COMMON PHASE ERROR DUE TO PHASE NOISE IN OFDM - ESTIMATION AND SUPPRESSION

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1 COMMON PHASE ERROR DUE TO PHASE NOISE IN OFDM - ESTIMATION AND SUPPRESSION Denis Petrovic, Wolfgang Rave and Gerhard Fettweis Vodafone Chair for Mobile Communications, Dresden University of Technology, Helmholtzstrasse 18, Dresden, Germany {petrovic,rave,fettweis}@ifn.et.tu-dresden.de Abstract - Orthogonal frequency division multiplexing (OFDM) has already become a very attractive modulation scheme for many applications. Unfortunately OFDM is very sensitive to synchronization errors, one of them being phase noise, which is of great importance in modern WLAN systems which target high data rates and tend to use higher frequency bands because of the spectrum availability. In this paper we propose a linear Kalman filter as a means for tracking phase noise and its suppression. The algorithm is pilot based. The performance of the proposed method is investigated and compared with the performance of other known algorithms. Keywords - OFDM, Synchronization, Phase noise, WLAN I. INTRODUCTION OFDM has been applied in a variety of digital communications applications. It has been deployed in both wired systems (xdsl) and wireless LANs (IEEE802.11a). This is mainly due to the robustness to frequency selective fading. The basic principle of OFDM is to split a high data rate data stream into a number of lower rate streams which are transmitted simultaneously over a number of orthogonal subcarriers. However this most valuable feature, namely orthogonality between the carriers, is threatened by the presence of phase noise in oscillators. This is especially the case, if bandwidth efficient higher order modulations need to be employed or if the spacing between the carriers is to be reduced. To compensate for phase noise several methods have been proposed. These can be divided into time domain [1][2] and frequency domain approaches [3][4][5]. In this paper we propose an algorithm for tracking the average phase noise offset also known as the common phase error (CPE)[6] in the frequency domain using a linear Kalman filter. Note that CPE estimation should be considered as a first step within more sophisticated algorithms for phase noise suppression [5] which attempt to suppress also the intercarrier interference (ICI) due to phase noise. CPE compensation only, can however suffice for some system design scenarios to suppress phase noise to a satisfactory level. For these two reasons we consider CPE estimation as an important step for phase noise suppression. II. SYSTEM MODEL An OFDM transmission system in the presence of phase noise is shown in Fig. 1. Since all phase noise sources can be mapped to the receiver side [7] we assume, without loss of generality that phase noise is present only at the front end of the receiver. Assuming perfect frequency and timing synchronization the received OFDM signal samples, sampled at frequency f s, in the presence of phase noise can be expressed as r(n) = (x(n) h(n))e jφ(n) + ξ(n). Each OFDM symbol is assumed to consist of a cyclic prefix of length N CP samples and N samples corresponding to the useful signal. The variables x(n), h(n) and φ(n) denote the samples of the transmitted signal, the channel impulse response and the phase noise process at the output of the mixer, respectively. The symbol stands for convolution. The term ξ(n) represents AWGN noise with variance σn. 2 The phase noise process φ(t) is modelled as a Wiener process [8], the details of which are given below, with a certain 3dB bandwidth f 3dB. OFDM Modulator X m, l l = 0,1,2... N -1 IFFT CP LPF Rm, l l = 0,1,2... N -1 FFT x( n) = x( nt s ) CP f s r( n) = r( nt s ) OFDM Demodulator x( t) e Upconversion e j2π fct ( j2 π fct φ ( t)) j ( t ) [ ] r( t) = x( t) h( t) e φ Downconversion Fig. 1 Block diagram of an OFDM transmission chain. Channel At the receiver after removing the N CP samples corresponding to the cyclic prefix and taking the discrete Fourier transform (DFT) on the remaining N samples, the demodulated carrier amplitude R m,lk at subcarrier l k (l k = 0, 1,...N 1) of the m th OFDM symbol is given as [4]: R m,lk = X m,lk H m,lk I m (0) + ζ m,lk + η m,lk (1) where X m,lk, H m,lk and η m,lk represent the transmitted symbol on subcarrier l k, the channel transfer function and

2 linearly transformed AWGN with unchanged variance σn 2 at subcarrier l k, respectively. The term ζ m,lk represents intercarrier interference (ICI) due to phase noise and was shown to be a gaussian distributed, zero mean, random variable with variance σici 2 = πn f 3dB 3f s [7]. The term I m (0) also stems from phase noise. It does not depend on the subcarrier index and modifies all subcarriers of one OFDM symbol in the same manner. As its modulus is in addition very close to one [9], it can be seen as a symbol rotation in the complex plane. Thus it is referred to in the literature as the common phase error (CPE) [6]. The constellation rotation due to CPE causes unacceptable system performance [7]. Acceptable performance can be achieved if one estimates I m (0) or its argument and compensates the effect of the CPE by derotating the received subcarrier symbols in the frequency domain (see Eq. (1)), which significantly reduces the error rate as compared to the case where no compensation is used. The problem of estimating the CPE was addressed by several authors [3][4][10]. In [3] the authors concentrated on estimating the argument of I m (0) using a simple averaging over pilots. In [10] the argument of I m (0) was estimated using an extended Kalman filter, while in [4] the coefficient I m (0) itself was estimated using the LS algorithm. Here we introduce an alternative way for minimum mean square estimation (MMSE)[11] of I m (0) using a linear scalar Kalman filter. The algorithm is as [4] pilot based. III. PHASE NOISE MODEL For our purposes we need to consider a discretized phase noise model φ(n) = φ(nt s ) where n N 0 and T s = 1/f s is the sampling period at the front end of the receiver. We adopt a Brownian motion model of the phase noise [8]. The samples of the phase noise process are given as φ(n) = 2πf c cb(n) where fc is the carrier frequency, c = f 3dB /πfc 2 [8] and B(n) represents the discretizied Brownian motion process, Using properties of the Brownian motion [12] the following holds: B(0) = 0 and B(n + 1) = B(n) + db n, n N 0 where each increment db n is an independent random variable and db n T s N (0, 1). Noting that φ(n) = 2πf c cb(n) we can write the discrete time phase noise process equation as φ(n + 1) = φ(n) + w(n) (2) where w(n) N (0, 4π 2 f 2 c ct s ) is a gaussian random variable with zero mean and variance σ 2 w = 4π 2 f 2 c ct s. IV. CPE ESTIMATION USING A KALMAN FILTER Since all received subcarriers within one OFDM symbol are affected by the same factor, namely I m (0), the problem at hand can be seen as an example of estimating a constant from several noisy measurements given by Eq. (1) for which purpose a Kalman filter is well suited [11]. For a Kalman filter to be used we need to define the state space model of the system. Define first the set L = {l 1, l 2, l 3,...l P } as a subset of the subcarrier set {0, 1,...N 1}. Using Eq. (1) one can write R m,lk = A m,lk I m,lk (0) + ε m,lk (3) where A m,lk = X m,lk H m,lk and I m,lk (0) = I m (0) for all k = 1, 2..., P. Additional indexing of the CPE terms is done here only for convenience of notation. On the other hand one can write I m,lk+1 (0) = I m,lk (0). (4) Equations (3) and (4) are the measurement and process equation of the system state space model, where A m,lk represents the measurement matrix, while the process matrix is equal to 1 and I m,lk (0) corresponds to the state of the system. The measuring noise is given by ε m,lk which combines the ICI and AWGN terms in Eq. (1), the variance of which for all l k equals σε 2 = (σici 2 + σ2 n). The process noise equals zero. Note that the defined state space model is valid only for one OFDM symbol. For the state space model to be fully defined, knowledge of the A m,lk = X m,lk H m,lk is needed. Here we assume to have ideal knowledge of the channel. On the other hand we define the subset L to correspond to the pilot subcarrier locations within one OFDM symbol so that X m,q, q L are also known. We assume that at the beginning of each burst perfect timing and frequency synchronization is achieved, so that the phase error at the beginning of the burst equals zero. After the burst reception and demodulation, the demodulated symbols are one by one passed to the Kalman filter. For a Kalman filter initialization one needs for each OFDM symbol an a priori value for Îm,l 1 (0) and an a priori error variance Km,1. At the beginning of the burst, when m = 1, it is reasonable to adopt Î 1,l 1 (0) = 1. Within each OFDM symbol, say m th, the filter uses P received pilot subcarriers to recursively update the a priori value Î 1 m,l 1 (0). After all P pilot subcarriers are taken into account Î m,lp (0) is obtained, which is adopted as an estimate of the CPE within one OFDM symbol, denoted as Îm(0). The Kalman filter also provides an error variance of the estimate of I m,lp (0) as K m,p. Îm,l P (0) and K m,p are then used as a priori measures for the next OFDM symbol. The detailed structure of the algorithm is as follows. Step 1: Initialization Î m,l 1 (0) = E{I m,l 1 (0)} = Îm 1(0) Km,1 = E{ I m(0) Îm 1(0) 2 } = E{ φ m ˆφ m 1 2 } = σcp 2 E + K m 1,P [ ] where σcp 2 E = 4π 2N 2 +1 f 3N + N 3dB CP f s (see [10]), K 0,P = 0 and φ m = arg{i m (0)}.

3 Repeat Step 2 and Step 3 for k = 1, 2,..., P Step 2: a-posteriori estimation (update) G m,k = K m,k HH m,l k H m,lk K m,k HH m,l k + (σ 2 ICI + σ2 n) Î m,lk (0) = Î m,l k (0) + G m,k [R m,lk H m,lk Î m,l k (0)] K m,k = (1 G m,k H m,lk )K m,k Step 3: State and error variance propagation K m,k+1 = K m,k (5) Î m,l k +1 (0) = Îm,l k (0) Note that no matrix inversions are required, since the state space model is purely scalar. V. CPE CORRECTION The easiest approach for CPE correction is to derotate all subcarriers l k of the received m th symbol R m,lk by φ m = arg{îm(0)}. Unambiguity of the arg{ } function plays here no role since any unambiguity which is a multiple of 2π rotates the constellation to its equivalent position in terms of its argument. The presented Kalman filter estimation algorithm is readily applicable for the decision feedback (DF) type of algorithm presented in [4]. The idea there was to use the data symbols demodulated after the first CPE correction in a DFE manner to improve the quality of the estimate since that is increasing the number of observations of the quantity we want to estimate. In our case that would mean that after the first CPE correction the set L = {l 1, l 2, l 3,...l P } of the subcarriers used for CPE estimation, which previously corresponded to pilot subcarriers, is now extended to a larger set corresponding to all or some of the demodulated symbols. In this paper we have extended the set to all demodulated symbols. The Kalman filter estimation is then applied in an unchanged form for a larger set L. VI. NUMERICAL RESULTS The performance of the proposed algorithm is investigated and compared with the proposal of [4] which is shown to outperform other known approaches. The system model is according to the IEEE802.11a standard, where modulation is used. We investigate the performance in AWGN channels and frequency selective channels using as an example the ETSI HiperLAN A-Channel (ETSI A). Transmission of 10 OFDM symbols per burst is assumed. A. Properties of an Estimator The quality of an estimation is investigated in terms of the mean square error (MSE) of the estimator for a range of phase noise bandwidths f 3dB [10 800]Hz. Table 1 can be used to relate the phase noise bandwidth with other quantities. Figures 2 and 3 compare the MSE of the LS estimator from [4] and our approach for two channel types and both standard correction and using decision feedback. Note that SNRs are chosen such that the BER of a coded system after the Viterbi algorithm in case of phase noise free transmission is around Kalman filter shows better performance in all cases and seems to be more effective for small phase noise bandwidths. As expected when DF is used the MSE of an estimator is smaller because we are taking more measurements into account. MSE MSE SNR = 22dB AWGN KF Wu et al. w/o DF with DF Fig. 2 MSE of an estimator for AWGN channel. SNR = 36dB Frequecy selective channel: ETSI A Channel KF Wu et al. w/o DF with DF Fig. 3 MSE of an estimator for ETSI A channel.

4 Table 1 Useful relations Quantity Symbol Relation Typical values for IEEE802.11a 1 Oscillator constant c[ radhz ] Oscillator 3dB bandwidth f 3dB [Hz] f 3dB = πfc 2 c f Relative 3dB bandwidth 3dB f 3dB N f car f s Phase noise energy E PN [rad] E PN = 4π f 3dB f car Subcarrier spacing f car f car = fs N Hz B. Symbol Error Rate Degradation Symbol error rate (SER) degradation due to phase noise is investigated also for a range of phase noise bandwidths f 3dB [10 800]Hz and compared for different correction algorithms. Ideal CPE correction corresponds to the case when genie CPE values are available. In all cases simple constellation derotation with φ = arg {Îm(0)} is used. SER 10-1 SNR = 36dB Frequecy selective channel: ETSI A Channel Ideal CPE cor. SER 10-1 SNR = 22dB AWGN KF w/o DF KF with DF Wu et al. w/o DF Wu et al. with DF Ideal CPE cor. no phase noise Fig. 4 SER degradation for AWGN channel. In Figs. 4 and 5 SER degradation for AWGN and ETSI A channels is plotted, respectively. It is interesting to note that as opposed to the ETSI A channel case in AWGN channel there is a gap between the ideal CPE and both correction approaches. This can be explained if we go back to Eq. (1) where we have seen that phase noise affects the constellation as additive noise. Estimation error of phase noise affects the constellation also in an additive manner. On the other hand the SER curve without phase noise in the AWGN case is much steeper than the corresponding one for the ETSI A channel. A small SNR degradation due to estimation errors will cause therefore large SER variations. This explains why the performance differs much less in the ETSI A channel case. Generally from this discussion a conclusion can be drawn that systems with large order of diversity are more sensitive to CPE estimation errors. Note that this is meant KF w/o DF KF with DF Wu et al. w/o DF Wu et al. with DF no phase noise Fig. 5 SER degradation for ETSI A channel. not in terms of frequency diversity but the SER vs. SNR having closely exponential dependence. It can be seen that our approach shows slightly better performance than [4] especially for small phase noise bandwidths. What is also interesting to note is, that DF is not necessary in the case of ETSI A types of channels (small slope of SER vs. SNR) while in case of AWGN (large slope) it brings performance improvement. VII. CONCLUSIONS We investigated the application of a linear Kalman filter as a means for tracking phase noise and its suppression. The proposed algorithm is of low complexity and its performance was studied in terms of the mean square error (MSE) of an estimator and SER degradation. The performance of an algorithm is compared with other algorithms showing equivalent and in some cases better performance. REFERENCES [1] R. A. Casas, S. Biracree, and A. Youtz, Time Domain Phase Noise Correction for OFDM Signals, IEEE Trans. on Broadcasting, vol. 48, no. 3, 2002.

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