Abstract

In order to track the changing channel in multiple-input multiple-output orthogonal frequency division multiplexing (MIMO-OFDM) systems, it is prior to estimate channel impulse response adaptively. In this paper, we proposed an adaptive blind channel estimation method based on parallel factor analysis (PARAFAC). We used an exponential window to weight the past observations; thus, the cost function can be constructed via a weighted least squares criterion. The minimization of the cost function is equivalent to the decomposition of third-order tensor which consists of the weighted OFDM data symbols. To reduce the computational load, we adopt a recursive singular value decomposition method for tensor decomposition; then, the channel parameters can be estimated adaptively. Simulation results validate the effectiveness of the proposed algorithm under diverse signalling conditions.

1. Introduction

The combination of MIMO with OFDM technique has become the most promising broadband wireless access scheme due to large system capacity and high data rates without any extra consumption of bandwidth and power [1]. In a MIMOOFDM system, the channel needs to be estimated accurately, and then, the transmitted signal can be obtained by channel equalization. Thus, an exact estimation of the changing channel impulse response is necessary [2–5].

In the past decades, many methods have been proposed for MIMO-OFDM systems to solve channel estimation problems. They were grouped into two categories. In one of the categories, the channel parameters can be estimated by sending training sequence or inserting pilot structure which is known to the receiver. Especially, one can realize adaptive channel estimation (ACE) by repeating this process periodically [6–13]. Linear channel estimation methods in [14, 15] use an adaptive filter to estimate channel information. Due to less computational complexity, linear channel estimation methods such as least squares (LS) algorithms are relatively easy to implement. The Least Mean Square (LMS) algorithm is one of the ACE methods with relatively low computational complexity, but the mean squared error (MSE) performance is poor [14]. Furthermore, simplified LMS algorithms like the Sign Data NLMS (SDNLMS) algorithm [15] can be used to decrease the complexity of the LMS algorithm. Sparse channel estimation methods [16–20] commonly use the technique of compressive sensing. However, these methods have great dependence on the number of nonzeros taps. The remaining category is semiblind or blind channel estimation [21]. Here, training sequences and knowledge of noise statistics are not necessary; channel impulse response can be estimated only by the received signals. Therefore, blind channel estimation methods have attracted wide attention due to its improved spectral efficiency. Blind channel estimation methods usually use the statistical properties of received signals; channel is considered static during the receiving time. When ACE process is concerned, existing blind OFDM channel estimation methods are not suitable for online applications.

Parallel factor (PARAFAC) analysis has become a useful tool for dealing with linear algebra of multiplexed arrays [22]. In the field of signal processing, PARAFAC has been widely used for blind signal detection and parameter estimation. In many recent works, tensor decompositions are used for blind channel estimation in OFDM systems [23–27]. [24] proposed a trilinear alternating least squares (TALS) algorithm to estimate channel parameters for SIMO-OFDM systems with Carrier Frequency Offset (CFO). However, the extension on MIMO-OFDM systems is missed. The TALS method can obtain good MSE, but it has high computational complexity and poor convergence rate. The DEterministic Blind REceiver (DEBRE) algorithm uses the received signal to form a 4-way tensor for MIMO-OFDM system in frequency domain; then, the parameters of the model are identified via an ALS algorithm [25]. Similar with the TALS algorithm, the computational complexity of the DEBRE method is very high. The least squares Khatri-Rao factorization (LS-KRF) algorithm and simplified closed-form PARAFAC (S-CFP) algorithm are established based on the tensor model for Tucker decomposition [26], and they achieve very similar performance with extra pilot overhead at high signal-to-noise ratio (SNR) conditions. The mentioned algorithms on PARAFAC are suitable for stationary channel environment; when the tensor model needed to be decomposed at each sampling instant, the computation complexity is too high.

There are some existing fast adaptive PARAFAC decomposition algorithms in mathematics [28, 29]. The two recursive update algorithms in [28] use a sliding or an exponential window to weight the observations, the computation complexities are very low. The 3DOPAST algorithm generalizes the orthonormal projection approximation subspace tracking (OPAST) method to track the loading matrix of third-order tensor [29]. However, none of them are used for adaptive channel estimation of MIMO-OFDM systems. In this paper, we proposed an online adaptive blind channel estimation method using simultaneous diagonalization tracking. We use an exponential window to weight past observations; thus, the cost function can be constructed via a least squares criterion. We construct a third-order tensor which is composed by all the observations; the third loading matrix of the tensor can span the same column space with the right singular matrix of the weighted matrix. Therefore, we can obtain a nonsingular matrix to link them. By updating the nonsingular matrix, the channel parameters can be estimated adaptively. Different from many existing methods that only use the last receiving symbols to estimate current channel, the proposed cost function involves the past information of the exponential windowed receiving signals; thus, we can obtain better performance than the existing methods.

The remainder of this paper is organized as follows. In Section 2, we introduced the PARAFAC-based MIMO-OFDM system model. In Section 3, an adaptive channel estimation algorithm is developed for tracking changing channels in MIMO-OFDM systems. Simulation results are presented in Section 4. Finally, our conclusions are given in Section 5. The main contribution of this paper is briefly summarized as follows: (1) we propose a PARAFAC model for time-varying MIMO-OFDM systems in time domain, where time is measured in OFDM symbols. For one OFDM symbol, by reshaping the received signal vector of all antennas, and letting the reshaped matrix be the lateral slice matrix one by one, we establish a third-order tensor model for time-varying MIMO-OFDM systems. (2) We develop an adaptive channel estimation algorithm using simultaneous diagonalization tracking (ABCE-SDT). Here, training sequences and knowledge of noise statistics are not necessary; channel impulse response can be estimated adaptively only by the received signals. In addition, our algorithm is a recursive update solution; the computation complexity is very low compared to other PARAFAC decomposition counterpart. (3) We compare the performance of the proposed algorithm with other existing channel estimation methods. Simulation results show that compared with the SDNLMS algorithm and adaptive sparse channel estimation (ASCE-NLMS) algorithm, the proposed algorithms can significantly improve the channel estimation performance of time-varying systems under different conditions.

Notation: scalars are denoted by lower case italic letters , vectors by lower case boldface letters , and matrices by boldface capitals . Italic capitals are used to denote index upper bound . The entry with row index and column index in a matrix , i.e., , is symbolized by . The columns of a matrix, say , are denoted by generically. The superscripts , , , and denote the transpose, conjugate transpose, inverse operators, and pseudo inverse, respectively. denotes the identity matrix; denotes the Euclid norm. denotes the Khatri-Rao product. denotes the expectation operator. The operator may either forms a diagonal matrix by a vector or forms a vector by collecting the diagonal entries of a matrix.

2. System Model

In this section, we describe the MIMO-OFDM system model with transmit antennas and receive antennas. symbols are transmitted in each OFDM block, so the th modulated information of th transmitter is . Suppose all the channel paths have the memory upper bounded by , and let denote the equivalent discrete channel response from the th transmit antenna to the th receive antenna. So the points DFT of the channel vector is . The received signal vector from the th receive antenna by the th transmit antenna is then represented aswhere denotes the normalized discrete Fourier transform matrix with its -th entry given by .

Letdenote the overall frequency-domain channel matrix. Therefore, the received symbol vector from the th receive antenna is

Constructing the matrix , can be represented aswhere , denote the diagonal matrix with its entries from the th row of the frequency-domain channel matrix . Now, we consider a time-varying MIMO-OFDM system. Suppose that the channel parameters stay unchanged during one OFDM symbol and vary for different OFDM symbols. Therefore, we define the channel matrix at the th symbols as . Considering all the receive antennas, the received signal vector of all antennas is then represented as

If we reshape to be a matrix, then we stack up corresponding matrices of OFDM symbols, a third-order tensor can be obtained, which is composed by all the receive symbols.

As we know, the PARAFAC decomposition of a third-order tensor is a decomposition of as a sum of minimal number of rank-1 tensorswhere , , and are the th columns of , , and , respectively [28]. , , and are named “loading matrices.” The matrix representations of the tensor decomposition are as follows:

Therefore, matrix representation of tensor decomposition of can be written aswhere , denote the continuous transmit symbols. Since the matrix and are both constant, we define for derivation clarity, then (8) can be rewritten as . Thus, the loading matrices are , , and , respectively. Therefore, we can obtain the channel information by solving the loading matrices of the tensor.

3. Blind Adaptive Channel Estimation Algorithm

3.1. Uniqueness Analysis

The CP decomposition of a tensor is unique up to scaling and permutation ambiguities under a mild condition [28, 30]. The PARAFAC decomposition of is said to be essentially unique if any other matrix triplet that also satisfy the model is related to viawith , , and arbitrary diagonal matrices satisfying , and is an arbitrary permutation matrix. A specific case is where two loading matrices are full column rank, and the third one does not contain collinear columns. In this case, PARAFAC is unique up to its trivial indeterminacies. This result is summarized in the following Theorem [28].

Theorem 1. Assume and are drawn from a jointly continuous distribution with respect to the Lebesgue measure in , and is full column rank. The PARAFAC decomposition of is essentially unique if

According to the Theorem 1, we can obtain a similar Lemma 2 about the CP decomposition of the tensor for MIMO-OFDM systems.

Lemma 2. Assume that and are full rank, and is full column rank. The CP decomposition in (8) is essentially unique if

The above assumptions are generically satisfied in our signal model. On one hand, the normalized discrete Fourier transform matrix is full rank, and then, the channel matrix is completely full rank surely. On the other hand, all the transmit signals are assumed to have independent continuous distribution; thus, the symbols matrix is full column rank, which practically satisfies Lemma 2.

3.2. The Algorithm Derivation

We use the first receiving symbols to build the initial observed tensor . As illustrated in Figure 1, let be the new observation after attaching a new received signal tensor as the side slice of , which means that . Similar to (8), the PARAFAC decomposition of iswhere .

Figure 1 

Appending a new slice in the tensor of observation.

In fact, the alternating least squares (ALS) algorithm can be used to do PARAFAC decomposition of [24]. However, the ALS algorithm needs to do pseudoinverse operation three times in each iteration, and the convergence speed is very slow even with a proper initialization. When it is necessary to decompose the tensor model at each sampling time, the computation complexity is too high. Here, we proposed a closed form solution to avoid the drawbacks of the iterative ALS algorithm. An exponential window least squares criterion is adopted, and the resulting algorithm does not require iterations for channel estimation in one OFDM symbol period. In addition, we avoid directly performing the optimization of the cost function; the minimization of the cost function is transformed to the decomposition of third-order tensor which consists of the weighted OFDM data symbols; thus, the computational complexity can be reduced.

Let be the new lateral slice appended to , such that . Suppose that there is a mild variation of channel parameters between to , which means . So the initial estimate of can be given in the least squares sense by ; then, the estimation of symbols can be built as the structure . So that the least squares update of is then given by

Given , then we can reupdate by substituting by . Finally, since is an estimate for , the estimation of can be obtained. To track the channel in a time-varying MIMO-OFDM system, the sufficient statistic data in (5) should be gradually forgotten. To this end, we use the following exponential window least squares criterion to equal the decomposition of third-order tensor .where is the forgetting factor. Thus, we define the windowed observed matrix aswhere is the weighting matrix. Thus, we can obtain based on the PARAFAC decomposition. The exponential window implies the following update rulethen we consider the SVD of the weighted matrix , where , , and . Since the PARAFAC decomposition of satisfies the Lemma 2, the weighted matrix has the same rank with , which means . To satisfy the condition, we restrict to and . Therefore, there exists a nonsingular matrix such thatwhere . It is easy to see that the matrix links equations (17) and (18). When the th symbols transmit to the receiver, the equations become

Hence, we can use the common block between and to update the matrix . To avoid SVD at every th time, we can perform the classical Bi-Iteration technique to do adaptive SVD of [31]. By this means, the computational complexity can be decreased. In addition, to avoid explicit computation of the pseudoinverse of , derivation of recursive updates of should be taken into consideration; thus, we define .

For derivation clarity, we define and rewrite aswhere is the last row of . Based on the structure of , we can obtain the following equation from equations (17) and (19)it follows thatthus, the computation of is

Obviously, computation of the pseudoinverse of and is not expected in our algorithm. Since is a unitary matrix, we have ; thus, the computation of can be rewritten as

To avoid explicit computation of the pseudoinverse of , derivation of recursive updates of should be taken into consideration. As the definition of before, we can obtain the following equation from the matrix inversion Lemma for rank-1 updates [28],

Substitute (26) into (23), the computation of is

Therefore, the matrix can be obtained. Let as the estimation matrix of the channel, where . Due to the Khatri-Rao product , we can obtain the following equation:where denotes the th column of , and denotes the th column of . We define the rank-one matrices , , as

Since the normalized discrete Fourier transform matrix and the correlation coefficient matrix are constant, we can obtain the equation of th column of

Therefore, estimation of channel and signals are obtained as and , respectively. We present the pseudocode of the proposed adaptive channel estimation algorithm for MIMO-OFDM systems in Algorithm 1.

4. Simulation Results

In this section, we evaluate the performance of the proposed method. We provide simulation results under three different scenarios. In the first simulation, we test the proposed algorithm and the other two existing algorithms under a stationary environment exhibiting abrupt changes; in the second one, we consider the performance of the proposed method under a more realistic case of a slowly changing environment. In all cases, we consider the MSE as a measure of performance. The MSE is defined aswhere and are true channel parameter matrix and their estimates, respectively. The th received signal at the th antenna writes in noisy environment, where represents the additive noise in the th antenna. Then, SNR is defined as

We drop the dependency of the SNR with in the definition, which is because we use the same SNR for all OFDM symbols.

4.1. Abruptly Changing Environment

Here, we employ a MIMO-OFDM system with antennas; the number of subcarriers is set to 32. The modulation scheme used is 16QAM; the length of Cyclic Prefix is set to 4. We use symbols to form the initial received symbols. The forgetting factor is 0.5. AWGN noise is added to the simulated MIMO-OFDM system and . We compare the channel estimation performance of the proposed ABCE-SDT algorithm and methods in [15, 20] with receive symbols vary from 200 to 1200, which can be seen in Figure 2. The step-size of the SDNLMS algorithm in [15] and the ASCE-NLMS algorithm in [20] is both set to 0.5. We use the IEEE 802.11 Model in the simulation. The simulated channel has the following structure; the power delay spectrum obeys negative exponential distribution.where . Firstly, we can obtain a random channel response by the simulated channel model. We start with the channel response, and at iteration 501, we abruptly switch a new response obtained by the simulated channel model. We use OFDM symbols to measure time, especially, during the intervals and ; we let the channel remains static.

Figure 2 

Channel estimation performances of three algorithms.

Figure 2 presents the channel tracking performance of the proposed ABCE-SDT algorithm, SDNLMS algorithm, and ASCE-NLMS algorithm. Obviously, the proposed ABCE-SDT method has higher MSE than other two methods at first, but it can achieve better performance soon, and it provides faster convergence. This is because the SDNLMS algorithm and ASCE-NLMS algorithm use training sequence to obtain the initial channel information whereas the ABCE-SDT method is blind. The window we used makes full use of all previously observed slices with different weights, which can speed up the convergence at low error. The other two algorithms only rely on the last observations; more symbols are needed to converge. The complexities of the ABCE-SDT, SDNLMS, and ASCE-NLMS algorithms for one iteration are , , and , respectively. Obviously, the proposed algorithm requires the least computational burden compared to the other two methods, which is satisfied with what simulation shows. When the tracking process gets stable, it can be seen that the above algorithms have stable MSE with the iterations. When we abruptly change parameters of channel, the SDNLMS and ASCE-NLMS algorithms obtain poorer MSE performance than the ABCE-SDT algorithm, and they need more iterations to converge, even when the tracking process get stable; the MSE of the two methods still keep higher value. That is because the performance of the two competitors has error accumulation; new estimations are obtained by updating the old ones. The saltation of channel parameters caused the two competitors hardly to obtain an accurate channel estimation, whereas the proposed algorithm can compute the parameters of channel directly.

Moreover, we use Cramer-Rao bound as a benchmark. We plot the performances of these three algorithms and the corresponding CRB [32] in Figure 3. This result indicates that MSEs of the proposed ABCE-SDT method are very close to the corresponding CRBs. In order to reveal the performance of the entire system from the perspective of error probability, after the channel can be tracked, Figure 4 illustrates the averaged BER performance of three algorithms over 100 independent trials. We can obtain that the BER of three competitors decreases with the growth of SNR. At BER of , the proposed algorithm achieves an SNR gain of about 6 dB compared with the existing algorithms, which means that the ABCE-SDT method achieves better BER performance than the other two existing methods.

Figure 3 

Channel estimation performances with abruptly changing environment.

Figure 4 

BER performances with abruptly changing environment.

4.2. Slowly Changing Environment

Now, we consider a more realistic scenario where the channel changes every OFDM symbol. The simulated MIMO-OFDM has antennas with subcarriers. We also use symbols to form the initial receive symbols. The simulated channel consists of five equal power taps; we create the initial parameters of channel by the IEEE 802.11 Model; then, we change the parameters linearly. The modulation scheme used is 16QAM. The forgetting factor is 0.5. AWGN noise is added to the simulated MIMO-OFDM system and .

Firstly, we let the channel parameters change linearly in every OFDM symbol. The simulated system has 6 paths; thus, we plot how the estimated channel tracked the real channel in Figures 5(a)–5(f). The labels of the -axis and -axis are real and imaginary of channel parameters, respectively. We can see the estimated channels have very close parameters with the real channels. In order to further verify the effectiveness, MSE performance of the proposed algorithm versus SDNLMS and ASCE-NLMS methods under the linear changed channel over 100 independent trials was shown in Figures 6 and 7. The step-size of the SDNLMS algorithm and ASCE-NLMS algorithm are both set as . In Figure 6, the ABCE-SDT method has higher MSE than the other two methods at first, but it can achieve better performance soon, and it provides faster convergence, which is close to the stationary environment with the stable channel. This is because the window we used makes full use of all previously observed slices with different weights, but the other two algorithms only rely on the last observations. It can also be observed that the SDNLMS and ASCE-NLMS algorithms perform a much higher fluctuation when channel can be tracked. Figure 8 represents the corresponding BER performance of the three competitors. Obviously, with the growth of SNR, the BER of three competitors decreases. At the BER of , the proposed algorithm outperforms the SDNLMS and the ASCE-NLMS algorithms by about 6 dB and 8 dB SNR gains, respectively. Hence, the proposed ABCE-SDT algorithm achieves better BER performance than the other two algorithms.

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Figure 5 

Estimated channel and real linearly changing channel (, , ).