Limits Of Phase Noise Suppression In Ofdm

Limits of Phase Noise Suppression in OFDM Denis Petrovic, Wolfgang Rave and Gerhard Fettweis Vodafone Chair Mobile Communications Systems, Dresden University of Technology, Dresden, Germany. e-mail: {petrovic, rave, fettweis}@ifn.et.tu-dresden.de. Abstract: We introduce two independent approaches for phase noise suppression. The dominant effects, responsible for the degradation of an OFDM system performance, if phase noise is present, are identified. We found out, that the system performance is strongly influenced by certain phase noise realizations, which cause burst errors, resulting in the performance error floor. Consideration of the system capacity in the presence of phase noise, motivates the idea, that bit interleaving can significantly improve the system performance. For systems where interleaving delay is a critical issue, we propose, as a second approach, an iterative phase noise suppression algorithm. Simulation results in terms of packet error rate (PER) show, that both bit interleaving and iterative algorithm are capable to significantly suppress the phase noise.

1.

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 [4]. Phase noise reflects imperfections of the local oscillator (LO), i.e. random drift of the LO phase from its reference. There are two effects that occur if the phase noise is present in an OFDM system [1]: 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 from its DC value, during one OFDM symbol. The problem of suppressing 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, it can be used 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 [13, 17, 19]. More advanced approaches try to estimate higher spectral components to get better approximation of phase noise waveform, thus reducing ICI [3, 11, 20]. Suppressing ICI is of large importance, especially if bandwidth efficient higher order modulations need to be employed or if the spacing between the carriers is to be reduced. In this paper we concentrate only on the effects of ICI on OFDM transmission. CPE is already corrected for in wireless standards using pilots [7]. Understanding ICI is a very important issue, because it can lead researchers to new algorithms for its suppression. Here we focus on identifying the dominant effects, that are responsible for the performance degradation of an OFDM system, if the phase noise is present. We have found out, that the system performance is strongly influenced by certain phase noise realizations, which cause

burst like errors, resulting in the performance error floor. Consideration of the system capacity in the presence of the phase noise in Sec. 3., motivates the idea, that bit interleaving can significantly improve the system performance. For systems where interleaving delay is a critical issue, we propose an iterative algorithm for phase noise suppression. We presented the idea to iteratively suppress the phase noise in [16]. In Sec. 4. one realization of this idea is presented and in Section 5. the algorithm performance is verified by numerical simulations.

2. System Model Consider an OFDM transmission over N subcarriers, as shown in Fig. 1. For simplicity, we assume the direct conversion approach where both up- and downconversion are done in one step [8]. In the case of perfect frequency and timing synchronization the received OFDM signal samples in the presence of phase noise can be expressed as r(n) = (x(n) ⋆ h(n))ejφ(n) + ξ(n). 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 [5] [12], with a certain 3dB bandwidth ∆f3dB . To characterize the quality of an oscillator in an OFDM system the relative phase noise bandwidth δP N = ∆f3dB /∆fcar is used, where the ∆fcar is the subcarrier spacing. The reason for this is that δP N parameter incorporates both the phase noise and system parameters. Since we use a discrete time model, we need the discrete time model of the phase noise. The discrete time equation for the Wiener phase noise process can be written as [6] [5]: φ(n + 1) = φ(n) + w(n)

(1)

where φ(n) denotes the phase noise process at sampling instant nTs at the receiver, n ∈ Z and w(n) is a Gaussian random variable w(n) ∼ N (0, 4π∆f3dB Ts ). The result after the discrete Fourier transform (DFT) at the receiver can be obtained by the following reasoning. Phase noise affects the received signal as an angular multiplicative distortion. Multiplication of two signals in the time domain is equivalent to convolving the spectra of the corresponding signals in the frequency domain. To be precise, since the discrete signals are considered here, in the frequency domain (discrete fourier transform domain) the spectra of two signals are circularly convolved [9]. Therefore, at the receiver, after removing the cyclic prefix and taking the DFT on the remaining samples, the demodulated carrier amplitudes Rm,s at subcar-

X m ,s s = 0,1, 2...N -1

Upconversion

OFDM Modulator IFFT

CP

LPF

x(t )

x(n) = x(nTs )

e j 2π f ct

e

fs

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FFT

− ( j 2π f c t −φ ( t ))

Channel

CP

r (n) = r (nTs ) OFDM Demodulator

r (t ) = [ x(t ) ∗ h(t ) ] e jφ (t ) Downconversion

Figure 1: OFDM Transmission in the presence of phase noise.

demodulator block at the receiver, i.e. OFDM modulation/demodulation, channel effects and phase noise. Capacity is chosen as the information theoretic parameter, to describe the system performance. Capacity is the ultimate limit for the data rate which can be achieved in a system. Consider the memoryless discrete input and continuous output channel, with input x from alphabet X , output y and transition distribution pY |X (y|x). Then the capacity under uniform inputs constraints and perfect channel state information is given by [2]: "

C = b − Ex,y log2 rier s (s = 0, 1, ...N − 1) of the m given as:

th

OFDM symbol are

Rm,s = Xm,s Hm,s Im (0) + | {z } CP E

N −1 X

Xm,v Hm,v Im (s − v) +ηm,s

(2)

v=0 v6=s

|

{z

ICI

}

where Xm,s , Hm,s and ηm,s represent transmitted symbols on the subcarriers, the sampled channel transfer function at subcarrier frequencies and transformed white noise which remains AWGN. The terms Im (i), i = −N/2, ..., N/2 − 1 correspond to the DFT of the realization of ejφ(n) during one OFDM symbol: Im (i) =

N −1 1 X −j2πni/N jφ(n) e e N n=0

(3)

In Eq. (2) the multiplicative distortion term Im (0) common to all subcarriers of one OFDM symbol, corresponds to the common phase error (CPE). The CPE equals the DC value of the phase noise and must be corrected for to obtain acceptable performance. The intercarrier interference (ICI) part is the additional error term caused by non-zero frequency components of the phase noise process. It is a mixture of channel transfer function coefficients, transmitted symbols and phase noise terms. It is found that the ICI term is non-Gaussian distributed 2 . Interrandom variable [10] [14] [15] of power σICI 2 carrier interference power σICI can be calculated in the closed form, using several different approaches [11] [14] [15] [18].

3.

Capacity of an OFDM System with Phase Noise

An equivalent representation of the coded OFDM transmission system, according to the IEEE802.11a standard [7], is shown in Fig. 2. The building blocks of this scheme are encoder, interleaver (π), symbol mapper, channel with transition probability density function (pdf) pY |X (y|x), demodulator (branch metric computer), branch metric deinterleaver (π −1 ) and decoder. Encoder, interleaver and mapper are the building blocks of the bit interleaved coded modulation (BICM) [2]. In our case pY |X (y|x) describes all effects of an equivalent ”channel” between mapper at the transmitter and the

P

x∈X

pY |X (y|x)

pY |X (y|x)

#

(4)

where b is the number of bits, transmitted over QAM symbol, which is mapped to one subcarrier. Increasing b, increases maximum throughput of the system. New wireless standards consider using more and more signal points in a signal constellation. Assuming that BICM is used [2], we have determined the capacity of an OFDM transmission over an AWGN channel with added phase noise. For calculating the capacity, the equivalent channel transition (pdf) pY |X (y|x) is required. We resorted to a Monte Carlo method to obtain this pdf and we have numerically calculated the capacity. Since we are interested only in the effects of ICI on OFDM transmission, we assume that CPE is ideally corrected for. In Fig. 3 the capacity for different MQAM OFDM signaling schemes over an AWGN channel as a function of the relative phase noise bandwidth δP N is plotted (the corresponding curves are denoted as: with real phase noise). This figure can be assumed as a limit since the SN R for which it is simulated is very large, namely SN R = 30 dB. From the capacity curves we see that the phase noise has a much stronger influence on the capacity of higher order constellations, as this dependence gets much steeper near the origin with an increase of the signal points. Even for very small δP N it is impossible to transmit 8 bits over the channel. Further, we compare this benchmark system with ”real” phase noise and ideal CPE correction, with the system, where the ICI term after DFT is replaced with a 2 Gaussian random variable of variance σICI before the demodulator. Note that in the latter case we use the standard formula for the calculation of the capacity over gaussian channels. Fig. 3 shows that the performance of the system with ”real” phase noise is different from that, where ICI is assumed to be gaussian distributed. This is to be expected considering the discussions in [10] [15] [14] which point out that the ICI is non-gaussian distributed. Indeed, if the ”real” phase noise is present, the capacity curves start to fall off earlier than in the case where ICI is assumed to be gaussian. However, one should notice that the differences of the capacity curves are not large, which motivates the discussion of the next section. 3.1. Phase noise ”suppression” using Interleaving In [14] the distribution of ICI and its influence on the symbol error rate has been investigated. It was con-

π

Encoder

Mapper

Demodulator

pY|X(y|x)

π -1

Decoder

Figure 2: Equivalent OFDM transmission block diagram.

8

64 QAM OFDM AWGN Channel

with real phase noise ICI gaussian approximation

256 QAM OFDM

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ICI modelled as Gaussian r.v. and variance σ 2ICI

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Figure 3: Phase noise influence on system performance: Capacity of the system with phase noise is compared to the capacity of the system where ICI is assumed to be gaussian distributed

cluded that the distribution of ICI has significant influence on the symbol error rate (SER) of the system. The statistics of the phase noise characterize this random process on the long term. It would be interesting at this point to concentrate on the specific realizations of the process, i.e. realizations of the phase noise during one OFDM symbol and relate it with ICI, within one OFDM symbol interval. If the phase noise does not change within one OFDM symbol, then the ICI term is zero. The more the phase noise changes the larger is the ICI. Normally phase noise oscillates around zero. However, within some OFDM symbols the phase noise can change dramatically (almost linearly) in one direction and in this case ICI is very large. These events spread the distribution of ICI, producing tails, which are much more pronounced than by Gaussian distribution [14]. The occurrence of such phase noise realizations cause burst errors, which degrade the performance of the system. Coming back to the discussion on the capacity of an OFDM system in the presence of phase noise, we note, that the expression for capacity evaluation Eq. (4) assumes ideal (infinite) interleaving. Therefore the capacity curves give us the result, which seems to be reasonable, namely: If one uses ideal interleaving, then the capacity of the system with phase noise is close to the capacity, where ICI is assumed to be gaussian distributed, and this means larger capacity. It follows, that one simple way to improve the performance of the OFDM system in the presence of phase noise is to increase the interleaving depth. 3.2. Numerical Experiment We have examined this expectation by using computer simulations. System parameters used throughout this pa-

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5 10

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10 15 E b/N0 [dB]

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Figure 4: Effect of the interleaving on the OFDM transmission in the presence of phase noise for AWGN channels per correspond to the IEEE802.11a standard. We use 64QAM modulation, standard convolutional code with rate r = 1/2 and random interleaving. One transmission block consists of 10 OFDM symbols which comprise one code word. Number of 10000 packets is transmitted, to assure valid statistics. Hard decision Viterbi decoder is used at the receiver. Fig. 4 shows the packet error rate (PER) dependence on the interleaving depth, for coded OFDM transmission over AWGN channels and phase noise with δP N = 2 · 10−3 . The interleaving depth is varied, i.e. it amounts 1, 5 and 10 OFDM symbols, within one code word. The performance of the system changes drastically as the interleaving depth varies. The reason for that is that the bad events, i.e. symbols which suffer from large ICI, are spread over a code word and the decoder tackles better with these events. The limit is the curve, where ICI is 2 assumed to be gaussian with variance σICI . For AWGN channels without phase noise, and for the case, where ICI is assumed Gaussian, interleaving depth does not play any role. Thus one approach for phase noise ”suppression” is to increase the interleaving depth. Capacity curves justify this approach as they show that OFDM transmission is not much sensitive to phase noise. Thus bit interleaving can be considered as a means for improving performance if the delay due to interleaving is not critical and the code words are not to short. Even though, this sounds to be reasonable approach to easily suppress the phase noise, to our knowledge it has not been considered in the literature as a means for the phase noise suppression. If the interleaving depth cannot be changed, then the phase noise suppression in standard systems is limited to a CPE correction, which is estimated using pilots.

In the next section, we review an approach for phase noise suppression we presented in [11], and present a new iterative algorithm for phase noise suppression based on this approach.

4.

Phase Noise Approximation and Correction

4.1. ICI Correction - Idea A phase noise compensation beyond the simple CPE correction will be possible only if one knows instantaneous realization of the phase noise process. The already introduced factors Im (i), i = −N/2, ..., N/2 − 1 (see Eq. (3)) represent the DFT coefficients (spectral components) of one realization of the random process ejφ(n) . The more spectral components Im (i) of the signal are known, the more is known of the signal waveform ejφ(n) , and thus φ(n). The signal ejφ(n) has the characteristics of a low-pass signal [5], with power spectral density of the form 1/(1 + f 2 ), where f denotes the frequency. Additionally, phase noise has a very small bandwidth compared with the subcarrier spacing. Due to the shape of the spectrum of ejφ(n) , very few low pass spectral components will suffice to give a ”good” approximation of the phase noise waveform. This is illustrated by the example in Fig. 5, where it can be seen that already second order approximation gives much better phase noise approximation than only DC value. Therefore knowledge of the coefficients Im (i) gives the possibility to approximate the phase noise waveform to a higher order, and allows a better compensation of it than with only CPE correction, i.e. 0th order approximation.

0.14 Phase noise trajectory (δ

angle [rad]

0.12

=1⋅ 10-3)

PN

0-th Order approx. (CPE) Im(0)

0.1 0.08

1st Order approx. Im(0), Im(1), Im(-1)

0.06

2nd Order approx. Im(0), Im(1), Im(-1),Im(2), Im(-2)

0.04

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samples

Figure 5: Phase noise waveform approximation using various orders of approximation.

4.2. ICI Correction Algorithm The details of an ICI suppression algorithm can be found in our previous work [11] [14]. Proposed ICI suppression algorithm estimates as many spectral components Im (i),i = −N/2, ..., N/2 − 1 as possible using MMSE estimation. The information about these spectral components is hidden in the ICI part of the signal at the output of the DFT demodulator Rm,s , s = 0, 1, ...N − 1 given by Eq. (2). Estimation algorithm is a decision feedback algorithm, since it requires transmitted sym-

bols. As transmitted symbols estimates, the symbols after necessary CPE correction are adopted. Once the DFT coefficients of the phase noise are known, one possesses enough information on the phase noise waveform, in order to suppress it. 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))ejφ(n) + ξ(n) with the estimate of e−jφ(n) . Multiplication in the time domain for discrete time systems is mapped to the circular convolution of DFT spectra in the frequency domain [9]. This means that the ICI cancellation for the mth OFDM symbol can be done in the frequency domain by circularly convolving the demodulated symbols vector of all subcarriers Rm,N = [Rm (0), . . . , Rm (N − 1)]T with the vector of estimated DFT coefficients of e−jφ(n) . The concrete realization of an algorithm is as follows: 1. Step 1: Perform standard CPE correction using least square (LS) estimation [19] [17]. 2. Step 2: Make a decisions on the transmitted symbols and use all available hard decisions for the MMSE estimation of the I˜m (i), i = −u...u, according to the method provided in [14]. Here u denotes the order of the phase noise approximation. 3. Step 3: Convolve the vector of the received symbols with the DFT coefficients of the conjugate of the phase noise waveform. 4.3. Iterative Phase Noise Suppression Described algorithm for ICI suppression is the decision feedback algorithm. It is to expect that falsely detected symbols after initial CPE correction, which are fed to the MMSE estimator in the Step 2 of the algorithm, influence the estimation process. The reduction of the symbol error of the symbols, which are fed back, will improve the quality of the phase noise estimation and thus the quality of the phase noise suppression. This can be achieved if the algorithm described in Sec. 4.2. is applied iteratively. As discussed in Sec. 4.1. it is to expect that the noniterative ICI suppression algorithm gives better performance, i.e. reduced symbol error rate, than the pure CPE correction. If this ”better” symbols are used, to again estimate the phase noise, it is to expect that the system performance will be improved. The proposed algorithms is realized in three steps: 1. Step 1: Perform standard CPE correction using least square (LS) estimation [19] [17]. 2. Step 2: Carry out ICI suppression algorithm described in the previous section (only Steps 2 and 3), 3. Step 3: Demodulate QAM symbols and feed them back to the Step 2. Iterate until the desired performance achieved. A block diagram of this scheme is presented in Fig. 6. The complexity of the algorithm is large, however the performance of the algorithm is dramatically improved as will be shown in Sec. 5..

Demodulator Estimate Im(i)

Deconv

Viterbi

64 QAM δ PN = 5⋅ 10-3

yes

Xˆ m,lk lk = 0,1, 2...N -1

10 Step 1

Step 2

-1

ICPE

3

ICI supp. alg genie ICI supp.

-2

1-iter 3-iter

Step 3

no PN

feedback Reconstruct Tx symbols

10

Xˆ m′ ,lk lk = 0,1, 2...N -1

Figure 6: Block diagram of the iterative phase noise suppression algorithm.

0

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Figure 7: Performance of the iterative ICI suppression algorithm for ETSI A channel and δP N = 5 · 10−3 . 10

Numerical Results

System parameters correspond to that described in Sec. 3.2.. Within simulations six scenarios are compared: 1) without phase noise (no PN); 2) with phase noise and genie CPE correction (ICPE); 3) with phase noise and CPE correction using least squares (LS) algorithm [19]; 4) with phase noise and genie ICI correction of certain order u; 5) with ICI correction of uth order and 6) with phase noise and an iterative phase noise suppression (number of iterations denoted by numbers). First set of simulation results in terms of PER is plotted in Figs. 7 and 8 for the ETSI A channel and an AWGN channel, respectively. Adopted relative phase noise bandwidth is δP N = 5 · 10−3 . The ICI correction order adopted is u = 3. The ICI correction algorithm shows better performance than the pure CPE correction, however, the results are much worse than the achievable genie correction of the specified order. This performance limitation is due to the decision feedback nature of the algorithm. Falsely detected symbols, from Step 2 of an algorithm, which are used for estimation of the phase noise DFT coefficients, will influence the estimation process. This problem is more pronounced if the phase noise bandwidth is large, because then ICI is large, which influences also the estimation of the CPE. To improve the performance of this algorithm, or in other words to reduce the error propagation problem, the iterative approach for phase noise suppression should be considered. This algorithm provides results that are very close to genie phase noise suppression of the corresponding order (see Figs. 7 and 8). However the complexity of the algorithm is quite large. Therefore, the quality of the phase noise suppression is the trade off between the complexity and performance. Fig. 9 shows an additional example for the same scenarios, ETSI A channel and δP N = 2 · 10−3 . The conclusions are similar as for the δP N = 5 · 10−3 . The ICI correction of order u = 7 has been performed. Simulation results show that for both frequency selective and AWGN channels, and both large and small phase noise bandwidths, performance very close to genie correction of certain order

3

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I m ( i ) , i = −u ,..., u

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Figure 8: Performance of the iterative ICI suppression algorithm for AWGN channel and δP N = 5 · 10−3 . can be achieved. The performance presented here is in terms of the packet error rate (PER). It is interesting to note that the bit error rate performance can even worsen with increasing number of iterations, while the PER decreases. For OFDM symbols, for which, after the initial CPE correction, many subcarriers are erroneously detected, the ICI estimation can produce additional errors. In the iterative algorithm this causes error propagation. However, for OFDM symbols with only few falsely detected subcarriers, ICI algorithm is capable of correcting these errors. Packets with few errors will be recovered by the algorithm, while packets with many errors after the initial CPE correction will probably have even more errors.

6. Conclusions In this paper we introduced two new approaches for phase noise suppression in OFDM. It was shown that the bit interleaving is a very simple, still effective means of suppressing the effects of the phase noise. For the applications where the interleaving depth cannot be large, e.g. for short code words, we propose an iterative algorithm for phase noise suppression. The performance of the algorithm is very close to the genie suppression of certain

10

[9] A.V. Oppenheim and R.W. Schafer. Discrete-Time Signal Processing. Prentice-Hall Inc., 1989.

0

64 QAM -3 δ PN = 2⋅ 10

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no PN

-2

[10] D. Petrovic, W. Rave, and G. Fettweis. Phase Noise Influence on Bit Error Rate, Cut-off Rate and Capacity of M-QAM OFDM Signaling. In Proc. Intl. OFDM Workshop (InOWo)02, 2002.

ETSI A Channel

PER

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-1

ICPE LS

ICI supp. alg genie ICI supp.

[11] D. Petrovic, W. Rave, and G. Fettweis. Phase Noise Suppression in OFDM including Intercarrier Interference. In Proc. Intl. OFDM Workshop (InOWo)03, pages 219–224, 2003.

1-iter 7 10

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Figure 9: Performance of the iterative ICI suppression algorithm for ETSI A channel and δP N = 2 · 10−3 . phase noise approximation order.

Acknowledgment This work was supported by the German ministry of research and education within the project Wireless Gigabit With Advanced Multimedia Support (WIGWAM) under grant 01 BU 370

REFERENCES [1] A.G. Armada. Understanding the Effects of Phase Noise in Orthogonal Frequency Division Multiplexing (OFDM). IEEE Trans. on Broadcasting, 47(2), June 2001. [2] G. Caire, G. Taricco, and E. Biglieri. BitInterleaved Coded Modulation. IEEE Trans. Commun., 44(3):927–946, May 1998. [3] R. A. Casas, S.L. Biracree, and A.E. Youtz. Time Domain Phase Noise Correction for OFDM Signals. IEEE Trans. on Broadcasting, 48(3), September 2002. [4] E. Costa and S. Pupolin. M-QAM-OFDM System Performance in the Presence of a Nonlinear Amplifier and Phase Noise. IEEE Trans. Commun., 50(3), March 2002. [5] A. Demir, A. Mehrotra, and J. Roychowdhury. Phase Noise in Oscillators: A Unifying Theory and Numerical Methods for Characterisation. IEEE Trans. Circuits Syst. I, 47(5), May 2000. [6] D. J. Higham. An Algorithmic Introduction to Numerical Simulation of Stochastic Differential Equations. SIAM Review, 43(3):525–546, 2001. [7] IEEE. Part11: Wireless LAN Medium Access Control (MAC) and Physical Layer (PHY) Specifications. High-speed Physical Layer in the 5GHz Band. IEEE Std 802.11a-1999, 1999. [8] S. Mirabbasi and K. Martin. Classical and modern receiver arcitectures. IEEE Communications Magazine, pages 132–139, November 2000.

[12] D. Petrovic, W. Rave, and G. Fettweis. Phase Noise Suppression in OFDM using a Kalman Filter. In Proc. WPMC, 2003. [13] D. Petrovic, W. Rave, and G. Fettweis. Common Phase Error due to Phase Noise in OFDM - Estimation and Suppression. In In Proc. of the 15th IEEE International Symposium on Personal, Indoor and Mobile Radio Communications (PIMRC), September 2004. [14] D. Petrovic, W. Rave, and G. Fettweis. Intercarrier Interference due to Phase Noise in OFDM - Estimation and Suppression. In Proc. IEEE VTC Fall, in press., September 2004. [15] L. Piazzo and P. Mandarini. Analysis of Phase Noise Effects in OFDM Modems. IEEE Trans. Commun., 50(10), October 2002. [16] W. Rave, D. Petrovic, and G. Fettweis. Iterative Correction of Phase Noise in Multicarrier Modulation. In In Proc. of the 9th International OFDM Workshop (InOWo)., September 2004. [17] P. Robertson and S. Kaiser. Analysis of the effects of phase noise in OFDM systems. In Proc. ICC, 1995. [18] S.Wu and Y.Bar-ness. Performance Analysis of the Effect of Phase Noise in OFDM Systems. In IEEE 7th ISSSTA, 2002. [19] S. Wu and Y. Bar-Ness. A Phase Noise Suppression Algorithm for OFDM-Based WLANs. IEEE Communications Letters, 44(3), May 1998. [20] 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. 2003.