σ 2 (tx) N CIR N P re (tx) σ2 n + σζ 2 ICIP re. (5) E s (tx) Here σζ 2 ICIP represents the variance of the intercarrier interference (ICI) due to higher phase noise harmonics in the re preamble, which is a weighted sum over all other symbols. Since we cannot estimate the contribution of the ICI at this stage we have to assume it to be Gaussian distributed. A general expression for computing the variance of the ICI is given in [9].

Per subcarrier l an estimated channel matrix Ĥ l has been determined. In order to increase readability we leave out the subcarrier index l from now on. Let the estimated channel transfer matrix be defined as Ĥ = H + H, where H models the remaining estimation error, caused by the AWGN and ICI noise due to PN in the preamble. From the viewpoint of the detector, which bases its equalization on Ĥ, the MIMO transmission of the data can be modeled as follows: Y = (Ĥ H ) E 0 Data S + ɛ (6) which corresponds to Eq. (3). In addition ɛ represents the effective noise consisting of remaining ICI noise and AWGN noise of the data symbols. Assuming that the transmitted symbols, phase noise and the AWGN are statistically independent [10] the covariance matrix of the noise due to the channel estimation error can be approximated by: Φ H S H S = V ar{e 0 Data } V ar{ } E N T x s N T x tx=1 E s N T x E{ H H H} (7) N CIR N P re (tx) σ2 n + σζ 2 ICIP re E s (tx) where I NRx is the identity matrix of dimension N Rx. I NRx III. PHASE NOISE CORRECTION AND DETECTION First let s assume perfect channel information at the receiver. A possible way to track the phase noise distortion during the data transmission is to use known pilots. We define the location of the pilots within the transmitted data symbols with the vector P. The received symbols taking only the CPE into account are given by: Finally Φ0 is used as an initial correction of the CPE. The remaining preamble CPE ( ) is part of the estimated channel. Thus, the overall CPE (E 0Data ) in the data symbols will inherently be compensated by performing the equalization step. However, one should know that this derivation is only valid for code and frequency orthogonally designed preambles. In these cases, the full channel matrix is estimated per preamble symbol and an average CPE is obtained which is constant for the whole channel matrix. In the case of time orthogonal design each single estimated column in the channel matrix will suffer from a different CPE realization. It is obvious that such a preamble design will lead to high distortion of the estimated channel matrix at the presence of PN. This results in a significant performance degradation as given in Sec. IV. A. Higher order PN mitigation After initial correction of the common phase error and decoding a first estimation of the transmitted symbols is available. However, this estimation is not very reliable, since each received symbol still suffers from a weighted sum over all other symbols, which again results in ICI. Therefore, we will perform an iterative estimation of higher order harmonics using the idea of joint LMMSE estimation [1], [4]. The detailed MIMO-OFDM transmission chain including phase noise correction is shown in Fig. 2 where we first discuss the linear MMSE estimation and phase noise correction block. Later we will briefly discuss the MIMO detection problem. Source Outer Encoder Π Interleaver M-QAM MIMO OFDM Y P = H P S P E 0Data + ɛ P (8) with ɛ P representing the effective measuring noise consisting of remaining ICI noise and AWGN noise. Using the least squares estimation (LSE) the common phase rotation Φ 0Data is given by [2]: e jφ 0 Data E0Data = (H P S P ) Y P (9) with ( ) representing the Moore Penrose pseudo inverse. However, in a real system the preamble also suffers from PN distortion and thus the channel estimation is not perfect anymore. As given in the previous section a preamble based channel estimation results in an effective channel Ĥ P including the real channel and an average CPE term: Ĥ P = H P + HP. The received pilots in the data symbols based on the channel estimation can now be written as: Y P = H P S P E0 Data HP E 0Data S P +ɛ P (10) where the second and the third term are considered as noise. Again applying a LSE, a delta common phase rotation is obtained: e j( Φ 0 ) = e j(φ0 Data Φ0 P re ) E 0Data / = E 0. (11) Sink SISO Decoder Π Symbol LLR Π 1 Subcar. Selection Symbol Mapper Detect. LMMSE PN Cor. CPE Cor. OFDM demod.+ Channel Estimat. Fig. 2. OFDM transmission chain with channel estimation and iterative Phase Noise correction The key idea of this approach is to estimate higher order phase noise components. In order to obtain a Bayesian estimator for the spectral components E k up to a certain order u (e.g. k { u,, u}) we rewrite Eq. (3) in combination with the estimated channel transfer matrix in a vector matrix notation for a subset of B 2u + 1 equations (see [5] for details): Y = A E + ε (12) where A is a B 2u+1 matrix containing the products of Ĥ S. One should note that this is the crucial point since the LMMSE is based on the estimated channel transfer matrix. Hence, a non optimally estimated transfer matrix will lead to significant

performance degradation. Furthermore, E is a (2u+1) 1 PN E vector and ε = ζ ICI +η 0Data H E 0P S represents the effective re noise consisting of remaining ICI noise of the data symbols, the AWGN noise and the remaining channel estimation error. The LMMSE estimate of the vector E is given by Ê = M Y (13) with M = Φ EE A H (AΦ EE A H + Φ εε ) 1 and Φ EE and Φ εε representing correlation matrices of the vector of Fourier components E and the remaining noise terms ε, respectively. The evaluation of the correlation matrices Φ EE and Φ εε can be found in [5], [9] for the case of perfect channel state information at the receiver. In the case of non perfect information the remaining noise term correlation matrix has to be modified according to: Φ εε = Φ εε + Φ H S H S. Having obtained the coefficients Ê, the complex conjugate of the Fourier expansion can be used to correct the received signal samples in time domain: ŷ rx (n) = y rx (n)e j ˆφ(n). B. MIMO Detection The aim of the MIMO detector is to provide soft information to the outer decoder for each bit. This information is based on the modified received signal Ŷ l, the estimated channel state information and the a-priori knowledge (P [X tx,m ]) from the outer decoder. In order to increase readability we drop the subcarrier index l in the following. To lower the complexity we concentrate in this work on a linear MMSE equalization. The solution of the mean square error minimization problem leads to the filter matrix W W = (Φ 1 SS + ĤH Φ εε 1 Ĥ) 1 Ĥ H Φ εε 1, (14) with Φ SS = E[S H S]. The MMSE solution provides a tradeoff between noise amplification and interference reduction and results in a biased estimation. To overcome the bias problem the filter matrix W has to be modified by a diagonal matrix that restores unit ( gain, leading ) to a new filter matrix given as: W UB = diag diag 1 (W Ĥ) W. Using the so called max- Log approximation, the L-values from the detector can now be approximated by a difference of two maximum operation based on the filter output Ỹtx L(X tx,m Ŷ) max X X +1 max X X 1 j γtx σ 2 S tx Ỹ tx S tx 2 + ln M Y m=1 j γtx 2 Y M σs 2 Ỹ tx S tx + ln tx m=1 ff P [X tx,m] ff P [X tx,m ]. (15) Using the MMSE approach in a coded environment, the knowledge of the SINR at each antenna is of essential importance for the computation of the L-values. For each antenna the SINR is given as the diagonal elements of the error covariance matrix 1 γ tx = [(Φ SS + Ĥ H Φ 1 εε Ĥ) 1 Φ 1 SS ] 1. (16) tx,tx IV. NUMERICAL RESULTS The performance of the iterative phase noise correction in a MIMO-OFDM environment was tested by simulating a 4 4 MIMO IEEE 802.11a system with N C = 64 carriers, where 4 are reserved for pilots and 12 guard carriers, M = 4 (16- QAM). For the preamble the training sequence can occupy a maximum number of 52 subcarriers per transmit antenna. For the channel we simulated for each antenna link an uncorrelated realization of an HiperLAN A model. As channel code we used a rate 1/2 convolutional code with generator polynomial G = [133, 171] 8. Fig. 3(a) shows the performance results in terms of bit error rate () for perfect channel information at the receiver as well as using channel estimation based on different preamble length using Hadamard sequences as code orthogonal design. In the following we will use those results as an assessment criterion for the case if PN is also present. Fig. Perfect CIR 1 OFDM Symb. 2 OFDM Symb. 4 OFDM Symb. (a) CE no PN, different No. of OFDM Symbols in preamble Fig. 3. 1st Iter. / CPE 2nd Iter. / ICI 1 3rd Iter. / ICI 2 4th Iter. / ICI 3 (b) Iter. PN correction using perfect channel knowledge, δ 3dB = Performance results ( 4 4, MIMO 16-QAM) 3(b) shows the of iterative PN correction for an oscillator quality of δ 3dB = 0.01 under the assumption of perfect channel state information at the receiver. We have chosen a rather bad oscillator in order to show the improvement which one can get by performing iterative correction of higher order PN components. The remaining error floor is due to the remaining non estimated PN harmonics. In the following we will present results where perfect channel knowledge is not available at the receiver. The channel information is obtained by performing LS channel estimation techniques. The CIR used in Fig. 4 is obtained by 4 time orthogonal designed OFDM symbols. One can see a drastic performance degradation if PN is present in the channel estimation part. Even a simple CPE correction performs quite bad. This can be explained in the following way. During the channel estimation part each column in the Ĥ suffers from a different CPE which results in a significant distortion. Therefore, the CPE correction as presented in Sec. III is not valid anymore. Furthermore, one can see a performance loss over the iterations. This is due to the fact that the wrongly estimated channel directly affects the PN correction algorithm (Eq. (12)). Hence, a wrong PN correction is performed. In order to overcome

1st Iter./CPE 2nd Iter./ICI 1 3rd Iter./ICI 2 4th Iter./ICI 3 1st Iter. / CPE 2nd Iter. / ICI 1 3rd Iter. / ICI 2 4th Iter. / ICI 3 Fig. 4. performance, CIR based on 4 OFDM symbols in the preamble, time orthogonal design 30 Fig. 6. performance, CIR based on 2 OFDM symbols in the preamble, code orthogonal design this problem the phase offsets between the OFDMS symbols in the preamble have to be compensated before the channel estimation is performed. One approach which allows such a compensation is a orthogonal preamble design in frequency domain assuming that adjacent subcarriers are almost identical. The result of such a preamble design given in Fig. 5. One 1st Iter./CPE 2nd Iter./ICI 1 3rd Iter./ICI 2 4th Iter./ICI 3 30 Fig. 5. performance, CIR based on 4 OFDM symbols in the preamble, freq. orthogonal design can see a much better behavior and resistance of the channel estimation against PN. Looking at the behavior per iterations a performance improvement is visible. However, the curves are leveling out at high SNR. This is due to the remaining ICI in the estimated channel and the non estimated PN harmonics in the data symbols. Finally performance results based on 2 code orthogonally designed OFDM symbols in the preamble are shown in Fig. 6. Again a performance improvement per iteration is noticeable. Regarding the decay of the curves and the remaining error floors we draw the following conclusions. The decay of the curves in the low SNR region is mainly determined by the AWGN noise and the number of OFDM symbols in the preamble. Whereas the error floor is specified by the ICI in the preamble and the remaining ICI in the data part. Furthermore, we found out that an optimal preamble design is of essential importance as it also discussed in [3]. V. CONCLUSIONS In this work we discussed the performance of preamble based channel estimation techniques for MIMO-OFDM under the presence of phase noise. We extended the MSE approximation of the channel estimation by the phase noise statistics. Furthermore, we used the MSE in the phase noise mitigation and data detection step. As a more general outcome of this work, we showed that it is of essential importance to not estimate the columns of the channel matrix during different preamble symbols like it is performed in time orthogonal preamble design. This is due to the fact that each symbol will suffer from a different phase noise realization leading to a strong distortion of the estimated channel matrix. Finally we showed that code and frequency orthogonal preamble design lead to good results in terms of floor. REFERENCES [1] S. Wu, P Liu, and Y. Bar-Ness, Phase Noise Estimation and Mitigation for OFDM Systems, IEEE Trans. Wireless Commun., vol. 5, no. 12, pp. 3616 3625, 2006. [2] T. C. W. Schenk, X.-J. Tao, P. F. M. Smulders, and E. R. Fledderus, Influence and Suppression of Phase Noise in Multi-Antenna OFDM, in VTC, 2004. [3] H. Minn, N. Al-Dhahir, and Y. Li, Optimal Training Signals for MIMO OFDM Channel Estimation in the Presence of Frequency Offset and Phase Noise, IEEE Trans. Commun., vol. 54, no. 10, pp. 1754 1759, 2006. [4] D. Petrovic, W. Rave, and G. Fettweis, Phase Noise Suppression in OFDM Including Intercarrier Interference, in InOWo, 2003. [5] S. Bittner, W. Rave, and G. Fettweis, Phase Noise Suppression in OFDM with Spatial Multiplexing, in VTC, spring 2007. [6] T. C. W. Schenk, X.-J. Tao, P. F. M. Smulders, and E. R. Fledderus, On the Influence of Phase Noise Induced ICI in MIMO OFDM Systems, IEEE Commun. Lett., vol. 9, no. 8, pp. 682 684, 2005. [7] A. Demir, A. Mehrotra, and J. Roychowdhury, Phase Noise in Oscillators: A Unifying Theory and Numerical Methods for Characterisation, IEEE Trans. Circuits Syst. I, vol. 47, no. 5, pp. 655 674, 2000. [8] G. Auer, Analysis of pilot-symbol aided channel estimation for OFDM systems with multiple transmit antennas, in IEEE International Conference on Communications, June 2004, vol. 6, pp. 3221 3225. [9] D. Petrovic and G. Fettweis W. Rave, Intercarrier Interference due to Phase Noise in OFDM: Estimation and Suppression, in VTC, 2004. [10] T. L. Marzetta, Blast training: Estimating channel characteristics for high capacity space-time wireless, in 37th Allerton Conference on Communications, Control and Computing, Sep. 1999.