In orthogonal frequency division modulation (OFDM) communication systems, channel state information (CSI) is required at receiver due to the fact that frequency-selective fading channel leads to disgusting intersymbol interference (ISI) over data transmission. Broadband channel model is often described by very few dominant channel taps and they can be probed by compressive sensing based sparse channel estimation (SCE) methods, for example, orthogonal matching pursuit algorithm, which can take the advantage of sparse structure effectively in the channel as for prior information. However, these developed methods are vulnerable to both noise interference and column coherence of training signal matrix. In other words, the primary objective of these conventional methods is to catch the dominant channel taps without a report of posterior channel uncertainty. To improve the estimation performance, we proposed a compressive sensing based Bayesian sparse channel estimation (BSCE) method which cannot only exploit the channel sparsity but also mitigate the unexpected channel uncertainty without scarifying any computational complexity. The proposed method can reveal potential ambiguity among multiple channel estimators that are ambiguous due to observation noise or correlation interference among columns in the training matrix. Computer simulations show that proposed method can improve the estimation performance when comparing with conventional SCE methods.

1. Introduction

In broadband wireless communication systems using orthogonal frequency division modulation (OFDM), frequency-selective fading is incurred by the reflection, diffraction, and scattering of the transmitted signals due to the buildings, large moving vehicles, mountains, and so forth. Such fading phenomenon distorts received signals and poses critical challenges in the design of communication systems for high-rate and high-mobility wireless communication applications. Hence, accurate channel estimation becomes a fundamental problem of such communication systems. In last several years, various linear estimation methods have been proposed based on the assumption of rich multipath channel model. However, recently, a lot of physical channel measurements verified that the channel taps exhibit sparse distribution [13] due to the broadband signal transmission. A typical example of sparse multipath channel is shown in Figure 1 where the length is 100 while the number of nonzero taps is 5 only. Note that different broadband transmission may incur different channel structures in wireless communication systems as shown in Table 1.

To improve the estimation performance, extra sparse structure information can be exploited as prior information. Thanks to the development of compressive sensing [4, 5], many sparse channel estimation (CCE) methods have been proposed for exploiting the channel sparsity. In [6], orthogonal matching pursuit (OMP) algorithm with application to sparse multipath channel estimation in the OFDM systems has been proposed. In [7, 8], sparse channel estimation methods have been proposed using compressive sampling matching pursuit (CoSaMP) algorithm [9] in frequency-selective and doubly-selective channel fading communication systems. In [10], to further reduce the computational complexity, sparse channel estimation using smooth -norm (SL0) algorithm [11] has been proposed. Compared to traditional linear methods, sparse channel estimation methods have two obvious advantages: spectral efficiency and lower performance bound. For one thing, improving the spectral efficiency by utilizing less training sequence can achieve the same estimation performance as linear methods. For another, the lower performance bound can be obtained by exploiting channel sparsity due to the fact that less active channel freedom of degree is acquired [12].

Conventional sparse channel estimation methods have a cardinal objective that try to probe the dominant channel taps as accurate as possible, while these methods neglect the posterior information report from additive noise received signal. These proposed channel estimation methods are termed as model selection or basis selection. Unfortunately, their estimation performances are often degraded due to the neglecting channel model uncertainty [13]. To mitigate the unexpected model uncertainty, Bayesian compressive sensing (BCS) [14] and a slight improved Bayesian compressive sensing using Laplace priors (BCS-LAP) [15] could be adopted for estimating sparse channel. The estimation performance could be improved effectively but at the cost of high computational complexity when comparing with existing simple algorithms (e.g., OMP [6] and SL0 [11]). Hence, it is impractical to employ this algorithm in real communication systems.

Unlike these aforementioned methods, in this paper, we propose an improved Bayesian sparse channel estimation (BSCE) method while its computational complexity is comparable with OMP and SL0. Our proposed Bayesian channel method can be divided into two steps: position detection of dominant channel taps and channel estimation using minimum mean square error (MMSE). In general, our proposed Bayesian estimation method provides model uncertainty which reveals uncertainty among multiple possible position sets of dominant channel taps that are ambiguous due to observation noise or correlation among columns in the training matrix. Furthermore, the complexity of the proposed method is relatively lower due to its smaller search space when compared to conventional methods. Simulation results are given to verify two folds: performance and complexity. Note that estimation performance is evaluated by two metrics: mean-square-error (MSE) and bit-error rate (BER), while computational complexity is measured coarsely by CPU time of computer.

The remainder of this paper is organized as follows. An OFDM system model is described and problem formulation is given in Section 2. In Section 3, the BSCE method is proposed in OFDM systems. Computer simulation results are given in Section 4 in order to evaluate and compare performance of the BSCE method with conventional methods. Finally, we conclude the paper in Section 5.

Notation 1. Throughout the paper, matrices and vectors are represented by boldface upper case letters (i.e., ) and boldface lower case letters (i.e., ), respectively; the superscripts , , , and denote the transpose, the Hermitian transpose, and the inverse and diagonal operators, respectively; denotes the expectation operator; is the -norm operator that counts the number of nonzero taps in ; and stands for the -norm operator which is computed by , where is considered in this paper.

2. System Model and Problem Formulation

Consider a frequency-selective multipath channel whose impulse response is given by where is the number of multipaths and and are the (complex) channel gain and the delay spread, respectively, of path at time . Hence, the -length discrete channel vector can be written as . Let the OFDM system use size- discrete Fourier transform (DFT), and its number of pilot subcarriers is . To avoid intersymbol interference (ISI), we assume that the length of the zero-padding cyclic prefix (CP) in the OFDM symbols is larger than maximum delay spread , where , . Suppose that denote th subcarrier in an OFDM symbol, where . If the coherence time of the channel is much larger than the OFDM symbol duration , then the channel can be considered quasistatic over an OFDM symbol. Let be the vector of received signal samples in one OFDM symbol after DFT; then where denotes diagonal subcarrier matrix, is the channel frequency response (CFR) in frequency-domain, and is assumed to be additive white Gaussian noise (AWGN) with variance . is an partial DFT matrix with its th row which is easily given by and denotes an equivalent time-domain signal matrix. In addition, denotes a time-domain channel vector. Since , hence, the frequency-domain channel impulse response lies in the time-delay spread domain.

Assume that a binary random vector denotes a taps’ position indicator of sparse channel vector which is generated from a Gaussian mixture density (GMD) function as where the covariance matrix is determined by position indicator . For a better understanding, we take to be diagonal element with for , implying that are independent with Gaussian distribution . Assume that the position indices are satisfied Bernoulli distribution with probability ; then the probability of nonzero and zero channel taps of channel vector can be written as for . According to (4), one can easily find . In real communication systems, broadband channels are often described by sparse models [16, 17]. Hence, we choose and , so that has relatively few dominant channel taps. In other words, sparseness of channel vector depends on the probability as shown in Figure 2. Smaller probability implies sparser channel vector and vice versa.

The research objective of this paper is to estimate the sparse channel vector using received signal vector and training signal matrix . Hence, the system model can be assumed to satisfy distribution as where is the covariance matrix of . That is, .

3. Compressive Sensing Based Bayesian Sparse Channel Estimation

In this section, compressive sensing based Bayesian sparse channel estimation is proposed in two steps: detect the position set of dominant channel taps and then estimate sparse channel   using MMSE algorithm. Obviously, how to find the dominant channel taps’ position is a key technique with low-complexity Bayesian method for estimating sparse channels.

3.1. Position Detection on Dominant Channel Taps

According to the well-known Bayesian rules, the posterior of position indicator can be written as where denotes all of possible position index sets of channel taps as shown in Figure 3. Equation (6) implies that estimating reduces to estimating . Due to the extremely computational complexity in (6), the huge size of makes it impractical to compute or for all in the case of high-dimensional broadband channels. By considering sparse structure in channels, only posteriors of dominant taps’ position are needed for sparse channel estimation. Assuming that the set is responsible for position indicator of dominant channel taps, then the search space in rather than can be quite small and therefore practical to compute. Hence, the posteriors of dominant channel taps can be approximated by for dominant channel set . Hence, exploiting the dominant channel taps set reduces to the search for which only computes the dominant values of in (7). First of all, the probability density function (PDF) for position indicator can be written as

By transforming it in log-domain for convenience, then the position indicator (PI) can be given by which is a metric of position indicator . According to in (9), one can easily find that the position indicator depends on received signal, channel length, position indicator, and probability of nonzero taps. Due to the positive exponent relationship , in (9) can also be considered as a measure function of on dominant channel taps. However, it is still unfeasible to get the position information of channel in practical system without considering channel estimation. According to [18], the mathematical expectation of can be given by

For a given pair , can be used to compare the mean and standard deviation in order to get a rough evaluation of .

To reduce the search space in position set, we resort to an efficient method [13] to determine as follows. The basic idea is that the position set of unknown channel yielding the dominant values of is equivalent to the high probability of . The search starts with and the initial position set is set as . If we change each element in , then it yields position indicators. Consider all of position indicators in a set and refer it to . The metrics for the PI vectors in are then computed by (9), and elements of with the largest value of the dominant channel tap are collected in . For each possible dominant taps’ set in , all positions of a second nonzero tap are then considered, yielding unique binary vectors to store in . The for all possible vectors in are then computed, and the elements of with the largest value are collected in . Then for each candidate vector in , all possibilities of a third dominant channel tap are considered, and those with the largest channel taps are stored in . The process continues until is computed, where can be chosen to make sufficiently small to exploit all of channel sparsity. Note that constitutes the algorithm’s final estimate of and later we denote as the final estimate. For better understanding of the PI update of dominant channel taps, an intuitive example is given in Figure 4, where the length of position indicator is set as ; the number of largest value of PI is chosen as , and the maximum number of nonzero taps is set as .

For use with the aforementioned Bayesian matching pursuit (BMP) algorithm, we consider a fast metric update which computes the change in that results from the activation of a position of nonzero tap. More precisely, if we denote by the vector identical to except for the th coefficient, which is active in but inactive in (i.e., and ), then it is defined as to track the change of active positions. Note that at the initial step is set as via (9) and the fact that . To obtain the fast PI update, we start with the property that, for any and , for which the matrix inversion lemma implies where and . Notice that the cost of computing in (14) is if standard matrix multiplication is used [13]. According to previous analysis, we can get which, combined with (5), yield

In summary, in (18) quantifies the change in due to the activation of the th position of .

Please note that the cost of computing via and is , if standard matrix multiplication is used. As we described, the complexity of this operation can be made linear in by exploiting the structure of . Say that contains the indices of active elements in g. Then from (14), we can get when activating the th position in . The key observation is that the coefficients need only to be computed once, that is, when index is active. Furthermore, only need to be computed for surviving indices . According to previous analysis in (20), the number of multiplications required by the algorithm is [13]. Moreover, the complexity of the proposed algorithm could be reduced if the smaller is adopted.

3.2. MMSE for Estimating Values of Dominant Channel Taps

By utilizing the dominant taps’ posteriors, the sparse channel can be estimated readily by MMSE algorithm as

According to the above introduction, compressive sensing based Bayesian sparse channel estimation could be implement by (20)–(22) with high estimation performance and low complexity.

4. Computer Simulations

In this section, the proposed BSCE estimator adopts 1000 independent Monte Carlo runs for averaging. The length of channel vector is set as . Values of dominant channel taps follow Gaussian distribution and their positions are randomly allocated within the length of which is subjected to . The received signal-to-noise ratio (SNR) is defined as , where .

The proposed method is compared to five conventional sparse channel estimation methods using algorithms OMP [19], CoSaMP [9], BCS [14], BCS-LAP [15], and SL0 [20]. It was worth noting that these simulation parameters were chosen in accordance with detailed communication environment in this paper. The stopping error criteria threshold is set as for all algorithms in Monte Carlo computer simulations. The initial noise variance for BSC and BSC-LAP is set as , where denotes standard derivation and . In addition, the Laplace prior for BCS-LAP is computed automatically which was suggested in [15]. The parameters of FBMP algorithm were initialized as , , , and . Computer simulation parameters are listed in Table 2.

4.1. MSE versus SNR

The estimation performance is evaluated by average mean square error (MSE) standard which is defined as where denotes expectation operator and and are the actual channel vector and its channel estimator, respectively. In Figures 5, 6, 7, and 8, we compare the average MSE performance of the proposed channel estimator with traditional sparse channel estimators with respect to different channel sparseness, and . As the four figures show, our proposed method can achieve better estimation performance than conventional methods. The lower bound is given by least square (LS) method (oracle) which utilized the channel position information. In this figure, it is easily found that the proposed method obtained lower MSE performance than conventional methods. In other words, if the proposed estimator is applied in data detection, smaller BER performance can be achieved when comparing with conventional methods.

4.2. BER versus SNR

By using the above channel estimators, signal transmission performances are evaluated as shown in Figures 9, 10, 11, and 12. From the four figures, average BER performance curves are depicted with respect to SNR for binary phase shift keying (BPSK) data. We can see that the BER performance of the proposed method is more close to lower bound which is given by ideal channel estimator whose nonzero taps’ positions are known. Here, only low signal modulation was considered for BER evaluation. It is very easy to predict that our proposed method could improve BER performance in case of high signal modulation.

4.3. Complexity Evaluation

To compare the computational complexity of the proposed method with other methods, CPU time is adopted for evaluation standard as shown in Figures 13, 14, 15, and 16. It is worth mentioning that although the CPU time is not an exact measure of complexity, it can give us a rough estimation of computational complexity. Our simulations are performed in MATLAB 2012 environment using a 2.90 GHz Intel i7 processor with 8 GB of memory and under Microsoft Windows 8 (64 bit) operating system. For comprehensive comparing between our proposed method and other methods in different length of training signal and different channel sparsity, we simulate their comparison results in Figures 1316. As the four figures shown, the complexity of the proposed method is close to OMP and SL0-based methods and lower than CoSaMP, BCS, and BCS-LAP based methods. It is well known that the complexity of OMP and SL0 is very low on sparse channel estimation [10, 22]. Hence, comparing with traditional methods, our proposed method can achieve better estimation performance and low complexity.

5. Conclusion

Traditional sparse channel estimation methods are vulnerable to noise and column coherence interference in training matrix. Their primary aim is an attempt to exploit sparse structure information without a report of posterior channel uncertainty. To improve the estimation performance, fast Bayesian matching pursuit algorithm with application to sparse channel estimation has not only exploited the channel sparsity but also mitigated the unexpected inferences in training matrix. In addition, the proposed method has revealed potential ambiguity among multiple channel estimators that are ambiguous due to observation of noise or correlation among columns in the training signal. Computer simulation results have showed that proposed method improved the estimation performance with comparable computational complexity when comparing with traditional methods.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper. The authors of the paper do not have a direct financial relation that might lead to a conflict of interests for any of the authors.