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Wiener filter preprocessing for OFDM systems in the presence of both nonstationary and stationary phase noises

Ke Zhong*, Xia Lei and Shaoqian Li

Author Affiliations

The National Key Laboratory of Science and Technology on Communications, University of Electronic Science and Technology of China, Chengdu 611731, P. R. China

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EURASIP Journal on Advances in Signal Processing 2013, 2013:7  doi:10.1186/1687-6180-2013-7


The electronic version of this article is the complete one and can be found online at: http://asp.eurasipjournals.com/content/2013/1/7


Received:20 May 2012
Accepted:24 December 2012
Published:23 January 2013

© 2013 Zhong et al; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Statistics-based 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 performance-improving 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 Long-Term Evolution [4]. However, OFDM is known to be very sensitive to radio frequency front-end imperfections such as carrier frequency offset (CFO), in-phase and quadrature-phase (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 [5-8]. 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 [9-11]. 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 [12-17]. More specifically, by trading the ICI as additional noise, Wu and Bar-Ness [12] use minimum mean square error criterion to cancel the CPE. By considering that phase noise can be modeled as a low-pass process, the authors of [13,14] resort to estimate a few spectral components of phase noise. In particular, the ICI-cancelation 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 quasi-static 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., <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M1">View MathML</a>) indicates an estimate of the variable. <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M2">View MathML</a> denotes the expectation. Superscripts [·]T, [·]−1, [·]H, and [·] denote the transpose, the matrix inversion, the Hermitian, and the complex conjugate operations, respectively. IN is an identity matrix with dimension N. 0N denotes an N×1 vector whose elements are all zeros. The symbol ⊛ denotes convolution. <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M3">View MathML</a> and <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M4">View MathML</a> are the real and imaginary parts of the quantity in the brackets, respectively. δ(·) denotes the Dirac delta function. The matrix Fis the normalized discrete Fourier transform (DFT) matrix with the (m,n)th element given by <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M5">View MathML</a>.

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)]Tis modulated onto N parallel subcarriers to obtain the time domain signal z = [z(0),z(1),…,z(N−1)]T = FHX. In general, the elements of X can be categorized into

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M6">View MathML</a>

(1)

where Id is the index set of subcarriers allocated for data symbols with Nd elements, and Ip is the index set of subcarriers allocated for pilot symbols with Npelements, respectively. Notice that N=Nd + Np. From (1), we have X=EdXd + EpXp, where the N×Ndmatrix Ed and N×Np matrix Ep denote matrices collecting columns of INcorresponding to Idand Ip, respectively, and Xd=[Xd(0),Xd(1),…,Xd(Nd−1)]T, Xp=[Xp(0),Xp(1),…,Xp(Np−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 Es, i.e., <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M7">View MathML</a>. A cyclic prefix (CP) with length Ncplonger than the delay spread of the channel is inserted at the beginning of each OFDM symbol to prevent intersymbol interference, i.e., the transmitted time domain signal is [z(" close="">NNcp),z(NNcp + 1),…,z(N − 1),z(0),z(1),…,z(N−1)T.

thumbnailFigure 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

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M8">View MathML</a>

(2)

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 σ2, representing the contribution of additive white Gaussian noise. The coefficients <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M9">View MathML</a> represent the equivalent multipath discrete-time channel impulse response (CIR) with length L, including transmit and receive filters, i.e., <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M10">View MathML</a>, where gTx(t), hPr(t), and gRx(t) denote the pulse shaping filter at the transmitter, the impulse response of the propagation medium, and the shaping filter at the receiver, respectively, and Tsis the sampling interval. We consider a slow fading frequency-selective channel where the CIR is assumed to remain constant during the transmission of one OFDM symbol period but can change randomly from symbol to symbol. The CIR is assumed perfectly known at the receiver. Powerful schemes for accurate channel estimation in the presence of phase noise have been proposed in [20-22]. The channel taps are assumed mutually independent and the correlation function of different taps is given by <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M11">View MathML</a>, where <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M12">View MathML</a> denotes the average power of the lth tap and without loss of generality, we assume <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M13">View MathML</a>.

After discarding the CP, the received signal after transforming back into the frequency domain is given by

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M14">View MathML</a>

(3)

where J(k), the DFT of the phase noise process, is given by

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M15">View MathML</a>

(4)

The DC coefficient J(0)acts on all subcarriers as a CPE in (3) and the second term on the right-hand side of (3) represents ICI that results from higher order of J(k). The channel frequency response H(k)is given by

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M16">View MathML</a>

(5)

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 zero-mean, nonstationary, infinite-power Wiener process. In this case, the phase noise is expressed as a free-running or Brownian process, i.e.,

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M17">View MathML</a>

(6)

where v(n) is an i.i.d zero-mean Gaussian variable with variance <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M18">View MathML</a>, with β representing the two-side 3-dB linewidth of phase noise, βTrepresenting the phase noise rate [9,10] and T being the OFDM symbol period. In the case of perfect synchronization at the beginning of the OFDM symbol, ϕ(−1)=0and therefore, (6) can be expressed alternatively as

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M19">View MathML</a>

(7)

Stationary phase noise

When the system is phase-locked, the resulting phase noise is low and modeled as a zero-mean, stationary, finite-power 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., <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M20">View MathML</a>, is determined by estimating a set of coefficients <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M21">View MathML</a>, in order to minimize the estimation mean square error (MSE) as

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M22">View MathML</a>

(8)

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M23">View MathML</a>

(9)

where γkN 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 γkN to zero, we obtain the optimal estimate of γkNas

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M24">View MathML</a>

(10)

Substituting (10) into (8), the MSE can be expressed as

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M25">View MathML</a>

(11)

where ϒk=[γk0,γk1,…,γkN−1]T, ΣYY is the covariance matrix of Y = [Y(0),Y(1),…,Y(N−1)]T, ΣYS(k) is the cross-covariance vector of Y and S(k), ΣS(k)S(k) is the variance of S(k). Setting the first derivative of (11) with respect to ϒk to zero, we obtain the optimal estimate of ϒkas

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M26">View MathML</a>

(12)

Substituting (10) and (12) into (9), the optimal estimate of S(k) can thus be obtained as

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M27">View MathML</a>

(13)

Stacking all N subcarriers, the optimal estimate of S=[S(0),S(1),…,S(N−1)]T can be written in matrix form as

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M28">View MathML</a>

(14)

where

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M29">View MathML</a>

(15)

and

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M30">View MathML</a>

(16)

In (15), <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M31">View MathML</a> is derived as

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M32">View MathML</a>

(17)

and in (16), <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M33">View MathML</a> is derived as

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M34','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M34">View MathML</a>

(18)

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 ΦPQ represents the correlation function between Pand Q.

We now derive each term required to compute <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M35">View MathML</a> in (14). From (3), the elements of ΦSSin (17) and (18) can be obtained as

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M36">View MathML</a>

(19)

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 <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M37">View MathML</a> can be derived as

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M38">View MathML</a>

(20)

Therefore, substituting (20) into (19) we obtain that if k1k2,

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M39">View MathML</a>

(21)

and if k1=k2,

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M40">View MathML</a>

(22)

Using (3) the elements of ΦIIin (18) can be derived as

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M41">View MathML</a>

(23)

and therefore, if k1k2,

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M42','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M42">View MathML</a>

(24)

and if k1=k2,

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M43','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M43">View MathML</a>

(25)

Similarly, the elements of ΦSIin (17) and (18) can be derived as

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M44','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M44">View MathML</a>

(26)

and therefore, if k1k2

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M45','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M45">View MathML</a>

(27)

and if k1=k2

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M46">View MathML</a>

(28)

It is noted from (19)–(28) that the correlation function <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M47">View MathML</a> plays a pivotal role in the computation of ΦSS, ΦII, and ΦSI.

Using (4), the correlation function <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M48">View MathML</a> can be derived as

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M49">View MathML</a>

(29)

Nonstationary phase noise

Substituting (7) into (29), we obtain

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M50">View MathML</a>

(30)

where sgn(n) represents the sign operation, i.e.,

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M51">View MathML</a>

(31)

Since v(n) is an i.i.d zero-mean Gaussian variable with variance <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M52">View MathML</a>, <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M53">View MathML</a> are also i.i.d. Gaussian variable with zero mean and variance given by

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M54">View MathML</a>

(32)

Therefore, the variance of <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M55">View MathML</a> can be derived as

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M56">View MathML</a>

(33)

Notice that for a Gaussian variable αwith mean μ and variance ψ2, its characteristic function is given by [23]

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M57">View MathML</a>

(34)

Substituting (33) into (34) and letting μ = 0 and t = 1, we obtain

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M58">View MathML</a>

(35)

Substituting (35) into (30), the correlation function <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M59">View MathML</a> for the nonstationary phase noise case can be obtained.

Stationary phase noise

Since ϕ(n) is modeled as a stationary Gaussian process with zero mean, ϕ(n1)−ϕ(n2)is also a stationary Gaussian process with zero mean and variance given by

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M60">View MathML</a>

(36)

Therefore, substituting (36) into (34) and letting μ = 0 and t = 1, we obtain

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M61">View MathML</a>

(37)

Similar to the nonstationary one, substituting (37) into (29), the correlation function <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M62">View MathML</a> for the stationary phase noise case can be obtained.

Finally, considering that

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M63">View MathML</a>

(38)

and

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M64">View MathML</a>

(39)

we have for <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M65">View MathML</a> in (15) as

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M66">View MathML</a>

(40)

and for <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M67">View MathML</a> in (15) and (16) as

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M68">View MathML</a>

(41)

Since the FFT does not change the noise distribution, ΦWW in (18) is given by

<a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M69">View MathML</a>

(42)

In summary, substituting the derived results (21) and (22), (24) and (25), (27) and (28), (40)–(42) into (14), the optimal estimate <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M70">View MathML</a> can be obtained. Notice that for the nonstationary phase noise case, (30) and (35) should be adopted; for the stationary phase noise case, (29) and (37) should be adopted.

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., [12-17]) can be performed based on the preprocessed received signal <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M71','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M71">View MathML</a> instead of on the received signal Y. We will show in the next section that by utilizing the correlation inherently exists in phase noise process, the proposed Wiener filter preprocessing algorithm can effectively mitigate the ICI which results from phase noise and lower the error floor, and therefore significantly improve the performances of previous researches phase noise cancelation, especially in severe phase noise case.

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 Tsis thus 50 ns. The data are modulated by 16QAM modulation. The channel has three taps (L=3) with an exponential power delay profile, namely <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M72">View MathML</a> with κ=1/3. Each tap coefficient follows a complex Gaussian distribution. Fourteen equally spaced OFDM subcarriers are allocated to pilot symbols, i.e., Np=14and Nd=114. This means that roughly 10.94% of the subcarriers are occupied by pilots. The pilots are used for the CPE estimation and cancelation algorithm given in [12] (marked as “conventional [12]” in Figures 2, 3, 4, and 5). The pilots are also used for the ICI-cancelation algorithm given in [13] with ICI correction of order 3 (marked as “conventional [13]” in Figures 2, 3, 4, and 5). The performance of [12,13] preprocessed by our proposed Wiener filter preprocessing algorithm is marked as “proposed [12]” and “proposed [13]” in Figures 2, 3, 4, and 5.

thumbnailFigure 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.

thumbnailFigure 3. BER performance: stationary phase noise case. BER versus SNR for stationary phase noise, where the standard deviations are 4° and 0.4°.

thumbnailFigure 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−5to 1e−1.

thumbnailFigure 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 ICI-cancelation 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 θrms = 4 and 0.4°. More specifically, the elements of the correlation matrix for stationary phase noise are given by <a onClick="popup('http://asp.eurasipjournals.com/content/2013/1/7/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://asp.eurasipjournals.com/content/2013/1/7/mathml/M73">View MathML</a>, where Ω0=1000 kHz is the 3-dB bandwidth of a single pole Butterworth filter. It is observed from Figure 3 that similar results can be obtained as the nonstationary phase noise case.

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 2011ZX03001-007-01, and by the Program for New Century Excellent Talents in University, NCET-11-0058.

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