Abstract: Compressive sensing is an emerging area which uses a relatively small number of non-traditional samples in the form of randomized projections to reconstruct sparse or compressible signals. This study considered the carrier frequency offset estimation problem for interleaved orthogonal frequency-division multiple-access uplink systems. A new carrier frequency offset estimation method based on the compressive sensing theory is proposed to estimate the carrier frequency offsets in interleaved OFDMA uplink systems. The presented method can effectively estimate all carrier frequency offsets of the active users by finding the sparest coefficients. Simulation results are presented to verify the efficiency of the proposed approach.
INTRODUCTION
Orthogonal frequency-division multiple access (OFDMA) has been or is being considered for various wireless communication systems, such as IEEE 802.16, 802.20 (Bolton et al., 2007), 802.22 (Cordeiro et al., 2006). However, similar to OFDM, OFDMA is extremely sensitive to the carrier frequency offsets (CFOs) caused by oscillator mismatch and Doppler shift. The uncompensated CFOs will result in loss of orthogonality among subcarriers and introduce intercarrier interference (ICI) (Abdellaoui et al., 2006; Kumar et al., 2008; Jiang et al., 2010) as well as Multiple Access Interference (MAI) (Venkatesan and Ravichandran, 2007; Ren et al., 2008; Wu et al., 2010), thus leading to severe performance degradation. Frequency synchronization is critically important for the OFDMA system. With imperfect synchronization, ICI is generated due to the loss of subcarrier orthogonality, contributing both co-channel and interchannel interference. In OFDMA downlink systems, each user could perform CFO estimation by exploiting the preamble from the Base Station (BS) at the beginning of each downlink time slot. This operation can be easily accomplished by using the algorithms employed in OFDM systems (Schmidl and Cox, 1997; Zhao et al., 2011). However, in OFDMA uplink systems, distinct users pass through different channels and experience distinct CFOs. All CFOs have to be simultaneously estimated via the received signal which is the superposition of signals from all active users. Thus, CFO estimation for uplink systems is a multiple-parameter estimation problem and hence, it is much more difficult than the downlink case. Therefore, reliable CFO estimation has been a challenging task and is critical for the data detection for OFDMA uplink systems.
CFO estimation schemes for uplink OFDMA systems can be categorized as data-aided and non-data-aided. Pun et al. (2006) proposed an effective iterative scheme which is referred to as the alternating-projection frequency estimator (APFE). However, it requires an exhaustive grid searching to estimate each CFO which results in unattractive complexity. To reduce the complexity, Sezginer and Bianchi (2008) reported a suboptimal approach by replacing the matrix inversion in APFE with an approximated matrix extension. However, it requires a large number of subcarriers. The estimator (Wu et al., 2011) utilizes constant amplitude zero autocorrelation (CAZAC) training sequences to obtain the CFOs. However, it causes an irreducible error floor. A simple iterative algorithm (Sun and Zhang, 2011) is derived via exploiting the fact that the tile structure in 802.16e provides inherent MAI power compression in the frequency domain.
When training sequences are not available, the CFO for each user cannot easily be obtained. Fortunately, this problem can be overcome with a proper Carrier Assignment Scheme (CAS), such as interleaved CAS. With the interleave-based CAS, the CFO estimators (Cao et al., 2004; Lee et al., 2007) exploit the inherent periodic structure of each user's signal. Due to the spectrum peak searching involved, the computational complexity of the MUSIC estimator (Cao et al., 2004) is not attractive. The ESPRIT estimator (Lee et al., 2007) can avoid the complex peak searching at the expense of the loss of estimation performance. Based on interleaved CAS, Hsieh and Wu (2011) proposed an optimum CFO estimator by truncating the series expansion of the correlation matrix in maximum likelihood function.
In most previous works, the CFO estimation is based on the Nyquist sampling. Compressed Sensing (CS) theory (Donoho, 2006; Candes et al., 2006; Wu et al., 2010) is a novel data collection and coding theory under the condition that signal is sparse or compressible. It has been shown that when the measurement matrix satisfies certain random properties, the original signal can be reconstructed even when the number of observation is far less than the number of Nyquist rate samples. It is noted that the CS theory quickly gains intensive attention in many areas including imaging, communications, analog-to-digital conversion, direction estimation as well as text classification, etc. (Berger et al., 2010; Deng et al., 2011; Cevher et al., 2008; Liu et al., 2011; Deng et al., 2012).
Motivated by the rich research results using compressive sensing in different areas, this paper focused on the CFO estimation problem for interleaved OFDMA uplink systems based on CS theory.
DATA MODEL FOR INTERLEAVED OFDMA UPLINK SYSTEMS
In this section, we introduce the uplink signal model of OFDMA (Fig. 1). Consider a K-user OFDMA uplink system with N subcarriers. Assume that the N sub-carriers are interleaved into Q subchannels, each of which has P = N/Q uniformly spaced subcarriers. The qth sub-channel consists of subcarriers with the index set {q, Q+q, ..., (P-1) Q+q}, (q = 0, 1, 2, ..., Q-1). Suppose that each user has only one subchannel and the P subcarriers of the qth subchannel are assigned to the kth user. After the Cyclic Prefix (CP) is removed, the signal sample of the nth subcarrier of one OFDMA block at the uplink receiver can be described as:
| (1) |
where, K denotes the total number of active users in the system. z(n) presents the spatially and temporally white additive Gaussian noise with zero-mean and equal variance σ2. rk(n) stands for the signal sample of the nth subcarrier from the kth user at the uplink receiver, it is given by:
where, the subscript (.)k denotes the kth user. Hk,p denotes the channel frequency response on the pth subcarrier of the kth user's channel during one OFDMA block. Xk,p denotes the data symbol on the pth subcarrier of the kth user. ξk = Δfk/Δf is the normalized CFO of the kth user. Δf denotes the subcarrier spacing and Δfk represents the CFO between the kth user and the uplink receiver. For practical purpose, the absolute value of Δfk is assumed to be less than half of OFDMA subcarrier spacing, that is, ξk ε (-0.5, 0.5).
| Fig. 1: | The OFDMA uplink model |
From (1), we construct a QxP matrix Y by a data stacking technology. The matrix Y has the following form:
| (2) |
where, the Vandermonde matrix:
(.)T denotes the transpose operation and:
being the effective CFO of the kth user.
Notice that the effective CFO has one important property, that is, different users have distinct effective CFOs. It can be shown that if one user occupies subchannel q, the range of its effective CFO is((q-0.5)/Q, (q+0.5)/Q), since the range of its normalized CFO is (-0.5, 0.5). Because different users occupy different subchannels, their effective CFOs fall in nonoverlapping ranges.
CFO ESTIMATION BASED ON CS THEORY
Assume that there exist a small number of users, the CFOs are sparse in the
effective CFO space, i.e.,
Sparse representation and atom dictionary: In order to accomplish the CFO estimation based on CS theory, sparse representation and over-complete atom dictionary need to be constructed for CFO estimation problem.
Let yl, sl, zl denote the lth column of Y, S, Z, (l = 1, 2, ..., P). Thus, we have the following equation:
| (3) |
where, V is defined in (2). Similarly, let vk denote the kth column
of the matrix V, namely,
The covariance matrix of vector yl is given by:
| (4) |
where E{.} and (.)H denote the expectation and the conjugate transpose,
respectively. The signal covariance matrix is
Consider each column
| (5) |
where (.)* denotes the complex conjugate operation. Yl,m is the
mth element of the vector yl. The atom dictionary
Observation matrix: In the normal circumstances, the Gaussian random matrix can act as the observation. The observation matrix Φ is the LxQ Gaussian random matrix, where, L presents the number of random sampling points. We can use the L points to accomplish the CFO estimation.
CFO estimation: In the noise environment, we can use the following formula to solve the problem according to the BP theory. The convex problem can be given as follows:
| (6) |
where, the down sampled data xm = Φrm. Φ presents
the observation matrix. The matrix
For the optimization problem (6), we can use the CVX toolbox (Grant and Boyd, 2009) to get the sparsest representation η, then figure out some biggest sparse coefficient components of η.
The positions of these K biggest sparse coefficient components located on the vector K should be recorded. Furthermore, according to the positions we should find the corresponding atom in dictionary. Thus, we can obtain the CFO estimation on the basis of the atom.
SIMULATION RESULTS
In this section, some simulations are conducted to validate the performance of the proposed CFO estimation based on CS theory.
Case 1: The considered OFDMA system has N = 1024 subcarriers. QPSK modulation scheme is used to generate the transmitted symbols. The wireless channel is modeled as a frequency-selective fading channel consisting of six independent Rayleigh multipaths, with an exponentially decaying power-delay profile. In this simulation, the numbers of subchannels and users are set to Q = 8 and K = 3, respectively. The normalized CFOs are set to -0.2, 0.1, 0.35, respectively. The signal-to-noise-ratio (SNR) is set to 5 dB. Figure 2 gives the CFO curves estimated by the proposed method and MUSIC, respectively. As shown in this figure, all interesting CFOs are successfully estimated by the proposed method and MUSIC. Figure 2 shows that the performance of the proposed method is similar to that of MUSIC.
Case 2: Except for the normalized CFOs of all active users (ξk, k = 1, 2, ..., K) are generated as random variables that are uniformly distributed in (-0.5, 0.5) with 500 random trial runs, other simulation conditions are similar to Case 1. Figure 3 gives the normalized Root Mean Square Error (RMSE) curves of the CFOs estimated by the aforementioned methods. The RMSE curves with SNRs ranging from 0 to 20 dB. The RMSE is defined as:
| (7) |
where, K is the number of all users, Π is the total number of Monte Carlo
trials,
Case 3: Except for the number of users, other simulation conditions are similar to Case 2. Figure 3 gives the normalized root mean square error (RMSE) curves of the CFOs estimated via the proposed algorithm when the number of users varies. As can be seen in Fig. 4, the proposed estimator can work well in different users. Meanwhile, as expected, the less the user number, the better performance we will obtain.
| Fig. 2(a-b): | Comparison of the CFO estimation between MUSIC and the proposed method |
| Fig. 3: | Comparison of the RMSE curves versus SNR for MUSIC, ESPRIT and the proposed method |
| Fig. 4: | Comparison of the RMSE curves versus SNR for various K |
CONCLUSIONS
In this study, a new CFO estimation method is proposed for the interleaved OFDMA uplink systems. According to the compressive sensing theory, the presented method can simultaneously estimate the CFOs of all active users by finding the sparest coefficients which can be obtained by solving the convex optimization problem. Simulation results verify that the proposed CFO estimation method is effective, practical and easily achieved.
ACKNOWLEDGMENTS
This study has been supported by the National Natural Science Foundation of China under Grant No. 61104005, by Directive Plan of Science Research from the Bureau of Education of Hebei Province, China, under Grant Nos.Z2011129 and by Science and Technology Support planning project of Hebei Province, China, under Grand No. 11213502D.