Estimation of Stationary Phase Noise by the
Autocorrelation of the ICI Weighting Function in OFDM Systems
Yi-Ching Liao, Student Member, IEEE, and Kwang-Cheng Chen, Senior Member, IEEE
Abstract— Phase noise resulting in Common Phase Error
(CPE) and Inter-Carrier Interference (ICI) is a critical challenge to the implementation of OFDM systems. Modeling phase noise as a stationary Gaussian random process with the specified power spectrum density, different from conventional approaches which mostly relay on pilots to provide CPE estimation, we explore the statistical characteristics of the sufficient statistics then propose a pilot-aided decision-directed approach according to maximum-likelihood criterion. Numerical results demonstrate that the proposed algorithm enjoys 2dB gain at moderate SNR and is quite robust against possible model mismatch.
Index Terms— Maximum-likelihood estimation, OFDM, phase
noise.
I. INTRODUCTION
O
RTHOGONAL frequency-division multiplexing (OFDM) is a spectrally efficient transmission technique which can combat dispersive channel by performing simple frequency domain equalization. However, OFDM is much more sensitive to synchronization errors than single carrier systems. Among them, phase noise which is mainly caused by the instability of the local oscillator is a crucial challenge. In [1], phase noise is classified into two categories. When the system is only 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. When the system is phase-locked, the resulting phase noise is low and modeled as a zero-mean, stationary, finite-power random process. Here, we concentrate on the suppression of the stationary phase noise process.To eliminate the effect of phase noise, we propose a Pilot-Aided Decision-Directed (PADD) approach based on maxi-mum likelihood criterion. Distinct from conventional phase noise suppression algorithms [2]–[4] which do not consider the statistical information of phase noise, we investigate the autocorrelation function of the ICI weighting function which is the kernel of the second order statistics of the ICI to enhance the performance. Under stationary phase noise model, the relation between the autocorrelation function of phase noise process and that of the ICI weighting function is analytically derived. Numerical results demonstrate the effectiveness of the proposed algorithm.
Manuscript received February 3, 2005; revised December 31, 2005; ac-cepted March 1, 2006. The associate editor coordinating the review of this letter and approving it for publication was F. Daneshgaran. This research was supported by the National Science Council under the contract NSC 94-2219-E-002-022.
The authors are with the Institute of Communication Engineering,
National Taiwan University, Taipei, Taiwan, 10617 (e-mail:
ycliao@santos.ee.ntu.edu.tw; chenkc@cc.ee.ntu.edu.tw). Digital Object Identifier 10.1109/TWC.2006.05071
II. SIGNALMODEL
We start considering a general OFDM system using N -point inverse fast Fourier transform (IFFT) for modulation. Assume the frequency domain subcarrier index set is composed of three mutually exclusive subsets defined by
D= {d 1, d2,· · · , dNd}
P=p1, p2,· · · , pNp
V= {v 1, v2,· · · , vNv} ,
(1)
where D denotes the set of indices for Nd data-conveying
subcarriers,P is the set of indices for Nppilot subcarriers and
V stands for Nv virtual subcarriers. Then, the set of indices
for Nu useful subcarriersU can be defined as
U = {u1, u2,· · · , uNu} = D ∪ P, (2)
where Nu= Nd+ Np. Let Xm(k) be the modulated symbol
on the k-th subcarrier of the m-th OFDM symbol. For k∈ U,
Xm(k) is taken from some constellation with zero mean and
average power σ2X ≡ E{|Xm(k)|2}. The output of the IFFT
has a duration of T seconds which is equivalent to N samples. A Ng-sample cyclic prefix longer than the channel impulse
response is preceded to eliminate the inter-symbol interference (ISI).
At the receiver, timing and frequency offset is assumed to be recovered. Considering the multiplicative phase noise and the additive white noise, the received n-th sample of the m-th OFDM symbol can be written as
rm(n) = [xm(n) ⊗ hm(n)] ejφm(n)+ ξm(n) (3)
in which⊗ is the circular convolution and {xm(n)}, {hm(n)}
and{φm(n)} with n ∈ {0, · · · , N −1} represent the
transmit-ted signal, the channel impulse response and the phase noise respectively, while{ξm(n)} denotes the AWGN. After
remov-ing the cyclic prefix and performremov-ing the FFT, the frequency domain symbol can be expressed by
Rm(k) =Φm(0)Hm(k)Xm(k) + l∈U l=k Φm(k − l)Hm(l)Xm(l) Im(k) +Zm(k) (4)
where Hm(k) is the channel frequency response and Zm(k)
denotes the frequency domain expression of ξm(n). Φm(q) is
the discrete Fourier transform of the phase noise process given by Φm(q) = 1 N N−1 n=0 ejφm(n)e−j2πqnN. (5) 1536-1276/06$20.00 c 2006 IEEE
It can be viewed as a weighting function on the transmitted frequency domain symbols. In particular, when q= 0, Φm(0)
is actually the time average of the phase noise process within one OFDM symbol duration. This term is usually known as the Common Phase Error (CPE) which causes the same phase rotation and amplitude distortion to each transmitted frequency domain symbol. On the other hand, when q= 0, the second term in (4) is the Inter-Carrier Interference (ICI) resulting from contributions of other subcarriers by the weighting ofΦm(q)
due to the loss of orthogonality. We call Φm(q) as the ICI
weighting function.
Based on (4), the received frequency domain vector can be given by rm= Φm(0)Hmxm+ ιm+ ζm = Φm(0)Hmxm+ εm, (6) where Hm= diag(Hm(0), Hm(1), · · · , Hm(N − 1)) xm= Xm(0) Xm(1) · · · Xm(N − 1) T ιm = Im(0) Im(1) · · · Im(N − 1) T ζm = Zm(0) Zm(1) · · · Zm(N − 1) T (7)
and diag(·) is a diagonal matrix. We add a second subscript to one of the vector or matrix variables defined in (6) and (7) to indicate its sub-vector or sub-matrix which is taken according to one of the subcarrier index sets in (1) and (2). The second subscript may be chosen from {p, d, v, u} which relates to
{P, D, V, U} respectively. For example,
rm,p=
Rm(p1) Rm(p2) · · · Rm(pNp)
T
(8) stands for the received pilot vector.
Conventionally, rm,p is utilized to obtain the channel
fre-quency response and the CPE to carry out equalization on
rm,d and finally send the equalized results to the detection
block to get the decisions. Since accurate channel estimation in OFDM systems can be obtained by either preambles or pilot symbols [5], we assume that the channel frequency response is acquired by the receiver in subsequent sections.
III. PHASENOISEMODEL
Accurate modeling of oscillator phase noise is a key factor to the analysis and simulation of the distortion and interference caused by phase noise. For a classical model of stationary phase noise, φm(n) can be modeled as a stationary Gaussian
process with zero mean and the specified power spectrum density (PSD). Two types of phase noise PSD are given in the literature, they differ in the decay of the transition band. The well-known model proposed by [3] which has a linear decay is given by Sφ(f) = 10−c+ ⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩ 10−a |f| ≤ f l 10−(f−fl)bfh−fl−a fl< f 10−(f+fl)bfh−fl −a f <−fl (9)
where c determines the white phase noise floor of the oscilla-tor. The parameter a gives the phase noise level near the center
−200 −150 −100 −50 0 50 100 150 200 −115 −110 −105 −100 −95 −90 −85 −80 −75 −70 −65 f (KHz) PDS (dB) Linear Decay Logarimic Decay
Fig. 1. PSD of phase noise processes.
frequency up to±fland b is the steepness of noise reduction
with increasing frequency distance up to±fhwhere the noise
floor becomes dominant. Another important characterization of stationary phase noise is its Auto-Correlation Function (ACF). The ACF of a phase noise process can be obtained by performing the inverse Fourier transform of its PSD. In [6], the ACF of (9) is given by
Rφ(τ) =10−cδ(τ) + 2 · 10−a fl· sinc(2flτ) +pcos 2πτfl− (2πτ) sin 2πτfl p2+ (2πτ)2 (10) where p= bln 10 fh− fl , q= bln 10 (fh− fl)Ts .
An alternative PSD model [7] which has a logarithmic decay is given by Sψ(f) = 10−c+ ⎧ ⎪ ⎨ ⎪ ⎩ 10−a |f| ≤ fl 10−a(f/fl)−b fl< f 10−a(−f/fl)−b f <−fl , (11)
and its corresponding ACF is
Rψ(τ) =10−cδ(τ) + 2 · 10−a· fl· sinc(2flτ) + 2fb l · 10−a ∞ fl f−bcos 2πfτdf. (12)
An example of the two phase noise PSD model where a= 7,
b= 4, c = 11, fl= 10 KHz and fh= 100 KHz is illustrated
in Fig. 1.
IV. DATA-AIDEDDECISION-DIRECTEDCPE ESTIMATION
Most OFDM systems employ pilots to facilitate receiver synchronization since data-aided estimation gives better and steadier estimate. However, since pilots cost system utilization, the number of pilots should be kept as small as possible which confines the performance of pilot-aided CPE correction algorithms. Comparatively, decision-directed approaches enjoy a larger observation space, however, to ensure the acceptable correctness of the decision, they can only be operated when
phase noise is small. Therefore, one can use pilots to acquire the CPE and provide an initial compensation, then enlarge the observation space by including the tentative decisions as the sufficient statistics to perform the final estimate which can benefit from the advantages of both approaches. We refer this method as the Pilot-Aided-Decision-Directed (PADD) approach.
Considering the pilot-aided decision-directed approach, the sufficient statistics isrm,u and can be modeled as a complex
Gaussian random vector with mean vectorΦm(0)Hm,uxm,u
and covariance matrix Crm,u. We will investigate of the statistical characteristics of the sufficient statisticsrm,uin next
section. Assume that we have acquired the covariance matrix, the log-likelihood function ofΦm(0) can be given by
Λ (Φm(0)) =2
xH
m,uH H
m,uC−1rm,urm,uΦ
∗ m(0) − xH m,uH H
m,uC−1rm,uHm,uxm,u|Φm(0)|
2.
(13) Differentiating (13) with respect toΦm(0), the ML estimation
of the common phase error can be given by ˆΦm(0) =
xH
m,uH H
m,uC−1rm,urm,u
xH
m,uH H
m,uC−1rm,uHm,uxm,u
. (14)
Please note that we need the covariance matrix which requires the second order statistics of the ICI to perform this estimation.
V. STATISTICALCHARACTERISTICS OF THE
SUFFICIENTSTATISTICS
In the PADD approach, since the data and the channel fre-quency response are acquired, the statistical characteristics of the sufficient statistics mentioned above depend on that of the ICI and AWGN noise. The AWGN noise on each subcarrier can be modeled as a zero mean complex Gaussian random variable with variance σ2Z. As for the ICI, we also approximate its distribution as complex Gaussian. The argument of this reasonable approach is confirmed by our simulation which is not illustrated here due to the limit of space. In the following, we first show that the ICI has a zero mean, then, the second order statistics of the ICI will be investigated.
1) The Mean of the ICI: To investigate the statistical
characteristic of Rm(k), we need to analyze the ICI, namely,
Im(k) first. The ICI can be modeled as a random variable
which is independent of Zm(k), since Xm(k) and Hm(k) are
assumed to be known in PADD circumstance, the mean of the ICI depends on the mean of the ICI weighting functionΦm(q)
which can be expressed as
E[Φm(q)] = 1 N N−1 n=0 E ejφm(n) ej2πqnN (15)
where E[·] is the expectation operation. Since φm(n) is
stationary, E[ejφm(n)] shall be a constant and can be taken
out of the summation such that
N−1
n=0
ej2πqnN = 0, ∀q = 0. (16)
Therefore, the mean of the ICI becomes zero.
2) The Second Order Statistics of the ICI: In a
pilot-aided decision-directed scenario, the sufficient statistics are the received useful symbols given by
rm,u= Φm(0)Hm,uxm,u+ ιm,u+ ζm,u. (17)
Similarly, we may denote the covariance matrix of ιm,u by
Cιm,u, then the covariance matrix ofrm,u can be expressed
as
Crm,u = Cιm,u+ σ
2
ZI. (18)
As for σ2Z, since it can be obtained by preamble signal and has
been proposed in the literature[8], therefore, we may assume that σZ2 is known by the receiver henceforth. Let the element ofCιm,u be denoted by σi(k1, k2), it can be expressed as
σi(k1, k2) = l1∈U l1=uk1 l2∈U l2=uk2 Hm(l1)Hm∗(l2)Xm(l1)Xm∗(l2) E[Φm(uk1− l1)Φ∗m(uk2− l2)] . (19) We can observe that σi(k1, k2) depends on the autocorrelation
function of the ICI weighting functionΦm(q) which is defined
as RΦ(q1, q2) ≡ E[Φm(q1)Φ∗m(q2)] = N−1 n1=0 N−1 n2=0 E[ej(φm(n1)−φm(n2))] N2 e −j2πn1q1−n2q2 N . (20) Since φm(n) can be modeled as a stationary Gaussian random
process, φm(n1)−φm(n2) is a Gaussian random variable with
zero mean and variance:
E[(φm(n1) − φm(n2))2] = 2Rφ(0) − 2Rφ(n1− n2), (21)
where Rφ(n) is the autocorrelation function of φm(n).
There-fore,
E[ej(φm(n1)−φm(n2))] = eRφ(n1−n2)−Rφ(0)
. (22)
Substitute (22) into (20), we have
RΦ(q1, q2) = N−1 n1=0 N−1 n2=0 eRφ(n1−n2)−Rφ(0) N2 e −j2πn1q1−n2q2 N . (23) Because Rφ(n) is already available from (10) and (12), the
ML CPE estimator (14) can be performed. VI. NUMERICALRESULTS
The proposed maximum-likelihood CPE estimator is evalu-ated in frequency selective slowly fading channels with 50 ns and 75 ns rms delay spread [9]. Channel impulse response remains static within a frame containing 16 symbols, but varies independently from frame to frame. Transmitted data is constructed according to IEEE 802.11a WLAN standard [10]. 16 QAM and 64 QAM which are more sensitive to phase noise than M-PSK, are used in the simulation to evaluate the performance of the proposed algorithm. Phase noise is generated using the linear-decay phase noise model with b= 4, fl = 100 KHz and fh = 10fl while a is chosen from
5 10 15 20 25 30 35 10−2 10−1 100 SNR (E b/N0, dB)
Probability of Symbol Error
No correction a=7, LS a=7, ML a=7.5, LS a=7.5, ML Ideal
(a) r.m.s. delay spread = 50 ns
5 10 15 20 25 30 35 10−2 10−1 100 SNR (E b/N0, dB)
Probability of Symbol Error
No correction a=7, LS a=7, ML a=7.5, LS a=7.5, ML Ideal (b) r.m.s. delay spread = 75 ns
Fig. 2. Symbol error rate with 16 QAM.
5 10 15 20 25 30 35
10−1 100
SNR (Eb/N0, dB)
Probability of Symbol Error
No correction a=7, LS a=7, ML a=7.5, LS a=7.5, ML Ideal
(a) r.m.s. delay spread = 50 ns
5 10 15 20 25 30 35
10−1 100
SNR (Eb/N0, dB)
Probability of Symbol Error
No correction a=7, LS a=7, ML a=7.5, LS a=7.5, ML Ideal (b) r.m.s. delay spread = 75 ns
Fig. 3. Symbol error rate with 64 QAM.
{7, 7.5} and c = a + 4. Pilot-aided approach based on
least-square criterion [4] is also simulated as a comparison. Each simulation point is generated using 3 · 105 OFDM symbols. The probability of symbol error (SER) with 16 QAM and 64 QAM in different channels are shown in Fig. 2 and Fig. 3 respectively.
About the effect of different channels, comparing Fig. 2(a) with Fig. 2(b), we can find that the shorter rms delay spread gives the better performance which is also evident in Fig 3. Observing Fig. 2 which corresponds to 16 QAM modulation, in general, the proposed ML estimator outperforms the LS estimator. The performance gaps between them at moderate SNR are 3 dB when a= 7 and 1 dB when a = 7.5 respec-tively. In Fig. 3 which corresponds to 64 QAM modulation, the performance gaps between the proposed ML estimator and the LS estimator at moderate SNR are both 2.5 dB when a= 7 and7.5 respectively. In particular, to see the composite effect of different modulation schemes and phase noise severity, we first observe that under severer phase noise (a = 7), the performance gaps between the proposed ML estimator and the LS estimator in 64 QAM become smaller compared to
16 QAM. This phenomenon is mainly caused by the reduced correctness of the tentative decisions. Contrarily, under mild phase noise (a = 7.5), the performance gaps in 64 QAM become larger compared to 16 QAM. This phenomenon results from the closer Gaussian approximation of the distribution of the ICI when employing high order modulation which makes the estimation more accurate while the tantative decisons are still reliable under mild phase noise.
To examine the robustness of our proposed algorithm with respect to model mismatch, we discuss the situation when a parameter mismatch presents in the analytical phase noise model used at the receiver. The dominant parameter a is inspected and the results are illustrated in Fig. 4. The mod-ulation used is 16 QAM and the channel r.m.s delay spread is 50 ns. The phase noise is generated by the aforementioned assumption except that a is fixed to 7. At the receiver, we vary the value of a which is used in the proposed CPE estimator to test the robustness. The operating SNR in the simulation are 15 dB, 20 dB and 25 dB. As we can see, under the parameter mismatch of a, our proposed algorithm is quite robust and can still outperform the conventional least-square based approach
5 5.5 6 6.5 7 7.5 8 8.5 9 10−3 10−2 10−1 a SER 15db,LS 15db,ML 20db,LS 20db,ML 25db,LS 25db,ML
Fig. 4. The effect of model mismatch ofa.
in the range of practical interest which justify the practicability of the proposed algorithm.
VII. CONCLUSION
In this letter, we propose a common phase error estimator based on maximum likelihood criterion to effectively remove the complex gain caused by stationary phase noise on the frequency domain transmitted symbols. We systematically de-rive the autocorrelation function of the ICI weighting function based on two stationary phase noise models. The second order statistics of the ICI can be calculated by this autocorrelation function. Then, different from conventional maximum like-lihood approach that ideally assumes the carrier inter-ference observed on different subcarriers to be independent
identically distributed, we combine pilot-aided and decision-directed approaches and investigate their covariances to yield the generalized maximum likelihood estimation scheme. The effectiveness of the proposed algorithm is manifested by computer simulations, moreover, the proposed algorithm is robust that in case of possible model mismatch, it still works and outperforms the conventional scheme.
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