Compressive sensing based joint frequency offset and channel estimation for OFDM
 Rıfat Volkan Şenyuva^{1}Email author,
 Güneş Karabulut Kurt^{2} and
 Emin Anarim^{1}
https://doi.org/10.1186/s1363801605828
© Şenyuva et al. 2016
Received: 3 June 2015
Accepted: 3 March 2016
Published: 22 March 2016
Abstract
We consider joint estimation of carrier frequency offset (CFO) and channel impulse response (CIR) for orthogonal frequency division multiplexing (OFDM) with pilot symbols. A new method based on compressed sensing is proposed. It has been shown that the CIR can be represented as a 1block sparse signal by using a dictionary constructed by concatenating subspaces of CFO values taken from a search space. Recovery of both CFO and CIR is accomplished by the block orthogonal matching pursuit algorithm. The proposed method uses only one OFDM training block and does not require any initialization. The performance of the proposed method is compared against the wellestablished pilot based estimators: Moose, Classen, the maximum likelihood estimator, and the palgorithm. Numerical results show that the performance of the proposed method does not depend on the value of the CFO. We also give worstcase upper bounds for the mean squared error of the CIR estimate for a sparse multipath channel.
Keywords
Carrier frequency offset Channel estimation Compressive sensing Blocksparsity OFDM1 Introduction
Orthogonal frequency division multiplexing (OFDM) has become a standard multicarrier modulation technique for broadband wireless communication networks due to its resistance to interblock interference (IBI) caused by frequencyselective multipath fading channels. The success of OFDM systems relies on the orthogonality property of its chosen subcarriers which transforms the frequency selective channel into a set of frequency flat fading channels simplifying the equalization task. However, it is known that OFDM is very sensitive to both frequency synchronization and channel estimation errors. The oscillator mismatches and/or Doppler shifts introduce carrier frequency offset (CFO) which destroys the orthogonality amongst the subcarriers. As a result, the signal constellation is rotated and intercarrier interference (ICI) occurs. The quality of the channel estimates, which is vital for coherent data detection, is also affected negatively under CFO. All of these effects reduce the effective signaltonoise ratio (SNR) at the receiver and degrade the performance of the OFDM system. Thus, accurate CFO and channel estimation is essential for exploiting the full potential of OFDM systems.
CFO and channel estimation can be done separately by first estimating the CFO and then performing channel estimation in the second step. Both training symbolbased and blind CFO estimators have been proposed in the literature [1–5]. The estimator proposed by Moose uses repeated data symbols [1]. Classen’s method inserts pilot subcarriers in OFDM blocks for CFO estimation and assume that the channel does not change for two consecutive OFDM blocks [2]. Both methods assume a sufficiently small CFO and also a not high SNR so that the ICI is much smaller than the noise and can be ignored. They are sensitive to the value of the CFO and are valid for small CFO values. The blind estimator developed by Beek exploits the redundancy in the cyclic prefix (CP) [3]. A blind estimator based on MUSIC subspace method is developed in [4] using the virtual subcarriers (VC). The palgorithm in [5] considers both VC and pilot carriers in each OFDM block but assumes that the channel remains constant for two OFDM blocks like the Classen method. Channel estimation can be performed after applying these wellestablished CFO synchronization methods. However, since perfect synchronization is not possible, the residual CFO will degrade the performance of the channel estimate significantly. Thus, better performance can be obtained when CFO and channel are estimated jointly. There exists a number of joint CFO and channel estimators that use pilot symbols [6–9]. Both [6] and [7] use the framework of the expectationmaximization (EM) algorithm. While [6] directly computes the channel parameters, [7] estimates the parameters of a basis expansionbased parametric channel model assuming the channel delays are known. The initialization of [6] requires a coarse CFO estimate such as provided by the Beek’s method [3]. A joint maximum likelihood estimate (MLE) of CFO and channel impulse response (CIR) using a training symbol is given in [8]. The estimator in [9] is an approximate MLE since the received signal samples of the OFDM system are assumed to be Gaussian. Also, both [6] and [9] require the secondorder statistics of the channel and noise. For [9], an initial estimate of both the channel and CFO is necessary to initialize the joint MLE.
A joint estimator of CFO and CIR using the compressive sensing (CS) framework is proposed in this paper. The framework of CS coined in [10] deals with the recovery of an unknown signal from an underdetermined system of linear equations. By exploiting the key property that the unknown signal is sparse, only a few entries of the signal are nonzero, the perfect reconstruction of the unknown signal is possible even if the system is underdetermined. Sparse signals may have additional structure in the form of nonzero coefficients occurring in clusters (such as in [11, 12]). Such signals are referred to as blocksparse signals [11, 12]. A similar structure can be created by concatenating dictionaries of OFDM training symbols perturbed by CFO values taken from a search space and with such a structure the CIR can be represented as a 1block sparse signal. The proposed framework allows solving for both CFO and CIR simultaneously by using the CS recovery algorithms for blocksparse signals. To the best of our knowledge, the proposed method is the first CSbased approach towards the joint estimation of both CFO and CIR for an OFDM system. There exist numerous implementations of the CSbased block sparse signal recovery methods for the estimation of the CIR of the OFDM systems [13–15]. In these works, block sparsity is achieved by either assuming that the several channel instantiations are groupsparse, locations of the nonzero channel coefficients are same, [15] or concatenating multiple CIRs of different antennas with common support in a block sparse structure [13, 14]. However, frequency offset is not considered in these papers, which assume perfect synchronization for the OFDM system and so only CIR can be estimated using these methods. The block orthogonal matching pursuit (BOMP) algorithm which has a computational complexity of O (d N N _{ g }), d is the length of the search space, N is the number of subcarriers, and N _{ g } is cyclic prefix length, is used as the recovery algorithm. The MLE makes use of an FFTbased search of complexity \(d(5\beta \log _{2}d+1)\) in flops, where β denotes the saving for skipping operations on the zeros in the FFT. However, the input of this implementation requires a matrix inversion step which cannot always be precomputed if the channel and noise statistics are not available [9]. As a result, the complexity of the MLE is determined by the most costly computation step that is the matrix inversion O (N ^{3}). The palgorithm does not involve matrix inversion and can be implemented in two ways: FFTbased search and polynomial rooting [5]. The complexity of the polynomial rooting which is recommended for smaller N and N _{ g } is approximated as O\(\left (\left (2N+N_{g}\right)^{3}\right)\) [5].

The proposed method only needs one OFDM block of training symbols unlike Moose, Classen, and the palgorithm. The use of multiple blocks makes the estimation susceptible to changes in the channel or the CFO. Also, the use of more blocks means an increase in pilot overhead.

The proposed method does not require any initialization or the secondorder statistics of the channel and noise unlike EMbased methods. Apriori knowledge about either the channel or the noise may not be available for every case.

The performance of the proposed method does not depend on the value of the CFO.

Our work makes use of the worstcase bounds of the perturbed CS recovery for sparse multipath channel estimation. These bounds provide a way to observe how the performance of the sparse channel estimation methods scales with the perturbation due to frequency offset.
2 System model
where m denotes the OFDM block index, w _{ N }=e ^{ j2π/N }, and N _{ t } is the total number of samples in a OFDM block including the CP samples, N _{ t }=N+N _{ g }. CP samples required to ensure that no IBI occurs and so the first N _{ g } samples of the OFDM block, {x _{ m }[ 0],…,x _{ m }[ N _{ g }−1]}, are taken same as the last N _{ g } samples of the OFDM block {x _{ m }[ N _{ t }−N _{ g }],…,x _{ m }[ N _{ t }−1]}.
As it is seen from (6), the received signal is affected by the CFO. While the magnitude of the signal H _{ m }[ k]X _{ m }[ k] is attenuated by \(\left  \frac {\sin (\pi \epsilon)} { N \sin (\pi \epsilon /N)} \right \), its phase is increased by π ε(1−1/N)+2π ε[ m(N _{ t }/N)+(N _{ g }/N)]. In addition to noise Z _{ m }[ k], ICI denoted as I _{ m }[ k] is added to the signal. We derive our proposed estimator by using the frequency domain model given in Eq. (6), as detailed below.
3 Compressive sensingbased joint frequency offset and channel estimation
where \({\tilde {\mathbf {A}}}=\mathbf {C}(\epsilon) \mathbf {X}\mathbf {F}_{N_{g}}\), z=[ Z[ 0]⋯Z[N−1]]^{ T }, and \(\mathbf {h}=\left \lbrack h[\!0] \cdots h[N_{g}1] \right \rbrack ^{T}\).
where the vector \({\tilde {\mathbf {h}}}\) is built by stacking d blocks of N _{ g }×1 channel vectors, \({\tilde {\mathbf {h}}}=[\mathbf {h}_{0} \ldots \mathbf {h}_{d1}]\), and the matrix D is constructed by concatenating d matrices each corresponding to a \(\mathbf {C}(\tilde {\epsilon }_{i})\). Each block of \({\tilde {\mathbf {h}}}\) is represented as \({\tilde {\mathbf {h}}}[\!i]=\mathbf {h}_{i}=\lbrack \! h[0], \ldots, h[\!N_{g}1] \rbrack ^{T}\). In this representation, \(\tilde {\mathbf {h}}\) can be considered as a 1block sparse vector.
where q≥0 and I(·) is the indicator function which is zero when its argument is zero and is one otherwise. The indicator function counts the number of nonzero blocks of a solution. However, solving (12) is an NP hard problem since it involves searching over all choices of a few blocks of D [11, 12]. A number of algorithms such as Group Lasso [19], mixed ℓ _{2}/ℓ _{1} program [20], block orthogonal matching pursuit (BOMP) algorithm [11], block sparse Bayesian learning [21], block iteratively reweighted least squares (BIRLS) [22], and block iterative support detection (blockISD) [14] have been shown to recover block sparse signals. In this paper, the BOMP algorithm [11] is implemented for the recovery of the block sparse channel vector.
BOMP algorithm is an extension of the orthogonal matching pursuit (OMP) algorithm [23, 24] used in conventional sparse recovery. The steps of the BOMP algorithm are shown in Algorithm 1.
For 1block sparsity, the BOMP algorithm runs only l=1 iteration. The computational complexity of the algorithm for 1block sparse signals is O (d N N _{ g }). The BOMP method [11] selects the block that is most correlated with the current residual and then applies LS. So at any given iteration, l, of the BOMP method, the solution is guaranteed to be l blocksparse. The performance of the BOMP method is shown to be better than the performance of the BIRLS method when the block sparsity is small [22]. BIRLS method [22] applies weighted LS in each iteration and depends on the weight matrix to make the solution sparse. In the first iteration of BIRLS method, the weight matrix is initialized to the identity matrix which means a regular LS is applied. In [14], blockISD algorithm is implemented to estimate the CIR of a MIMO OFDM system with multiple transmit antennas and no CFO. Block sparse equivalent CIR is generated by assuming that the CIRs from different antennas share a common support. As it can be seen from Eq. (11), the block sparsity structure of our method is exploited by concatenating matrices corresponding to different frequency offsets and so does not use the common support assumption. The proposed dictionary (11) allows the joint estimation of both CFO and CIR while the dictionary generation shown in [14] does not allow the estimation of CFO. BlockISD method [14] is based on the iterative support detection (ISD) reconstruction algorithm [25]. Like the matching pursuit methods, i.e., OMP [23, 24], a support set containing the locations of the detected nonzero elements is maintained in each iteration of the ISD method. At each iteration, this support set is used to solve a truncated basis pursuit (BP) problem by ℓ _{1} minimization. Unlike OMP, the support set of the ISD method can be updated with more than one elements in a given iteration.
4 Perturbation analysis for sparse multipath channel
holds for any Ksparse vector h. It is observed that \(\\mathbf {A}\^{(K)}_{2}=\sigma ^{(K)}_{\text {max}}(\mathbf {A}) \le \sqrt {1+\delta _{K}}\) and \(\sigma ^{(K)}_{\text {min}}(\mathbf {A}) \ge \sqrt {1\delta _{K}}\), where \(\ \mathbf {A} \^{(K)}_{2}\) denotes the largest spectral norm, the largest singular value, taken over all Kcolumn submatrices of A while \(\sigma _{\text {min}}^{K}(\mathbf {A})\) denotes the smallest nonzero singular value over all Kcolumn submatrices of A.
In order for A to satisfy Eq. (32), the RIC must be nonnegative 0<δ _{2K } and so the relative perturbation must be less than \(\varepsilon _{\mathbf {A}}^{(2K)}<\sqrt [4]{2}1\approx 0.1892\).
where the complete solution, h ^{ # }, is obtained by extending \(\mathbf {h}_{S}^{\#}\) with zeropadding on the complement of the support set S and \(\mathbf {A}_{S}^{\dagger }=\left (\mathbf {A}_{S}^{H}\mathbf {A}_{S}\right)^{1}\mathbf {A}_{S}^{H}\) is the pseudoinverse of A _{ S } [26].
5 Numerical results
Numerical results are presented for three scenarios: timeinvariant sparse multipath channel, timevarying sparse multipath channel, and timevarying sparse multipath with varying CFO. The number of carriers of the OFDM system is chosen as N=128. The length of the sparse multipath channel is fixed to L=20 with K=6 nonzero coefficients for all cases. The nonzero coefficients of the channel are generated from independent complex Gaussian with variances set according to an exponential decaying power delay profile. The CP length is set to N _{ g }=25 to prevent IBI. The mean squared error of the frequency estimates (MSE CFO) and channel estimates (MSE CIR) are given for 1000 Monte Carlo iterations. The pilots symbols are randomly chosen from 16QAM constellation and then are fixed for each Monte Carlo iteration.
5.1 Timeinvariant sparse multipath channel
5.2 Timevarying sparse multipath channel
Depending on the mobility of the receiver, the channel may remain essentially constant over the duration of the block, or may be slowly time varying. For timevarying channels, the CIR cannot be assumed to be static over two consecutive OFDM blocks.
5.2.1 Correlated timevarying sparse multipath channel
5.2.2 Uncorrelated timevarying sparse multipath channel
5.3 Uncorrelated timevarying sparse multipath channel with varying CFO
6 Conclusions
We introduced a novel CS based framework for the joint estimation of CFO and CIR in OFDM systems. It is shown that the CIR can be represented as a 1block sparse signal if a dictionary is built by concatenating subspaces of CFO values within a search grid. CS theory allows the recovery of signals that are given in such representation. The BOMP algorithm is used to reconstruct the CIR coefficients. The proposed estimator uses only one block of training symbols and no initialization is needed. Worst case analysis using perturbation bounds from CS theory are applied to sparse channel estimation. Numerical results show that the proposed estimator gives the same performance as the MLE.
Declarations
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License(http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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