Intercarrier Interference due to Phase Noise in OFDM - Estimation and Suppression

Transcription

1 Intercarrier Interference 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 In this paper we provide an analysis of the intercarrier interference (ICI) due to phase noise in OFDM systems and present an algorithm for its suppression. We examine the general case where phase noise can take any values, thus the small phase noise model is dropped. The statistical properties of the intercarrier interference are analyzed, showing that the ICI is generally a non-gaussian random process which has a large impact on the system performance. Closed form expressions which describe the correlation properties of the constituents of ICI are calculated. An MMSE approach for suppressing ICI in the frequency domain is presented. This approach avoids error propagation to which our previously proposed algorithm was prone. The performance of the suppression algorithm is shown, pointing out the limits for the ICI suppression algorithms in general. I. INTRODUCTION OFDM has been applied in a variety of digital communications applications due to its robustness to frequency selective fading. However, OFDM is very sensitive to synchronization errors one of them being phase noise [1]. 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. There are two effects that occur if the phase noise is present in an OFDM system [2]: rotation of all demodulated subcarriers of an OFDM symbol by a common angle, called common phase error (CPE) and the occurrence of the intercarrier interference (ICI). The CPE results from the DC value of the phase noise and the ICI comes from the deviations of the phase noise during one OFDM symbol from its DC value. The problem of suppressing the phase noise in OFDM systems can be understood as getting as much information on the phase noise waveform as possible. Once one has this information one can use it to remove the effects the phase noise. The simplest approach would be to approximate the phase noise with a constant value i.e. its mean [3] [5]. More advanced approaches try to estimate higher spectral components to get a better approximation of phase noise thus reducing ICI [6] [7] [8]. Understanding ICI is a very important issue. Different models of phase noise lead to different models of the ICI that may not be realistic i.e. modelling of the phase noise as a colored gaussian noise. In this paper we use the Wiener process phase noise model which is found to be an appropriate description for the phase noise in oscillators [9]. We also drop the small phase noise model of [1]. Under this assumptions we analyze the properties of the ICI in Section III. In Section IV the algorithm based on MMSE is presented, which suppresses ICI in the frequency domain. At the end, the limits of the ICI suppression are discussed. II. SYSTEM MODEL Consider an OFDM transmission as shown in Fig. 1. Assuming perfect frequency and timing synchronization the received OFDM signal samples at the receiver side 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 N subcarriers. 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. The phase noise process φ(t) is modelled as a Wiener process [9] [10], with a certain 3dB bandwidth f 3dB. To characterize the quality of an oscillator in an OFDM system the relative phase noise bandwidth f 3dB,rel = f 3dB / f car is more practical measure, where the f car is the subcarrier spacing. The discrete time equation for the Wiener phase noise process equation can be written as [11] [9] φ(n + 1) = φ(n) + w(n) (1) where φ(n) denotes the phase noise process at sampling instant nt s, n Z and w(n) is a gaussian random variable w(n) N(0,4π f 3dB T s ). At the receiver after removing the X lk = N OFDM Modulator m, lk 0,1..., -1 IFFT CP LPF Rm lk = N, l 0,1, k Fig. 1. FFT x( n) = x( nt s ) CP f s r( n) = r( nt s ) OFDM Demodulator x( t) e Upconversion j2 fct e π j(2 π fct φ ( t )) [ ] ( ) j t r( t) = x( t) h( t) e φ Downconversion Block diagram of an OFDM transmission chain Channel cyclic prefix and taking the discrete Fourier transform (DFT)

2 on the remaining samples, the demodulated carrier amplitudes R m,lk at subcarrier l k (l k = 0,1,...N 1) of the m th OFDM symbol are given as [3]: R m,lk = X m,lk H m,lk I m (0) + X m,nh m,ni m (l k n) }{{} n=0 CPE n l k }{{} ICI (2) where X m,lk, H m,lk and η m,lk represent transmitted symbols on the subcarriers, the sampled channel transfer function at subcarrier frequencies and transformed white noise which remains AWGN. The terms I m (i) i = N/2,...,N/2 1 correspond to the DFT of one realization of e jφ(n) during one OFDM symbol: I m (i) = 1 N n=0 e j2πni/n e jφ(n) (3) In Eq. (2) the multiplicative distortion term I m (0) common to all subcarriers of one OFDM symbol, corresponds to the common phase error (CPE). In the sequel the properties of ICI and its impact on system performance will be analyzed. Throughout this work we assume wireless LAN system parameters based on the IEEE802.11a standard [12]. If not otherwise stated 64-QAM modulation and standard convolutional code of rate r = 1/2 are used. The channel model assumed is a passive two-path rayleigh channel model with an impulse response h(t) = q 1 + aq 2 δ(t τ), where q 1,q 2 CN(0,1), τ = T s and a is a constant which determines the depths of the channel fades. III. PROPERTIES OF THE INTERCARRIER INTERFERENCE The properties of the intercarrier interference term in Eq. (2) have been addressed in several publications [1] [13] [7]. The ICI term analysis in [1] has been performed assuming that the phase noise φ(t) is very small leading to a closed form expressions for the ICI power. In [13] the assumption of the small phase noise is dropped and the variance of the ICI term is calculated. Also, the variances of the DFT coefficients of the phase noise, precisely of e jφ(t), I m (i) i = N/2,...,N/2 1 are obtained. In addition, the system performance is analyzed in terms of the SNR degradation at the demodulator due to phase noise. In [7] the alternative method for obtaining the variance of I m (i) i = N/2,...,N/2 1 is presented. In our opinion for the analysis and performance prediction of the OFDM system in the presence of the phase noise e.g. bit error rate, not only the variance of the ICI terms is required but rather their statistical characterization. ICI is assumed so far to be gaussian distributed justifying this by the central limit theorem. This work is motivated by the observation that this does not hold and that it has implications on the system performance. To understand ICI one needs to characterize the coefficients I m (i) and interactions that arise when they are scaled with the channel coefficients and transmitted symbols and summed up as in Eq. (2). To this end we first derive a closed form +η m,lk expression for the cross-correlation matrix of the vector of DFT coefficients I m (i). Define a vector I m = [I m ( N/2)...I m (N/2 1)] T, as a vector of the DFT coefficients of one realization of e jφ(n) during one OFDM symbol. The correlation matrix of this vector is defined as R ImI m = E { I m Im} H where H stands for Hermitian operator. Using Eq. (3) the (n,p) th element of the correlation matrix R ImI m is calculated as: E{I m (n)im(p)} = { = 1 N 2 E = 1 N 2 k=0 l=0 k=0 e j(φ(k) φ(l)) e j 2π N (nk pl) } E { e j φ kl} e j 2π N (nk pl) (4) l=0 where φ kl denotes the cumulative phase noise increment between the l th and k th samples of the received signal. From Eq. (1) the increments of the phase noise from sample to sample are i.i.d. gaussian random variables of variance σw 2 = 4π f 3dB T s. φ kl, as a sum of gaussian i.i.d. random variables is also a gaussian random variable φ kl N(0, k l σw). 2 In order to evaluate Eq. (4) the expectation E { e kl} j φ has to be calculated for each k,l = 0,1,...N 1. This expression can be calculated using the definition of a characteristic function. The characteristic function of the random variable φ kl is defined as Φ kl (ω) = E { e kl} jω φ. It follows that E { } e j φ kl = Φkl (1) = e k l σ2 w 2 since φ kl is a gaussian random variable. Finally one obtains that: R ImI m (n,p) = E{I m (n)i m(p)} = = 1 N 2 k=0 l=0 e k l σ 2 w 2 e j 2π N (nk pl) = 1 F(n, p) (5) N2 where F(n, p) represents the two-dimensional discrete fourier transform [14] of e k l σ2 w 2. We have evaluated the matrix R ImI m for a relative phase noise bandwidth f 3dB,rel = which corresponds to the phase noise bandwidth f 3dB = 200Hz in a IEEE802.11a and the results are presented in Fig. 2. Several curves which present the crosscorrelation between some of the DFT coefficients of the phase noise are shown. It can be seen that the cross-correlation between the terms cannot be neglected when compared to the terms E { I m (i) 2}. Considering ICI term in Eq. (2), the correlation between the I m (i)s is destroyed due to the randomization by data and channel coefficients. The total ICI

3 also larger then σici 2 which is due to the correlation of I m(i). The variance of Z can be calculated as a sum of the elements of the matrix R ImI m excluding the correlation terms which incorporate I m (0). Fig. 4 shows the effect that the distribution N 1 m, n m, n m n= n 0` l X, X = X H I ( l n) ICI Y Y N σ ICI 2, (0, ) Fig. 2. Correlation between DFT components of the phase noise. = N / 2 1 Z, Z I ( i) m i= N / 2 power can then be easily calculated as [3]: σ 2 ICI = E{ = = ν 0 ν 0 ν 0 X m,q ν H m,q ν I m (ν) 2 } E{ X m,q ν 2 }E{ H m,q ν 2 }E{ I m (ν) 2 } E{ I m (ν) 2 } (6) which equals the sum of the diagonal elements of the correlation matrix R ImI m. It is assumed that E{ X m,q ν 2 } = 1 and E{ H m,q ν 2 } = 1. Using computer simulations we have observed that the ICI term (see Eq. (6)) is not gaussian distributed. We have performed extensive nonparametric hypothesis testing, Kolmogorov-Smirnov and Jarque-Bera tests [15], to test ICI term for gaussianity. We have tested the hypothesis that the ICI term in Eq. (2), denoted here as X, is complex gaussian variable X N(0,σICI 2 ) or equivalently that its amplitude X is Rayleigh distributed with the corresponding variance. First we note that the variance of X is equal to σici 2 of Eq. (6), which confirms the assumption that the correlation between the I m (i) coefficients is destroyed by multiplying them with random transmitted symbols and channel coefficients. The Fig. 3 plots the complementary cumulative probability density function CCDF of the ICI term amplitude obtained by simulating our benchmark system in the presence of the phase noise. The CCDF of X deviates from the expected Rayleigh distribution which is also plotted. The distribution of X has much broader tails. This points out that the distribution of X is not complex gaussian. Fig. 3 also shows the amplitude distribution of the plain sum of the I m (i) denoted by Z. The distribution deviates strongly from the Rayleigh distribution. The variance of Z is Fig. 3. ICI distribution. of ICI has on the OFDM system symbol error rate (SER). The benchmark system with phase noise is compared to the system where the ICI term is replaced with a gaussian random variable of variance σici 2 before the demodulator. Note that in terms of the SNR loss, this is a fair comparison. If the phase noise is present we assume ideal common phase error correction in all cases. The relative phase noise bandwidth is f 3dB,rel = For AWGN channel the performance of the system with real phase noise is much worse than for the gaussian noise model. For frequency selective channels we observe an opposite behavior. This is in our opinion explained by the fact, which distribution is advantageous for which types of channels. For AWGN the tails of the ICI distribution are dominant. For frequency selective channels the channel fades are dominant. IV. ICI CORRECTION ALGORITHM The phase noise suppression algorithm presented here, is a modified version of the ICI cancellation in the frequency domain which was presented in [7]. Here we avoid the part error propagation due to the falsely detected symbols, when the ICI is cancelled as it was the case in [7]. The estimation of the DFT coefficients I m (i) still suffers form the falsely detected symbols. Also, the full correlation matrix R JmJ m calculated in Section III is used, instead of using simplified assumption in [7] that the correlation matrix is diagonal. The idea of the algorithm is the following: The factors I m (i), i = N/2,...,N/2 1 represent the DFT coefficients (spectral components) of one realization of the random process e jφ(n). Further, e jφ(n) has the characteristics of a low-pass signal with power spectral density of the form 1/(1 + f 2 ), where f denotes the frequency [7] [9]. Due to the shape of the spectrum of e jφ(n), very few low pass spectral components

4 Fig. 4. Phase noise influence on system performance. The vector Ũm of estimated DFT coefficients of e jφ(n) can then be formed as described above. The demodulated symbols with suppressed intercarrier interference up to order u are obtained as : R m,n = R m,n Ũm (7) V. ESTIMATION OF SPECTRAL COMPONENTS I m (i) Define the set L = {l 1,l 2,l 3,...l k } as a subset of the set {0,1,...N 1}. Let further R m = [R m,l1,r m,l2,...r m,lk ] T be the vector of the received symbols at subset L of all subcarriers. Each of the components of the vector R m can be expressed as in Eq. (2) and thus R m can be expressed in a matrix form as given by R m = A m J m + ε m = a l1 a l2.. J m + ε m (8) will suffice to give a good approximation of the phase noise waveform [7]. The phase noise suppression in the time domain would be a logical approach. One should multiply the received signal r(n) = (x(n) h(n))e jφ(n) + ξ(n) with e jφ(n), in a real case only with its estimation. Multiplication in the time domain for discrete time systems is mapped to the circular convolution of DFT spectra in the frequency domain [14]. This means that the ICI cancellation for the m th OFDM symbol in the frequency domain can be done by circularly convolving the demodulated symbols vector of all subcarriers R m,n = [R m (0),...,R m (N 1)] T with the vector of the DFT coefficients of e jφ(n). Using properties of the DFT [14], if I m = DFT{e jφ(n) }, then the spectrum of the complex conjugate signal e jφ(n) reads as U m (i) = I m( i), i = N/2,...,N/2 1 where U m = DFT{e jφ(n) }. Therefore the ICI suppression in the frequency domain can be done by circularly convolving the vectors R m,n and U m. The task of the ICI suppression algorithm is to estimate the DFT components of the phase noise and suppress ICI by performing deconvolution in the frequency domain. The details of it are presented here: 1) Perform standard OFDM demodulation and obtain an estimate I m (0) of I m (0), using a Kalman filter [4] or some other method [3] [5] and derotate the demodulated signal constellation. 2) Using such a derotated constellation make a decision on the transmitted symbols and use these hard decisions for the estimation of the J m i.e.ĩm(i), i = u...u according to the method provided in Section V. The justification for using these decisions for I m (i) estimation is that even for very high phase noise bandwidth just a fraction of the decided symbols will be erroneous [7]. 3) The estimated DFT coefficients Ĩm(i), i = u...u comprise the vector Ĩm which denotes the vector of estimated DFT coefficients of the phase noise. All unestimated terms Ĩm(i), i > u are adopted to be zero.. H m,li+ux m,li+u a lk }{{} A m In Eq. (8) the vectors a li and J m are defined as: T H m,li X m,li H m,li 1X m,li 1 a li = H m,li+1x m,li+1 and J m = I m (0) I m (1). I m (u) I m ( u) (9) where the vector J m is the vector of spectral components, up to order u,u N/2, we wish to estimate. The term ε m = ζ ICI + η m is composed of the additive noise term η m = [η m,l1,η m,l2,...,η m,lk ] T and the vector ζ ICI = [ζ ICI,l 1 ζ ICI,l 2...ζ ICI,l k ] T which represents the residual ICI vector. The components of the vector ζ ICI, namely ζ ICI,l i,l i L, incorporate all the remainder addends in a sum of Eq. (2) for a corresponding li th carrier which are nor incorporated in the product a li J m. The system model represented by Eq. (8) is a linear model with respect to the vector J m and we choose the minimum mean square estimation (MMSE) to estimate this vector from the demodulated received symbols vector R m. The MMSE estimate of the vector J m is given by J m = MR m (10) where M = R JmJ m A H m(a m R JmJ m A H m + R εmε m ) 1, and R JmJ m and R εmε m represent correlation matrices of J m and ε m respectively. Note that the estimation of J m assumes the knowledge of one part of the transmitted symbols apart from the channel knowledge. These symbols are obtained in a decision feedback (DF) manner described in Section IV. The correlation matrix R JmJ m can be obtained form R ImI m by selecting the required columns and rows of R ImI m, which was calculated in Section III.

5 Further we have R εmε m = diag(e{ ζ ICI (l 1) 2 } + σn,...,e{ ζ 2 ICI (l k) 2 }+σn) 2 where ζ ICI (q) denote the residual ICI power as discussed above. Using Eqs. (2) and (8) ζ ICI (q), q L can be expressed as ζ ICI(q) = ν >u X m,q ν H m,q ν I m (ν) (11) The power of the ζ ICI (q) is calculated similarly as in Eq. (6) as E{ ζ ICI(q) 2 } = ν> u E{ I m (ν) 2 }. VI. NUMERICAL RESULTS We address the following question: What are the limiting factors in suppressing the phase noise? Fig. 5 shows the performance of the OFDM system in the presence of the phase noise. The channel profile with a = 0.8 is adopted. Four scenarios are compared: 1) without phase noise 2) with phase noise and genie CPE correction 3) with phase noise and genie ICI correction of different orders 4) with phase noise and ICI correction of different orders using the proposed algorithm, assuming that for the CPE correction a Kalman filter is used [4]. As opposed to the expectations in [7] even very good approximation of the phase (20 order) is not enough for significant performance improvement for frequency selective rayleigh fading channels. Consider Eq. (2) which can be visualized as in Fig. 6. As it is noticed in [7], in AWGN Fig. 5. ICI suppression performance. channels, most of the interference comes from the neighboring subcarriers. Due to the Lorentzian spectrum of the phase noise most of the energy of the phase noise is concentrated, in the spectral components I m (i) around zero frequency (see Fig. 2). In the frequency selective channels however the channel premultiplies each of the I m (i) i 0. That means that the channel is shaping the spectrum of the phase noise and thus significant ICI term can come from very distant subcarriers. Therefore in general a very small components I m (i) of the phase noise can have very significant contribution to ICI due to channel. This explains why one needs very high spectral components of the phase noise to get significant performance improvement. We want to note that the estimation of the phase noise waveform we obtain is significantly better than its DC. The conclusion is that the performance of one ICI suppression algorithms depends on the both phase noise characteristics and the channel profile. X I (2) m, l 2 Hm, l 2 m X m, l+ 2 Hm, l+ 2 I m( 2) X m, l uh m, l u I m( u) I (1) I m (0) X m, l 1 H m, l 1 m X I ( 1) m, l + 1 H m, l + 1 m X m, l+ uh m, l+ u I m( u) l 2 l 1 l l + 1 l + 2 Frequency index Fig. 6. ICI explanation. ACKNOWLEDGMENT This work was supported by the German ministry of research and education within the project (WIGWAM) under grant 01 BU 370 REFERENCES [1] E. Costa and S. Pupolin, M-QAM-OFDM System Performance in the Presence of a Nonlinear Amplifier and Phase Noise, IEEE Trans. Commun., vol. 50, no. 3, Mar [2] A. Armada, Understanding the Effects of Phase Noise in Orthogonal Frequency Division Multiplexing (OFDM), IEEE Trans. on Broadcasting, vol. 47, no. 2, June [3] S. Wu and Y. Bar-Ness, A Phase Noise Suppression Algorithm for OFDM-Based WLANs, IEEE Communications Letters, vol. 44, no. 3, May [4] D. Petrovic, W. Rave, and G. Fettweis, Common Phase Error due to Phase Noise in OFDM - Estimation and Suppression, in Proc. PIMRC, [5] P. Robertson and S. Kaiser, Analysis of the effects of phase noise in OFDM systems, in Proc. ICC, [6] R. A. Casas, S. Biracree, and A. Youtz, Time Domain Phase Noise Correction for OFDM Signals, IEEE Trans. on Broadcasting, vol. 48, no. 3, Sept [7] D. Petrovic, W. Rave, and G. Fettweis, Phase Noise Suppression in OFDM including Intercarrier Interference, in Proc. Intl. OFDM Workshop (InOWo)03, 2003, pp [8] S. Wu and Y. Bar-Ness, A New Phase Noise Mitigation Method in OFDM Systems with Simultaneous CPE and ICI Correction, in Proc. MCSS, Germany, Sep [9] 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, May [10] D. Petrovic, W. Rave, and G. Fettweis, Phase Noise Suppression in OFDM using a Kalman Filter, in Proc. WPMC, [11] D. J. Higham, An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations, SIAM Review, vol. 43, no. 3, pp , [12] IEEE, Part11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications. High-speed Physical Layer in the 5GHz Band, IEEE Std a-1999, [13] S.Wu and Y.Bar-ness, Performance Analysis of the Effect of Phase Noise in OFDM Systems, in IEEE 7 th ISSSTA, 2002.

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