Abstract
Statisticsbased intercarrier interference (ICI) mitigation algorithm is proposed for orthogonal frequency division multiplexing systems in presence of both nonstationary and stationary phase noises. By utilizing the statistics of phase noise, which can be obtained from measurements or data sheets, a Wiener filter preprocessing algorithm for ICI mitigation is proposed. The proposed algorithm can be regarded as a performanceimproving technique for the previous researches on phase noise cancelation. Simulation results show that the proposed algorithm can effectively mitigate ICI and lower the error floor, and therefore significantly improve the performances of previous researches on phase noise cancelation, especially in the presence of severe phase noise.
Introduction
Due to its high data rate transmission capability and its robustness to multipath delay spread, orthogonal frequency division multiplexing (OFDM) has been adopted in most parts of modern wireless communication systems, such as wireless local area networks [1], digital audio and video broadcasting [2], and so is OFDM being standardized for the future wireless communication systems, such as Worldwide Interoperability for Microwave Access [3] and 3GPP’s LongTerm Evolution [4]. However, OFDM is known to be very sensitive to radio frequency frontend imperfections such as carrier frequency offset (CFO), inphase and quadraturephase (IQ) imbalance and phase noise. These imperfections, if not properly estimated and compensated, will severely degrade the performance of OFDM systems. CFO and IQ imbalance are constant over an OFDM symbol period and have widely been investigated and well solved in the literature [58]. However, unlike the constant CFO and IQ imbalance, phase noise is a stochastic process during an OFDM symbol period and therefore causes a greater challenging problem. This is because each OFDM sample in an OFDM symbol period suffers from different phase noise.
The detrimental effect of phase noise on the performance of OFDM systems has extensively been studied in [911]. The effect of phase noise on OFDM is commonly categorized as common phase error (CPE) and intercarrier interference (ICI). Several techniques have been proposed to estimate and compensate for phase noise in OFDM systems [1217]. More specifically, by trading the ICI as additional noise, Wu and BarNess [12] use minimum mean square error criterion to cancel the CPE. By considering that phase noise can be modeled as a lowpass process, the authors of [13,14] resort to estimate a few spectral components of phase noise. In particular, the ICIcancelation scheme proposed in [13] stems from iterative detection principle, while [14] resorts to interpolation between the CPE estimates of two consecutive OFDM symbols in a noniterative way. Zou et al. [15,16] propose to estimate a few phase noise components in the time domain and obtain the rest components by interpolation. In [17], each OFDM symbol is partitioned into several subblocks where phase noise process is assumed quasistatic over each subblock and therefore only the CPE of each subblock needs to be estimated.
In this article, we are concerned with ICI mitigation at the receiver side of OFDM systems using the statistics of phase noise, which can be obtained from measurements or data sheets (see [9,10,13,18,19] and references therein). Based on the framework of Wiener filter, a Wiener filter preprocessing algorithm for ICI mitigation is proposed. The proposed algorithm performs directly on the received signal without changing the structure of conventional OFDM systems. Subsequently, the algorithms of previous researches on phase noise cancelation can be performed on the preprocessed received signal. Simulation results show that by utilizing the correlation inherently exists in phase noise, the proposed Wiener filter preprocessing algorithm can effectively mitigate the ICI and lower the error floor, and therefore significantly improve the performances of previous researches on phase noise cancelation, especially in the severe phase noise case.
The remainder of this article is organized as follows. Section 2 describes the OFDM system and phase noise models. The proposed Wiener filter preprocessing algorithm is presented in Section 3. Section 4 gives some simulation results that demonstrate the effectiveness of the proposed algorithm. Finally, conclusions are drawn in Section 5.
Notation: Vectors and matrices are boldface letters. A hat over a variable (e.g.,
System model
OFDM model
A model of the OFDM system in the presence of phase noise is depicted in Figure 1. In an OFDM system, the source data in the frequency domain X = [X(0),X(1),…,X(N−1)]^{T}is modulated onto N parallel subcarriers to obtain the time domain signal z = [z(0),z(1),…,z(N−1)]^{T} = F^{H}X. In general, the elements of X can be categorized into
where I_{d} is the index set of subcarriers allocated for data symbols with N_{d} elements, and I_{p} is the index set of subcarriers allocated for pilot symbols with N_{p}elements, respectively. Notice that N=N_{d} + N_{p}. From (1), we have X=E_{d}X_{d} + E_{p}X_{p}, where the N×N_{d}matrix E_{d} and N×N_{p} matrix E_{p} denote matrices collecting columns of I_{N}corresponding to I_{d}and I_{p}, respectively, and X_{d}=[X_{d}(0),X_{d}(1),…,X_{d}(N_{d}−1)]^{T}, X_{p}=[X_{p}(0),X_{p}(1),…,X_{p}(N_{p}−1)]^{T} denote the data and pilot vectors, respectively. We assume that the OFDM subcarrier
signals are mutually independent random variables with zero mean and variance E_{s}, i.e.,
Figure 1. The OFDM system in the presence of phase noise.
At the receiver side, assuming perfect timing and frequency synchronization are achieved, the nth sample of the received signal is given by
where ϕ(n) represents the phase noise, w(n)’s are independent and identically distributed (i.i.d.) complex random variables
with zero mean and variance σ^{2}, representing the contribution of additive white Gaussian noise. The coefficients
After discarding the CP, the received signal after transforming back into the frequency domain is given by
where J(k), the DFT of the phase noise process, is given by
The DC coefficient J(0)acts on all subcarriers as a CPE in (3) and the second term on the righthand side of (3) represents ICI that results from higher order of J(k). The channel frequency response H(k)is given by
W(k) represents the corresponding noise in the frequency domain. It is observed that the effect of phase noise, i.e., CPE and ICI, distorts the received signal Y(k)in a multiplicative and additive manner, respectively. Due to the detrimental effect of phase noise on the performance of OFDM systems, the phase noise should be compensated for.
Phase noise model
Nonstationary phase noise
When the system is frequency locked, the resulting phase noise is slowly varying but not limited, and it is modeled as a zeromean, nonstationary, infinitepower Wiener process. In this case, the phase noise is expressed as a freerunning or Brownian process, i.e.,
where v(n) is an i.i.d zeromean Gaussian variable with variance
Stationary phase noise
When the system is phaselocked, the resulting phase noise is low and modeled as a zeromean, stationary, finitepower stochastic process. For a classical model of stationary phase noise, ϕ(n)is modeled as a stationary Gaussian process with zero mean and a specified power spectrum density [13,18,19].
The proposed Wiener filter preprocessing algorithm
It is observed from (3) that we wish to estimate the desired term S(k), from Y(k),k=0,1,…,N−1. In the framework of Wiener filter, the estimate of S(k), i.e.,
where γ_{kN} is a bias term that allows for nonzero means of S(k)and Y(k),k=0,1,…,N−1. Substituting (9) into (8) and setting the first derivative of the resulting (8) with respect to γ_{kN} to zero, we obtain the optimal estimate of γ_{kN}as
Substituting (10) into (8), the MSE can be expressed as
where ϒ_{k}=[γ_{k0},γ_{k1},…,γ_{kN−1}]^{T}, Σ_{YY} is the covariance matrix of Y = [Y(0),Y(1),…,Y(N−1)]^{T}, Σ_{YS(k)} is the crosscovariance vector of Y and S(k), Σ_{S(k)S(k)} is the variance of S(k). Setting the first derivative of (11) with respect to ϒ_{k} to zero, we obtain the optimal estimate of ϒ_{k}as
Substituting (10) and (12) into (9), the optimal estimate of S(k) can thus be obtained as
Stacking all N subcarriers, the optimal estimate of S=[S(0),S(1),…,S(N−1)]^{T} can be written in matrix form as
where
and
In (15),
and in (16),
where we have assumed that the desired term Sand the ICI I=[I(0),I(1),…,I(N−1)]^{T} are independent of the noise W=[W(0),W(1),…,W(N−1)]^{T}, and Φ_{PQ} represents the correlation function between Pand Q.
We now derive each term required to compute
where the fact that the transmit signal, the wireless channel, and the phase noise
are independent from each other has been used. Using (5), the correlation function
Therefore, substituting (20) into (19) we obtain that if k_{1}≠k_{2},
and if k_{1}=k_{2},
Using (3) the elements of Φ_{II}in (18) can be derived as
and therefore, if k_{1}≠k_{2},
and if k_{1}=k_{2},
Similarly, the elements of Φ_{SI}in (17) and (18) can be derived as
and therefore, if k_{1}≠k_{2}
and if k_{1}=k_{2}
It is noted from (19)–(28) that the correlation function
Using (4), the correlation function
Nonstationary phase noise
Substituting (7) into (29), we obtain
where sgn(n) represents the sign operation, i.e.,
Since v(n) is an i.i.d zeromean Gaussian variable with variance
Therefore, the variance of
Notice that for a Gaussian variable αwith mean μ and variance ψ^{2}, its characteristic function is given by [23]
Substituting (33) into (34) and letting μ = 0 and t = 1, we obtain
Substituting (35) into (30), the correlation function
Stationary phase noise
Since ϕ(n) is modeled as a stationary Gaussian process with zero mean, ϕ(n_{1})−ϕ(n_{2})is also a stationary Gaussian process with zero mean and variance given by
Therefore, substituting (36) into (34) and letting μ = 0 and t = 1, we obtain
Similar to the nonstationary one, substituting (37) into (29), the correlation function
Finally, considering that
and
we have for
and for
Since the FFT does not change the noise distribution, Φ_{WW} in (18) is given by
In summary, substituting the derived results (21) and (22), (24) and (25), (27) and
(28), (40)–(42) into (14), the optimal estimate
It is noted that without changing the structure of conventional OFDM systems, the
proposed Wiener filter preprocessing algorithm is based on the statistics of phase
noise (which can be obtained from measurements or data sheets) and performs directly
on the received signal Y(see Figure 1 for illustration). Subsequently, the algorithms of previous researches on phase noise
cancelation (e.g., [1217]) can be performed based on the preprocessed received signal
Simulation results and discussions
In this section, the performance of the proposed Wiener filter preprocessing algorithm
for ICI mitigation is demonstrated by Monte Carlo simulations. In the simulations,
each OFDM symbol has 128 subcarriers (N=128) and communicates over a bandwidth of 20 MHz. The sampling interval T_{s}is thus 50 ns. The data are modulated by 16QAM modulation. The channel has three taps
(L=3) with an exponential power delay profile, namely
Figure 2. BER performance: nonstationary phase noise case. BER versus SNR for nonstationary phase noise, where the phase noise rates are 0.1 and 0.01.
Figure 3. BER performance: stationary phase noise case. BER versus SNR for stationary phase noise, where the standard deviations are 4° and 0.4°.
Figure 4. BER performance: nonstationary phase noise case when SNR = 30 dB. BER versus phase noise rate for nonstationary phase noise, where the range of phase noise rate is from 1e^{−5}to 1e^{−1}.
Figure 5. BER performance: stationary phase noise case when SNR = 30 dB. BER versus standard deviation for stationary phase noise, where the range of standard deviation is from 0.0004° to 4°.
For the nonstationary phase noise case, it can be seen from Figure 2 that if the detrimental effect of phase noise, i.e., CPE and ICI, are left uncompensated (marked as “no phase noise correction” in Figures 2, 3, 4, and 5), the performance is totally unacceptable for both cases where phase noise rate are 0.1 and 0.01.
For the case that phase noise rate is 0.01 (dashed lines in Figure 2), the CPE dominates the effect of phase noise. Therefore, as can be seen from Figure 2, the performance can significantly be improved if only CPE is compensated. If [12] is preprocessed by our proposed Wiener filter preprocessing algorithm, there is a minor performance improvement for “proposed [12]” compared to “conventional [12]”, as can be observed from Figure 2. Although ICI is small in this case, if ICI is also mitigated, a minor performance improvement can be obtained especially in high SNR region, where the decision on the transmitted symbols are more accurate than that in low SNR region and therefore, the ICI can be more accurately estimated and canceled according to [13]. It is observed from Figure 2 that the performance of “proposed [13]” is almost the same as that of the “conventional [13]” (only with minor improvement) for the reason that ICI is small in this case.
However, it can be observed from Figure 2 that for the severe case where phase noise rate is 0.1 (solid lines in Figure 2), the ICI dominates the effect of phase noise. Since [12] only aims to cancel CPE without canceling any of the ICI terms, the performance of “conventional [12]” is not satisfactory. After [12] is preprocessed by our proposed Wiener filter preprocessing algorithm, a performance improvement can be observed from the “proposed [12]” compared to the “conventional [12]”. However, it is observed that the performance gap between the “proposed [12]” and “no phase noise” case is still large, which means that if an ICIcancelation technique is employed, the performance may be further improved. As can be observed that in this severe phase noise case, the performance of the “proposed [13]” is significantly better than that of the “conventional [13]” for the reason that ICI is not negligible and contributes significantly to the overall effect of phase noise. It is noted that ICI can effectively be mitigated and the error floor is obviously lowered through our proposed algorithm.
For the stationary phase noise case, it is generated according to [18] with the standard deviation θ_{rms} = 4^{∘} and 0.4°. More specifically, the elements of the correlation matrix for stationary
phase noise are given by
Figures 4 and 5 show the BER performance of the proposed algorithm and that of comparisons [12,13] versus phase noise rate and standard deviation when SNR is set to 30 dB, for the nonstationary and stationary phase noise cases, respectively. The performance advantage of the proposed algorithm over that of conventional [12,13] can be observed especially for severe phase noise cases.
Conclusions
In this article, we focused on ICI mitigation for OFDM systems in the presence of phase noise. By utilizing the correlation inherently exists in phase noise, a Wiener filter preprocessing algorithm based on the statistics of phase noise has been proposed, which performs directly on the received signal without changing the structure of conventional OFDM systems. Subsequently, the algorithms of previous researches on phase noise cancelation can be performed on the preprocessed received signal. Simulation results showed that for both nonstationary and stationary phase noise cases, the proposed algorithm can effectively mitigate ICI and lower the error floor, and therefore significantly improve the performances of previous researches on phase noise cancelation, especially in the presence of severe phase noise.
Competing interests
The authors declare that they have no competing interests.
Acknowledgements
This study was supported in part by the National Science Foundation of China under Grant numbers 61032002, 60902026, and 60972029, the Chinese Important National Science & Technology Specific Projects under Grant 2011ZX0300100701, and by the Program for New Century Excellent Talents in University, NCET110058.
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