Pilot-Aided Sequential Monte Carlo Estimation of Phase Distortions and Transmitted Symbols in Multicarrier Systems

Abstract

We address the challenging problem of the joint estimation of transmitted symbols and phase distortions in standardized multicarrier systems, including pilot or virtual subcarriers. These subcarriers create time correlation on the useful transmitted OFDM signal that we propose to take into account by an autoregressive model. Because the phase distortions are nonlinear, we set the joint estimation algorithm on the framework of the Sequential Monte Carlo methods. Simulation results are provided in terms of bit error rate (BER) and mean square error (MSE); they highlight the efficiency and the robustness of the estimator.

1. Introduction

Regarding the“Digital Subscriber lines’’ (DSLs), the “Digital Audio Broadcast’’ (DAB), the IEEE 802.11 “Wireless Local Area Network’’ (WLAN), or the IEEE 802.16, most of the recent communications systems are based on orthogonal multicarrier technologies. Unfortunately, such technologies are sensitive to phase noise (PHN) and carrier frequency offset (CFO) coming from the defaults of the oscillators. These phase distortions destroy the orthogonality between subcarriers and lead after Fourier Transform to both a common rotation of all the symbols and intercarrier interferences. In this paper, we propose to take into account the time evolution of the OFDM symbols for the joint estimation of the transmitted symbols, the phase noise and the frequency offset. The Sequential Monte-Carlo-based algorithm we have proposed in [1] for this joint estimation achieves unequaled performance when dealing with orthogonal multicarrier systems without virtual and pilot subcarriers. But, in multicarrier standards, virtual and pilot subcarriers are used, taking up spectral resources and leading time correlation of the useful transmitted OFDM signal. This correlation has not been considered in the algorithm developed in [1], making it suboptimal in practical cases. As a consequence, we propose here to take into account the correlation of the unknown OFDM signal by an autoregressive model (AR) and to deduce a new dynamic state space model on which the SMC estimation algorithm is built. The rest of the paper is organized as follows. Section 2 is devoted to the formulation of the problem. In Section 3, the estimation of the AR parameters is given followed by the SMC estimator of the a posteriori distribution of the unknown OFDM symbols, the PHN and the CFO. Section 4 presents the performance of our algorithm and compares it to a perfect Common Phase Error (CPE) correction scheme which corresponds to the ideal case of [2]. Section 5 concludes this paper.

In this paper, 𝑁, 𝑁_cp, and 𝑇 denote, respectively, the number of subcarriers, the cyclic prefix length, and the useful OFDM symbol duration. Let 𝒩(𝑥;𝜇,𝚺) and 𝒩_𝑐(𝑥;𝜇,𝚺) represent, respectively, the real and circularly symmetric complex Gaussian random vectors with mean 𝜇, and let covariance matrices 𝚺. 𝐈_𝑛 and 𝟎_𝑛×𝑚 be, respectively, the 𝑛×𝑛 identity matrix and the 𝑛×𝑚 matrix of zeros. Finally, lower case bold letters are used for column vectors and capital bold letters are used for matrices; (⋅)^∗, (⋅)^𝑇, and (⋅)^𝐻 denote, respectively, conjugate, transpose, and Hermitian transpose.

2. Problem Formulation

2.1. Equations of the Phase Distortions and of the Received Signal

First, input i.i.d. bits are encoded into M-QAM symbols 𝑑_𝑛,𝑖, where 𝑖 denotes the 𝑖th subcarrier and 𝑛 denotes the 𝑛th OFDM symbol. In standard systems, some of the subcarriers located at the edges of the OFDM block are not modulated. These 𝑁_𝑔 subcarriers are referred to as virtual subcarriers (VSCs) and belong to the set Ω_𝑔. Moreover, a set of 𝑃 pilot subcarriers are used. They are located on Ω_𝑝, with {Ω_𝑝∩Ω_𝑔}=∅ and Ω_𝑝∪Ω_𝑔=Ω.

After the Inverse Discrete Fourier Transform (IDFT), the samples of the transmitted signal can be written, for 𝑙=0,…,𝑁−1, as follows: 𝑠_𝑛,𝑙=1√𝑁^𝑁−1_𝑖=0𝑑_𝑛,𝑖𝑒^{𝑗2𝜋𝑖𝑙/𝑁}. (1) Before transmission, a cyclic prefix (CP) of length 𝑁_cp is introduced to remove intersymbol interference (ISI). By convention, negative time index 𝑡 is used for the prefix cyclic as follows: 𝑠_𝑛,𝑡=𝑠_{𝑛,𝑁+𝑡}if−𝑁_cp𝑠≤𝑡≤−1_{𝑛−1,𝑡+𝑁+𝑁cp}if𝑡<−𝑁_cp, (2) where 𝑠_𝑛,𝑡 can be decomposed into a known data signal 𝑘_𝑛,𝑡 and an unknown data signal 𝑢_𝑛,𝑡 so that 𝑠_𝑛,𝑡=𝑘_𝑛,𝑡+𝑢_𝑛,𝑡 (3) with 𝑘_𝑛,𝑡=1√𝑁_𝑖∈Ω𝑑_𝑛,𝑖𝑒^{𝑗2𝜋𝑖𝑡/𝑁}𝑢_𝑛,𝑡,=1√𝑁^𝑁−1_𝑖=0∉Ω𝑑_𝑛,𝑖𝑒^{𝑗2𝜋𝑖𝑡/𝑁}. (4)(5)

The resulting signal is transmitted in a time varying frequency selective channel ℎ(𝑡,𝜏) which is assumed both to be invariant over one OFDM symbol and to be characterized by 𝐿_𝑝 independent propagation paths. Imperfect oscillators at the transmitter and at the receiver introduce a phase noise and a carrier frequency offset Δ𝑓 so that, at the sampling rate 𝑁/𝑇 of the receiver, the discrete form of the carrier delivered by the noisy oscillator on the 𝑡th sample of the 𝑛th OFDM symbol is 𝑝_𝑛,𝑡=exp(𝑗𝜙_𝑛,𝑡)∀𝑡=0,…,𝑁+𝑁_cp−1, (6) where 𝜙_𝑛,𝑡 is the phase distortion which includes the Brownian phase noise and the CFO 𝜙_𝑛,𝑡=𝜈_𝑛,0,𝜙_{𝑛,𝑡−1}+2𝜋𝜖𝑁+𝜈_𝑛,𝑡∀𝑡=1,…,𝑁+𝑁_cp−1 (7) with 𝜖 being the normalized CFO. 𝜖 is a noninformative variable which is uniformly distributed on the support [(Δ𝑓𝑇)_min,(Δ𝑓𝑇)_max]. 𝜈_𝑛,𝑡 is a white Gaussian noise with variance 𝜎²_𝜈=2𝜋𝛽(𝑇/𝑁) where 𝛽 is the bandwidth of the Brownian phase noise normalized with respect to the OFDM symbol rate 1/𝑇, namely, the parameter 𝛽𝑇.

Under matrix notation, the received signal 𝑟_𝑛,𝑡 corrupted by phase noise and frequency offset can be written before removal of the cyclic prefix as follows: 𝑟_𝑛,𝑡=𝑒^{𝑗𝜙𝑛,𝑡}𝐡^𝑇_𝑛(𝐤_𝑛,𝑡+𝐮_𝑛,𝑡)+𝑤_𝑛,𝑡, (8) where 𝐮_𝑛,𝑡=𝑢_{𝑛,𝑡−𝑁cp}⋯𝑢_{𝑛,−𝑁cp−𝐿𝑝+1}𝟎_{1×(𝑁+𝑁cp−𝑡−1)}^𝑇,𝐤_𝑛,𝑡=𝑘_{𝑛,𝑡−𝑁cp}⋯𝑘_{𝑛,−𝑁cp−𝐿𝑝+1}𝟎_{1×(𝑁+𝑁cp−𝑡−1)}^𝑇,𝐡_𝑛=ℎ_𝑛,0⋯ℎ_{𝑛,𝐿𝑝−1}𝟎_{1×(𝑁+𝑁cp−1)}^𝑇. (9)

At the receiver, the cyclic prefix is first discarded, then, assuming 𝑁_cp≥𝐿_𝑝 and perfect timing synchronization, a Discrete Fourier Transform is performed on the 𝑁 remaining samples of the received signal. The useful signal is then corrupted by two types of distortions: a multiplicative one which is common to all subcarriers and known as Common Phase Error (CPE) and an additive intercarrier interference (ICI). The larger the phase noise rate 𝛽𝑇 is, the worse the SINR (signal to noise plus interference ratio) is. Furthermore, on a same bandwidth, a larger number of subcarriers lead to worse system performance due to the shorter subcarrier spacing for a same two-sided 3 dB bandwidth of the phase noise [3].

2.2. Autoregressive Modeling of the Unknown Data Signal Process

Due to the independence of the process between two different OFDM symbols, the process at time 𝑡 can be modeled via the time domain recursion by the complex AR model of order (𝑡−1) (i.e., AR(𝑡−1)) 𝑢_𝑛,𝑡=−^𝑡−1_𝑖=1𝑎_𝑡,𝑖𝑢_{𝑛,𝑡−𝑖}+𝑏_𝑛,𝑡, (10) where 𝑏_𝑛,𝑡 is a circular white Gaussian noise. The AR model parameters consist of the filter coefficients {𝑎_𝑡,1,𝑎_𝑡,2,…,𝑎_{𝑡,𝑡−1}} and the driving noise variance 𝜎²_{𝑏𝑛,𝑡}.

They are obtained by solving both the noise variance equation 𝜎²_{𝑏𝑛,𝑡}=𝐶_𝑢𝑢(0)+𝐚^𝑇_𝑡𝐱_𝑛,𝑡 (11) and the Yule-Walker equation 𝐂_𝑛,𝑡𝐚_𝑡=−𝐱_𝑛,𝑡 (12) with 𝐚_𝑡=𝑎_𝑡,1𝑎_𝑡,2⋯𝑎_{𝑡,𝑡−1}^𝑇,𝐱_𝑛,𝑡=𝐶_𝑢𝑢(1)𝐶_𝑢𝑢(2)⋯𝐶_𝑢𝑢(𝑡−1)^𝑇,𝐂_𝑛,𝑡=𝐸[𝐮_𝐧,𝐭𝐮^𝐇_𝐧,𝐭]. (13)

The correlation function 𝐶_𝑢𝑢(𝑡−𝑙) is obtained by using the cyclostationarity with period 𝑁 of the process 𝑢_𝑛,𝑡 as follows: 𝐶_𝑢𝑢(𝑡−𝑙)=𝑁−𝑃−𝑁_𝑔𝑁−1if𝑡=𝑙,𝑁_𝑖∈Ω𝑒^{𝑗2𝜋𝑖(𝑡−𝑙)/𝑁}if𝑡≠𝑙. (14)

This correlation function clearly shows that, for (𝑡−𝑙)<𝑁, 𝑢_𝑛,𝑡 is colored. Moreover, in (5), we can denote that the term 𝑢_𝑛,𝑡 is the sum of (𝑁−𝑃−𝑁_𝑔) independent and unit power random variables. Based on the central limit theorem, a rigorous proof given in [4] establishes that the complex envelope of a bandlimited uncoded OFDM signal converges to a Gaussian random process. Consequently, 𝑢_𝑛,𝑡 is modeled by a Gaussian autoregressive process. Using (10), the state equation of the vector 𝐮_𝑛,𝑡 can finally be written in the matrix form as 𝐮_𝑛,𝑡=𝐀_𝑡𝐮_{𝑛,𝑡−1}+𝐛_𝑛,𝑡, (15) where the transition matrix 𝐀_𝑡 is defined as: 𝐀_𝑡=𝝃_𝑡^𝑇𝐈_{(𝑁+𝑁cp+𝐿−2)}𝟎_{(𝑁+𝑁cp+𝐿−2)×1} (16) with, according to the cyclostationarity of 𝑢_𝑛,𝑡 and (10), 𝝃_𝑡=𝐚_𝑡𝟎_{1×(𝑁+𝑁cp+𝐿−𝑡)}^𝑇𝟎if0≤𝑡≤𝑁−1,_{1×(𝑁−1)}1𝟎_{1×(𝑁cp+𝐿−1)}^𝑇if𝑁≤𝑡≤𝑁+𝑁_cp−1 (17) and 𝐛_𝑛,𝑡 is a (𝑁+𝑁_cp+𝐿−1)-by-1 zero mean Gaussian noise vector with the covariance matrix 𝐸[𝐛_𝑛,𝑡𝐛^𝐻_𝑛,𝑡𝜎]=^{2𝑏𝑛,𝑡}0⋮0⋯0, (18) where, using (11), 𝜎²_{𝑏𝑛,𝑡}=𝐶_𝑢𝑢(0)+^𝑘−1_𝑖=1𝑎_𝑡,𝑖𝐶_𝑢𝑢(−𝑖)if0≤𝑡≤𝑁−1,0if𝑁≤𝑡≤𝑁+𝑁_cp−1. (19)

2.3. Dynamic State Space Model

By using the dynamic evolution of the PHN (7) and the proposed AR modeling of the unknown OFDM signal (15), we obtain the following DSS model: 𝜙_𝑛,𝑡=𝜈_𝑛,0𝜙_{𝑛,𝑡−1}+2𝜋𝜖𝑁+𝜈_𝑛,𝑡∀𝑡=1,…,𝑁+𝑁_cp𝐮−1,_𝑛,𝑡=𝐀_𝑡𝐮_{𝑛,𝑡−1}+𝐛_𝐧,𝐭,𝑟_𝑛,𝑡=𝑒^{𝑗𝜙𝑛,𝑡}𝐡^𝑇_𝑛(𝐤_𝑛,𝑡+𝐮_𝑛,𝑡)+𝑤_𝑛,𝑡. (20)

In order to jointly estimate 𝜙_𝑛,𝑡, 𝜖, and 𝐮_𝑛,𝑡, we need the joint posterior probability density function (p.d.f.) 𝑝(𝜙_𝑛,𝑡,𝜖,𝐮_𝑛,𝑡∣𝑟_𝑛,𝑡). Unfortunately, this p.d.f. is analytically intractable, so we propose to numerically approximate 𝑝(𝜙_{𝑛,0∶𝑡},𝜖,𝐮_{𝑛,0∶𝑡}∣𝑟_{𝑛,0∶𝑡}) via the SMC methodology [5].

3. SMC Method for Joint OFDM Signal, CFO, and PHN Estimation

The dynamic state space model (20) depends not only on the hidden state process 𝜙_𝑛,𝑡, 𝐮_𝑛,𝑡 and the hidden variable 𝜖 but also on the parameters of the AR process which have to be evaluated before the estimation of the processes of interest. The first subsection is devoted to the identification of the unknown OFDM symbol by the AR process. Then the SMC-based joint estimation of the OFDM signal, the CFO, and the PHN is deduced.

3.1. AR Parameters of the Unknown OFDM Signal

The correlation of the unknown OFDM signal 𝑢_𝑛,𝑡 can easily be calculated from (14) so that the Yule-Walker equation (12) can be solved efficiently by the Levinson-Durbin recursion. However, as the process 𝑢_𝑛,𝑡 consists of a sum of (𝑁−𝑃−𝑁_𝑔) sinusoids, a necessary condition is that the order of the AR process satisfies 𝑡−1≤(𝑁−𝑃−𝑁_𝑔−1) [6]. With this assumption, the inverse matrix 𝐂⁻¹_𝑛,𝑡 exists, and the Yule-Walker equation has a unique solution: 𝐚_𝑡=−𝐂⁻¹_𝑛,𝑡𝐱_𝑛,𝑡. Nevertheless, as shown in [7], adding a small value 𝜏₀ to its principle diagonal enables the stability and the accuracy of larger-order AR models. This strategy is equivalent to the introduction of white noise of variance 𝜏₀ to the original process. The addition of this spectral bias removes the bandlimitation of the original spectrum and creates a nondeterministic or regular process that in some sense closely approximates the original process. Consequently using this approach, the order of the AR process can be chosen up to 𝑁−1 (the largest possible order in our context). The choice of 𝜏₀ thus represents a good tradeoff between the improvement of the AR modeling of 𝑢_𝑛,𝑡 and the bias introduced in the zeroth autocorrelation lag (i.e., in the signal power).

3.2. Joint Estimation of the Phase Distortions and of the Transmitted Symbols

We notice that the DSS model (20) is similar to the one obtained when the multicarrier system has no virtual or pilot subcarriers. Only the matrices 𝐀_𝑡 (16) and 𝜎²_{𝑏𝑛,𝑡} (11) differ from the DSS model used in [1] to build the joint SMC estimation without pilot and virtual subcarriers. As a consequence, we do not have details on the JSCPE-MPF (joint signal, CFO, and PHN estimation using marginalized particle filter) algorithm in this paper and suggest that the reader refer to [1].

4. Results

With regard to the system parameters, 16-QAM modulation is assumed and we have chosen 𝑁=64 subcarriers with a cyclic prefix of length 𝑁_cp=8. A Rayleigh frequency selective channel with 𝐿=4 paths and a uniform power delay profile, perfectly known by the receiver, has been generated for each multicarrier symbol. In the following simulations, the CFO term 𝜖 is generated from a uniform distribution in [−0.4;0.4] for each multicarrier symbol which severely degrades the received signal.

Figure 1 shows the BER performance of the proposed algorithm for different PHN rates and also different numbers of null subcarriers with and without the AR modeling of the multicarrier signal. The BER performance of a multicarrier system in the absence of phase distortions, using the classical frequency domain MMSE equalizer is also depicted and denoted below by MMSE-FEQ. From this figure, it can be firstly denoted that the perfect CPE correction, which corresponds to the ideal case of [2], achieves unsatisfactory BER performance due to a large ICI induced by the severe phase impairments. Then, we can remark that the proposed scheme exhibits very good results. Performance of our algorithm increases obviously with the number of null subcarriers, especially for high level of phase distortions. In fact, the proposed AR model of the multicarrier signal gives more prior information if the number of pilots or null subcarriers increases and thus improves the robustness of the proposed estimator. For 𝛽𝑇=10⁻² and 𝐸_𝑏/𝑁₀=30 dB, BER performance of the JSCPE-MPF is, respectively, 4×10⁻³ and 1.8×10⁻³ without and with the proposed AR model. In order to illustrate the gain obtained by the use of the proposed signal model, the MSE of the multicarrier signal obtained using the JSCPE-MPF with or without our AR modeling is shown in Figure 2. The performance of the proposed estimator is compared to the Posterior Cramér-Rao bound (PCRB) of a multicarrier system without phase distortions derived in [1]. The performance gap with and without the AR model really becomes significant when the number of null subcarrier increases. All these results clearly highlight the benefit from the proposed AR model which makes the SMC estimator more robust to severe phase distortions in practical system configurations.

(a) With AR model

(b) Without AR model

Figure 1 

BER performance of the proposed JSCPE-MPF versus 𝐸𝑏/𝑁0 for different PHN rates 𝛽𝑇 and numbers of null subcarriers (𝑃=4, 𝜖=0.4).

(a) 𝑁𝑔=0

(b) 𝑁𝑔=8

Figure 2 

MSE of the multicarrier signal estimate using the proposed AR model (solid lines) and without the AR model (dashed lines) versus 𝐸𝑏/𝑁0 for different PHN rates 𝛽𝑇 (𝑃=4, 𝜖=0.4).

5. Conclusion

In this paper, we address the difficult problem of data detection in pilot-aided multicarrier systems that suffer from the presence of phase noise and carrier frequency offset. The originality of this work consists in an autoregressive modeling of the OFDM signal from which we have deduced an SMC method for time domain processing of the nonlinear received signal. Numerical simulations show that even with significant PHN rates, the JSCPE-MPF achieves good performance in terms of both the phase distortion estimation and BER performance; moreover, it offers a significant performance gain in comparison to existing methods. Thus the JSCPE-MPF algorithm with AR modeling can be efficiently used with the channel estimator proposed in [8] for the design of a complete multicarrier receiver in wireline and wireless communication systems.